Representations of Lie algebra of vector fields on a torus and chiral de Rham complex

The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra - the Lie algebra of vector fields on an N-dimensional torus for N>1. The case N=1 gives a famous Virasoro algebra (or its centerless version - the Witt algebra). The algebra of vector fields has an important class of tensor modules parametrized by finite-dimensional modules of gl(N). Tensor modules can be used in turn to construct bounded irreducible modules for the vector fields on N+1-dimensional torus, which are the central objects of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible modules for the semidirect product of vector fields with the quotient of 1-forms by the differentials of functions. These modules remain irreducible when restricted to the subalgebra of vector fields, unless they belongs to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob.


Introduction.
In this paper we study the representation theory of a classical infinite-dimensional Lie algebra -the Lie algebra VectT N of vector fields on a torus. This algebra has a class of representations of a geometric nature -tensor modules, since vector fields act on tensor fields of any given type via Lie derivative. Tensor modules are parametrized by finite-dimensional representations of gl N , with the fiber of a tensor bundle being a gl N -module.
Irreducible gl N -modules yield tensor modules that are irreducible over VectT N , with exception of the modules of differential k-forms. In the latter case, the gl N -module is irreducible, yet the modules of k-forms are reducible, which follows from the fact that the differential of the de Rham complex is a homomorphism of VectT N -modules. In the present paper we give a vertex algebra analogue of this result.
In case of a circle, a conjecture of Kac, proved by Mathieu [21], states that for the Lie algebra of vector fields on a circle an irreducible weight module with finite-dimensional weight spaces is either a tensor module or a highest/lowest weight module. There is a generalization of this conjecture to an arbitrary N due to Eswara Rao [7]. The analogues of the highest weight modules in this case are defined using the technique introduced by Berman-Billig [1]. These modules are bounded with respect to one of the variables. It follows from a general result of [1] that irreducible bounded modules for the Lie algebra of vector fields on a torus have finite-dimensional weight spaces, however the method of [1] yields no information on the dimensions of the weight spaces. This is the question that we solve in the present paper -we find explicit realizations of the irreducible bounded modules, using which the dimensions of the weight spaces may be readily determined.
A partial solution of this problem for the 2-dimensional torus was given by Billig-Molev-Zhang [3] using non-commutative differential equations in vertex algebras. The algebra of vector fields on T 2 contains the loop algebra sl 2 = C[t 0 , t −1 0 ] ⊗ sl 2 . This subalgebra plays an important role in representation theory of VectT 2 . According to the results of [3], some of the bounded modules for VectT 2 remain irreducible when restricted to the subalgebra sl 2 . Futorny classified in [11] irreducible generalized Verma modules for sl 2 . Such generalized Verma modules admit the action of the much larger algebra VectT 2 .
This relation between representations of sl 2 and VectT 2 suggests that for the Lie algebra of vector fields on T N , an important role is played by its subalgebra sl N . An unexpected twist here is that it is not the generalized Verma modules for sl N that admit the action of VectT N for N > 2, but rather the generalized Wakimoto modules. The generalized Wakimoto modules are sl N -modules that have the same character as the generalized Verma modules, but need not to be isomorphic to them.
The generalized Wakimoto modules for sl N that we use here were constructed in [2] in the context of the representation theory of toroidal Lie algebras, however their special properties with respect to the loop subalgebra sl N were not previously recognized.
Let us outline the result of [2] that we use here. Since one of the variables plays a special role, it will be more convenient to work with an (N + 1)-dimensional torus. To construct a full toroidal algebra, one begins with the algebra ofġ-valued functions on T N +1 : 0 , t ±1 1 , . . . , t ±1 N ] ⊗ġ, whereġ is a finite-dimensional simple Lie algebra. Next we take the universal central extension of this multiloop algebra, with the center realized as the quotient of 1-forms on the torus by differentials of functions [14]: Finally, one adds the Lie algebra of vector fields on the torus: 0 , . . . , t ±1 N ] ⊗ġ ⊕ K ⋊ VectT N +1 . Irreducible bounded modules for this Lie algebra were constructed in [2] using vertex algebra methods. Note that the results of [2] admit a specialization toġ = (0). The multiloop algebra then disappears, leaving behind, like the smile of the Cheshire Cat, the space of its central extension: It turns out that it is easier to study representations of this Lie algebra, rather than of vector fields alone, because of the duality between vector fields and 1-forms. Representation theory of this larger Lie algebra is controlled by a tensor product of three vertex algebras: a subalgebra of a hyperbolic lattice vertex algebra, the affine gl N vertex algebra at level 1 and the Virasoro vertex algebra of rank 0. The tensor product of the first two components, V + Hyp ⊗ V glN is one of the bounded modules for K ⋊ VectT N +1 , and in fact it is a generalized Wakimoto module for the subalgebra sl N +1 ⊂ VectT N +1 . Then the results of [3] suggest that there is a chance that K ⋊ VectT N +1 -modules constructed in [2] remain irreducible when restricted to VectT N +1 . The study of this question is the main part of the present paper. The answer that we get is remarkably parallel to the classical picture with the tensor modules. We prove that a bounded irreducible K⋊VectT N +1 -module remains irreducible when restricted to the subalgebra of vector fields, unless it belongs to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob [20] (for arbitrary manifolds).
It is only in very special situations an irreducible module remains irreducible when restricted to a subalgebra. A prime example of this is the basic module for an affine Kac-Moody algebra, which remains irreducible when restricted to the principal Heisenberg subalgebra. This exceptional property of the basic module leads to its vertex operator realization and is at heart of several spectacular applications of this theory.
The space of the chiral de Rham complex is the vertex superalgebra where V Z N is the lattice vertex superalgebra associated with the standard euclidean lattice Z N . The vertex superalgebra V Z N is graded by fermionic degree: are irreducible K ⋊ VectT N +1 -modules. Yet for these modules the restriction to the Lie algebra of vector fields is no longer irreducible since the differential of the chiral de Rham complex is a homomorphism of VectT N +1 -modules.
In fact it was noted in [18] that these components admit the action of the Lie algebra C[t 0 , t −1 0 ]⊗ VectT N , but here we prove a much stronger result.
In conclusion, we make two curious observations. The Lie algebra of vector fields on a torus has a trivial center, yet its representation theory is described in terms of vertex algebras V + Hyp and V glN that involve non-trivial central extensions. The central charges of these tensor factors cancel out to give a vertex algebra of total rank 0.
The final remark is that the chiral de Rham complex is an essentially super object, whereas the Lie algebra of vector fields we started with, is classical.
The structure of the paper is as follows. We introduce the main objects of our study in Sections 2, 3 and 4. We discuss vertex algebras and their applications to the representation theory of VectT N +1 in Sections 5 and 6. In Section 7 we introduce the generalized Wakimoto modules for the loop algebra sl N . We construct a non-degenerate pairing for the bounded modules in Section 8. We prove the main result on irreducibility in Section 9 and make a connection with the chiral de Rham complex in the final section of the paper.

Lie algebra of vector fields and its tensor modules
We begin with the algebra of Fourier polynomials on an N -dimensional torus T N . Introducing the variables t j = e ixj , j = 1, . . . , N , we realize the algebra of functions as Laurent polynomials The Lie algebra of vector fields on a torus is It will be more convenient for us to work with the degree derivations d p = t p ∂ ∂tp as the free generators of VectT N as a C[t ±1 1 , . . . , t ±1 N ]-module: The Lie bracket in VectT N is then written as Here we are using the multi-index notations t r = t r1 1 . . . t rN N for r = (r 1 , . . . , r N ) ∈ Z N . The Cartan subalgebra d 1 , . . . , d N acts on VectT N diagonally and induces on it a Z N -grading. The Lie algebra of vector fields (on any manifold) has a class of representations of a geometric nature. Vector fields act via Lie derivative on the space of tensor fields of a given type. The resulting tensor modules are parametrized by representations of gl N . Let us describe the construction of tensor modules in case of a torus T N .
Fix a finite-dimensional gl N -module W . In case when W is irreducible, the identity matrix acts as multiplication by a scalar α ∈ C. Let γ ∈ C N . We define the tensor module T = T (W, γ) to be the vector space where r ∈ Z N , µ ∈ γ + Z N , a = 1, . . . , N and E pa is the matrix with 1 in (p, a)-position and zeros elsewhere.
The middle terms in this complex are reducible VectT N -modules, while the terms q γ Ω 0 (T N ) and q γ Ω N (T N ) are reducible whenever γ ∈ Z N .
Note that de Rham differential d is a homomorphism of VectT N -modules. Let us specify irreducible gl N -modules that correspond to the tensor modules in de Rham complex. The modules of functions Ω 0 and the module of differential N -forms Ω N correspond to 1-dimensional gl Nmodules W on which the identity matrix acts as multiplication by α = 0 and α = N respectively. The remaining modules Ω k , k = 1, . . . , N − 1, are the highest weight modules for sl N with the fundamental highest weights ω k and α = k (see e.g. [6]). Even though they correspond to irreducible gl N -modules, tensor modules of differential forms are reducible since the kernels and images of the differential d are obviously the submodules in Ω k .

Bounded modules
Our goal is to generalize to an arbitrary N the category of the highest weight modules over VectT N . In our constructions one of the coordinates will play a special role. From now on, we will be working with the N +1-dimensional torus and will index our coordinates as t 0 , t 1 , . . . , t N , where t 0 is the "special variable". We would like to construct modules for the Lie algebra D = VectT N +1 in which the "energy operator" −d 0 has spectrum bounded from below.
Let us consider a Z-grading of D by degrees in t 0 . This Z-grading induces a decomposition into subalgebras of positive, zero and negative degrees in t 0 . The degree zero part is In particular, D 0 is a semi-direct product of the Lie algebra of vector fields on T N with an abelian ideal C[t ±1 1 , . . . , t ±1 N ]d 0 . We begin the construction of a bounded module by taking a tensor module for D 0 . Fix a finite-dimensional irreducible gl N -module W , β ∈ C and γ ∈ C N . We define a D 0 -module T as a space T = q γ C[q ±1 1 , . . . , q ±1 N ] ⊗ W with the tensor module action (2.1) of the subalgebra VectT N ⊂ D 0 and with Next we let D + act on T trivially and define M (T ) as the induced module The module M (T ) has a weight decomposition with respect to the Cartan subalgebra d 0 , . . . , d N and the (real part of) spectrum of −d 0 on M (T ) is bounded from below. However the weight spaces of M (T ) that lie below T are all infinite-dimensional.
It turns out that the situation improves dramatically when we pass to the irreducible quotient of M (T ). One can immediately see that the Lie algebra D belongs to the class of Lie algebras with polynomial multiplication (as defined in [1]), whereas tensor modules belong to the class of modules with polynomial action. A general theorem of [1] (see also [4]) yields in this particular situation the following In [3] these problems were solved for some of the modules L(T ) in case of a 2-dimensional torus (N = 1). In the present paper we will give a solution in full generality for any N .

Toroidal Lie algebras
For a finite-dimensional simple Lie algebraġ we consider a multiloop algebra Its universal central extension has a realization with center K identified as the quotient space of 1-forms by differentials of functions [14], The Lie bracket in is the Killing form onġ and denotes the projection We will set 1-forms k a = t −1 a dt a , a = 0, . . . , N as generators of Ω 1 (T N +1 ) as a free C[t ±1 0 , . . . , t ±1 N ]module. We will use the same notations for their images in K.
The Lie algebra D = VectT N +1 acts on the universal central extension of the multiloop algebra with the natural action on C[t ±1 0 , . . . , t ±1 N ] ⊗ġ, and the action on K induced from the Lie derivative action of vector fields on Ω 1 : The full toroidal Lie algebra is a semi-direct product In fact [2] treats a more general family of Lie algebras, where the Lie bracket in g is twisted with a 2-cocycle τ ∈ H 2 (D, Ω 1 /dΩ 0 ). However for the purposes of the present work we need to consider only the semi-direct product, i.e., set τ = 0.
A category of bounded modules for the full toroidal Lie algebra is studied in [2] and realizations of irreducible modules in this category are given. The constructions of [2] admit a specializatioṅ g = (0), which yields representations of the semi-direct product The approach of the present paper is to look at the representations of this semidirect product, constructed in [2], and to study their reductions to the subalgebra D of vector fields on T N +1 . Surprisingly, as we shall see below, most of the irreducible modules for D ⋉ K remain irreducible when restricted to D.
In order to describe here the results of [2], we will need to present a background material on vertex algebras.

Vertex superalgebras: definitions and notations
Let us recall the basic notions of the theory of the vertex operator (super) algebras. Here we are following [12] and [17].
Definition 5.1. A vertex superalgebra is a Z 2 -graded vector space V with a distinguished vector 1l (vacuum vector) in V , a parity-preserving operator D (infinitesimal translation) on the space V , and a parity-preserving linear map Y (state-field correspondence) such that the following axioms hold: (V1) For any a, b ∈ V, a (n) b = 0 for n sufficiently large; and Y (a, z)1l| z=0 = a for any a ∈ V (self-replication); (V5) For any a, b ∈ V , the fields Y (a, z) and Y (b, z) are mutually local, that is, A vertex superalgebra V is called a vertex operator superalgebra (VOA) if, in addition, V contains a vector ω (Virasoro element) such that (V6) The components L n = ω (n+1) of the field satisfy the Virasoro algebra relations: where C Vir acts on V by scalar, called the rank of V .
This completes the definition of a VOA. As a consequence of the axioms of the vertex superalgebra we have the following important commutator formula: As usual, the delta function is δ(z) = n∈Z z n .
By (V1), the sum in the left hand side of the commutator formula is actually finite. The commutator in the left hand side of (5.2) is of course the supercommutator. Let us recall the definition of a normally ordered product of two fields. For a formal field a(z) = j∈Z a (j) z −j−1 define its positive and negative parts as follows: Then the normally ordered product of two formal fields a(z), b(z) of parities p(a), p(b) ∈ {0, 1} respectively, is defined as The following property of vertex superalgebras will be used extensively in this paper: z) : , for all a, b ∈ V.
Let L be a Lie superalgebra with the basis {u (n) , c (−1) |u ∈ U, c ∈ C, n ∈ Z} (U, C are some index sets). Define the corresponding fields in L[[z, z −1 ]]: Let F be a subspace in L[[z, z −1 ]] spanned by all the fields u(z), c(z) and their derivatives of all orders.
Definition 6.1. A Lie superalgebra L with the basis as above is called a vertex Lie superalgebra if the following two conditions hold: (VL1) for all x, y ∈ U, where f j (z) ∈ F , n ≥ 0 and depend on x, y, (VL2) for all c ∈ C, the elements c (−1) are central in L.
The universal enveloping vertex algebra V L of a vertex Lie superalgebra L is defined as the induced module V L = Ind L L (+) (C1l) = U (L (−) )⊗1l, where C1l is a trivial 1-dimensional L (+) module. (a) V L has a structure of a vertex superalgebra with the vacuum vector 1l, infinitesimal translation D being a natural extension of the derivation of L given by D(u (n) ) = −nu (n−1) , D(c (−1) ) = 0, u ∈ U, c ∈ C, and the state-field correspondence map Y defined by the formula: where a j ∈ U, n j ≥ 0 or a j ∈ C, n j = 0.
(b) Any bounded L-module is a vertex superalgebra module for V L .
(c) For an arbitrary character χ : C → C, the quotient module is a quotient vertex superalgebra.
(d) Any bounded L-module in which c (−1) act as χ(c)Id, for all c ∈ C, is a vertex superalgebra module for V L (χ).
The value χ(c) is referred to as central charge or level.
The vertex algebra that controls representation theory of D ⋉ K is the tensor product of three VOAs: a subalgebra V + Hyp of a hyperbolic lattice vertex algebra, an affine gl N vertex algebra V glN at level 1, and the Virasoro vertex algebra V Vir of rank 0. In order to apply the results of [2] to representation theory of D ⋉ K, we use specializationsġ = (0) and τ = 0. In this specialization one has to fix the following values for the various central charges appearing in ( [2], Theorems 5.3 and 6.4): , c ′ Vir = 0. Let us briefly review the constructions of these three vertex algebras. Hyperbolic lattice VOA. Consider a hyperbolic lattice Hyp, which is a free abelian group on 2N generators {u a , v a |a = 1, . . . , N } with the symmetric bilinear form We complexify Hyp to get a 2N -dimensional vector space and extend the bilinear form by linearity on H. Next, we affinize H by defining a Heisenberg Lie algebra Here and in what follows, we are using the notation The algebra H has a triangular decomposition Let Hyp + be an isotropic sublattice of Hyp generated by {u a |a = 1, . . . , N }. We consider its group algebra C[Hyp + ] = C[e ±u 1 , . . . , e ±u N ] and define the action of H 0 ⊕ H + on C[Hyp + ] by x (0) e y = (x|y)e y , Ke y = e y , H + e y = 0.
To be consistent with our previous notations, we set q a = e u a , a = 1, . . . , N .
Finally, let V + Hyp be the induced module We coordinatize V + Hyp as a Fock space over H: where H acts by operators of multiplication and differentiation: for p = 1, . . . , N, j = 1, 2, . . . . The space V + Hyp has the structure of a vertex algebra -it is a vertex subalgebra in the vertex algebra corresponding to lattice Hyp. We give here the values of the state-field correspondence map on the generators of this vertex algebra: The Virasoro element of V + Hyp is and the Virasoro field is .
.. ] has a natural structure of a simple module for V + Hyp (see e.g. [1] for details).
Affine gl N VOA. The second vertex algebra that we will need is the affine gl N vertex algebra at level 1. Since gl N is reductive, but not simple, it has more than one affinization. Here we consider a particular version of gl N : We note that gl N is a vertex Lie algebra and consider its universal enveloping vertex algebra V glN at level 1 (i.e., χ(C) = 1).
Let us give the value of the state-field correspondence map on the generators of this affine vertex algebra: Since gl N has a decomposition gl N = sl N ⊕ CI, where I is the identity N × N matrix, the affine gl N vertex algebra is the tensor product of the affine sl N vertex algebra and a Heisenberg vertex algebra. The Virasoro element ω glN of V glN can be thus written as a sum of the Virasoro elements ω slN for the affine sl N vertex algebra and ω Hei for the Heisenberg vertex algebra. The usual formula for the Virasoro element in affine vertex algebra gives the following explicit expression: The rank of the affine sl N vertex algebra at level 1 is N − 1.
For the Heisenberg vertex algebra we choose a non-standard Virasoro element (see [2], (4.33)): The rank of this Heisenberg VOA is 1 − 3N . Adding the two Virasoro elements, we get the Virasoro element for V glN : The corresponding Virasoro field is : E ij (z)E ji (z) : + : I(z)I(z) : Let W be a finite-dimensional simple module for gl N . Let C act on W as the identity operator and set (tC[t] ⊗ gl N ) W = 0. Construct the generalized Verma module for the Lie algebra gl N as the induced module from W , and consider its irreducible quotient L glN (W ). Then L glN (W ) is a simple module for the vertex algebra V glN .
Virasoro VOA. The last vertex algebra that we need to introduce is the Virasoro vertex algebra V Vir of rank 0. The Virasoro Lie algebra (5.1) is a vertex Lie algebra with U = {ω Vir } and Let V Vir be its universal enveloping vertex algebra of zero central charge, χ(C Vir ) = 0. Let L Vir (h) be the irreducible highest weight module for the Virasoro Lie algebra with central charge 0 with the highest weight vector v h , satisfying The vertex algebra that controls representation theory of D ⋉ K is the tensor product of the sub-VOA V + Hyp of the hyperbolic lattice vertex algebra, affine gl N vertex algebra V glN at level 1, and the Virasoro VOA V Vir of rank 0 is a module for the Lie algebra D ⋉ K with the action given as follows: for a = 1, . . . , N .
(ii) The module is an irreducible module over the Lie algebra D ⋉ K.

Generalized Wakimoto modules.
In [3] the structure of irreducible modules L(T ) over the Lie algebra of vector fields was determined in case of a 2-dimensional torus (N = 1). It turned out that the situation was analogous to the case of a basic module for an affine Kac-Moody algebra, which remains irreducible when restricted to the principal Heisenberg subalgebra [16], [13]. For the Lie algebra of vector fields on T 2 this role is played by its loop subalgebra This extends to an embedding The following theorem was proved in [3]: , and let γ ∈ C. Then the module L(T ) = L(α, β, γ) over the Lie algebra VectT 2 remains irreducible when restricted to subalgebra sl 2 .
Note that [3] uses a different convention for the sign of α.
]d 1 and thus acts on T . The positive part sl + 2 acts on T trivially. We can form the generalized Verma module over sl 2 : Ind sl2 By the results of [11], this generalized Verma module is irreducible over sl 2 if and only if α ∈ 1 2 Z. This gives the following Then the VectT 2 -module L(α, β, γ) when restricted to sl 2 is isomorphic to the generalized Verma module over sl 2 : . These results show that the loop subalgebra sl 2 plays a crucial role in representation theory of the Lie algebra of vector fields on T 2 . It is natural to conjecture that in the representation theory of D = VectT N +1 such a role is played by the loop algebra sl N +1 . Indeed, D 0 has VectT N as a subalgebra and VectT N contains sl N +1 (see e.g. [22]). Thus VectT N +1 contains the loop algebra The modules T (W, β, γ) may be viewed as sl N +1 -modules, and we can form the generalized Verma module over sl N +1 : . It turns out, however, that in general, the action of sl N +1 on the generalized Verma module can not be extended to the action of the bigger algebra VectT N +1 . Instead one should use certain generalized Wakimoto modules. We define the generalized Wakimoto modules in the following way: is an sl N +1 -module that contains T as an sl N +1 -submodule with sl + N +1 acting on T trivially and having the property that the character of M coincides with the character of the generalized Verma module for sl N +1 : The generalized Verma module is by definition a generalized Wakimoto module. In case when the generalized Verma module is irreducible, it is the only generalized Wakimoto module with the given top T . As we mentioned above, for sl 2 this happens for the tops T (α, β, γ) with α ∈ 1 2 Z [11]. For N > 1 and any top T (W, β, γ) with a finite-dimensional gl N -module W , the generalized Verma module over sl N +1 is always reducible.
We shall now see that Theorem 6.3 yields a construction of a generalized Wakimoto module for sl N +1 . These modules admit the action of the whole algebra of vector fields on T N +1 . Proposition 7.4. Let M glN (W ) be the generalized Verma module for gl N at level 1, induced from an irreducible finite-dimensional gl N -module W . Then the module Proof. Applying Theorem 6.3 with a trivial 1-dimensional Virasoro module, we see that M as a module for the Lie algebra VectT N +1 . By restriction, view M as an sl N +1 -module. The top of the module M is the tensor module This completes the proof of the proposition. Theorem 6.3 describes irreducible D ⋉ K-modules. We would like to study their reductions to subalgebra D. In general, when reduced to a subalgebra, modules become reducible. Here, however, the link with generalized Wakimoto modules for sl N +1 and the result of [3] for N = 1, give us hope that the situation may be better than one would expect a priory.

Duality for modules over the Lie algebra of vector fields
In this section we will establish a duality for the class of modules described in Theorem 6.3 (ii), that will be useful for the study of their irreducibility as modules over Lie algebra D. We begin by looking at this question in a general setup.
Let L be a Z-graded Lie algebra L = ⊕ n∈Z L n with an anti-involution σ such that σ(L n ) = L −n .
Extend σ to the universal enveloping algebra U (L) by σ(ab Suppose T 1 , T 2 be two L 0 -modules with a bilinear pairing For an L 0 -module T we let L + act on T trivially and construct the generalized Verma modules for L: The generalized Verma module M (T ) inherits the Z-grading (assuming degree of T to be zero). Define the radical of a generalized Verma module M (T ) as the maximal homogeneous submodule trivially intersecting with the top T . If T is an irreducible L 0 -module then the quotient L(T ) of M (T ) by its radical is an irreducible module for L.
Consider the Shapovalov projection This proposition is standard (cf., [23], Proposition 2.8.1) and its proof is left to the reader as an exercise.
Next we will apply this proposition to establish the duality for the bounded modules described in Theorem 6.3 (ii).
We consider the following anti-involution on D ⋉ K: For a finite-dimensional simple gl N -module W , on which the identity matrix I acts as multiplication by α ∈ C, let W * be the dual space to W with sl N -module structure of the dual module, but with I acting as scalar N − α. The natural pairing between W and W * satisfies for all x ∈ D ⋉ K, u ∈ L(W, γ, h), v ∈ L(W * , γ, h).
For the proof of this theorem we will use an alternative construction of the simple D ⋉K-module L(W, γ, h), which is discussed in [2]. These modules may be abstractly defined using approach of Theorem 3.1. The top of the module L(W, γ, h) is the space is a module for the zero degree component D 0 ⋉ K 0 of D ⋉ K with respect to its Z-grading by degree in t 0 . The positive part of D ⋉ K acts on T (W, γ, h) trivially, and we can consider the induced D ⋉ K-module M (T ). The induced module has a unique irreducible quotient, which is isomorphic to L(W, γ, h).
The action of D 0 ⋉ K 0 on T (W, γ, h) can be derived from Theorem 6.3 (i) (see Theorem 6.4 in [2] for details) and T is essentially a tensor module that we discussed above: where Here Ω W is the scalar with which the Casimir operator of sl N acts on W . The claim of Theorem 8.2 follows from Proposition 8.1 and the following lemma.
Lemma 8.3. Let W be a simple finite-dimensional gl N -module on which the identity matrix acts as scalar α, and let W * be a gl N -module which is dual to W as a sl N -module, and on which the identity matrix acts as scalar N − α. Then the pairing given by Proof. Using (8.4) we get In case of t r k a , a = 1, . . . , N , both left and right hand sides are zero. Let us verify the contragredient property for the action of t r d a , a = 1, . . . , N : Finally, to check the case of t r d 0 , we note that the constant β in (8.8) is the same for T (W, γ, h) and T (W * , γ, h). This follows from the fact that the Casimir operator for sl N acts with the same scalar on W and W * , while the last term in (8.8) is invariant under the substitution α → N − α. Thus the computation in the case of t r d 0 is analogous to the case of t r k 0 . This completes the proof of the lemma.
with respect to appropriate anti-involutions of corresponding Lie algebras.
Remark 8.5. The duality of Theorem 8.2 can be alternatively constructed via vertex algebra approach, using the definition of the contragredient module over a vertex algebra (see section 5.2 in [10]).
One of the goals of this paper is to analyze which of the modules defined in Theorem 6.3 (ii) remain irreducible after restriction to D. First of all, let us look at the question of irreducibility of the top T (W, γ, h) as a module over D 0 . Proof. Clearly, if T (W, γ, h) is reducible as a D 0 -module, it must also be reducible as a VectT Nmodule. By Theorem 2.1, all such modules appear in the de Rham complex. Note that 1 , . . . , t ±1 N ]d 0 , and by (8.7), t r d 0 acts on T (W, γ, h) as multiplication by βq r . It is well known that in the modules of differential forms there are no proper submodules that are C[q ±1 1 , . . . , q ±1 N ]-invariant. Thus for T (W, γ, h) to be reducible as a D 0 -module it is necessary and sufficient that it is reducible as a VectT N -module and the value of β given by (8.8) is zero. Let us analyze the values of β for the modules in the de Rham complex. For the modules Ω 0 (T N ) and Ω N (T N ) the sl N -module W is trivial, so the Casimir operator acts with constant Ω W = 0, while the identity matrix acts on W with scalars α = 0 and α = N respectively. Simplifying the expression in (8.8) we get in this case that β = −h. In case of the modules of k-forms Ω k (T N ), k = 1, . . . , N − 1, the highest weight of the corresponding sl N -module W is the fundamental weight ω k . A standard computation shows that in this case the Casimir operator acts with the scalar Since the identity matrix acts with the scalar α = k, the formula (8.8) again simplifies to β = −h. This implies the claim of the Lemma.
Consider now an irreducible D ⋉K module L(W, γ, h) described in Theorem 6.3 (ii), and assume that its top T (W, γ, h) is irreducible as a D 0 -module. To show that L(W, γ, h) remains irreducible as a module over D, it is sufficient to establish two properties: (C) Every critical vector of L(W, γ, h) (i.e., annihilated by D + ) belongs to its top T (W, γ, h). Proof. We use the existence of a non-degenerate contragredient pairing of D ⋉ K-modules: If L(W, γ, h) has a vector annihilated by D + , which does not belong to the top, it also has a homogeneous vector with this property. Suppose u is a critical vector of degree s ∈ Z N +1 . Since the pairing is non-degenerate, there exists a vector v in L(W * , γ, h) of the same degree, such that u|v = 0. If property (G) holds for L(W * , γ, h), v can be written as Applying the contragredient property, we get The last expression is zero since σ(x i ) ∈ D + and u is a critical vector. This gives a contradiction, which implies that property (G) does not hold for L(W * , γ, h).
To prove the converse, assume that the component of degree s in L(W * , γ, h) is not generated by D − acting on the top. Let V be the intersection of that homogeneous component with the space U (D − )T (W * , γ, h). Since the pairing is non-degenerate, we can find a non-zero vector u in the degree s component of L(W, γ, h), such that u|V = 0. If property (C) holds for the module L(W, γ, h), there exists a homogeneous y ∈ U (D + ), such that z = yu is a non-zero vector in T (W, γ, h). Let z ′ ∈ T (W, γ, h) be such that z|z ′ = 0. Then But σ(y) ∈ U (D − ), thus σ(y)z ′ ∈ V , which leads to a contradiction. The lemma is now proved. Lemma 8.7 reduces the question of irreducibility of the family of modules L(W, γ, h) to the question of existence of critical vectors. If both L(W, γ, h) and L(W * , γ, h) satisfy condition (C), then they are both irreducible as modules over D.

Critical vectors
In this section we will establish a necessary condition for the existence of non-trivial critical vectors in the modules L(W, γ, h), which together with Lemma 8.7 will give a sufficient condition for the irreducibility of such modules.
We will call a module L(W, γ, h) exceptional if h = 0, the identity matrix acts on W by an integer k ∈ Z and W is a trivial one-dimensional sl N -module when k = 0 mod N or has a fundamental highest weight ω k ′ with 1 ≤ k ′ ≤ N − 1 and k = k ′ mod N .
Theorem 9.2 is an immediate consequence of Theorem 9.1 and Lemma 8.7. The proof of Theorem 9.1 will be split into a sequence of lemmas. Proof. The algebra D + contains the elements t j 0 d p with p = 1, . . . , N , j ≥ 1, which act as ∂ ∂upj . The condition (t j 0 d p )g = 0 implies the claim of the lemma.
For a formal series a(z) we denote by a(z) − its part that only involves negative powers of z. Recalling that d a (r, z) = j∈Z t j 0 t r d a z −j−1 , we have (zd a (r, z)) − g = 0. Using (6.11) for the action of d a (r, z) and taking into account that g does not depend on {u pj }, we get Let us project to the subspace (9.1), setting u pj = 0 in the above equality. Also, since the operator of multiplication by q r is invertible, we can drop it. We then get P a (r, z) − g = 0, a = 1, . . . , N, At this point we find it convenient to make a change of variables x aj = jv aj . In these notations P a (r, z) takes form Let us expand P a (r, z) in a formal series in variables r = (r 1 , . . . , r N ): It is easy to see that for any j ∈ Z and any vector g ′ , there are only finitely many s ∈ Z N + such that the coefficient at z j in P as (z)g ′ is non-zero. Thus the coefficient at z j in s∈Z N + r s P as (z)g is a polynomial in r. Since for each j < 0 these polynomials vanish for all r ∈ Z N , we conclude that for all s ∈ Z N + , a = 1, . . . , N , P as (z) − g = 0. Note that for s = 0 this equation is trivial. Let us consider the case s ∈ Z N + , with s p = 1 and s i = 0 for i = p. This gives us an equation Our module is Z N +1 -graded via the action of operators d 0 , . . . , d N , and without loss of generality we may assume that g is homogeneous relative to Z N +1 -grading. We will call the eigenvalue of β Id − d 0 the degree of g. We use the negative sign here to make the degree non-negative. In fact the Z-grading by degree may be defined on each of the tensor factors C[x pj | p=1,...,N j=1,2,... ], L glN and L Vir by On the space C[x pj | p=1,...,N j=1,2,... ] we will also consider a refinment of Z-grading by degree, where we will compute the degree in each of the N families of variables. In addition to the degree of monomials, we will consider another Z N -grading by length, where len a (x pj ) = δ ap and define the total length to be len(y) = N a=1 len a (y).

Let us fix homogeneous bases {y
..,N j=1,2,... ], L glN and L Vir respectively. Then we can expand g into a finite sum Note that in the above decomposition deg(g) = deg(y ′ i ) + deg(y ′′ j ) + deg(y ′′′ k ). Since equations (9.2), (9.4) and (9.5) do not involve any operators acting on the component L Vir , we conclude that these must be satisfied not only by g, but also by each of the components Lemma 9.4. Let g be a homogeneous non-zero critical vector. Then in the decomposition (9.6) there exist y ′′ j ∈ W and y ′′′ k ∈ Cv h with α ijk = 0 for some i. Proof. Let us rewrite (9.6) as Consider the smallest degree n 0 of y ′′′ k for which α ijk = 0 for some i, j. We claim that n 0 = 0. Otherwise, since L Vir is irreducible, there exists a raising operator ω Vir (n) , n ≥ 2, such that Consider now the Virasoro operator acting on all three factors of the tensor product: ω (n) = ω Hyp (n) + ω glN (n) + ω Vir (n) . The part of ω (n) g involving the terms of the smallest degree in the component L Vir will be Thus ω (n) g = 0. However operator ω (n) represents −t n−1 0 d 0 ∈ D + and must annihilate g since g is a critical vector. This is a contradiction, which implies that n 0 = 0.
Let g = q µ ij α ij y ′ i ⊗ y ′′ j ⊗ v h be the projection of g to the space We just proved that g = 0, and it was noted earlier that it satisfies equation (9.4). Let n 1 be the smallest degree of y ′′ j such that α ij = 0 for some i. To complete the proof of the lemma, we need to show that n 1 = 0. If n 1 > 0 using the same argument as above we see that there exists a raising operator E pa (n) , n ≥ 1 such that E pa (n) g will have a non-zero component with terms in L glN (W ) of degree n 1 − n. However the equation (9.4) implies that all factors from L glN (W ) that appear in E pa (n) g have degrees at least n 1 . This contradiction implies n 1 = 0, and the lemma is proved.
Let g be the projection of the critical vector g to the space By the above Lemma, g = 0. Let us take the projection of the equation (9.5) to the space (9.7). This yields P ′′ a (r, z) − g = 0, Again we decompose P ′′ a (r, z) in a formal power series in r, P ′′ a (r, z) = s∈Z N + r s P ′′ as (z), and we have P ′′ as (z) − g = 0 for all s ∈ Z N + . Next we will consider the grading of g by (total) length. Note that the operator P ′′ as (z) has two homogeneous components with respect to the length grading -one that decreases the length by Our goal is to solve the system of equations (9.9) in the space C[x pj | p=1,...,N j=1,2,... ]⊗W . The solutions we are looking for are homogeneous in both degree and length. Let deg(f ) = m, len(f ) = ℓ. The equation (9.9) has trivial solutions with ℓ = 0. These correspond to the critical vectors in the top of the module. We are going to show that non-trivial solutions of (9.9) must have length ℓ = 1.
To establish this claim we will first analyze the equation (9.9) in cases N = 1 and N = 2. The case of the general N will follow from the following simple observation. Consider a proper subset S ⊂ {1, . . . , N }. Let us take a solution f of (9.9) and specialize all variables x pj with p ∈ S to scalars, we will get a solution for (9.9) with a smaller N . To see this, set in (9.9) r p = 0 for all p ∈ S and restrict a to the set S. The information about the solutions of (9.9) with N = 1 and N = 2 may be used to establish properties of solutions of this equation for a general N . Lemma 9.5. Let N = 1, and let W be a 1-dimensional gl 1 -module with identity matrix I = E 11 acting by scalar α ∈ C. Let f be a non-constant homogeneous (in both length and degree) solution of (9.9) with N = 1. Then len(f ) = 1 and α = 1 − deg(f ).
Proof. First of all, let us rewrite (9.10) for the case N = 1. To simplify notations we will drop redundant indices and write x i instead of x pi , etc., (9.11) Let f satisfy Q(r, z) − f = 0. We may view f as a polynomial of length ℓ > 0 and degree m ≥ ℓ in C[x 1 , x 2 , . . .]. Choose a natural number s such that  Let us consider the coefficient at r s+1 in the equation Q(r, z) − f = 0: Consider an expansion R(z) = R n z −n + R n+1 z −n−1 + . . . + R m z −m with R n = 0. Let us look at the coefficient at z −n in the above equation: αR n + nR n − sR n = 0, which implies α = s − n ∈ Z − . Applying the operator z d dz to (9.12) we get Let us prove by induction that for all j ≥ 0 Suppose (9.15) holds for all j ′ ≤ j, j ≥ 1. Applying the operator ∞ k=1 k j z −k ∂ ∂x k to (9.14) and using the induction assumption, we get which establishes the inductive step. In the above calculation we used the fact that k−1 p=1 p j is a polynomial in k with the leading term k j+1 j+1 . Using the Vandermonde determinant argument we conclude from (9.15) that for all k. This implies that all R i are scalars, but since they can't have equal degrees, we conclude that R n is a non-zero scalar, while all other coefficients are zero. Without the loss of generality we may thus assume Thus s = len(f ) = ℓ and n = deg(f ) = m, and then α = len(f ) − deg(f ). It remains to prove that len(f ) = 1. We will reason by contradiction. Let us suppose that ℓ = len(f ) > 1 and consider Since every term in S(z) has length 1, we can write Now we look at the coefficient at r ℓ in the equation Q(r, z) − f = 0: Taking out the factor of (−1) ℓ ℓ! we get We stress that the above calculation is only valid when ℓ > 1. If ℓ > 2 we immediately get a contradiction since the coefficient at z −1 yields x m−1 = 0. It only remains to rule out the case ℓ = 2. In the latter case the equation simplifies to the following: Let us specialize this equation to x 1 = . . . = x m−1 = 1, z = 1. Then we get which implies m = 1, which is impossible since m = deg(f ) can not be less than ℓ = len(f ). Thus ℓ = len(f ) = 1 and the lemma is proved.
Next we are going to show that for a general N , a homogeneous non-trivial solution of (9.9) must have total length 1. The previous lemma implies that such a solution may have only two components with respect to len a grading for each a, where len a may be either 0 or 1. To prove the general case, it is sufficient to consider N = 2, since if a monomial has len a + len b at most 1 for any pair of distinct indices, then its total length does not exceed 1.
Lemma 9.6. Let N = 2 and let W be a finite-dimensional gl 2 -module. Then any homogeneous (in both length and degree) non-constant solution f of (9.9) has total length 1.
Proof. It is sufficient to consider the case of W being irreducible since the equation (9.9) is compatible with the gl N -module homomorphisms. Let us fix a basis {w n , w n−1 , . . . , w −n } of W , where n ∈ 1 2 Z + and (E 11 − E 22 )w i = 2iw i . Assuming that the identity matrix acts on W by scalar α, we get It follows from the previous lemma that every monomial in the decomposition of f has length at most 1 with respect to each of the two indices. Thus we only need to prove that f can not have total length 2. We will reason by contradiction. If len(f ) = 2 then for each monomial in f both len 1 and len 2 are 1. Suppose deg(f ) = m. Let us write By Lemma 9.5 we have Then f may be written as where we set β i = 0 whenever b i ≤ 0 or c i ≤ 0. Let us take the equation derived from (9.9) by taking the coefficient at r 1 r 2 with a = 1 and substitute (9.16) in it. We get Note that only the last sum contains variables x 2,j . By equating this sum to zero, we conclude that β i E 21 w i = 0 for all i = −n, . . . , n. It follows that β i = 0 for all i = −n. Similarly, taking the same equation with a = 2, we will get that β i = 0 for all i = n. Thus the only possibility for a non-zero solution is when n = 0, which means that W is 1-dimensional and m is even. In this case b 0 = c 0 = m 2 and f = x 1, m 2 x 2, m 2 ⊗ w, and the equation (9.17) becomes which gives a contradiction. Thus the total length of f must be 1 and the lemma is proved.
Now we return to the general case. We proved that a non-trivial homogeneous solution f of (9.3) must have length 1. Suppose deg(f ) = m. Then f can be written as The equation (9.9) then simplifies as follows: Consider a new action ρ ′ of gl N on W : This gives the same structure of W as an sl N -module, but now the identity matrix acts with scalar α ′ = α + (m − 1)N . Then (9.18) is equivalent to the system of equations where a, b, c = 1, . . . , N . We will also use a third gl N -action ρ ′′ on W : The identity matrix here acts with scalar α ′′ = α + mN . For this action the equation (9.19) may be written as We also have We are going to classify gl N -modules W for which the system (9.19) has non-trivial solutions. We will do this indirectly, linking this system with reducibility of tensor modules. Lemma 9.7. Let (W, ρ ′′ ) be a finite-dimensional irreducible gl N -module. Let P be the set of all solutions (w 1 , . . . , w N ) ∈ W × . . . × W of the system of equations (9.20). Then the subspace is a VectT N -submodule in the tensor module T (W ) = C[q ±1 1 , . . . , q ±1 N ]⊗W , associated with (W, ρ ′′ ). Proof. Let (w 1 , . . . , w N ) ∈ P. Then using the tensor module action and (9.21) we get To complete the proof of the lemma, it is sufficient to show that ( w 1 , . . . , w N ) ∈ P. Instead of working with (9.20), it will be easier to check an equivalent condition (9.19). Note that w p = ρ ′ (E pa )w i + δ pa w i . Then using the fact that (w 1 , . . . , w N ) satisfies (9.19), we obtain Lemma is now proved.
If L(W, γ, h) has a critical vector of degree m ≥ 1 then either W has a fundamental highest weight ω k , 1 ≤ k ≤ N − 1, with respect to sl N -action, with identity matrix acting with scalar α = k − mN , or W is a 1-dimensional module with identity matrix acting with scalar α = N − mN .
Proof. If the system (9.20) has a non-trivial solution then the submodule P in the tensor module T (W ) corresponding to (W, ρ ′′ ) is non-zero. It is a proper submodule since its component at q 0 is trivial. Using the classification of reducible tensor modules (Theorem 2.1), we conclude that T (W ) is one of the de Rham modules Ω k (T N ), k = 1, . . . , N . Taking into account the relation α = α ′′ − mN , we obtain the claim of the corollary.
To complete the proof of Theorem 9.1 it remains to establish the following Lemma 9.9. If L(W, γ, h) has a critical vector that does not belong the top then h = 0.
Proof. Our strategy will be the same as in derivation of equation (9.9). A critical vector g is annihilated by t j 0 t r d 0 for j ≥ 0. Thus (z 2 d 0 (r, z)) − g = 0.
We will project this equation to the subspace (9.7) in order to derive an equation on f . Finally, we will take r a -component of the resulting equation. The action of d 0 (r, z) is given by (6.12), which has three summands. We will analyze the contribution of each summand in z 2 d 0 (r, z) separately.
Consider the first summand Thus the corresponding terms in the above expression may be dropped. We also recall that . We further split (9.22) into three summands corresponding to this decomposition. For the case of the Virasoro field of the hyperbolic lattice component we have The first summand in (9.23) does not contribute to the projection to (9.7) since it contains multiplications by u pj , while Y (q r , z) does not involve differentiations in these variables. Note that we are only interested in powers z j in ω Hyp (z) with j ≥ −2. Thus the only terms that will contribute are: In operator Y (q r , z) we may then drop the factors containing u pj when taking the projection to (9.7). The contribution that we get will be Let us now take the r a -coefficient of the expansion in powers of r: the first summand in (9.24) vanishes, while the second simplifies to  Next, let us consider the contribution of the Virasoro field of V glN : The operators ω gl N (j) with j ≤ 0 increase the degree in the component L glN and thus do not contribute to the projection to the space (9.7), and only the term with ω gl N (1) will contribute. The Virasoro field of gl N is a sum of the Virasoro fields of sl N (6.5) and the Heisenberg algebra (6.6).
Using (6.5) we can write and the terms that contribute to the projection are which is a multiple of the Casimir operator for sl N . If W corresponds to the tensor module of k-forms, k = 1, . . . , N , this operator will act on the space (9.7) with scalar k(N −k)

2N
(see (8.9), this also includes the case of a trivial sl N module W when k = N ).
Analogously, for the Virasoro field (6.6) of the Heisenberg algebra ω Hei (1) will be acting on f with the scalar 1 2N Going back to (9.26), we get the contribution of the Virasoro field in V glN .
and its r a -term will yield Now let us deal with the Virasoro field of L Vir . The corresponding term is Since the operators ω Vir (j) with j ≤ 0 increase the degree in the component L Vir , the only term that contributes to the projection to (9.7) is ω Vir (1) , which acts on (9.7) with scalar h. Thus the r a -term of (9.28) gives the contribution  Next we shall look at the the summand − N a,b=1 r a u b (z)E ab (z)Y (q r , z) in (6.12). Its r a -term is When we consider the projection to (9.7), we can drop terms with multiplications by u pj and E ap (k) with k ≤ −1, while for E ap (k) with k ≥ 1 we may use (9.4), which yields Taking into account that f is linear in v pm , this can be simplified to the following Lemma 9.7 provides a relation between components (w 1 , . . . , w N ) and submodules in the tensor modules Ω k (T N ). Using computations in the tensor module of k-forms one can show that The r a -coefficient of the last summand N p=1 and its projection to (9.7) yields Finally, collecting (9.25), (9.27), (9.29), (9.31) and (9.32) together, we get Since a is arbitrary, we can choose it so that w a = 0. Thus h = 0, which was to be demonstrated.

Chiral de Rham complex
Chiral de Rham complex was introduced by Malikov et al. in [20]. In case a of torus T N the space of this differential complex is a tensor product of two vertex (super) algebras Here V Z N is the lattice vertex superalgebra of the standard euclidean lattice Z N . Before we define the differential of this complex, let us review the structure of V Z N . The vertex superalgebra V Z N has two main realizations -the bosonic realization and the fermionic one, with boson-fermion correspondence being an isomorphism between the two models. For our purposes it will be more convenient to use the fermionic realization of V Z N .
Consider the Clifford Lie superalgebra Cl N of "charged free fermions" with basis {ϕ p (j) , ψ p (j) |p = 1, . . . , N , j ∈ Z} of its odd part and a 1-dimensional even part spanned by a central element K. The Lie bracket in Cl N is given by With this choice of fields Cl N becomes a vertex Lie superalgebra since the only non-trivial relation between these fields is The lattice vertex superalgebra V Z N is isomorphic to the universal enveloping vertex algebra of Cl N at level 1. As a vector space it is the unique Cl N -module generated by vacuum vector 1l, satisfying K1l = 1l, ϕ p (j) 1l = ψ p (j) 1l = 0 for j ≥ 0, p = 1, . . . , N. In its fermionic realization V Z N is the exterior algebra with generators {ϕ p (j) , ψ p (j) | p=1,...,N j≤−1 } and is irreducible as a module over Cl N . The state-field correspondence map Y is given by the standard formula (6.1).
We fix the Virasoro element in V Z N : The rank of this VOA is −2N . It is well-known that vertex superalgebra V Z N contains a level 1 simple gl N vertex algebra. The fields generating this subalgebra are It is easy to check that these satisfy relations (6.4) and the central element of gl N acts as identity operator. It is also straightforward to verify that the Virasoro element (6.7) in the gl N vertex algebra maps to ω fer under this embedding.
Let us define two Z-gradings on V Z N . The fermionic degree is defined by deg fer (ϕ p (j) ) = 1, deg fer (ψ p (j) ) = −1, deg fer (K) = deg fer (1l) = 0. The bosonic grading is defined as follows: deg bos (ϕ p (j) ) = −j − 1, deg bos (ψ p (j) ) = −j, deg bos (K) = deg bos (1l) = 0. Let V k Z N be the subspace of the elements of fermionic degree k. We have a decomposition Note that each subspace V k Z N is a gl N -submodule, which is graded by the bosonic degree. Its structure is described by the following well-known result (see e.g. [9] or [15]): then as an sl N -module V k Z N has the fundamental highest weight ω k ′ . The identity matrix of gl N acts on V k Z N as k Id. Combining this result with Theorem 6.3, we get has a structure of a module for the Lie algebra VectT N +1 of vector fields.
For these modules the Virasoro tensor factor L Vir (h) is 1-dimensional (h = 0). The modules in this family are precisely the exceptional modules L(W, γ, h) for which Theorem 9.2 does not claim irreducibility. We are going to see below that these modules are in fact reducible.
Let us express the action (6.11), (6.12) of the Lie algebra VectT N +1 on M Hyp (γ) ⊗ V k Z N using the fermionic realization: Following [20], let us now introduce the differential Proof. Since a (0) b = 0, the commutator formula (5.2) yields However the right hand side does not contain terms with z −1 1 and the claim of the lemma follows.
Let us continue with the proof of the theorem. We need to show that Q (0) d a (r) = 0 and Q (0) d 0 (r) = 0. Since : v i (z)ϕ i (z) :, we have It is easy to see that v i (j) d a (r) = 0 for j ≥ 1 and ϕ i (j) d a (r) = 0 for j ≥ 1. Thus r p v a (−1) ϕ p (−1) q r = 0.
Let us now show that Q (0) d 0 (r) = 0. Since v i (j) d 0 (r) = 0 for j ≥ 2 and ϕ i (j) d 0 (r) = 0 for j ≥ 1, we get Let us compute each of three terms in the right hand side separately: The last summand in (10.1) vanishes since it is antisymmetric in {a, i}. Next, And finally, r a v i (−1) u i (−1) ϕ a (−1) q r . Let us present here a diagram of the Chiral de Rham complex for N = 2. On the diagram, the fermionic degree increases in the horizontal direction and bosonic in vertical.
The tops of the modules M Hyp (γ)⊗V k Z N with 0 ≤ k ≤ N are the spaces q γ Ω k (T N ) of differential k-forms that form the classical de Rham complex. Non-trivial VectT N -submodules in these tops generate non-trivial VectT N +1 submodules in corresponding modules M Hyp (γ) ⊗ V k Z N . It was proved in [20] that the cohomology of the chiral de Rham complex coincides with the classical de Rham cohomology. This implies, in particular, that for k < 0 or k > N the short sequences are exact. Using this fact, we get Corollary 10.5. (i) For k ≤ 0, VectT N +1 -modules M Hyp (γ) ⊗ V k Z N have non-trivial critical vectors.
(ii) For k ≥ N , VectT N +1 -modules M Hyp (γ) ⊗ V k Z N are not generated by their top spaces. Proof. We can see from the above diagram that for k < 0 the images of the top vectors in M Hyp (γ) ⊗ V k Z N are non-trivial critical vectors in M Hyp (γ) ⊗ V k+1 Z N . For k ≥ N , the top spaces of M Hyp (γ) ⊗ V k Z N are in the kernel of d. Thus the submodules generated by the tops are annihilated by d as well. Since the map d is non-zero, these submodules are proper.
As a result we see that all modules that belong to the chiral de Rham complex are reducible. The claim of Corollary 10.5 is consistent with the existence of the contragredient pairing given by Theorem 8.2: For the chiral de Rham complex this duality was constructed in [19].