The Vertex Algebra $M(1)^+$ and Certain Affine Vertex Algebras of Level -1

We give a coset realization of the vertex operator algebra $M(1)^+$ with central charge $\ell$. We realize $M(1)^+$ as a commutant of certain affine vertex algebras of level -1 in the vertex algebra $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$. We show that the simple vertex algebra $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$ can be (conformally) embedded into $L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$ and find the corresponding decomposition. We also study certain coset subalgebras inside $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$.


Introduction
In the last few years various types of W-algebras have been studied in the framework of vertex operator algebras (see [Ar], [AM2], [AM3], [DLY], [KW2]). In this paper we will be focused on W-algebras which admit coset realization. To any vertex algebra V and its subalgebra U , one can associate a new vertex algebra Com(U, V ) = {v ∈ V | u n v = 0 for all u ∈ U, n ≥ 0} called the commutant (or coset) of U in V . This is a very important construction in the theory of vertex operator algebras, because it gives a realization of a large family of W-algebras. Another important construction is the orbifold construction, where a new vertex operator algebra is obtained as invariants in a given vertex operator algebra with respect to the finite automorphism group. As we shall see in our paper, some vertex algebras admit both realizations, coset and orbifold.
(1.1) Although there are no precise general results (to the best of our knowledge) about the structure of these cosets, it is believed that these vertex operator algebras are finitely generated and rational.
In [AP2], we present a vertex-algebraic proof of the fact that in the case k = 1 and affine Lie algebras of types D (1) n and B (1) n , the coset (1.1) is isomorphic to the rational vertex operator algebra V + L . The situation is even more complicated for general k, m ∈ C, such that k, m, k + m = −h ∨ . Then one has the coset vertex operator algebra where L X (1) n ((k +m)Λ 0 ) is a certain affine vertex operator algebra associated to X (1) n of level k + m (not necessarily simple). In this paper we identify some special cases of such cosets, and it turns out that they are not rational.
The construction in [AP2] is based on fermionic construction of vertex algebras and certain conformal embeddings. In the present paper we use bosonic construction of vertex algebras and construct new conformal embeddings of affine vertex algebras at level −1. By applying the bosonic realization of the affine vertex algebras L A (1) ). It is interesting that these cosets have central charge 1. We show that these cosets are isomorphic to M (1) + , where M (1) is the Heisenberg vertex operator algebra of rank 1, and M (1) + is the Z 2 -orbifold vertex algebra studied in [DN1]. The structure theory of M (1) + shows that these cosets are irrational vertex operator algebras and isomorphic to W (2, 4)-algebra with central charge c = 1.
By combining results from [A2] and the present paper, we classify irreducible ordinary L C (1) ℓ (−Λ 0 )-modules. We believe that the (tensor) category of L C (1) ℓ (−Λ 0 )-modules is related to the (tensor) category of M (1) +modules. We plan to address this correspondence in our forthcoming publications.
Generalizing (1.3), we use a natural realization of the vertex operator algebra is isomorphic to M (1) + , where M (1) is the Heisenberg vertex operator algebra of rank ℓ.
Our construction is based on a new, interesting conformal embedding of affine vertex operator algebras at level −1 which can be of independent interest. We show that By using conformal embeddings we study certain categories of A This result is an affine analogue of the isomorphism of finite-dimensional C ℓmodules: Using these conformal embeddings we also show that the coset is the Heisenberg vertex operator algebra of rank ℓ − 1.
Let h be a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form ·, · , considered as an abelian Lie algebra. Let h = h ⊗ C[t, t −1 ] ⊕ CK be its affinization with the center K. Then the free bosonic Fock space where {h (1) , . . . , h (ℓ) } is any orthonormal basis of h (cf. [FLM], [LL]). We shall also use the notation M h (1) to emphasize the associated vector space h.
Vertex algebra M (1) has an order 2 automorphism which is lifted from the map h → −h, for h ∈ h. Denote by M (1) + (or M h (1) + ) the subalgebra of invariants of that automorphism. The irreducible modules for M (1) + were classified in [DN1] and [DN2]. For ℓ = 1, it was proved in [DG] that M (1) + is generated by ω and one primary vector of conformal weight 4, so it is isomorphic to a W (2, 4)-algebra with central charge 1.
Let g be the simple Lie algebra of type X n , andĝ the associated affine Lie algebra of type X (1) n . For any weight Λ ofĝ, denote by L X (1) n (Λ) the irreducible highest weightĝ-module. Denote by Λ i , i = 0, . . . n the fundamental weights ofĝ (cf. [K1]). We shall also use the notation V Xn (µ) for a highest weight g-module of highest weight µ, and ω i , i = 1, . . . n for the fundamental weights of g.
For any k ∈ C, denote by N X (1) n (kΛ 0 ) the generalized Vermaĝ-module with highest weight kΛ 0 . Then, N X (1) n (kΛ 0 ) is a vertex operator algebra of central charge k dim g k+h ∨ , for any k = −h ∨ , with Virasoro vector obtained by Sugawara construction: ..,dim g is an arbitrary basis of g, and {b i } i=1,...,dim g the corresponding dual basis of g with respect to the symmetric invariant bilinear form, normalized by the condition that the length of the highest root is √ 2. (cf. [FZ], [L], [FB], [K2], [LL]).
It follows that any quotient of N X (1) n (kΛ 0 ) is a vertex operator algebra, for k = −h ∨ . Specially, L X (1) n (kΛ 0 ) is a simple vertex operator algebra, for any k = −h ∨ .
3. Simple Lie algebras of type C ℓ and A 2ℓ−1

Consider two 2ℓ-dimensional vector spaces
The Weyl algebra W 2ℓ is the complex associative algebra generated by A and non-trivial relations The normal ordering on A is defined by Then the Lie algebra g 1 generated by the set [Bou] and [FF]). The Cartan subalgebra h 1 is spanned by Let θ be the automorphism of W 2ℓ of order two given by and similarly for root vectors associated to negative roots.
The subalgebra g of g 1 generated by for i, j = 1, . . . , ℓ, i < j, is the simple Lie algebra of type C ℓ . The Cartan subalgebra h is spanned by Clearly, θ acts as 1 on g. Furthermore, is a highest weight vector for g, which generates the irreducible g-module V C ℓ (ω 2 ). Clearly, θ acts as −1 on V C ℓ (ω 2 ). We obtain the decomposition one easily concludes that the irreducible g 1 -module V A 2ℓ−1 (nω 1 ) remains irreducible when restricted to g. Thus, We will consider certain affine analogues of relations (3.7)-(3.9).
In what follows we shall need the following decompositions of g-modules: (3.10)

Weyl vertex algebras and symplectic affine Lie algebras
The Weyl algebra W ℓ ( 1 2 + Z) is a complex associative algebra generated by The fields a ± i (z), i = 1, . . . , ℓ generate on M ℓ the unique structure of a simple vertex algebra (cf. [K2], [FB]). Let us denote the corresponding vertex operator by Y .
We have the following Virasoro vector in M ℓ : The following result is well-known.
Theorem 4.1 ( [FF]). We have In the case ℓ = 1 we have Remark 4.1. The highest weights of modules from Theorem 4.1 are admissible in the sense of [KW1]. Representations of vertex operator algebras associated to affine Lie algebras of type A (1) 1 and C (1) ℓ with admissible highest weights were studied in [AM1] and [A1].
We shall now consider the vertex algebra M 2ℓ and its subalgebra L C (1) be the automorphism of order two of the vertex algebra M 2ℓ which is lifted from the following automorphism of the Weyl algebra for i = 1, . . . , ℓ and s ∈ 1 2 + Z. If we have a subalgebra U ⊂ M 2ℓ which is θ-invariant, we define Define the following vectors in M 2ℓ : and for ℓ = 1: In this section we use Weyl vertex algebra M 2ℓ to study certain subalgebras of L C (1) Then h can be considered as an abelian Lie algebra, and the components of the vertex operators define a representation of the associated Heisenberg algebra h. Moreover, h generates the subalgebra of the M 2ℓ which is isomorphic to the Heisenberg vertex algebra M h (1) with central charge ℓ. We also note that Using relations (5.12) one can show that the Virasoro vector (4.11) (in M 2ℓ ) can be written in a form: is the Virasoro vector in U and Proposition 5.1. We have: Proof. It is clear that W = Com(U, M 2ℓ ) contains a subalgebra isomorphic to M h (1). Assume now that M h (1) = W . Then there is a vector w ∈ W, wt(w) > 0 such that (Note that each H (i) (0) acts semisimply on M 2ℓ with eigenvalues in Z.) By definition of W we have that L 1 (0)w = 0. Therefore This contradicts the fact that wt(w) > 0. So, W = M h (1).
Theorem 5.1. We have: and Theorem 5.2. We have: In the next section we generalize Theorem 5.2 in another way.

Commutant of level
In this section we study another subalgebra of L C (1) Set h 1 = CH ⊂ h. Let M h 1 (1) be the Heisenberg vertex algebra generated by H. Clearly H, H = −2ℓ. Since Proposition 6.1. The Virasoro vector (4.11) in M 2ℓ can be written in a form: where ω 1 is the Virasoro vector (2.6) in L C (1) ℓ (−Λ 0 ) obtained by the Sugawara construction and is the Virasoro vector in M h 1 (1).
Using Proposition 6.1 and applying similar arguments as in the proof of Proposition 5.1, we obtain: Proposition 6.2. We have: (1) + . By using Proposition 6.2 we obtain the following theorem.

The classification of ordinary modules for
In this section we obtain a classification of irreducible L C (1) ℓ (−Λ 0 )-modules, which we use in the following sections.
Since a + 1 (−1/2) n 1 ∈ M 2ℓ is a singular vector of highest weight −(n + 1)Λ 0 + nΛ 1 , and is a singular vector of highest weight −2Λ 0 + Λ 2 , we have that every module from the set (7.18) is a module for these vertex operator algebras. The proof is now complete.
The subalgebra of M 2ℓ generated by is level −1 affine vertex operator algebra associated to the affine Lie algebrâ . Letĝ be the affine Lie algebra of type C Proof. Similarly as in Proposition 6.1, one can show that which implies the claim of Proposition.
Furthermore, the vector e * ǫ 1 +ǫ 2 from the proof of Proposition 7.1 is a singular vector forĝ in L A (1) Clearly, θ acts as 1 on L C (1) ℓ (−Λ 0 ) and as −1 on L C (1) Proof. It suffices to prove the lemma for u and v from top component R(0) of L C (1) ℓ (−2Λ 0 + Λ 2 ). Then the statement will follow from the associator formulae.
First we notice that (since vectors of conformal weight 1 with bracket [u, v] = u 0 v span Lie algebra g 1 of type A 2ℓ−1 , and fixed point subalgebra g is a Lie algebra of type C ℓ ). Assume now that for certain u, v ∈ R(0) and n 0 ∈ Z. Take maximal n 0 with this property. Then u n 0 v has nontrivial component in some highest weight L C (1) ℓ (−Λ 0 )module W of highest weight −Λ 0 + µ, and therefore there is a nontrivial intertwining operator of type One can associate to this intertwining operator, a non-trivial g-homomorphism In particular, V C ℓ (µ) must appear in the decomposition of tensor product First consider the case ℓ = 2. Using the decomposition of tensor product V C 2 (ω 2 )⊗V C 2 (ω 2 ) from (3.10) and the fact that the lowest conformal weights of modules of highest weights −Λ 0 + 2ω 2 and −Λ 0 + 2ω 1 are 5 2 and 3 2 , respectively, we conclude that these modules cannot appear inside L A (1) Thus, u n 0 v ∈ L C (1) ℓ (−Λ 0 ). Now, let ℓ ≥ 3. Relation (3.10) implies that the only cases of weights −Λ 0 + µ (aside from µ = 0) from Proposition 7.1 such that µ appears in the decomposition of V C ℓ (ω 2 ) ⊗ V C ℓ (ω 2 ) are when µ = 2ω 1 or µ = ω 2 . Since On the other hand, the lowest conformal weight of module of highest weight −Λ 0 + 2ω 1 is ℓ+1 ℓ , which is not an integer. Thus, this module cannot appear Theorem 8.1. We have: In particular, Theorem 8.2. Assume that ℓ ≥ 3, n ∈ Z >0 . Then we have the following isomorphism of L C (1) ℓ (−Λ 0 )-modules: 2ℓ−1 (−(n + 1)Λ 0 + nΛ 1 ) and let w denote the highest weight vector. Then the top component W (0) is isomorphic to the irreducible highest weight g 1 -module V A 2ℓ−1 (nω 1 ) ∼ = U (g 1 )w. Relation (3.8) implies that W (0) ⊂ W 1 = U (ĝ).w.
Since generators H