Vertex Algebroids over Veronese Rings

We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.

graded algebra resolution R → A. This gives rise to a graded Courant algebroid functor A → V poiss (A) • .
The problem of combining the two, that is to say, finding a quantization, V(A) • , of V poiss (A) • appears to be intellectually attractive and important for applications. If A is a complete intersection, then the resolving algebra R can be chosen to be a super-polynomial ring on finitely many generators, the corresponding resolution R → A being none other than the standard Koszul complex, and the quantization, a differential graded vertex algebroid V(R), is immediate; this observation has been used in a number of physics and mathematics papers. If, however, A is not a complete intersection, then any resolving algebra R is infinitely generated in which case defining a vertex algebroid V(R) becomes problematic because of various divergencies. A regularization procedure for some of these divergencies was suggested in [4] and elaborated on in [17].
Here is what we do in the present paper. Let V N be the (N + 1)-dimensional irreducible sl 2module, O N ⊂ P(V N ) the highest weight vector orbit, and A N the corresponding homogeneous coordinate ring. All of this is a representation theorist's way of saying that P 1 ∼ −→ O N ⊂ P N is a Veronese curve, and A N is a Veronese ring.
A N is a quadratic algebra, in fact it is Koszul [3,19,6], but it is not a complete intersection. The main result of the paper, Theorem 5.1.1, asserts that V poiss (A N ) • admits a unique quantization. It is no surprise that this quantization, V(A N ), contains a vertex algebroid, V(sl 2 ) k , attached to sl 2 with some central charge k. What is more important is that the vertex algebroid attached to gl 2 enters the fray. The latter, V(gl 2 ) k 1 k 2 , depends in general on two central charges, k 1 , k 2 , and we find that the quantization conditions imply, first, that k 1 + k 2 = −2 and, second, that k 1 = −N − 2.
Theorem 5.1.1 and its proof appear in Section 5, and it is for the sake of this section that the paper was written. Section 4 is to a large extent an exposition of Hinich's result (see also [2]) with some extensions (Sections 4.3,4.4,4.5) that are needed in Section 5. Sections 2 and 3 are an attempt, perhaps futile, to make the paper self-contained -except Sections 3.7.3, 3.7.4, where sheaves of vertex algebroids over C 2 \0 are classified. The classification obtained is instrumental in proving Theorem 5.1.1; in particular, the vertex algebroid V(gl 2 ) k 1 ,k 2 with the compatibility condition k 1 + k 2 = −2 makes appearance in Section 3.7.4.
An obvious generalization of A N is provided by the homogeneous coordinate ring of a higher dimensional Veronese embedding P(C n ) → P(S N (C n )). We show (Theorem 5.3.1) that if n > 2 and N > 1, then no quantization exists.
Much of the above carries over to an arbitrary simple g, where A N is replaced with the homogeneous coordinate ring of the highest weigh vector orbit in the projectivization of a simple g-module. For example, C 2 \ 0 becomes the Bernstein-Gelfand-Gelfand base affine space, G/N . Constructed in [16] is the 1-parameter family of sheaves of vertex algebroids H ∞/2 (Ln, V G,k ) over G/N , where V G,k , k ∈ C, is a family of vertex algebroids over G, [1,10,14,16]. There is little doubt that the family H ∞/2 (Ln, V G,k ), k ∈ C, is universal in that it classifies vertex algebroids over G/N equipped with V(g) k -structure. This is a higher rank analogue of the classification obtained in Section 3.7.3 and alluded to above. Note that just as G/N is a G × Tspace, the maximal torus acting on the right, so there is a diagram of embeddings V(g) k 1 ֒→ H ∞/2 (Ln, V G,k ) ←֓ V(t) k 2 with k 1 + k 2 = −hˇ.
Therefore, V(g) k ⊕ V(t) −k−hˇi s a higher rank analogue of V(gl 2 ) k,−k−2 ; here g, n and t are the Lie algebras of the Lie groups G, N , and T resp. We elaborate on these remarks in Section 6, where we use the technique of semi-infinite cohomology to compute CDO-s on some homogeneous spaces including the spaces of pure spinors punctured at a point. In the latter case, this gives an approach alternative to that of the original result by Nekrasov [24].
Some aspects of the sl 2 -case, however, are not that easy to generalize. As they say, we hope to return to this subject in a separate paper.
We would like to conclude by saying that a major source of inspiration was provided to us by the work of Berkovits and Nekrasov [5,4], where similar problems are analyzed in the case of the spinor representation of the spinor group.
2 Vertex algebras 2.1. Conventions. Underlying all the constructions in this paper will be the category of Zgraded vector superspaces and grading preserving linear maps over C. This grading will be called (and should be thought of as) the homological degree grading. More often, though, the attribute 'graded' will be skipped. Thus the phrase 'let V be a vector space' will mean that V = ⊕ n∈Z V n , V even = ⊕ n∈Z V 2n , V odd = ⊕ n∈Z V 2n+1 . Likewise, the prefix 'super-' will be usually omitted so that commutative will mean super-commutative, algebra super-algebra, bracket super-bracket: [a, b] = ab − (−1) ab ba.
If V and W are vector spaces, then V ⊗ W is also a vector space with homological degree grading defined in the standard way so that (V ⊗ W ) n = ⊕ i∈Z V i ⊗ W n−i . Various bilinear operations ('multiplications') to be used below will be morphisms of graded vector spaces V ⊗ W → U .
Along with the homological degree grading, the grading by conformal weight will play a prominent role. The latter will be indicated by a subindex; thus, for example, the phrase 'a graded (by conformal weight) vertex algebra' will mean, in particular, a vector space V with a direct sum decomposition V = ⊕ n≥0 V n valid in the category of graded vector spaces.
Most of the definitions and constructions in this and the following section are well known, and their graded versions are always straightforward. We recommend [20] and [12] as an excellent introduction to vertex things and a guide to further reading.

Def inition 2.2.
A vertex algebra is a collection (V, 1, T, (n) , n ∈ Z), where V is a vector space, 1 ∈ V is a distinguished element known as the vacuum vector, T : V → V is a linear operator known as the translation operator, each (n) is a multiplication that is subject to the following axioms: (1) (vacuum) (2.4) (5) (quasi-associativity or normal ordering) The collection of axioms we used in Definition 2.2 is a little redundant but makes the exposition a little more transparent. It emphasizes the fact that the notion of a vertex algebra is a mixture of (appropriate analogues of) that of an associative algebra and a Lie algebra. Extracting the Lie part of the definition one arrives at the notion of a vertex Lie algebra.
There is an obvious forgetful functor Its left adjoint functor (the vertex enveloping algebra functor) is well known to exist, see [25]; it also appears in [20] as 'the vertex algebra attached to a formal distribution Lie superalgebra'. Note a canonical map that is the image of Id ∈ Hom(U L, U L) under the identification Hom(U L, U L) ∼ −→ Hom(L, ΦU L).
Example 2.4. Let L ′ be a free C[T ]-module on one generator L and let L(Vir) c = L ′ ⊕ C, where C is considered as a trivial C[T ]-module. L(Vir) c carries a unique vertex Lie algebra structure such that Upon quotienting out by the relation 1 = 1, the vertex enveloping algebra U L(Vir) c becomes the vacuum representation of the Virasoro algebra of central charge c.
Example 2.5. Let g be a Lie algebra with an invariant bilinear form (·, ·). Let This space carries an obvious action of T , where again we consider C as a trivial C[T ]-module, and a unique vertex Lie algebra structure such that Upon quotienting out by the relation 1 = 1, the vertex enveloping algebra U L(g) k becomes the vacuum representation of the corresponding affine Lie algebra of central charge k.
If g is chosen to be gl N = sl N ⊕ C · I, then this construction has the following version: we let (a, b) = tr(a · b), L(gl N ) k 1 ,k 2 = L(sl N ) k 1 ⊕ C[t] ⊗ C · I and extend (2.10) by In order to handle the case of the trivial bilinear form (·, ·), or more generally the case where (·, ·) is not unique even up to proportionality, we will change the notation and denote by L(g) (·,·) the vertex Lie algebra which is precisely L(g) k except that the last of conditions (2.10) is replaced with for some (·, ·).
A passage to the quasiclassical limit is a gentler way to blend the Lie and commutative/associative algebra parts of the structure.
Vertex Poisson algebras are to vertex algebras what Poisson algebras are to noncommutative algebras. The following construction (cf. [21]) is meant to illustrate this point.

2.7.
Vertex algebras with f iltration. Suppose a vertex algebra V carries an exhaustive increasing filtration by vector spaces By focusing on symbols one discovers that the vertex algebra structure on V defines the following on the corresponding graded object Gr V = ⊕ p F p V /F p−1 V : It is then immediate to check that (Gr V, 1 gr , T gr , (n) gr , n ≥ −1) is a vertex Poisson algebra. For example, commutativity of the product (−1) follows from the n = −1 case of (2.3) and associativity from the n = −1 case of (2.5).
The vertex algebras reviewed in Examples 2.4, 2.5 possess a filtration with the indicated properties -as does any vertex enveloping algebra: • in the case of U L(Vir) c the filtration is determined by assigning degree one to ι(L(Vir) c ), see (2.9); • in the case of U L(g) k the filtration is determined by assigning degree one to ι(L(g) k ).

Courant and vertex algebroids
Such grading is usually referred to as conformal, V n is called the conformal weight n component, and v ∈ V n is said to have conformal weight n; the conformal weight of v ∈ V n is usually denoted by ∆(v).
All the vertex algebras we have seen are graded: U L(Vir) c is graded by letting V 0 = C, A graded vertex (vertex Poisson) algebra structure on V induces the following structure on the subspace V 0 ⊕ V 1 : These data satisfy a list of identities obtained by inspecting those listed in Definitions 2.2 and 2.6 and choosing the ones that make sense. The meaning of 'make sense' is clear in the case of identities, such as the Jacobi, involving only operations (n) with n ≥ 0, because V 0 ⊕ V 1 is closed under these operations.
are said to make sense if either they are compositions of operations (3.2)-(3.5) or because and both afford the left adjoints U alg and U poiss alg , which are analogous to (2.8).
When applied to the algebras of (2.14) and of Examples 2.4, 2.5, they give us first examples of Courant and vertex algebroids resp.
Here is an example of geometric nature.
Since all the operations recorded in (3.10) are of geometric nature, there arises, for each scheme X, a sheaf of vertex algebroids (3.11) Note that in keeping with our convention we assume that X is graded, that is to say, O X is a sheaf of graded algebras; consequently, T X and Ω X are sheaves of graded O X -modules. If V = V 0 ⊕ V 1 is a vertex algebroid, then part of its structure coincides with that of V poiss (A). For example, the triple (V 0 , (−1) , 1) is a commutative associative algebra with unit, and a moment's thought will show that, absolutely analogously to Section 2.7, the corresponding Gr V carries a canonical Courant algebroid structure.
Def inition 3.5. Note that in the case of V poiss (A), filtration (3.12) implies the following exact sequence of vector spaces The problem of quantizing V poiss (A) is not trivial and was studied in [15]. Call A suitable for chiralization if Der(A) is a free A-module on generators τ 1 , . . . , τ N s.t. [τ i , τ j ] = 0. If A is suitable for chiralization and a basis τ 1 , . . . , τ N is fixed, then V poiss (A) can be quantized by letting V(A) = V poiss (A) as a vector space and requiring all of the relations (3.10) except the last two; the latter are to hold only for the basis vector fields: It is easy to show, using axioms (2.1)-(2.5), at least that these choices determine a vertex algebroid structure. Let us point out the differences between the Courant and vertex algebroid structures thus obtained: in the Courant case operation (1) is an A-bilinear pairing and in the Courant case multiplication (−1) is associative, e.g., in the vertex case it is not as here f, g ∈ A and axioms (2.3)-(2.5) along with simplifications due to grading have been used. Furthermore, if A is suitable for chiralization, then and where the automorphism of V(A) attached to ω ∈ Ω 2,cl (A) 0 is defined by the assignment π being defined in (3.13). Note that we had to pay the price for relentlessly working in the graded setting by extracting the homological degree 0 subspace in (3.17), (3.18). Since any smooth algebraic variety X can be covered by the spectra of rings suitable for chiralization, (3.17), (3.18) create an avenue to define sheaves of vertex algebroids over X, to be denoted Def inition 3.6. Let X be a graded scheme.
(a) Call a sheaf of vertex algebroids over X a quantization of V poiss X , see (3.11), if for each affine U ⊂ X its space space of sections over U is a quantization of Γ(U, V poiss X ). The characteristic property of V X ∈ Vert X is the existence of the sequence of sheaves this is a sheaf analogue of (3.13).
Here are the (obvious graded versions of the) main results of [15]: • there is a gerbe of vertex algebroids over a manifold X such that the space of sections over each U = Spec(A) ⊂ X, A being suitable for chiralization, is the category of quantizations of V poiss (A); • in the case where O X = O 0 X this gerbe possesses a global section, i.e., a sheaf V X ∈ Vert X , if and only if the 2nd component of the Chern character vanishes; if this class vanishes, then • the forgetful functor (3.6) has a left adjoint functor a sheaf of chiral differential operators, CDO for short.
Note that proving (3.22) amounts to covering X by open sets that are suitable for chiralization and re-gluing a given sheaf by composing the old gluing functions with automorphisms (3.18), (3.19).

Further examples and constructions.
3.7.1. Localization. For any quantization V(A) and an ideal a ⊂ A a natural quantization V(A a ) is defined [15], the reason being that all the (n) -products on V(A) are in fact certain differential operators. For example, at the quasiclassical level, all the operations recorded in (3.10) are differential operators of order ≤ 1. Furthermore, (3.15), (3.16) provide examples of genuine quantum operations being order ≤ 2 differential operators. Therefore, given an A and V(A), there arises a sheaf of vertex algebroids This construction underlies the above discussion of gerbes of vertex algebroids.
A little more generally, if X and Y are manifolds and p : X → Y is a covering, then there is a functor The reason for this to be true is that the story of quantizing V poiss (A), Spec(A) ⊂ Y , starts with a choice of an Abelian basis {τ i } ⊂ Der(A), and any such choice is canonically lifted to an Abelian basis of Γ(V, T X ) for any affine V ⊂ p −1 (Spec(A)).
In particular, if X carries a free action of a finite group G, then there arises an equivalence of categories where Vert G X is a full subcategory of G-equivariant vertex algebroids. The inverse functor is, essentially, that of G-invariants where p * is the push-forward in the category of sheaves of vector spaces.
Here is a version of localization called a push-out in [15,16]. If a Lie group acts on A by derivations, then is a vertex algebroid.
. More explicitly, D ch (C N ) can be defined to be the vertex algebra generated freely by the vector space C N ⊕ (C N ) * and relations 3.7.3. Punctured plane. Let X = C 2 \(0, 0) and choose the trivial grading where O X = O 0 X . The obstruction (3.21) vanishes, because one sheaf, say, the restriction of V C 2 to C 2 \(0, 0), exists. Isomorphism classes of sheaves of vertex algebroids over X are easy to classify. Indeed, consider the affine covering and the Cech 1-cocycle It is known (and easy to check) that Hence the isomorphism classes of sheaves of vertex algebroids over X are in 1-1 correspondence with linear combinations of ω ab . Here is an explicit construction of the sheaf attached to kω ab : let V X be the restriction of the standard V C 2 to X, V U j its pull-back to U j , j = 1, 2; now glue V U 1 and V U 2 over the intersection U 1 ∩ U 2 , cf. (3.18), (3.19), as follows: It is immediate to generalize this to the case of an arbitrary ω in the linear span of {ω ab }. Denote the sheaf sheaf thus defined by V X,ω . This simple example will be essential for our purposes.
A quantization of a Courant algebroid morphism ρ poiss : Similarly, if g operates on X, that is, there is a Lie algebra morphism then a quantization of the corresponding Courant algebroid morphism ρ poiss : commutes.
To see an example of importance for what follows, let us consider the tautological action of gl 2 on C 2 . If we let X = C 2 \ (0, 0), then there arises and we ask if this map can be quantized to a map V( Lemma 3.7.5. Quantization of (3.32), Proof . We shall use the notation of Section 3.7.3. In terms of the coordinates y 1 , y 2 the morphism ρ is this If we consider y i ∂ j as an element of Γ(U 1 , V X,ω ), then over U 2 it becomes, according to (3.29), (y i ∂ j ± kT (y j±1 )y i )/y a 1 y b 2 and hence may develop a pole. To compensate for it, we can choose a different lift of E ij to Γ(U 1 , V X,ω ) by replacing y i ∂ j with y i ∂ j + α, where α ∈ Γ(U 1 , Ω X ). Over U 2 this element becomes (y i ∂ j ± T (y j±1 )y i )/y a 1 y b 2 + α. Since α may have at most a pole along {y 1 = 0}, for this element to extend to a section over U 2 , one of the following two things must happen: either T (y j±1 )y i /y a 1 y b 2 has no pole along {y 1 = 0}, in which case no α is needed, or T (y j±1 )y i /y a 1 y b 2 has no pole along {y 2 = 0}, in which case a desired α can be found. For a favorable event to occur for i = 1 and i = 2, both a and b must be at most 1. But by definition, see Section 3.7.3, a and b are at least 1; therefore a linear map gl 2 → Γ(X, V X,ω ) may exist only if ω = kdy 1 ∧ dy 2 /y 1 y 2 . On the other hand, if ω = kdy 1 ∧ dy 2 /y 1 y 2 , then the mapρ defined so that It is easy to see the vertex algebroid morphism condition determines the map uniquely.
Note that the top row of (3.31), unlike that of (3.30), does not have to be exact in general. In the case at hand, however, it is precisely when ω = kω 11 : Corollary 3.7.6. The sequence is exact if and only if ω = kω 11 for some k ∈ C.
Proof . Notice that T C 2 \(0,0) is generated by ρ(gl 2 ) over functions. The "if" part is then seen to be an immediate consequence of Lemma 3.7.5. The "only if" part was actually proved at the beginning of the proof of the lemma cited.
3.7.7. Conformal structure. If x 1 , . . . , x N are coordinates on C N , ∂ j = ∂/∂ j , then there is a vertex (Poisson) algebra morphism where the latter is a quantization of the former. A little more generally, if A is an algebra suitable for chiralization with τ 1 , . . . , τ N an Abelian basis of Der(A), then one can find a coordinate system, i.e., {x 1 , . . . , x N } ⊂ A s.t. τ i (x j ) = δ ij , and thus obtain In this case, N is the Krull dimension of A. Another example is provided by the twisted sheaves V C 2 \0,ω of Section 3.7.3. Somewhat unexpectedly, the same definition (3.39), which in the present case becomes L → T (y 1 )∂ 1 + T (y 2 )∂ 2 , applied locally over both both charts U 1 and U 2 survives the twisted gluing transformation (3.29) for any ω and defines a global morphism (3.40) 4 A graded Courant algebroid attached to a commutative associative algebra 4.1. Modules of dif ferentials. Even though the assumption that all the vector spaces in question are Z-graded has been kept since the very beginning of Section 2, it has been barely used. From now on it will be essential and referred to explicitly as in the following definition.
Def inition 4.1.1. A differential graded algebra (DGA) R is a pair (R # , D), where R # = ⊕ ∞ n=0 R n is a graded supercommutative associative algebra with R even = ⊕ ∞ n=0 V 2n , R odd = ⊕ ∞ n=0 R 2n+1 , and D is a square 0 degree (−1) (hence odd) derivation. Call a DGA (R # , D) quasi-free if there is a graded vector superspace is a a graded supercommutative associative algebra.
For any commutative associative algebra A there is a quasi-free DGA R and a quasi-isomorphism that is to say, a DGA morphism (A being placed in homological degree 0 and equipped with a zero differential) that delivers a graded algebra isomorphism H • D (R) ∼ −→ A. If A is finitely generated, then a DGA resolution R can be chosen so that each V j from Definition 4.1.1 is finite dimensional. These two finiteness assumptions will be made throughout.
A DGA resolution of A is not unique, but for any two such resolutions there is a homotopy equivalence [2] f : If R is a quasi-free DGA, denote by Ω(R) the module of Kähler differentials of R. It is canonically a differential-graded (DG) free R-module with derivation d : R → Ω(R) and differential Lie D , which we choose to denote by D, too. The correspondence R → Ω(R) is functorial in that naturally associated to an algebra morphism f : R 1 → R 2 there is a map of DG R 1 -modules: and an isomorphism where R is a quasi-free DGA resolution of A.
The assignment A → Ω(A) • defines a functor from the category of algebras to the category of graded vector spaces.
Note that is a graded A-module, and Ω(A) 0 is the module of Kähler differentials of A, Ω(A).

Modules of derivations.
If R is a quasi-free DGA, we denote by Der(R) the Lie algebra of derivations of R. Like Ω(R), it is a DG R-module, but unlike Ω(R) it is graded in both directions: and, which is more serious, not free; in fact, each component Der(R) n is a direct product where V j is one of the ingredients of Definition 4.1.1 assumed to be finite dimensional.
The derivation [D, ·] : Der(R) → Der(R) is a differential because D ∈ Der(R) −1 is odd. Hence a Lie algebra H • [D,·] (Der(R)) arises. The assignment R → Der(R) is not quite functorial, because even if f : R 1 → R 2 is a quasiisomorphism, a Lie algebra morphism Der(f ) : Der(R 1 ) → Der(R 2 ) does not quite exist. It does exist though at the level of the corresponding homotopy categories. This remark and what follows belongs to Hinich [18,Section 8].
Decompose f : R 1 → R 2 as follows where i is a standard acyclic cofibration, and p is an acyclic fibration. (Recall that i being a standard cofibration means S being obtained by adjoining variables to R 1 , and being a fibration means being an epimorphism.) In the case of i, there arises a diagram of quasiisomorphisms where Der(i) = {τ ∈ Der(S) s.t. τ (R 1 ) ⊂ R 1 }, π i is the obvious projection, and in i is the obvious embedding.
Analogously, in the case of p, there is a diagram of quasiisomorphisms where Der(p) = {τ ∈ Der(R 1 ) s.t. τ (Ker(p)) ⊂ Ker(p)}, in p is the obvious embedding, and π p is the obvious projection. Hinich defines  4) that, for any ξ ∈ V poiss (R), ξ (0) ∈ End(V poiss (R)) is a derivation of all products. Identity (2.2) implies that ξ (0) commutes with T . If, in addition, ξ is odd and ξ (0) ξ = 0, then (ξ (0) ) 2 = 0 as another application of (2.4) shows. Therefore, a pair (V poiss , ξ) is a differential Courant  If f : R 1 → R 2 is a homotopy equivalence, then is a vector space isomorphism by virtue of (4.6) and (4.16). In fact, (4.18) is a graded Courant algebroid isomorphism. This follows from the fact that the Courant algebroid structure on V poiss (R) consists of classical differential geometry operations, such as the tautological action of Der(R) on R and the action of Der(R) on Ω(R) by means of the Lie derivative. An inspection of maps (4.11)-(4.16) shows that Hinich's construction respects all these operations.
where R is a quasi-free DGA resolution of A.  Proof of Lemma.   A spectral sequence (E r pq , d r ) ⇒ Der(A) p+q arises so that Since R = (R # , D) is quasi-isomorphic to A placed in degree 0, we have It follows at once that the spectral sequence collapses and Der(A) −n is the n-th cohomology of the complex Item (a) of the lemma is thus proven. In order to prove item (b), we have to write down a formula for the differential of complex (4.23). The resolution R → A gives an exact sequence We shall regard an element τ ∈ (V j ) * as a derivation of R = S • V . The differential, D, of R can be written thus: D = f j ∂ j + ξ, where ∂ j ∈ (V 1 ) * , {f j } generate J, and ξ ∈ F 2 Der(R) −1 . It easily follows from the construction of the spectral sequence that if τ ∈ (V 0 ) * , then It follows at once that Ker{d 1 : (V 0 ) * ⊗ A → (V 1 ) * ⊗ A} is precisely the algebra of derivations of R 0 that preserve the ideal J modulo those derivations whose image is J, and this is Der(A) 0 by definition.
5 Quantization in the case of a Veronese ring 5.1. Set-up. Consider the Veronese ring It is known that Spec(A N ) is the cone over the highest weight vector orbit in the projectivization of the (N + 1)-dimensional representation of sl 2 . Hence the canonical Lie algebra morphism An explicit formula for this morphism will appear in Section 5.2.3 below. Being a cone, Spec(A N ) carries the Euler vector field j x j ∂ j . This allows us to extend (5.2) to an action of gl 2 : Remark. The normalizing factor of N is not particularly important but can be justified by the geometry of the base affine space SL 2 /N .

As in Section 3.7.4, this gives a Courant algebroid morphism
The following theorem, the main result of this paper, uses the concept of quantization of a Courant algebroid, see Definition 3.5, and the notion of quantization of a Courant algebroid map, see Section 3.7.4, (3.30), (3.31).     (3.12) in the present situation becomes Now suppose that only V(A N ) 0 is given.
The homological degree of the l.h.s. of these is non-positive, see (5.7), of the r.h.s. is non-negative; therefore, the operations may be non-zero only if both ξ, τ ∈ Der(A) 0 ; the uniqueness follows.
(ii) To prove the existence, note that Definition 3.2 differs from Definition 3.3 in the following two respects only: • the associativity of (−1) in the former is replaced with quasi-associativity (2.5) in the latter; • the requirements of the former that multiplications (n) , n ≥ 0, be derivations of multiplication (−1) and that multiplication (−1) be commutative are simultaneously replaced with the Jacobi identity (2.4) with m or n equal to −1.
Upon choosing a splitting as at the beginning of the proof, in each of the cases the identities of Definition 2.2 exhibit quantum corrections, i.e., the terms that measure the failure of a quantum object to be a classical one. It is easy to notice, by inspection, that in our situation the quantum corrections may be non-zero only if all the terms involved belong to V = V(A N ) 0 .
One such example is provided by formula (3.16), where the failure of multiplication (−1) to be associative is measured by −τ (g)df −τ (f )dg; both the summands vanish unless τ ∈ V, f , g being in V automatically.
Another example deals with the commutativity of (−1) . Let f ∈ A N , τ ∈ Der(A N ) • ; then (2.4) reads which is 0 unless τ ∈ V. We leave it to the untiring reader to check the validity of all the remaining requirements of Definition 3.3.
In order to prove Theorem 5.1.1, it remains to quantize V poiss (A N ) 0 . We shall do this in a somewhat roundabout manner.

Localization.
To return to the hypothetical vertex algebroid V(A N ) • . By virtue of Section 5.2.1, it is enough to consider V(A N ) (3.25) in Section 3.7.1, so as to get a sheaf V C ∈ Vert C . LetČ be C \ {0} and VČ the restriction of V C toČ. Apparently, VČ ∈ VertČ and, the manifoldČ being smooth, our strategy will be to use the classification of the objects of VertČ, Section 3.7.3, so as to identify those vertex algebroids overČ that may have come from C as above.
We begin by realizingČ as a quotient of a manifold w.r.t. a finite group action. Consider the action The map is an isomorphism; hence isomorphisms There arises a projection p : C 2 \ 0 →Č (5.11) and a faithful functor It is an equivalence of categories where Vert Z N C 2 \0 is the full subcategory of Z N -equivariant vertex algebroids; the inverse functor is that of Z N -invariants: V C 2 \0 → V Z N C 2 \0 ; cf. Section 3.7.1, (3.26), (3.27). The objects of the category Vert C 2 \0 were classified in Section 3.7.3 to the effect that there is a 1-1 correspondence between isomorphism classes of vertex algebroids and linear combinations ω = a,b>0 k ab ω ab , where ω ab is the 2-form dy 1 ∧ dy 2 /y a 1 y b 2 . It follows from the construction that the vertex algebroid, V C 2 \0,ω is Z N -equivariant if and only if ω is, hence if and only if ω = a,b>0,N divides a+b−2 k ab ω ab . (5.14) It is from this list that we have to select.

5.2.3.
Conclusion of the proof. By definition, our hypothetical sheaf V C must fit, for some ω, in the following commutative diagram: Note that the vertical arrows are all the restriction (from C toČ) maps. Furthermore, the rightmost vertical arrow is an equality. To see this, note that Γ(Č, TČ) = Γ(C 2 \ 0, The latter is generated, over functions, by the tautological action of gl 2 , see Section 3.7.4, formulas (3.32), (3.33). (Indeed, an element of Γ(C 2 \ 0, T C 2 \0 ) Z N is a linear combination of f (y 1 , y 2 )∂ 1 and g(y 1 , y 2 )∂ 2 , where N divides deg(f ) − 1 and deg(g) − 1. This implies that f (y 1 , y 2 )∂ 1 is proportional to either ρ(E 11 ) or to ρ(E 21 ) and g(y 1 , y 2 )∂ 2 is proportional to either ρ(E 12 ) or to ρ(E 22 ), see (3.33).) Therefore, so is the former. But this action is precisely the action of gl 2 on C = Spec(A N ) described somewhat implicitly in (5.2), hence the equality Γ(C, T C ) = Γ(Č, TČ). Contrary to this, the leftmost vertical arrow is not an equality; e.g. it is easy to check that This simple remark is the reason why the quantization of V poiss (A N ) is unique. Now, the upper row of (5.15) is exact by virtue of Definition 3.5. This and the fact that the rightmost arrow is an equality imply that the lower row must also be exact, at least on the right. By virtue of Corollary 3.7.6, ω = kω 11 . (5.17) Now our task is to determine k. Define W to be the vertex subalgebroid of Γ(Č, VČ ,kω 11 ) generated by A N = Γ(Č, OČ ) and ρ(V(gl 2 ) −k−1,k−1 ), whereρ is the one from Lemma 3.7.5.
Since Γ(C, T C ) is generated by ρ(gl 2 ) over A N , W = Γ(C, V C ). It is clear that W = A N ⊕ Γ(C, Ω C ) + A N (−1)ρ V ch (gl 2 ) −k−1,k−1 and were the elements ρ(E ij ) independent over A N , we would be done: W would be the sought after quantization for any k. (In fact, were that true, we could equivalently define V(A N ) to be the push-out A N ⊗ V(gl 2 ) −k−1,k−1 , see (3.28).) But they are not, and the problem with this is that an element of A N (−1)ρ V ch (gl 2 ) −k−1,k−1 may belong to Γ(Č, ΩČ) and not to Γ(C, Ω C ), cf. (5.16).
In fact, A N is a quadratic algebra, see (5.1), and Γ(C, T C ) is a quadratic A N -module generated by {E ij , 1 ≤ i, j ≤ 2}. The relations, in terms of y 1 , y 2 , are as it easily follows from (3.33). Our task then is to ensure that Let us consider for the sake of definiteness the case of i = 1, k = j = 2. We have to compute the following section of Γ(Č, VČ ,kω 11 (Č)):
The second one will likewise give Adding one to another makes expression (5.20) into The latter equals the total derivative and is therefore an element of Γ(C, Ω C ), precisely when k = N + 1. The case where i = j = 1, k = 2 works out similarly and gives the same answer k = N + 1. This concludes the proof of item (a) of Theorem 5.1.1.
As to item (b), (5.5) follows from the k = N + 1 case of Lemma 3.7.5, the assertion that has been instrumental for the proof anyway, and (5.6) follows from (3.40). be the classical Veronese embedding. By A nN let us denote the homogeneous coordinate ring of ι N (P(V )). It is clear that if n = 2, then A nN is the algebra A N we dealt with above. It is now natural to ask if V poiss (A nN ) affords a quantization. The result is a bit disheartening. Proof . Consider the action Analogously to (5.9), (5.10), we obtain isomorphisms Thus we are led to the question, "How many vertex algebroids are there on (V \ 0)/Z N ?" That such vertex algebroids exist is obvious because V Z N V \0 is one; here V V \0 is the pull-back of the standard V V , cf. Section 3.7.2, on V \ 0. Note that if x 1 , . . . , x n is a basis of C n -remember that we think of C n as the space of linear functions on V -then the assignment defines a vertex algebroid morphism The next differential is . Hence any such quantization can be equal only to the vertex subalgebroid of Γ(V \ 0, V V \0 ) Z N generated by A nN andρ(V(gl n ) −1,−1 , see (5.24-5.25). But this subalgebroid necessarily contains elements from Γ((V \ 0)/Z N , Ω (V \0)/Z N ) \ Ω(A nN ). Indeed, a computation analogous to the one performed at the end of Section 5.2.3 shows that if n ≥ 3, then This concludes the proof of Theorem 5.3.1.

Chiral Hamiltonian reduction interpretation
We will now interpret some of the constructions above in the language of semi-infinite cohomology. Our exposition will be brief and almost no proofs will be given. In some respects, the material of this section is but an afterword to [16].
Since we will be mostly concerned with smooth varieties, we will find it convenient to work not with vertex algebroids, such as V(g) k , V X , but with the corresponding vertex algebras or CDO-s, such as U alg V(g) k = U L(g) k , D ch X = U alg V X .
Introduce the Clifford vertex algebra built on Π(g ⊕ g * ), Π being the parity change functor. This vertex algebra is nothing but the space of global sections of the standard CDO on superspace Π(g⊕g * ), see Section 3.7.2 for a discussion of a purely even analogue. Denote this vertex algebra by D ch (Π(g ⊕ g * )).
By definition, if we let {x i } be a basis of g, {φ i } the corresponding basis of Π(g), {φ * i } the dual basis of Π(g * ), then D ch (Π(g ⊕ g * )) is the vertex algebra generated by the vector space Π(g ⊕ g * ) and relations There arises the vertex algebra V ⊗ D ch (Π(g ⊕ g * )).
If {c k ij } are the structure constants of g relative to {x i }, that is to say, if [x i , x j ] = k c k ij x k , then following [13] one considers the element .
A direct computation shows that where K(·, ·) is the Killing form on g: K(a, b) = tr(ad a · ad b ).
If condition (6.2) is satisfied, then we obtain a DGVA C ∞/2 (Lg; V) (0) ). The cohomology vertex algebra H ∞/2 (Lg; V) is due to Feigin [13] and well known as either semi-infinite or BRST cohomology of the loop algebra Lg with coefficients in V. If one chooses to think of V as an algebra of ('chiral') functions on a symplectic manifold with gstructure, then H ∞/2 (Lg, V) is to be thought of as an algebra of functions on the symplectic quotient M//g, hence the title of this section.
One similarly defines the relative version H ∞/2 (Lg, g; V), see [16] for some details; the condition (6.2) remains the same in this case.
6.2. The sl 2 case. Let us return to the set-up of Section 5.2.2, where we had the Veronese cone C = Spec(A N ),Č = C \ 0, and consider L N , the degree N line bundle over P 1 , anď L N = L N \ {the zero section}. We obtain the commutative square where the upper horizontal map is a surjective birational isomorphism, a blow-up of the vertex of the cone. We have seen thatČ carries a family of CDO-s, D cȟ C,ω , ω ∈ H 1 (Č, Ω 2Č ). Denote the coorresponding family of CDO-s onĽ N by D cȟ L N ,ω . Theorem 5.1.1 says that D cȟ C,ω is a pull-back of a CDO on C iff ω = (N − 1)ω 11 , in which case it contains V(sl 2 ) −N −2 . Now a question arises, "For what, if any, ω is D cȟ L N ,ω a pull-back of a CDO from L N ?" The existence of such ω depends on the vanishing of the characteristic class ch 2 (L N ), (3.21). A simple way to prove the vanishing result, and to compute a possible ω, is provided by the semi-infinite cohomology.
To conclude, (1) the manifoldĽ N carries a 1-parameter family of CDO-s, D ch C 2 \0,−(k+1)ω 11 , with U alg V(sl 2 ) (·,·) -structure, see Section 3.7.4; (2) the condition that a CDO onČ extends to one on C picks a unique (·, ·); the latter depends on N linearly, see Theorem 5.1.1; (3) the family D ch C 2 \0,−(k+1)ω 11 contains at least one representative that extends to a CDO on L N ; the condition that a CDO onĽ N extends to L N and affords a realization via the chiral Hamiltonian reduction picks a unique (·, ·); the latter depends on N quadratically, (6.9).
In fact, there is a third way to fix a (·, ·). This one amounts to carrying a regularization procedureà la Lambert, used in [4] in a similar but different situation, and gives another quadratic dependence on N . The importance of this approach is yet to be worked out. Indeed, there is at least one CDO on G/Q that can be defined via the chiral Hamiltonian reduction as follows. By analogy with Section 6.2, since (6.4) holds true with SL 2 replaced with an arbitrary simple complex Lie group G, we observe that if q = Lie(Q), then H ∞/2 (Lq, q; D ch G,(·,·) ) is well defined for precisely one choice of (·, ·). In fact, in this case condition (6.2) amounts to the requirement that the restriction (·, ·) to M 1 be equal to the Killing form on M 1 , and there is one and only one way to achieve that by appropriately rescaling (·, ·) -this is where assumption (6.10) is crucial. It is rather clear that H ∞/2 (Lq, q; D ch G,(·,·) ) is a CDO on G/Q. As a consequence, we obtain a vertex algebra morphism way to prove the vanishing of the 2nd component of the Chern character, originally verified by Nekrasov [24].
Needless to say, our discussion is very close in spirit to the definition of Wakimoto modules due to Wakimoto and Feigin-Frenkel, see [11] and references therein. In fact, it is easy to see that the spaces of sections over 'the big cell' of the sheaves constructed in Section 6.2 contain Wakimoto modules over sl 2 , and those of the present section contain the so-called generalized Wakimoto modules corresponding to R.