arXiv:hep-th/9405049v1 9 May 1994 dnie steetne oooia ofra ler fnon-c of algebra conformal topological extended the as identified sl ipero ytmo h ffieLesuperalgebra Lie pa affine a the on of based system is root construction simple The reduction. hamiltonian quantum of 2 medn.Atrtitn n iiaiytasomto,this transformation, similarity a and twisting After embedding. (2) eoti e refil elzto of realization field free new a obtain We xeddTplgclCnomlSymmetry Conformal Topological Extended nttt fPyis nvriyo skb,Iaai35 J 305, Ibaraki Tsukuba, of University Physics, of Institute eateto ahmtc,HrsiaUniversity, Hiroshima Mathematics, of Department nNon-critical in i ueagbaand Superalgebra Lie iah-iohm 2,Japan 724, Higashi-Hiroshima iok Kanno Hiroaki assiIto Katsushi Abstract and N super 2 = W sl (3 3 | 2) Strings (1) W soitdwt non-standard a with associated 3 ler sn h technique the using algebra ritical tclrcoc fthe of choice rticular W apan W 3 ler a be can algebra tigtheory. string UTHEP-277 a 1994 May Many attempts have been done to construct non-critical string theories with W algebra symmetry (non-critical W strings)[1], which may be defined beyond the c = 1 barrier of non-critical string theory. Physical states of non-critical W -strings can be characterized by the BRST cohomology. It is known that the BRST current has quite non-trivial structure due to non-linearity of W -algebra [2]. It is recently understood that the BRST algebra in non-critical string theory may be enlarged to the twisted N = 2 superconformal algebra i.e. topological conformal algebra[3]. Bershadsky, Lerche, Nemeschansky and Warner[4] found the topological W -symmetry in the non-critical W3-string. This fact would be universal in a class of non-critical W -string theories and essential for investigation of their properties as topological strings, which is a clue to a non-perturbative formulation of string theories in higher dimensions. Practical computations involving (topological) W algebra often become complicated due to its non-linearity. In this paper we regard the topological WN symmetry as a result of the quantum Hamiltonian reduction of an affine Lie superalgebra sl(N N | − 1)(1). We believe that the viewpoint of Lie superalgebra is helpful in more systematic understanding of the algebraic structure of topological W symmetry, which is obscured by the non-linearity. For example, the existence of the BRST current which has completely vanishing nilpotent OPE relation is most clearly understood from the hidden symmetry of sl(N N 1)(1) [4]. The Lie superalgebra may also explain the origin of the screening | − operators which play an essential role in investigating the physical spectrum, especially the problem of W gravitational dressing. This implies the Lie superalgebra sl(N N 1)(1) | − is important for geometrical aspects of the theory, such as W moduli. In a previous paper[5], we have shown the relation between the topological conformal algebra and the Lie superalgebra sl(2 1)(1) using the hamiltonian reduction. In this article | we will examine the quantum hamiltonian reduction of an affine Lie superalgebra sl(3 2)(1) | and study a free field realization of the N = 2 super-W3 algebra [6] relevant to the non-critical W3-string theory. This reduction at classical level has been discussed in [4]. However, the fermionic ghosts that they employed are not free fields due to ghost number violation term in the U(1) current, which cannot be expected from the standard hamiltonian reduction. We will show that by a similarity transformation the fermionic ghosts of Bershadsky et.al. are related to the genuine free fields which naturally appear

1 in the quantum hamiltonian reduction. Let us define an affine Lie superalgebra gˆ = sl(N N 1)(1). The algebra gˆ at level K | − is generated by bosonic currents J (z)(i, j =1,...,N), J (z)(a, b =1,...,N 1) i,j N+a,N+b − and fermionic currents j (z), j (z) (i =1,...,N,a =1,...,N 1). The diagonal i,N+a N+a,i − part of the bosonic currents satisfies the super-traceless condition:

N N−1 J (z) J (z)=0. (1) i,i − N+a,N+a Xi=1 aX=1 The operator product expansions for these currents are given by K(δ δ δ δ ) δ J (w) δ J (w) J (z)J (w) = j,k i,l − i,j k,l + j,k i,l − i,l k,i + , i,j k,l (z w)2 z w ··· − − K(δ δ + δ δ ) δ J (w) δ J (w) J (z)J (w) = − b,c a,d a,b c,d + d,a N+c,N+b − b,c N+a,N+d + , N+a,N+b N+c,N+d (z w)2 z w ··· − − Kδ δ J (z)J (w) = − i,j a,b + , i,j N+a,N+b (z w)2 ··· − δ j (w) J (z)j (w) = j,k k,N+a + , i,j k,N+a z w ··· δ j− (w) J (z)j (w) = − i,k N+a,j + , i,j N+a,k z w ··· δ j − (w) J (z)j (w) = a,c k,N+b + , N+a,N+b k,N+c z w ··· δ j− (w) J (z)j (w) = − b,c N+a,k + , N+a,N+b N+c,k z w ··· Kδ δ− δ J (w) δ J (w) j (z)j (w) = i,j a,b + a,b i,j − i,j N+b,N+a + . (2) i,N+a N+b,j (z w)2 z w ··· − − The Lie superalgebra g = sl(N N 1) admits several choices of the simple root system. | − In the present paper, we use the following simple root system: α = e e (i = i i − i+1 1,...,N 1), α = e δ , α = δ δ (a =1,...,N 2), where we introduced an − N N − 1 a+N a − a+1 − orthonormal basis e (i =1,...,N) and δ (a =1,...,N 1), which satisfies e e = δ , i a − i · j i,j δ δ = δ . This simple root system, which is characterized as having only one odd root, a· b − a,b corresponds to a non-standard sl(2) embedding. The standard sl(2) embedding gives the purely fermionic simple root system, which corresponds to the manifestly supersymmetric N = 2 super-W algebra [7]. The bosonic currents J (J ) for i = j (a = b) N i,j N+a,N+b 6 6 correspond to even roots e e (δ δ ). The fermionic currents j (j ) correspond i − j a − b i,N+a N+a,i to odd roots e δ (δ e ). The positive roots are e e (i < j), δ δ (a < b) and i − a a − i i − j a − b 2 e δ . The Cartan currents for simple roots H = α H (i = 1,..., 2n) are defined by i − a i i · H = J J . i i,i − i+1,i+1 Based on this algebra, we construct the reduced phase space by imposing the con- straints. The Lie superalgebra demands the second class constraints. In such a case it is convenient to introduce auxiliary free fermions which change the second class con- straints to the first class[8]. Let us introduce N 1 pairs of fermionic fields (ψ† , ψ ) − N+a N+a (a = 1,...,N 1) with weights (1/2+ N a, 1/2 N + a). The operator product − − − expansions are given by δ ψ (z)ψ† (w)= a,b + . (3) N+a N+b z w ··· − The constraints which are consistent with the affine Lie superalgebra are

Ji,j(z) = δi,j−1, for i < j, † ji,N+a(z) = δi,N ψN+a(z), † JN+a,N+b(z) = δa,b−1 + ψN+aψN+b(z), for a < b. (4)

Let us consider the BRST quantization of this constraint system. Introduce fermionic ˜ ˜ ghosts (˜ci,j(z), bi,j(z)) (i < j) and (˜cN+a,N+b(z), bN+a,N+b(z)) (a < b) for the constrains

for bosonic currents. Bosonic ghosts (˜γi,N+a(z), β˜i,N+a(z)) for fermionic constraints. The operator product expansions are given by δ δ c˜ (z)˜b (w) = i,k k,l + , i,j k,l z w ··· δ −δ c˜ (z)˜b (w) = a,c b,d + , N+a,N+b N+c,N+d z w ··· δ −δ β˜ (z)˜γ (w) = i,j a,b + . (5) i,N+a j,N+b z w ··· − The BRST current is defined as

J (z) = c˜ (J δ )+ c˜ (J δ ψ ψ† ) BRST i,j i,j − i+1,j N+a,N+b N+a,N+b − a+1,b − N+a N+b Xi

3 dz Indeed the BRST charge QBRST = 2πi JBRST (z) satisfies the nilpotency condition: 2 1 QBRST = 0. We wish to investigate theH BRST cohomology on the space of operator algebra : Atot = Ugˆ Cl † Cl˜ H ˜ , (7) Atot ⊗ ψ,ψ ⊗ b,c˜ ⊗ β,γ˜

where Ugˆ is the universal enveloping algebra of gˆ, Clψ,ψ† the Clifford algebra of auxiliary † ˜ fermionic fields (ψN+a, ψN+a), Cl˜b,c˜ the Clifford algebra of fermionic ghosts (bi,j, c˜i,j), ˜ Hβ,˜ γ˜ the Heisenberg algebra of bosonic ghosts (βi,N+a, γ˜i,N+a). The algebra of BRST cohomology H ( ) may be identified with the quantum W algebra. According QBRST Atot to Feigin-Frenkel [9], we can simplify the full BRST complex by decomposing the BRST currents into a constraint part associated with an sl(2) embedding and the canonical coboundary operator of the nilpotent subalgebra. This double complex can be analyzed by using the spectral sequence technique. Recently, a quite interesting observation was made by de Boer and Tjin. They found that the components of non-trivial cohomology which have zero gradation in total degree of the double complex form a closed algebra and the generators are nothing but a free field realization of the quantum W -algebra[10]. In presence of the second class constraints, however, this standard decomposition does not work in general [11], which makes the analysis more complicated. 0 In the present paper, we propose the following decomposition JBRST (z)= JBRST (z)+ 1 JBRST (z), where

1 † JBRST (z)= c˜i,i+1 c˜N+a,N+a+1 γ˜N,N+aψN+a, (8) − − a − a Xi X X and J 0 (z) = J (z) J 1 (z). Define two BRST charges Q and Q by con- BRST BRST − BRST 0 1 0 1 0 tour integration of JBRST (z) and JBRST (z), respectively. Note that JBRST is differ- ent from the canonical coboundary operator for the nilpotent subalgebra by the term † a

It is convenient to introduce new currents Ji,j, JN+a,N+b, ji,N+a and jN+a,i (see Appendix 1 This BRST cohomology should not be confusede withe the BRST cohomoloe gy whiche defines the physical spectrum of non-critical strings.

4 A) modified by the ghosts, which satisfy

Q0(˜bi,j) = Ji,j, i

Q ( β˜ ) = je , 0 − i,N+a i,N+a Q0(˜bN+a,N+b) = JeN+a,N+b, a

The pairs (J , ˜b )(i < j), (J , ˜b e)(a < b) and (j , β˜ ) form BRST i,j i,j N+a,N+b N+a,N+b i,N+a − i,N+a doublets ande decouple from thee non-trivial cohomology. Therefore e we consider only the reduced complex , which spanned by other modified currents and ghosts. In the Ared following we shall take N = 3; the sl(3 2)(1) case for simplicity. | Let us consider the Q0 action on the reduced complex. The action on the modified currents is given by

Q0(J2,1) = (H1c˜1,2)+(K + 1)∂c˜1,2, † Q (Je ) = (Hf c˜ )+(K + 1)∂c˜ (˜c (ψ ψ )), 0 3,2 2 2,3 2,3 − 4,5 4 5 Q (Je ) = ((fH + H )˜c )+(J c˜ ) (J c˜ )+(K + 2)∂c˜ , 0 3,1 1 2 1,3 2,1 2,3 − 3,2 1,2 2,3 Q (Je ) = (fH c˜ f) K∂c˜ ,e e 0 5,4 − 4 4,5 − 4,5 † † Q (je ) = (Hfγ˜ )+(K + 1)∂γ˜ + ((ψ ψ )˜γ ) ((ψ ψ )˜γ ), 0 4,3 3 3,4 3,4 5 5 3,4 − 4 5 3,5 † † Q (ej ) = ((fH + H )˜γ )+(K + 2)∂γ˜ +(J γ˜ )+(j c˜ ) ((ψ ψ )˜γ ) ((ψ ψ )˜γ ), 0 4,2 2 4 2,4 2,4 3,2 3,4 4,3 2,3 − 4 4 2,4 − 4 5 2,5 Q0(ej4,1) = ((Hf1 + Hf2 + H3)˜γ1,4)+(K + 3)∂γ˜1,e4 +(J2,1γ˜2,e4)+(J3,1γ˜3,4) † † e +(fj c˜ f)+(fj c˜ ) ((ψ ψ )˜γ ) ((ψe ψ )˜γ ), e 4,2 1,2 4,3 1,3 − 4 4 1,4 − 4 5 1,5 Q0(j5,1) = ((He1 + H2 + He 3 + H4)˜γ1,5)+(K + 2)∂γ˜1,5 +(J2,1γ˜2,5)+(J3,1γ˜3,5) † † e +(fj c˜ f)+(fj c˜ f) (J γ˜ ) ((ψ ψ )˜γ )e ((ψ ψ )˜γe ), 4,1 4,5 5,2 1,3 − 5,4 1,4 − 5 4 1,4 − 5 5 1,5 Q (j ) = ((He + H + He )˜γ )+(Ke + 1)∂γ˜ +(J γ˜ ) (J γ˜ ) 0 5,2 2 3 4 2,5 2,5 3,2 3,5 − 5,4 2,4 † † e +(fj c˜ f)+(fj c˜ ) ((ψ ψ )˜γ ) ((ψe ψ )˜γ ), e 4,2 4,5 5,3 2,3 − 5 4 2,4 − 5 5 2,5 † † Q (j ) = ((He + H )˜γ e)+ K∂γ˜ +(j c˜ ) (J γ˜ ) ((ψ ψ )˜γ ) ((ψ ψ )˜γ ). (11) 0 5,3 2 3 3,5 3,5 4,3 4,5 − 5,4 3,4 − 5 4 3,4 − 5 5 3,5 Heree we definef the normalf ordered producte (AB)(z)e for two operators A(z) and B(z) dw A(w)B(z) by (AB)(z) = z 2πi w−z . We take the Q1-cohomology first. It is easy to see that H ( ) is generatedR by Q1 Ared (0) = 2H +3H +6H +3H , U 1 2 3 4 f f f f 5 − (0) = ψ†, G 5 + (0) = j + j , G 4,1 5,2 (0) † † = eJ +eJ J + j ψ +(j ψ )ψ , T2 2,1 3,2 − 5,4 5,3 5 4,3 − 5 4 (0) = Je , e e e e W2 5,4 † − (0) = ψe , W 4 + (0) = j , W 5,1 (0) † † = eJ +(j + j )ψ + j ψ . (12) W3 3,1 4,1 5,3 4 4,3 5 e From the standard procedure of the spectrale e sequencee [12], we need to solve the descent equation Q ( (n))= Q ( (n+1)). (13) 0 O 1 O The generators of the total BRST cohomology are given by

= (0) (1) + (2) + , (14) O O − O O ··· For a moment we shall restrict our concern to the N = 2 superconformal sector. The spin one operator (0) satisfies Q ( (0)) =0. We get = (0). For a spin two operator (0), U 0 U U U T we find = (0) (1) where T T − T 1 (1) = (J 2 + J 2 + J 2 J 2 J 2 ) (H (ψ ψ†)) ((H + H )(ψ ψ†)) T −2 1,1 2,2 3,3 − 4,4 − 5,5 − 3 4 4 − 3 4 5 5 † † † 2((ψe ψ )(eψ ψ ))+((e Je ψ )ψe )) f f f − 4 4 5 5 5,4 4 5 † † † † (ψ ∂ψ ) (2 + K)(∂ψe ψ ) (ψ ∂ψ ) (1 + K)(∂ψ ψ ) − 4 4 − 4 4 − 5 5 − 5 5 K 1 (1 + K)∂H (2 + K)∂H 3H + − ∂H . (15) − 1 − 3 − 3 2 4 f f f f Next we consider the fermionic part. For − (0), we get − = − (0). For the operator G G G +, we find + = + (0) + (1) + + (2) + (3). In the present paper we do not want G G G − G G − G to write down explicit formulae for each term in the above expression due to complexity. Compared to the sl(2 1)(1) case, all of the last terms of solutions to the descent equation | cannot be written in terms of the modified Cartan currents H’s and the auxiliary fermions.

We note that the argument of ref. [10] is not valid in thisf case, because we have made a non-standard decomposition of the BRST current. The gradation of the free fields in the double complex depends on the way of decomposing the BRST operator. Nevertheless

6 we find that we can obtain a free field realization O(z) of N = 2 superconformal algebra by collecting the terms which contain H’s and ψ, ψ†’s from the corresponding generator (z) of the total BRST cohomology. O f We now derive the free field realization from the quantum hamiltonian reduction described above. Introduce four free bosons φ =(φ1,φ2,φ3,φ4) and bosonize the modified Cartan currents H = iaα ∂φ with a = √K + 1. In the following we use more explicit i i · expressions: f ia 3 H1 = ∂φ1 ia ∂φ2, √2 − s 2 f ia 3 H2 = ∂φ1 + ia ∂φ2, √2 s2 f ia ia a a H3 = ∂φ1 ∂φ2 + ∂φ3 + ∂φ4, −√2 − √6 √6 √2 Hf = a√2∂φ . (16) 4 − 4 f The generators of N = 2 superconformal algebras are found to be (v i∂φ ) k ≡ k U = (ψ ψ†)+(ψ ψ†) ia√6v , 4 4 5 5 − 3 + + + + G = G1 + G2 + ∂(∆G ), − † G = ψ5, 4 N=2 1 2 1 3 1 1 1 T = v + √2(a )∂v1 i ∂v3 i (a + )∂v4 2 i − a − s 2 a − √2 a Xi=1 3 5 1 3 (∂ψ†)ψ ψ†∂ψ (∂ψ†)ψ ψ†∂ψ , (17) −2 4 4 − 2 4 4 − 2 5 5 − 2 5 5 where

+ 1 G = ψ5(TM + TL + T † + T † ), 1 2 ψ5,ψ5 ψ4,ψ4 + 2 2 † a 2a 1 † G2 = ψ4 (1 + a )(∂ ψ4ψ4) WM v+TM +(a + )∂TM √6av+(∂ψ4ψ4) ( − √3bM − √6 a − 4 2 a 3 1 3(2a a + 2) 2 + 4iv+ +6√6(a + )v+∂v+ i − ∂ v+ , 3√6 a − a2 )   ∆G+ = ψ (i√6av (ψ ψ†))+(1 3a2)∂ψ + (1 2a2)ψ (∂ψ ψ†) 5 3 − 4 4 − 5 − 4 4 4 2 2 2 3 3 (6a 1)(a 1) 2 a TM ψ4 + i ∂(v+ψ4)+ i√6a (∂v+)ψ4 + − − ∂ ψ4. (18) − s2 6

7 Here we define the energy momentum tensors for W3-minimal model (M), W3-gravity (L) and two ghost systems with spins (3, 2) and (2, 1); − − 1 1 1 T = v2 + v2 + √2(a )∂v , M 2 1 2 2 − a 1

1 2 1 2 3 1 1 1 TL = v + v i (a + )∂v3 i (a + )∂v4, 2 3 2 4 − s 2 a − √2 a † † T † = 2(∂ψ4)ψ4 3ψ4∂ψ4, ψ4,ψ4 − − † † T † = (∂ψ5)ψ5 2ψ5∂ψ5. (19) ψ5,ψ5 − − We also define a spin 3 current W for the matter sector [13]: 6i W = √2W , where M bM M M f 3 2 3i 9i 2 2 WM = iv 3iv v2 α0v2∂v1 α0v1∂v2 3iα ∂ v2, (20) − 2 − 1 − √2 − √2 − 0 f 1 16 2 and α0 = a , bM = , cM =2 24α . Then we have − a 22+5cM − 0

N=2 1 T + ∂U = TM + TL + T † + T † . (21) 2 ψ5,ψ5 ψ4,ψ4 The chiral supercurrent G+ agrees with that used in ref. [14] up to total derivative, after changing the variables φ φ , φ φ , and a ia . 1 ↔ 3 2 ↔ 4 → The present realization looks different from [4]. However if we transform the free fields by a similarity transformation [15], we can get the expression similar to ref. [4]. Define a homotopy transformation (A)R(z) of an operator A(z) by

∞ 1 (A)R(z) eRA(z)e−R = (A)R(z), (22) ≡ i! i Xi=0 where (A)R =( (A)R ) , (A)R = A and the generator is i R i−1 −1 0 dz R = (z). (23) Z 2πiR

(AB)−1(z) means the simple pole term in the OPE between two operators A(z) and B(w). We wish to choose such that the mixing term T v ψ does not appear in (G+)R. Then R M + 4 we find 2ia † (z)= v+(ψ4ψ ) (24) R −√6 5

8 is such an operator. This transformation was considered in ref. [15] in the classical case. Under the transformation, the free fields transform as

R 2 (ψ5) = ψ5 av+ψ4, − s 3

† R † 2 † (ψ ) = ψ + i av+ψ , 4 4 s3 5 (v )R = v ia2∂(ψ ψ†). (25) + + − 4 5 Other fields are invariant. Note that these fields are no longer free fields after this transfor- mation. In fact, there appear mixing terms in the U(1) current and the energy momentum tensor;

U R = U a2∂(ψ ψ†), − 4 5 a2 (T N=2)R = T N=2 (1 + )∂2(ψ ψ†). (26) − 2 4 5 G− is invariant. G+ transforms as 1 (G+)R = G+ +(G+)R + (G+)R. (27) 1 2 2 Again (G+)R takes the form

(G+)R = R + ∂(∆(G+)R), (28) Q where

R 1 = ψ5(TM + TL + T † + T † ) Q 2 ψ5,ψ5 ψ4,ψ4 a 1+ a2 ia 1+ a2 +ψ4 WM + ∂TM + WL ∂TL −√3bM 2 √3bL − 2 ! a3 + (T T )(ψ (∂ψ ψ†)) 3 M − L 4 4 5 +a2(ψ (∂2ψ ψ†)) (ψ (ψ ∂2ψ†)) 2(ψ (∂ψ ∂ψ†)) 4 5 5 − 4 5 5 − 4 5 5 a2 2 a2 2a2 +( 2+ + a4)(ψ (∂ψ ∂2ψ†))+( )(ψ (∂3ψ ψ†)) − 2 4 4 5 −3 − 2 − 3 4 4 5 +( 2 a2 a4)(ψ (∂2ψ ∂ψ†)), (29) − − − 4 4 5

∆(G+)R = ψ (i√6av (ψ ψ†))+(1 3a2)∂ψ 5 3 − 4 4 − 5 9 2 2 2 a 2 3a 2 3 3 3i a TM ψ4 + v v + i (a + a )∂v3 + ∂v4 ψ4 −  2 3 − 2 4 s 2 √2    2 3 † † 2 † i a v3(ψ4(∂ψ4ψ ))+2(ψ4(ψ5∂ψ )) a (ψ4(∂ψ5ψ )) − s 3 5 5 − 5 3a2 +( + a4)(ψ (∂2ψ ψ†)). (30) 2 4 4 5 We have introduced another spin 3 current; 6i W = i√2W , where bL L L f 3 2 3 3√3 2 2 3 2 2 WL = v +3v3v +3√2β0v3∂v4 +3 β0(v4∂v4 v3∂v3)+ β ∂ v4 β ∂ v3 (31) − 3 4 s2 − 2 0 − 2 0

f 1 16 2 where β0 = i(a + ), bL = , cL =2 24β . − a 22+5cL − 0 Although the ghost coupling term is different from [4], the ghost fields coupled to the W -currents give the same expression. But this comes from the fact that they use the same expression of the energy momentum tensor as the untransformed one. Next we argue the higher spin current part of the topological W algebra, although we do not go into the detailed structure in the present paper. From (0) we get = W2 W2 (0) (1). We can read off the free field realization of another spin two current W (z) W2 − W2 2 from . The result is W2 1 1+ K W (z)= (ψ ψ†)+ (H )2 + ∂H . (32) 2 − 5 4 4 4 2 4 f N=2 f The W2 is a quasi-primary field with respect to T . But the current 1+2a2 (1+2a2)(1 3a2) W = W + (UU) − T N=2 (33) 2 2 4(5 18a2) − 5 18a2 − − f becomes N = 2 primary and obeys the OPE relation of N = 2 super W3 algebra: 1 β2cN=2 A (w) ∂A (w) W (z)W (w)= 2 + 2 + 2 + , (34) 2 2 (z w)4 (z w)2 z w ··· − − − where f f 2 2 2 N=2 6(1 5a )(1 2a ) 2β c N=2 1 A2(z)= − − W2 + T (UU) (35) 5 18a2 5 18a2 − 4(1 3a2) ! − − − 2 2 2 f 2 (1+2a )(2−3a )(1−4a ) and β = 3(5−18a2 ) . Other currents in the N = 2 multiplet are defined by + + − − + − 2 W = G−1/2W2, W = G−1/2W2 and W3 = G−1/2G−1/2W . It is a straightforward task

tof check thatf theyf form the Nf= 2 super-f W3 algebra. f

10 In the present paper we have established a new free field realization of N = 2 super-

W3 algebra, which cannot be written in superspace formalism. To study the degenerate

representation of the N = 2 super-W3 algebra, we need the screening operators associated with the simple roots αi of g

1 + iaαj ·φ − −i a αj ·φ Sαj = e , Sαj = e , (j =1, 2), 1 4 4 + −iaα ·φ − −i a α ·φ Sα4 = e , Sα4 = e , i † 3 − a α ·φ Sα3 (z) = ψ4e . (36)

The singular vectors in the Fock modules of bosons α φ for i = 1, 2 and 4 can be i · characterized by using the associated screening operators. We take the irreducible rep- resentation for these bosons. There is a single boson λ φ = φ which is orthogonal to 3 · 3 these bosons, where λ denotes the fundamental weight of g: α λ = δ . Contrary to i i · j i,j the other bosons, the Hilbert space of φ3 remains to be the Fock module. In the case of sl(2 1)(1), the corresponding free field is identified as the Liouville field, which plays an | essential role in the analysis of the BRST cohomology of non-critical string theory. One

might expect that φ3 plays a similar role in the W -string theory. But this cannot be the end of story. This N = 2 model would effectively reduce to a model of non-critical string theory, if only the Liouville dressing was considered. Therefore, if we want to introduce W -dressing, which is indispensable to W gravity coupling, we need one more Fock module of a free boson. This can be achieved, for example, by ignoring the screening operator ± Sα4 in the degenerate representation. This suggests that the representation theory of a non-compact Lie superalgebra sl(3 2) with the bosonic subalgebra sl(3) sl(2, R) u(1) is | ⊕ ⊕ preferable for the investigation of the non-trivial observable in the non-critical W3-string theory. It would be interesting to examine the recent analysis of the physical spectrum

of the non-critical W3-string[16][14] from such a viewpoint of Lie superalgebra.

The work of K.I. is partially supported by University of Tsukuba Reserach Projects.

11 Appendix A: Definition for currents modified by ghost fields

J N c˜ ˜b + i−1 c˜ ˜b N−1 γ˜ β˜ , for i < j, i,j − k=j+1 j,k i,k k=1 k,i k,j − a=1 j,N+a N+a  P P P  i−1 ˜ N ˜ N−1 ˜  Ji,i + k=1 c˜k,ibk,i k=i+1 c˜i,kbi,k a=1 γ˜i,N+aβi,N+a Ji,j(z) =  N−1 − † −  δ ψ ψ , for i = j,  − i,NP a=1 N+a PN+a P e P  i−1 ˜ N ˜ N−1 ˜  Ji,j + k=1 c˜k,ibk,j k=j+1 c˜j,kbi,k a=1 γ˜j,N+aβi,N+a, for i > j.  − −  N ˜ N−1 ˜  JN+a,NP+b i=1 γ˜i,NP+aβi,N+b + c=b+1Pc˜N+b,N+cbN+a,N+c a−1 − ˜ †  c=1 c˜N+c,NP +abN+d,N+b ψN+PaψN+b, for a < b  − −   P  N ˜ †  JN+a,N+a i=1 γ˜i,N+aβi,N+a ψN+aψN+a JN+a,N+b(z) =  a−1 − −N−1  c˜ ˜b + c˜ ˜b , for a = b  − c=1 N+c,NP+a N+c,N+a c=a+1 N+a,N+c N+a,N+c e P P  N b−1  J γ˜ β˜ c˜ ˜b  N+a,N+b i=1 i,N+a i,N+b c=1 N+c,N+a N+c,N+b  N−1 − ˜ −†  + c=a+1 c˜NP+b,N+cbN+a,N+c + ψNP+bψN+a, for a > b.   i−1 N−1  P j (z) = j c˜ β˜ c˜ β˜ , i,N+a i,N+a − k,i k,N+a − N+a,N+b i,N+b kX=1 b=Xa+1 e i−1 N−1 j (z) = j γ˜ ˜b + γ˜ ˜b . N+a,i N+a,i − k,N+a k,i i,N+b N+a,N+b kX=1 b=Xa+1 e e

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