Formality theorems: from associators to a global formulation

Let M be a differential manifold. Let Φ be a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting from Φ. More precisely, following Tamarkin’s proof, we construct a Lie homomorphism “up to homotopy" between the Lie algebra of Hochschild cochains on C∞(M) and its cohomology (Γ(M, ΛTM), [−,−]S). This paper is an extended version of a course given 8 12 March 2004 on Tamarkin’s works. The reader will find explicit examples, recollections on G∞-structures, explanation of the Etingof-Kazhdan quantizationdequantization theorem, of Tamarkin’s cohomological obstruction and of globalization process needed to get the formality theorem. Finally, we prove here that Tamarkin’s formality maps can be globalized.


Introduction
Let M be a differential manifold and A = C ∞ (M ) the algebra of smooth differential functions over M . Formality theorems link commutative objects with their non commutative analogs. More precisely, one has two graded Lie algebra structures: -The space T poly = Γ(M, ΛT M ) of multivector fields on M . It is endowed with a graded Lie bracket [−, −] S called the Schouten bracket (see [20]), extending the Lie bracket of vector fields (see Example 2.3 in section 1).
-The space D poly = C(A, A) = k≥0 C k (A, A), of regular Hochschild cochains (generated by differential k-linear maps from A k to A and support preserving). This vector space D poly is also endowed with a differential graded Lie algebra structure given by the Gerstenhaber bracket [−, −] G [9] and coHochschild differential b (see Example 2.4 in section 1).
We have: Theorem 1.1. The cohomology H * (D poly , b) of D poly with respect to b is isomorphic to the space T poly ( [15]).
This map ϕ 1 is not a differential Lie algebra morphism but it is "up to (higher) homotopy". Formality maps are the collection of those homotopies: they are maps, ϕ 1,...,1 : Λ n T poly → D poly , for n ≥ 0, such that where we have "extended" the Lie bracket [−, −] S to a coderivation d 1,1 T : Λ · T poly → Λ · T poly , the Lie bracket [−, −] G and the differential b to coderivations d 1,1 D and d 1 D : Λ · D poly → Λ · D poly and the maps ϕ 1,...,1 to morphisms of coalgebras : Λ · T poly → Λ · D poly on the corresponding cofree cocommutative coalgebras. In the first section of this paper we will recall precise definitions of Λ · E for E a graded vector spaces, of the above maps and of their "extension".
Existence of such homotopies was proven for M = R d by Kontsevich (see [18] and [19]) and Tamarkin (see [22]). They use different methods in their proofs. Kontsevich proved also that those maps can be globalized on a general manifold. When M is a Poisson manifold equipped with a Poisson bracket corresponding to a Poisson 2-tensor field π (such that [π, π] S = 0), one can deduce the existence of a star-product m on M , i.e. an associative product on A[[ ]] for a formal parameter: m = m + ϕ 1 (π) + n≥2 n n! ϕ n (πΛ · · · Λπ).
Notice that until the end of the paper, we will use the notation Λ for the product on the exterior algebra Λ · E and ∧ for the exterior product on T poly . The fact that m is associative, i.e. [m , m ] G = 0, follows from equation (1.1) and one has ϕ 1 (π) = {−, −}, the Poisson bracket.
We will follow Tamarkin's proof and show how to build such homotopies. In the first three sections, we will suppose that M = R d .
The paper is organized as follows: • In Section 2, we will make precise definitions of L ∞ and G ∞structures and morphisms used to define the formality maps. Explicit formulas will be given.
• In Section 3, we will show that the space D poly can be endowed with a G ∞ -structure. This is where associators and Etingof-Kazhdan theorem will be needed. We will outline proofs by Etingof and Kazhdan and also by Enriquez.
• In Section 4, we will construct the formality maps when the manifold M = R d . To do so, we will describe obstructions to such a construction and show that they vanish when M = R d .
• In Section 5, we will prove that those maps can be globalized when M is an arbitrary manifold. To do so, we will follow Dolgushev's approach ( [2]) where the globalization process was done to local Kontsevich's maps.
I would like to thank D. Manchon and D. Arnal for their invitation in Dĳon, D. Calaque, V. Dolgushev, G. Ginot and B. Keller for many useful suggestions and B. Enriquez and P. Etingof for helping better understand the Etingof-Kazhdan dequantization theorem.
In particular, we have Let us recall the two examples T poly and D poly : The space T poly is a graded Lie algebra (and so a L ∞algebra) with 0 differential and Schouten bracket [−, −] S defined as follows we set [α, f ] S = α · f , the action of the vector field α on f . The grading on T poly is defined by |α| = n ⇔ α ∈ Γ(M, Λ n+1 T M ) and the exterior product is graded commutative: Let us denote d T the associated coderivation (d 1,1 T is corresponding to Example 2.4. Similarly, D poly is a differential graded Lie algebra (and so a L ∞ -algebra). Its bracket, the Gerstenhaber bracket [−, −] G , is defined, for D, E ∈ D poly , by The space D poly has a grading defined by | D |= k ⇔ D ∈ C k+1 (A, A). Finally, its differential is the coHochschild differential One can now define the generalization of Lie algebra morphisms: is a morphism of differential cofree coalgebras, of degree 0, In particular ϕ • d 1 = d 2 • ϕ. As ϕ is a morphism of cofree cocommutative coalgebras (i.e. ∆ 2 ϕ = (ϕ ⊗ ϕ)∆ 1 where ∆ 1 and ∆ 2 are the coproducts on E 1 and E 2 ), ϕ is determined by its image on the cogenerators, i.e., by its components: ϕ 1,...,1 : Λ k E 1 → E 2 [1]. Again one gets a general formula for ϕ: where the signs are Quillen's signs corresponding to permutations of odd elements. Now equation (1.1) can be rewritten as follows: let d T and d D correspond respectively to the Lie algebra structure on T poly and to the differential Lie algebra structure on D poly . We want to construct a L ∞morphism ϕ such that ϕ 1 is the Hochschild-Kostant-Rosenberg map and: If one tries to construct the maps ϕ 1,...,1 : Λ n T poly → D poly by induction on n, one will find obstructions in the non acyclic Chevalley Eilenberg complex Hom(Λ T poly , T poly , [d T , −]).
Tamarkin's idea was then to extend the structure (or increase the constrains) to reduce the obstructions. Indeed, T poly has a Gerstenhaber structure. It would be convenient to find such a structure on D poly (we will see that D poly has actually a G ∞ -structure i.e. an "up to homotopy" Gerstenhaber structure) and to construct a G ∞ -morphism between them (that restricts to a L ∞ -morphism on the corresponding Lie algebra structures). Thanks to the addition of those extra operations, we will see that obstructions to the construction of G ∞ -morphisms will vanish in the case M = R d . Let us end this section by some recollections on G ∞ -structures. We will follow works of Ginot ([10]).
To define a G ∞ -structure on E, we will need a bigger space than ΛE. Let us denote c T (E) the cofree tensor coalgebra of E (with coproduct ∆ ). We will sometimes use the notation E ⊗· . Equipped with the shuffle product • (defined on the cogenerators c T (E) ⊗ c T (E) → E as pr ⊗ε + ε ⊗ pr, where pr : c T (E) → E is the projection and ε is the counit), it is a bialgebra. Let c T (E) + be the augmentation ideal. We note c T (E) = c T (E) + /( c T (E) + • c T (E) + ) the quotient by the shuffles. It has a graded cofree coLie coalgebra structure (with coproduct δ = ∆ − ∆ op ), see [12] for example. Then S( c T (E) [1]) has a structure of cofree coGerstenhaber algebra (i.e., equipped with cofree coLie and cofree cocommutative coproducts δ and ∆ satisfying compatibility condition). One can write δ explicitly: for γ i ∈ E ⊗p i , , where the sum is over all integers 1 ≤ k ≤ n, 1 ≤ j ≤ p k and all permutations ε fixing k which are (i, n−1−i)-shuffles on {1, . . . , n}−{k}. We have denoted γ k = α k 1 · · · α k j α k j+1 · · · α k p k and the sign s k = (−1) (|α 1 |+···+|α j |)(p k −j) . Moreover, we still have: where the sum is over (i, n − i)-shuffles. We use the notation c T m (E) for the elements of degree m, and, for γ 1 1 , . . . , γ pn n ∈ E, we have Definition 2.6. A vector space E is endowed with a G ∞ -algebra (Gerstenhaber algebra "up to homotopy") structure if there are degree one linear maps d p 1 ,...,p k : c T p 1 (E)Λ · · · Λ c T p k (E) ⊂ Λ kc T E → E [1] such that the associated coderivations (extended with respect to the cofree coGerstenhaber structure on Λ c T (E)) d: More details on G ∞ -structures are given in [10]. In particular we have Applying this remark to the spaces T poly and D poly we get Let us, as an exercise, extend maps d 1,1 T and d 2 T for degree 0 elements α, β, γ in T poly .
. and so the condition d 2 T . In the same way, we have: by compatibility between [−, −] S and ∧. So all the identities defining the Gerstenhaber algebra structure on T poly can be summarized into the unique relation (d 1,1 The space D poly is not a (graded) Gerstenhaber algebra when equiped with the product of cochains ∪ defined, for D, E ∈ D poly and x 1 , . . . , x |D|+|E|+2 ∈ A, by where γ = (|E| + 1)(|D| + 1). The projection of this product on the cohomology of (D poly , b) is the exterior product ∧, but unfortunately (D poly , [−, −] G , ∪, b) is not a Gerstenhaber algebra: one can see, for example, that ∪ is not a graded commutative product and thus can not be defined as a map D poly ⊗2 → D poly . More generally, Gerstenhaber's cachain structure have the same "failure", only the cohomology behaves well.
We will show in Section 2 that it can be equiped with a G ∞ -structure. One can now define the generalization of Gerstenhaber morphisms: is a morphism of differential coGerstenhaber coalgebras, of degree 0, In particular ϕ • d 1 = d 2 • ϕ. As ϕ is a morphism of cofree coGerstenhaber coalgebras, ϕ is determined by its image on the cogenerators, i.e., by its

A G ∞ -structure on the space of cochains
The objective of this section is to prove the following proposition ( [22]).
(3) d 2 D is the cup product ∪, up to a Hochschild coboundary.

Construction of the G ∞ -structure
We first reformulate this problem: let L D = ⊕ D poly ⊗n be the cofree coLie coalgebra on D poly (see Section 2 for the notation). Since L D is a cofree coLie coalgebra, a differential Lie bialgebra structure on L D is uniquely determined by the restriction to cogenerators of the Lie bracket and the differential (which are coderivations on L D ) and so by degree one maps l n D : D poly ⊗n → D poly (for the differential L D → L D ), and maps l p 1 ,p 2 D : D poly ⊗p 1 ΛD poly ⊗p 2 → D poly (for the Lie bracket L D ΛL D → L D ). The following lemma is well known.

Lemma 3.2.
Suppose we have a differential Lie bialgebra structure on the coLie coalgebra L D , with differential and Lie bracket respectively determined by maps l n D and l p 1 ,p 2 D as above. Then D poly has a G ∞ -structure given, for all p, q, n ≥ 1, by Thus to obtain the desired G ∞ -structure on D poly , it is enough to define a differential Lie bialgebra structure on L D given by maps l n D and l p 1 ,p 2 up to homotopy". Let us now give an equivalent formulation of our problem, which is stated in terms of the associated operads in [22]: Proposition 3.3. Suppose we have a differential bialgebra structure on the cofree tensorial coalgebra T D = ⊕ n≥0 D poly ⊗n with differential and multiplication given respectively by maps a n D : D poly ⊗n → D poly and a p 1 ,p 2 D : Then we have a differential Lie bialgebra structure on the coLie coalgebra L D = ⊕ n≥0 D poly ⊗n , with differential and Lie bracket respectively determined by maps l n D and l p 1 ,p 2 D where l 1 D = a 1 D , l 1,1 D is the anti-symmetrization of a 1,1 D and l 2 D = a 2 D "up to homotopy".
A differential bialgebra structure on the cofree tensorial coalgebra ⊕V ⊗n associated to a vector space V is often called a B ∞ -structure on V , see [1].
Proof. The proof relies on the existence of a quantization/dequantization functor, that we will recall in the next subsection. Let V be a finitedimensional vector space and V * be the dual space. A differential bialgebra structure on the cofree coalgebra c T V = ⊕ n≥0 V ⊗n is defined on the cogenerators by maps a n : V ⊗n → V (n ≥ 2), corresponding to the differential n≥0 a n : c T V → c T V , and maps a p 1 , We can define dual maps of those maps to get again a differential bialgebra with differential D:T →T and coproduct ∆:T →T⊗T , whereT is the completion of the tensor algebra ⊕ n≥0 V * ⊗n . The differential and coproduct D and ∆ are defined now on the generators of the free algebraT by maps a n * : V * → V * ⊗n and a p 1 ,p 2 * : V * → V * ⊗p 1 ⊗ V * ⊗p 2 . The tensor algebra ⊕ n≥0 V * ⊗n is graded as follows: |x| = p when x ∈ V * ⊗p .
Similarly, if we consider a differential Lie bialgebra structure on the cofree coLie coalgebra L = ⊕ n≥0 V ⊗n , the dual maps d and δ of the structure maps n≥0 l n and p 1 ,p 2 ≥0 l p 1 ,p 2 induce a differential Lie bialgebra structure onL, the completion of the free Lie algebra We now replace formally each element x of degree n inT (resp.L) by h n x, where h is a formal parameter. Letting |h| = −1, we easily see that it is equivalent to define • a differential associative (respectively Lie) bialgebra structure on the associative (resp. Lie) algebras with the product and coproduct being of degree zero • or a differential associative (resp. Lie) bialgebra structure on the associative (resp. Lie) algebraT (resp.L).
Note that those two bullets are dual. Thus we have a differential free We can apply now Etingof-Kazhdan's dequantization theorem for graded differential bialgebras ( [7] and Appendix in [11]) to our particular case where we start from a differential bialgebra free as an algebra (T , ∆, D): this proves that  depending only on ∆ and the product ofT ), if we apply Etingof-Kazhdan's quantization functor (see [6]) The last condition implies thatL is free as a Lie algebra becauseT is free as an algebra. Moreover the structure maps l p * D and l p,q * D onL satisfy is the anti-symmetrization of a 1,1 * D ans l 2 * D = a 2 * D "up to homotopy". Taking now dual maps, we get the result.
Remark 3.5. Here one strongly used the quantization/dequantization theorem. Indeed, if one only takes the anti-symmetrization and the classical limit to get the wanted Lie algebra structure on L D , one will lose the information on degree 2 maps and in particular the information on l 2 D . Recall that we wanted l 2 D = ∪ "up to homotopy" and by taking the naive classical limit one would get l 2 D = 0 which will then only give the Lie algebra structure on D poly that we started with ! By Proposition 3.3, the problem of defining a differential Lie bialgebra structure on L D given by maps l n D and up to homotopy" is now equivalent to defining a differential bialgebra structure on T D given by maps a n D : D poly ⊗n → D poly and a p 1 ,p 2 D : D poly in Section 0 and a 2 D = ∪ "up to homotopy". Indeed, the anti-symmetrization of {−|−} is by definition [−, −] G . The latter can be achieved using the braces operations (defined in [9]) acting on the Hochschild cochain complex D poly = C(A, A) for any algebra A. The braces operations are maps a 1,p D : D poly ⊗ D poly ⊗p → D poly (p ≥ 1) defined, for all homogeneous D, E 1 , . . . , E p ∈ D poly ⊗p+1 and x 1 , . . . , [23] asserts (see also [9] and [17]) that: • The maps a 1,p D : D poly ⊗ D poly ⊗p → D poly , a q≥2,p D = 0 and the degree 0 shuffle product determine a coderivation = a p,q D on the cofree tensorial coalgebra T D = ⊕ n≥0 D poly ⊗n which turns T D into a bialgebra.
• Similarly taking a 1 D to be the Hochschild coboundary b and a 2 D to be the cup-product ∪, and a q≥3 D = 0, the coderivation d = a n D defines a differential structure on the tensor coalgebra T D .
• These maps yield a differential bialgebra structure (T D , , d) on the cofree coalgebra T D .
Actually, one only need to prove the associativity condition as the differential is given by the commutator (with respect to the product ) [m, −] with the multiplication m on A. Let us prove the three points for the first orders with respect to the degree: • Let us check that a 1 D + a 2 D is a differential. For A, B in D poly one gets: • Let us check the associativity of = a 1,1 D + a 1,2 D + · · · up to order 2. For A, B, C in D poly , one gets (here we forget the signs): • Let us check the compatibility condition between = a 1,1 D + a 1,2 D + · · · and the differential d = a 1 D + a 2 D up to order 2. For A, B in D poly , one gets (here again we forget the signs):

The quantization/dequantization fonctor
Let us recall the definition of a Drinfeld associator (cf [4]): Let T n be the algebra generated by elements t ij , 1 ≤ i, j ≤ n, i = j, with defining relations t ij = t ji , [t ij , t lm ] = 0 for i, j, l, m distincts and [t ij , t ik + t jk ] = 0 for i, j, k distincts. Let P 1 , . . . , P n be disjoint subsets of {1, . . . , m}. There exists a unique homomorphism ρ P 1 ,...,Pn : T n → T m defined by For any X ∈ T n , we denote ρ P 1 ,...,Pn (X) by X P 1 ,...,Pn . Let Φ ∈ T 3 . The relation The relations are called the hexagon relations.
An element Φ ∈ T 3 satisfying the pentagon and hexagon relations is called a Drinfeld associator. Such associators exist over C ( [3]). They are obtained from the KZ equations. Drinfeld also prove that such associators exist over Q.
In this subsection we recall the following theorem (see appendix in [11]) which gives, as a consequence, Proposition 3.4 (here is where an associator Φ is used): Theorem 3.6. There exists an equivalence of categories from the category of differential graded quantized universal enveloping graded algebras to that of differential graded Lie graded bialgebras such that if U ∈ Ob(DGQUE) and a = DQ Φ (U ), then U/ U = U(a/ a), where U is the universal algebra functor, taking a differential graded Lie graded algebra to a differential graded graded Hopf algebra.
This theorem is a consequence of the Etingof-Kazhdan quantization theorems. The key point is that the quantization theorem is "universal" and so will be valid for any symmetric category and so for complexes (V · , d · ). A right way to understand the "universality" is to use the language of operads and props. We will not recall the definitions in this paper.
Let us outline the construction of the quantization functor starting with an associator Φ. Let (g, δ) be a Lie bialgebra. Let D = g ⊕ g * be its associated Lie bialgebra double. Let r ∈ g ⊗ g * ∈ D ⊗2 be the canonical rmatrix (corresponding to the identity map) and t = r +r 2,1 ∈ S 2 (D) D . Let us consider the homomorphism T n → U (D) ⊗n sending t ij to t i,j (where components of t are put in the i-th and j-th place in the tensor product). We will still denote by Φ the image of Φ by this homomorphism.
We get that (U (D)[[ ]], m 0 , ∆ 0 , R 0 = e t/2 , Φ) is a quasi-triangular quasi-Hopf algebra ( [4]). Quasi-triangular means that for all a ∈ U (D) and quasi-Hopf means that the coproduct ∆ 0 is quasicoassociative, that is to say for all a ∈ U (D). To make this quasi-Hopf algebra into a Hopf algebra, one has to twist Φ into the identity, that is to say one has to construct J ∈ U (D) ⊗2 such that and so (U (g), Ad(J) • ∆ 0 ) is then a quantization of (g, δ). Notice that the product in U (g) is not the same as the one in U (D) (and so the product in U (D)) as the algebra isomorphism U (g) U (g)[[ ]] is not the identity (which itself is not an algebra morphism).
Let us end this subsection showing how one can construct the twist J. In [6], the construction was done using the "categorical yoga" and one gets a general formula: where M + and M − are respectively the Verma module Ind D g 1 and Ind D g * 1, 1 + , and 1 − are respectively the generators of those module over U (g * ) and U (g) and φ is the isomorphism U (D) → M + ⊗ M − generated by the assignment 1 → 1 + ⊗ 1 − . Finally, s is the twist in the tensor product. As an exercise, let us calculate the first terms of J. Let {a i } be a basis of g and {b i } its dual basis, a basis of g * . So r = a i ⊗ b i . Let us write the structure constants: Starting from an asso- To get universal formulas, one has then to reorder the terms in J.
In [5], Enriquez proposed a cohomological construction of the twist J. He looks for this element in a "universal" algebra U univ made from the r-matrix. The definition is rather complicated and uses the language of props. We will retain that it is generated by the components of r, i.e. words in {a i } and {b j } (with as many a's as b's) and the relations (the r-matrix relations): This allows to write all the a's on the left hand side and all the b's in the right hand side. In the same spirit one can define where FA N is the free algebra with generators x i , i = 1, . . . , N , graded by ⊕ i Nδ i (x i has degree i). We view Φ as an element of U ⊗3 univ and we will build J = 1 + r 2 + · · · ∈ U ⊗2 univ such that equation (3.1) is fulfiled in U ⊗3 univ . The construction is made by induction. Suppose we have built J = 1 + · · · + n J n + · · · up to order n − 1. Equation (3.1) at order n is equivalent to It is well known that ker d coHo n = Im d coHo n−1 ⊕Λ n (D univ ) (this is true for any enveloping algebra). For any choice of J k , k ≤ n − 1, Φ n + J 1 , . . . , J n−1 is in Ker d coHo 2 . Moreover, one can always replace J n−1 with J n−1 + λ n−1 (λ n−1 ∈ Λ 2 (D univ )) so that we still have a solution up to order n − 1. The equation we want to solve now is the following equation with unknown (J n , λ n−1 ): Actually, we have a complex, making |[r, −]| into a differential: when 0 ≤ k ≤ n − 1, let us define Then we have a complex ((Λ · (D)) univ , ∂ · ) where It turns out that the 3-rd cohomology group of that complex is 0 if the "degree" in a's and b's is greater than 3 and is spanned by the class of [t 1,2 , t 2,3 ] otherwise. Moreover, one checks that Alt((∂ ⊗Id ⊗2 ) univ (µ n )) = 0 so there exists λ n−1 ∈ Λ 2 (D univ ) such that which gives the induction step and allows us to construct J.
Remark 3.7. Following Enriquez's proof, it seems that the term J n−1 in the -series of J is built from terms Φ n−1 and Φ n in the -series of Φ, as we had to correct J n−1 by λ n−1 which seems to be dependent from Φ n . On the other hand, it is clear, from the Etingof-Kazhdan's formula that their J n−1 do not depend from Φ n . This is not surprising: in Enriquez's construction, the correcting term λ n−1 only depends on an anti-symmetrization of Φ n which is unique (it is an easy check).

A G ∞ -morphism between chains and tensor fields
4.1. A differential d T on Λ · T poly ⊗· and G ∞ -morphism The objective of this section is to prove the following proposition: Proof. For i = T or D and n ≥ 0, let us set i . Let d p 1 ,...,p k D : D poly ⊗p 1 Λ · · · ΛD poly ⊗p k → D poly be the components of the differential d D defining the G ∞ -structure of D poly (see Definition 2.6) and denote d D .

G. Halbout
D . In the same way, we define d

[n]
T and d [≤n] T . We know from Section 2 that a morphism ψ: is uniquely determined by its components Similarly we set We have to build both the differential d T and ψ, the morphism of differential. In fact we will build the maps d Suppose we have built maps (d . These conditions are enough to insure that d T is a differential and ψ a morphism of differential coalgebras. If we reformulate the identity ψ T , we get (4.1) If we take now into account that d [1] T = 0, and that on V D • ψ [l] = 0 for k + l > n + 1, the identity (4.1) becomes T and A = d D ψ [≤n−k+1] (we now omit the composition sign •). The term d  D (B − A) = 0 which is true by direct computation (see [11]). We also have to show that for any choice of those maps, we have Again this is always true by direct computation (see again [11]).
Thus d T [2] is the image of d [2] D through the projection on the cohomology of D poly and as the Hochschild-Kostant-Rosenberg map ψ [1] is injective from T poly =H(D poly , b = d T .

Remark 4.2.
The main tool we have used here is the existence of a quasiisomorphism between the complexes (T poly , 0) and (D poly , b). Since we know explicit homotopy formulas for such a quasi-isomorphism (see [21], [13]), we can obtain explicit formulas for d T

A G
In this subsection, we will prove the following proposition. such that the induced map ψ [1] : T poly → T poly is the identity.
We will use the same notations for V Proof. We will build the maps ψ [n] by induction as before. For ψ [1] we have to set: ψ [1] = Id (the identity map).

Suppose we have built maps
If we now take into account that d We have seen in the previous section that d T Notice that d [2] T = d 1,1 T +d 2 T . By the acyclicity of the complex (End(Λ · T poly ⊗· ), [d [2] T , −]), the construction of ψ [≤n] will be possible when is a cocycle in this complex, which is true by direct computation (see [11] In this section the manifold M is supposed to be the Euclidian space R d for m ≥ 1. We prove the following proposition: Proof. Since morphism of coalgebras Λ · T poly ⊗· → Λ · T poly ⊗· are in one to one correspondence with maps Λ · T poly ⊗· → T poly , we are left to check that the cochain complex is acyclic. Firstly, we introduce an "external" bigrading on the cochain complex induced by the following bigrading on Λ · T poly ⊗· : if x ∈ T poly ⊗p 1 Λ · · · ΛT poly ⊗pn , |x| e = (p 1 −1+· · ·+p n −1, n−1). This grading gives a bicomplex structure on the vectorial space Hom(Λ · T poly ⊗· , T poly ), [d 1,1 We now use the fact that (T poly , d 2 T ) = (Γ(M, ΛT M ), ∧) is a polynomial algebra to show that In particular, the differential d CE is induced by the usual exterior derivative (see [15]) on Hom T poly (T poly ⊗Λ · T poly ⊗· , T poly ). Proposition 4.5 can be proved using spectral sequences but can also be obtained directly.
Proof. We have explicit quasi-isomorphisms and homotopies between T poly ⊗·+1 and Λ · Ω T poly : J : T poly ⊗·+1 → Λ · Ω T poly sending γ 0 ⊗ · · · ⊗ γ n to γ 0 dγ 1 · · · dγ n , I : Λ · Ω T poly → T poly ⊗·+1 , the anti-symmetrization given by and explicit homotopies s : T poly ⊗·+1 → T poly ⊗·+2 described in [13] such that J • I = Id and I • J = Id + d • s + s • d. One can extend those maps to have quasi-isomorphisms and homotopies between T poly ⊗T poly ⊗· and Λ · Ω T poly . Finally, since Λ · T poly T poly ⊗T poly ⊗· is a bicomplex with dif- [16], Section 3 that there exists a map u : Λ · T poly T poly ⊗T poly ⊗· and a (degree one) map T poly ⊗T poly ⊗· [1] such that pu = Id To finish the proof of Proposition 4.4, we proceed as in [22] and [14]. Recall from the introduction that A = C ∞ (R d ) is the algebra of smooth functions on R d . Let Der(A) = Ω * A be the space of smooth derivations on A. Since T poly is a A-module, by transitivity of the space of Kähler differentials for smooth manifolds, one has Since T poly ∼ = Λ * A Der(A), we find that Ω T poly /A ∼ = T poly ⊗Der(A) (with grading shifted by minus one on Der(A)). Hence (see [22].3.5) there is an isomorphism where d dR is de Rham's differential (the degree on the left hand of the isomorphism is the one induced by the inner degree of T poly ). When T poly = Γ(R d , ΛR d ) this complex is acyclic.

Globalization process
In this section, we recall the process of globalization of formality maps. Globalization was proven by Kontsevich in [18]. Here we will present Dolgushev's approach which uses Fedosov methods. This approach is actually very similar to the one of Kontsevich but maybe more explicit. The idea is to first write formality theorem locally on bundles that can be seen as bundles of the Taylor expansion (in the neighbourhood of the base points) of the considered objects. Let us define those bundles as done in [8] by Fedosov: ..i l are coefficients of a tensor with symmetric covariant part (indices i 1 , . . . , i l ) and antisymmetric contravariant part (indices j 0 , . . . , j k ).
• D · := W ⊗ T ·+1 (SE) is the graded bundle of formal fiberwise polydifferential operators. Local homogeneous sections of degree k look like as follow where α s are multi-indices, and P α 0 ...α k i 1 ...i l are coefficients of a tensor with symmetric covariant part (indicies i 1 , . . . , i l ) which is also symmetric in indices α 1 s , . . . , α d s for any s = 0, . . . , k. From now on, and until the end of this section, B denotes any of these three bundles. For our purpose, we need to tensor B by the exterior algebra bundle ΛT * M (in other words we consider differential forms with values in B). These new bundles B := ΛT * M ⊗ B carrie natural fiberwise algebraic structures; namely • W is a bundle of graded commutative algebras with grading given by the exterior degree of forms, which is also filtered (as an algebra) by the polynomial degree in the fibers.
• T and D are endowed with fiberwise dgla-structures respectively induced by those of T poly and D poly . Grading is given by the sum of the exterior degree and the degree in B · .
In what follows, and when it does not lead to any confusion, we denote the same operations on bundles B by the same letters. We also use dual local basis Proof. Let us introduce the operator δ * = y i ι(e i ) of contraction with the Euler vector field Θ = y i e i . Then we define the homotopy operator κ to be 1 k+l δ * on k-differential forms with value in B and l-polynomial in the fibers for k + l > 0, and 0 on sections of B constant in the fibers. Then by a direct computation one obtains where Hu ∈ F 0 B is the harmonic part of u, that is to say its homogeneous part of zero exterior degree and constant in the fibers.
Suppose now that we have a torsion free connection ∇. Such a connection, which always exists, defines a derivation of W, that we denote by the same symbol ∇. Namely, let Γ k ij (x) :=< ξ k , ∇ e i e j > be Christoffel's symbols of ∇, then locally It obviously extends to derivations of the graded Lie algebras T and D. Namely ij y j ∂ ∂y k , m] G = 0, then ∇m = 0 and thus ∇ (anti)commutes with b in D. Since the connection is torsion free one can also show by a direct computation that ∇ and δ (anti)commute.
The standard curvature tensor of ∇ induces an operator R on B which is given locally by Eventhough ∇ is not nilpotent in general, we use it to deform the differential δ on B. Namely This equation has a unique solution and using Bianchi's identity ∇R = δR = 0, homotopy property (5.1), κA = HA = 0, and the fact that κ raises the polynomial degree in the fiber one can show that D 2 = 0.
In what follows we refer to the nilpotent differential D as the Fedosov differential.
The following theorem states that the δ-cohomology described in proposition 5.1 is equal to the cohomology given by Fedosov differential D. Proof. This follows essentially from a spectral sequence argument. Namely, let us denote by F p B the sheaf of homogeneous sections of polynomial degree p in the fibers; then remark that D(F ≥p+1 B) ⊂ F ≥p B and that D = −δ mod F ≥p+1 B. Thus there is a spectral sequence with E p,q 1 ∼ = H p+q (F p B, δ) which converges to H * (B, D); then we conclude using proposition 5.1.
Following [2], one can define explicitly an isomorphism ϑ : F 0 B → Z 0 (B, D): it is the linear map that assigns to any section u 0 of F 0 B the unique section u of B satisfying the equation It is proved in [2] (proof of theorem 3) that this defines a bĳective linear We will now define a L ∞ -morphism ϕ: . We will suppose that the L ∞ -morphism ϕ define in the previous sections satisfies the following conditions: (1) The L ∞ -morphism is local and it can be made equivariant with respect to linear transformations of the coordinates on R d 0 .
(2) For any set of vector fields is linear in the coordinates on R d 0 , then for any set of multivector fields γ i ∈ Γ(R d 0 , ΛT R d 0 ): ϕ 1,1,...,1 (αΛγ 2 Λ · · · Λγ m ) = 0. (5.4) Thanks to the first conditions, it is obvious that such a morphism naturally extends to a morphsim (T, d T ) → (D, d D ). Moreover, it commutes with the differential d. Let us now write ∇ = d +[B, −] and defineφ, the twit of ϕ by B as follows: It is a well known fact (see [11] for example) thatφ is a L ∞ -isomorphism from (T, . Thanks to the second condition, we get ϕ(BΛ · · · ΛB) = B. Finally, one can prove (see [2] for example) that the term in B that depends on the choice of the local trivialization is linear in the fiber coordinates soφ does not depend on a choice of local coordinate thanks to the third condition. Finally, we have the following diagram: To end the proof, one has to show that the morphismφ • ϕ T can be deformed into a map T poly → Z 0 (D, D) D poly · . This can be done using general arguments on L ∞ -isomorphisms or explicitally as in [2]

Existence of globalizable formality maps
In this part, we will show that one can construct a G ∞ -morphism which, when reduced to a L ∞ -morphism is globalizable that is to say satisfies the three conditions described in the previous subsection. Here is our main theorem: Theorem 5.5. Suppose M = R d and we are given a G ∞ -structure on D poly given by a differential d D as in Section 2. One can construct a G ∞morphism ϕ: T poly → D poly satisfying the extra conditions: (1) The G ∞ -morphism is local (one can replace R d by its formal completion R d 0 at the origin, or in other words, one can replace the functions with their Taylor expansion) and it can be made equivariant with respect to linear transformations of the coordinates on R d 0 .
Proof. By construction, the maps d p,q D are invariant under the action of linear vector fields and even quadratic functions and constant 2-vector fields. In other words, those maps are invariant under the action of gl d,d . Let us prove the lemma by induction on p. Supposoe the result is true for p > 1. Let us write γ = γ 1 · · · γ p+1 . For α ∈ gl d,d , let us write α· for the action of α. Then invariance under the action of gl d,d implies that, for any α, β ∈ gl d,d , one has β · d 1,p+1 D (α, γ) = d 1,p+1 D (β · α, γ) + d 1,p+1 D (α, β · γ).
Let us now write the Jacoby identity for β, α and γ. Using the induction hypothesis, we get: The theorem will now follow if we prove that points 2 and 3 of Tamarkin's construction are still true with ψ and ψ satisfying the extra conditions of Theorem 5.5 and d T satisfying conditions (5.6) for n ≥ 3 or n = 2 and p 2 > 1.
• We want first to construct the maps d Note that ϕ 1 satisfies the first conditions of Theorem 5.5. Now suppose the construction is done for n − 1 (n ≥ 2), i.e., we have built maps (d We have proved that for any such (d T and ψ [n] such that condition (5.9) is true for n instead of n − 1, as this last statement is equivalent to ϕ 1 d T [n] = bψ [n] + A where A is always a Hochschild cocycle.
-It is obvious (use homotopy formulas of [13]) that the first condition in Theorem 5.5 can then be satisfied for those maps d T T and ψ [n] when they are supposed to be true by induction for k ≤ n−1. Using the induction hypothesis in Equation (5.9) and the fact that d p 1 ,...,pn D = 0 for n > 2 and d 1,p D (α, γ 1 · · · γ p ) = 0 for p > 1 and any linear vector field α, one can see that those conditions are equivalent to [X, ψ [n−1] (· · · Λx 1 i ⊗ · · · ⊗ x p i i Λ · · · )] G = ±ψ [n−1] (· · · Λ· · · ⊗ [X, x n ij i ] S ⊗ · · ·Λ · · · ), (5.10) H ⇔ i X U = L X U = 0. It suffices now to show H is acyclic which is true because so is H and H is quasi-isomorphic to the relative cochain complex C * (gl d [ ], gl d ; H).
To prove this quasi-isomorphism, split gl d -equivariantly T poly = gl d ⊕ h; this induces an isomorphism of gl d [ ]-modules H ∼ = i hom(Λ i gl d , H ).
Let us discuss the differential on the right hand side of this formula corresponding to that on H under our identification. Let F be the filtration of H given by F k H = H ∩F k H, where in turn, F k H consists of all elements which vanish on T poly ⊗p 1 Λ · · · ΛT poly ⊗p i as long as p 1 + · · · + p i < k. The differential is induced by that in C * (gl d , H ) ∼ = i hom(Λ i gl d , H ) modulo a term which increases F . An easy spectral sequence argument implies then the statement.