On chiral differential operators over homogeneous spaces

In this note we study algebras of chiral differential operators over an algebraic group and over homogeneous spaces $G/G'$ where $G$ is simple and $G'$ is unipotent or parabolic.


Introduction
The notion of an algebra of chiral differential operators (cdo for short) over a smooth algebraic variety X has been studied in [GII]. (This notion has been invented and first studied, in a different language, by Beilinson and Drinfeld,cf. [BD1], Chapter 3, §8). In the present note we consider some examples in more detail. We shall work over the ground field C.
First, we give a classification of cdo over X in the following cases: X = G is an affine algebraic group; X = G/N or G/P where N is a unipotent subgroup and P is a parabolic subgroup and G is simple (the extension to the case of a semisimple G being straightforward).
Before we describe the result, let us explain some terminology and notation. For a smooth algebraic variety X, an algebra of cdo over X is by definition a Zariski sheaf V of Z ≥0 -graded vertex algebras on X such that (a) if Alg(V) = (A, T , Ω, ∂, γ, , , c) is the sheaf of vertex algebroids associated with V (see [GII], §2), then the corresponding extended Lie algebroid (A, T , Ω, ∂) (op. cit., 1.1, 1.4) is identified with (O X , Θ X , Ω 1 X , d DR ) where Θ X denotes the tangent bundle and d DR the de Rham differential; (b) the adjunction morphism U Alg(V) −→ V is an isomorphism. Here U is the functor of vertex envelope defined in op. cit., §9.
For each Zariski open U ⊂ X we can consider the category (a groupoid in fact) of cdo over U , or, what is the same, the groupoid of vertex algebroids (defined in [GII], §3) over U satisfying (a) above. When U varies, we get a sheaf of groupoids Dif f ch X over X -the gerbe of chiral differential operators. As usual, Γ(U ; Dif f ch X ) will denote the sections over U ; a generic object of this category will be sometimes denoted D ch U ; the set of isomorphism classes of cdo over U will be denoted by π 0 (Γ(U ; Dif f ch X )). Let G be an affine connected algebraic group, g the corresponding Lie algebra. For each symmetric ad-invariant bilinear form (, ) ∈ (S 2 g * ) g we construct a cdo D ch G; (,) over G such that if G is semisimple, then the correspondence (, ) → D ch G; (,) gives rise to a bijection We have a canonical embedding of vertex algebras where V g;(,) denotes the vacuum module of the Kac-Moody algebra g at level (, ). This embedding is induced by the embedding of g into T G := Γ(G; Θ G ) as left invariant vector fields.
It is characterised by the requirement that the images of i (,) and i o (,) commute in an appropriate sense (see Theorem 2.5 and Corollary 2.6). This beautiful fact was communicated to us by B.Feigin, E.Frenkel and D.Gaitsgory. We give a proof using the language of [GII].
Let us pass to homogeneous spaces. Assume that G is simple. Let N ⊂ G be a unipotent group. The classification of cdo over G/N is the same as over G; namely, for each level (, ) one can define a cdo D ch G/N ;(,) such that the correspondence (, ) → D ch G/N ;(,) induces a bijection The sheaves D ch G/N ;(,) are constructed using the BRST (or quantum Hamiltonian) reduction of the corresponding cdo's on G. More precisely, where the rhs denotes the BRST cohomology of the loop algebra Ln := n[T, T −1 ], n := Lie(N ). For the precise definition see §3.
Let B ⊂ G be a Borel subgroup. We show that there exists a unique, up to a unique isomorphism, cdo D ch G/B on the flag space G/B. Again this cdo may be constructed using the BRST reduction. Namely, Here D ch G;crit is by definition the cdo D ch G;(,)crit on the critical level (, ) crit = −(, ) g;(K) /2. For the definition of the relative BRST cohomology in the rhs we again refer the reader to the main body of the note, see §4. (A more explicit construction of the sheaf D ch G/B for G = SL(n), using vertex operators, has been suggested in [MSV],.) The embeddings (I4) induce canonical morphisms of vertex algebras Taking the spaces of sections over a big cell, we get another construction of Feigin-Frenkel Wakimoto modules, cf. [FF1] - [FF3].
Finally, if P ⊂ G is parabolic but not Borel, we show that Γ(G/P ; Dif f ch G/P ) is empty. The classification of cdo over homogeneous spaces is exactly reflected in the BRST world: namely the square of the corresponding BRST charge is zero at all levels for G/N , only at the critical level for G/B and is never zero for G/P . This introduction would not be complete without mentioning that this note relies heavily on the ideas of B.Feigin and E.Frenkel. This note started from our attempts to find a proof of 2.5 and 2.6. Our sincere gratitude goes to D.Gaitsgory who had communicated these facts to us and told us that he had known their proofs. We are also grateful to H.Esnault for a crucial remark 4.1.1.
This work was done while the authors were visiting Institut des HautesÉtudes Scientifiques in Bures-sur-Yvette and Max-Planck-Institut für Mathematik in Bonn. We are grateful to these institutes for the hospitality. §1. Chiral differential operators over an algebraic group

Perfect vertex algebroids over constants
The discussion below is nothing but the specification of [GII], § §1 -4 to the case A = C.
1.1. Let g be a Lie algebra. We shall need two complexes connected with g, both concentrated in nonnegative degrees. The first one, C · (g) = C · (g; C), is the cochain complex of g with trivial coefficients. Thus, by definition C i (g) = (Λ i g) * = the space of skew symmetric polylinear maps f : g i −→ C, i ≥ 0.
The second complex,C · (g), is the shifted by 1 and augmented cochain complex of g with coefficients in the coadjoint representation g * . By definition,C 0 (g) = C andC i (g) = Hom k (Λ i−1 g, g * ) = the space of skew symmetric polylinear maps h : g i−1 −→ g * for i ≥ 1.
One checks that the embeddings (1.1.3) are compatible with the differentials, so that one has an embedding of complexes C · (g) ֒→C · (g).
1.2. Let us consider the groupod Alg g of vertex algebroids of the form A = (C, g, g * , ∂, γ, , , c) where T = (C, T, Ω, ∂) = (C, g, g * , 0) is a perfect extended Lie algebroid over C, in the sense of [GII], 1.2, with T = g. Note that the last object is uniquely defined by the Lie algebra g = T ; we must have Ω = g * , the "Lie derivative" action of T on Ω must be the coadjoint one, and a C-linear derivation ∂ : C −→ Ω must be zero.
Turning to the axioms of a vertex algebroid, op. cit., 1.4, we see that for A as above, , : g × g −→ C is a symmetric bilinear map (which may be regarded as an element ofC 2 (g)), c ∈C 3 (g), (A1) implies that γ = 0, (A2) and (A3) hold true automatically, (A4) takes the form and (A5) takes the form where d is the differential inC · (g) given by (1.1.2).
see [GII], Theorem 3.5. The composition of morphisms is induced by the addition inC 2 (g).
1.3. As a corollary, we have a canonical bijection More precisely, for a 3-cocycle c ∈ C 3,cl (g) we have a vertex algebroid and the correspondence c → A g;c induces the bijection (1.3.1).
The enveloping algebra V g;c := U A g;c is generated by the same fields as in 1.2, subject to OPE and (1.2.6).
Let us define another interesting class of objects of Alg g . Namely, each symmetric ad-invariant bilinear form (, ) ∈ (S 2 g * ) g gives rise to an object The enveloping algebra V g;c := U A g;c is generated by the same fields as in 1.2, subject to OPE and (1.2.6).
It is easy to see that we have an isomorphism given by a map h (,) : g −→ g * where and the cocycle c (,) is defined by Cf. [GII], Theorem 4.5.
The correspondence τ · T n → τ (n) defines on V g;(,) a structure of the vacuum module over the Kac-Moody algebra g = g[T, T −1 ] ⊕ C · 1 at level (, ).
We have an obvious embedding of vertex algebroids A g;(,) ֒→ A g;0 which induces an embedding of vertex algebras Therefore, in this case the algebroids A g;(,) form a complete set of representatives of isomorphism classes in Alg g . In other words, Passing to a group 1.6. Let G = Spec(A) be an affine algebraic group, g the corresponding Lie algebra . The tangent bundle Θ G is trivial, so the obstruction c(Dif f ch G ) to the existence of an algebra of chiral do over G, D ch G ∈ Γ(G; Dif f ch G ) (cf. [GII], Corollary 7.11) vanishes.
By op. cit., §4, the set of isomorphism classes of chiral do over G, π 0 (Γ(G; Dif f ch G )) is a nonempty torseur under the "Chern-Simons group" H 3 DR (G) = H 3 (G; C). In fact the groupoid Γ(G; Dif f ch G ) has a distinguished object D ch G;0 , so that we have a canonical bijection This is a consequence of the following general construction.
1.7. Let A g; , ,c be an arbitrary object of Alg g . Let us apply to it the pushout construction of [GII], 1.10 with respect to the structre morphism C −→ A. Here the morphism g −→ T := Der C (A) is defined as the embedding of left invariant vector fields, and the map γ : A × g −→ Ω := Ω 1 (A) is set to be zero. This way we get a vertex A-algebroid A G; , ,c . Its enveloping algebra We have a canonical embedding We shall use the notations A G;(,) := A G;(,),0 , A G;c := A G;0,c , A G;0 := A G;0,0 and D ch G;(,) , etc. for the corresponding enveloping algebras. If A G;0 = (A, T, Ω, d DR , γ 0 , , 0 , c 0 ) and ω ∈ Ω 3,cl (A) is a closed 3-form then we can form a vertex algebroid The correspondence ω → A G;ω induces the bijection (1.6.1).
If c ∈ C 3,cl (g) is a 3-cocycle with trivial coefficients then by definition where ω c ∈ Ω 3,cl (A) is the left invariant 3-form on G corresponding to c.
Indeed, one knows that for a reductive group the correspondence c → ω c gives rise to an isomorphism H 3 (g) This follows from 1.8 and the remarks 1.5.
1.10. Note that for an arbitrary G and (, ) ∈ (S 2 g * ) g one has a canonical embedding It is the composition of (1.4.2) and (1.7.2). §2. Dual embedding 2.1. Let G = Spec(A) be a smooth affine connected algebraic group with the Lie algebra g. Pick a symmetric ad-invariant bilinear form ("level") (, ) ∈ (S 2 g * ) g .
Let (, ) (K) denote the Killing form on g, Let us pick a base {τ i } of g. In terms of structure constants If we want to stress the dependence on g, we shall write (, ) g;(K) , (, ) g;crit .
2.2. We have two commuting left actions of G on itself: the left multiplication, (g, x) → gx and the right one, (g, x) → xg −1 .
Let T = Der k (A) denote the Lie A-algebroid of vector fields over G. The above two actions induce two embeddings of Lie algebras Below we shall identify g with its image under i L , i.e. write simply x instead of i L (x). We shall also use the notation Embedding i L induces an isomorphism of left A-modules in coordinates. We have for all τ, τ ′ ∈ g. such that (a) the composition of (2.5.1) with the canonical projection D ch G;(,)1 −→ T is equal to i R ; (b) for all τ, τ ′ ∈ g and n ≥ 0 for each τ, τ ′ ∈ g.
2.6. Corollary. (B.Feigin -E.Frenkel, D.Gaitsgory) The map (2.5.1) induces an embedding of chiral algebras The images of j L and j R commute in the following sense: for each x ∈ V g;(,) , y ∈ V g;(,) o and n ≥ 0.
Evidently τ i(n) j R (τ j ) = 0 for n ≥ 2. This proves part (i) of the theorem.
2.9. Let us compute j R (τ i ) (1) j R (τ j ). We have This is a sum of four terms.
I := τ R i(1) τ R j = a ip τ p , a js τ s = using [GII] ( Using (2.2.6) and (2.1.4) we see that −a ip τ s τ p (a js ) = −a js τ p τ s (a ip ) = τ p (a js )τ s (a ip ) = (τ p , τ s ) (K) a ip a js , Next, Similarly, III := τ R i(1) (τ s , τ u ) o a js ω u = II and evidently Let us differentiate this expression. We have ) o a ip a jv = 0 Therefore, (2.9.1) is a constant. It may be computed by noticing that the matrix (a ij ), considered as a function on the group G, is equal to the identity at the identity of the group. Hence (2.9.1) is equal to (τ i , τ j ) o , which proves (2.5.3).

Let us compute
Let us compute the first summand. Using (2.8.1), we have by (2.8) and (2.7).
On the other hand which proves (2.5.4) and part (ii) of the theorem. △ 11. Proof of (2.6). The first claim is a reformulation of (2.5.3) and (2.5.4).
Let a be a finite dimensional Lie algebra. Choose a base {a i } in a; denote the structure constants [a i , a j ] = c ij p a p (3.1.1) Recall that the Killing form (, ) (K) : a × a −→ C is given by Let Πa be the space a with the reversed parity; denote by {φ i = Πa i } the corresponding base and by {φ * j } the dual base of Πa * given by Let C BRST (La) denote a graded vertex superalgebra generated by odd fields φ i (z) of conformal dimension 1 and odd fields φ * i (z) of conformal dimension 0 with OPE We shall identify the spaces Πa and Πa * with their obvious images in C BRST (La) 1 and C BRST (La) 0 respectively.
Let us introduce an odd element D a ∈ C BRST (La) of conformal dimension 1 by Thus, we have the corresponding field D a (z) = D a;n z −n−1 and we set d a := D a;0 (3.1.6) The pair (C BRST (La), d a ) may be regarded as a chiral analogue of the Chevalley cochain complex C(a). However, in the chiral case the square d 2 a may be nonzero. It is easy to compute it. Namely, let us write down the OPE D a (z)D a (w) using Wick theorem. We have 3.2. Corollary. If the Lie algebra a is nilpotent then d 2 a = 0.
Indeed, the Killing form of a nilpotent Lie algebra is zero.
3.3. Let (, ) : a × a −→ C be an arbitrary symmetric invariant bilinear form ("level"). Recall that the vertex algebra V a;(,) is generated by even fields a i (z) of conformal weight 1, subject to OPE defines an embedding of vertex algebras According to the previous lemma this space is canonically a graded (by conformal weight) V a;0 -supermodule. This space is also graded by "fermionic charge" where we assign to φ i (resp. φ * i , m ∈ M) the charge −1 (resp., 1, 0). Introduce an odd element D a;M ∈ C BRST (La; M) of conformal weight 1 and fermionic charge 1 by It follows from (3.1.7) that The pair (C BRST (La; M), d a;M ) is called the BRST complex of La with coefficients in M, and its cohomology H * BRST (La; M) is called the BRST cohomology. 3.6. Example. Let N be a unipotent algebraic group with the Lie algebra n. Consider a V n;0 -module D ch N ;0 (note that according to 1.6 this algebra represents a unique isomorphism class of cdo's over N ). Inside the loop algebra Ln = n[T, T −1 ] consider two Lie subalgebras: n − and n + , generated by all elements τ T n (τ ∈ n) with n < 0 and n ≥ 0 respectively. Then D ch N ;0 is a free n − -module and a cofree (i.e. the dual module is free) n + -module. cf. [FF3], p. 178. §4. Homogeneous spaces 4.1. Let X be a smooth variety. We have an exact triangle

4.2.
Let G be a simple algebraic group. In this section we shall discuss the chiral differential operators on homogeneous spaces G/G ′ where G ′ = N -a unipotent subgroup, G ′ = P -a parabolic but not minimal parabolic, or G ′ = B -a Borel subgroup.
The projection π is an affine morphism which is a Zariski locally trivial bundle with fiber N isomorphic to an affine space, so π * : H * (X; C) We have a short exact sequence (4.3.4) and the vector bundles Θ G , Θ G/X are trivial (a base of global sections of Θ G/X is given by left invariant vector fields coming from the Lie algebra n := Lie(N )).
Therefore we have π * c(Dif f ch X ) = π * ch 2 (Θ X ) = ch 2 (Θ G ) = 0 (4.3.5) hence c(Dif f ch X ) = 0 (4.3.6) The Killing form on g restricts to zero on n (since the trace of a nilpotent endomorphism is zero). Therefore we have the canonical embedding of vertex algebras so that the sheaf D ch G;(,) becomes a sheaf of V n;0 -modules. Applying the BRST construction 3.5 to M = π * D ch G;(,) and a = n we get a sheaf of BRST complexes C BRST (Ln; π * D ch G;(,) ) and BRST cohomology sheaves H * BRST (Ln; π * D ch G;(,) ) over X. They are sheaves of Z ≥0 -graded vertex superalgebras.
Note that higher direct images R i π * D ch G;(,) are trivial for i > 0 since the morphism π is affine and the sheaves D ch G;(,) admits a filtration whose qutients are coherent (in fact, locally free) O G -modules. 4.7. Proof (sketch). Locally on X the projection π : G −→ X is isomorphic to the direct product U × N −→ U . If D is an algebra of chiral differential operators on U × N then D ∼ −→ D U ⊠ D N where D U (resp. D N is an algebra of differential operators on U (resp. N ). Now the first claim of the theorem follows from (3.6.1).
The second claim is a corollary of 4.4. Indeed, we know that (4.8.1) commutes with the left action of g hence with the BRST differential.
In particular, the Kac-Moody algebra g at level (, ) o acts canonically on Wakimoto modules which may be defined as the spaces of sections Γ(U ; D ch X,(,) ) where U is a big cell. This is a result due to Feigin-Frenkel obtained in [FF1] - [FF3] in a different way.
The case G/B 4.9. Let B ⊂ G be a Borel subgroup, π : G −→ X := G/B. X is a smooth projective variety and we have H p (X; Ω q X ) = 0 (p = q) (4.9.1) and H i (X; Ω i X ) = H 2i (X; C) (4.9. 2) It follows that H 2 (X; Ω [2 ) = H 4 (X; C), and [GII], Theorem 7.5 says that the image of c(Dif f ch X ) in H 4 (X; C) is equal to where c i (Θ X ) ∈ H 2i (X; C) are the Chern classes of the tangent bundle Θ X .
4.9.1. Lemma. We have ch 2 (Θ X ) = 0 (4.9.1.1) Proof. The space H 2 (X; C) may be identified with the complexification of the root lattice of G. The classical theorem by J. Leray says that that the cohomology algebra H * (X; C) is equal to the quotient of the symmetric algebra of the space H 2 (X; C) modulo the ideal generated by the subspace of invariants of the Weyl group W having positive degree, cf. [L], Théoreme 2.1 b).
The class [Θ X ] in the Grothendieck group of vector bundles is equal to the sum [L α ] where α runs through all negative roots, and L α is the line bundle with c 1 (L α ) = α. Hence ch 2 (Θ X ) = α 2 ; this element is invariant under the action of W , therefore its image in H 4 (X; C) is zero. △

This lemma implies that
c(Dif f ch X ) = 0 (4.9.4) by 4.1.1. On the other hand, it follows from 4.1.1 and (4.9.1) that Thus, by [GII], loc. cit. we get 4.10. Theorem. The groupoid Γ(X; Dif f ch X ) is nonempty and trivial. In other words, there exists a unique, up to a unique isomorphism, algebra of chiral differential operators D ch X over X. △ 4.11. Let us construct the algebra D ch X using BRST reduction. Consider the sheaf D ch G;(,) as in 4.5. Let (, ) b denote the restriction of (, ) to b := Lie(B). We have a canonical embedding of vertex algebras We have (, ) g;(K) | b×b = 2(, ) b;(K) (4.11.2) Therefore, to get the minus Killing on b we have to start from −(, ) g,(K) /2, i.e. from the critical level on g. Thus, by construction 3.5, 4.11.1. the BRST complex C BRST (Lb; π * D ch G;(,) ) is defined iff (, ) = (, ) g;crit , i.e. on the critical level, cf. (2.1.5).
Let us denote the algebra D ch G;(,)g;crit by D ch G;crit . 4.12. Let V b;crit denote the vacuum module on the critical level V b;(,) b;crit . Let M be a vertex algebra equipped with a morphism of vertex algebras V b;crit −→ M.
By definition, the subspace C BRST (Lb, h; M) is spanned by all monomials (4.12.3) such that (a) all j p ≥ 1; (b) α p − α ′ q + µ = 0. It is a vertex subalgebra. One checks that the BRST differential d in C BRST (Lb; M) preserves C BRST (Lb, h; M).
We define the relative BRST cohomology H * BRST (Lb, h; M) as the cohomology of C BRST (Lb, h; M) with respect to this differential. It is canonically a vertex algebra.

4.13.
Applying the previous definition to M = π * D ch G;crit we get the BRST cohomology sheaves H * BRST (Lb, h; π * D ch G;crit ) on X. 4.14. Theorem. We have H i BRST (Lb, h; π * D ch G;crit ) = 0 (i = 0) (4.14.1) and the sheaf of vertex algebras H 0 BRST (Lb, h; π * D ch G;crit ) is canonically isomorphic to D ch X . The proof is the same as that of Theorem 4.6.
4.15. Corollary. We have a canonical isomorphism of vertex algebras H * (X; D ch X ) = H * BRST (Lb, h; Γ(G; D ch G;crit )) (4.15.1) Indeed, as we have already remarked, sheaves of cdo D ch G on G have a filtration whose graded quotients are vector bundles; but the vatiety G and the morphism π are affine, hence H i (G; D ch G ) = R i π * D ch G = 0 for i > 0.  Indeed, we know that (4.16.1) commutes with the left action of g hence with the BRST differential.
In particular, the Kac-Moody algebra g on the critical level acts canonically on Wakimoto modules which may be defined as the spaces of sections Γ(U ; D ch X ) where U is a big cell. This is a result of Feigin-Frenkel proven in [FF1] - [FF3] in a different way.
The case G/P 4.17. Let P ⊂ G be a parabolic but not Borel, p = Lie(P ), π : G −→ X := G/P . The discussion of 4.9 applies as it is, except for Lemma 4.9.1, which is replaced by 4.17.1. Lemma. We have ch 2 (Θ X ) = 0 (4.17.1.1) Hence c(Dif f ch X ) = 0, and we get 4.18. Theorem. The groupoid Γ(X; Dif f ch X ) is empty. In other words, there is no cdo over X. △ 4.19. Let us see how this fact is reflected in the BRST world. We would like to get a sheaf of cdo over X as the BRST cohomology of Lp with coefficients in some sheaf π * D ch G; (,) . However, no form (, ) on G restricts to the Killing form on p, which implies that the square of the BRST differential in C BRST (Lp; π * D ch G;(,) ) is always nonzero. Thus, the BRST cohomology is not defined, which is compatible with 4.18.