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 repre- sentations. 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. Adamovi´c / 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.Adamovi´c / 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).