Chiral de Rham Complex

Commun. Math. Phys. 204, 439 – 473 (1999)

Communications in

Mathematical Physics

© Springer-Verlag 1999

Chiral de Rham Complex Fyodor Malikov1,2 , Vadim Schechtman3 , Arkady Vaintrob4 1 Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn, Germany.

E-mail: [email protected]

2 Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA.

E-mail: [email protected]

3 Department of Mathematics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK.

E-mail: [email protected]

4 Department of Mathematics, New Mexico State University, Las Cruces, NM 88003-8001, USA.

E-mail: [email protected] Received: 29 April 1998 / Accepted: 7 February 1999

Abstract: We define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi-infinite de Rham complexes of certain D-modules over the loop spaces. For Calabi–Yau manifolds they admit N = 2 supersymmetry. Connection with Wakimoto modules is discussed. Introduction 0.1. The aim of this note is to define certain sheaves of vertex algebras on smooth manifolds. In this note, “vertex algebra” will have the same meaning as in Kac’s book [K]. Recall that these algebras are by definition Z/(2)-graded. “Smooth manifold” will mean a smooth scheme of finite type over C. For each smooth manifold X, we construct a sheaf ch X , called the chiral de Rham complex of X. It is a sheaf of vertex algebras in the Zarisky topology, i.e. for each open U ⊂ X, 0(U ; ch X ) is a vertex algebra, and the restriction maps are morphisms of vertex algebras. It comes equipped with a Z-grading by fermionic charge, and the chiral de ch , which is an endomorphism of degree 1 such that (d ch )2 = 0. Rham differential dDR DR One has a canonical embedding of the usual de Rham complex ch (X , dDR ) ,→ (ch X , dDR ).

(0.1)

The sheaf ch X has also another Z≥0 -grading, by conformal weight, compatible with ch respects conformal weight, and the subfermionic charge one. The differential dDR complex X coincides with the conformal weight zero component of ch X . The wedge multiplication on X may be restored from the operator product in ch X , see 1.3. The map (0.1) is a quasi-isomorphism, cf. Theorem 4.4. Each component of ch X of fixed conformal weight admits a canonical finite filtration whose graded factors are symmetric and exterior powers of the tangent bundle TX and of the bundle of 1-forms 1X .

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Similar sheaves exist in complex-analytic and C ∞ settings. If X is Calabi–Yau, then the sheaf ch X has a structure of a topological vertex algebra (i.e. it admits N = 2 supersymmetry, cf. 2.1), cf. 4.5. (For an arbitrary X, the obstruction to the existence of this structure is expressible in terms of the first Chern class of TX , cf. Theorem 4.2.) One may hope that the vertex algebra R0(X; ch X ) defines the conformal field theory which is Witten’s “A-model” associated with X. The intuitive geometric picture behind our construction is as follows. Let LX be the space of “formal loops” on X, i.e. of the maps of the punctured formal disk to X. Let L+ X ⊂ LX be the subspace of loops regular at 0. Note that we have a natural projection L+ X −→ X (value at 0). We have a functor p : (Sheaves on LX) −→ (Sheaves on X), namely, if F is a sheaf on LX, then 0(U ; p(F)) = 0(LU ; F ) for an open U ⊂ X. Now, the sheaf ch X is the image under p of the semi-infinite de Rham complex of the D-module of δ-functions along L+ X. This sheaf is a particular case of a more general construction which associates with every D-module M over X its “chiral de Rham complex” ch X (M) which is a sheaf of . Its construction is sketched in §6, cf. 6.11. vertex modules over the vertex algebra ch X ch of vertex algebras, which could 0.2. One can also try to define a purely even sheaf OX be called chiral structure sheaf. Here the situation is more subtle than in the case of ch X , where “fermions cancel the anomaly”. One can define this sheaf for curves, cf. 5.5. If dim(X) > 1, then there exists a non-trivial obstruction of cohomological nature to the ch . This obstruction can be expressed in terms of a certain homotopy construction of OX Lie algebra, cf. §5 A. ch for the flag manifolds X = G/B (G being a simple However, one can define OX algebraic group and B a Borel subgroup), cf. Sect. 5 B, C. This sheaf admits a structure of ab g-module at the critical level (here g = Lie(G) andb g is the corresponding affine Lie ch ) is the irreducible vacuum b g-module algebra). The space of global sections 0(X; OX ch for g = sl(2); conjecturally, it is also true for any g, cf. 5.13. The sheaf OX may be regarded as localization of Feigin-Frenkel bosonization. More generally, if we start from an arbitrary D-module M on X = G/B correponding to some g-module M, then we can define its “chiralization” Mch which is an Och module. It seems plausible that the space of global sections 0(X; Mch ) coincides with the Weyl module over b g corresponding to M (on the critical level).

0.3. A few words about the plan of the note. In Sect. 1 we recall the basic definitions, to fix terminology and notations. In Sect. 2, some “free field representation” results are described. No doubt, they are all well known (although for some of them we do not know the precise reference). They are a particular case of the construction of ch X , given in Sect. 3. In Sect. 4, we discuss the topological structure, and in Sect. 5 the chiral structure sheaf. In Sect. 6 we outline another construction of our vertex algebras, and some generalizations.

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1. Recollections on Vertex Algebras For more details on what follows, see [K]. For a coordinate-free exposition, see [BD1]. 1.1. For a vector space V , V [[z, z−1 ]] will denote the space of all formal sums P i i∈Z ai z , ai ∈ V . V ((z)) will denote the subspace of Laurent power series, i.e. the sums as above, with ai = 0 for i > 0. The meaning of this equality is explained in [K]. Given two fields a(z), b(z), their operator product expansion is defined as in [K], (2.3.7a) N X cj (w) + : a(z)b(w) :, a(z)b(w) = (z − w)j j =1

where

˜

: a(z)b(w) := a(z)+ b(w) + (−1)a˜ b b(w)a(z)− .

Here

a(z)− = P

X n≥0

z−n−1 ,

a(n) z−n−1 ; a(z)+ =

X

(1.2)

a(n) z−n−1

n 0. Let HN be the Lie algebra which as a vector space has the base ani , bni , i = 1, . . . , N; n ∈ Z, and C, all these elements being even, with the brackets j

[am , bni ] = δij δm,−n C,

(1.9)

all other brackets being zero. Our vertex algebra, to be denoted VN , as a vector space is the vacuum representation of HN . As an HN -module, it is generated by one vector 1, subject to the relations i 1 = 0 if m ≥ 0; C1 = 1. bni 1 = 0 if n > 0; am

The mapping

j

(1.10)

j

P (bni , am ) 7 → P (bni , am ) · 1 j

identifies VN with the ring of commuting polynomials on the variables bni , am , n ≤ 0, m < 0, i, j = 1, . . . , N. We will use this identification below. Let us define the structure of a vertex algebra on VN . The Z/(2) grading is trivial: everything is even. The vacuum vector is 1. The fields corresponding to the elements of VN are defined by induction on the degree of a monomial. First we define the fields for the degree one monomials by setting X X j j bni z−n ; a−1 (z) = an z−n−1 . (1.11) b0i (z) = n∈Z

n∈Z

j

Here bni , an in the right hand side are regarded as operating on VN by multiplication. We set j i i (z) = ∂z(n) b0i (z), a−n−1 (z) = ∂z(n) a−1 (z) (1.12) b−n for n > 0. The fields corresponding to the monomials of degree > 1 are defined by induction, j using the normal ordering. Let us call the operators bni , n > 0, and an , n ≥ 0, acting on VN , annihilation operators. j For x = bni or an , n ∈ Z, and b ∈ End(VN ), the normal ordered product : xb : is given by  bx if x is an annihilation operator : xb := . (1.13) xb otherwise Define by induction : x1 · . . . · xk :=: x1 · (: x2 · . . . · xk :) :,

(1.14)

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F. Malikov, V. Schechtman, A. Vaintrob j

for xp = bni or an , p = 1, . . .P , k. P Given two series x(z) = n∈Z xn z−n+p and y(z) = n∈Z yn z−n+q , with xn as above, we set X : x(z)y(w) := : xn ym : z−n+p w −m+q . (1.15) n,m∈Z

For any finite sequence x1 , . . . , xp , where each xj is equal to one of ani or bni , we define the series : x1 (z)x2 (z) · . . . · xp (z) :∈ End(VN )[[z, z−1 ]] by induction, as in (1.14). This expression does not depend on the order of xi ’s. Given a monomial x1 · . . . · xp 1 ∈ VN , with xi as above, we define the corresponding field by (1.16) x1 · . . . · xp (z) =: x1 (z) · . . . · xp (z) : . Since every element of VN is a finite linear combination of monomials as above, this completes the definition of the mapping (1.1). We will use the shorthand notations j

bi (z) = b0i (z); a j (z) = a−1 (z).

(1.17)

The operator products of these basic fields are a j (z)bi (w) =

δij + (regular), z−w

bi (z)bj (w) = (regular); a i (z)a j (w) = (regular),

(1.18a) (1.18b)

where “(regular)” means the part regular at z = w. These operator products are equivalent to the commutation relations (1.9). Other operator products are computed by differentiation of (1.18), and using the Wick theorem, cf. [K], 3.3. One can say that the vertex algebra VN is generated by the even fields bi (z), a j (z), of conformal weights 0 and 1 respectively, subject to the relations (1.18). The Virasoro field is given by L(z) =

N X

: bi (z)0 · a i (z) : .

(1.19)

i=1

The central charge is equal to 2N . Let us check this. Assume for simplicity that N = 1, and let us omit the index 1 at the fields a, b. Thus, we have L(z) =: b(z)0 a(z) : . Let us compute the operator product L(z)L(w) using the Wick theorem. We have : b(z)0 a(z) :: b(w)0 a(w) := [b(z)0 a(w)][a(z)b(w)0 ] +[b(z)0 a(w)] : a(z)b(w)0 : +[a(z)b(w)0 ] : b(z)0 a(w) : (we have b(z)0 a(w) = a(z)b(w)0 = 1/(z − w)2 ) =

2 : b(w)0 a(w) : : b(w)00 a(w) : + : b(w)0 a(w)0 : 1 . + + (z − w)4 (z − w)2 z−w

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Hence,

2L(w) L(w)0 1 + + , 4 2 (z − w) (z − w) z−w

L(z)L(w) = which is (1.6) with c = 2.

1.6. Clifford algebra. Let ClN be the Lie algebra which as a vector space has the base φni , ψni , i = 1, . . . , N, n ∈ Z, and C, all these elements being odd, with the brackets j

[φm , ψni ] = δij δm,−n · C.

(1.20)

Clifford vertex algebra. 3N is defined as in the previous subsection, starting with the odd fields X X j ψ i (z) = ψni z−n−1 ; φ j (z) = φn z−n (1.21) n∈Z

n∈Z

and repeating the definitions of loc. cit., with a (resp. b) replaced by ψ (resp. φ). One must put the obvious signs in the definition of the normal ordering. Thus, 3N is generated by the odd fields φ i (z), ψ i (z), subject to the relations φ i (z)ψ j (w) =

δij + regular; z−w

φ i (z)φ j (w) = regular; ψ i (z)ψ j (w) = regular.

(1.22a) (1.22b)

The Virasoro field is given by L(z) =

N X

: φ i (z)0 · ψ i (z) : .

(1.23)

i=1

The central charge is equal to −2N . 1.7. If A and B are vertex algebras then their tensor product A ⊗ B admits a canonical structure of a vertex algebra, cf. [K], 4.3. The Virasoro element is given by LA⊗B = LA ⊗ 1 + 1 ⊗ LB .

(1.24)

We will use in the sequel the tensor product of the Heisenberg and Clifford vertex algebras N = VN ⊗ 3N . Its Virasoro central charge is equal to 0. 1.8. Let A be a vertex algebra. A linear map f : A −→ A is called derivation of A if for any a ∈ A, f (a)(z) = [f, a(z)]. (1.25) Note that an invertible map g : A −→ A is an automorphism of A iff g(a)(z) = ga(z)g −1

(1.26)

and (1.25) is the infinitesimal version of (1.26). R It follows from Borcherds formula that for every a ∈ A, the Fourier mode Ra(z) is a derivation, cf. Lemma 1.3 from [LZ]. Consequently, if the endomorphism exp( a(z)) is well defined, it is an automorphism of A.

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2. De Rham Chiral Algebra of an Affine Space 2.1. A topological vertex algebra of rank d is a conformal vertex algebra A of Virasoro central charge 0, equipped with an even element J of conformal weight 1, and two odd elements, Q, of conformal weight 1, and G, of conformal weight 2. The following operator products must hold: L(z)L(w) = J (z)J (w) =

L(w)0 2L(w) + , (z − w)2 z−w

(2.1a)

d J (w) J (w)0 d ; L(z)J (w) = − + + , (z − w)2 (z − w)3 (z − w)2 z−w

(2.1b)

G(w) G(w)0 2G(w) ; J (z)G(w) = − , (2.1c) + 2 (z − w) z−w z−w

G(z)G(w) = 0; L(z)G(w) = Q(z)Q(w) = 0; L(z)Q(w) = Q(z)G(w) =

Q(w) Q(w)0 Q(w) ; J (z)Q(w) = , + 2 (z − w) z−w z−w

(2.1d)

d J (w) L(w) . + + 3 2 (z − w) (z − w) z−w

(2.1e)

Note the following consequence of (2.1e), [Q0 , G(z)] = L(z).

(2.2)

2.2. In this subsection, we will intorduce a structure of a topological vertex algebra of rank N on the vertex algebra N from 1.6. This topological vertex algebra will be called the de Rham chiral algebra of the affine space AN . Recall that the Virasoro element is given by L=

N X i=1


i i i i a−1 + φ−1 ψ−1 b−1 .

(2.3a)

Define the elements J, Q, G by J =

N X i=1

i φ0i ψ−1 ; Q=

N X i=1

i a−1 φ0i ; G =

N X i=1

i i ψ−1 b−1 .

The corresponding fields are X
L(z) = : bi (z)0 a i (z) : + : φ i (z)0 ψ i (z) : and J (z) = G(z) =

X X

: φ i (z)ψ i (z) :, Q(z) =

X

: a i (z)φ i (z) :,

: ψ i (z)bi (z)0 : .

The relations (2.1) are readily checked using the Wick theorem.

(2.3b)

(2.4a)

(2.4b)

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2.3. Let us define the fermionic charge operator acting on N , by XX i F := J0 = : φni ψ−n :. i

We have and We set Obviously,

(2.5)

n

F1 = 0

(2.6)

[F, φni ] = φni ; [F, ψni ] = −ψni ; [F, ani ] = [F, bni ] = 0.

(2.7)

p

N = {ω ∈ N | F ω = pω}. p

N = ⊕p∈Z N .

We define an endomorphism d of the space N by X i : ani φ−n : d := −Q0 = −

(2.8) (2.9)

(2.10)

i,n

(we could omit the normal ordering in the last formula since the letters a and φ commute anyway). We have d 2 = 0. Indeed, by the Wick theorem Q(z)Q(w) = regular, hence all Fourier modes of Q(z) (anti)commute. The map d is called a chiral de Rham differential. The map d increases the fermionic charge by 1, by (2.7). Thus, the space N equipped with the fermionic charge grading and the differential d, becomes a complex (infinite in both directions), called a chiral de Rham complex of AN . p N Consider the usual algebraic de Rham complex (AN ) = ⊕N p=0  (A ) of the affine space AN . We identify the coordinate functions with the letters b01 , . . . , b0N , and their differentials with the fermionic variables φ01 , . . . , φ0N . Thus, we identify the commutative dg algebras (AN ) = C[b01 , . . . , b0N ] ⊗ 3(φ01 , . . . , φ0N ),

(2.12)

the second factor being the exterior algebra. The grading is defined by assigning to the j letters b0i (resp. φ0 ) the degree 0 (resp. 1). The usual de Rham differential is given by X a0i φ0i , (2.13) dDR = i

as follows from the relations (1.9). Theorem 2.4. The obvious embedding of complexes i : ((AN ), dDR ) −→ (N , d)

(2.14)

is compatible with the differentials, and is a quasi-isomorphism. We identify the space N with the space of polynomials in the letters bni , φni (n ≤ 0) i and ani , ψni (n < 0). One sees that on the subspace C[b0i , φ0i ], all the summands ani φ−n with n 6 = 0 act trivially. It follows that the map i is compatible with the differentials.

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Proof. To prove that it is a quasi-isomorphism, let us split d in two commuting summands d = d+ + d− , where XX XX i i ani φ−n , d− = ani φ−n . (2.15) d+ = i

n≥0

i

n0 , and our space as C[bj , φ j ] ⊗ this purpose, split d+ once again as d+ = dDR + d+ 0 0 j j >0 acts trivially on the first factor, and on the second C[bm , φm ]m0 kills all non-zero modes, and we are left precisely with the usual de Rham complex. d+ Alternatively, it follows from (2.2) that [G0 , d] = L0 .

(2.16)

The operator G0 commutes with L0 , and it follows from (2.16) that it gives a homotopy to 0 for the operator d on all the subcomplexes of non-zero conformal weight. Therefore, all cohomology lives in the conformal weight zero subspace. u t 2.5. The vertex algebra N satisfies the assumptions of 1.3. The subspace (AN ) coincides with the conformal weight zero component of it. If we apply the definition of 1.3, we get the structure of a commutative algebra on (AN ) which is given by the usual wedge product of differential forms. 3. Localization 3.1. Consider the Heisenberg vertex algebra VN defined in 1.4. As in loc. cit., we will i , aj ] identify the space VN with the space of polynomials C[b−n −m n≥0, m>0 . To simplify i the notations below, let us denote the zero mode variables b0 by bi . Let AN denote the algebra of polynomials C[b1 , . . . , bN ]. The space AN is iden(0) tified with the subspace VN ⊂ VN of conformal weight zero. The space VN has an bN denote the algebra of formal power series obvious structure of an AN -module. Let A C[[b1 , . . . , bN ]]. Set bN ⊗AN VN . bN = A (3.1) V bN . We are going to introduce a structure of a conformal vertex algebra on the space V Let us define the map (1.1). bN . We claim that the expression f (b1 (z), Let f (b1 , . . . , bN ) be a power series from A N bN )[[z, z−1 ]]. (We are grateful to Boris . . . , b (z)) makes sense as an element of End(V Feigin who has shown us a particular case of the following construction.) Let us express the power series bi (z) as (3.2) bi (z) = bi + 1bi (z). Thus,

1bi (z) =

X n>0

i (bni z−n + b−n zn ).

(3.3)

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Let us define f (b1 (z), . . . , bN (z)) by the Taylor formula X 1b1 (z)i1 · . . . · 1bN (z)iN ∂ (i1 ,... ,iN ) f (b1 , . . . , bN ), (3.4) f (b1 (z), . . . , bN (z)) = where ∂

(i1 ,... ,iN )

=

∂bi11 i1 !

· ... ·

∂biNN

iN !

.

(3.5)

bN )[[z, z−1 ]]. We will show that the series (3.4) gives a well-defined element of End(V Let us write X i ,... ,i N −k ck1 z . 1b1 (z)i1 · . . . · 1bN (z)iN = k

The coefficient

cki1 ,... ,iN

is an infinite sum of the monomials j

j

bk11 · . . . · bkII ,

(3.6)

bN . There exists M with I = i1 + . . . + iN , k1 + . . . + kI = k. Pick an element v ∈ V P j1 jN li > M. such that bl1 · . . . · blN v = 0 if We have X X X k= ki = + |ki | − − |ki |, P P (resp. − ) denotes the sum of all positive (resp. negative) summands. If where + j1 jI bk1 · . . . · bkI v 6 = 0, then X + |ki | ≤ M. (3.7a) On the other hand, −

X

|ki | =

+

X

|ki | − k ≤ M − k.

(3.7b)

There exists only a finite number of tuples (k1 , . . . , kI ) satisfying (3.7a) and (3.7b). bN . Therefore, cki1 ,... ,iN are well-defined endomorphisms of V We have  X  X i ,... ,i N (i1 ,... ,iN ) ck1 ∂ f (b1 , . . . , bN ) z−k . (3.8) f (b1 (z), . . . , bN (z)) = k

i1 ,... ,iN

number of positive (resp. negative All numbers ki are non-zero. P Let I+ (resp., I− ) be theP ki ≥ I+ = I −I− ≥ I −M +k. ki ’s). We have I− ≤ − ki ≤ M −k, hence M ≥ + Therefore, I ≤ 2M − k. (3.9) Therefore, when we apply the series (3.8) to the element v, only a finite number of terms in the sum over (i1 , . . . , iN ) survives. Therefore, the series (3.8) is a well-defined bN )[[z, z−1 ]]. element of End(V bN is a finite sum of products g(a)f (b), where g(a) is a polynomial Every element of V in the letters a and f (b) is a power series as above. We have already defined f
(b)(z). The definition of g(a)(z) is the same as in the case of VN . We define g(a)f (b) (z) by
g(a)f (b) (z) =: g(a)(z)f (b)(z) :, (3.10)

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where the normal ordering is defined in (1.2). This completes the definition of the mapping bN )[[z, z−1 ]], bN −→ End(V (3.11) Y : V The following version of the definition of the map (3.11) is helpful in practice. Every bN is a limit of the elements of ci ∈ VN (in the obvious topology). We can element c ∈ V bN )[[z, z−1 ]]. The field c(z) is the limit regard the fields ci (z) as the elements of End(V of the fields ci (z). bN as the image of the correWe define the vacuum and Virasoro element 1, L ∈ V bN . sponding element of VN under the natural map VN −→ V Theorem 3.2. The construction of 3.1 defines a structure of a conformal vertex algebra bN . on the space V This follows from [K], Theorem 4.5. b 3.3. Let Aan N ⊂ AN denote the subalgebra of power series, convergent in a neighbourhood of the origin. Set b (3.12) VNan = Aan N ⊗AN VN ⊂ VN .

It is clear from the inspection of the Taylor formula (3.4) that for f (a, b) ∈ VNan , bN ), respect the subspace V an . the Fourier modes of f (a, b)(z) which belong to End(V N bN defined in 3.1 induces the Therefore, the conformal vertex algebra structure on V structure of a conformal vertex algebra on VNan . In this argument, Aan N can be replaced by any algebra of functions containing AN and closed under derivations. More precisely, one has the following general statement. Let A0 be an arbitrary commutative AN -algebra, given together with an action of the Lie algebra T = Der(AN ) by derivations, extending the natural action of T on AN . Then the space VA0 := A0 ⊗AN VN admits a natural structure of a vertex algebra. For the details, see 6.9. ∞ For example, let Asm N denote the algebra of germs of smooth (C ) functions. Then we get a vertex algebra (3.13) VNsm = Asm N ⊗AN VN . Another natural example is that of localization of A. It is treated in the next subsection. 3.4. Zariski localization. Let f ∈ AN be a nonzero polynomial. Let AN ;f denote the localization AN [f −1 ]. Set (3.14) VN;f = AN;f ⊗AN VN . Consider the Taylor formula (3.4) applied to the function f −1 . We have evidently   ∂ (i1 ,... ,iN ) f −1 (b1 , . . . , bN ) ∈ AN ;f . P In more concrete terms, let f (z) = fn z−n be the field correponding to f , then we want to define the field corresponding to f −1 as
−1 = f (z)−1 = (f0 + f−1 z + f1 z−1 + . . . )−1 = f0−1 1 + f0−1 (f−1 z + f1 z + . . . ) (we use the geometric series) = f0−1 (1 + f0−2 (2f−1 f1 + 2f−2 f2 + . . . ) + . . . )

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(we started to write down the coefficient at z0 ). Now, in the right hand side, the coefficient at each power of z is an infinite sum, but as an operator acting on AN ;f it is well defined since only a finite number of terms act nontrivially. We need only to invert f0 = f . Therefore, the construction 3.1 provides a conformal vertex algebra structure on the space VN;f . ch be the O -quasicoherent sheaf Let X denote the affine space Spec(AN ). Let OX X corresponding to the AN -module VN . We have just defined the structure of a conformal ch ), where U = Spec(A vertex algebra on the spaces VN;f = 0(Uf ; OX f N ;f ). If Uf ⊂ of conformal vertex algebras. Ug then the restriction map VN;g −→ VN;f is a morphism S If U ⊂ X is an arbitrary open, we have U = Uf , and Y
−→ Y ch ) = Ker VN ;f −→ VN ;f g . 0(U ; OX Using this formula, we get a structure of a conformal vertex algebra on the space ch ). Therefore, O ch gets a structure of a sheaf of conformal vertex algebras. 0(U ; OX X bN ⊗AN N , bN = A 3.5. We can add fermions to the picture. Consider the spaces  an ⊗ sm = Asm ⊗ b = A  ⊂  ,   . The construction 3.1 provides a an AN N N AN N N N N N structure of a topological vertex algebras on these spaces. Let X be as in 3.4; let ch X denote the OX -quasicoherent sheaf associated with the AN -module N . The construction 3.4 provides a structure of a sheaf of topological vertex algebras of rank N on ch X. 3.6. Now we want to study coordinate changes in our vertex algebras. Let X be the formal scheme Spf(C[[b1 , . . . , bN ]]). Consider the formal N|N -dimensional superscheme X˜ = 5T X (here T X is the total space of the tangent bundle, 5 is the parity change functor). Thus, X˜ has the same underlying space as X, and the structure sheaf of X˜ ˜ we have N even coincides with the de Rham algebra of differential forms X. On X, coordinates b1 , . . . , bN and odd ones φ 1 = db1 , . . . , φ N = dbN . ˜ with the above coordinates, we have assigned a (super)vertex To this superscheme X, bN , generated by the fields bi (z), a i (z) (even ones) and φ i (z), ψ i (z) (odd ones). algebra  ˜ The fields a i (z) (resp. ψ i (z)) correspond to the vector fields ∂bi (resp. ∂φ i ) on X. These fields satisfy the relations (cf. (1.18), (1.22)) a i (z)bj (w) =

δij , z−w

bi (z)bj (w) = (regular); a i (z)a j (w) = (regular), φ i (z)ψ j (w) =

δij , z−w

(3.15a) (3.15b) (3.15c)

φ i (z)φ j (w) = (regular); ψ i (z)ψ j (w) = (regular),

(3.15d)

bi (z)φ j (w) = (regular); bi (z)ψ j (w) = (regular),

(3.15e)

a i (z)φ j (w) = (regular); a i (z)ψ j (w) = (regular).

(3.15f)

Consider an invertible coordinate transformation on X, b˜ i = g i (b1 , . . . , bN ); bi = f i (b˜ 1 , . . . , b˜ N ),

(3.16a)

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where g i ∈ C[[bj ]]; f i ∈ C[[b˜ j ]]. It induces the transformation of the odd coordinates φ i = dbi , φ˜ i =

∂g i j ∂f i j φ˜ φ ; φi = j ∂b ∂ b˜ j

(3.16b)

(the summation over the repeating indices is tacitly assumed). The vector fields transform as follows, ∂f j ∂ 2f k ∂g l (g(b))∂bj + (g(b)) · r · φ r ∂φ k (3.16c) ∂b˜ i = ∂b ∂ b˜ i ∂ b˜ i ∂ b˜ l and ∂φ˜ i =

∂f j (g(b))∂φ j . ∂ b˜ i

(3.16d)

We want to lift the transformation (3.16a) to the algebra N . Define the tilded fields by b˜ i (z) = g i (b)(z), φ˜ i (z) =

 ∂g i j φ (z), ∂bj

  2 k  ∂ f ∂g l r k ∂f j (g(b)) (z) + (g(b)) r φ ψ (z), a˜ (z) = a ∂b ∂ b˜ i ∂ b˜ i ∂ b˜ l i





j

ψ˜ i (z) =



 ∂f j (g(b))ψ j (z). ∂ b˜ i

(3.17a) (3.17b) (3.17c) (3.17d)

Theorem 3.7. The fields b˜ i (z), a˜ i (z), φ˜ i (z) and ψ˜ i (z) satisfy the relations (3.15). Proof. We will use the relations h(b)(z)a i (w) = −

∂h/∂bi (w) i ∂h/∂bi (w) ; a (z)h(b)(w) = z−w z−w

(3.18)

bN , which follow from (3.15a) and the Wick theorem. Let us check (3.15a) for each h ∈ A for the tilded fields. We have k

∂f (g(b))(w) b˜ i (z)a˜ j (w) =g i (b)(z)a k ∂ b˜ j ∂ 2f k ∂g l (g(b)) r ψ k φ r (w). − g i (b)(z) ∂b ∂ b˜ j ∂ b˜ l By the Wick theorem, the first summand is equal to −

δij 1 ∂g i ∂f k (g(b))(w) · = , k ∂b ∂ b˜ j z−w z−w

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by (3.18). The second summand is zero. Let us check (3.15b). The first identity is clear. We have ∂f k ∂f n (g(b))(z)a n (g(b))(w) ∂ b˜ i ∂ b˜ j ∂ 2f n ∂g l ∂f k (g(b))(z) (g(b)) r ψ n φ r (w) + + ak ∂b ∂ b˜ i ∂ b˜ j ∂ b˜ l 2 k l ∂g ∂f n ∂ f (g(b)) r ψ k φ r (z)a n (g(b))(w) − ∂b ∂ b˜ i ∂ b˜ l ∂ b˜ j ∂g l ∂ 2f n ∂g l ∂ 2f k (g(b)) r ψ k φ r (z) (g(b)) r ψ n φ r (w). + ∂b ∂b ∂ b˜ i ∂ b˜ l ∂ b˜ j ∂ b˜ l

a˜ i (z)a˜ j (w) = a k

When we compute each term using the Wick theorem, there appear single and double pairings. The part corresponding to the single pairings coincides with the expression of the bracket [∂b˜ i , ∂b˜ j ] in old coordinates bp , so it vanishes. The “anomalous” part comes from the double pairings. One double pairing appears in the first term and is equal to −

    ∂ ∂f n 1 ∂ ∂f k (g(b)) (z) (g(b)) (w) · , n k ∂b ∂ b˜ i ∂b ∂ b˜ j (z − w)2

another one appears in the fourth term and equals ∂g l ∂ 2f n ∂g p 1 ∂ 2f k (g(b)) r (z) (g(b)) m (w) · i l j p ˜ ˜ ˜ ˜ ∂b ∂b (z − w)2 ∂b ∂b ∂b ∂b ∂ 2f k ∂g l ∂ 2f n ∂g p 1 = (g(b)) n (z) (g(b)) k (w) · . i l j p ˜ ˜ ˜ ˜ ∂b ∂b (z − w)2 ∂b ∂b ∂b ∂b

δkm δrn

We see that these terms cancel out. The remaining relations, (3.15c–f), contain only single pairings, and are easily checked. t u Thus, for each automorphism g = (g 1 , . . . , g N ) of C[[b1 , . . . , bN ]], (3.16a), the formulas (3.17) determine a morphism of vertex algebras bN . bN −→  g˜ : 

(3.19)

i , bj , bN is an (infinite) sum of finite products of a−n More precisely, each element c of V −m
i , ψ j . We have c = c(z)1 (0). By definition, ψ−n −m





g(c) ˜ = g(c(z))1 ˜ (0).

(3.20)

bN . If c is one of the generators Thus, we have to define the field g(c(z)) ˜ for each c ∈ V i , bi , φ i , ψ i , we define g(c(z)) ˜ by formulas (3.17). We set a−1 0 0 −1 i i (z)) = ∂z(n) g(a ˜ −1 (z)), g(a ˜ −1−n

(3.21)

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and the same with b, φ, ψ, cf. (1.12). Finally, if c = c1 c2 · . . . · cp , where each ci is one of the letters a, b, φ or ψ, we set ˜ 2 (z)) · . . . · g(c ˜ p (z)) :, g(c(z)) ˜ =: g(c ˜ 1 (z))g(c

(3.22)

where the normal ordered product of two factors is defined by (1.2), and if p > 2, we use the inductive formula (1.14). j Equivalently, if cj = xkj where x j = a i , bi , φ i or ψ i , we have  1   p  ˜ (z)) k · . . . · g(x ˜ (z)) k 1. g(c ˜ 1 · . . . · cp 1) = g(x 1

p

(3.23)

P (Here we use the following notation. If a(z) = i ai z−i−n is a field corresponding to an element of conformal weight n, we denoted the Fourier mode ai by a(z)i .) Let GN denote the group of automorphisms (3.16a). Theorem 3.8. The assignment g 7 → g˜ defines the group homomorphism GN −→ bN ). Aut( Proof. Let us consider two coordinate transformations, 0 bi = g1i (b), and 00 bi = g2i (0 b). Let fj denote the transformation inverse to gj . We have to check that g ] 2 g1 = g˜ 2 g˜ 1 .

(3.24)

By Theorem 4.5 from [K], it suffices to check this equality on the generators. Let us i 1 is expressed in the coordinates 0 a, etc., as follows begin with a i . The element 00 a−1 00 i a−1 1

 =

j 0 j ∂f2 a−1 00 i (g2 (0 b0 )) − ∂ b

 ∂ 2 f2k ∂g2l 0 0 0 k 0 r (g2 ( b0 )) 0 r ( b0 ) ψ−1 φ0 1. ∂ 00 bi ∂ 00 bl ∂b

Expressing it in the coordinates a, etc., we get the element g˜ 2 g˜ 1 (a i )   p p q j ∂f2 ∂ 2 f1 ∂g1 p s p ∂f1 (g g (b))(z)0 1 = a 0 j (g1 (b))(z) + 0 j 0 q (g1 (b)) s ψ φ (z) 00 i 2 1 ∂b ∂b ∂b ∂b −1 ∂ b   r   p ∂g l ∂f ∂g1 q ∂ 2f k φ (z) 1 − 00 i 200 l (g2 g1 (b)) 0 2r (g1 (b))(z)0 0 1k (g1 (b))ψ p (z) q ∂ b∂ b ∂b ∂b −1 ∂b 0 (3.25) (we have used (3.23)). Now, the action of our group GN on the classical de Rham i complex is associative. It follows that the expression (3.25) is equal to g ] 2 g1 (a ) plus two anomalous terms:  p j ∂f2 ∂f1 (g1 (b))(z) (g g (b ))1 00 i 2 1 0 ∂ 0 bj −1 ∂ b   p j ∂f1 ∂ ∂f2 = + 0 j (g1 (b))(z) p 00 i (g2 g1 (b0 ))1 ∂b −1 ∂b0 ∂ b

p a0



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coming from the first summand, and  p  ∂g1r ∂g2l ∂f1 ∂ 2 f2k p q (g g (b )) (g (b )) (g (b))(z) (b )ψ φ 1 2 1 0 1 0 1 q 0 0 0 ∂ 00 bi ∂ 00 bl ∂ 0 br ∂ 0 bk ∂b −1  p  ∂g1r ∂g2l ∂f1 ∂ 2 f2k (b )1 = − 00 i 00 l (g2 g1 (b0 )) 0 r (g1 (b0 )) 0 k (g1 (b))(z) p 0 ∂ b∂ b ∂b ∂b −1 ∂b

−

coming from the second one. One sees that these two terms cancel out, which proves t (3.24) for a i . For the generators b, ψ, and φ, the anomaly does not appear at all. u 3.9. Theorems 3.7 and 3.8 allow one to define the sheaf of conformal vertex algebras ch X for each smooth manifold X in an invariant way, by gluing the sheaves defined in 3.5. This can be done in each of the three settings: in algebraic, complex analytic or smooth one. In the complex analytic situation, we have our sheaves of vertex algebras over the coordinate charts, and the formulas (3.17) allow to glue these sheaves in a sheaf over X. In the algebraic situation, Theorem 3.8 ensures the existence of our sheaves by the standard arguments of “formal geometry” of Gelfand and Kazhdan, cf. [GK]. Consider bN is a GN -module. Therefore, the Lie the formal situation. By 3.8, the vertex algebra  bN ) of formal vector fields vanishing at 0 acts on  bN by derivations. algebra WN0 = Lie( In fact, since the proof of Theorem 3.8 never uses the fact that the automorphisms in question preserve the origin, the infinitesimal version of formulas (3.17) shows that the bN . (Alternatively, this can be entire algebra WN of formal vector fields operates on  bN the anomaly shown by the computation similar to the one from 5.1 (in the case of  bN is a (WN , GN )-module (cf. [BS, BFM]). Now, the vanishes!)). In other words,  standard result, [GK], says that such a module defines naturally a sheaf ch X on each smooth algebraic variety X. They are sheaves of vertex algebras since GN (resp., WN ) acts by vertex algebra automorphisms (resp., derivations). A more direct construction of these sheaves is outlined in Sect. 6, see 6.10. bN admits a canonical 3.10. Consider the formal situation. Let us show that the algebra  filtration whose graded factors are standard tensor fields. Placing the formulae (3.16) and (3.17), (3.23) on the desk next to each other, one realizes that the “symbols of fields” transform in the same way as the corresponding geometric quantities: functions, 1-forms, and vector fields. To be more precise, introduce bN as follows. a filtration on  bN -module with a base consisting of monomials in letters bN is a free A The space  i i i i an , ψn (n < 0); bm , φm (m ≤ 0). Define a partial ordering on this base by j (a) a > φ, a > ψ, a > b, ψ > φ, ψ > b, φ > b; xni > xm if n < m, x being a, b, φ or ψ; (b) extending this order to the whole set of monomials lexicographically. This partial order on the base naturally determines an increasing exhausting filtration on the spaces of fixed conformal weight, (i)

(i)

b ⊂ F1  b ⊂ ... . F0  N N

(3.27)

bN , F0  b i b(0) = A b(−1) = ⊕N A For example, F0  N N i=1 N b−1 , etc. A glance at (3.17), (3.23) (i)

shows that the corresponding graded object GrF• N is a direct sum of symmetric powers

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of the tangent bundle, symmetric powers of the bundle of 1-forms, and tensor products i (n > 0) is a b(0) is a function, that of b−n thereof. For example, the image of b0i in GrF•  N i is a vector field, etc. 1-form, that of a−n This filtration is stable under coordinate changes. Therefore all the sheaves ch X acquire the natural filtration with graded factors being the bundles of tensor fields. 4. Conformal and Topological Structure 4.1. Let us return to the formal setting 3.6. Recall that we have in our vertex algebra bN the fields L(z), J (z), Q(z) and G(z), defined by the formulas (2.4), which make it  a topological vertex algebra. Let us study the effect of the coordinate changes (3.16a) on these fields. ˜ Let us denote by L(z), etc., the field L(z), etc., written down using formulas (2.4), in terms of the tilded fields a˜ i (z), etc., and then expressed in terms of the old fields a i (z), etc. Theorem 4.2. We have ˜ L(z) = L(z),
0 J˜(z) = J (z) + Tr log(∂g i /∂bj (z)) , 0   ∂  i j ˜r ˜ ˜ Tr log(∂f /∂ b ) φ (z) , Q(z) = Q(z) + ∂ b˜ r ˜ G(z) = G(z). i 1. Therefore, (cf. (3.23)), Proof. We have J = φ0i ψ−1

  k   i   k  ∂f ∂g ∂f ∂g i j k φ (z) ψ (z) 1 = J + δj k (z) (z) 1 j i i ˜ ∂bj ∂b 0 ∂b −1 −1 ∂ b˜ 0  i 0  j  ∂f ∂g (z) (z) 1, =J + i ∂bj 0 0 ∂ b˜

J˜ =



i bi 1. Therefore, which implies (4.1b). We have G = ψ−1 −1

  j   i  j    i ∂f ∂f ∂g k 0 j j ˜ ψ (z) ψ (z) b (z) 1 g (b)(z) −1 1 = G= k ∂ b˜ i ∂ b˜ i −1 −1 ∂b 0 j

k 1 = G. = δj k ψ−1 b−1 i φ i 1. Therefore, This proves (4.1d). We have Q = a−1 0

  i   j ∂ 2 f k ∂g l k r ∂g q j ∂f ˜ (z) − ψ φ (z) φ (z) 1. Q= a q ∂ b˜ i ∂ b˜ i ∂ b˜ l ∂br −1 ∂b 0 The classical terms:  2 k  j   i   ∂ f ∂g l ∂f ∂g q j k r − ψ−1 φ0 φ 1 = Q, a−1 r i i l ˜ ˜ ˜ ∂bq ∂b 0 ∂ b ∂ b ∂b 0 0

(4.1a) (4.1b) (4.1c) (4.1d)

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k and φ . Quantum since the second summand is zero, due to the anticommutation of ψ−1 0 corrections (anomalous terms):

 2 k l   i    2 i  ∂ f ∂g r ∂ g ∂g ∂f j r φ + φ 1 0 j ∂bq r k i i l ˜ ˜ ˜ ∂b ∂b ∂b ∂ b −1 ∂b b 0 −1 0  2 k   i   j   2 i  ∂ f ∂g l ∂ g ∂g ∂f + φ0r = j r r i i l ∂ b˜ −1 ∂b ∂b 0 ∂ b˜ ∂ b˜ ∂b −1 ∂bk 0  2 k     ∂ f ∂g l ∂g i ∂f k ∂g l ∂g i r r + φ−1 1 = φ 1, ∂ b˜ i ∂ b˜ l ∂br ∂bk 0 ∂ b˜ i ∂ b˜ l ∂br ∂bk −1 

since



∂ 2 f k ∂g l ∂ b˜ i ∂ b˜ l ∂br

  0

∂g i ∂bk

(4.2)

 k 2 l   i ∂f ∂ g ∂f t ∂g 1=− 1 k l ∂bt ∂br ∂ b˜ i ˜ ∂b ∂ b −1 0 −1  k 2 l   t  i ∂f ∂g ∂f ∂ g 1 = t r l i ˜ ˜ ∂ b ∂b ∂b 0 ∂ b −1 ∂bk 0  2 i   t ∂f ∂ g 1. = ∂bt ∂br 0 ∂ b˜ i −1



Returning to (4.2), we have   ∂ 2 f k ∂g i l ∂ ∂ 2 f k ∂g l ∂g i r i j ˜ ˜ φ = φ = Tr log(∂f /∂ b ) φ˜ l , ∂ b˜ i ∂ b˜ l ∂br ∂bk ∂ b˜ l ∂ b˜ i ∂g k ∂ b˜ l which proves (4.1c). It follows from (4.1c) that the operator Q0 is invariant. Hence, (4.1a) follows from (4.1d) and (2.2). u t 4.3. It follows from (4.1a) that for an arbitrary smooth manifold X, the field L(z) is ch −1 a well-defined global section of the sheaf End(ch X )[[z, z ]], i.e. X is canonically a sheaf of conformal vertex algebras. It follows from (4.1b) and (4.1c) that the Fourier modes F = J0 (“fermionic charge”) ch = Q (“BRST charge”) are well-defined endomorphisms of the sheaf ch . and dDR 0 X ch ) becomes a complex of sheaves, graded by F . This is a localization of , d Thus, (ch DR X Definition 2.3. Theorem 4.4. For any smooth manifold X, the obvious embedding of complexes of sheaves ch i : (X , dDR ) −→ (ch X , dDR )

(4.3)

is a quasi-isomorphism. This is true in algebraic, analytic and C ∞ settings. Indeed, the problem is local along X, and we are done by Theorem 2.4. 4.5. If X is Calabi–Yau, i.e. c1 (TX ) = 0, then the fields J (z) and Q(z) are globally well defined, by (4.1b) and (4.1c). Here TX denotes the tangent bundle. Therefore, in this case the sheaf ch X is canonically a sheaf of topological vertex algebras.

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5. Chiral Structure Sheaf A. Obstruction bN of “chiral 5.1. Consider the formal setting 3.1, 3.2, 3.6. We have the vertex algebra V bN ), where A bN = C[[b1 , . . . , bN ]]. Let WN functions” over the formal disk DN = Spf(A bN ) denote the module denote the Lie algebra of vector fields f i (b)∂bi on DN . Let 1 (A bN and 1 (A bN ) are naturally WN -modules. Recall of one-forms f i (b)dbi . The spaces A bN ) is given by that the action of WN on 1 (A f i ∂bi · g j dbj = f i ∂bi g j dbj + g j df j .

(5.1)

bN ) is compatible with the WN -action. bN −→ 1 (A The de Rham differential d : A Let us define a map bN ). (5.2) π : WN −→ End(V For a vector field τ = f i (b)∂bi , let τ (z) denote the field f i (b)a i (z) (of conformal weight bN ) denote the Fourier mode bN . Let π(τ ) ∈ End(V 1) of our vertex algebra V Z (5.3) π(τ ) := τ (z) = τ (z)0 . bN . Note that by 1.7, the maps π(τ ) are derivations of V The mapping π does not respect the Lie bracket. Let us compute the discrepancy. Let τ1 = f i (b)∂bi , τ2 = g i (b)∂bi be two vector fields. We have the operator product ∂bj f i (b(z))∂bi g j (b(w)) (z − w)2 i j j f (b(w))∂bi g (b(w))a (w) − g j (b(w))∂bj f i (b(w))a i (w) (5.4) + z−w   0 ∂bj f i (b(w)) ∂bi g j (b(w)) ∂ j f i (b(w))∂bi g j (b(w)) [τ1 , τ2 ](w) + − . =− b (z − w)2 z−w z−w τ1 (z)τ2 (w) = −

It follows that  0 [π(τ1 ), τ2 (w)] = [τ1 , τ2 ](w) − ∂bj f i (b(w)) ∂bi g j (b(w)).

(5.5)

In particular, Z [π(τ1 ), π(τ2 )] = π([τ1 , τ2 ]) −


0 ∂bj f i (b(w)) ∂bi g j (b(w)).

(5.6)

bN ), let us denote by ω(z) the field f i (b)bi (z)0 of our 5.2. For ω = f i (b)dbi ∈ 1 (A R vertex algebra. Denote by π(ω) the Fourier mode ω(z)0 = ω(z). bN . Its conformal bN , let f (z) denote the corresponding field of V For f = f (b) ∈ A weight is 0, and f (z)0 = f . We have df (z) = f (z)0 .

(5.7)

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bN ), we have the operator product Given τ = f i ∂bi ∈ WN , ω = g j dbj ∈ 1 (A f i (b(z))g i (b(w)) f i (b(w))∂bi g j bj (w)0 + (z − w)2 z−w i i i j f (b(w))g (b(w)) f ∂bi g (w)bj (w)0 + f i (b(w))0 g i (b(w)) . (5.8) + = (z − w)2 z−w

τ (z)ω(w) =

It follows that [π(τ ), ω(z)] = (τ ω)(z)

(5.9)

[π(τ ), π(ω)] = π(τ ω).

(5.10)

and bN ) generated by the Fourier modes π(τ ) (τ ∈ Let W˜ N denote the linear subspace of End(V 1 b WN ) and π(ω) (ω ∈  (AN )). Let IN ∈ W˜ N be the linear subspace generated by the Fourier modes π(ω). It follows from (5.7) that if ω is exact then π(ω) = 0. Thus, π induces an epimorphic map, bN )/d A bN −→ IN . (5.11) 1 (A

Lemma 5.3. The map (5.11) is an isomorphism. This can be proved by writing down the Fourier mode as an infinite sum of monomials the coefficients of like terms. In fact, a more general statement, in bni and comparing R namely, that Q(z) = 0 if and only if Q = const · 1 or Q(z) = P (z)0 for some P , seems to be valid for a broad class of vertex algebras, cf. a similar statement in [FF3]. From our point of view, this phenomenon has topological nature. It is amusing to exhibit an example of a vertex algebra, for which the lemma above isRfalse. Namely, take b−1 b−1 ∈ A1 [b−1 ], see 3.4; then (b−1 b−1 )(z) = b(z)−1 b(z)0 and b(z)−1 b(z)0 = 0, but b(z)−1 b(z)0 is not a total derivative. 4 bN ). It follows from (5.6) and 5.4. Obviously, ω1 (z)ω2 (w) = 0 for all ω1 , ω2 ∈ 1 (A ˜ (5.10) that WN is a Lie algebra, IN is its abelian ideal, and we have the canonical extension (5.12) 0 −→ IN −→ W˜ N −→ WN −→ 0. The action of WN on IN arising from this extension coincides with the canonical action bN )/d A bN , by (5.10). Note that we have defined this extension together of WN on 1 (A bN )/d A bN ) of with its splitting (5.2). It is given by the two-cocycle c ∈ Z 2 (WN ; 1 (A bN )/d A bN , read from (5.5), WN with values in 1 (A bN ). c(f i ∂bi , g j ∂bj ) = −∂bi g j d(∂bj f i )(mod d A

(5.13)

5.5. Let us consider the truncated and shifted de Rham complex bN −→ 1 (A bN ) −→ 0, • : 0 −→ A

(5.14)

bN ) in degree zero. It is a complex of WN -modules. We have an where we place 1 (A obvious map of complexes of WN -modules

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bN )/d A bN , • −→ 1 (A

(5.15)

where the target is regarded as a complex sitting in degree zero. Let us write down a two-cocycle c˜ ∈ Z 2 (WN ; • ) which is mapped to c, (5.13), under the map (5.14). Such a cocycle is by definition a pair (c2 , c3 ), where c2 ∈ bN )) is a two-cochain, and c3 ∈ C 3 (WN ; A bN ) is a three-cochain, such C 2 (WN ; 1 (A that dLie (c2 ) = dDR (c3 ),

(5.16a)

dLie (c ) = 0.

(5.16b)

3

Let us define and

c2 (f i ∂i , g j ∂j ) = ∂i g j d(∂j f i ) − ∂j f i d(∂i g j )

(5.17)

c3 (f i ∂i , g j ∂j , hk ∂k ) = ∂j f i ∂k g j ∂i hk − ∂k f i ∂i g j ∂j hk .

(5.18)

We write for brevity ∂i instead of ∂bi . One checks the compatibilities (5.16) directly. Thus, we have defined c. ˜ One sees that c˜ is mapped to −2c under (5.15). bN )/d A bN is trivial. This allows one to define the sheaf Och For N = 1 the space 1 (A X bN instead of  bN . for curves, acting as in 3.9, and starting from V Assume that N > 1. Using the computations of Gelfand–Fuchs, cf. [F], Theorems 2.2.7 and 2.2.4, one can show that the map in cohomology bN )/d A bN ) H 2 (WN ; • ) −→ H 2 (WN ; 1 (A

(5.19)

induced by (5.15) is an isomorphism. We have the canonical short exact sequence bN )) −→ H 2 (WN ; • ) −→ H 3 (WN ; A bN ) −→ 0. 0 −→ H 2 (WN ; 1 (A

(5.20)

the left- and right-most terms being one-dimensional. Under the second map of this sequence, our cocycle c˜ is mapped to its second component c3 which is a canonical bN ), cf. [F], Theorem 2.2.70 and representative of a generator of the space H 3 (WN ; A Chapter 2, Sect. 1, no. 4. In particular, our cocycle c˜ is non-trivial. It follows that the cocycle c defining the extension (5.12) is also non-trivial. What kind of an object does the cocycle c˜ define? Recall that a homotopy Lie algebra L• is a complex of vector spaces equipped with a collection of brackets [ , . . . , ]i : 3i L• −→ L• [−i + 2], i ≥ 2,

(5.21)

satisfying certain compatibility conditions, cf. for example [HS], Sect. 4. In particular [ , ]2 is a skew symmetric map, satisfying the Jacobi identity up to the homotopy (given by the third bracket). bN ), L−1 = A bN , the Let us define a complex L• as follows. Set L0 = WN ⊕ 1 (A other components being trivial. The differential L−1 −→ L0 is the composition of the bN ) ,→ L0 . de Rham differential and the obvious embedding 1 (A Let the second bracket [ , ]2 be given by the usual bracket of vector fields, and bN ). Define the third bracket with the only bN and 1 (A the action of vector fields on A nontrivial component being the three-cocycle c3 , (5.18). We set the other brackets equal to zero. This way we get a structure of a homotopy Lie algebra on L• . We have a canonical extension of homotopy Lie algebras 0 −→ • −→ L• −→ WN −→ 0.

(5.22)

Chiral de Rham Complex

461

Here • is an abelian ideal in L• (all brackets are zero). This is a refinement of extension (5.12). B. Projective Line 5.6. Let X be the projective line P1 . Let us fix a coordinate b on P1 , and consider the open covering X = U0 ∪ U1 , where U0 = Spec(C[b]), U1 = Spec(C[b−1 ]). ch on U with coordinate b, and O ch on U with coordinate Consider the sheaves OU 0 1 U1 0 −1 b , which were defined in 3.4. Let us glue them on the intersection U01 = U0 ∩ U1 using the transition function ˜ b(z) = b(z)−1 ,

(5.23a) 0

a(z) ˜ = b a(z) + 2b(z) . 2

(5.23b)

ch . In this way, we get the sheaf on the X, to be denoted OX ch ) admits a natural structure of an Theorem 5.7. The space of global sections 0(X; OX b irreducible vacuum sl2 -module on the critical level. ch ∩ O ch where both O ch are regarded as subspaces of O ch . It is We have to compute OU U1 Ui U01 0 the essence of the Wakimoto construction, [W], that the fields a(z)b(z)2 + 2b(z)0 , a(z) ch (resp., on O ch ), and under ˜ 0 , a(z)) ˜ 2 +2b(z) ˜ generate an b sl2 -action on OU (resp., a(z) ˜ b(z) U1 0 ch , i = 1, 2, become the restricted Wakimoto module with zero highest this action, OU i weight. (Restricted here means that the level is critical, and the Sugawara operators act ch ) and, therefore, by zero.) It follows from (5.23) that the b sl2 -action comes from 0(X; OX ch ∩ O ch is also an b ch contains a sl2 -module. It follows from [FF1] or [M] that each OU OU U1 0 i unique proper submodule which is isomorphic to the irreducible vacuum representation. ch ch ch To complete the proof, it remains to show that OU0 6= OU0 ∩ OU1 , and this is obvious. 4 ch ) is also isomorphic to the same 5.8. In fact, the first cohomology space H 1 (X; OX irreducible b sl2 -module. To prove this, let us compute the Euler character ch )= ch(X; OX

∞ X N=0

ch(N)

χ (X; OX

) · qN

in two different ways. First, by definition ch ch ch ) = ch(0(X; OX )) − ch(H 1 (X; OX )). ch(X; OX

By Theorem 5.7 and [M], ch )) = (1 − q)−1 ch(0(X; OX

∞ Y

(1 − q N )−2 .

N =1

ch carries a filtration F such that the On the other hand, formulas (5.23) imply that OX F image of a−n (resp., b−n ) (n ≥ 1) in Gr is a vector field (resp., a 1-form). It follows that

462

F. Malikov, V. Schechtman, A. Vaintrob ch(N )

each monomial · . . . · a−nr b−m1 · . . . · b−ms contributes 2s − 2r + 1 in χ(X; OX P a−n1 P where N = ni + mj . Therefore, ch )= ch(X; OX

∞ Y

),

(1 − q N )−2 ,

N=1

hence ch ch )) = q · ch(0(X; OX )). ch(H 1 (X; OX ch ) has the same (up to the shift by q) character as 0(X; O ch ). In other words, H 1 (X; OX X Again by [M], these two spaces are isomorphic as b g-modules.

C. Flag Manifolds 5.9. Let G be a simple algebraic group, B ⊂ G a Borel subgroup, N ⊂ B the maximal nilpotent subgroup. The manifold N is isomorphic to the affine space, and is a (g, B)-scheme, where B acts by conjugation. Consider the Heisenberg vertex algebra V associated with the affine space N. According to [FF2], V admits a structure of a b g-module R (Wakimoto module); in particular, V is a (g, B)-module. Note that x ∈ g acts on V as X(z) for some X ∈ V . Consequently, considered as an affine space, V admits a structure of a (g, B)-scheme. Let M be the algebra of functions on V . Proceeding as in 3.9, with K = B, Xˆ = G, and X = G/B, we get the sheaf of ind-schemes U 7 → Spec(H∇ (1(M))) on X. The sheaf of its C-points is called the chiral structure sheaf of X and denoted by ch . OX ch admits a more explicit construction, using charts 5.10. If G = SL(n) then the sheaf OX and gluing functions. In this case X = GL(n)/(B × C∗ ). The Weyl group is identified with the symmetric group Sn and can be realized as the subgroup of GL(n) consisting of permutation matrices. One checks that in terms of the Lie algebra gl(n), the simple permutation ri (interchanging i and i + 1) can be written as follows:

√ ri = exp(π −1Eii ) exp(Ei+1,i ) exp(−Ei,i+1 ) exp(Ei+1,i ).

(5.24)

where Eij (1 ≤ i, j ≤ n) form the standard base of gl(n). The manifold X is covered by |S| = n! charts, the chart associated with an element w ∈ Sn being Uw = wNw0 B, where N ⊂ B is the unipotent subgroup consisting of all upper-triangular matrices and w0 ∈ Sn is the element of maximal length. Let us identify Uw with N using the bijection n 7 → wnw0 B. Under this identification, if x ∈ Uw1 ∩Uw2 , then the change from the coordinates determined by Uw1 to the ones determined by Uw2 , is given by x 7 → w2−1 w1 x. ch , we declare Each Uw may be identified with the affine space Cn(n−1)/2 . To define OX ch ch that OX U = OUw , where the last sheaf is defined in 3.4. Now we have to glue these w sheaves over the pairwise intersections in a consistent manner.

Chiral de Rham Complex

463

ch ). First, we extend the b Let V denote the vertex algebra Vn(n−1)/2 = 0(Uw ; OU sl(n)w module structure on V to a gbl(n)-module structure. For that, in addition to the formulae in [FF1], p. 279, define X X bij a ij (z) + bj i a j i (z). (5.25) Eii (z) = − j >i

j