PSU(2,2|4) Exchange Algebra of N=4 Superconformal Multiplets
aa r X i v : . [ h e p - t h ] D ec PSU(2,2 |
4) Exchange Algebraof N =4 Superconformal Multiplets Shogo Aoyama ∗ Department of PhysicsShizuoka UniversityOhya 836, ShizuokaJapanNovember 10, 2017
Abstract
It is known that the unitary representation of the D = 4 , N = 4 supercon-formal multiplets and their descendants are constructed as supercoherent statesof bosonic and fermionic creation oscillators which covariantly transform underSU(2,2 | | { SO(1,4) ⊗ SO(5) } which is reparametrized by the D = 10 supercoordi-nates ( X, Θ). We consider a D = 2 non-linear σ model on the coset superspace andset up Poisson brackets for X and Θ on the light-like line. It is then shown thatthe non-linearly realized creation oscillators satisfy the classical exchange algebrawith the classical r-matrix of PSU(2,2 | D = 4 , N = 4 superconformal multiplets and their de-scendants on the light-like line. It is because they are supercoherent states of theoscillators. The arguments are straightforwardly extended to the case where thosequantities are endowed with the U( N ) YM gauge symmetry. Keywords: Quantum Groups, Sigma Models, Extended Supersymmetry ∗ e-mail: [email protected] Introduction
The gauge/string duality between the D = 4 , N = 4 SUSY YM theory and the IIBstring theory on AdS × S [1] is one of the subjects which have been discussed with greatinterest in recent years. The integrability and the superconformal symmetry PSU(2,2 | D = 4 , N = 4 SUSY YM theory on one side was casted to a D = 2 spin-chain sys-tem with the superconformal symmetry PSU(2,2 | |
4) is also hypothetical, since it is broken by the Bethe ansatzto two copies of the subgroup PSU(1,1 |
2) with central charges. The appearance of centralcharges makes the purely algebraic construction of the universal R-matrix for a simplegroup[4, 5] unreliable. That is, the plug-in formula for the universal R-matrix works onlyif we concern a simple (super)group G and its Yangian generalization Y(G)[4, 5]. Thisunusual feature of the R-matrix attracted a particular interest as a challenging subject[6].It gives a clue to study the anomalous scaling dimension of the N = 4 SUSY YM theoryby means of the R-matrix of a D = 2 spin-chain.The IIB string theory on the other side was effectively described by a D = 2 non-linear σ -model on the coset superspace PSU(2,2 | { SO(1,4) ⊗ SO(5) } [7]. It is integrableat the classical level admitting an infinite number of conserved currents[8]. Quantumextension of the integrability was argued by the Bethe ansatz[9]. The Bethe ansatz is acommon language to understand the gauge/string duality on both of the sides. In thisapproach the origin of the integrability and the superconformal symmetry PSU(2,2 |
4) areclear because of the Poisson structure of the non-linear σ -model and the resemblance tothe Green-Schwarz superstring respectively[10].In this paper we pursue the approach of the string side. For the non-linear σ -model onPSU(2,2 | { SO(1,4) ⊗ SO(5) } we set up Poisson brackets for the basic fields on the light-like line x + = const instead of the equal-time line x = const . In [11] the consistency andthe virtue for doing this were shown for the non-linear σ -model on the general bosonic cosetspace G/H. Namely, the Poisson brackets satisfy the three conditions. (i) They satisfiesthe Jacobi identities. (ii) The energy-momentum tensor T −− generates diffeomorphismon the light-like line by means of the Poisson brackets. (iii) At the origin of the cosetspace they coincide with the Poisson brackets of the free boson theory. There exists aquantity Υ, called Killing scalar, which transforms as a linear representation vector of Gby the Killing vectors of G/H. It exists in any representation of G and obeys the classicalexchange algebra { Υ( x ) ⊗ , Υ( y ) } = − hr xy Υ( x ) ⊗ Υ( y ) , (1.1)on the light-like line with the Poisson brackets for the basic fields. Here r xy is the classicalr-matrix of G. If G is a simple group, we may have recourse to the plug-in formula topromote it to the universal R-matrix R xy , which is expressed purely in terms of the2enerators of G[4, 5]. Then (1.1) becomes the quantum exchange algebra R xy Υ( x ) ⊗ Υ( y ) = Υ( y ) ⊗ Υ( x ) , (1.2)Its classical correspondence to (1.1) can be seen by R xy = 1 + hr xy + O ( h ) . Correlation functions of Υs arrayed on the light-like line, may be obtained by using thequantum exchange algebra to braid Υs at adjacent positions successively.In this paper we apply all of these arguments to the non-linear σ -model on PSU(2,2 | { SO(1,4) ⊗ SO(5) } [7]. Now the basic fields of the coset space are the D = 10 supercoor-dinates ( X, Θ). For this non-linear σ -model the exchange algebra (1.1) or (1.2) appearswith the r- or R-matrix of the superconformal group PSU(2,2 | N = 4 superconformal multiplet. But PSU(2,2 |
4) is non-compact. Hence the uni-tary representation contains infinitely many descendants. We call them as a whole thesuperconformal multiplet V . In the previous work [11] the exchange algebra (1.1) or (1.2)of the non-linear σ -model on G/H was discussed in an arbitrary representation. But thedimension of the unitary representation was finite by assuming that G is a compact group.It is awkward to simply apply the arguments in [11] to the case where the dimension ofthe unitary representation is necessarily infinite. It is our main concern to make a bridgeover this gap.To this end we remember that the N = 4 superconformal multiplet V can constructedover a supercoherent space of bosonic and fermionic creation oscillators[12]. The oscilla-tors form a 8-d vector, say ψ , transforming covariantly under the superconformal groupPSU(2,2 | ψ as an 8 × e iM . Then weshow that the unitary representation is given by ˆ U = e i ¯ ψMψ , which is an infinite dimen-sional representation of PSU(2,2 | e iM on the 8-d vector ψ , as shown by the state-operator relations (3.3) and (3.4).Therefore the Killing scalar Υ which we want to let satisfy the classical exchangealgebra (1.1) is not necessarily the N = 4 superconformal multiplet V , but may be the 8-d covariant vector ψ . The Killing scalar Υ transforming identically with ψ can be readilyconstructed on the coset space PSU(2,2 | { SO(1,4) ⊗ SO(5) } , following [11]. Once this isdone, the whole arguments in [11] can be applied to the non-linear σ -model on this cosetspace as well. That is, this Killing scalar Υ satisfies the classical exchange algebra (1.1)with the r-matrix in the 8 × | |
4) is a simplegroup. Hence the finite-dimensional r-matrix can be quantized to the universal R-matrixby means of the plug-in formula[4, 5]. Thus we get the quantum exchange algebra (1.2) forthe Killing scalar Υ or equivalently for the covariant vector ψ . From this we can calculatethe quantum exchange algebra for the N = 4 superconformal multiplet V , because V consists of ψ as shown in table 2. The R-matrix for V is infinite-dimensional and yetalgebraically the same as for the covariant vector ψ owing to the operators-state relations The R-/S-matrix discussed in [2] is not the one for PSU(2,2 | | ⋉R . Further comments on this will be made at the end of this paper. N = 4 superconformalalgebra of PSU(2,2 |
4) in terms of bosonic and fermionic oscillators forming an 8-d covari-ant vector ψ . It is done by following [2] closely. In section 3 we construct the unitaryrepresentation of the superconformal group PSU(2,2 |
4) over a supercoherent space of theoscillators, following [12]. In particular we focus on the N = 4 field strength multipletappearing as a half-BPS state in the unitary representation of PSU(2,2 | V are given inappendix A. The reader who is familiar the subjects may skip sections 2 and 3. In section4 we discuss the 8 × | |
4) is non-linearly realized on the coset space PSU(2,2 | { SO(1,4) ⊗ SO(5) } , ina way independent of the representation. Embedding the subgroup SO(1,4) ⊗ SO(5) inPSU(2,2 |
4) is carefully studied. The salient feature of this coset space is that the ba-sic fields of the coset space are the D = 10 supercoordinates ( X, Θ). In section 6 theoscillators, forming the 8-d covariant vector ψ of PSU(2,2 | | { SO(1,4) ⊗ SO(5) } as the Killing scalar Υ. In section 7, we consider thenon-linear σ -model on the coset space and impose Poisson brackets for ( X, Θ), accordingto [11]. Then we get the classical exchange algebra for the non-linearly realized oscillatorsΥ and discuss its implication for correlation functions when the non-linear σ -model isquantized on the light-like line. Appendix A is devoted to complete the argument on theunitary(oscillator) representation of PSU(2,2 |
4) in section 3. Superconformal multipletsother than the field strength multiplet appear as larger BPS multiplets. Though theywere argued in various works [13, 14, 15], here we straighten the arguments by unifyingthe notations. Finally in appendix B we explain how to calculate the Killing vectors ofthe general coset space G/H in a way independent of the representation, i.e., by usingonly the Lie-algebra. The unitary(oscillator) representation of PSU(2,2 |
4) as well as thematrix one require central charges as shown in section 3 and 4. The algebraic calculationin appendix B dispenses us with meeting central charges. It is desirable since PSU(2,2 | N =4 SUSY YM theory and PSU(2,2 | The N =4 SUSY YM theory is described by a set of fundamental fields A µ , Ψ αa , Ψ a ˙ α , Φ [ a,b ] . Our index convention is as follows: µ refers to vector indices of the Lorentz group SO(1,3),taking four values. α, ˙ α refer to two independent spinor indices of SU(2) ⊗ SU(2)( ∈ SU(2,2)). They respectively takes two values. a, b refer spinor indices of the R-symmetry4U(4), taking four values. [ , ] indicates anti-symmetrization of them. Complex conju-gation of the spinor representation is indicated by raising or lowering indices. The N =4SUSY field strength multiplet is constructed out of these fundamental fields as shown intable 1. There F indicates the field strength F µν , which has been split into F { αβ } and F { ˙ α ˙ β } by using the spinor indices of SU(2) ⊗ SU(2). { , } indicates symmetrization of theindices. D indicates space-time derivative D µ , which may be written as D α ˙ β . The repre-sentation of SU(2) ⊗ SU(2) and SU(4) are indicated by the Dynkin labels for the highestweight as [ s , s ] and [ r , r , r ] respectively. The Young tableau representing the SU(4)representation is drawn in figure 1. Their dimensions are given by dim [ s , s ] = ¯ s ¯ s ,dim [ r , r , r ] = 112 ¯ r ¯ r ¯ r (¯ r + ¯ r )(¯ r + ¯ r )(¯ r + ¯ r + ¯ r ) , with ¯ s i = s i + 1 and ¯ r a = r a + 1.field SU(2) ⊗ SU(2) SU(4)h.w. h.w. D k F [ k + 2 , k ] [0,0,0] D k Ψ [ k + 1 , k ] [1,0,0] D k Φ [ k, k ] [0,1,0] D k ˙Ψ [ k, k + 1] [0,0,1] D k ˙ F [ k, k + 2] [0,0,0]Table 1: N = 4 SUSY field strength multiplet. · · · · · · · · ·· · · · · ·· · · | {z } r | {z } r | {z } r Figure 1: The SU(4) Young tableau for the representation with the Dynkin label [ r , r , r ].The N =4 SUSY YM theory has the superconformal symmetry defined by the super-group PSU(2,2 | | |
4) is decomposed as T − ⊕ T ⊕ T + , where T represents generators of the compact subgroup U(2,2) ⊗ U(4) ⊗ U(1) and T − ⊕ T + represents non-compact ones such that[ T , T ± ] = T ± , [ T ± , T ± } = T ± . , } is a graded commutator understood as an anti-commutator betweenfermionic generators, and as a commutator otherwise. We introduce two set of bosonicoscillators ( a α , a † α ) and ( b ˙ α , b † ˙ α ) and one set of fermionic ones ( c a , c † a ) to realize thesegenerators. The non-trivial commutation relations are[ a α , a † β ] = δ αβ , [ b ˙ α , b † ˙ β ] = δ ˙ α ˙ β , { c a , c † b } = δ ab . (2.1)To be explicit, T consists of the generators L αβ = a † β a α − δ αβ a † γ a γ ,L ˙ α ˙ β = b † ˙ β b ˙ α − δ ˙ α ˙ β b † ˙ γ b ˙ γ , (2.2) R ab = c † b c a − δ ab c † c c c , and the three U(1) generators D = 1 + 12 a † γ a γ + 12 b † ˙ γ b ˙ γ = 12 a † γ a γ + 12 b ˙ γ b † ˙ γ ,C = 1 − a † γ a γ + 12 b † ˙ γ b ˙ γ − c † c c c = − a † γ a γ + 12 b ˙ γ b † ˙ γ − c † c c c , (2.3) B = − a † γ a γ − b † ˙ γ b ˙ γ = 12 a † γ a γ − b ˙ γ b † ˙ γ . The generators in T + are given by Q aα = a † α c a , ˙ Q ˙ αa = b † ˙ α c † a , P ˙ αβ = b † ˙ α a † β , while those in T − by S αa = c † a a α ˙ S ˙ αa = b ˙ α c a , K α ˙ β = a α b ˙ β . Then the generators in (2.2) form the subalgebra SU(2) ⊗ SU(2) ⊗ SU(4) of U(2,2 | L αβ , L γδ ] = − δ γβ L αδ + δ αδ L γβ , [ L ˙ α ˙ β , L ˙ γ ˙ δ ] = − δ ˙ γ ˙ β L ˙ α ˙ δ + δ ˙ α ˙ δ L γβ , [ R ab , R cd ] = − δ cb R ad + δ ad R cd . (2.4)The algebra [ T ± , T ± } = T ± is nilpotent in the sense that [ T ± [ T ± , T ± }} = 0, and is givenby { ˙ Q ˙ αb , Q aβ } = δ ab P ˙ αβ , { ˙ S a ˙ β , S αb } = δ ab K β ˙ α , (2.5) If Q aα = a † α c a , S αa = c † a a α are replaced by Q aα = a † α c † a , S αa = c a a α , they form the algebra[ T , T ± ] = T ± , { T + , T − } = T , { T ± , T ± } = 0 . This form of the Lie-algebra U(2,2 |
4) was used to discuss the unitary representation in refs [16] T + , T − ] = T by[ K α ˙ β , P ˙ γδ ] = δ ˙ β ˙ γ L αδ + δ αδ ˙ L ˙ β ˙ γ + δ ˙ β ˙ γ δ αδ D, { S αb , Q aβ } = δ ab L αβ + δ αβ R ab + 12 δ ab δ αβ ( D − C ) , (2.6) { ˙ S a ˙ β , ˙ Q ˙ αb } = δ ab ˙ L ˙ β ˙ α − δ ˙ β ˙ α R ab + 12 δ ab δ ˙ β ˙ α ( D + C ) . Finally the algebra [ T + , T − ], which does not close into T , is given by[ S αb , P ˙ αβ ] = δ αβ ˙ Q ˙ αb , [ K α ˙ β , Q ˙ αb ] = δ ˙ β ˙ α S αb , [ ˙ S a ˙ β , P ˙ αβ ] = δ ˙ β ˙ α Q aβ , [ K α ˙ β , Q aβ ] = δ αβ ˙ S a ˙ β . (2.7)We omit the algebra [ T , T ± ] = T ± , which can be easily written down. Altogether thealgebrae (2.4) ∼ (2.7) define the Lie-algebra of U(2,2 | ψ † = ( a † β , c † b , b ˙ β ) , ψ = a α c a b † ˙ α , (2.8)as ψ † ⊗ ψ = L αβ S αb K α ˙ β Q aβ R ab ˙ S a ˙ β P ˙ αβ ˙ Q ˙ αb ˙ L ˙ β ˙ α + δ αβ ( D + B ) 0 00 − ( C + B ) 00 0 δ ˙ β ˙ α ( D − B ) . (2.9)Here use was made of (2.3). P ˙ αβ , Q aβ , ˙ Q ˙ αb in the lower-left blocks are generators of (su-per) translation while K α ˙ β , S αb , ˙ S a ˙ β in the upper-right blocks are generators of (super)boost. In the diagonal blocks L αβ , ˙ L ˙ β ˙ α , R ab are generators of the Lorentz subsymmetrySU(2) ⊗ SU(2)( ∈ SU(2,2)) and the R-symmetry SU(4), and
D, C, B are three U(1) charges. D is the dilatation. B never appears in the above superalgebrae of U(2,2 | ∼ (2.7).All the generators commute with C . Hence C is a central charge.Finally we get the quadratic Casimir in the form L αβ L βα − R ab R ba + L ˙ α ˙ β L ˙ β ˙ α + D − { P ˙ αβ , K β ˙ α } − [ Q aα , S αb ] − [ ˙ Q ˙ αa , ˙ S a ˙ α ] , (2.10)as can be checked by a direct calculation. | The superconformal transformations act on the N = 4 SUSY field strength multiplet givenin table 1. In quantum field theory they are represented as unitary linear transforma-tions in the Hilbert space. Hence the unitary representation of the superconformal group7SU(2,2 |
4) is the primary concern for quantization of the N =4 SUSY YM theory. SincePSU(2,2 |
4) is non-compact , the unitary representation is necessarily infinite-dimensional.The N =4 SUSY field strength multiplet is one of infinitely many multiplets in the uni-tary representation of PSU(2,2 | N =4 superconformalmultiplets, are known by a systematic analysis of the unitary representation[13, 14, 15].They are given in appendix A.A unitary operator ˆ U representing U(2,2 |
4) may be given byˆ U = e i ¯ ψMψ , (3.1)with ¯ ψ = ψ † γ [12]. Here M and γ are 8 × M = V θ Xθ † W ǫ − X † − ǫ † Z , γ = − , (3.2)in which V, W, Z are Hermitian matrices, X is a complex matrix, but θ (or ǫ ) is a 2 ⊗ ⊗
2) matrix of which elements are Grassmannian numbers. The unitarity of ˆ U followsfrom the Hermiticity of ¯ ψM ψ , i.e., ( ¯ ψM ψ ) † = ¯ ψM ψ . The vector ψ transforms covariantlyby the action of U(2,2 |
4) as ˆ U † ψ ˆ U = e iM ψ, (3.3)and ¯ ψ contravariantly as ˆ U † ¯ ψ ˆ U = ¯ ψe − iM . (3.4)The minus sign in M is a hallmark of non-compactness of U(2,2 | ψ in (2.8) as having creation and annihilation oscillators mixed.For representing the compact supergroup U(4 | ψ by annihilationoscillators alone. Consequently the minus sign is not needed for the block matrices X † and ǫ † in M . Then M is Hermitian in itself and γ is not needed either.We explain this point of the unitary(oscillator) representation by taking much simplergroups SU(1,1) and SU(2) as examples. Both Lie-algebrae are realized by using two pairsof oscillators ( a, a † ), and ( b, b † ). The non-trivial commutation relations are[ a, a † ] = 1 , [ b, b † ] = 1 . Then SU(1,1) is realized by the unitary operator (3.1) with ψ † = ( a † , b ) , ψ = (cid:18) ab † (cid:19) , M = (cid:18) V φ − φ ∗ − V (cid:19) , γ = (cid:18) − (cid:19) , while SU(2) by the unitary operator e iψ U † M U ψ U with ψ U † = ( a † , b † ) , ψ U = (cid:18) ab (cid:19) , M U = (cid:18) V φφ ∗ − V (cid:19) . ψM ψ and ψ U † M U ψ U we read the generators of the respective group asSU(1 ,
1) : T + = a † b † , T − = ab, T = 12 ( a † a + b † b ) , SU(2) : T U + = a † b, T U − = ab † , T U = 12 ( a † a − b † b ) , which satisfy the algebrae[ T + , T − ] = − T , [ T , T ± ] = ± T ± , and [ T + U , T − U ] = 2 T U , [ T U , T ± U ] = ± T ± U . Let | > to be the vacuum of the Fock space. Then we have( T + ) n a † | > = 0 , ( T + U ) n a † | > = 0 , for a positive integer n . Thus by means of the unitary operator (3.1) we can realize thenon-compact group U(1,1) in an infinite dimensional representation.We return to the main arguments on PSU(2,2 | |
4) acts on a Fock space given by all possible oscillator excitations Y α =1 ∞ Y n aα =1 ( a † α ) n aα Y ˙ α =1 ∞ Y n b ˙ α =1 ( b † ˙ α ) n b ˙ α Y a =1 ∞ Y n ca =1 ( c † a ) n ca | > . Thus it is the unitary representation of U(2,2 | |
4) is represented in a subsectorof the Fock space constrained by C = 1 − X α n a α + 12 X ˙ α n b ˙ α − X a n c a = 0 , (3.5)with C given in (2.3). If we have a α | > = 0 , b ˙ α | > = 0 c a | > = 0 , the vacuum | > is not in this subsector because C = 1. Hence we define a new physicalvacuum Z which has C = 0. It may be realized by Z = c † c † | > . (3.6)It is convenient to rename the whole fermionic oscillators c a , c † a , a = 1 , , , c , c ) ≡ c ¯ a , ( c , c ) = ( d † , d † ) ≡ d † ˙ a , ( c † , c † ) ≡ c † ¯ a , ( c † , c † ) = ( d , d ) ≡ d ˙ a . (3.7)9hen Z satisfies a α | > = 0 , b ˙ α | > = 0 , c ¯ a | > = 0 , d ˙ a | > = 0 . The physical Fock space is built up on this Z as Y α =1 ( a † α ) n aα Y ˙ α =1 ( b † ˙ α ) n b ˙ α Y ¯ a =1 ( c † ¯ a ) n c ¯ a Y ˙ a =3 ( d † ˙ a ) n d ˙ a Z. (3.8)The constraint (3.5) becomes C = X α =1 n a α − X ˙ α =1 n b ˙ α + X ¯ a =1 n c ¯ a − X ˙ a =3 n d ˙ a = 0 . According to this redefinition all the generators representing U(2,2 |
4) in (2.9) get thecentral charge C = 0. Among them the following generators non-trivially act on Z , Q ˙ aβ , ˙ Q ˙ α ¯ b , P α ˙ β , R ˙ a ¯ b , with the renamed indices by (3.7). To be explicit, they are a † β d † ˙ a , b † ˙ α c † b , a † α b † ˙ β , c † ¯ b d † ˙ a . Acting on Z the fermionic generators create the states as shown in table 2[13]. Theyexactly correspond to the fundamental fields of the N =4 SUSY field strength multi-plet in table 1. Furthermore acting on those states P α ˙ β = a † α b † ˙ α and R ˙ a ¯ b = c † ¯ b d † ˙ a createSU(2) ⊗ SU(2) excited states with the Dynkin label [ k, k ] respectively. The former excita-tion implies space-time derivative D of the N = 4 SUSY field strength multiplet. (Seetable 1.) The latter excitation occurs in the representation space of the R-symmetrySU(4). All of these states have the central charge C = 0, so that they are indeed inthe infinite-dimensional unitary representation of PSU(2,2 | Z . In particular the fermionic ones are given by Q ¯ aβ , ˙ Q ˙ α ˙ b , (3.9)which are a † β c ¯ a , b † ˙ α d ˙ b . They are half of the 16 supercharges. Thus the states in table 2 form a half-multiplet[13].They are the smallest BPS multiplet. The vacuum Z is the highest weight vector of themultiplet, which is denoted by the SU(2,2) Dynkin label [0,1,0].Larger BPS multiplets for the N = 4 SUSY theory, i.e., other N = 4 superconformalmultiplets, can be also constructed by generalizing the above construction. It will be donein appendix A to complete the argument. 10eld states F a † α a † β d † d † Z Ψ a † α d † ˙ a Z, a † α c † ¯ a d † d † Z Φ Z , c † ¯ a d † ˙ a Z , c † c † d † d † Z ˙Ψ b † ˙ α c † ¯ a Z, b † ˙ α d † ˙ a c † c † Z ˙ F b † ˙ α b † ˙ β c † c † Z Table 2: Oscillator representation of the N = 4 SUSY field strength multiplet. | So far we have considered the unitary(oscillator) representation of U(2,2 |
4) taking a baseobtained by the tensor product ψ † ⊗ ψ . In this section we discuss a matrix representationof U(2,2 |
4) which is induced from the unitary(oscillator) representation by (3.3) and (3.4).To this end we put 64 generators in a base which manifests U(2,2 |
4) more faithfully than(2.9), i.e., T αβ T αb T α ˙ β T aβ T ab T a ˙ β T ˙ αβ T ˙ αb T ˙ α ˙ β . (4.1)Using an 8 × t γδ t γd t γ ˙ δ t cδ t cd t c ˙ δ t ˙ γδ t ˙ γd t ˙ γ ˙ δ , we write the generators as T αβ = δ αδ δ γβ , T aβ = δ ad δ γβ
00 0 00 0 0 , T ˙ αβ = δ γβ δ ˙ α ˙ δ ,T αb = δ αδ δ cb , T ab = δ ad δ cb
00 0 0 , T ˙ αb = δ cb δ ˙ α ˙ δ , (4.2) T α ˙ β = − δ αδ δ ˙ γ ˙ β , T a ˙ β = − δ ˙ γ ˙ β δ ad , T ˙ α ˙ β = − δ ˙ α ˙ δ δ ˙ γ ˙ β . minus sign in the last line which accounts for non-compactnessof U(2,2 | ⊗ U(4) [ T αβ , T γδ ] = − δ γβ T αδ + δ αδ T γβ , [ T ˙ α ˙ β , T ˙ γ ˙ δ ] = δ ˙ γ ˙ β T ˙ α ˙ δ − δ ˙ α ˙ δ T γβ , [ T ab , T cd ] = − δ cb T ad + δ ad T cd , (4.3)[ T α ˙ β , T ˙ γδ ] = δ ˙ γ ˙ β T αδ − δ αδ T ˙ γ ˙ β . Anti-commuting fermionic generators in the off-diagonal blocks with each other yields { T αb , T aβ } = δ αβ T ab + δ ab T αβ , { T a ˙ β , T ˙ αb } = − δ ab T ˙ α ˙ β + δ ˙ α ˙ β T ab , { T αb , T a ˙ β } = δ ab T α ˙ β , { T aβ , T ˙ αb } = δ ab T ˙ αβ . (4.4)Commuting these fermionic generators with bosonic generators yields[ T αb , T γδ ] = δ αδ T γb , [ T αb , T cd ] = − δ cb T αd , [ T αb , T ˙ γ ˙ δ ] = 0 , [ T aβ , T γδ ] = − δ γβ T aδ , [ T aβ , T cd ] = δ ad T cβ , [ T aβ , T ˙ γ ˙ δ ] = 0 , [ T a ˙ β , T γδ ] = 0 , [ T a ˙ β , T cd ] = δ ad T c ˙ β , [ T a ˙ β , T ˙ γ ˙ δ ] = δ ˙ γ ˙ β T a ˙ δ , (4.5)[ T ˙ αb , T γδ ] = 0 , [ T ˙ αb , T cd ] = − δ cb T ˙ αd , [ T ˙ αb , T ˙ γ ˙ δ ] = − δ ˙ α ˙ δ T ˙ γb . while commuting them with bosonic generators[ T αb , T γ ˙ δ ] = 0 , [ T αb , T ˙ γδ ] = δ αδ T ˙ γb , [ T aβ , T γ ˙ δ ] = − δ γβ T a ˙ δ , [ T aβ , T ˙ γδ ] = 0 , [ T a ˙ β , T γ ˙ δ ] = 0 , [ T a ˙ β , T ˙ γδ ] = δ ˙ γ ˙ β T aδ , (4.6)[ T ˙ αb , T γ ˙ δ ] = − δ ˙ α ˙ δ T γb , [ T ˙ αb , T ˙ γδ ] = 0 . All other (anti-)commutation relations are vanishing. The diagonal blocks contain thegenerators of the subgroup SU(2) ⊗ SU(2) ⊗ SU(4) given by L αβ = T αβ − δ αβ T γγ , ˙ L ˙ α ˙ β = T ˙ α ˙ β − δ ˙ α ˙ β T ˙ γ ˙ γ , R ab = T ab − δ ab T cc , (4.7)and three U(1) generators defined by D = 12 δ γδ − δ ˙ γ ˙ δ = 12 ( T αα + T ˙ α ˙ α ) , − C = 12 δ γδ δ cd
00 0 δ ˙ γ ˙ δ = 12 ( T αα − T ˙ α ˙ α + T aa ) , (4.8)12 = 12 δ γδ δ ˙ γ ˙ δ = 12 ( T αα − T ˙ α ˙ α ) . They are identical to the one given in the unitary(oscillator) representation (2.3). U(2,2 | |
4) or PU(2,2 |
4) when the U(1) generators are constrained by B = 0 or C = 0 respectively. When imposed both constraints, it becomes PSU(2,2 | ∼ (4.5). The firstthree algebrae in (4.3) remain in the same form[ L αβ , L γδ ] = − δ γβ L αδ + δ αδ L γβ , [ ˙ L ˙ α ˙ β , ˙ L ˙ γ ˙ δ ] = δ ˙ γ ˙ β ˙ L ˙ α ˙ δ − δ ˙ α ˙ δ ˙ L ˙ γ ˙ β , [ R ab , R cd ] = − δ cb R ad + δ ad R cd . (4.9)Other algebrae in (4.3) ∼ (4.5) also do not change significantly the forms, except for thelast algebra in (4.3) and the first two in (4.4). Those are found to be[ T α ˙ β , T ˙ γδ ] = δ ˙ γ ˙ β L αδ + δ αδ ˙ L ˙ γ ˙ β + δ ˙ γ ˙ β δ αδ D, { T αb , T aβ } = δ αβ R ab + δ ab L αβ + 12 δ αβ δ ab ( D − C ) , (4.10) { T a ˙ β , T ˙ αb } = δ ab ˙ L ˙ α ˙ β − δ ˙ α ˙ β R ab + 12 δ ab δ ˙ α ˙ β ( D + C ) . The quadratic Casimir is given by L αβ L βα − R ab R ba + ˙ L ˙ α ˙ β ˙ L ˙ β ˙ α + D − { T α ˙ β , T ˙ βα } − [ T aβ , T βa ] − [ T ˙ βa , T a ˙ β ] . (4.11)Now we compare the algebrae (4.3) ∼ (4.6) with (2.4) ∼ (2.7) in the unitary(oscillator)representation. We find them to be equivalent by redefining the generators as ǫ ˙ α ˙ δ ǫ ˙ β ˙ γ ˙ L ˙ γ ˙ δ = ˙ L ˙ α ˙ β = − ˙ L ˙ α ˙ β , ǫ ˙ β ˙ δ T α ˙ δ = K α ˙ β , ǫ ˙ α ˙ γ T ˙ γβ = P ˙ αβ ,ǫ ˙ β ˙ δ T a ˙ δ = ˙ S a ˙ β , ǫ ˙ α ˙ γ T ˙ γb = ˙ Q ˙ αb , L αβ = L αβ , R ab = R ab , S αb = T αb , T aβ = Q aβ , D = D. The redefinition does not change the form of the quadratic Casimir (4.11). It coincideswith the quadratic Casimir (2.10), given in the unitary(oscillator) representation. But theredefintion changes the sign of the algebrae linearly containing ˙ L ˙ α ˙ β in (4.3) ∼ (4.6). Forinstance, the second one in (4.9) becomes that of (2.4).We compare also the algebra in (4.3) ∼ (4.6) with those of U(4 |
4) and U(8). If thematrices (4.2) get all entries with plus sign, i.e., ( T AB ) CD = δ AD δ CB , they become the gener-ators of U(4 | ∼ (4.6) where T α ˙ β , T a ˙ β , T ˙ α ˙ β get the signchanged. Accordingly the quadratic Casimir (4.11) changes the form as X B = β, ˙ β,b T αB T Bα + X B = β, ˙ β,b T ˙ αB T B ˙ α − X B = β, ˙ β,b T aB T Ba = ( − g ( A ) T AB T BA . g ( A ) to the index A in such a way g ( A ) = 1 for afermionic index and otherwise g ( A ) = 0. So T AB has the grading g ( A ) g ( B ). If we do notassign the grading, T AB satisfies the Lie-algebra of U(8)[ T AB , T CD ] = − δ CB T AD + δ AD T CB . being defined as ( T AB ) CD = δ AD δ CB , The the quadratic Casimir of U(8) is simply X B = β, ˙ β,b T αB T Bα + X B = β, ˙ b T ˙ αB T B ˙ α + X B = β, ˙ β,b T aB T Ba = T AB T BA . Or we had better formulate the superalgebrae of U(2,2 |
4) and U(4 |
4) in a converse way,i.e., starting with this form of the algebra of U(8) instead of the graded form of (4.3) ∼ (4.6). | Both the unitary(oscillator) representation and the matrix one allow linear realization ofPSU(2,2 |
4) only as a subgroup of its centrally extended group SU(2,2 | |
4) is a simple group so that we donot need the central extension at the algebraic level. In this section we want to discussa purely algebraic method to non-linearly realize PSU(2,2 | | |
4) in a common form by which wecan freely change the unitary(oscillator) representation to the matrix one and vice versa .Then using that algebra we give general accounts of non-linear realization of PSU(2,2 | | ⊗ SO(5), which is themain concern in this paper.
Let us put the generators of PU(2,2 |
4) in a row and denote them by { T Ξ } . That is, 62generators in the unitary(oscillator) representation, discussed in section 3, are denoted by { T Ξ } = { L αβ , ˙ L ˙ α ˙ β , R ab , D, P ˙ αβ , K β ˙ α , S αb , Q aα , ˙ S a ˙ α , ˙ Q ˙ αa } , (5.1)while the corresponding generators in the matrix representation, discussed in section 4,by { T Ξ } = {L αβ , ˙ L ˙ α ˙ β , R ab , D , P ˙ αβ , K β ˙ α , S αb , Q aα , ˙ S a ˙ α , ˙ Q ˙ αa } . (5.2)Using either set of these 62 generators we represent PSU(2,2 |
4) in a common form as e iM Ξ T Ξ ∈ PSU(2 , | , (5.3)14n which M Ξ are 62 elements of the supermatrix M given in (3.2) { M Ξ } = n V αβ − δ αβ , Z ˙ α ˙ β − δ ˙ α ˙ β , W ab − δ ab ,D, − X † ˙ αβ , X β ˙ α , θ αb , θ † aα , ǫ a ˙ α , − ǫ † ˙ αa o . We find explicit forms of M Ξ T Ξ for the respective representations, expanding ¯ ψM ψ and M in terms of the generators (5.1) and (5.2). The expansion of the former reads¯ ψM ψ ≡ X B = β, ˙ β,b ψ † α ( γM ) αB ψ B X B = β, ˙ β,b ψ † ˙ α ( γM ) ˙ αB ψ B + X B = β, ˙ β,b ψ † a ( γM ) aB ψ B = h V αβ L βα + X α ˙ β P ˙ βα + θ α a Q aα i + h ( − Z ˙ α ˙ β L ˙ β ˙ α + Z ˙ γ ˙ γ ) + X † ˙ αβ K β ˙ α + ǫ † ˙ αa ˙ S a ˙ α i + h W ab R ba − θ † aα S αa − ǫ a ˙ α ˙ Q ˙ αa i + 12 ( V γγ − Z ˙ γ ˙ γ ) D + 12 ( V γγ + Z ˙ γ ˙ γ ) B − W cc ( B + C ) ≡ M Ξ T Ξ + 12 ( V γγ + Z ˙ γ ˙ γ ) B − W cc ( B + C ) , (5.4)by using the commutation relations (2.1) and the generators defined by (2.2) and (2.3).On the other hand the expansion of the latter reads M = X B = β, ˙ β,b M αB · γT Bα + X B = β, ˙ β,b M ˙ αB · γT B ˙ α + X B = β, ˙ β,b M aB · γT Ba = h V αβ L βα + X α ˙ β P ˙ βα + θ α b Q aα i + h Z ˙ α ˙ β ( − ˙ L ˙ β ˙ α ) − X † ˙ αβ ( −K β ˙ α ) − ǫ † ˙ αa ( − ˙ S a ˙ α ) i + h W ab R ba + θ † aα S αb + ǫ a ˙ α ˙ Q ˙ αa i + 12 ( V γγ − Z ˙ γ ˙ γ ) D + 12 ( V γγ + Z ˙ γ ˙ γ ) B − W cc ( B + C ) ≡ M Ξ T Ξ + 12 ( V γγ + Z ˙ γ ˙ γ ) B − W cc ( B + C ) , (5.5)by using the generators defined by (4.7) and (4.8) and noting ( γT AB ) CD = δ AD δ CB . A signdifference in the third square brackets [ · · · ] of the the respective expansions (5.4) and (5.5)does not indicate anything wrong. This is due to the different prescription in grading T Ξ in both representations. In (5.4) we have employed the prescription T Ξ M Φ = ( − g (Ξ) g (Φ) M Φ T Ξ . (5.6)Here the grading of T Ξ is the same as M Ξ , i.e., g (Ξ) = g ( A ) g ( B ) when { T Ξ } is put inthe tensor form { ψ † A ψ B } as (2.9). Hence ¯ ψM ψ is a bosonic operator acting on the Fockspace (3.8). On the other hand, in (5.5) we have employed the prescription T Ξ M Φ = M Φ T Ξ , (5.7)15ssigning no grading to T Ξ . This is also reasonable because the generators (5.2) consist ofbosonic elements as (4.2) and commute any element of M . The reader may see appendixB for more arguments on these prescriptions.It is the fact that exp( i ¯ ψM ψ ) and exp( iM ) with (5.4) and (5.5) are related by theoperator-state relations (3.3) and (3.4). Note that owing to these relations the multi-plication exp( i ¯ ψM ψ ) × exp( i ¯ ψM ψ ) in the Fock space induces that of supermatrices as M M . Thus we are now in a position to discuss the coset space PSU(2,2 | |
4) , given by either (5.1) or (5.2), under a subgroup H as { T Ξ } = { T i , H I } , (5.8)in which H I are generators of H, while T i coset ones. Then we consider a coset element e iφ T + iφ T + ··· ≡ e iφ · T . (5.9)Here φ , φ , · · · , are coordinates reparametrizing the coset space, denoted by φ ¯ i . (Wekeep the index i for indicating a vector component in the tangent frame as (5.8) .) Ofcourse they have the same grading as the coset part of M Ξ , i.e., g (¯ i ) = g (Ξ) | Ξ= i . For leftmultiplication of an element e iM Ξ T Ξ ∈ G, (5.3), the coset element changes as e iM Ξ T Ξ e φ · T = e φ ′ ( φ ) · T e iρ ( φ,M ) , (5.10)with an appropriate compensator e iρ ( φ,M ) . This defines a transformation of the coordinates φ ¯ i → φ ′ ¯ i ( φ ). When M Ξ are infinitesimally small, this relation defines the Killing vectors R Ξ¯ i as δφ ¯ i = φ ′ ¯ i ( φ ) − φ ¯ i ≡ M Ξ R Ξ¯ i ≡ M Ξ δ Ξ φ ¯ i . (5.11)They satisfy the Lie-algebra of PSU(2,2 | R Ξ¯ i ∂∂φ ¯ i R Φ¯ j − ( − g (Ξ) g (Φ) R Φ¯ i ∂∂φ ¯ i R Ξ¯ j = f ΞΦΣ R Σ¯ j , (5.12)with the structure constants f ΞΦΣ of PSU(2,2 | R Ξ¯ i works at thealgebraic level, once given the Lie-algebra of the generators T Ξ . We give a demonstrationfor this in appendix B. Hence the forms of the Killing vectors R Ξ¯ i are the same if tworepresentations take the same form of the Lie-algebra, like the unitary(oscillator) repre-sentation (5.1) and the matrix one (5.2). Moreover the Killing vectors R Ξ¯ i are free fromany extra U(1) factor of the central charge C , since the calculation is purely algebraic. On16he contrary, if the construction is done by using the unitary(oscillator) representation orthe matrix one, in (5.10) the compensator acquires an extra U(1) factor as e ρ ( φ,M ) = e ρ ( φ,M ) I H I + c ( φ,M ) C ∈ H ⊗ U (1) , even though e iM Ξ T Ξ does not have it. Here ρ ( φ, M ) I and c ( φ, M ) are appropriate functionsof φ ¯ i . This is due to the fact that the Lie-algebra of PSU(2,2 |
4) is merely realized byembedding it in SU(2,2 | R Ξ a realize the Lie-algebra of PSU(2,2 | | { SO(1,4) ⊗ SO(5) } So far non-linear realization of PSU(2,2 |
4) has been discussed on the coset space PSU(2,2 | ⊗ SO(6) to proceedwith our discussions. First of all we note thatSU(2 , ⊗ SU(4) ∼ = SO(2 , ⊗ SO(6) ⊃ SO(1 , ⊗ SO(5) . The matrix representation of SU(2,2) ⊗ SU(4) so far discussed can be identified with thechiral spinor representation of SO(2,4) ⊗ SO(6). The Dirac algebrae of SO(2,4) and SO(5)respectively read { Γ p , Γ q } = 2 η pq = 2( − , , , , , − , p, q = 0 , , · · · , , { ˆΓ ˆ p , ˆΓ ˆ q } = 2ˆ η ˆ p ˆ q = 2( 1 , , , , , , ˆ p, ˆ q = 0 , , · · · , . In the chiral spinor representation the Dirac matrices of SO(2,4) are, for example, givenby Γ = − ! , Γ m = γ m γ m ! , (5.13)with 4 × γ matrices satisfying { γ m , γ n } = 2 η mn = 2( − , , , , , m, n = 0 , , , · · · , . On the other hand the Dirac matrices of SO(6) are given byˆΓ = ! , ˆΓ ˆ m = i ˆ γ ˆ m − i ˆ γ ˆ m ! , (5.14)with 4 × γ matrices satisfying { ˆ γ ˆ m , ˆ γ ˆ n } = 2ˆ η ˆ m ˆ n = 2( 1 , , , , , ˆ m, ˆ n = 1 , , · · · , .
17y using these Dirac matrices the generators of SO(2,4) and SO(6) are given byΓ pq = 14 [Γ p , Γ q ]P + , ˆΓ ˆ p ˆ q = 14 [ˆΓ ˆ p , ˆΓ ˆ q ] ˆP + . The chiral projectors take the diagonalized formsP + = 12 (1 + Γ ) = ! , ˆP + = 12 (1 + ˆΓ ) = ! , when we choose the Weyl representation γ = − ! , γ = σ σ ! , γ = σ σ ! ,γ = σ σ ! , γ = iγ γ γ γ , for γ m in (5.13) and a similar representation for γ ˆ m in (5.14). We have one to one corre-spondence between the generators of SU(2,2) ⊗ SU(4) in (5.2) and those of SO(2,4) ⊗ O(6)as {L αβ } = { Γ , Γ , Γ } , { P α ˙ β } = { Γ µ + Γ µ } , { ˙ L ˙ α ˙ β } = { Γ , Γ , Γ } , { K ˙ αβ } = { Γ µ − Γ µ } , D = Γ , { R ab } = { ˆΓ ˆ p ˆ q } with µ = 0 , , , ⊗ SO(6) under SO(1,4) ⊗ SO(5) asΓ pq = Γ mn Γ m Γ n , m, n = 0 , , · · · , , ˆΓ ˆ p ˆ q = ˆΓ ˆ m ˆ n Γ ˆ m ˆΓ n , ˆ m, ˆ n = 0 , , · · · , . Using this basis we rewrite the generators of PSU(2,2 |
4) in the matrix representation,given by (5.2), as { T Ξ } = { Γ mn , Γ m , ˆΓ ˆ m ˆ n , ˆΓ ˆ m , S αb , Q aα , ˙ S a ˙ α , ˙ Q ˙ αa } . (5.15)Now we are in a position to construct the coset superspace PSU(2,2 | ⊗ SO(5),following the general method given previously. The coset element e iφ · T , given by (5.9),takes an explicit form with { T i } = { Γ m , ˆΓ ˆ m , S αb , Q aα , ˙ S a ˙ α , ˙ Q ˙ αa } . | ⊗ SO(5) transitively. They are identifiedwith the corresponding generators of the D = 10 Poincar´e superalgebra at the origin ofthe coset superspace. After this identification it is natural to rename the generators of { T i } as { Γ m , ˆΓ ˆ m } = { Γ M } , {S αb , Q aα , ˙ S a ˙ α , ˙ Q ˙ αa } = {Q M } , (5.16)with M = 0 , , , · · · , M = 1 , , · · · ,
32. Correspondingly the coordinates φ ¯ i reparametrizing PSU(2,2 | ⊗ SO(5) are renamed as { φ ¯ i } = { X M , Θ M } . (5.17)They are identified with supercoordinates in the D = 10 curved spacetime. By usingthem we may write the coset element (5.9) in a form looking like a vertex operator of theGreen-Schwarz string theory as e iφ · T = e i ( X · Γ+Θ ·Q ) . (5.18)The Killing vectors defined by (5.11) are found as functions of the the D = 10 superco-ordinates, δ Ξ φ ¯ i = R Ξ¯ i ( φ ) = (cid:16) R Ξ M ( X, Θ) , R Ξ M ( X, Θ) (cid:17) . (5.19) In the previous section we have discussed that the coset element (5.18) looks like a vertexoperator and it transforms according to (5.10), i.e., e i ( X · Γ+Θ ·Q ) −→ e iX ′ ( X, Θ) · Γ+Θ ′ ( X, Θ) ·Q ) = e iM Ξ T Ξ e i ( X · Γ+Θ ·Q ) e − iρ ( X, Θ ,M ) , (6.1)in which the non-linear transformations X ′ ( X, Θ) and Θ ′ ( X, Θ) are generated by theKilling vectors (5.19). The arguments have been given in an algebraic way which doesnot relies on either of the unitary(oscillator) representation and the matrix one. Howeverlet us now choose the matrix representation. Then the transformation (6.1) is written byan 8 × η ( X, Θ) transforming as η ( X, Θ) −→ η ′ ( X ′ , Θ ′ ) = e iρ ( X, Θ ,M ) η ( X, Θ) , (6.2)by the non-linear transformations X ′ ( X, Θ) and Θ ′ ( X, Θ), then (6.1) becomes e i ( X · Γ+Θ ·Q ) η ( X, Θ) −→ e iM Ξ T Ξ e i ( X · Γ+Θ ·Q ) η ( X, Θ) . (6.3)It implies that e i ( X · P +Θ ·Q ) η ( X, Θ) is a covariant vector under PSU(2,2 | X, Θ) = e i ( X · Γ+Θ ·Q ) η ( X, Θ)) . (6.4)19he transformation is exactly the same as for the 8-d column vector ψ = a α c a b † ˙ α , (6.5)which was defined by (2.8). Making the identificationΥ( X, Θ) = ψ, (6.6)we claim that this is a non-linear realization of the oscillators.The remaining question is whether the quantity η with the transformation property(6.2) really exists. In [11] the existence was shown for the general bosonic coset space G/Hin an arbitrary, but finite representation of the coset element e iφ · T . We have chosen thematrix representation to discuss the coset space PSU(2,2 | { SO(1,4) ⊗ SO(5) } . Therefore η exists for this case similarly. Here we recall only of the point of the arguments andexplain the quantity more explicitly for the coset space PSU(2,2 | { SO(1,4) ⊗ SO(5) } .First of all we consider the Cartan-Maurer 1-form g − dg = i ( e i ¯ j T i + ω I ¯ j H I ) dφ ¯ j , (6.7)denoting the coset element (5.18) as g and using the index notation (5.17). This definesthe vielbein e i ¯ j and the connection ω I ¯ j in the tangent frame of the coset space. Underthe transformation (6.1) they transform as e i ¯ j dφ ¯ j −→ [ e iρ ( φ,M ) ] ik e k ¯ j dφ ¯ j ,ω I ¯ j H I dφ ¯ j −→ e − iρ ( φ,M ) [ d − iω I ¯ j H I dφ ¯ j ] e iρ ( φ,M ) . Then we have the Wilson line-operator W ( φ, φ ) = P exp i Z φφ ω I ¯ j H I dφ ¯ j , which transforms as W ( φ, φ ) −→ e iρ ( φ,M ) W ( φ, φ ) e − iρ ( φ ,M ) . The compensator e iρ ( φ,M ) becomes a constant element at the origin φ = 0 of the cosetspace, i.e., e iρ ( φ,M ) (cid:12)(cid:12)(cid:12) φ =0 = e iM I H I ∈ H = SO(1 , ⊗ SO(5) . (6.8)Let e iM I H I η to be a linear representation vector with M I parametrizing the subgroup H.Then it transforms by the compensator (6.8) at the origin as e iM I H I η −→ e iM I H I e M I H I η . η is a constant vector fixed in the representation space of H. To be concrete for thecase of PSU(2,2 | { SO(1,4) ⊗ SO(5) } , we have e iM I H I η = e i ( M mn Γ mn + M ˆ m ˆ n ˆΓ ˆ m ˆ n ) η . by using the generators in (5.15). Hence η is now a constant chiral spinor of SO(1,4) ⊗ SO(5).As the result we find the quantity η ( X, Θ) η ( φ ) = W ( φ, e iM I H I η , with { φ ¯ i } = { X M , Θ M } , which has the transformation property (6.2). Thus we havejustified the identification (6.4) with η ( X, Θ) of this form.
In the previous section we have identified the Killing scalar Υ( X, Θ) of the coset spacePSU(2,2 | { SO(1,4) ⊗ SO(5) } with the 8-d column vector ψ given by (6.5). In [11] thegeneral accounts for the Killing scalar were given for the ordinary coset space G/H, i.e.,G is not a supergroup. It was shown that it satisfies the classical exchange algebra of Gin the non-linear σ -model on G/H with the Poisson brackets set up on the light-like line.For this it was essential to have the linear transformation property (6.3), i.e., δ Ξ Υ = T Ξ Υ , (7.1)by the Killing vectors (5.19). In this section we show that this is also true for the Killingscalar (6.4) of the non-linear σ -model on PSU(2,2 | { SO(1,4) ⊗ SO(5) } . The identifica-tion (6.6) implies that the 8-d covariant vector ψ given by (6.5) satisfies the classicalexchange algebra of PSU(2,2 | σ -model on PSU(2,2 | { SO(1,4) ⊗ SO(5) } S = 12 Z d x η + − ( e i ¯ j ∂ + φ ¯ j )( e i ¯ k ∂ − φ ¯ k )( − g ( i ) , (7.2)with the vielbein e i ¯ j defined by (6.7) and the supercoordinates φ ¯ i given by (5.17). Herewe have the graded summation for the index i according the quadratic Casimir (4.11).We set up the Poisson brackets on the light-like line x + = y + { φ ¯ i ( x ) ⊗ , φ ¯ j ( y ) } = − h θ ( x − y ) t +ΦΞ δ Ξ φ ¯ i ( x ) ⊗ δ Φ φ ¯ j ( y ) − θ ( y − x ) t +ΦΞ δ Ξ φ ¯ j ( y ) ⊗ δ Φ φ ¯ i ( x )( − g (¯ i ) g (¯ j ) ( − g (Ξ) g (Φ) i . (7.3)The notation is as follows. θ ( x ) is the step function. δ Ξ φ ¯ i ( x ) are the Killing vectors definedby (5.11). More correctly they should be written as δ Ξ φ ¯ i (( φ ( x )), but the dependence of21 ¯ i ( x ) was omitted to avoid an unnecessary complication. The quantity t +ΞΦ is the mostcrucial in our arguments. It is a modified Killing metric of t ΞΦ . By means of it wedefine the classical r-matrix satisfying the classical Yang-Baxter equation. To explainthis quantity let us remember the definition of the classical r-matrices for the ordinarygroup r ± = X α ∈ R sgn αE α ⊗ E − α ± X A,B t AB T A ⊗ T B ≡ t ± AB T A ⊗ T B . (7.4)Here T A denote the generators of the group G with t AB the Killing metric. They aregiven in the Cartan-Weyl basis as { E ± α , H µ } with sgn α = ± according as the roots arepositive or negative. Note the relation t + AB = − t − BA . The r-matrix satisfies the classicalYang-Baxter equation [ r xy , r xz ] + [ r xy , r yz ] + [ r xz , r yz ] = 0 . Here the r-matrix acts at on a tensor product of the Killing scalars Υ( φ ( x )) ⊗ Υ( φ ( y )) ⊗ Υ( φ ( z )) but only at the designated positions[11, 18]. For the supergroup PSU(2,2 | X ΞΦ t ΦΞ T Ξ T Φ ≡ L αβ L βα − R ab R ba + ˙ L ˙ α ˙ β ˙ L ˙ β ˙ α + D − { T α ˙ β , T ˙ βα } − [ T aβ , T βa ] − [ T a ˙ β , T ˙ βa ] . Correspondingly to this expression the r-matrix of PSU(2,2 |
4) is given by r + ≡ X ΞΦ t +ΦΞ T Ξ T Φ = X α>β L αβ L βα − X a>b R ab R ba + X ˙ α> ˙ β ˙ L ˙ α ˙ β ˙ L ˙ β ˙ α − X all α, ˙ β T α ˙ β T ˙ βα − X all aβ T aβ T βa − X all a ˙ β T a ˙ β T ˙ βa + X ΞΦ t ΦΞ T Ξ T Φ . That is, t +ΞΦ in (7.3) is a simple generalization of the quantity in (7.4) for the case ofPSU(2,2 | r xy , r xz } + [ r xy , r yz } + [ r xz , r yz } = 0 , (7.5)with the graded commutator [ , } . Note that now we have t ΞΦ = ( − g (Ξ) g (Φ) t ΦΞ , t +ΞΦ = − ( − g (Ξ) g (Φ) t − ΦΞ . Then it follows that { φ ¯ i ( x ) ⊗ , φ ¯ j ( y ) } = ( − g (¯ i ) g (¯ j ) { φ ¯ j ( y ) ⊗ , φ ¯ i ( x ) } , |
2) and OSP(2 |
2) in [19, 20]. Therethe r-matrix of the respective supergroup appeared as showing integrability of the D = 2,(1 ,
0) and (2,0) effective gravity.Finally we can show the consistency of the Poisson brackets (7.3). First of all itsatisfies the Jacobi identities owing to the classical Yang-Baxter equation for the r-matrix.Secondly the energy-momentum tensor of the non-linear σ -model (7.2) reproduces thediffeomorphism δ diff φ ¯ i ( x + , x − ) ≡ ǫ ( x − ) ∂ − φ ¯ i ( x + , x − )= Z dy − ǫ ( y − ) { φ ¯ i ( x ) , T −− ( φ ( y )) } (cid:12)(cid:12)(cid:12) x + = y + , Thirdly the Poisson brackets tend to those of the free boson and fermion theory as { φ ¯ i ( x ) ⊗ , φ ¯ j ( y ) } = −
14 [ θ ( x − y ) δ ¯ i ¯ j − θ ( y − x ) δ ¯ j ¯ i ( − g (¯ i ) g (¯ j ) ] . These statements can be verified in the same way as for the ordinary non-linear σ -model.With the Poisson brackets (7.3) let us calculate { Υ( x ) ⊗ , Υ( y ) } for the Killing scalarΥ using the property { φ ¯ i ( x ) ⊗ , Υ( y ) } = { φ ¯ i ( x ) ⊗ , φ ¯ j ( y ) } δ Υ( y ) δφ ¯ j ( y ) , together with (7.1). We then get the classical exchange algebra in the form { Υ( x ) ⊗ , Υ( y ) } = −
14 [ θ ( x − y ) r + + θ ( y − x ) r − ]Υ( x ) ⊗ Υ( y ) , (7.6)on the light-like plane x + = y + . Here Υ( x ) should be understood with an abbreviatednotation for Υ( φ ( x )). It is a non-linear realization of the oscillators by the identification(6.4). Thus the oscillators obey the classical exchange algebra (7.6).The supergroup PSU(2,2 |
4) is a simple group. Hence we may use the plug-in formulato promote the r-matrix to the universal R-matrix R xy . It is expressed purely in terms ofgenerators of G. Then (7.6) becomesΥ( x ) ⊗ Υ( y ) = θ ( x − y ) R + xy Υ( y ) ⊗ Υ( x ) + θ ( y − x ) R − xy Υ( y ) ⊗ Υ( x ) . (7.7)Here the universal R-matrix satisfies the quantum Yang-Baxter equation R xy R xz R yz = R yz R xz R xy . (7.8)(7.6) and (7.5) are the respective classical correspondents of (7.7) and (7.8) obtained by R ± xy = 1 + hr ± xy + O ( h ) , h = − . For G=SU(2) or SU(1,1) the plug-in formula of the universal R-matrix isgiven in a rather simple form as R = q H ⊗ H ∞ X n =0 q n ( n − ( q − q − ) n [ n ] q ! ( E + ) n ⊗ ( E − ) n = exp q [( q − q − ) E + ⊗ E − ] , (7.9)by using the notation (7.4) for the Lie-algebra. Here the q -exponential is defined byexp q ( x ) = ∞ X n =0 q n ( n − x n [ n ] q ! , with q = e h and [ n ] q = q n − q − n q − q − . The formula (7.9) can be generalized for the general simple group. The generalized plug-in formula for the ordinary group can be found in [4]. That for supergroups was given in[5]. We consider a correlation function < V ( x ) ⊗ V ( x ) ⊗ · · · ⊗ V i ( x i ) ⊗ V i +1 ( x i +1 ) ⊗ · · · ⊗ V N ( x N ) > . (7.10)Here V i ( x i ) are the N = 4 SUSY field strength multiplet and their descendants in table1, i.e., {V ( x ) } = { D k F ( x ) , D k Ψ( x ) , D k Φ( x ) , D k ˙Ψ( x ) , D k ˙ F ( x ) } , k = 0 , , , · · · . They are arrayed on the light-like line x +1 = x +2 = · · · = x + N as shown in figure 2.Figure 2: Arrows stand for SU(2)-spins of the oscillators a α , b ˙ α , c ¯ a , d ˙ a . Encircling themindicates a component of the N = 4 field strength multiplet V i ( x i ).Each of the multiplets is given in terms of the oscillators as in table 2. Let V i ( x i ) and V i +1 ( x i +1 ) at the adjacent positions to be one of the components belonging to D α ˙ β Φ andΨ respectively. For instance, we have V i ( x i ) = ( a † α ( x i ) b † ˙ α ( x i ))( c † ¯ a ( x i ) d † ˙ a ( x i )) , V i +1 ( x i +1 ) = a † α ( x i +1 ) c † ¯ a ( x i +1 ) d † ( x i +1 ) d † ( x i +1 ) , D α ˙ β = P α ˙ β = a † α b † ˙ β . The quantum exchange algebra for the field strength multiplet V i ( x i ) follows from (7.7), because the oscillators are identified with the Killing scalar Υas (6.6). That is, the quantum exchange algebra is obtained by braiding the respectiveoscillators in V i ( x i ) and V j ( x j ) one by one. The R-matrix is now in an infinite-dimensionalrepresentation.The N = 4 SUSY field strength multiplet V is the simplest one. The arguments canbe similarly applied to other multiplets than N = 4 SUSY field strength multiplet V . Asummary of the general superconformal multiplets is given in appendix A. Thus we are led to conclude that the supercoherent space of the unitary(oscillator) repre-sentation of PSU(2,2 |
4) becomes non-commutative. To show this, we have constructed theoscillators as the Killing scalar Υ in the non-linear σ -model on PSU(2,2 | { SO(1,4) ⊗ SO(5) } .It was argued that they satisfy the exchange algebra when the model is quantized on thelight-like line. They took a suggestive form of the vertex operator of the Green-Schwarzsuperstring as given by (6.4), i.e.,Υ = e i ( X · P +Θ ·Q ) η ( X, Θ)) . We comment on the D = 10 flat space-time limit of the non-linear σ -model onPSU(2,2 | { SO(1,4) ⊗ SO(5) } . Taking a naive limit where the AdS ⊗ S radius tendsto ∞ does not give the desired D = 10 super-Poincar´e invariance to the non-linear σ -model. According to [10] a correct way to go to the the flat space-time limit is to rescalethe structure constants in the algebrae (2.6 ) as { S αb , Q aβ } = 1 r δ ab L αβ + 1 r δ αβ R ab + 12 δ ab δ αβ ( D − C ) , { ˙ S a ˙ β , ˙ Q ˙ αb } = 1 r δ ab ˙ L ˙ β ˙ α − r δ ˙ β ˙ α R ab + 12 δ ab δ ˙ β ˙ α ( D + C ) . Then in the limit r → ∞ the vielbein defined by (6.7) tends to e M ¯ j dφ j = dX M − Θ · Γ M d Θ , e M ¯ j dφ ¯ j = d Θ M , (8.1)by using (5.16) and (5.17). We no longer need distinguish the coordinates of the cosetspace and those of the tangent space, so that X M = X M , Θ M = Θ M . The vielbeins in (8.1) are invariant under the global supertransformation δX M = α · Γ M Θ , δ Θ M = α M . We also comment on the fact that our correlation function (7.10) is independent ofthe position x i of the observables V i ( x i ) on the light-like line. It might be considered as25 correlation function of the similar kind to the one in the topological field theory. Theexchange algebra of OSP(2 |
2) for the (2,0) topological gravity was discussed in such acontext in [20]. We may think of position-dependence for the correlation function such as e ip x e ip x · · · e ip N x N < V ( x ) ⊗ V ( x ) ⊗ · · · · · · ⊗ V N ( x N ) >, (8.2)in which p , p , · · · , p N are momenta excited along the light-like line. But PSU(2,2 |
4) is toorestrictive to allow for such a dependence. Therefore we think of breaking the symmetryof PSU(2,2 |
4) to a subgroup symmetry which contains a certain number of central chargesas factor groups. In [2] they took such a subgroup to be PSU(1,1 | ⋉R . The generators D, P, K in (2.9) are reduced to three U(1) charges acting on the correlation function (8.2).The position-dependent R-matrix acting on the correlation function (8.2) was given forthis residual subgroup in [2]. It played a crucial role in discussing the duality betweenthe D = 2 spin-chain and the D = 4 , N = 4 SUSY YM theory. It is interesting toinvestigate how such a position-dependent R-matrix occurs as symmetry breaking of theuniversal R-matrix of PSU(2,2 |
4) in the non-linear σ -model. The issue will be discussedin a forthcoming publication[21]. A - and -BPS multiplets In section 3 we have argued that the N = 4 SUSY field strength multiplet is the smallestBPS multiplet of the unitary(oscillator) representation of PSU(2,2 | N = 4 superconformal multiplets. Our arguments on the exchange algebra in this papercan be straightforwardly applied to those multiplets as well.In order to enlarge the BPS multiplet discussed in section 3, we retain only half of thesupercharges in (3.9) Q β , ˙ Q ˙ α , as fermionic generators annihilating the vacuum. They break the R-symmetry SU(2) ⊗ SU(2) of the physical vacuum of the -BPS multiplet to U(1) ⊗ U(1). To be explicit, theyare a † β c , b † ˙ β d . Then the following states
Z, Y = c † d † Z, ♯Q, ♯ ˙ Q ) SU(2) ⊗ SU(2) SU(2) ⊗ SU(2) ⊗ SU(4) h.w.h.w. h.w. dimension ∆ -BPS (4,4) [ j + 2 , ¯ j + 2] [ j, ¯ j ][0 , k, , k > k -BPS (2,2) [ j + 3 , ¯ j + 3] [ j, ¯ j ][ l, k, l ] , l > k + 2 l -BPS (2,0) [3 , ¯ j + 4] [0 , ¯ j ][ l, k, l + 2 m ] , m > k + 2 l + 3 m Table 3: The BPS multiplets. ♯Q and ♯ ˙ Q are the numbers of supercharges annihilat-ing the highest weight state. The third column indicates the highest weight state ofSU(2) ⊗ SU(2), excited from the highest state in the fourth column by applying the re-maining supercharges. Further excitation is possible by applying the space-time derivative P α ˙ β . ∆ in the last column is the conformal dimension defined by the dilatation in (2.3).are annihilated by them and satisfies the constraint C = 0. This defines the physicalvacuum of the -BPS multiplet. If we further halve the number of the annihilatingsupercharges and retain only ˙ Q ˙ α the R-symmetry gets enlarged as SU(3) ⊗ U(1). The states which are annihilated by themand satisfy C = 0 are Z, Y = c † d † Z, X = c † d † Z. (A.1)This defines the vacuum of the -BPS multiplet.Let us denote the fermionic oscillators belonging to the representation ∗ of the R-symmetry SU(4) by the Dynkin label as c † = [ − , , , c † = [1 , − , , d = [0 , , − , d = [0 , , . Then the states
X, Y, Z in (A.1) are denoted by the Dynkin label as Z = [0 , , , Y = [1 , − , , X = [ − , , , which form the representation ∗ of the R-symmetry SU(3). The states of the represen-tation are Z ∗ = [0 , − , , Y ∗ = [ − , , − , X ∗ = [1 , , − . In terms of the oscillators they are given by Z ∗ = c † c † d † d † Z, Y ∗ = c † d † Z, X ∗ = c † d † Z. Note that these six states were given by the field Φ in table 2.By a systematic analysis of the unitary representation of PSU(2,2 |
4) [13, 14, 15] theBPS multiplets are known as given in table 3[14]. The physical vacua of the BPS multiplets27re the highest weight states of the unitary representation of PSU(2,2 | Z k for 12 BPS ,Y l Z k + l for 14 BPS , (A.2) X m Y l + m Z l + m + k for 18 BPS , with (anti-)symmetrization indicated by the SO(6) Young tableau in figure 3. Here wehave to take k copies of the oscillators a α , b ˙ α , c a and a † α , b † ˙ α , c † a . Correspondingly the Fockspace which they act on is the tensor product of k copies of the Fock space (3.8). Forinstance, we understand Z , given by (3.6), as P ks =1 c ( s ) † c ( s ) † . It is easy to verify that thephysical vacua in (A.2) are indeed the SU(4) highest weight states and the conformaldimension ∆, respectively given in table 3.The BPS multiplets with ∆ ≥
2, given in table 3, are other N = 4 SUSY super-conformal multiplets than the field strength multiplet in table 1. So far our argumentshave been done by assuming the gauge group to be Abelian. All the fields of N = 4superconformal multiplet are in the adjoint representation of a gauge group. Let themto be Lie-valued in the gauge algebra. Then the physical vacua in (A.2) are given as thegauge singlets tr [ Z k ] for 12 -BPS ,tr [ Y l ] tr [ Z k + l ] for 14 -BPS , (A.3) tr [ X m ] tr [ Y l + m ] tr [ Z l + m + k ] for 18 -BPS . Here tr denotes trace over the gauge algebra. Therefore we have k, l, m ≥ · · · · · · | {z } k · · · · · ·· · · | {z } l | {z } k · · · · · · · · ·· · · · · ·· · · | {z } m | {z } l | {z } k Figure 3: The SO(6) Young tableaux representing symmetrization and anti-symmetrization for the -, -, -BPS states. 28auge group. We may consider other multitrace states obtained by replacements such as tr [ Z k ] −→ tr [ Z k − l ] tr [ Z l ] , tr [ Y k ] tr [ Z l ] −→ tr [ Y k Z l ] , etc (A.4)since they do not change the dimension ∆. However we have tr [ Z k ] = tr [ Z k − l ] tr [ Z l ] , by the symmetrization indicated by the SO(6) Young tableau in figure 3. Therefore theformer replacement gives nothing new[]. On the other hand the anti-symmetrization bythe SO(6) Young tableau implies that tr [ Y l ] tr [ Z k ] = tr [ Y l Z k ] . That is, the latter replacement of (A.4) changes the physical vacuum into a non-BPSvacuum, Thus the multitrace states with multiplicity greater than 3 are irrelevant for theBPS multiplets in table 3.Finally it is interesting to understand the vacua of the BPS multiplets so far discussedin a way which manifests the SO(6) symmetry of the Young tableau[15]. We consider thegenerators Q aα and ˙ Q ˙ αa and make three sets as { Q α , Q α , ˙ Q ˙ α , ˙ Q ˙ α } , { Q α , Q α , ˙ Q ˙ α , ˙ Q ˙ α } , { Q α , Q α , ˙ Q ˙ α , ˙ Q ˙ α } , in each of which all the generators are anti-commuting. Then the states X, Y, Z in (A.1)satisfy constraints as Q α Z = Q α Z = ˙ Q ˙ α Z = ˙ Q ˙ α Z = 0 ,Q α Y = Q α Y = ˙ Q ˙ α Y = ˙ Q ˙ α Y = 0 ,Q α X = Q α X = ˙ Q ˙ α X = ˙ Q ˙ α X = 0 . Consequently the physical vacua (A.3) to be constrained by Q α tr [ Z k ] = Q α tr [ Z k ] = ˙ Q ˙ α tr [ Z k ] = ˙ Q ˙ α tr [ Z k ] = 0 for 12 -BPS ,Q α (cid:18) tr [ Y l ] tr [ Z k + l ] (cid:19) = ˙ Q ˙ α (cid:18) tr [ Y l ] tr [ Z k + l ] (cid:19) = 0 for 14 -BPS , ˙ Q ˙ α (cid:18) tr [ X m ] tr [ Y l + m ] tr [ Z l + m + k ] (cid:19) = 0 for 18 -BPS . They are nothing but the constraints we have so far discussed.
B Algebraic calculation of the Killing vectors
The Killing vectors of the coset space G/H are defined when G is simple or most gen-erally speaking semi-simple. The supergroup PSU(2,2 |
4) is simple. But either of the29nitary(oscillator) and the matrix representation discussed in this paper realizes it as asubgroup of SU(2,2 |
4) which is not simple. In this appendix we present a purely alge-braic way to calculate the Killing vectors G/H, which is free from the central charge ofSU(2,2 | M are infinitesimally small, we may write the transformation(5.10) as e i [ φ · T + M Ξ R Ξ · T + O ( M )] = e iM Ξ T Ξ e iφ · T e − i [ M Ξ ρ Ξ · H ] , (B.1)Here use was made of the definition of the Killing vectors (5.11) and the notation R Ξ ( φ ) · T ≡ R Ξ¯ i ( φ ) T i , ρ Ξ ( φ ) · T ≡ ρ Ξ¯ I ( φ, M ) H I . (B.1) becomes e i [ φ · T + M Ξ R Ξ · T + O ( M )] = e i [ φ · T + P ∞ n =0 α n ( ad iφ · T ) n ( M Ξ T Ξ ) − P ∞ n =0 ( − n α n ( ad iφ · T ) n ( M Ξ ρ Ξ · H )+ O ( ǫ ))] , (B.2)by using the following formulae: for matrices E and X exp E exp X = exp (cid:16) X + ∞ X n =0 α n ( ad X ) n E + O ( E ) (cid:17) , (B.3)exp X exp E = exp (cid:16) X + ∞ X n =0 ( − n α n ( ad X ) n E + O ( E ) (cid:17) , (B.4)if E ≪
1. Here α n are the constants recursively determined by α n + α n −
2! + · · · + α ( n + 1)! = 0 , for n = 1 , , · · · , (B.5)as α = 1 , α = − , α = 112 , α = 0 , α = − , · · · · · · . (B.3) ∼ (B.5) will be proved at the end of this appendix. In (B.2) ( ad iφ · T ) n is a mappingdefined by the n -ple commutator( ad iφ · T ) n O = [ iφ · T, · · · , [ iφ · T, [ iφ · T, O ]] · · · ] . (B.6)As has been discussed in subsection 5.1, the grading of the commutator may differ depend-ing on the representation. We employed the prescription (5.6) for the unitary(oscillator)representation, but the one (5.7) for the matrix representation. For the respective case(B.6) reads i n M Ξ φ ¯ i n φ ¯ i n − · · · φ ¯ i [ T i , · · · , [ T i n , T Ξ }} · · ·} i n φ ¯ i φ ¯ i · · · φ ¯ i n M Ξ [ T i , · · · , [ T i n , T Ξ }} · · ·} . Here [ , } can be automatically read as a commutator or anti-commutator from the gradingof φ ¯ i , φ ¯ i , · · · . Note that the difference of the two prescriptions reflects merely on theordering of φ ¯ i s. For a simple case where n = 1 and M I = 0 it reads( ad φ · T )( T Ξ M Ξ ) (cid:12)(cid:12)(cid:12) M I =0 = X M M M [ T M , T M ] + X M M M [ T M , T M ]+ Θ M M M [ T M , T ¯ M ] ± Θ M M M { T M , T M } , with the sign taken to be + for the unitary(oscillator) representation, but - for the matrixrepresentation. Here use was made of the index notation (5.17) for φ ¯ i .We expand R Ξ ( φ ) and ρ Ξ ( φ ) in series of φ ¯ i : R Ξ ( φ ) = R Ξ(0) ( φ ) + R Ξ(1) ( φ ) + · · · + R Ξ( n ) ( φ ) + · · · ,ρ Ξ ( φ, M ) = ρ Ξ(0) ( φ, M ) + ρ Ξ(1) ( φ, M ) + · · · + ρ Ξ( n ) ( φ, M ) + · · · . The n -th order terms R Ξ( n ) ( φ ) and ρ Ξ( n ) ( φ, M ) of the expansion obey the recursive relationsobtained by comparing the powers of both sides of (B.2) order by order. We findto the 0-th order of φ R Ξ(0) ( z ) · T + α (cid:16) ρ Ξ(0) ( φ, M ) · H (cid:17) = iα T Ξ , to the first order of φR Ξ(1) ( φ ) · T + α (cid:16) ρ Ξ(1) ( φ, M ) · H (cid:17) = iα h φ · T, T Ξ i − α h φ · T, ρ
Ξ(0) ( φ, M ) · H i , to the second order in zR Ξ(2) ( φ ) · T + α (cid:16) ρ Ξ(2) ( φ, M ) · H (cid:17) = iα h φ · T, h φ · T, T Ξ ii − α h φ · T, ρ
Ξ(1) ( φ, M ) · H i , and so on. These recursion relations are solved for R Ξ¯ i ( n ) ( φ ) and ρ Ξ( n ) ( φ, M ) order by order,by setting R Ξ(0) ( φ ) = iα δ Ξ¯ i as an initial condition. Thus the Killing vectors are determinedonce the algebra of { T Ξ } given with the decomposition (5.8).Finally we will prove (B.3). (B.4) can be proved similarly. First of all, from theHausdorff formula e X e Y = exp h X m X pi ≥ ,pj ≥ pi + qi> ( − n m ( adX ) p ( adY ) q · · · · · · ( adX ) p n ( adY ) q n − Yp ! q ! · · · · · · p n ! q n !( p + q + · · · · · · + p n + q n ) i . we know that (B.3) holds with some coefficients α n , n = 0 , , , · · · if E ≪
1. To determinethe coefficients we expand the exponents of both sides. The expansion of the l.h.s. issimply l.h.s = (1 + E ) ∞ X n =0 n ! X n . X and E , retaining the terms of O ( E ): r.h.s = ∞ X n =0 n ! (cid:16) X + ∞ X n =0 α n ( ad X ) n E (cid:17) n + O ( E )= 1 + (cid:16) X + ∞ X n =0 α n ( ad X ) n E (cid:17) + 12! (cid:16) X + X ∞ X n =0 α n ( ad X ) n E + ∞ X n =0 α n ( ad X ) n E · X (cid:17) + 13! (cid:16) X + X ∞ X n =0 α n ( ad X ) n E + X ∞ X n =0 α n ( ad X ) n E · X + ∞ X n =0 α n ( ad X ) n E · X (cid:17) + O ( E )Expanding this in X furthermore we findto the 0-th oder in X : 1 + α E , to the first order in X : (1 + α E ) X + ( α + α
2! )( ad X ) , to the second order in X :12! (1+ α E ) X + ( α + α
2! + α
3! )( ad X ) E + ( α + α
2! )( ad X ) E · X, to the third order in X :13! (1+ α E ) X + ( α + α
2! + α
3! + α
4! )( ad X ) E + ( α + α
2! + α ad X ) E · X + 12! ( α + α
2! )( ad X ) E · X . If α = 1 and the recursion relation (B.5) holds up to n = 3, then both sides of (B.3 ) areequal up to the third order of X . The calculation may be inductively generalized to anyorder of X . Thus (B.3 ) was proved. References [1] D. Berenstein, J. Maldacena, H. Nastase, “Strings in flat space and pp waves from N = 4 Super Yang Mills”, JHEP (2002) 013, arXiv:hep-th/0202021;32. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators fromnon-critical string theory”, Phys. Lett. B428 (1998)105, arXiv:hep-th/9802109;J. A. Minahan and K. Zarembo, “The Bethe-ansatz for N = 4 super Yang-Mills”,JHEP (2003)103, arXiv:hep-th/0212208.[2] N. Beisert, “The SU(2 |
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