Current Superalgebra and Twisted Conformal Field Theory

Transcription

1 Commun. Theor. Phys. Beijing, China pp c International Academic Publishers Vol. 47, No. 1, January 15, 007 On gl Current Superalgebra and Twisted Conformal Field Theory DING Xiang-Mao, 1, WANG Gui-Dong, 1, and WANG Shi-Kun 1,3 1 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, P.O. Box 734, Beijing , China Graduate School of the Chinese Academy of Scisences, Beijing , China 3 KLMM, AMSS, CAS, Beijing , China Received March 17, 006 Abstract Motivated by the recently discovered hidden symmetry of the type IIB Green Schwarz superstring on certain bacground, the non-semisimple Kac Moody twisted superalgebra gl is investigated by means of the vector coherent state method and boson-fermion realization. The free field realization of the twisted current superalgebra at general level is constructed. The corresponding Conformal Field Theory CFT has zero central charge. According to the classification theory, this CFT is a nonunitary field theory. After projecting out a U1 factor and an outer automorphism operator, we get the free field representation of psl, which is the algebra of gl modulo the Z 4-outer automorphism, the CFT has central charge. PACS numbers: 11.5.Hf, 0.0.-a, Pb, q Key words: superalgebra, differential operator realization, free field realization, conformal field theory 1 Introduction Virasoro algebra and current algebras are algebraic structures in Conformal Field Theories CFTs [1 3] and string theory. [4] According to Zamolodchiovs c-theorem [5] a unitary CFT corresponds to a fixed point of the renormalization group flow. The unitary CFTs and current algebras also play an important role in the study of D critical behaviors of statistical mechanics models. So the unitary CFTs have been studied extensively in the past two decades. On the other hand, our nowledge on nonunitary CFT is in very preliminary state by the two facts: the nonunitary CFT has been regarded as a physical meaningless artificiality in a quite long time, and it is complex in mathematics. But the story becomes quite different for the recent progresses of the study of type II string theory on AdS n S n bacgrounds. The first example is the Sigma models which were considered in Refs. [6] and [7], in which the target spaces of the Sigma models are certain supergroup manifold P SLn 1 n 1, and the exactly conformal property of the models were discussed. The most interesting case of them is the P SL Sigma model, which describes the superstring on AdS 3 S 3 bacground. P SL has a larger outer automorphism, and with AdS/CFT correspondence, the correlation function can be computed by the boundary theory. The super-sigma model with other interesting bosonic geometry was considered recently in Ref. [8]. The second example is the type IIB superstring on AdS 5 S 5 bacground, the duality structure implies the integrability of the theory, [9 11] the description is simplified by adding extra U1 factors. [1] In these stories, the Lie superalgebra cannot be neglected. In Ref. [13], the finite Lie superalgebra is extended to an infinite dimensional Kac Moody Lie superalgebra by introducing a continuous parameter, it describes the symmetry which conserves the area of the string worldsheet. From this sense, it is worth while to investigate the Kac Moody Lie superalgebra in more details. On the other hand, because more and more of their applications are founded, much of interesting research was brought to the Kac Moody Lie superalgebras and their corresponding nonunitary CFTs. These applications range from the high energy physics to condensed matter physics. Particular interesting are applications in the topological field theories, [14,15] logarithmic CFTs, disordered systems and the integer quantum Hall effects. [16 19] In such contexts, the vanishing of the Virasoro central charge and superdimension of the Kac Moody Lie superalgebras are crucial. In these case, the energy-momentum tensors become spin- primary fields with respect to themselves. According to the classification theory, these CFTs are nonunitary. If we regard these nonunitary CFT as Wess Zumino Witten WZW models, or Sigma models with Wess Zumino WZ-term, then the models should be defined on certain supergroup manifolds. The most interesting supergroup manifolds with such properties are Ospn n and Gln n, and their current algebras are ospn n and gln n, respectively. For these two infinite series, the dual Coxeter number vanishes and therefore the one-loop beta function is zero. On the other hand, the twisted current algebra and the twisted CFT have much more applications in the string theory, the brane could be described by the boundary state. The twisted brane is interesting in string theory. The twisted GLn n WZW model is a nice candidate to be investigated with these features. Another special feature The project supported by National Key Basic Research Program of China under Grant No. 004CB31800, 006CB and National Natural Science Foundation of China under Grant Nos and Corresponding author, xmding@amss.ac.cn

2 70 DING Xiang-Mao, WANG Gui-Dong, and WANG Shi-Kun Vol. 47 of the twisted GLn n WZW model is that it includes both the Ramond sector and the Neveu Schwarz sector, and there exists a current, such as the H z in our manuscript, and it exchanges fermions on the Ramond sector and fermions on the Neveu Schwarz sector. Beside the applications mentioned above, new features will arise in the representation theory of the Kac Moody Lie superalgebra. For example, for Lie superalgebra there are two inds of representations, which are named as the typical representation and the atypical representation, respectively. There is no counter part for the atypical representation in the ordinary Lie algebra. The importance of the highest weight representation of the superalgebra in physics has been announced in Ref. [7]. It was pointed out that the short representation of the representation was essential for physics. The short series is the atypical representation. So it is worth while to study the Lie superalgebra in more details. Representations of osp 1 and gl 1 were investigated in Refs. [0] [3]. The Kac Moody superalgebra with twisted structure was considered only for the simplest superalgebra osp with the vanishing of the super-dimension in Ref. [1]. In this paper, we will consider the CFT based on the twisted current superalgebra gl at general level. The important ingredient of CFT is the primary field, or the highest weight representation of the Kac Moody superalgebra. We believe that for the twisted superalgebra gl, only the atypical representation is surviving, which is the chiral primary fields of Virasoro algebra with conformal dimension zero in the boundary theory. For any Lie superalgebra, it has a Z -grading. For convenience, we call this Z structure as superalgebra structure. However, for a certain ind of the Lie superalgebra, it has a larger finite order cyclic symmetry. Our aim of this paper is to investigate Kac Moody superalgebra gl, the Z -twisted version of the non-semisimple Kac Moody superalgebra gl 1, which means that beside the ordinary Z -grading of the Lie superalgebra. We will consider another Z -symmetry, which is defined by an order- out-automorphism of its Dynin diagram. We denominate it as the twist structure to distinguish it from the previous one. The twisted or the orbifold version of the gl is obtained by folding the ordinary Dynin diagram with certain symmetry. In fact, for the Lie superalgebra gln n, there is a larger finite order Z Z or Z 4 automorphism, but the second Z factor cannot be fixed by the folding process mentioned above. We will consider the subtlety in the paper. The free field realization, which is an infinite dimensional generalization of the Heisenberg algebra, is a powerful tool in many physical applications. This approach was first given by Waimoto for the simplest case sl, [4] and extended to sln in Ref. [6], and other simple Lie algebras in Ref. [5]. For twisted algebra, the free field approach was first realized in Refs. [7] and [8], and extended to higher ran in Ref. [9]. In Refs. [0] and [1], gl in details. The the method was generalized to the Kac Moody superalgebra. Using this approach, the correlation function can be easily carried out. The aim of this paper is to investigate the free field realization of the twisted algebra paper is organized as follows. For conveniences, in Sec., we give some notations which will be used in the paper. In Sec. 3, we define a WZW-model on the supergroup manifold GL. In Sec. 4, we derive a differential operator realization of the twisted Lie superalgebra gl, and then give the free field realization of the twisted Kac Moody superalgebra gl, and construct an energymomentum tensor of the current superalgebra, which is CFT with zero central. As we have mentioned, the twisted Kac Moody superalgebra gl is non-semisimple, so in Sec. 5, we project out a U1 factor, and an outer automorphism generator from the algebra, then we obtain a new Kac Moody superalgebra psl, and the corresponding coset CFT. For this coset CFT, the central charge is, which agrees with Refs. [6] and [7] for untwisted case. The representation of the Kac Moody superalgebra is very complicated and very different from the ordinary Lie algebras. For the untwisted superalgebra gl, its representation was only discussed case by case for the first few levels in Ref. [30]. In a much recent paper of Schomerus and Saleur, [31] they considered the representation and correlation function of GL1,1 WZW model. Even for this much simpler case, the subtle analysis and great job were needed, and the representation and correlation function was much involved. In fact, it needs the free field realization of the algebra to obtain the highest weight representation and to calculate the correlation function. So the present wor is just the first step on this. At present, we are dealing with the representation and the correlation function for the gl case, but the results are uncompleted yet. We will discussed it in the next paper. Notations In this section, we will recall some basic notations of Lie superalgebras. For more details please see Refs. [3] and [33]. A Lie superalgebra is a Z -graded algebra g = g 0 g 1 with an operation [, ] satisfying the following axiom: [a, b] = 1 deg adeg b [b, a], 1 [a, [b, c]] = [[a, b], c] + 1 deg adeg b [b, [a, c]], where g 0 is an ordinary Lie algebra and the multiplication on the left by elements of g 0 determines a structure of a g 0-module on g 1. Now we give the definitions of dimension and superdimension of the representation of a Lie superalgebra. See Ref. [31] for more details. Let V = V 0 V 1 be a Z -graded vector space for a given Lie superalgebra, and consider the superalgebra End V = End 0V End 1V of endomorphism of V. A linear representation π of g is a homomorphism of g into End V, that is, παx = απx, πx + Y = πx + πy,

3 No. 1 On gl Current Superalgebra and Twisted Conformal Field Theory 71 π[x, Y ] = [πx, πy ], πg 0 End 0V, πg 1 End 1V, 3 for all X, Y g, α C. The dimension resp. superdimension of the representation π is the dimension resp. graded dimension of the vector space V : dim π = dim V 0 + dim V 1, s dim π = dim V 0 + dim V 1. 4 The classical Lie superalgebras can be described as matrix superalgebras as follows. [30,31] Consider the Z -graded vector space V = V 0 V 1 with dim V 0 = m and dim V 1 = n. Then the algebra End V acquires naturally a superalgebra structure by End V = End 0V End 1V, where End ī V = {φ End V φv j V i+j }. 5 The Lie superalgebra lm, n is defined as the superalgebra End V supplied with the Lie superbracet. lm, n is spanned by m n matrices of the form A B X =, C D where A and D are m m and n n bosonic matrices, and B and C are m n and n m fermionic matrices. The supertrace of a matrix X is str X = Tr A Tr D, 6 and superdimension is sdim lm, n = m + n mn = m n. 7 For m = n =, tae E ij, i, j = 1,, 3, 4 be the matrices with entry 1 at the i-th row and j-th column, and zero elsewhere. Then gl can be spanned by matrices E ij, i, j = 1,, 3, 4, and its superalgebra structure is introduced by defining the Lie anti-bracet for any two matrices E ij and E l : [E ij, E l ] = δ j E il 1 [i]+[ j]+[]+[l] δ il E j, 8 where the Z -grading superalgebra structure is defined as [1] = [] = 0, [3] = [4] = 1. Now let us give some basic concepts of twisted affine algebras. [,30] Let g be a simple finite-dimensional Lie algebra and σ be an automorphism of g satisfying σ r = 1 for a positive integer r, then g can be decomposed into the form: g = r 1 j=0 g j, where g j is the eigenspace of σ with eigenvalue e jπi/r, and [ g i, g j ] g i+j mod r, then r is called the order of the automorphism the r = 1 is trivial case for any algebra. In this wor, we only consider the twisted superalgebra that has an automorphism σ on the superalgebra with finite order. It is well nown that unlie the ordinary Lie algebra, for a given Lie superalgebra, its Dynin diagram is not unique, and not all of them have a finite-order automorphism. For the given Lie superalgebra, if one can choose an appropriate Dynin diagram, which has a finite-order automorphism, the Lie superalgebra can be twisted by folding the Dynin diagram. In the gl case, there are three inds of Dynin diagrams, if we choose the following Dynin diagram for other choices, the process is similar: ε 1 δ 1 δ 1 δ δ ε in which ε i ε j = δ ij, and δ i δ j = δ ij. So in the diagram represents a null vector of fermionic type simple root. Now, we define an order- transformation on the Dynin diagram, τ : ε i ε 3 i, δ i δ 3 i. 9 Obviously, the Dynin diagram is invariant under the action of τ. Identically, we give the explicit action on the matrices as follows: τe 11 = E, τe = E 11, τe 33 = E 44, τe 44 = E 33, τe 13 = E 4, τe 4 = E 13, τe 34 = E 34, τe 31 = E 4, τe 4 = E 31, τe 43 = E 43, τe 14 = E 3, τe 3 = E 14, τe 1 = E 1, τe 41 = E 3, τe 3 = E 41, τe 1 = E It is easy to chec that τ is an automorphism of the superalgebra gl with order r =. So we obtain the twisted superalgebra gl, and we have gl = gl 0 gl 1, 11 in which gl 0 is spanned by e 1 = 1 E 13 + E 4, e = E 34, e 3 = 1 E 14 E 3, H 1 = E 11 E + E 33 E 44, f 1 = 1 E 31 E 4, f = E 43, f 3 = 1 E 41 + E 3, H = E 11 E E 33 + E 44. Here gl 0 is a fixed point sub-superalgebra under the automorphism. Obviously, this subalgebra is just the osp, or sl 1. If we twist the fixed subalgebra once more, we will get its fixed subalgebra osp 1. [1] So twisting is an appropriate way to get a smaller algebra from a larger algebra. And the generators of gl 1 are ē 1 = 1 E 13 E 4, ē = E 1, ē 3 = 1 E 14 + E 3, H1 = E 11 + E + E 33 + E 44, f 1 = 1 E 31 + E 4, f = E 1, f3 = 1 E 41 E 3, H = E 11 + E E 33 E 44.

4 7 DING Xiang-Mao, WANG Gui-Dong, and WANG Shi-Kun Vol. 47 The anti-commutation relations of g in this basis are as follows: [e 1, e ] = e 3, {e 1, f 1 } = 1 H 1, {e 1, f 3 } = f, [H, e 1 ] = e 1, [e, f ] = 1 H 1 H, [e, f 3 ] = f 1, [H 1, e ] = e, [H, e ] = e, [H 1, e 3 ] = e 3, {e 3, f 1 } = e, [e 3, f ] = e 1, {e 3, f 3 } = 1 H, [f 1, f ] = f 3, [H, f 1 ] = f 1, [H 1, f ] = f, [H, f ] = f, [H 1, f 3 ] = f 3, 1 for [ gl 0, gl 0 ], and {ē 1, f 1 } = 1 H 1, [ē 1, f ] = f 3, {ē 1, f 3 } = f, [ H, ē 1 ] = e 1, [ē, f 1 ] = e 3, [ē, f ] = 1 H 1 + H, [ē, f 3 ] = e 1, {ē 3, f 1 } = e, [ē 3, f ] = f 1, {ē 3, f 3 } = 1 H, [ H, ē 3 ] = e 3, [ H, f 1 ] = f 1, [ H, f 3 ] = f 3, for [gl 1, gl 1 ]. At last 13 {e 1, ē 3 } = ē, {e 1, f 1 } = 1 H 1, [e 1, f ] = f 3, [ H, e 1 ] = ē 1, [e, ē 1 ] = ē 3, [e, f 3 ] = f 1, {e 3, ē 1 } = ē, [e 3, f ] = f 1, {e 3, f 3 } = 1 H 1, [ H, e 3 ] = ē 3, [H 1, ē ] = ē, [H 1, ē 3 ] = ē 3, [H 1, f ] = f, [H 1, f 3 ] = f 3, [H, ē 1 ] = ē 1, [H, ē ] = ē, [H, f 1 ] = f 1, [H, f ] = f, {f 1, ē 1 } = 1 H 1, [f 1, ē ] = ē 3, {f 1, f 3 } = f, [ H, f 1 ] = f 1, [f, ē 3 ] = ē 1, [f, f 1 ] = f 3, [f 3, ē ] = ē 1, {f 3, ē 3 } = 1 H 1, {f 3, f 1 } = f, [ H, f 3 ] = f 3, 14 for [ gl 0, gl 1 ]. All the other relations are identical to zero. So gl 0 and gl 1 satisfy [ gl i, gl j ] gl i+j mod. 15 Please note that the bosonic generator H 1 indeed is the identity generator of the algebra, while changing its minus under the bosonic generator H is special in the algebra. It cannot be obtained from any anticommutator of other generators. But it must be added into the algebra to balance the superdimension, in our case, the superdimension is zero, say that the number of bosonic generators equals the ones of the fermionic generators. It turns out that H = I. 16 So it is an order- outer automorphism, commuting with the bosonic generators and rotating the two fermionic copies of. In fact, in untwisted case with n = m =, it has a larger outer automorphisms SL, R for the superalgebra see Refs. [6] and [7], but here we will only consider the discrete order- outer automorphisms. It is easy to see that the restriction of the adjoint representation of gl to gl 0 induces a linear representation of the subalgebra gl 0 on the subspace gl 1. In fact this is the nontrivial smallest representation of the subalgebra gl 0, and the dimension of the smallest representation is 8. For the twisted algebra gl there is an endomorphism of any representation. For a given Killing form, we can construct the quadratic Casimir of gl as C = i,j 1 [ j] E ij E ji = 1 H 1H e 1 f 1 + f 1 e 1 e f f e e 3 f 3 + f 3 e H 1 H ē 1 f1 + f 1 ē 1 + ē f + f ē ē 3 f3 + f 3 ē In fact, the quadratic Casimir is independent of the choice of the basis. It is useful to constructing the energymomentum operator. According to the definition of superalgebra, gl 0 is a Lie algebra itself. By a simple calculation we can obtain the Casimir operator of this Lie fixed subalgebra C 0 = 1 H 1H e 1 f 1 + f 1 e 1 e f f e e 3 f 3 + f 3 e It is obviously that the Casimir element C 0 is just the part of the Casimir element C corresponding to the fix subalgebra. From the expression of quadratic Casimir of twisted gl, we now that if we project out any one of the two generators H 1 or H, the term H 1 H in the quadratic Casimir of twisted gl will disappear, so that we get the quadratic Casimir of twisted psl. For consistence, we must project out the two bosonic generators H 1 and H simultaneously from the twisted gl to obtain the twisted psl. So that the superdimension of the twisted algebra psl is, but we will not repeat the algebraic relations of psl here. 3 Super-WZW Model and Twisted CFT In the last section, we introduced the twisted superalgebra gl. In this section we define a Wess Zumino Witten WZW model on a supergroup, whose algebra is the twisted algebra gl. WZW-model is a CFT, which is obtained from a Sigma model by adding the Wess Zumino WZ-term. [3] It has the current algebra as its symmetry. If we regard the Kac Moody superalgebra gl as the symmetry of certain WZW model,

5 No. 1 On gl Current Superalgebra and Twisted Conformal Field Theory 73 then the model should be defined on a supermanifold, or more precisely a super orbifold GL. It is needed to point out that the convenience of GL is that it has the fundamental -dimensional representation, so one can define an invariant quadratic form by a supertrace in this -dimensional representation, x, y = Str xy. 19 For the Lie supergroup GL with generators Ta, which are the generators of P SL and H1, H, we introduce coordinates Φ a = x a ; θ a and φ 1, φ parametrize an element of the group near the identity as g = exp φ 1 H1 + φ H + Φ a T a expφ A T A. 0 Lie the discussion of untwisted super Sigma model in Refs. [6] and [7], up to a possible multiplicative constant, the Sigma model action for a G-valued field g can be defined as 1 S = d σ f Str g 1 dg. 1 f is a constant that is interpreted as the inverse radius of AdS 3 S 3. Similar to the untwisted case, one might introduce another possible constant, by adding a term which is proportional to an invariant but degenerate quadratic form Str x Str y, but the concrete expression does not concern us. We will not use the action in the following discussion. It is nown that any local lagrangian cannot be added while preserving its all symmetries, the remaining option is to add a topological term to the Sigma model action. [34] To express interaction as the integration S = d 3 z ɛ ij g 1 i g, [g 1 j g, g 1 g]. Σ 3 The integration is carried out over a two-dimensional boundary of Σ 3. The general G G invariant action that is conformally invariant at classical level is I = S + is. 3 is a constant. Up to a constant, the corresponding conserved currents of the left L and right R moving parts are J L = dgg 1, J R = g 1 dg. 4 If one introduces the Gauss decomposition of the supergroup element, and following the standard approach, the currents should satisfy the OPEs only consider the left part, similar for the right part, Jz = J A zt A, J A zj B w = Str T A T B z w + [T A, T B }w, 5 z w where [T A, T B } denotes the anti-commutator if both T A and T B are fermionic fields or the commutator for other cases. If we expand J A z as J A z = J A,n z n A, 6 n then n Z for J A g 0, and n Z + 1/ for J A g 1, and the modes J A,n satisfy the level- Kac Moody superalgebra gl. The energy-momentum tensor of the WZW-model associated with the current superalgebra is obtained through the Sugawara construction. It should be subject to the relation T zt w = c/ T w T w + + z w 4 z w z w, 7 in which the central charger c is c = sdim G. 8 + g is the level of the representation, g is the dual Coxeter number, which is zero for gln n. Thus, c = sdim G. 9 sdim is the super dimension of Lie algebra gln n. It is zero for gln n. If we project out the outer automorphism generator H from the algebra, then the identity operator H 1 will disappear, and the invariant metric of the resulted algebra is non-degenerate, so that one gets a twisted psl algebra. The CFT with this current algebra has central charge. For these CFTs, the one-loop β-function, which is proportional to the dual Coxeter number, is identical zero. 4 Coherent State and Free Field Realization In the last section, we consider a WZW-model on a supergroup manifold. Gaussian decomposing the group elements, with the definition of the conserved current, one can get the current algebra expressed in terms of the fields on-diagonal and off-diagonal. But in this section we will adopt an alternative approach to get an explicit expression of the current algebra in terms of the free field theory. To obtain a free field realization, we first construct Foc space representation of gl. The Foc space is constructed by the actions of the lowering operators f 1, f, f 3, f1, f, and f 3 on the highest weight state. Define the highest weight state Λ of twisted algebra gl by e 1 Λ = e Λ = e 3 Λ = ē 1 Λ = ē Λ = ē 3 Λ = 0, H i Λ = Λ i Λ, Hi Λ = Λ i Λ, i = 1,. 30 Then the supergroup action of the operator e ρ with vector ρ = θ 1 e 1 + x e + θ 3 e 3 + θ 1 ē 1 + x ē + θ 3 ē 3, 31 on the highest state Λ generates a coherent state of the algebra, where x and x are bosonic coordinates satisfying [ x, x ] = [ x, x ] = 1, and θ i and θ j are fermionic coordinates obeying θ i θ j = θ j θ i, θi θj = θ j θi and { θi, θ j } = { θi, θj } = δ ij, i, j = 1,, 3. This can be viewed as the super extension of the ordinary coherent state method. Please see Ref. [35] for more applications of coherent state method on representation theory and physics. Now we define T A e ρ Λ = d T A e ρ Λ, 3 where T A is any generator of gl and d T A is the corresponding differential operator realization of the generator T A. By using the defining relations of twisted gl and the Baer Campbell Hausdorff BCH formula, after

6 74 DING Xiang-Mao, WANG Gui-Dong, and WANG Shi-Kun Vol. 47 a long algebraic calculation we obtain the differential operators representation of the twisted algebra. For positive simple roots, the expressions are very simple, d e1 = θ1 1 x θ3 + 1 θ 3 x x θ 1 x, d e = x + 1 θ 1 θ3 + 1 θ 1 θ3 1 6 θ 1 θ 1 x, d e3 = θ3 1 θ 1 x, dē1 = θ1 1 x θ3 1 θ 3 x 1 1 x θ 1 x, dē = x, dē3 = θ3 + 1 θ 1 x, 33 and the generators of the Cartan parts are D H1 = Λ 1 θ 3 θ3 x x θ 3 θ3 x x, D H = Λ θ 1 θ1 + x x θ 1 θ1 x x, D H1 = Λ 1, D H = Λ θ 1 θ1 θ 3 θ3 θ 1 θ1 θ 3 θ3. 34 But for the negative roots operators, the expressions of their realization are much more involved, d f1 = 1 θ 1Λ θ 1 Λ1 + θ 3 1 θ 1x x 1 θ 1θ 3 θ3 1 θ 1 x θ 1θ 3 θ1 x 1 θ 1 θ3 + x 1 6 θ 1x θ1 θ3, d f = 1 x Λ 1 Λ + θ 3 θ1 + θ 3 θ1 1 6 x θ 1 θ3 x + θ 3 θ1 x x θ 1 θ3 + θ 1 θ3 + 1 x θ 1 θ1 x x θ 3 θ3 + θ 1 θ1 θ 3 θ3, d f3 = 1 θ 3Λ θ 1x Λ 1 Λ + 1 θ x θ 1 Λ1 + x θ1 + 1 [θ 1θ 3 θ1 + x θ 3 x + θ 1 θ3 θ 3 θ1 θ1 θ 3 x x x x θ3 ] 1 1 [6θ 1x x θ 1 x θ1 θ1 + θ 1 x x + 3θ 1 θ 3 θ3 x + 3θ 1 x θ3 + x θ 3 θ1 θ3 ] θ 1x θ 1 θ3 3θ 1 x θ 3 θ1 x, d f1 = 1 θ 1 Λ θ 1 Λ 1 + x θ3 + θ 3 x 1 1 θ 1 θ 1 x θ3 θ 3 x 1 θ 1 x x + θ 3 θ3 + x x + θ 3 θ3, d f = 1 x Λ 1 + Λ 1 4 θ 1θ 3 + θ 3 θ1 Λ 1 Λ + 1 θ 1θ 3 θ 1 θ3 Λ θ 1x θ1 Λ 1 Λ θ 3 θ3 1 x θ 3 θ1 1 θ 1x θ θ 1x θ 1 x θ 1 x + 1 θ 1θ 3 θ1 θ1 θ 3 x θ 1x θ 3 θ1 θ3 θ1 x 1 θ θ 1 1 θ3 θ1 x 1 3 θ 1θ 3 θ1 θ3 x x θ3 1 1 θ 1x θ1 θ3 θ3, d f3 = 1 θ 3 Λ x θ 1 1 Λ 1 Λ + θ θ 1x Λ1 + θ 1 θ3 + θ1 x + 1 θ θ θ θ θ 1x θ1 θ1 + x θ3 1 x θ 1 1 x + x x θ 1x θ x θ 3 θ1 1 6 θ 1x θ 1 θ3 + 1 x θ θ θ 3 1 θ3 1 1 x θ 1 x θ 1x θ1 θ3 x. 35 It is long but straightforward to chec that the above differential operators satisfy the algebraic relations of twisted algebra gl. This realization is a higher-ran and super-version extension of the ordinary Heisenberg algebra representation of the Lie algebra. Please note that, the differential operators realization here and the free fields realization that will be considered will only depend on the algebraic relations, but not on the matrices representation of the algebra. With the help of the differential operator representation of finite Lie algebra g, we can find the Waimoto realization of the corresponding Kac Moody algebra ĝ. This approach was first achieved by Waimoto for the simplest case sl, [4] and extended to sln in Ref. [5], and other Lie algebra by Ref. [6]. For twisted algebra, the free field approach was first realized in Refs. [7] and [8]. Furthermore, in Refs. [0] and [1], the method was generalized to the Kac Moody superalgebra. For gl Kac Moody superalgebra, the free field realization for arbitrary will be given in terms of sixteen fields, namely two bosonic β-γ pairs, four fermionic b-c type pairs and four free scalar fields φ. The free fields obey the following OPEs: β zγ w = γ zβ w = 1 z w, β z γ w = γ z β w = 1 z w, ψ i zψ + j w = ψ+ j zψ iw =, i, j = 1, 3, z w ψ i z ψ + j w = ψ + j z ψ i w =, i, j = 1, 3, z w φ 1 zφ w = φ 1 z φ w = 4 lnz w, 36 and all other Operator Product Expansions OPEs are trivial. The conformal weights of β i z, βi z, ψ + i ψ z, + i z, φ iz and φ i z are 1, and 0 for else. The free field realization of the gl current superalgebra is obtained by substitution of the differential δ ij δ ij

7 No. 1 On gl Current Superalgebra and Twisted Conformal Field Theory 75 operators with certain inds of fields. Concretely, the relations are, x i γ i z, xi β i z, x i γ i z, xi β i z, θ i ψ i z, θi ψ + i z, θ i ψ i z, θi ψ + i z, Λ i φ i z, Λi φ i z. 37 The modes expansion of the fields is, J i z = n Z J i;n z n i, 38 for J i z = β z, γ z, ψ i z, ψ + i z or φ iz, and i is the conformal weight of J i z. Similarly, J i z = J i;n z n i, 39 n Z+1/ for J i z = β z, γ z, ψi z, ψ+ i z or φ i z, and i is the conformal weight of Ji z. As we can see, Ji zs are supercurrents coming from the superaffinization, so the index n taes values in Z + 1/. And for the currents J i zs mentioned above, the index n just taes values in Z. For simplicity, we denote the generating functions of the Kac Moody superalgebra as T A z, if the finite Lie superalgebra generator is D T A. The expressions for the positive and Cartan currents are straightforward, e 1 z = ψ + 1 z 1 γ zψ + 3 z + 1 ψ 3 z β z γ z ψ 1 z β z, e z = β z + 1 ψ 1zψ + 3 z + 1 ψ 1 z ψ + 3 z 1 6 ψ 1z ψ 1 z β z, e 3 z = ψ + 3 z 1 ψ 1 z β z, ē 1 z = ψ + 1 z 1 γ z ψ + 3 z 1 ψ 3z β z 1 1 γ zψ 1 z β z, ē z = β z, ē 3 z = ψ + 3 z + 1 ψ 1z β z, H 1 z = φ 1 z ψ 3 zψ + 3 z γ z β z ψ 3 z ψ + 3 z γ zβ z, H z = φ z ψ 1 zψ + 1 z ψ 1 z ψ + 1 z + γ zβ z γ z β z, H 1 z = φ 1 z, H z = φ z ψ 1 z ψ + 1 z ψ 3z ψ + 3 z ψ 1 zψ 1 + z ψ 3 zψ 3 + z. 40 For the negative currents, the results are not so naive. Additional terms are needed to satisfy the Kac Moody algebra, these terms are governed by the central extension. The negative currents are f 1 z = 1 ψ 1z φ 1 z+ 1 ψ 1 z φ 1 z+ ψ 3 z 1 ψ 1zγ z ψ 1zψ 3 z ψ 1 z β z 1 γ z + ψ 1 z ψ 3 z 1 6 ψ 1zγ z ψ 1 z ψ+ 3 z + ψ 1 z, β z 1 ψ 1zψ 3 zψ + 3 z 1 f z = 1 φ1 z φ zγ z + ψ 3 zψ + 1 z + ψ 3 z ψ + 1 z γ zψ 1 zψ + 3 z + ψ 1 z ψ + 3 z + 1 γ zψ 1 zψ + 1 z γ zβ z ψ 3 zψ + 3 z + ψ 1 z ψ + 1 z ψ 3 z ψ + 3 z 1 6 γ zψ 1 z ψ 3 z β z + ψ 3 z ψ 1 z β z + 1 γ z, f 3 z = 1 ψ 3z φ z ψ 1zγ z φ 1 z φ z + 1 ψ 3 z γ z ψ 1 z φ1 z + γ z ψ + 1 z + 1 [ψ 1zψ 3 zψ + 1 z + γ zψ 3 zβ z + ψ 1 z ψ 3 z ψ 3 z ψ 1 z ψ + 1 z ψ 1 z γ z ψ 3 z γ z β z γ z γ z ψ + 3 z] 1 1 [6ψ 1zγ zβ z ψ 1 zγ z ψ 1 z ψ + 1 z + ψ 1zγ z γ z + 3ψ 1 zψ 3 z ψ 3 z β z + 3ψ 1 zγ z ψ 3 z + γ zψ 3 z ψ 1 z ψ 3 + z] ψ 1zγz ψ 1 z ψ 3 + z 3ψ 1 zγ zψ 3 z ψ 1 z β z ψ 3 z γ z ψ 1 z ψ 1 z γ z, 6 3 f 1 z = 1 ψ 1 z φ 1 z + 1 ψ 1z φ 1 z + γ zψ + 3 z + ψ 3 zβ z 1 1 ψ 1z ψ 1 zγ zψ + 3 z ψ 3 z β z 1 ψ 1 zγ zβ z + ψ 3 zψ + 3 z + γ z β z + ψ 3 z ψ + 3 z + ψ 1 z, f z = 1 γ z φ 1 z + φ z 1 4 ψ 1z ψ 3 z + ψ 3 z ψ 1 z φ 1 z φ z + 1 ψ 1zψ 3 z ψ 1 z ψ 3 z φ 1 z ψ 1zγ z ψ 1 z φ 1 z φ z

8 76 DING Xiang-Mao, WANG Gui-Dong, and WANG Shi-Kun Vol. 47 ψ 1 z γ z + 1 ψ 1zψ 3 z ψ 1 z ψ 1 + z ψ 3 z ψ 3 z 1 γ zψ 3 z ψ 1 z 1 ψ 1zγ z ψ 3 z ψ 1zγz ψ 1 z β z ψ 3 z γ z ψ 1zγ zψ 3 z ψ 1 z ψ 3 + z ψ1 z γ z 1 ψ 1z ψ 1 z ψ 3 z ψ+ 1 z γ z 1 3 ψ 1zψ 3 z ψ 1 z ψ 3 z β z γ z ψ 3 z 1 1 ψ 1zγ z ψ 1 z ψ 3 z ψ+ 3 z + γ z + 1 ψ 3z ψ 1 z 1 1 γ z 6 ψ 1 z ψ 1 z 1 ψ 3z ψ 1 z γ zψ 1 z 6 ψ 1 z ψ 1z ψ 1 z γ z ψ 1 z ψ 3 z ψ 1z ψ 3 z, f 3 z = 1 ψ 3 z φ z γ z ψ 1 z 1 φ 1 z φ z + ψ 3z ψ 1zγ z φ1 z + ψ 1 z ψ 3 z ψ 1 γ + z + z + 1 ψ 1z ψ 3 z 1 ψ 3z ψ 1 z 1 6 ψ 1zγ z ψ 1 z ψ 1 + z + γ z ψ 3 z 1 γ z ψ 1 1 z β z + γ z γ z ψ 1zγ z ψ 3 z γ zψ 3 z ψ 1 z 1 6 ψ 1zγz ψ 1 z ψ 3 + z + 1 γ z ψ 3 z ψ 3z ψ 1 z ψ 3 z 1 1 γ z ψ 1 z γ z ψ 1zγ z ψ 1 z ψ 3 z β z ψ 3 z γ z 6 ψ 1 z ψ1 z γ z One can chec them directly that they are indeed the gl current superalgebra at general level. Obviously we have H zψ i w = ψ i w z w, H z ψ i w = ψ iw z w. 4 So the current H z generates the spectral flow of the theory, and we can regard ψ i z and ψ i z as a mirror pair. The energy-momentum tensor corresponding to the quadratic Casimir C is given by T z = 1 : 1 [j] E ij ze ji z : ij = 1 4 [ φ 1z φ z + φ 1 z φ z] + β z γ z + β z γ z ψ + 1 z ψ 1z ψ + 1 z ψ 1 z ψ 3 + z ψ 3z ψ 3 + z ψ 3 z 1 φ 1 z. 43 One can easily chec that, for tensor T z and all currents here we denote them as Jz, we have the following relations: T w T w T zt w = + z w z w, T zjw = Jw z w + Jw z w. 44 We can see that T z is the energy-momentum tensor of the gl current superalgebra, with zero central charge. In this sense, the energy-momentum tensor is a spin- primary filed with respect to itself, and respect to this tensor, all currents are spin-1 primary fields. Similarly, from the Casimir element C 0 of Lie algebra gl 0, by the same calculation we can get energymomentum tensor T 0 for the fixed subalgebra. Then the coset energy-momentum tensor T G/H = T T 0, which in terms of the fields is as follows: T G/H = 1 4 φ 1 z φ z + 1 β z γ z + β z γ z 1 ψ+ 1 z ψ 1z ψ 1 + z ψ 1 z 1 ψ+ 3 z ψ 3z ψ+ 3 z ψ 3 z. 45 But it does not satisfy the Virasoro algebraic relation. This is quite different from the untwisted algebra. For a untwisted algebra, we can construct a coset model G/H from any group G with a normal subgroup H by using the GKO projection, [36] and the energy-momentum tensor of the coset model G/H is expressed in terms of the difference of G and H. As discussed above, we cannot construct a coset model from the fixed subgroup by using the usual GKO projection. But we still have freedom to construct a coset model from the twisted CFT. As mentioned above, the algebra gl is non-semisimple, so if we remove the Abelian generator, we will obtain a new semisimple Lie algebra. In our case, we just need to remove the generator H 1 and H of gl, then the algebra is psl. As we now that the operator realization of D H1 is just a constant, it commutes with all generators, and D H will never appear in the right-hand side of any commutator, so the deletion of them has no effect to the algebraic relations. So if we project out the generators, we get a differential operators realization of the algebra psl. In fact, we just

9 No. 1 On gl Current Superalgebra and Twisted Conformal Field Theory 77 need to set the operator D H1 = 0 in the gl differential operators realization, and we will get the differential operators realization of psl, and then we will get the corresponding free field realizations too. Here we will omit the expressions. 5 Summery and Outloo In this paper, we consider the differential operators realization of Lie superalgebras psl and gl, and the free field realization of Kac Moody superalgebras psl and gl. The algebras can be viewed as the symmetries of the WZW-model on certain supergroup manifold. The CFTs corresponding to them are also considered. The supergroup P SL can be used to describe the bosonic geometry of Ads 3 S 3. If we project out other two U1 generators, it might be useful to probe the bosonic geometry of Ads S. There are several interesting issues that do not consider in this paper. The first one is the highest weight representation, or the primary field of the current algebra is not given. The higher spin primary fields, or the W -symmetry has not considered yet. The Knizhni Zamolodchiov KZ equation, and the correlation functions are also interesting problems, but we do not mention it in the present paper. According to the Drinfeld conjecture, the CFT at critical level = ĥ the dual Coxeter number will correspond to a central element of a completion of the universal enveloping algebra Ûg ĥ, and it has a canonical Poisson structure. In psln n or gln n case, the critical level is zero. It is still an open problem to deal with the system with zero dual Coxeter number. We hope to consider some aspects of them in the future wors. Acnowledgments Ding thans Prof. A. Bellen for his warm invitation and great help while Ding was staying in Trieste, where the wor was partially complected. Thans also to Prof. G. Lindi for his indness. And the wor is partially supported by Inistero degli Affari Esteri Direzione Generale per la Promozione la Cooperazione Culturale, and by Istituto Nazionale di Alta Matematica, francesco severi INdAM, Roma. References [1] A.A. Belavin, A.M. Polyaov, and A.B. Zamolodchiov, Nucl. Phys. B [] V.G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge [3] Ph. Di Francesco, P. Mathieu, and D. Senehal, Conformal Field Theory, Springer, Berlin [4] J. Polchinsi, String Theory, Cambridge University Press, Cambridge [5] A.B. Zamolodchiov, JETP Lett [6] N. Berovits, C. Vafa, and E. Witten, JHEP [7] M. Bershadsy, S. Zhuov, and A. Vaintrob, Nucl. Phys. B [8] N. Mann and J. Polchinsi, Bethe Ansatz for a Quantum Supercoset Sigma Model, hep-th/ [9] R. Metsaev and A. Tseytlin, Nucl. Phys. B [10] I. Bena, J. Polchinsi, and R. Roiban, Phys. Rev. D [11] A. M. Polyaov, Mod. Phys. Lett. A [1] R. Roiban and W. Siegel, JHEP [13] B.Y. Hou, D.T. Peng, C.H. Xiong, and R.H. Yue, The Affine Hidden Symmetry and Integrability of Type IIB Superstring in AdS 5 S 5, hep-th/ [14] L. Rozansi and H. Saleur, Nucl. Phys. B [15] J.M. Isidro and A.V. Ramallo, Nucl. Phys. B [16] K. Efetov, Adv. Phys [17] Z. Maassarani and D. Serban, Nucl. Phys. B [18] Z.S. Bassi and A. LeClair, Nucl. Phys. B [19] S. Guruswamy, A. LeClair, and A.W. Ludwig, Nucl. Phys. B [0] X.M. Ding, M.D. Gould, and Y.Z. Zhang, Phys. Lett. A [1] X.M. Ding, M.D. Gould, C.J. Mewton, and Y.Z. Zhang, J. Phys. A [] Y.Z. Zhang and M.D. Gould, J. Math. Phys [3] Y.Z. Zhang, Phys. Lett. A [4] M. Waimoto, Commun. Math. Phys [5] B.L. Feigin and E. Frenel, Commun. Math. Phys [6] J.L. Petersen, J. Rasmussen, and M. Yu, Nucl. Phys. B [7] M. Szczesny, Math. Res. Lett [8] X.M. Ding, M.D. Gould, and Y.Z. Zhang, Phys. Lett. B [9] L. Feher and B.G. Pusztai, Nucl. Phys. B [30] Y.Z. Zhang, X. Liu, and W.L. Yang, Nucl. Phys. B [31] V. Schomerus and H. Saleur, Nucl. Phys. B [3] V.G. Kac, Ad. Math [33] L. Frappat, P. Sorba, and A. Sciarrino, Dictionary on Lie Superalgebras, hep-th/ [34] E. Witten, Commun. Math. Phys ; Nucl. Phys. B [35] A. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin [36] P. Goddard, A. Kent, and D. Olive, Phys. Lett. B ; Commun. Math. Phys