0 v 1 1 8 D ec 1 99 5 REPRESENTATION THEORY OF THE VERTEX ALGEBRA

In our paper [KR] we began a systematic study of representations of the universal central extension D̂ of the Lie algebra of differential operators on the circle. This study was continued in the paper [FKRW] in the framework of vertex algebra theory. It was shown that the associated to D̂ simple vertex algebra W1+∞,N with positive integral central charge N is isomorphic to the classical vertex algebra W (glN ), which led to a classification of modules over W1+∞,N . In the present paper we study the remaining non-trivial case, that of a negative central charge −N . The basic tool is the decomposition of N pairs of free charged bosons with respect to glN and the commuting with glN Lie algebra of infinite matrices ĝl. Introduction In this paper we study representations of a central extension D̂ = D ⊕ CC of the Lie algebra D of differential operators on the circle. This central extension appeared first in [KP] and its uniqueness was subsequently established in [F] and in [Li]. In our paper [KR] we began a systematic study of representations of the Lie algebra D̂. In particular, we classified the irreducible “quasi-finite” highest weight representations and constructed them in terms of irreducible highest weight representations of the central extension ĝl of the Lie algebra of infinite matrices. This study was continued in [FKRW] in the framework of vertex algebra theory. The advantage of such an approach is twofold. From the mathematical point of view it picks out the most interesting representations and equips them with a rich additional structure. From the physics point of view it provides building blocks of two-dimensional conformal field theories. We just mention here applications to integrable systems [ASM], to W -gravity [BMP], and to the quantum Hall effect [CTZ1,CTZ2]. In more detail, let us consider the subalgebra P of D consisting of the differential operators that can be extended in the interior of the circle. The cocycle of our central extension (see (6.2)) restricts to a zero cocycle on P, hence P is a subalgebra of D̂. Given c ∈ C, we denote by Mc the representation of D̂ induced from the 1dimensional representation of the subalgebra P ⊕CC defined by P 7→ 0, C 7→ c. It was shown in [FKRW] that Mc carries a canonical vertex algebra structure. The Supported in part by NSF grant DMS–9103792. Typeset by AMS-TEX 1 2 VICTOR KAC AND ANDREY RADUL D̂-module Mc has a unique irreducible quotient which carries the induced simple vertex algebra structure. Following physicists, we denote this simple vertex algebra by W1+∞,c. The highest weight representations of the vertex algebra Mc are in a canonical 1–1 correspondence with that of the Lie algebra D̂. As usual, the situation is much more interesting when we pass to the simple vertex algebra W1+∞,c. The cocycle (6.2) is normalized in such a way that Mc is an irreducible D̂-module (i.e., Mc = W1+∞,c) iff c / ∈ Z. Thus, we are led to the problem of classification of irreducible highest weight representations of the vertex algebra W1+∞,c with c ∈ Z. This is a highly nontrivial problem since the field corresponding to every singular vector of Mc must vanish in a representation of W1+∞,c which gives rise to an infinite set of equations. The problem has been solved in [FKRW] in the case of a positive integral c = N by the use of an explicit isomorphism of W1+∞,N and the classical W -algebra W (glN ). This approach cannot be used for c negative since probably W1+∞,−N is a new structure, which is not isomorphic to any classical vertex algebra. Another important result of [FKRW] is an explicit construction of a “model” of “integral” representations of W1+∞,N by decomposing the vertex algebra of N charged free fermions with respect to the commuting pair of Lie algebras glN and ĝl. One of the main results of this paper is an analogous decomposition of the vertex algebra of N charged free bosons with respect to a commuting pair glN and ĝl, which produces a large class of irreducible modules of the vertex algebra W1+∞,−N (Theorems 3.1 and 6.1). We conjecture that all irreducible modules of W1+∞,−N can be obtained from these by restricting to D̂ and taking their certain tensor products (Conjecture 6.1). This explicit construction allows one to also derive explicit character formulas (formula (3.7)). Partial results in this direction were previously obtained in [Mat] and [AFMO]. Our basic tool is the suitably modified theory of dual pairs of Howe [H1], [H2]. Namely one has the following general irreducibility theorem (Theorem 1.1): if a Lie algebra g acts completely reducibly on an associative algebra A and if V is an irreducible A-module with an equivariant action of g such that V is a direct sum of at most countable number of irreducible finite-dimensional modules, then the pair g and A = {a ∈ A | ga = 0} acts irreducibly on each isotypic component of g in V . This result allows us not only to decompose both free charged fermions and bosons with respect to the pair glN and ĝl, but also to interpret the vertex algebra W1+∞,−N as a subalgebra of the vertex algebra of N free charged bosons killed by glN (formula (4.4)). In the same way, one identifies W1+∞,N with a subalgebra of the vertex algebvra of N free charged fermions killed by glN , a result previously obtained in [FKRW]. This result is interpreted in that paper as an isomorphism W1+∞,N ≃ W (glN) due to the connection to affine Kac-Moody algebras. It seems, however, that there is no such interpretation in the case of negative central charge. It is interesting to note that while the decomposition of N free charged fermions produces all unitary (with respect to the compact involution) irreducible highest weight representations of ĝl with central charge N , the decomposition of N free charged bosons produces a very interesting class of irreducible highest weight representations of ĝl with central charge −N . This allows us to compute their characters (formula (3.7)). We also show that the subcategory of the category O of representations all of whose irreducible subquotients are members of this class is a semisimple REPRESENTATION THEORY OF THE VERTEX ALGEBRA W1+∞ 3 category (Theorem 4.1). (Of course, in the first case a similar result goes back to H. Weyl.) Provided that Conjecture 6.1 is valid, this implies semisimplicity of the category of positive energy W1+∞,−N -modules. The paper is organized as follows. In Section 1 we give a proof of the general irreducibility theorem (Theorem 1.1). In Section 2 we use Theorem 1.1 and classical invariant theory (as in [H2]) to prove irreducibility of each isotypic component of glN in the metaplectic representation of the infinite Weyl algebra WN (N free charged bosons in physics terminology) with respect to the commuting pair glN and ĝl (Theorem 2.1). By a somewhat lengthy combinatorial argument, we derive an explicit highest weight correspondence in Section 3 (Theorem 3.1) and a character formula for the above-mentioned highest weight representations of ĝl (formula (3.7)). In Section 4 we prove the complete reducibility Theorem 4.1. Section 5 is a brief digression on vertex algebras and their twisted modules which we conclude by a construction of twisted modules over N free charged bosons which become untwisted modules with respect to W1+∞,−N . In Section 6 we construct a large family of representations of the vertex algebra W1+∞,−N using the above mentioned modules (Theorem 6.2) and conjecture that these are all its irreducible representations (Conjecture 6.1). In Section 7 we apply similar methods to N free charged fermions to recover most of the results of [FKRW]. 1. Representations of associative g-algebras Let A be an associative algebra over C and let DerA denote the Lie algebra of derivations of A. Let g be a Lie algebra over C and let φ : g → DerA be a Lie algebra homomorphism. The triple (A, g, φ) is called an associative g-algebra. An A-module V is called a (g, A)-module if V is given a structure of a g-module such that the A-module structure is equivariant, i.e. g(av) = (φ(g)a)v + a(gv), g ∈ g, a ∈ A, v ∈ V . Let A = {a ∈ A | φ(g)a = 0 for all g ∈ g} be the centralizer of the action of g on A. Given a g-submodule U of V and a ∈ A, the map U → aU , given by u 7→ au, is clearly a g-module homomorphism. Given an irreducible g-module E, denote by VE the sum of all g-submodules of V isomorphic to E. This is called the E-isotypic component of the g-module V . By the above remark, VE is a A -submodule of V . Choose a 1-dimensional subspace f ⊂ E. Then, due to Schur’s lemma, this gives us a choice of a 1-dimensional subspace in each of the irreducible g-submodules of VE . We denote the sum of all of these 1-dimensional subspaces by V E (it depends on the choice of f). Clearly, V E is a A-submodule of VE , and we have a (noncanonical) (g, A)-module isomorphism: VE ≃ E ⊗ V E . The Lie algebra g (resp. associative algebra A) acts on E⊗V E by g(e⊗v) = ge⊗v (resp. a(e⊗ v) = e⊗ av). Thus, if V is a (g, A)-module, which is a semisimple g-module, we have the following isomorphism of (g, A)-modules: