ON W-ALGEBRA EXTENSIONS OF (2,p) MINIMAL MODELS: p > 3 DRAZˇENADAMOVIC´ AND ANTUNMILAS 1 1 0 Abstract. Thisisacontinuationof[AM4],where,amongotherthings,weclassifiedirreducible 2 representations of the triplet vertex algebra W2,3. In this part we extend the classification to n W2,p, for all odd p > 3. We also determine the structure of the center of the Zhu algebra a A(W2,p) which implies the existence of a family of logarithmic modules having L(0)–nilpotent J ranks 2 and 3. A logarithmic version of Macdonald-Morris constant term identity plays a key 4 role in the paper. ] A Q 1. Introduction . h t Recently, there has been a stream of research on W-algebras extensions of Virasoro mini- a mal models from several different points of view [AM3], [AM4], [FGST1], [FGST2], [GRW1]. m [GRW2], [PRZ], [R], [W], etc. As shown in [FGST1], (see also [AM4]), there is a remarkable [ vertex algebra W , (p,q) = 1, an extension of the Virasoro vertex algebra L(c ,0) with cen- q,p p,q 1 tral charge c = 1− 6(p−q)2. Studying the category of W -modules is interesting for several v p,q pq q,p 3 reasons. On one hand, we expect to get new examples of C -cofinite, non self-dual conformal 2 0 vertex algebras, which give rise to finite tensor categories [Hu] [HLZ] (although perhaps not 8 necessarily rigid [Miy]). More interestingly, the algebra W is expected to be in Kazhdan- 0 q,p . Lusztig duality with a certain quantum group U explicitly described in [FGST2]. In addition, 1 q,p 0 representations of Wq,p are rich in combinatorics and their considerations should most likely 1 lead to new parafermionic bases and q-series identities. 1 The main object of study in this paper are W-algebras W , p ≥ 2. In [AM4], we start 2,p : v to investigate these W–algebras in the framework of vertex algebras, where we proved their Xi C2–cofiniteness and irrationality. We also classified irreducible W2,3-modules (this exactly cor- respond to the c = 0 triplet model from [GRW1] [GRW2]). In the present paper we extend r a most of the results from [AM4] to every odd p. An important role in our proofs is played by the doublet vertex superalgebra V , with parity decomposition L V = W ⊕M, L 2,p whereM isacertainsimpleW -module. AnotherkeyingredientistheproofofConjecture10.1 2,p from [AM4]. We obtained this via a logarithmic deformed version of Macdonald-Morris-Dyson constant term identities (see below). Let us state the main results first. Theorem1.1. ThevertexalgebraW hasprecisely4p+p−1 (inequivalent)irreducible modules, 2,p 2 explicitly constructed in Section 4. This result, together with explicit formulas of irreducible characters gives the following useful fact 1 2 DRAZˇENADAMOVIC´ ANDANTUNMILAS Theorem 1.2. SL(2,Z)-closure of the space of irreducible W characters is 15p−5-dimensional 2,p 2 Having Theorem 1.1 handy it is natural to ask for a complete structure of the Zhu algebra of W . In our very recent work [AM5], we introduced a new method for the description of Zhu’s 2,p algebraforcertainvertexalgebras. Thismethodwasusedtocompletelydescribethestructureof the Zhu algebra A(W ). As an important consequence, we proved in [AM5] that W admits 2,3 2,3 a logarithmic module of L(0)–nilpotent rank 3, conjectured previously in the physics literature. This module is then used in the detailed analysis of the c = 0 triplet model in [GRW2]. In the present paper we apply the results from [AM5] for the Zhu algebra A(W ). Although we 2,p cannot precisely describe A(W ), we can still show 2,p Theorem 1.3. The center Z(A(W )) is isomorphic to 2,p C[x]/hf (x)i, 2,p where f (x) is a certain polynomial of degree 15p−5. 2,p 2 Equality of dimensions in Theorem 1.2 and 1.3 is far from accidental. As shown in [FGST2] the center of U is also 15p−5-dimensional. The previous result implies that W admits 2,p 2 2,p p−1 (non-isomorphic) logarithmic modules of L(0)–nilpotent rank three. Our forthcoming work 2 [AM6] will provide explicit constructions of some logarithmic modules of L(0)-nilpotent rank three, including several new modules. Finally, we finish with a constant term identity, expressed as a residue identity. In a special case this identity is needed for purposes of proving Theorem 1.1. Theorem 1.4. Let k ≥ 0, and let also p ≥ 1 be odd. Then ∆(x ,...,x )p k x 2k+1 Res 1 2k+1 ln 1− 2i (1+x )t x1,...,x2k+1(x ···x )(2k+1)p x i 1 2k+1 i=1 (cid:18) 2i−1(cid:19) i=1 Y Y 2k t+ pi = λ 2 , k,p (k+1)p−1 i=0(cid:18) (cid:19) Y where ∆(x ,...,x )= (x −x ), 1 2k+1 i j 1≤i<j≤2k+1 Y and λ 6= 0 is a constant not depending on t. k,p If we assume Conjecture 7.1, the constant λ can be computed exactly. k,p 2. The vertex algebra W 2,p We assume the reader is familiar with vertex algebra theory as in say [LL] of [DL]. In this section we shall consider the triplet vertex algebra W introduced in [FGST1] and [AM4] , a 2,p certain subalgebra of the rank one lattice vertex algebra. Assume that p is an odd natural number, p ≥ 3, and let L = Zα, hα,αi = p. As usual, we extend the scalars of L and define an abelian algebra h = L⊗C. We also write C[L] for the group algebra of L. Consider the usual affinization hˆ of h, with brackets defined ON W ALGEBRA EXTENSIONS OF (2,p) MINIMAL MODELS: p>3 3 in the standard way by using the bilinear form on L. The Fock space of hˆ is denoted by M(1). Then V ∼= M(1)⊗C[L], L has a natural vertex superalgebra structure [DL], [LL], with vertex operator map Y(u,x) = u x−n−1. n n∈Z X Fix the Virasoro vector 1 ω = (α(−1)2 +(p−2)α(−2)) ∈ M(1) ⊂ V L 2p and (screening) operators −2α Q = eα, Q = e p , 0 0 ∞ ∞ 1 e G = eα eα, Gtw = 1 eα eα . i −i i i+1/2 −i−1/2 i+1/2 i=1 i=0 X X The action of G (resp. Gtw) is well-defined on any V -module (resp. Z –twisted V –module). L 2 L For details see [AM4]. Although W was originally defined as the intersection of two screening operators [FGST1], 2,p we showed in [AM4] that it can be realized as a subalgebra of V generated by ω and three L primary vectors F = Qe−3α, H = GF, E = G2F. The algebra W is of course Z–graded whose charge zero component is the singlet ver- 2,p tex algebra M(1) generated by ω and H. Let A(M(1)) = M(1)/O(M(1)) and A(W ) = 2,p W /O(W ) be the associated Zhu algebras of M(1) and W , respectively (see [AM4] for 2,p 2,p 2,p details). We recall further results from [AM4] on the structure of the Zhu algebra A(M(1)). Let (pr−2s)2−(p−2)2 h = . r,s 8p Theorem 2.1. The Zhu algebra A(M(1)) is isomorphic to the commutative associative algebra C[x,y] A(M(1))∼= hP(x,y)i where 2p−1 2p−1 P(x,y) = y2−C (x−h )2 (x−h ) (C 6= 0). p 1,i 2,i p i=1 i=1 Y Y (Here x corresponds to [ω] and y to [H].) 4 DRAZˇENADAMOVIC´ ANDANTUNMILAS 3. Some modules for the doublet vertex superalgebra V L As in [AM4], we consider the vertex superalgebra V generated by ω, L a− = Qe−2α and a+ = Ga−. In this section we shall describe some twisted and untwisted modules for V . Some proofs in L this section are analogous to the proofs of similar results in Section 5 of [AM4], so we omit some details for brevity. We should also say that these results are in agreement with the structural resultsobtainedin[FGST1]. Maindifferenceinourapproachisinthefactthatsomecomplicated screening operators constructed in [TK] are now replaced by exponents of screening operators G and Gtw. The proof of the following theorem is completely analogous to that of Theorem 4.1 of [AM4], so we only indicate the main steps of the proof. Theorem 3.1. Let 1 ≤ k ≤ p. The space of intertwining operators L(c ,h) 2,p (3.1) I L(c ,h ) L(c ,h ) (cid:18) 2,p 5,1 2,p n,k (cid:19) is nontrivial only if (3.2) h ∈ {h ,h ,h ,h ,h }. n−4,k n−2,k n,k n+2,k n+4,k Proof. (Sketch) As in [AM4], we have a singular vector v ∈ M(c ,h ) of (relative) degree sing 2,p 5,1 five, which generates a submodule M ⊂ M(c ,h ). Again, as in [AM4], we use the Frenkel- 1 2,p 5,1 Zhu’s formula to show that if space I L(c2,p,h) is nontrivial then h must be in M(c2,p,h5,1)/M1 L(c2,p,hn,k) the given range. But then the same holds for any quotient of M(c ,h )/M and in particular (cid:0) (cid:1) 2,p 5,1 1 for L(c ,h ) . The proof follows. (cid:3) 2,p 5,1 We will need another result from [AM4] Proposition 3.1. Assume that v is a lowest weight vector in M(1)–module M(1,λ) such that λ α(0)v = λ(α)v . Let t = λ(α). Then we have: λ λ t+p t t+p/2 H(0)v = D v (D 6= 0). λ p λ p 2p−1 2p−1 2p−1 (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) Define now the following cyclic V –modules: L k−1 α k−1 α−α − + α A (h ):= V .Qe p , A (h ):= V .e 2 p (k = 1,...,p). p 1,2p−k L p 2,k L Then A (h ) is an untwisted and A (h ) is a Z –twisted V –module. Let us denote by p 1,2p−k p 2,k 2 L Y(·,z) and Ytw(·,z) the associated vertex operators. The key point for the description of these modules are in the following lemmas which gives non-triviality of certain singular vectors. The proof of the first lemma is identical to that of Lemma 5.1 of [AM4]. Lemma 3.1. Assume that 1 ≤ k ≤ p. We have: k−1 (i) Y(a−,x)Qe p −(n+1)α ∈ W((x)), where k−1 W = U(Vir).Qe p α−(n+2)α ∼= LVir(c ,h ). 2,p 2n+5,k ON W ALGEBRA EXTENSIONS OF (2,p) MINIMAL MODELS: p>3 5 k−1 k−1 (ii) GnQe p α−(n+1)α 6= 0 for n ∈ Z , GjQe p α−(n+1)α = 0 for j > n. ≥0 k−1 (iii) GnQe p α−(n+1)α ∈ A (h ). p 1,2p−k Lemma 3.2. Assume that 1 ≤ k ≤ p. We have: α k−1 (i) Ytw(a−,x)e−2+ p α−nα ∈ W((x)), where α k−1 W = U(Vir).e−2+ p α−(n+1)α ∼= LVir(c ,h ). 2,p 2n+4,k α k−1 α k−1 (ii) (Gtw)ne−2+ p α−nα 6=0 for n ∈ Z , (Gtw)je−2+ p α−nα = 0 for j > n. ≥0 α k−1 (iii) (Gtw)ne−2+ p α−nα ∈ A (h ). p 2,k Proof. Firstweshallproveassertion(i). SinceV isasimplevertexsuperalgebraandV L k−1 L−α/p+ α p is its simple twisted module we conclude that α k−1 Ytw(a−,x)e−2+ p α−nα 6=0. Therefore, we can find j ∈ 1 +Z, such that 0 2 α k−1 α k−1 a−e−2+ p α−nα 6= 0, a−e−2+ p α−nα = 0 for j > j . j0 j 0 α k−1 By usingfusionrulesfrom Theorem3.1 we concludethata−e−2+ p α−nα mustbeasingular j0 α k−1 − + α−(n+1)α vector in M(1)⊗e 2 p of conformal weight h (there are no singular vectors 2n+4,k of weight h in this Fock space). Therefore j = −(n−3/2)p+k−3 and 2n+6,k 0 α k−1 α k−1 a−e−2+ p α−nα = µ e−2+ p α−(n+1)α (µ 6= 0), j0 n n α k−1 a−e−2+ p α−nα ∈W for j ≤ j . j 0 In this way we have proved assertion (i). We shall prove the assertions (ii) and (iii) by induction on n ∈ Z . >0 a k−1 − + α−α Since H(0)e 2 p 6= 0 (see Proposition 3.1), we conclude that a k−1 α k−1 Gtwe−2+ p α−α = µ a+ e−2+ p α 6= 0. 0 3p/2+k−3 So the assertion holds for n = 1. Assume now that assertions (ii)-(iii) hold for certain n ∈ Z . Since V is a simple vertex >0 L operator algebra we have that α k−1 Y(a+,z)(Gtw)ne−2+ p α−nα 6= 0, (for the proof see [LL]). So there is k ∈ 1 +Z such that 0 2 α k−1 α k−1 a+(Gtw)ne−2+ p α−nα 6= 0 and a+(Gtw)ne−2+ p α−nα = 0 for j > k . k0 j 0 6 DRAZˇENADAMOVIC´ ANDANTUNMILAS Since α k−1 α k−1 a+(Gtw)ne−2+ p α−nα = ν (Gtw)n+1(a−e−2+ p α−(n+1)α), k0 2 k0 for certain non-zero constant ν , then by using assertion (i) and the fact that Gtw is a screening 2 operator we conclude that α k−1 α k−1 a+(Gtw)ne−2+ p α−nα ∈ U(Vir)(Gtw)n+1e−2+ p α−(n+1)α. k0 α k−1 Therefore (Gtw)n+1e−2+ p α−(n+1)α 6= 0 and (Gtw)n+1e−α2+k−p1α−(n+1)α = 1 a+(Gtw)ne−α2+k−p1α−nα ∈ A (h ). ν µ j0 p 2,k 2 n The proof follows. (cid:3) By usingfusionrulesfromTheorem3.1, Lemmas 3.1 and3.2 andsimilar methodsas in[AM4] we can describe these modules: Theorem 3.2. (1) A (h ) is a completely reducible module for the Virasoro algebra generated by singular p 1,2p−k vectors: k−1 v(1) = GjQe p α−(n+1)α, n ≥ 0, 0 ≤ j ≤ n. k,n We have the following decomposition: ∞ A (h ) = (n+1)L(c ,h ). p 1,2p−k 2,p 2n+3,k n=0 M (2) A (h ) is a completely reducible module for the Virasoro algebra generated by singular vec- p 2,k tors: α k−1 v(2) = (Gtw)je−2+ p α−nα, n ≥ 0, 0≤ j ≤ n. k,n We have the following decomposition: ∞ A (h )= (n+1)L(c ,h ). p 2,k 2,p 2n+2,k n=0 M We have the following useful description of the structure of some V –modules constructed L above. Proposition 3.2. We have: (3.3) A (h )= Ker Q, p 1,p V p−1 L+ p α (3.4) A (h )= Ker Q, p 2,1 V α L−2 (3.5) A (h )= V . p 2,p α p−e1 L− + α 2 p ON W ALGEBRA EXTENSIONS OF (2,p) MINIMAL MODELS: p>3 7 4. Some W -modules 2,p In this part we introduce some W -modules and describe their structure as Vir-module. In 2,p a special case we obtain Proposition 5.4 in [AM4], a decomposition of W . 2,p LetV beasbefore. WealsoletD = 2Zα,sothatV isavertexsubalgebraofV . Thereare4p L D L irreducible V -modules, correspondingto 4p cosets in the quotient L◦/L, which we conveniently L describe here. For k = 1,...,p consider: V , V , V , V . D+(k−1)α/p D+(k−1)α/p−α D−α/2+(k−1)α/p D−α/2+(k−1)α/p−α By restriction, these are also modules for the triplet W , usually called ”Verma modules”. 2,p The singular vectors in V generate Soc (V ). We let D+γ Vir D+γ X+ := Soc (V ), 1,k Vir D+(k−1)α/p X+ := Soc (V ), 2,k Vir D−α/2+(k−1)α/p X− := Soc (V ), 1,k Vir D+(k−1)α/p−α X− := Soc (V ), 2,k Vir D−α/2+(k−1)α/p−α Our notation clearly follows parametrization from [FGST1]. We also let M(c,h) denote the Virasoro Verma module of lowest weight h and central charge c, and denote by L(c,h) the corresponding irreducible quotient. Bycombiningresultsfrom[FGST1]andactionofscreeningoperatorsGandGtw fromLemmas 3.1 and 3.2 (see also [AM4]) we get the following result: Theorem 4.1. As a Vir-modules X± are generated by the family of singular vectors Sing± , s,r s,k where s ∈ {1,2}, k ∈ {1,...,p} and we have: k−1 Sing+ = {GjQe p α−(2n+1)α | n ∈ Z , 0≤ j ≤ 2n}, 1,k ≥0 k−1 Sing− = {GjQe p α−2nα | n ∈ Z , 0≤ j ≤ 2n−1}, 1,k >0 1 k−1 Sing+ = {(Gtw)je−2α+ p α−2nα | n ∈ Z , 0 ≤ j ≤ 2n}, 2,k ≥0 1 k−1 Sing− = {(Gtw)je−2α+ p α−(2n−1)α | n ∈ Z , 0 ≤ j ≤ 2n−1}. 2,k >0 We have the following decompositions: X+ = ⊕∞ (2n+1)L(c ,h ) 1,k n=0 2,p 4n+3,k X+ = ⊕∞ (2n+1)L(c ,h ) 2,k n=0 2,p 4n+2,k X− = ⊕∞ (2n)L(c ,h ) 1,k n=1 2,p 4n+1,k X− = ⊕∞ (2n)L(c ,h ). 2,k n=1 2,p 4n,k In order to see that X± has the structure of W –modules we shall use results from pre- s,r 2,p vious section and construction of V –modules. The vertex superalgebra V has a canonical L L Z –automorphism σ and the fixed point subalgebra is exactly the triplet vertex algebra W . 2 2,p 8 DRAZˇENADAMOVIC´ ANDANTUNMILAS Moreover every (twisted) V –module from previous section is a Z –graded and decomposes into L 2 direct sum of two ordinary W –modules. We have: 2,p A (h ) =W (h )⊕W (h ), p 1,2p−k p 1,2p−k p 1,3p−k A (h ) = W (h )⊕W (h ). p 2,k p 2,k p 2,3p−k In this way we have obtained four families of ordinary W –modules. By using previous results 2,p on the structure of V –modules one can easily obtains that the constructed W –modules are L 2,p cyclic and k−1 (4.6) W (h ) = W .Qe p α−α, p 1,2p−k 2,p k−1 (4.7) W (h ) = W .Qe p α−2α, p 1,3p−k 2,p α k−1 (4.8) W (h ) = W .e−2+ p α, p 2,k 2,p α k−1 (4.9) W (h ) = W .e−2+ p α−α. p 2,3p−k 2,p Now by using Theorem 3.2 and Theorem 4.1 we get the following result. Corollary 4.1. X± , X± are W –modules and 1,k 2,k 2,p X+ = W (h ), X− = W (h ), X+ = W (h ), X− = W (h ). 1,k p 1,2p−k 1,k p 1,3p−k 2,k p 2,k 2,k p 2,3p−k We also let (p−2r)2−(p−2)2 (4.10) K+ = V(c , )+X+ , 1,r 2,p 8p 1,r where V(c ,(p−2r)2−(p−2)2) is the Vir-module generated by e(r−1)α/p. 2,p 8p (p−2r)2−(p−2)2 (p−2r)2−(p−2)2 Moreprecisely,V(c , )isaquotientoftheVermamoduleM(c , ), 2,p 8p 2,p 8p and it combines in an exact sequence (3p−2r)2−(p−2)2 (p−2r)2−(p−2)2 0 → L(c , )→ V(c , ) 2,p 2,p 8p 8p (p−2r)2−(p−2)2 → L(c , ) → 0. 2,p 8p If we restrict ourself in the range 1≤ r ≤ p−1, this way we obtain (2,p)-minimal models. 2 Proposition 4.1. k−1 (1) We have that K+ = W .e p α. 1,k 2,p (2) W (h ) is an maximal submodule of K+ and p 1,2p−k 1,k K+ p−1 1,k (4.11) W (h ) := L(c ,h )= (k = 1,··· , ). p 1,k 2,p 1,k W (h ) 2 p 1,2p−k is isomorphic to the Virasoro minimal model with lowest conformal weight h . 1,k ON W ALGEBRA EXTENSIONS OF (2,p) MINIMAL MODELS: p>3 9 k−1 α−α Proof. Assertion (1) follows from the fact that Qe p is a singular vector in V(c ,h ) 2,p 1,k which generates its maximal submodule. Let X ∈{E,F,H}. Set X(n) = X . First we notice that 6p−4+n L(n+1)Qek−p1α−α = X(n)Qek−p1α−α = 0 (n ∈ Z≥0). The first relation is clear. To see that F(n)Qek−p1α−α = 0, for n ∈ Z≥0, it is sufficient to observe Q2e−3α = 0 and Q2e(k−1)/pα−α = 0. For X = H and X = F, apply Lemma 3.1 (ii). Therefore ek−p1α ∈/ Wp(h1,2p−k) and Wp(h1,2p−k) is a maximal submodule of K1+,k. The proof follows. (cid:3) 5. Classification of irreducible W -modules 2,p In [AM4] we showed that classification of irreducible W –modules is related to a constant 2,p term identities. Infact Conjecture10.1. from [AM4]essentially gives classification of irreducible modules. In Section 7 below we shall prove that this conjecture holds. Now we shall briefly discuss the classification of irreducible modules. The results from Section 10 of [AM4] give the following relation in A(W ): 2,p Theorem 5.1. In the Zhu algebra A(W ) we have: 2,p f ([ω]) = 0, 2,p where 3p−1 3p−1 2 3p−1 f (x) = (x−h ) (x−h ) (x−h ) 2,p 1,i  1,2p−i  2,i ! ! i=1 i=1 i=1 Y Y Y   p−21 3 2p−1 2p−1 2 = (x−h ) (x−h ) (x−h )  1,i  1,i 2,i   ! i=1 i=p i=1 Y Y Y       3p−1 3p−1 (5.12) ·(x−h ) (x−h ) (x−h ) . 2,p 1,i 2,i    i=2p i=2p Y Y    Following approaches in [AM2] and [AM4] we can now classify all irreducible W . We shall 2,p omit some details. Let S = {h ,h ,h ,h | 1 ≤ i ≤ p−1, p ≤ j ≤ 3p−1, 1 ≤ k ≤ p, 2p ≤ l ≤ 3p−1}. 2,p 1,i 1,j 2,k 2,l 2 Theorem 5.2. We have: (1) W (h ) is an irreducible Z –graded W –module with lowest weight h . Its top p 1,2p−k ≥0 2,p 1,2p−k k−1 component is1-dimensional irreduciblemodulefortheZhualgebraA(W )spanned byQe p α−α. 2,p 10 DRAZˇENADAMOVIC´ ANDANTUNMILAS (2) W (h ) is an irreducible Z –graded W –module with lowest weight h . Its top p 1,3p−k ≥0 2,p 1,3p−k k−1 component is2-dimensional irreduciblemodulefortheZhualgebraA(W )spanned byQe p α−2α 2,p k−1 α−2α and GQe p . (3) W (h ) is an irreducible Z –graded W –module with lowest weight h . Its top compo- p 2,k ≥0 2,p 2,k α k−1 nent is 1-dimensional irreducible module for the Zhu algebra A(W ) spanned by e−2+ p α 2,p (4) W (h ) is an irreducible Z –graded W –module with lowest weight h . Its top p 2,3p−k ≥0 2,p 2,3p−k α k−1 component is2-dimensional irreduciblemodulefortheZhualgebraA(W )spanned bye−2+ p α−α 2,p α k−1 and Gtwe−2+ p α−α. Proof. Let us prove assertion (1). In Proposition 4.1 we proved that W (h ) is Z –graded p 1,2p−k ≥0 k−1 α−α and that its top component is spanned by Qe p . Next we notice that the set conformal weights of singular vectors appearing in the decomposition of W (h ) is {h |n ≥ 0} p 1,2p−k 4n+3,k and that (5.13) h ∈/ S for n ≥ 1. 4n+3,k 2,p Assume now that N ⊂ W (h ) is a non-trivial submodule. Then N is a Z –graded, and p 1,2p−k ≥0 Theorem 5.1 gives that the top component N(0) must have conformal weight h ∈ S . On the 2,p other hand every non-trivial vector from the top component is a singular vector for the Virasoro algebra and therefore h = h for certain n ≥ 1. This contradicts (5.13). The proof of other 4n+3, assertions are similar. (cid:3) By using Theorems 5.1 and 5.2 and the same proof as in [AM2] and [AM4] we get: Theorem 5.3. The set {W (h), h ∈ S } p 2,p provides, up to isomorphism, all irreducible W –modules. 2,p As in [AM5] we have a natural homomorphism Φ : A(M(1))→ A(W ) 2,p a+O(M(1))7→ a+O(W ) (a ∈ M(1)). 2,p By using the description of the Zhu algebra A(M(1)) from [AM4] and the methods from [AM5] we get obtain the following result: Proposition 5.1. Ker(φ) is contained in the following ideal in A(M(1)) A(M(1)).{p([ω])∗[H],f ([ω])} = {A([ω])p([ω])[H]+B([ω])f ([ω]), A,B ∈ C[x]}, 2,p 2,p where 3p−1 3p−1 p(x) = (x−h ) (x−h ). 1,i 2,i i=2p i=2p Y Y Theorem 5.4. The center of the Zhu algebra A(W ) is isomorphic to 2,p C[x]/hf (x)i 2,p