Vertex Algebras $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$ and Constant Term Identities

We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^\Gamma$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453; Internat. J. Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.


Introduction and notation
Orbifold theory of vertex algebras has an interesting and rich development. Although orbifolds have been studied even earlier in string theory, their first mathematical treatment goes back to already classical work by Frenkel, Lepowsky and Meurman on the Moonshine module, which is constructed as a Z 2 -orbifold [12]. We should also mention, closely related to our investigation, an important construction of c = 1 orbifolds by Ginsparg [13]. In the language of vertex algebra, Ginsparg was basically considering what we now call ADE-type orbifolds of the rank one lattice vertex algebra of central charge one and associated orbifold characters and the partition function. This line of work was later brought to even firmer footing in the VOA literature by Dong, Griess and others (see [10] and references therein). Our current line of work is an attempt to lift these classical results from rational to the setup of irrational vertex algebras with central charge 1 − 6(p−1) p 2 , p ≥ 2.
This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is available at http://www.emis.de/journals/SIGMA/LieTheory2014.html Arguably, the most famous constant term identity is the one due to F. Dyson who discovered it in connection to what we now call the circular ensembles model in random matrix theory. His conjectural identity, later proved by Gunson, Wilson and others, concerned the constant term of 1≤i =j≤n which can be elegantly expressed as a ratio of factorials; observe that the rational function is up to a monomial term just the p-th power of the discriminant. Dyson's identities are later generalized by Morris, Kadell, Aomoto and others by adding extra terms to the discriminant part. For example, one includes additional variable x 0 (e.g. Morris and Aomoto's identities) or perhaps symmetric functions in x i (as in Kadell's identitiy). All these identities are in a way related to Selberg's integral (in fact, Morris' identity seems to be equivalent to it). For a good review of this subject see [11]. Constant term identities are also related to computation of correlation functions arising from vertex operators in 2-dimensional CFT. This can be either done in the integral form leading to Selberg integral, or purely formally by using constant terms of generating functions -the method that we advertise here. There are three main computational ingredients that lead to aforementioned expressions and eventually to considerations of constant term identities • Various calculations with products of bosonic vertex operators Y (e α i , x 1 ) · · · Y (e αn , x n ) and their normal ordering. Already such considerations lead to Dyson-type expressions (see for instance [4]).
• Action of certain distinguished vectors (e.g. singular vectors for the Virasoro algebra in the Fock space) on highest weight modules. The key method is based on a simple observation that the zero mode operator of the vector exp − n>0 h(−n) n z n 1, which lives in the completed Fock space, acts on the highest weight Fock module vector v λ semisimply with eigenvalue (1 + z) h,λ .
• Extra terms coming from Zhu's algebra multiplication: a * b = Res x 0 Combination of these three methods for purposes of proving C 2 -cofiniteness and description of Zhu's algebra has led to many constant term evaluations and identities. Already a sample of new identities can be obtained from the triplet vertex algebra [5,6,7,8,9]. It is interesting that classification of modules for the triplet algebra can be reduced to a single constant term identity [5].
In this paper, we continue our investigation of the ADE subalgebras of W(p) initiated in [2] in connection to constant term identities.
We begin with a short review of the triplet algebra W(p), its orbifold subalgebras W(p) Γ , and status of classifications of their modules; more details can be found in [2,3,5].
Let L = Zα be a rank one lattice with α, α = 2p (p ≥ 1). Let be the corresponding lattice vertex operator algebra [14], where h is the affinization of h = Cα, and let C[L] be the group algebra of L. Let M (1) be the Heisenberg vertex subalgebra of V L generated by α(−1)1. With conformal vector the space V L has a vertex operator algebra structure of central charge for more details see [14]. This vertex algebra admits two (degree one) screenings: Q = e α 0 and Q = e −α/p 0 , where we used e β to denote vectors in the group algebra of the dual lattice of L. The triplet vertex algebra W(p) is defined to be the kernel of the "short" screening Q on V L . It is strongly generated by ω and three primary vectors We use M (1) to denote the singlet vertex algebra (cf. [1,4]). It is realized as a vertex subalgebra of W(p) generated by ω and H. Set acting on W(p). Let f ∈ End Vir (W(p)) be the unique operator defined by It was proved first in [2] that W(p) admits an action of sl 2 by the above three operators. The integration of the action of sl 2 gives rise to an action of PSL(2, C) on the vertex operator algebra W(p). Vertex algebra W(p) Am (resp. W(p) Dm ) is defined as the invariant subalgebras with respect to the cyclic group A m of order m (reps. dihedral group D m of order 2m). It is important to notice that W(p) Dm is the Z 2 -orbifold of W(p) Am with respect to the automorphism Ψ which is uniquely determined by the property Note also that Ψ is also an automorphism of M (1) such that Ψ(H) = −H and its fixed point subalgebra M (1) + is a subalgebra of W(p) Dm (for details see [2,3]).
In [2,3], based on investigation of modules and twisted modules of W(p), we gave a conjectural list of 2m 2 p irreducible W(p) Am -modules and a list of (m 2 + 7)p irreducible W(p) Dm -modules. Moreover, for m = 2, we showed that these two lists are complete under the assumption of validity of certain constant term identities which have been verified by computer for small value p.
In this paper we first give a detailed investigation of Zhu's algebras of W(p) Dm in connection to classification of irreducible W(p) Dm -modules. Results from [3] show that Zhu's algebra A(W(p) Dm ) is a commutative, finite-dimensional algebra such that dim A(W(p) Dm ) ≥ (m 2 + 8)p − 1. In order to prove that modules constructed in [3] provide a complete list of irreducible W(p) Dm -modules, it suffices to prove inequality We apply the methods similar to those used in [2] (see also [1,5,9]) and evaluate certain relations in Zhu's algebras on modules for Heisenberg vertex algebra M (1). This leads to a new series of constant term identities which we list in Appendix A.
The structure of Zhu's algebra A(W(p) Am ) is discussed in Section 3. We first show that the classification of irreducible modules for W(p) Am is equivalent to proving that Zhu's algebra A(W(p) Am ) has dimension (2m 2 + 4)p − 1. The results from [2] implies that dim A(W(p) Am ) ≥ (2m 2 +4)p−1. In Section 3 we present a detailed discussion of the proof of the opposite inequality, and finally show that it can be reduced to the proof of certain combinatorial identities which we checked for small values of m and p.
Throughout this paper we use to parameterize lowest conformal weights of modules.

On classif ication of irreducible W(p) D m -modules
In [3], we initiated the study of representation theory of the vertex operator algebra W(p) Dm , m ≥ 2. We proved C 2 -cofiniteness of these vertex algebras and showed that the associated Zhu's algebra is a finite-dimensional commutative algebra. In the representation theory of W(p) Dm , the singlet vertex algebra M (1) + has played an important role. In the same paper, we classified all irreducible modules for M (1) + and W(p) D 2 modulo the same constant term identity.
In this section we shall slightly extend results from [3] so we shall introduce two new combinatorial conjectures which will imply the classification of irreducible modules.
First we recall some results we obtained in [3].
Since W(p) Dm has p−1 logarithmic modules constructed in [9], we conclude that A(W(p) Dm ) also has p − 1 indecomposable 2-dimensional modules. So we have: In order to classify irreducible modules it is sufficient to prove the inequality This will imply that the dimension of Zhu's algebra is of course (m 2 + 8)p − 1 and therefore the list of irreducible modules constructed in [3] (see also Tables 1 and 2) will be a complete list of irreducible W(p) Dm -modules, up to equivalence. We obtained this result in [3] for m = 2.
Theorem 2.3. Assume that Conjecture 7.6 of [3] holds (verified by computer for small p). Then and W(p) D 2 has exactly 11p irreducible modules constructed explicitly in [3].
Next we shall see that for general m the inequality (2.1) is also related to certain constant term combinatorial identities which are more complicated than the identities which appear in [3].
Note that W(p) Dm contains a subalgebra generated by ω and H (2) . As in [3], we denoted this subalgebra by M (1) + since it is a Z 2 orbifold of the singlet vertex algebra M (1) [1], which is generated by ω and H = Qe −α . We also determined the structure of Zhu's algebra A(M (1) The next result is again taken from [3].
(1) Inside the Zhu algebra A(M (1) + ) we have the following relations: Let A 0 (W(p) Dm ) be the subalgebra of A(W(p) Dm ) generated by [ω] and [H (2) ]. Let Proof . First we notice that For technical reasons we need to recall some informations on lowest weights of irreducible modules. Since Zhu's algebra A(W(p) Dm ) is commutative (see below), its irreducible modules are 1-dimensional. Then applying Zhu's algebra theory we see that all irreducible A(W(p) Dm )modules should be parameterized by its lowest weights with respect to (L(0), H (2) (0), U (m) (0)). The lowest weights of irreducible W(p) Dm -module constructed in [3] can be found in the following two tables, where and the number is defined as in [2, Lemma 4.8].
By the above two tables and standard arguments we infer: Lemma 2.6. We have the following relation in A(W(p) Dm ): where h p (x) is a polynomial of degree at most 3p − 1, and satisfies the following interpolation conditions: for i = 1, . . . , p, j = 1, . . . , 2p. In particular, Zhu's algebra A(W(p) Dm ) is commutative.
Proof . We use the above tables and apply both vectors in the equation on lowest weight vectors of modules.
Next, for a, b ∈ W(p) we define In particular, Proof . We assume that m ≥ 3, noticing that the case m = 2 has been treated in [3]. From the structure of W(p) as a Virasoro module We see that there exists a polynomial g p (x) of degree at most 3p, such that in A(M (1)). By evaluating the left hand side on known irreducible W(p) Dm -modules we see that for some constant C p . As in our previous papers we shall relate evaluation of the constant C p with action of certain elements of M (1) on lowest weight M (1)-modules. These highest weight modules are realized as modules M (1, λ) for the Heisenberg vertex algebra M (1) (remember that M (1) ⊂ M (1)).
So let v λ be a lowest weight vector in the M (1, λ) and let t = λ, α . Then we get It follows from Conjecture A.3 that C p is nonzero.
By the proof of [15, Lemma 2.1.3], we get and Hence We can show that for small values of (m, p) where h p,m andh p,m are given in the table (up to a scalar factor): This lemma and the structure of Zhu's algebra A(M (1) + ) imply: Proposition 2.11. Assume that Conjectures 2.9 and A.3 hold. Then Now we are in a position to give the first main result of this paper. (1) Tables 1 and 2 give a complete list of irreducible W(p) Dm -modules. In particular, W(p) Dm has (m 2 + 7)p non-isomorphic irreducible modules, (2) Zhu's algebra A(W(p) Dm ) is a commutative algebra of dimension (m 2 + 8)p − 1.
Proof . As we discussed above it is enough to prove inequality (2.1). We proved in [3] that A(W(p) Dm ) is commutative. We will compute the dimension of A(W(p) Dm ). By Lemma 2.7 and Proposition 2.11, we get that The proof follows.

On classif ication of irreducible W(p) A m -modules
In our paper [2] we constructed 2m 2 p irreducible W(p) Am -modules and conjectured that these modules provide a complete list of irreducible W(p) Am -modules. In the case m = 2 we presented a proof which is based on certain constant term identities. These identities are difficult to prove in general, but using Mathematica they can be verified for p small. So our approach can be considered as an algorithm that reduces problems in representation theory to checking something purely computational.
In this section we shall extend our results of [2] to general m and thus provide an algorithm for a classification of irreducible W(p) Am -modules.

s algebra A(W(p) Am ) is generated by [ω], [H], [E (m) ] and [F (m) ].
It is also important to notice that the restriction of the automorphism Ψ of W(p) (cf. [2]) to W(p) Am gives an automorphism of order two such that In [2] we constructed 2m 2 p irreducible W(p) Am -modules. A list of irreducible W(p) Ammodules and their lowest weights with respect to (L(0), H(0)) are given in the following tables: Table 3. Irreducible W(p) Am -modules: m = 2k. Table 4. irreducible W(p) Am -modules: m = 2k + 1.
Remark 3.1. When m = 2k, the action of A(W(p) Am ) on the lowest weight space of Λ(i) m is given by where a m,p , b m,p are nonzero constants. Similar result holds for Π(i) m for m = 2k + 1.
So we constructed (2m 2 −1)p irreducible modules whose lowest components are 1-dimensional and p irreducible modules with 2-dimensional lowest components. Since these lowest components are irreducible modules for Zhu's algebra A(W(p) Am ) and since A(W(p) Am ) has (p − 1) 2-dimensional indecomposable modules constructed from logarithmic W(p)-modules we get: So if we prove that in the above relation equality holds, we get the classification of irreducible modules.
Lemma 3.2. Assume that dim A(W(p) Am ) ≤ (2m 2 + 4)p − 1. Then the above two tables give a complete list of irreducible W(p) Am -modules.

generated by [ω] and [H], and let
A ω (W(p) Am ) be the subalgebra generated by [ω]. Let As in Lemma 2.5 we have: In particular, Proof . The first assertion follows directly by evaluating this relation on lowest weight spaces of irreducible W(p) Am -modules. The second assertion follows by applying the automorphism Ψ ∈ Aut(W(p) Am ) such that Similarly, we have with deg v p ≤ p. By evaluating this relation on lowest components of modules Λ(i) − , we get that It follows from Conjecture A.2 that k p is nonzero. Hence we have Proposition 3.5. Assume that Conjecture A.2 holds. Then in A(W(p) Am ), we have In particular, On the other hand, by evaluating this relation on lowest components of W(p) Am -modules we get Lemma 3.6. , , and and use it to obtain (after we switch to [ω] polynomials) A Several conjectures about constant term identities In this section we list several constant term identities, although strictly speaking they are written as residues. Let's start by recalling our identity from [2]: Conjecture A.1 (constant term identity of type A m , I).
In Sections 2 and 3, we need the following two conjectures: Conjecture A.2 (constant term identity of type A m , II). where µ p,m is a nonzero constant.