gl(4|4) current algebra: free field realization and screening currents
aa r X i v : . [ h e p - t h ] M a y July 2006 gl (4 | current algebra: free field realization andscreening currents Wen-Li Yang a,b , Yao-Zhong Zhang b,c and Xin Liu b a Institute of Modern Physics, Northwest University, Xian 710069, P.R. China b Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia c Physikalisches Institut, Universit¨at Bonn, D-53115 Bonn, Germany
Abstract
The gl (4 |
4) current algebra at general level k is investigated. Its free field represen-tation and corresponding energy-momentum tensor are constructed. Seven screeningcurrents of the first kind are also presented. PACS:
Keywords : WZNW model; current algebra; free field realization.
Introduction
Two-dimensional non-linear sigma models with supermanifold target space naturally appearin the quantization of superstring theory on the AdS-type backgrounds. It was shown thatthe sigma model on the supergroup
P SU (1 , |
2) can be used for quantizing the superstringtheory on the
AdS × S background with Ramond-Ramond (RR) flux [1, 2]. It has beenbelieved [3, 2] that the sigma model on P SU (2 , |
4) is related to the string theory on the
AdS × S background and that the understanding of the P SU (2 , |
4) sigma model (or itsgeneralization, GL (4 | AdS × S background. The study of the non-linear sigma modelwith supermanifold target spaces has also believed to be relevant for disordered systems andthe integer quantum Hall transition [4, 5, 6, 7, 8].In most circumstances, models of interest are believed to be more complicated than theWZNW models on supergroups. However, even the WZNW models on supergroups are farfrom being understood [9]. This is largely due to technical reasons (such as indecomposabilityof the operator product expansion (OPE), appearance of logarithms in correlation functionsand continuous modular transformations of the irreducible characters [10]), combined withthe lack of “physical intuition”.Free field realization [11] has been proved to be a powerful method in the study theconformal field theory (CFT) such as WZNW models. In this letter, motivated by theapplications both to superstring theory and condensed matter physics, we investigate the gl (4 |
4) current algebra associated with the GL (4 |
4) WZNW model at general level k . gl (4 | current algebra Let us start with some basic notation of the gl (4 |
4) current algebra, i.e. the \ gl (4 |
4) affinesuperalgebra [12, 13]. Let { E i,j | i, j = 1 , . . . , } be the generators of the finite dimensionalsuperalgebra gl (4 | E i,j , E k,l ] = δ j k E i,l − ( − ([ i ]+[ j ])([ k ]+[ l ]) δ i l E k,j . (2.1)Here and throughout, we adopt the convention: [ a, b ] = ab − ( − [ a ][ b ] ba . The Z grading ofthe generators is [ E i,j ] = [ i ] + [ j ] , . . . = [4] = 0 , [5] = . . . = [8] = 1. Due to the non-simplicity of gl (4 | C = X i,j =1 ( − [ j ] E i,j E j,i , (2.2)there exists another independent quadratic Casimir element C = X i,j =1 E i,i E j,j = X i =1 E i,i ! . (2.3)These two Casimir elements are useful in the following for the construction of the energy-momentum tensor.Let V be a Z -graded (4+4)-dimensional linear space with the orthonormal basis {| i i , i =1 , . . . , } . The Z -grading is chosen as: [1] = . . . = [4] = 0 , [5] = . . . = [8] = 1. In fact V spans the fundamental representation of gl (4 | V by E i,j = e i,j , i, j = 1 , . . . , , where e i,j is the matrix with entry 1 at the i -th row and j -th column, and zero elsewhere.We remark that the Casimir element C vanishes on V , while C acts as a multiple ofidentity on V . From the fundamental representation of gl (4 | E i,j , E k,l ) = str ( e i,j e k,l ) . (2.4)Here str denotes the supertrace, i.e., str ( a ) = P i ( − [ i ] a i i .The gl (4 |
4) current algebra is generated by the currents J i,j ( z ) which are associated withthe generators E i,j of gl (4 | k obeys the followingOPEs [11], J i,j ( z ) J l,m ( w ) = k ( E i,j , E l,m )( z − w ) + 1( z − w ) (cid:0) δ j l J i,m ( w ) − ( − ([ i ]+[ j ])([ l ]+[ m ]) δ i m J l,j ( w ) (cid:1) . (2.5) Free field realization of the gl (4 |
4) currents, in principle , can be obtained by a general methodoutlined in [14, 15, 16, 17, 18], where differential realizations of the corresponding finite3imensional Lie (super) algebras play a key role. However, their constructions become verycomplicated for higher-rank algebras [16, 18, 19, 20]. We have found a way to obtain explicitexpressions of the differential realizations of gl ( m | n ). In our approach, the constructionbecomes much simpler. In this letter we will restrict our attention mainly to gl (4 | gl ( m | n ),more details and proofs, will be given elsewhere [21].Let us introduce 12 bosonic coordinates { x i,j , x i, j | ≤ i < j ≤ } with the Z -grading: [ x i,j ] = 0, and 16 fermionic coordinates { θ i, j | i, j = 1 , . . . , } with the Z -grading:[ θ i, j ] = 1. These coordinates satisfy the following (anti-)commutation relations:[ x i,j , x k,l ] = 0 , [ ∂ x i,j , x k,l ] = δ ik δ jl , [ θ i, j , θ k, l ] = 0 , [ ∂ θ i, j , θ k, l ] = δ ik δ jl , and the other commutation relations are vanishing. Then the generators corresponding tothe simple roots in the standard (distinguished) basis [13] can be realized by the followingdifferential operators: E j,j +1 = X k ≤ j − x k,j ∂ x k,j +1 + ∂ x j,j +1 , ≤ j ≤ , (3.1) E , = X k ≤ x k, ∂ θ k, + ∂ θ , , (3.2) E j, j = X k ≤ θ k, j ∂ θ k, j + X k ≤ j − x k, j ∂ x k, j + ∂ x j, j , ≤ j ≤ , (3.3) E j,j = X k ≤ j − x k,j ∂ x k,j − X j +1 ≤ k ≤ x j,k ∂ x j,k − X k ≤ θ j, k ∂ θ j, k + λ j , ≤ j ≤ , (3.4) E , = X k ≤ x k, ∂ x k, − X k ≤ θ , k ∂ θ , k + λ , (3.5) E j, j = X k ≤ θ k, j ∂ θ k, j + X k ≤ j − x k, j ∂ x k, j − X j +1 ≤ k ≤ x j, k ∂ x j, k + λ j , ≤ j ≤ , (3.6) E j +1 ,j = X k ≤ j − x k,j +1 ∂ x k,j − X j +2 ≤ k ≤ x j,k ∂ x j +1 ,k − X k ≤ θ j, k ∂ θ j +1 , k − x j,j +1 X j +1 ≤ k ≤ x j,k ∂ x j,k + X k ≤ θ j, k ∂ θ j, k ! + x j,j +1 X j +2 ≤ k ≤ x j +1 ,k ∂ x j +1 ,k + X k ≤ θ j +1 , k ∂ θ j +1 , k ! x j,j +1 ( λ j − λ j +1 ) , ≤ j ≤ , (3.7) E , = X k ≤ θ k, ∂ x k, + X ≤ k ≤ θ , k ∂ x , k − θ , X ≤ k ≤ (cid:0) θ , k ∂ θ , k + x , k ∂ x , k (cid:1)! + θ , ( λ + λ ) , (3.8) E j, j = X k ≤ θ k, j ∂ θ k, j + X k ≤ j − x k, j ∂ x k, j − X j +2 ≤ k ≤ x j, k ∂ x j, k − x j, j X j +1 ≤ k ≤ x j, k ∂ x j, k − X j +2 ≤ k ≤ x j, k ∂ x j, k ! + x j, j ( λ j − λ j ) , ≤ j ≤ , (3.9)where { λ j | j = 1 , . . . , } are c -numbers which label the lowest weight vector of gl (4 | E i,j = [ E i,k , E k,j ] , ≤ i < k < j ≤ ≤ j − i, (3.10) E j,i = [ E j,k , E k,i ] , ≤ i < k < j ≤ ≤ j − i. (3.11)One can prove that the differential realization (3.1)-(3.9) of gl (4 |
4) satisfies the commutationrelations (2.1).With the help of the differential realization we can obtain the free field realization (Waki-moto construction) of the gl (4 |
4) current algebra in terms of 12 bosonic β - γ pairs (( β i,j , γ i,j )and ( ¯ β i,j ¯ γ i,j ), for 1 ≤ i < j ≤ b - c pairs (( ψ † i,j , ψ i,j ), for 1 ≤ i, j ≤
4) and 8free scalar fields ( φ i , for i = 1 , . . . , β i,j ( z ) γ k,l ( w ) = − γ k,l ( z ) β i,j ( w ) = δ ik δ jl ( z − w ) , ≤ i < j ≤ ≤ k < l ≤ , (3.12)¯ β i,j ( z ) ¯ γ k,l ( w ) = − ¯ γ k,l ( z ) ¯ β i,j ( w ) = δ ik δ jl ( z − w ) , ≤ i < j ≤ ≤ k < l ≤ , (3.13) ψ i,j ( z ) ψ † k,l ( w ) = ψ † k,l ( z ) ψ i,j ( w ) = δ ik δ jl ( z − w ) , ≤ i, j ≤ ≤ k, l ≤ , (3.14) φ i ( z ) φ j ( w ) = ( − [ i ] δ ij ln( z − w ) , ≤ i, j ≤ , (3.15)and the other OPEs are trivial.The free field realization of the gl (4 |
4) current algebra (2.5) is obtained by the following5ubstitution: x i,j −→ γ i,j ( z ) , ∂ x i,j −→ β i,j ( z ) , ≤ i < j ≤ ,x i, j −→ ¯ γ i,j ( z ) , ∂ x i, j −→ ¯ β i,j ( z ) , ≤ i < j ≤ ,θ i, j −→ ψ † i,j ( z ) , ∂ θ i, j −→ ψ i,j ( z ) , ≤ i, j ≤ ,λ j −→ √ k∂φ j ( z ) − ( − [ j ] √ k X l =1 φ l ( z ) , ≤ j ≤ , in the differential realization (3.1)-(3.9) of gl (4 |
4) and a subsequent addition of anomalousterms linear in ∂ψ † ( z ), ∂γ ( z ) and ∂ ¯ γ ( z ) in the expressions of the currents. Here we presentthe realization of the currents associated with the simple roots, J j,j +1 ( z ) = X l ≤ j − γ l,j ( z ) β l,j +1 ( z ) + β j,j +1 ( z ) , ≤ j ≤ , (3.16) J , ( z ) = X l ≤ γ l, ( z ) ψ l, ( z ) + ψ , ( z ) , (3.17) J j, j ( z ) = X l ≤ ψ † l,j ( z ) ψ l,j +1 ( z ) + X l ≤ j − ¯ γ l,j ( z ) ¯ β l,j +1 ( z ) + ¯ β j,j +1 ( z ) , ≤ j ≤ , (3.18) J j,j ( z ) = X l ≤ j − γ l,j ( z ) β l,j ( z ) − X j +1 ≤ l ≤ γ j,l ( z ) β j,l ( z ) − X l ≤ ψ † j,l ( z ) ψ j,l ( z )+ √ k∂φ j ( z ) − √ k X l =1 ∂φ l ( z ) , ≤ j ≤ , (3.19) J , ( z ) = X l ≤ γ l, ( z ) β l, ( z ) − X l ≤ ψ † ,l ( z ) ψ ,l ( z ) + √ k∂φ ( z ) − √ k X l =1 ∂φ l ( z ) , (3.20) J j, j ( z ) = X l ≤ ψ † l,j ( z ) ψ l,j ( z ) + X l ≤ j − ¯ γ l,j ( z ) ¯ β l,j ( z ) − X j +1 ≤ l ≤ ¯ γ j,l ( z ) ¯ β j,l ( z )+ √ k∂φ j ( z ) + 12 √ k X l =1 ∂φ l ( z ) , ≤ j ≤ , (3.21) J j +1 ,j ( z ) = X l ≤ j − γ l,j +1 ( z ) β l,j ( z ) − X j +2 ≤ l ≤ γ j,l ( z ) β j +1 ,l ( z ) − X l ≤ ψ † j,l ( z ) ψ j +1 ,l ( z ) − γ j,j +1 ( z ) X j +1 ≤ l ≤ γ j,l ( z ) β j,l ( z ) + X l ≤ ψ † j,l ( z ) ψ j,l ( z ) ! + γ j,j +1 ( z ) X j +2 ≤ l ≤ γ j +1 ,l ( z ) β j +1 ,l ( z ) + X l ≤ ψ † j +1 ,l ( z ) ψ j +1 ,l ( z ) ! √ kγ j,j +1 ( z ) ( ∂φ j ( z ) − ∂φ j +1 ( z )) + ( k + j − ∂γ j,j +1 ( z ) , ≤ j ≤ , (3.22) J , ( z ) = X l ≤ ψ † l, ( z ) β l, ( z ) + X ≤ l ≤ ψ † ,l ( z ) ¯ β ,l ( z ) − ψ † , ( z ) X ≤ l ≤ (cid:16) ψ † ,l ( z ) ψ ,l ( z ) + ¯ γ ,l ( z ) ¯ β ,l ( z ) (cid:17)! + √ kψ † , ( z ) ( ∂φ ( z ) + ∂φ ( z )) + ( k + 3) ∂ψ † , ( z ) , (3.23) J j, j ( z ) = X l ≤ ψ † l,j +1 ( z ) ψ l,j ( z ) + X l ≤ j − ¯ γ l,j +1 ( z ) ¯ β l,j ( z ) − X j +2 ≤ l ≤ ¯ γ j,l ( z ) ¯ β j +1 ,l ( z ) − ¯ γ j,j +1 ( z ) X j +1 ≤ l ≤ ¯ γ j,l ( z ) ¯ β j,l ( z ) − X j +2 ≤ l ≤ ¯ γ j +1 ,l ( z ) ¯ β j +1 ,l ( z ) ! + √ k ¯ γ j,j +1 ( ∂φ j ( z ) − ∂φ j ( z )) − ( k + 5 − j ) ∂ ¯ γ j,j +1 ( z ) , ≤ j ≤ . (3.24)Here normal ordering of the free field expressions is implied. The free field realization forother currents associated with the non-simple roots can be obtained from the OPEs of thesimple ones. It is straightforward to check that the above free field realization of the currentssatisfy the OPEs of the gl (4 |
4) current algebra given in the last section.
In order to apply the free field realization of the gl (4 |
4) currents to compute conformal blocksof the associated WZNW-conformal field theory, we need to calculate the energy-momentumtensor of the associated CFT. The Sugawara tensor corresponding to the quadratic Casimir C is given by T ( z ) = 12 k X i,j =1 ( − [ j ] : J i,j ( z ) J j,i ( z ) := 12 X l =1 ( − [ l ] ∂φ l ( z ) ∂φ l ( z ) + 12 √ k ∂ X i =1 (2 i −
1) ( φ i ( z ) + φ − i ( z )) ! + X i 1) ( φ i ( z ) + φ − i ( z )) ! + X i 4) current algebra. An important object in applying the free field realization to the computation of correlationfunctions of the associated CFT is screening currents. A screening current is a primary fieldwith conformal dimension one and has the property that the singular part of the OPE withthe affine currents is a total derivative. These properties ensure that integrated screeningcurrents (screening charges) may be inserted into correlators while the conformal or affineWard identities remain intact. This in turn makes them very useful in computation of thecorrelation functions [22, 23]. For the present case, we find seven screening currents S j ( z ) = X j +2 ≤ l ≤ γ j +1 ,l ( z ) β j,l ( z ) + X l =1 ψ † j +1 ,l ( z ) ψ j,l ( z ) + β j,j +1 ( z ) ! ˜ s j ( z ) , ≤ j ≤ , (5.1) S ( z ) = X ≤ l ¯ γ ,l ( z ) ψ ,l ( z ) + ψ , ( z ) ! ˜ s ( z ) , (5.2) S j ( z ) = X j +2 ≤ l ≤ ¯ γ j +1 ,l ( z ) ¯ β j,l ( z ) + ¯ β j,j +1 ( z ) ! ˜ s j ( z ) , ≤ j ≤ , (5.3)where˜ s j ( z ) = e − √ k ( φ j ( z ) − φ j +1 ( z )) , ˜ s ( z ) = e − √ k ( φ ( z )+ φ ( z )) , ˜ s j ( z ) = e √ k ( φ j ( z ) − φ j ( z )) . (5.4)The normal ordering of the free field expressions is implicit in the above equations. Thenontrivial OPEs of the screening currents with the energy-momentum tensor and the gl (4 | T ( z ) S j ( w ) = S j ( w )( z − w ) + ∂S j ( w )( z − w ) = ∂ w (cid:26) S j ( w )( z − w ) (cid:27) , ≤ j ≤ , (5.5) J i +1 ,i ( z ) S j ( w ) = ( − [ i ]+[ i +1] δ ij ∂ w (cid:26) k ˜ s j ( w )( z − w ) (cid:27) , ≤ i, j ≤ . (5.6)The screening currents obtained here are associated with the simple roots and correspondto the first kind screening currents [24]. Moreover, S ( z ) is fermionic screening current andthe others are all bosonic ones. We have studied the gl (4 | 4) current algebra at general level k . We have constructed itsWakimoto free field realization (3.16)-(3.24) and the corresponding energy-momentum tensor(4.3). We have also found seven screening currents, (5.1)-(5.4), of the first kind.To fully take the advantage of the CFT method, one needs to construct its primary fields.It is well-known that there exist two types of representations for the underlying finite dimen-sional superalgebra gl (4 | cknowledgements The financial support from the Australian Research Council is gratefully acknowledged. YZZwas also supported by the Max-Planck-Institut f¨ur Mathematik (Bonn) and by the Alexandervon Humboldt-Stiftung. XL has been supported by IPRS and UQGSS scholarships of theUniversity of Queensland. YZZ would like to thank the Max-Planck-Institut f¨ur Mathematik,where part of this work was done, and Physikalisches Institut der Universit¨at Bonn, especiallyG¨unter von Gehlen, for hospitality. Note added : We became aware that free field realizations of sl ( N | N ) (or gl ( N | N )) currentalgebra were investigated previously in [25, 26]. There, the sl ( N | N ) k (or gl ( N | N ) k ) currentswere expressed in terms of the sl ( N ) k − N and sl ( N ) − k − N currents and some b - c pairs. Aspart of the results of our paper, we give the explicit expressions of gl (4 | k currents in termsof free fields, by using a different method. References [1] N. Berkovits, C. Vafa and E. Witten, JHEP , (1999), 018.[2] M. Bershadsky, S. Zhukov and A. Vaintrob, Nucl. Phys. B 559 (1999), 205.[3] R. R. Metsaev and A. A. Tseytlin, Nucl. Phys. (1998), 109.[4] D. Bernard, hep-th/9509137 .[5] C. Mudry, C. Chamon and X. -G. Wen, Nucl. Phys. B 466 (1996), 383.[6] Z. Massarani and D. Serban, Nucl. Phys. B 489 (1997), 603.[7] Z. S. Bassi and A. LeClair, Nucl. Phys. B 578 (2000), 577.[8] S. Guruswamy, A. LeClair and A. W. W. Ludwig, Nucl. Phys. B 583 (2000), 475.[9] V. Schomerus and H. Saleur, Nucl. Phys. B 734 (2006), 221.[10] A. M. Semikhatov, A. Taormina and I. Yu Timpunin, math.QA/0311314 .1011] P. Di Francesco, P. Mathieu and D. Senehal, Conformal Field Theory , Springer Press,Berlin, 1997.[12] V. G. Kac, Adv. Math. (1977), 8.[13] L. Frappat, P. Sorba and A. Scarrino, hep-th/9607161 .[14] B. Feigin and E. Frenkel, Commun. Math. Phys. (1990), 161.[15] P. Bouwknegt, J. McCarthy and K. Pilch, Prog. Phys. Suppl. (1990), 67.[16] K. Ito, Phys. Lett. B 252 (1990), 69.[17] E. Frenkel, hep-th/9408109 .[18] J. Rasmussen, Nucl. Phys. B 510 (1998), 688.[19] X. -M. Ding, M. Gould and Y. -Z. Zhang, Phys. Lett. A 318 (2003), 354.[20] Y. -Z. Zhang, X. Liu and W. -L. Yang, Nucl. Phys. B 704 (2005), 510.[21] W. -L. Yang, Y. -Z. Zhang and X. Liu, J. Math. Phys. (2007), 053514.[22] VI. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 240 (1984), 312; Nucl. Phys. B 251 (1985), 3691.[23] D. Bernard and G. Felder, Commun. Math. Phys. (1990), 145.[24] M. Bershsdsky and H. Ooguri, Commun. Math. Phys. (1986), 49.[25] I. Bars, Phys. Lett. B 255 (1991), 353.[26] J. M. Isidro and A. V. Ramallo, Nucl. Phys.