A Family of Regular Vertex Operator Algebras with Two Generators

Home , Operator algebra, Superalgebra, Superconformal algebra

DOI: 10.2478/s11533-006-0045-2 Research article CEJM 5(1) 2007 1–18

A family of regular vertex operator algebras with two generators

Draˇzen Adamovi´c∗

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia

Received 7 September 2006; accepted 20 November 2006

Abstract: For every m ∈ C \{0, −2} and every nonnegative integer k we deﬁne the vertex operator 3m (super)algebra Dm,k having two generators and rank m+2 .Ifm is a positive integer then Dm,k can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that Dm,k is a regular vertex operator (super)algebra and ﬁnd the number of inequivalent irreducible modules. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.

Keywords: Vertex operator algebras, vertex operator superalgebras, rationality, regularity, lattice vertex operator algebras MSC (2000): 17B69

1 Introduction

In the theory of vertex operator (super)algebras, the classification and construction of rational vertex operator (super)algebras are important problems. These problems are connected with the classification of rational conformal field theories in physics. The rationality of certain familiar vertex operator (super)algebras was proved in papers [1– 3, 7, 8, 14, 20, 25]. It is natural to consider rational vertex operator (super)algebras of certain rank. In particular, in the rank one case for every positive integer k we have√ the well-known rational vertex operator (super)algebra Fk associated to the lattice kZ. These vertex operator (super)algebras are generated by two generators. In the present paper we will be concentrated on vertex operator (super)algebras of rank

∗ E-mail: [email protected] 2D.Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18

3m ∈ C\{ − } cm = m+2 , m 0, 2 . This rank has the vertex operator algebra L(m, 0) associated to the irreducible vacuum slˆ2–module of level m and the vertex operator superalgebra

Lcm associated to the vacuum module for the N = 2 superconformal algebra with central charge cm (cf.[2, 3, 11, 15–17]). In the case m = 1 these vertex operator (super)algebras ∼ ∼ are included into the family Fk, k ∈ N,sinceL(1, 0) = F2 and Lc1 = F3.Themain purpose of this article is to include L(m, 0) and Lcm into the family Dm,k, k ∈ Z≥0,of rational vertex operator (super)algebras of rank cm for arbitrary positive integer m. In fact, for every m ∈ C \{0, −2} we define the vertex operator (super)algebra Dm,k as a subalgebra of the vertex operator (super)algebra L(m, 0) ⊗ Fk (cf. Section 4). In the special case k =1,Dm,1 is in the N = 2 vertex operator superalgebra Lcm constructed ∼ by using the Kazama-Suzuki mapping (cf. [15, 19]). We also have that Dm,0 = L(m, 0) ∼ and D1,k = Fk+2. Moreover, we shall demonstrate that Dm,k has many properties similar to those of affine and N = 2 superconformal vertex algebras. When m is not a nonnegative integer, then Dm,k has infinitely many irreducible representations. Thus, it is not rational (cf. Section 4). In order to construct new examples of rational vertex operator (super)algebras we shall consider the case when m is a positive integer. Then Dm,k can be embedded into a lattice vertex algebra (cf. Section 5). In fact, we shall prove that ∼ Dm,k ⊗ F−k = L(m, 0) ⊗ F− k (mk+2) (k even), (1) ∼ 2 Dm,k ⊗ F−k = L(m, 0) ⊗ F−2k(mk+2) ⊕ L(m, m) ⊗ MF−2k(mk+2) (k odd). (2)

These relations completely determine the structure of Dm,k ⊗ F−k as a weak L(m, 0)– module. In [9] the notion of a regular vertex operator algebra was introduced, i. e. rational vertex operator algebra with the property that every weak module is completely reducible. The relations (1)and(2), together with the regularity results from [9]and[21]implythat Dm,k is a simple regular vertex operator algebra if k is even, and a simple regular vertex operator superalgebra if k is odd. It was shown in [5] that regularity is equivalent to rationality and C2–cofiniteness. Therefore, vertex operator (super)algebras Dm,k are also rational and C2–cofinite. Let us here discuss the case k =2n,wheren is a positive integer. The relation (1) suggests that one can study the dual pair (Dm,2n,F−2n) directly inside L(m, 0) ⊗ F−2n(nm+1). This approach requires many deep results on the structure of the vertex operator algebra L(m, 0) and deserves to be investigated independently. Instead of this approach, we realize the vertex algebra Dm,2n ⊗F−2n inside a larger lattice vertex algebra. Then the formulas for the generators are much simpler (cf. Section 6). The similar analysis can be done when k is odd (cf. Section 7). This approach was also used in [3] for studying the fusion rules for the N = 2 vertex operator superalgebra Dm,1. Our results show that for every m ∈ N, there exists an infinite family of rational vertex operator algebras of rank cm. We believe that these algebras will have an important role in the classification of rational vertex operator algebras of this rank. As an example, in this paper we shall consider in detail the vertex operator (super)algebras of rank c4 =2. D. Adamović / Central European Journal of Mathematics 5(1) 2007 1–18 3

Then our vertex operator (super)algebras Dm,k admit nice realizations. In Section 8 we show that D4,k is a Z2–orbifold model of a lattice vertex operator superalgebra under an automorphism of order two. This paper is a slightly modiﬁed version of the preprint math.QA/0111055.

2 Preliminaries

In this section we recall the deﬁnition of vertex operator superalgebras their modules (cf. [12, 13, 18, 20]). We also recall the basic properties of regular vertex operator superalgebras. Let V = V0¯ ⊕ V1¯ be any Z2–graded vector space. Then any element u ∈ V0¯ (resp. u ∈ V1¯) is said to be even (resp. odd). We deﬁne |u| = 0if¯ u is even and |u| = 1if¯ u is odd. Elements in V0¯ or V1¯ are called homogeneous. Whenever |u| is written, it is understood that u is homogeneous.

Deﬁnition 2.1. A vertex superalgebra is a triple (V,Y,1)whereV = V0¯ ⊕ V1¯ is a Z2– graded vector space, 1 ∈ V0¯ is a speciﬁed element called the vacuum of V ,andY is a linear map

· → −1 Y ( ,z): V (End V )[[z, z ]]; −n−1 −1 a → Y (a, z)= anz ∈ (End V )[[z, z ]] n∈Z

satisfying the following conditions for a, b ∈ V : (V1) |anb| = |a| + |b|. (V2) anb =0forn sufficiently large. d (V3) [D, Y (a, z)] = Y (D(a),z)= dz Y (a, z), where D ∈ End V is defined by D(a)=a−21. (V4) Y (1,z)=IV (the identity operator on V ). (V5) Y (a, z)1 ∈ (End V )[[z]] and limz→0 Y (a, z)1 = a. (V6) The following Jacobi identity holds − − −1 z1 z2 − − |a||b| −1 z2 z1 z0 δ Y (a, z1)Y (b, z2) ( 1) z0 δ − Y (b, z2)Y (a, z1) z0 z0 −1 z1 − z0 = z2 δ Y (Y (a, z0)b, z2). z2 A vertex superalgebra V is called a vertex operator superalgebra if there is a ∈ special element ω V0¯(called the Virasoro element) whose vertex operator we write −n−1 −n−2 in the form Y (ω,z)= n∈Z ωnz = n∈Z L(n)z , such that − m3−m ∈ C (V7) [L(m),L(n)] = (m n)L(m + n)+δm+n,0 12 c, c =rankV . (V8) L(−1) = D. 1 ⊕ 1 Z ¯ ⊕ ¯ ⊕ 1 (V9) V = n∈ ZV (n)isa 2 –graded so that V0 = n∈ZV (n), V1 = n∈ +ZV (n), 2 2 L(0) |V (n)= nIV |V (n),dimV (n) < ∞,andV (n)=0forn sufficiently small. 4D.Adamović / Central European Journal of Mathematics 5(1) 2007 1–18

We shall sometimes refer to the vertex operator superalgebra V as quadruple (V,Y,1,ω).

Remark 2.2. If in the deﬁnition of vertex (operator) superalgebra the odd subspace V1¯ = 0 we get the usual deﬁnition of vertex (operator) algebra.

We will say that the vertex operator superalgebra is generated by the set S if { 1 ··· r | 1 r ∈ ∈ Z ∈ Z } V =spanC un1 unr 1 u ,...,u S, n1,...,nr ,r ≥0 .

A subspace I ⊂ V is called an ideal in the vertex operator superalgebra V if anI ⊂ I for every a ∈ V and n ∈ Z. A vertex operator superalgebra V is called simple if it does not contain any proper non-zero ideal. There is a canonical automorphism σV of the vertex operator superalgebra V such that σV |V0¯ =1andσV |V1¯ = −1.

Deﬁnition 2.3. Let V be a vertex operator superalgebra. A weak V –module is a pair (M,YM ), where M = M0¯ ⊕ M1¯ is a Z2–graded vector space, and YM (·,z) is a linear map −1 −n−1 YM : V → End(M)[[z, z ]],a → YM (a, z)= anz , n∈Z satisfying the following conditions for a, b ∈ V and v ∈ M: (M1) |anv| = |a| + |v| for any a ∈ V . (M2) YM (1,z)=IM . (M3) anv =0forn suﬃciently large. (M4) The following Jacobi identity holds − − −1 z1 z2 − − |a||b| −1 z2 z1 z0 δ YM (a, z1)YM (b, z2) ( 1) z0 δ − YM (b, z2)YM (a, z1) z0 z0 −1 z1 − z0 = z2 δ YM (Y (a, z0)b, z2). z2

Aweak V –module (M,YM ) is called a V –module if (M5) M = n∈C M(n); (M7) L(0)u = nu, u ∈ M(n); dim M(n) < ∞; (M8) M(n)=0forn suﬃciently small.

We recall the deﬁnition of regular vertex operator algebra introduced by C. Dong, H. Li and G. Mason in [9].

Deﬁnition 2.4. The vertex operator superalgebra V is called regular if every weak V – module is a direct sum of irreducible modules.

If vertex operator superalgebra V is regular, then V is also a rational vertex operator superalgebra, meaning that V has only ﬁnitely many irreducible modules and that every V –module is completely reducible. D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 5

A vertex operator superalgebra V is called C2–coﬁnite if the subspace C2(V )= spanC{u−2v| u, v ∈ V } has ﬁnite codimension in V . This condition is important in the representation theory of vertex operator superalgebras.

Proposition 2.5. ([5, 23]) The vertex operator superalgebra V is regular if and only if V is rational and C2–coﬁnite.

Remark 2.6. A regularity result for the aﬃne, Virasoro and lattice vertex operator algebras was obtained in [9]. Regularity of vertex operator superalgebras associated to minimal models for the Neveu-Schwarz and N = 2 superconformal algebra was proved in [3, 4].

3 Lattice and aﬃne vertex algebras

In this section, we shall recall the lattice construction of vertex superalgebras from [8, 18]. Let L be a lattice. Set h = C ⊗Z L and extend the Z-form ·, · on L to h.Let hˆ = C[t, t−1] ⊗ h ⊕ Cc be the aﬃnization of h. We also use the notation h(n)=tn ⊗ h for h ∈ h,n∈ Z. Set hˆ+ = tC[t] ⊗ h; hˆ− = t−1C[t−1] ⊗ h. Then hˆ+ and hˆ− are abelian subalgebras of hˆ.LetU(hˆ−)=S(hˆ−) be the universal enveloping algebra of hˆ−.Letλ ∈ h.Consider the induced hˆ-module

− M(1,λ)=U(hˆ) ⊗U(C[t]⊗h⊕Cc) C S(hˆ ) (linearly), where tC[t] ⊗ h acts trivially on C, t0 ⊗ h acting as h, λ for h ∈ h and c acts on C ∈ ∈ Z as multiplication by 1. We shall write M(1) for M(1, 0). For h h and n write n −n−1 h(n)=t ⊗ h.Seth(z)= n∈Z h(n)z . Then M(1) is a vertex operator algebra which is generated by the ﬁelds h(z), h ∈ h, and M(1,λ), for λ ∈ h, are irreducible modules for M(1). Let Lˆ be the canonical central extension of L by the cyclic group ±1:

1 → ±1→Lˆ →¯ L → 1(3) with the commutator map c(α, β)=(−1)α,β+α,αβ,β for α, β ∈ L.Lete : L → Lˆ be asectionsuchthate0 =1and : L × L → ±1 be the corresponding 2-cocycle. Then (α, β)(β,α)=(−1)α,β+α,αβ,β,

(α, β)(α + β,γ)=(β,γ)(α, β + γ)(4) and eαeβ = (α, β)eα+β for α, β, γ ∈ L. Form the induced Lˆ-module

C{L} = C[Lˆ] ⊗±1 C C[L] (linearly), where C[·] denotes the group algebra and −1actsonC as multiplication by −1. For a ∈ Lˆ, write ι(a)fora ⊗ 1inC{L}. Then the action of Lˆ on C{L} is given by: a · ι(b)=ι(ab) and (−1) · ι(b)=−ι(b)fora, b ∈ Lˆ. 6D.Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18

Furthermore we define an action of h on C{L} by: h·ι(a)= h, a¯ι(a)forh ∈ h,a∈ Lˆ. Define zh · ι(a)=zh,a¯ι(a). The untwisted space associated with L is defined to be

− VL = C{L}⊗C M(1) C[L] ⊗ S(hˆ ) (linearly).

h Then L,ˆ hˆ,z (h ∈ h) act naturally on VL by acting on either C{L} or M(1) as indicated above. Deﬁne 1 = ι(e0) ∈ VL. We use a normal ordering procedure, indicated by open colons, which signify that in the enclosed expression, all creation operators h(n)(n<0), a ∈ Lˆ are to be placed to the left of all annihilation operators h(n),zh (h ∈ h,n ≥ 0). For a ∈ Lˆ,set

Ê −1 Y (ι(a),z)=:e (¯a(z)−a¯(0)z )aza¯ : .

Let a ∈ Lˆ; h1, ··· ,hk ∈ h; n1, ··· ,nk ∈ Z (ni > 0). Set

v = ι(a) ⊗ h1(−n1) ···hk(−nk) ∈ VL.

Deﬁne vertex operator Y (v, z)with

1 d n1−1 1 d nk−1 : ( ) h1(z) ··· ( ) hk(z) Y (ι(a),z):. (5) (n1 − 1)! dz (nk − 1)! dz

This gives us a well-deﬁned linear map

· → −1 Y ( ,z): VL (EndVL)[[z, z ]] −n−1 v → Y (v, z)= vnz , (vn ∈ EndVL). n∈Z

Let { hi | i =1, ··· ,d} be an orthonormal basis of h and set

d 1 ω = hi(−1)hi(−1) ∈ VL. 2 i=1 −n−2 Then Y (ω,z)= n∈Z L(n)z gives rise to a representation of the Virasoro algebra on VL with the central charged d and

⊗ − ··· − L(0) (ι(a) h1( n1) hn( nk)) 1 = a,¯ a¯ + n1 + ···+ nk (ι(a) ⊗ h1(−n1) ···hk(−nk)) . (6) 2

The following theorem was proved in [8]and[18].

Theorem 3.1. (i) The structure (VL,Y,1) is a vertex (super)algebra. (ii) Assume that L is a positive deﬁnite lattice. Then the structure (V,Y,1,ω) is a vertex operator (super)algebra. D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 7

Deﬁne the Schur polynomials pr(x1,x2, ···)invariablesx1,x2, ··· by the following equation: ∞ ∞ xn n r exp y = pr(x1,x2, ···)y . (7) n=1 n r=0

n1 n2 nr n1 n2 nr For any monomial x1 x2 ···xr we have an element h(−1) h(−2) ···h(−r) 1 in both M(1) and VL for h ∈ h. Then for any polynomial f(x1,x2, ···), f(h(−1),h(−2), ···)1 is a well-deﬁned element in M(1) and VL. In particular, pr(h(−1),h(−2), ···)1 for r ∈ N are elements of M(1) and VL. Suppose a, b ∈ Lˆ such thata ¯ = α,¯b = β.Then ∞ α(−n) Y (ι(a),z)ι(b)=zα,β exp zn ι(ab) n=1 n ∞ r+α,β = pr(α(−1),α(−2), ···)ι(ab)z . (8) r=0

Thus

ι(a)iι(b)=0 fori ≥− α, β. (9)

Especially, if α, β≥0, we have ι(a)iι(b)=0fori ≥ 0, and if α, β = −n<0, we get

ι(a)i−1ι(b)=pn−i(α(−1),α(−2), ···)ι(ab)fori ∈{0,...,n}. (10)

Let n ∈ Z, n =0,and β,β = n. Deﬁne

Ln = Zβ, Fn = VLn .

Then Fn is a simple vertex algebra if n is even, and a simple vertex superalgebra if n i 0 is odd. For i ∈ Z,leti = i + nZ ∈ Z/nZ. We deﬁne Fn = VZβ+ i β. Clearly Fn = Fn .Itis n i well-known (cf. [7, 8, 26]) that the set {Fn}i=0,...,|n|−1 provides all irreducible Fn–modules. In particular, Fn has |n| inequivalent irreducible modules. β k ˜ Z ˜ If n =2k is even, we deﬁne L2k = 2 + β, and MF2k = VL2k = F2k.ThenF2k is a vertex algebra, and MF2k is a F2k–module. We shall also need the following result from [9].

Proposition 3.2. [9] Assume that n ∈ Z, n =0 . Then the vertex (super)algebra Fn is regular, i.e., any (weak) Fn–module is completely reducible.

Let g be the Lie algebra sl2 with generators e, f, h and relations [e, f]=h,[h, e]=2e, [h, f]=−2f.Letgˆ = g ⊗ C[t, t−1] ⊕ CK be the corresponding aﬃne Lie algebra of (1) n type A1 .Asusualwewritex(n)forx ⊗ t where x ∈ g and n ∈ Z.LetΛ0,Λ1 denote the fundamental weights for gˆ. For any complex numbers m, j,letL(m, j)= 8D.Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18

L((m − j)Λ0 + jΛ1) be the irreducible highest weight slˆ2–module with the highest weight (m − j)Λ0 + jΛ1 .ThenL(m, 0) has a natural structure of a simple vertex operator algebra. Let 1m denote the vacuum vector in L(m, 0). If m is a positive integer then L(m, 0) is a regular vertex operator algebra, and the set {L(m, j)}j=0,...,m provides all inequivalent irreducible L(m, 0)–modules. We shall now recall the lattice construction of the vertex operator algebra L(m, 0). Deﬁne the following lattice

A1,m = Zα1 + ···+ Zαm

αi,αj =2δi,j,

∈{ } ˜ α1+···+αm for every i, j 1,...,m . Deﬁne also A1,m = 2 + A1,m.Wehave:

Lemma 3.3. [8] The vectors E = ι(eα1 )+···+ ι(eαm ), F = ι(e−α1 )+···+ ι(e−αm ), generate a subalgebra of VA1,m isomorphic to L(m, 0). Moreover, L(m, m) is a L(m, 0) ˜ submodule of VA1,m .

4 The deﬁnition of Dm,k

In this section we give the deﬁnition of the vertex operator (super)algebra Dm,k.Letthe vertex (super)algebras L(m, 0) and Fk be deﬁned as in Section 3.

Deﬁnition 4.1. Let m ∈ C \{0, −2},andletk be a nonnegative integer. Let Dm,k be the vertex subalgebra of the vertex operator (super)algebra L(m, 0) ⊗ Fk generated by the vectors:

X¯ = e(−1)1m ⊗ ι(eβ)andY¯ = f(−1)1m ⊗ ι(e−β).

Let 1m,k = 1m ⊗ 1 ∈ Dm,k ⊂ L(m, 0) ⊗ Fk. Deﬁne the following elements of Dm,k:

¯ ¯ ¯ − ⊗ ⊗ − H = XkY = h( 1)1m 1 + m1m β( 1)1, 1 1 − k 2 ωm,k = X¯k−1Y¯ + Y¯k−1X¯ + H¯ 1m,k . 2(m +2) mk +2 −1

Assume that mk +2= 0. Then the components of the ﬁeld −n−2 Y (ωm,k,z)= L(n)z n∈Z

3m give rise a representation of the Virasoro algebra of central charge cm = m+2 .Weshall now investigate the conformal structure on Dm,k deﬁned by the Virasoro element ωm,k. For n ≥ 0 one has

k k L(n)X¯ = δn,0(1 + )X¯ and L(n)Y¯ = δn,0(1 + )Y.¯ (11) 2 2 D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 9

¯ ¯ k Therefore the generators X and Y of Dm,k are primary vectors of conformal weight 1 + 2 for the Virasoro algebra. Moreover, the operator L(0) defines on Dm,k a Z≥0–gradation 1 Z if k is even and a 2 ≥0–gradation if k is odd. Assume first that k is even. Then Dm,k is a subalgebra of the vertex algebra L(m, 0)⊗ Fk. Since the operator L(0) defines on Dm,k a Z≥0–gradation we have that Dm,k is a vertex operator algebra. If k is odd, then ι(eβ)andι(e−β)areoddelementsinFk, which implies that X¯ and Y¯ are also odd elements in the vertex superalgebra L(m, 0) ⊗ Fk. Therefore Dm,k carries the structure of a vertex operator superalgebra which is generated by the odd elements X¯ and Y¯ of half-integer conformal weight. In this way we get the following theorem.

Theorem 4.2. Let m ∈ C \{0, −2},andletk be a nonnegative integer. Assume that mk +2=0 .ThenDm,k is a vertex operator algebra if k is even and a vertex operator superalgebra if k is odd. The Virasoro element is ωm,k, the vacuum vector is 1m,k and the rank is cm.

Let k =0.ThenDm,0 is isomorphic to the slˆ2 vertex operator algebra L(m, 0). Note also that the vector 1 1 2 ωm,0 = X¯−1Y¯ + Y¯−1X¯ + H¯ 1m,0 2(m +2) 2 −1 coincides with the Virasoro element in L(m, 0) constructed using the Sugawara construction. For k =1,Dm,1 is in fact the vertex operator superalgebra associated to the vacuum representation of the N = 2 superconformal algebra constructed using the Kazama-Suzuki mapping (cf. [15, 19]). The Virasoro element in Dm,1 is 1 ωm,1 = (X¯0Y¯ + Y¯0X¯). 2(m +2)

Its representation theory was studied in [2, 3, 11]. Itwasprovedin[3]thatifm is a positive integer, then Dm,1 is a regular vertex operator superalgebra and that the vertex superalgebra Dm,1 ⊗ F−1 is a simple current extension of the vertex algebra L(m, 0) ⊗ F−2(m+2).Whenm is not a nonnegative integer then Dm,1 is not rational. In Theorem 4.4 we will generalize this fact for every positive integer k.

The deﬁnition of Dm,k implies that for every weak L(m, 0)–module M, M ⊗ Fk is a weak module for Dm,k. Thus, the representation theory of Dm,k is closely related to the representation theory of the vertex operator algebra L(m, 0). The case when m is a nonnegative integer will be studied in following sections. When m = −2andm is not an admissible rational number, then every highest weight slˆ2–module of level m is a module for the vertex operator algebra L(m, 0). This easily gives that Dm,k is not rational. In the case when m is an admissible rational number, by using the similar arguments to that of 10 D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18

[2], and by using the representation theory of the vertex operator algebra L(m, 0) in this case (cf. [6]) one can construct inﬁnitely many inequivalent irreducible Dm,k–modules. In order to be more precise, we shall state the following lemma.

Lemma 4.3. Assume that m is not a nonnegative integer and m = −2, mk +2=0 . Let k ≥ 1. Then for every t ∈ C there is an ordinary Dm,k–module Nt such that Nt =

⊕n∈ 1 Z Nt(n), and the top level Nt(0) satisﬁes 2 ≥0

Nt(0) = Cw, L(n)w = tδn,0w for n ≥ 0.

Proof. The proof will use a similar consideration to that in [2], Section 6. Assume that m is not a positive integer and t ∈ C. The results from [6]givesthat ∈ C Z ⊕ for every q there is a ≥0–graded L(m, 0)–module Mq = n∈Z≥0 Mq(n)andaweight vector vq ∈ Mq(0) such that (m +2)m Ω(0)|Mq(0) ≡ Id,h(0)vq = qvq, 2 1 2 where Ω(0) = e(0)f(0)+f(0)e(0)+ 2 h(0) is the Casimir element acting on the sl2–module Mq(0). Then Mq ⊗ Fk is a weak Dm,k–module. Choose q ∈ C such that m k − q2 = t. 4 4(mk +2)

Let Nt be the Dm,k–submodule of Mq ⊗ Fk generated by the vector w = vq ⊗ 1.Thenfor n ≥ 0wehavethat

m k 2 L(n)w = δn,0( − q )w = δn,0tw. 4 4(mk +2) 1 Z Now it is easy to see that Nt is an ordinary 2 ≥0–graded Dm,k–module with the top level Nt(0) = Cw and that L(0)|Nt(0) ≡ tId. Thus, the lemma holds.

In fact, Lemma 4.3 gives that there is uncountably many inequivalent irreducible Dm,k–modules. Thus, we conclude that the following theorem holds.

Theorem 4.4. Let k be a nonnegative integer. Assume that m is not a nonnegative integer and that m = −2, mk +2 =0 . Then for every positive integer k, the vertex operator (super)algebra Dm,k is not rational.

Remark 4.5. In what follows we will prove that if m is a positive integer, then Dm,k is rational. In fact, we will establish more general complete reducibility theorem, which will imply that Dm,k is regular in the sense of [9].

5 The lattice construction of Dm,k for m ∈ N

In this section we give the lattice construction of the vertex operator (super)algebra Dm,k. This construction is a generalization of the lattice constructions of the vertex operator D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 11 algebra L(m, 0) (cf. [8] and our Lemma 3.3) and of the N=2 vertex operator superalgebra

Lcm (cf. [3]). Let m ∈ N and k ∈ Z≥0. Deﬁne the lattice

Γm,k = Zγ1 + ···+ Zγm,

γi,γj =2δi,j + k for every i, j ∈{1,...,m}.

Then VΓm,k is a vertex operator algebra if k is even and a vertex operator superalgebra if k is odd.

Proposition 5.1. Let m ∈ N and k ∈ Z≥0. The vertex operator (super)algebra Dm,k is isomorphic to the subalgebra of the vertex operator (super)algebra VΓm,k generated by the vectors

¯ X = ι(eγ1 )+···+ ι(eγm ), ¯ Y = ι(e−γ1 )+···+ ι(e−γm ).

Set H¯ = X¯kY¯ . Then the Virasoro element in Dm,k is given by 1 1 − k 2 ω¯m,k = X¯k−1Y¯ + Y¯k−1X¯ + H¯ 1 2(m +2) mk +2 −1 m 1 − 2 1 = γi( 1) 1 + ι(eγi−γj )+ 2(m +2) i=1 m +2 i =j m 2 1 − k + γi(−1) 1. 2(m +2)(mk +2) i=1

Proof. Deﬁne the lattice Γ1 by

Γ1 = Zα1 + ···+ Zαm + Zβ,

αi,αj =2δi,j, αi,β =0, β,β = k.

For i =1,...,m set γi = αi + β. It is clear that the lattice Γm,k can be identiﬁed with Z ··· Z the sublattice γ1 + + γm of the lattice Γ1.InthesamewayVΓm,k can treated as a subalgebra of the vertex operator (super)algebra VΓ1 . Lemma 3.3 implies that E =

ι(eα1 )+···+ ι(eαm ), F = ι(e−α1 )+···+ ι(e−αm ), generate a subalgebra of VΓ1 isomorphic to L(m, 0), and the elements ι(eβ), ι(e−β) generate a subalgebra isomorphic to Fk.Since

X¯ = E−1ι(eβ)andY¯ = F−1ι(e−β),

¯ ¯ ∈ ⊂ we conclude that the vertex subalgebra generated by the elements X,Y VΓm,k VΓ1 is isomorphic to the vertex operator (super)algebra Dm,k. This concludes the proof of the theorem. 12 D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18

The previous result implies that we can identify the generators of Dm,k in L(m, 0)⊗Fk with the generators of Dm,k in VΓm,k . We shall also prove an interesting proposition which identiﬁes some regular subalgebras of Dm,k.

Proposition 5.2. For every positive integer n we have that

ι(en(γ1+···+γm)),ι(e−n(γ1+···+γm)) ∈ Dm,k.

In particular, Dm,k has a vertex subalgebra isomorphic to Fn2m(mk+2).

Proof. Using relations (9)and(10), it is easy to prove that: ¯ ··· ¯ ··· ¯ ··· ¯ · X−(nm−1)k−2n+1 X−(n−1)mk −2n+1 X−(2m−1)k−3 X−mk−3 · ¯ ··· ¯ ¯ X−(m−1)k−1 X−k−1X−11 = Cι(en(γ1+···+γm))

for some nontrivial constant C.Thusι(en(γ1+···+γm)) ∈ Dm,k. Similarly we prove that ∈ ι(e−n(γ1+···+γm)) Dm,k. The second assertion of the proposition follows from the fact

2 that the vectors ι(e±n(γ1+···+γm)) generate a subalgebra of VΓm,k isomorphic to Fn m(mk+2).

6 Regularity of the vertex operator algebra Dm,2n

In this section we study the vertex algebra L(m, 0) ⊗ F−2n(mn+1) where m, n are positive integers. We know that L(m, 0) ⊗ F−2n(mn+1) is a simple regular vertex algebra. Its irreducible modules are: Z L(m, r) ⊗ F s¯ ,r∈{1,...,m}, s¯ ∈ . −2n(mn+1) −2n(mn +1)Z

The fusion rules can be calculated easily from the fusion rules for L(m, 0) and F−2n(mn+1). Our main goal is to show that the vertex operator algebra Dm,2n is isomorphic to a subalgebra of L(m, 0) ⊗ F−2n(mn+1). In order to do this, we shall ﬁrst give the lattice construction of the vertex algebra L(m, 0) ⊗ F−2n(mn+1). Deﬁne the following lattice:

L = Zα1 + ···+ Zαm + Zβ,

αi,αj =2δi,j, αi,β =0, β,β = −2n(mn +1) for every i, j ∈{1,...,m}. We shall now give another description of the lattice L. For i =1,...,m, we deﬁne

δ = nα1 + ···+ nαm + β,

γi = αi + δ.

Since αi = γi − δ, β =(nm +1)δ − n(γ1 + ···+ γm), D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 13

we have that

L = Zγ1 + ···+ Zγm + Zδ,

γi,γj =2δi,j +2n, γi,δ =0, δ, δ = −2n

for every i, j ∈{1,...,m}. In fact, we have proved that ∼ ∼ L = Γm,2n + L−2n = A1,m + L−2n(mn+1), (12)

which implies that ∼ ∼ VL = VΓm,2n ⊗ F−2n = VA1,m ⊗ F−2n(mn+1). (13)

Deﬁne the following vectors in the vertex algebra VL:

E = ι(eα1 )+···+ ι(eαm );

F = ι(e−α1 )+···+ ι(e−αm ).

These vectors generate a subalgebra of VL isomorphic to L(m, 0). As in Section 5 we deﬁne:

¯ X = ι(eγ1 )+···+ ι(eγm ); ¯ Y = ι(e−γ1 )+···+ ι(e−γm ).

Clearly X,¯ Y¯ generate a subalgebra isomorphic to Dm,2n. In fact, the deﬁnition of elements E, F, X¯, Y¯ together with relations (12)and(13) imply the following lemma.

Lemma 6.1. (1) Let V be the subalgebra of VL generated by the vectors

E, F,ι(eβ),ι(e−β). ∼ Then V = L(m, 0) ⊗ F−2n(mn+1). (2) Let W be the subalgebra of VL generated by the vectors

X,¯ Y,ι¯ (eδ),ι(e−δ). ∼ Then W = Dm,2n ⊗ F−2n.

Now using standard calculations in lattice vertex algebras one easily gets the following important lemma.

Lemma 6.2. In the vertex algebra VL the following relations hold: ¯ E (1) X =( −2n−1ι(en(α1+···+αm)))−1ι(eβ); ¯ (2) Y =(F−2n−1ι(e−n(α1+···+αm)))−1ι(e−β); 14 D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18

(3) ι(eδ)=ι(en(α1+···+αm))−1ι(eβ);

(4) ι(e−δ)=ι(e−n(α1+···+αm))−1ι(e−β); (5) E = X¯−1ι(e−δ); (6) F = Y¯−1ι(eδ);

(7) ι(eβ)=ι(e(nm+1)δ )−1ι(e−n(γ1+···+γm));

(8) ι(e−β)=ι(e−(nm+1)δ )−1ι(en(γ1+···+γm)).

Theorem 6.3. The vertex subalgebras V and W coincide. In particular, we have the following isomorphism of vertex algebras: ∼ L(m, 0) ⊗ F−2n(mn+1) = Dm,2n ⊗ F−2n. (14)

Proof. Using the same arguments as in the proof of Proposition 5.2 we get ∈ ∈ ι(e±n(α1+···+αm)) V, ι(e±n(γ1+···+γm)) W.

Then the relations (1) - (4) in Lemma 6.2 implies that X,¯ Y,ι¯ (e±δ) ∈ V .ThusW ⊂ V . Similarly, the relations (5) - (8) in Lemma 6.2 gives that V ⊂ W . Hence, V = W .Then ∼ Lemma 6.1 implies that L(m, 0) ⊗ F−2n(mn+1) = Dm,2n ⊗ F−2n.

The next result follows from [9, 10]and[12].

Proposition 6.4. Let V be a vertex operator (super) algebra and s ∈ Z, s =0 .Thenwe have: (1) V ⊗ Fs is a simple vertex superalgebra if and only if V is a simple vertex operator (super)algebra. (2) V ⊗ Fs is a regular vertex superalgebra if and only if V is a regular vertex operator (super)algebra.

Theorem 6.5. Let m, m1,...,mr be positive integers and let k, k1,...,kr be positive even integers. (1) The vertex operator algebra Dm,k is simple and regular. In particular, Dm,k is rational and C2–coﬁnite.

(2) The vertex operator algebra Dm1,k1 ⊗···⊗Dmr,kr is simple and regular.

Proof. Since L(m, 0) and F−2n(nm+1) are simple regular vertex algebras, Proposition 6.4 implies that L(m, 0) ⊗ F−2n(nm+1) is also simple and regular. Since ∼ L(m, 0) ⊗ F−2n(nm+1) = Dm,2n ⊗ F−2n,

using again Proposition 6.4 we get that the vertex operator algebra Dm,2n is simple and regular. This gives (1). The proof of (2) is now standard (cf. [9]).

Since L(m, 0) has (m + 1) inequivalent irreducible modules, and for every k ∈ Z, k =0, Fk has |k| inequivalent irreducible modules, we get: D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 15

Corollary 6.6. The vertex operator algebra Dm,2n has exactly (m+1)(nm+1) inequivalent irreducible representations.

7 Regularity of the vertex operator superalgebra Dm,k for k odd

In this section, we shall consider the case when k is an odd natural number. When k =1,thenDm,1 is the vertex operator superalgebra associated to the unitary vacuum representation for the N = 2 superconformal algebra. This case was studied in [3]. First we see that the following relation between lattices holds: ∼ Γm,k + L−k = (A1,m + L−2k(mk+2)) ∪ (A˜1,m + L˜−2k(mk+2)), (15) which implies the following isomorphism of vertex algebras: ∼ ⊗ ⊗ ⊕ ˜ ⊗ VΓm,k F−k = (VA1,m F−2k(mk+2)) (VA1,m MF−2k(mk+2)). (16) Using (15), (16) and a completely analogous proof to that of Theorem 7.1 in [3], we get the following result.

Theorem 7.1. We have the following isomorphism of vertex superalgebras: ∼ Dm,k ⊗ F−k = L(m, 0) ⊗ F−2k(km+2) ⊕ L(m, m) ⊗ MF−2k(km+2).

In other words, the vertex superalgebra Dm,k ⊗ F−k is a simple current extension of the vertex algebra L(m, 0) ⊗ F−2k(km+2).

By using Proposition 6.4,Theorem7.1 and the fact that a simple current extension of a simple regular vertex algebra is a simple regular vertex (super)algebra (cf. [21]) we get the following theorem.

Theorem 7.2. Let m, m1,...,mr be positive integers and let k, k1,...,kr be positive odd integers. (1) The vertex operator superalgebra Dm,k is simple and regular. In particular, Dm,k is rational and C2–coﬁnite.

(2) The vertex operator superalgebra Dm1,k1 ⊗···⊗Dmr,kr is simple and regular.

We also have:

(m+1)(km+2) Corollary 7.3. The vertex operator superalgebra Dm,k has exactly 2 inequivalent irreducible representations.

Proof. The results from [21] imply that the extended vertex superalgebra

L(m, 0) ⊗ F−2k(km+2) ⊕ L(m, m) ⊗ MF−2k(km+2) 1 has exactly 2 (m +1)k(km + 2) inequivalent irreducible representations (see also [3, 22]). Since the vertex superalgebra F−n has n inequivalent irreducible representations, we con- (m+1)(km+2) clude that Dm,k has to have 2 inequivalent irreducible representations. 16 D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18

8 Realization of the vertex operator algebra D4,k

The lattice construction of Dm,k in Section 5 is based on a very general lattice realization of the vertex operator algebra L(m, 0). Since in some special cases L(m, 0) admits other realizations, one can apply them in the theory of our vertex operator algebras Dm,k.As an example, in this section we shall consider the case m = 4. We will show that the vertex operator (super)algebra D4,k is the ﬁxed point subalgebra of an automorphism g of the lattice vertex operator (super)algebra VPk . Our construction generalizes the fact that the vertex operator algebra L(4, 0) can be constructed as a subalgebra of the lattice vertex operator algebra VA2 . For every k ∈ Z≥0, we deﬁne the following lattice

Pk = Zγ1 + Zγ2, γ1,γ1 = γ2,γ2 = k +2, γ1,γ2 = k − 1.

Then VPk is a vertex operator algebra if k is even and a vertex operator superalgebra if k is odd. Set P = Pk + L−k,whereL−k = Zδ and

δ, γ1 = δ, γ2 =0, δ, δ = −k.

Deﬁne α1 = γ1 + δ, α2 = γ2 + δ, β = k(γ1 + γ2)+(2k +1)δ.

It is easy to see that P = A2 + Zβ,whereA2 = Zα1 + Zα2 is the root lattice of type A2. Since β,β = −k(2k + 1) we get that the following relation between lattices holds: ∼ Pk + L−k = A2 + L−k(2k+1).

Therefore, we have the following isomorphism of vertex (super)algebras:

∼ ⊗ ∼ ⊗ VP = VPk F−k = VA2 F−k(2k+1). (17)

Let g be the automorphism VP which is uniquely determined by

g(ι(e±γ1 )) = ι(e±γ2 ),g(ι(e±γ2 )) = ι(e±γ1 ),g(ι(e±δ)) = ι(e±δ). g is the automorphism of order two of the vertex (super)algebra VP anditisliftedfrom the automorphism γ1 → γ2, γ2 → γ1, δ → δ of the lattice P . The deﬁnition of g implies that

k g(ι(e±α1 )) = ι(e±α2 ),g(ι(e±α2 )) = ι(e±α1 ),g(ι(e±β)) = (−1) ι(e±β).

Let W be one of the subalgebras VPk , VA2 or VZβ.ThenW is g–invariant and W = W 0 ⊕ W 1,where

W 0 = {w ∈ W | gw = w},W1 = {w ∈ W | gw = −w}. D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 17

We have the following isomorphism of vertex algebras ⎧ ⎨⎪ 0 V ⊗ F−k(2k+1) if k is even 0 ∼ A2 V ⊗ F−k = . Pk ⎪ ⎩ 0 ⊗ ⊕ 1 ⊗ VA2 F−4k(2k+1) VA2 MF−4k(2k+1) if k is odd

Next we recall the important fact (see Note 7.3.2 of [24]) that

0 ∼ 1 ∼ VA2 = L(4, 0),VA2 = L(4, 4). (18)

Combining (18), Theorem 6.3 and Theorem 7.1 we get that

0 ⊗ ∼ ⊗ VPk F−k = D4,k F−k.

∼ 0 This implies that D4,k = VPk . In this way we have proved the following result.

Theorem 8.1. We have: ∼ 0 D4,k = VPk .

Under this isomorphism, the generators of D4,k are mapped to √ √ ¯ ¯ X → 2(ι(eγ1 )+ι(eγ2 )), Y → 2(ι(e−γ1 )+ι(e−γ2 )).

Acknowledgment

We would like to thank the referee for his valuable comments.

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