The Homotopy Class of twisted L_\infty-morphisms
aa r X i v : . [ m a t h . QA ] F e b The Homotopy Class of twisted L ∞ -morphisms Andreas Kraft ∗ ,Dipartimento di MatematicaUniversità degli Studi di Salernovia Giovanni Paolo II, 12384084 Fisciano (SA)Italy Jonas Schnitzer † ,Department of MathematicsUniversity of FreiburgErnst-Zermelo-Straße, 1D-79104 FreiburgGermanyFebruary 23, 2021 Abstract
The global formality of Dolgushev depends on the choice of a torsion-free covariant derivative.We prove that the globalized formalities with respect to two different covariant derivatives arehomotopic. More explicitly, we derive the statement by proving a more general homotopy equiv-alence between L ∞ -morphisms that are twisted with gauge equivalent Maurer-Cartan elements. Contents L ∞ -algebras . . . . . . . . . . . . . . . . . . . . . . . . 4 L ∞ -morphisms as Maurer-Cartan Elements . . . . . . . . . . . . . . . . . . . . . . 83.2 Homotopy Classification of L ∞ -algebras . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Homotopy Equivalence between Twisted Morphisms . . . . . . . . . . . . . . . . . 12 L ∞ -morphism . . . . . . . . . . . . . . . . 154.3 Homotopic Global Formalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ∗ [email protected] † [email protected] Final Remarks 20Bibliography 20
The celebrated formality theorem by Kontsevich [15] provides the existence of an L ∞ -quasi-isomorphism from the differential graded Lie algebra (DGLA) of polyvector fields T poly ( R d ) tothe DGLA of polydifferential operators D poly ( R d ) . In [6, 7] Dolgushev globalized this result togeneral smooth manifolds M using a geometric approach. Being a quasi-isomorphism, this for-mality induces a bijective correspondence U : Def ( T poly ( M )[[ ~ ]]) −→ Def ( D poly ( M )[[ ~ ]]) (1.1)between equivalence classes Def ( T poly ( M )[[ ~ ]]) of formal Poisson structures ~ π ∈ Γ ∞ (Λ T M )[[ ~ ]] on M and equivalence classes Def ( D poly ( M )[[ ~ ]]) of star products ⋆ on M , see also [4,18] for moredetails on deformation theory. In particular, this associates to a classical Poisson structure π cl aclass of deformation quantizations U ([ ~ π cl ]) in the sense of the seminal paper [1]. On the otherhand, it also gives a way to assign to each star product a class of formal Poisson structures, theso-called Kontsevich class of the star product.However, the above mentioned globalization procedure of the Kontsevich formality from R d toa general manifold M discussed in [6] depends on the choice of a torsion-free covariant derivative.More explicitly, it uses the covariant derivative to obtain Fedosov resolutions of the polyvectorfields and polydifferential operators between which one has a fiberwise Kontsevich formality. Re-cently, in [2, Theorem 2.6] it has been shown that the map U from (1.1) does not depend on thechoice of the connection. In this paper we investigate the role of the covariant derivative at thelevel of the formality and not at the level of equivalence classes of Maurer-Cartan elements.The key point is that changing the covariant derivative corresponds to twisting by a Maurer-Cartan element that is equivalent to zero, see [2, Appendix C] for this observation and [6,7,10,11]for more details on the twisting procedure. This corresponds to a more general observation: Let F : ( g , d , [ · , · ]) → ( g ′ , d ′ , [ · , · ]) be an L ∞ -morphism between DGLAs with complete descendingand exhaustive filtrations F • g resp. F • g ′ . Moreover, let π ∈ F g be a Maurer-Cartan elementequivalent to zero via π = exp([ g, · ]) ⊲ with g ∈ F g . The element π ′ = P ∞ k =1 1 k ! F k ( π ∨· · ·∨ π ) ∈ F g ′ is a Maurer-Cartan element in g ′ equivalent to zero. Let the equivalence be given by g ′ ∈ F g ′ , then one obtains Proposition 3.10: Proposition
The L ∞ -morphisms F and e [ − g ′ , · ] ◦ F π ◦ e [ g, · ] from ( g , d , [ · , · ]) to ( g ′ , d ′ , [ · , · ]) arehomotopic, where F π denotes the L ∞ -morphism F twisted by π . By homotopic we mean here that the two L ∞ -morphisms are equivalent Maurer-Cartan elementsin the convolution DGLA, compare [8, Definition 3], see also [9] for a comparison of differentnotions of homotopies between L ∞ -morphisms.This general statement can be applied to the globalization of the Kontsevich formality. Ourmain result here is the following theorem, see Theorem 4.12: Theorem
Let ∇ and ∇ ′ be two different torsion-free covariant derivatives. Then the two globalformalities constructed via Dolgushev’s globalization procedure are homotopic. This immediately implies that they induce the same map on the equivalence classes of formalMaurer-Cartan elements, i.e. [2, Theorem 2.6].Note that there are many other similar globalization procedures of formalities based on Dolgu-shev’s globalization of the Kontsevich formality [6,7], e.g. [3] for Lie algebroids, [16] for differentialgraded manifolds and [5] for Hochschild chains. The above technique can be adapted to thesecases and we plan to pursue them in further works.Finally, we want to mention that in [15, Section 7] there is also a globalization procedureexplained, using the language of ∞ -jet spaces of polyvector fields and polydifferential operators,respectively. However, these ∞ -jet spaces are (non-canonically) isomorphic as vector bundles to he formally completed fiberwise polyvector fields and polydifferential operators, respectively. Thecorresponding isomorphisms are constructed by the choice of a connnection. We strongly believethat the globalization procedure proposed by Kontsevich in [15] is homotopic to the globalizationfrom Dolgushev [6, 7] we are using in this note.The paper is organized as follows: In Section 2 we recall the basics concerning Maurer-Cartanelements in DGLAs and L ∞ -algebras, the notions of gauge and homotopy equivalence as wellas the twisting procedure. Then we recall in Section 3 the interpretation of L ∞ -morphisms asMaurer-Cartan elements and the notion of homotopic L ∞ -morphisms. We show that pre- andpost-compositions of homotopic L ∞ -morphisms with an L ∞ -morphism are again homotopic, astatement that is probably well-known to the experts, but that we could not find in the literature.Moreover, we prove here Proposition 3.10, i.e. that the twisted L ∞ -morphisms are homotopic forequivalent Maurer-Cartan elements. Finally, we apply these general results to the globalizationof Kontsevich’s formality theorem, proving Theorem 4.12 and also an equivariant version for Liegroup actions with invariant covariant derivatives. Acknowledgements:
The authors are grateful to Chiara Esposito, Ryszard Nest and BorisTsygan for the idea leading to this letter and for many helpful comments. This work was supportedby the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA– INdAM). The second author is supported by the DFG research training group "gk1821: Coho-mological Methods in Geometry".
We want to recall the basics concerning differential graded Lie algebras (DGLAs), Maurer-Cartanelements and their equivalence classes. In order to make sense of the gauge equivalence we considerin this context DGLAs ( g • , d , [ · , · ]) with complete descending filtrations · · · ⊇ F − g ⊇ F − g ⊇ F g ⊇ F g ⊇ · · · , g ∼ = lim ←− g / F n g (2.1)and d( F k g ) ⊆ F k g and [ F k g , F ℓ g ] ⊆ F k + ℓ g . (2.2)In particular, F g is a projective limit of nilpotent DGLAs. In most cases the filtration will bebounded below, i.e. bounded from the left with g = F k g for some k ∈ Z . If the filtration isunbounded, then we assume always that it is in addition exhaustive, i.e. that g = [ n F n g , (2.3)even if we do not mention it explicitly. Moreover, we assume that the DGLA morphisms arecompatible with the filtrations. Example 2.1
One motivation to consider the case of filtered DGLAs are formal power series g [[ ~ ]] of a DGLA g with filtration F k ( g [[ ~ ]]) = ~ k ( g [[ ~ ]]) . Definition 2.2 (Maurer-Cartan elements)
Let ( g , d , [ · , · ]) be a DGLA with complete descend-ing filtration. Then π ∈ F g is called Maurer-Cartan element if it satisfies the Maurer-Cartanequation d π + 12 [ π, π ] = 0 . (2.4) The set of Maurer-Cartan elements is denoted by MC ( g ) . Maurer-Cartan elements π lead to twisted DGLA structures ( g , d + [ π, · ] , [ · , · ]) and one has agauge action on the set of Maurer-Cartan elements. roposition 2.3 (Gauge action) Let ( g , d , [ · , · ]) be a DGLA with complete descending filtra-tion. The gauge group G ( g ) = { Φ = e [ g, · ] : g −→ g | g ∈ F g } defines an action on MC ( g ) via exp([ g, · ]) ⊲ π = ∞ X n =0 ([ g, · ]) n n ! ( π ) − ∞ X n =0 ([ g, · ]) n ( n + 1)! (d g ) = π − exp([ g, · ]) − id[ g, · ] (d g + [ π, g ]) . (2.5) The set of equivalence classes of Maurer-Cartan elements in g is denoted by Def ( g ) = MC ( g )G ( g ) . (2.6)Note that the gauge action is well-defined since g ∈ F g and as the filtration is complete. Def ( g ) is the transformation groupoid of the gauge action and also called Goldman-Millson groupoid [14].It plays an important role in deformation theory [18]. In particular, the definition implies thattwisting with gauge equivalent Maurer-Cartan elements leads to isomorphic DGLAs.
Corollary 2.4
Let ( g , d , [ · , · ]) be a DGLA with complete descending filtration and with gaugeequivalent Maurer-Carten elements π ′ , π via g ∈ G ( g ) . Then one has d + [ π ′ , · ] = exp([ g, · ]) ◦ (d + [ π, · ]) ◦ exp([ − g, · ]) . (2.7) In other words, exp([ g, · ]) : ( g , d + [ π, · ] , [ · , · ]) → ( g , d + [ π ′ , · ] , [ · , · ]) is an isomorphism ofDGLAs. L ∞ -algebras Let us recall the basics of L ∞ -algebras and L ∞ -morphisms. Proofs and further details can befound in [6, 7, 11]. Note that in this work we only consider L ∞ -morphisms between DGLAs.An L ∞ -algebra ( L, Q ) is a graded vector space L together with a degree +1 codifferential Q on the graded cocommutative cofree coalgebra (S( L [1]) , ∆) without counit cogenerated by L [1] .We always consider a vector space over a field K of characteristic zero. The codifferential Q isuniquely determined by the Taylor components Q n : S n ( L [1]) −→ L [2] for n ≥ . Sometimeswe also write Q k = Q k and following [4] we denote by Q in the component of Q in : S n ( L [1]) → S i ( L [1])[1] of Q . The property Q = 0 implies in particular that Q : L → L [1] is a cochaindifferential. Let us consider two L ∞ -algebras ( L, Q ) and ( L ′ , Q ′ ) . A degree coalgebra morphism F : S( L [1]) −→ S( L ′ [1]) such that F Q = Q ′ F is called L ∞ -morphism . Just like the codifferentialalso the morphism F is also uniquely determined by its Taylor components F n : S n ( L [1]) −→ L ′ [1] ,where n ≥ . We write again F k = F k and we get coefficients F jn : S n ( L [1]) → S j ( L ′ [1]) of F .Note that F jn depends only on F k = F k for k ≤ n − j + 1 . In particular, the first structure mapof F is a map of complexes F : ( L, Q ) → ( L ′ , ( Q ′ ) ) and one calls F L ∞ -quasi-isomorphism if F is a quasi-isomorphism of complexes. Example 2.5 (DGLA)
A DGLA ( g , d , [ · , · ]) is an L ∞ -algebra with Q = − d and Q ( γ ∨ µ ) = − ( − | γ | [ γ, µ ] , where | γ | denotes the degree in g [1] .In order to generalize the definition of Maurer-Cartan elements we consider again L ∞ -algebraswith complete descending and exhaustive filtrations on L . We assume again that L ∞ -morphismsare compatible with the filtrations. Definition 2.6 (Maurer-Cartan elements II)
Let ( L, Q ) be an L ∞ -algebra with compatiblecomplete descending filtration. Then π ∈ F L [1] is called Maurer-Cartan element if it satisfiesthe Maurer-Cartan equation X n> n ! Q n ( π ∨ · · · ∨ π ) = 0 . (2.8) The set of Maurer-Cartan elements is again denoted by MC ( L ) . Note that the sum in (2.8) is well-defined for x ∈ F L because of the completeness of L . Werecall some useful properties from [7, Prop. 1]: emma 2.7 Let F : ( g , Q ) → ( g ′ , Q ′ ) be an L ∞ -morphism of DGLAs and π ∈ F g .i.) dπ + [ π, π ] = 0 is equivalent to Q (exp( π )) = 0 , where exp( π ) = P ∞ k =1 1 k ! π ∨ k .ii.) F (exp( π )) = exp( S ) with S = F (exp( π )) = P n> n ! F n ( π ∨ · · · ∨ π ) .iii.) If π is a Maurer-Cartan element, then so is S . We recall the generalization of the gauge action to an equivalence relation on the set of Maurer-Cartan elements of L ∞ -algebras. We follow [4, Section 4] but adapt the definitions to the caseof L ∞ -algebras with complete descending and exhaustive filtrations as in [9]. Let therefore ( L, Q ) be such an L ∞ -algebra with complete descending and exhaustive filtration and consider L [ t ] = L ⊗ K [ t ] which has again a descending and exhaustive filtration F k L [ t ] = F k L ⊗ K [ t ] . We denote its completion by d L [ t ] and note that since Q is compatible with the filtration it extendsto d L [ t ] . Similarly, L ∞ -morphisms extend to these completed spaces. Remark 2.8
Note that one can define the completion as space of equivalence classes of Cauchysequences with respect to the filtration topology. Alternatively, the completion can be identifiedwith lim ←− L [ t ] / F n L [ t ] ⊂ Y n L [ t ] / F n L [ t ] ∼ = Y n L/ F n L ⊗ K [ t ] consisting of all coherent tuples X = ( x n ) n ∈ Q n L [ t ] / F n L [ t ] , where L [ t ] / F n +1 L [ t ] ∋ x n +1 x n ∈ L [ t ] / F n [ t ] under the obvious surjections. Moreover, F n d L [ t ] corresponds to the kernel of lim ←− L [ t ] / F n L [ t ] → L [ t ] / F n L [ t ] and thus d L [ t ] / F n d L [ t ] ∼ = L [ t ] / F n L [ t ] . Since L is complete, we can also interpret d L [ t ] as the subspace of L [[ t ]] such that X mod F n L [[ t ]] ispolynomial in t . In particular, F n d L [ t ] is the subspace of elements in F n L [[ t ]] that are polynomialin t modulo F m L [[ t ]] for all m .By the above construction of d L [ t ] it is clear that differentiation dd t and integration with respectto t extend to it since they do not change the filtration. Sometimes we write also ˙ X instead of dd t X and, moreover, the evaluation δ s : d L [ t ] ∋ X X ( s ) = X (cid:12)(cid:12) t = s ∈ L is well-defined for all s ∈ K since L is complete. Example 2.9
In the case that the filtration of L comes from a grading L • , the completion isgiven by d L [ t ] ∼ = Q i L i [ t ] , i.e. by polynomials in each degree. A special case is here the case offormal power series L = V [[ ~ ]] with d L [ t ] ∼ = ( V [ t ])[[ ~ ]] as in [2, Appendix A].Now we can introduce a general equivalence relation between Maurer-Cartan elements of L ∞ -algebras. Definition 2.10 (Homotopy equivalence)
Let ( L, Q ) be a L ∞ -algebra with a complete de-scending filtration. The homotopy equivalence relation on the set MC ( L ) is the transitive closureof the relation ∼ defined by: π ∼ π if and only if there exist π ( t ) ∈ F [ L [ t ] and λ ( t ) ∈ F [ L [ t ] such that dd t π ( t ) = Q ( λ ( t ) ∨ exp( π ( t ))) = ∞ X n =0 n ! Q n +1 ( λ ( t ) ∨ π ( t ) ∨ · · · ∨ π ( t )) ,π (0) = π and π (1) = π . (2.9) The set of equivalence classes of Maurer-Cartan elements of L is denoted by Def ( L ) = MC ( L ) / ∼ . ote that in the case of nilpotent L ∞ -algebras it suffices to consider polynomials in t as thereis no need to complete L [ t ] , compare [13]. We check now that this is well-defined and even yieldsa curve π ( t ) of Maurer-Cartan elements. Proposition 2.11
For every π ∈ F L and λ ( t ) ∈ F [ L [ t ] there exists a unique π ( t ) ∈ F [ L [ t ] such that dd t π ( t ) = Q ( λ ( t ) ∨ exp( π ( t ))) and π (0) = π . If π ∈ MC ( L ) , then π ( s ) ∈ MC ( L ) forall s ∈ K . Proof:
The proof for the nilpotent case can be found in [4, Prop. 4.8]. In our setting of completefiltrations we only have to show that the solution π ( t ) = P ∞ k =0 π k t k in the formal power series F L ⊗ K [[ t ]] is an element of F [ L [ t ] . By Remark 2.8 this is equivalent to π ( t ) mod F n L [[ t ]] ∈ L [ t ] for all n . Indeed, we have inductively dd t π ( t ) mod F L [[ t ]] = Q ( λ ( t )) mod F L [[ t ] ∈ L [1] . For the higher orders we get dd t π ( t ) mod F n L [[ t ]] = n − X k =0 k ! Q k ( λ ( t ) ∨ ( π ( t ) + F n − ) ∨ · · · ∨ ( π ( t ) + F n − )) mod F n L [[ t ]] and thus π ( t ) mod F n L [[ t ]] ∈ L [ t ] . (cid:3) One can show that for DGLAs with complete filtrations the two notions of equivalences areequivalent, see e.g. [18, Thm. 5.5].
Theorem 2.12
Two Maurer-Cartan elements in ( g , d , [ · , · ]) are homotopy equivalent if and onlyif they are gauge equivalent. This theorem can be rephrased in a more explicit manner in the following proposition.
Proposition 2.13
Let ( g , d , [ · , · ]) be a DGLA with complete descending filtration. Consider π ∼ π with equivalence given by π ( t ) ∈ F d g [ t ] and λ ( t ) ∈ F d g [ t ] . The formal solution of λ ( t ) = exp([ A ( t ) , · ]) − id[ A ( t ) , · ] d A ( t )d t , A (0) = 0 (2.10) is an element A ( t ) ∈ F d g [ t ] and satisfies π ( t ) = e [ A ( t ) , · ] π − exp([ A ( t ) , · ] − id[ A ( t ) , · ] d A ( t ) . (2.11) In particular, for g = A (1) ∈ F g one has π = exp([ g, · ]) ⊲ π . (2.12) Proof:
As formal power series in t Equation 2.10 has a unique solution A ( t ) ∈ F g ⊗ K [[ t ]] .But one has even A ( t ) ∈ F d g [ t ] since d A ( t )d t ≡ λ ( t ) − n − X k =1 k + 1)! [ A ( t ) , · ] k d A ( t )d t mod F n g [[ t ]] ≡ λ ( t ) − n − X k =1 k + 1)! [ A ( t ) mod F n − g [[ t ]] , · ] k (cid:18) d A ( t )d t mod F n − g [[ t ]] (cid:19) mod F n g [[ t ]] is by induction polynomial in t . Note that one has dd t e [ A ( t ) , · ] = (cid:20) exp([ A ( t ) , · ] − id[ A ( t ) , · ] d A ( t )d t , · (cid:21) ◦ exp([ A ( t ) , · ]) . ( ∗ ) ur aim is now to show that π ′ ( t ) = e [ A ( t ) , · ] π − exp([ A ( t ) , · ] − id[ A ( t ) , · ] d A ( t ) satisfies d π ′ ( t )d t = − d λ ( t ) + (cid:20) λ ( t ) , e [ A ( t ) , · ] π − exp([ A ( t ) , · ]) − id[ A ( t ) , · ] d A ( t ) (cid:21) . Then we know π ′ ( t ) = π ( t ) ∈ F d g [ t ] since the solution π ( t ) is unique by Proposition 2.11, whichimmediately gives π ′ (1) = π . At first we compute d λ ( t ) = exp([ A ( t ) , · ]) − id[ A ( t ) , · ] d d A ( t )d t + ∞ X k =0 k − X j =0 k + 1)! (cid:18) kj + 1 (cid:19)(cid:20) ad jA d A ( t ) , ad k − − jA d A ( t )d t (cid:21) and using ( ∗ ) we get d π ′ ( t )d t = (cid:20) exp([ A ( t ) , · ]) − id[ A ( t ) , · ] d A ( t )d t , exp([ A ( t ) , · ]) π (cid:21) − exp([ A ( t ) , · ]) − id[ A ( t ) , · ] d d A ( t )d t − ∞ X k =0 k − X j =0 k + 1)! (cid:18) kj + 1 (cid:19)(cid:20) ad jA d A ( t )d t , ad k − − jA d A (cid:21) = − d λ ( t ) + (cid:20) λ ( t ) , e [ A ( t ) , · ] π − exp([ A ( t ) , · ]) − id[ A ( t ) , · ] d A ( t ) (cid:21) and the proposition is proven. (cid:3) Remark 2.14
There are also different notions of homotopy resp. gauge equivalences for Maurer-Cartan elements in L ∞ -algebras: e.g. the above definition, sometimes also called Quillen homo-topy , and the gauge homotopy where one requires λ ( t ) = λ to be constant, compare [8]. In [9] itis shown that these notions are also equivalent for complete L ∞ -algebras, extending the result forDGLAs.One important property is that L ∞ -morphisms map equivalence classes of Maurer-Cartanelements to equivalence classes, see [4, Prop. 4.9]. Proposition 2.15
Let F : ( L, Q ) → ( L ′ , Q ′ ) be an L ∞ -morphism between L ∞ -algebras with com-plete filtrations, and π , π ∈ MC ( L ) with π ∼ π via π ( t ) ∈ F d g [ t ] and λ ( t ) ∈ F d g [ t ] . Then F is compatible with the homotopy equivalence relation, i.e. one has F (exp π ) ∼ F (exp π ) via π ′ ( t ) = F (exp( π ( t ))) and λ ′ ( t ) = F ( λ ( t ) ∨ exp( π ( t ))) . If F is an L ∞ -quasi-isomorphism, then it is well-known that it induces a bijection on theequivalence classes of Maurer-Cartan elements. Finally, recall that also the twisting with Maurer-Cartan elements can be generalized to L ∞ -algebras, see e.g. [5, Section 2.3]. Lemma 2.16
Let ( L, Q ) be an L ∞ -algebra and π ∈ F L [1] a Maurer-Cartan element. Then themap Q π given by Q π ( X ) = exp(( − π ) ∨ ) Q (exp( π ∨ ) X ) , X ∈ S( L [1]) (2.13) defines a codifferential on S( L [1]) . One can not only twist the DGLAs resp. L ∞ -algebras, but also the L ∞ -morphisms betweenthem. Below we need the following result, see [5, Prop. 2] and [7, Prop. 1]. Proposition 2.17
Let F : ( g , Q ) → ( g ′ , Q ′ ) be an L ∞ -morphism of DGLAs, π ∈ F g a Maurer-Cartan element and S = F (exp( π )) ∈ F g ′ .i.) The map F π = exp( − S ∨ ) F exp( π ∨ ) : S( g [1]) −→ S( g ′ [1]) defines an L ∞ -morphism between the DGLAs ( g , d + [ π, · ]) and ( g ′ , d + [ S, · ]) . i.) The structure maps of F π are given by F πn ( x , . . . , x n ) = ∞ X k =0 k ! F n + k ( π, . . . , π, x , . . . , x n ) . (2.14) iii.) Let F be an L ∞ -quasi-isomorphism such that F is not only a quasi-isomorphism of filteredcomplexes L → L ′ but even induces a quasi-isomorphism F : F k L −→ F k L ′ for each k . Then F π is an L ∞ -quasi-isomorphism. Here we prove the main results about the relation between twisted L ∞ -morphisms. More explicitly,consider an L ∞ -morphism F : ( g , Q ) → ( g ′ , Q ′ ) between DGLAs and let π , π ∈ F g be twoequivalent Maurer-Cartan elements via π = exp([ g, · ]) ⊲ π . We show that F π and F π can beinterpreted as homotopic in the sense of [8, Definition 3]. L ∞ -morphisms as Maurer-Cartan Elements At first, recall that we can interpret L ∞ -morphisms as Maurer-Cartan elements in the convolutionalgebra. More explicitly, let ( L, Q ) , ( L ′ , Q ′ ) be two L ∞ -algebras and denote the graded linearmaps by Hom(S( L [1]) , L ′ ) . If L and L ′ are equipped with complete descending filtrations, thenwe require the maps to be compatible with the filtration. The L ∞ -structures on L and L ′ leadto an L ∞ -structure on this vector space of maps, see [8, Proposition 1 and Proposition 2] andalso [2] for the case of DGLAs. Proposition 3.1
The coalgebra
S(Hom(S( L [1]) , L ′ )[1]) can be equipped with a codifferential b Q with structure maps b Q F = Q ′ ◦ F − ( − | F | F ◦ Q (3.1) and b Q n ( F ∨ · · · ∨ F n ) = ( Q ′ ) n ◦ ∨ n − ◦ ( F ⊗ F ⊗ · · · ⊗ F n ) ◦ ∆ n − . (3.2) It is called convolution L ∞ -algebra and its Maurer-Cartan elements are identified with L ∞ -morphisms. Here | F | denotes the degree in Hom(S( L [1]) , L ′ )[1] . Example 3.2
Let g , g ′ be two DGLAs. Then Hom(S( g [1]) , g ′ ) is in fact a DGLA with differential ∂F = d ′ ◦ F + ( − | F | F ◦ Q (3.3)and Lie bracket [ F, G ] = − ( − | F | ( Q ′ ) ◦ ( F ⊗ G ) ◦ ∆ . (3.4)Here | F | denotes again the degree in Hom(S( g [1]) , g ′ )[1] . This DGLA is also called convolutionDGLA .We note that the convolution L ∞ -algebra H = Hom(S( L [1]) , L ′ ) is equipped with the followingcomplete descending filtration: H = F H ⊃ F H ⊃ · · · ⊃ F k H ⊃ · · · F k H = n f ∈ Hom(S( L [1]) , L ′ ) | f (cid:12)(cid:12) S Two L ∞ -morphisms F, F ′ from ( L, Q ) to ( L ′ , Q ′ ) are called homotopic if theyare homotopy equivalent Maurer-Cartan elements in the convolution L ∞ -algebra H . e collect a few immediate consequences: Proposition 3.4 Let F, F ′ be two homotopic L ∞ -morphisms from ( L, Q ) to ( L ′ , Q ′ ) .i.) F and ( F ′ ) are chain homotopic.ii.) If F is an L ∞ -quasi-isomorphism, then so is F ′ .iii.) If L = g , L ′ = g ′ are two DGLAs equipped with complete descending filtrations, then F and F ′ induce the same maps from Def ( g ) to Def ( g ′ ) .iv.) In the case of DGLAs g , g ′ , compositions of homotopic L ∞ -morphisms with a DGLA mor-phism of degree zero are again homotopic. Proof: The first three points are proven in [2] and the last one follows directly. (cid:3) We now aim to generalize the last point of the previous proposition to compositions with L ∞ -morphisms. We start with the post-composition: Proposition 3.5 Let F , F be two homotopic L ∞ -morphisms from ( L, Q ) to ( L ′ , Q ′ ) . Let H bean L ∞ -morphism from ( L ′ , Q ′ ) to ( L ′′ , Q ′′ ) , then HF ∼ HF . Proof: For F ∈ Hom(S( L [1]) , L ′ ) we define b H ( F ) via ( b H ( F )) n = ( HF ) n = n X ℓ =1 H ℓ F ℓn = H ℓ (cid:18) ℓ ! F ∨ · · · ∨ F (cid:19) ◦ ∆ ℓ − . Here the ∨ -product of maps is given by F ∨ G = ∨◦ ( F ⊗ G ) : S( L [1]) ⊗ S( L [1]) → S( L ′ [1]) . Writing ∆ • = P ∞ k =0 ∆ k and defining all maps to be zero on the domains on which they where previouslynot defined, we can rewrite this as b HF = H ◦ exp F ◦ ∆ • . Let F ( t ) ∈ \ (Hom(S( L [1]) , L ′ )[1]) [ t ] and λ ( t ) ∈ \ (Hom(S( L [1]) , L ′ )[1]) − [ t ] describe the homotopyequivalence between F and F . Then b HF ( t ) ∈ \ (Hom(S( L [1]) , L ′′ )[1]) − [ t ] satisfies dd t b HF ( t ) = ∞ X ℓ =1 H ℓ dd t (cid:18) ℓ ! F ( t ) ∨ · · · ∨ F ( t ) (cid:19) ◦ ∆ l − = H ◦ (cid:16) b Q ( λ ( t ) ∨ exp( F ( t )) ∨ exp( F ( t )) (cid:17) ◦ ∆ • . As in [4, Lemma 4.1] one can check b Q ( λ ( t ) ∨ exp( F ( t ))) = exp( F ( t )) ∨ b Q ( λ ( t ) ∨ exp( F ( t ))) − λ ( t ) ∨ exp( F ( t )) ∨ b Q (exp( F ( t )))= exp( F ( t )) ∨ b Q ( λ ( t ) ∨ exp( F ( t ))) since F ( t ) is a Maurer-Cartan element. This allows us to compute dd t b HF ( t ) = H ◦ (cid:16) b Q ( λ ( t ) ∨ exp( F ( t ))) (cid:17) ◦ ∆ • = H ◦ Q ′ ◦ ( λ ( t ) ∨ exp( F ( t ))) ◦ ∆ • + H ◦ ( λ ( t ) ∨ exp( F ( t ))) ◦ ∆ • ◦ Q = ( Q ′′ ) ◦ H ◦ ( λ ( t ) ∨ exp( F ( t ))) ◦ ∆ • + H ◦ ( λ ( t ) ∨ exp( F ( t ))) ◦ ∆ • ◦ Q = ( b Q ′ ) (cid:16) H ◦ ( λ ( t ) ∨ exp( F ( t ))) ◦ ∆ • (cid:17) + ∞ X ℓ =2 ( Q ′′ ) ℓ ◦ H ℓ ◦ ( λ ( t ) ∨ exp( F ( t ))) ◦ ∆ • . Concerning the last term we have omitting the t -dependency since F and H are of degree zero k ! H ℓk +1 ◦ ( λ ( t ) ∨ F ( t ) ∨ · · · ∨ F ( t )) ◦ ∆ k ( X ) k ! ℓ ! ( H ∨ · · · ∨ H ) ◦ ∆ ℓ − ◦ ( λ ( t ) ∨ F ( t ) ∨ · · · ∨ F ( t )) ◦ ∆ k ( X )= 1 k ! ℓ ! ( H ∨ · · · ∨ H ) ◦ X i + ··· + i ℓ = k +1 i j ≥ X σ ∈ Sh ( i ,...,i ℓ ) σ ⊳ (cid:16) ( λ ( t ) ∨ F ( t ) ∨ · · · ∨ F ( t )) ◦ ∆ k ( X ) (cid:17) = ℓk ! ℓ ! ( H ∨ · · · ∨ H ) ◦ X i + ··· + i ℓ = k +1 i j ≥ X σ ∈ Sh ( i ,...,i ℓ ) σ (1)=1 σ ⊳ (cid:16) ( λ ( t ) ∨ F ( t ) ∨ · · · ∨ F ( t )) ◦ ∆ k ( X ) (cid:17) = 1( ℓ − X i + ··· + i ℓ = k +1 ,i j ≥ (cid:16) i − H i ( λ ∨ F · · · ∨ F ) ◦ ∆ i − ∨ i ! H i ( F ∨ · · · ∨ F ) ◦ ∆ i − ∨ · · · ∨ i ℓ ! H i ℓ ( F ∨ · · · ∨ F ) ◦ ∆ i ℓ − (cid:17) ◦ ∆ ℓ − ( X ) . Here we wrote σ ⊳ ( x ∨ · · · ∨ x k +1 ) = ǫ ( σ ) x σ (1) ∨ · · · ∨ x σ ( i ) ⊗ · · · ⊗ x σ ( k +1 − i ℓ +1) ∨ · · · ∨ x σ ( n ) with Koszul sign ǫ ( σ ) . Therefore, it follows dd t b HF ( t ) = ( b Q ′ ) (cid:16) H ◦ ( λ ( t ) ∨ exp( F ( t ))) ◦ ∆ • (cid:17) + ∞ X ℓ =2 ( b Q ′ ) ℓ ◦ (cid:16) ( H ◦ ( λ ( t ) ∨ exp F ) ◦ ∆ • ) ∨ exp( b HF ) (cid:17) and the statement is shown. (cid:3) Analogously, we have for the pre-composition: Proposition 3.6 Let F , F be two homotopic L ∞ -morphisms from ( L, Q ) to ( L ′ , Q ′ ) . Let H bean L ∞ -morphism from ( L ′′ , Q ′′ ) to ( L, Q ) , then F H ∼ F H . Proof: Let F ( t ) ∈ \ (Hom(S( L [1]) , L ′ )[1]) [ t ] and λ ( t ) ∈ \ (Hom(S( L [1]) , L ′ )[1]) − [ t ] describe thehomotopy equivalence between F and F . Then we consider ( F ( t ) H ) = F ( t ) ◦ H = F ( t ) ◦ exp H ◦ ∆ • ∈ \ (Hom(S( L ′′ [1]) , L ′ )[1]) [ t ] in the notation of the above proposition. We compute dd t ( F ( t ) H ) = b Q ( λ ( t ) ∨ exp( F ( t ))) ◦ H = ( Q ′ ) ◦ λ ◦ H + λ ◦ Q ◦ H + ∞ X ℓ =2 ℓ − Q ′ ) ℓ ◦ ( λ ∨ F ∨ · · · ∨ F ) ◦ ∆ ℓ − ◦ H = ( Q ′ ) ◦ λ ◦ H + λ ◦ H ◦ Q ′′ + ∞ X ℓ =2 ℓ − Q ′ ) ℓ ◦ ( λH ∨ F H ∨ · · · ∨ F H ) ◦ ∆ ℓ − = b Q ( λ ( t ) H ∨ exp( F ( t ) H )) since H is a coalgebra morphism intertwining Q ′′ and Q and of degree zero. Finally, since λ ( t ) H ∈ \ (Hom(S( L ′′ [1]) , L ′ )[1]) − [ t ] the statement follows. (cid:3) L ∞ -algebras The above considerations allow us to understand better the homotopy classification of L ∞ -algebrasfrom [4, 15], which will help us in the application to the global formality. efinition 3.7 Two L ∞ -algebras ( L, Q ) and ( L ′ , Q ′ ) are said to be homotopy equivalent if thereare L ∞ -morphisms F : ( L, Q ) → ( L ′ , Q ′ ) and G : ( L ′ , Q ′ ) → ( L, Q ) such that F ◦ G ∼ id L ′ and G ◦ F ∼ id L . In such case F and G are said to be quasi-inverse to each other. This definition coincides indeed with the definition of homotopy equivalence via L ∞ -quasi-isomorphisms from [4]. Lemma 3.8 Two L ∞ -algebras ( L, Q ) and ( L ′ , Q ′ ) are homotopy equivalent if and only if thereexists an L ∞ -quasi-isomorphism between them. Proof: Due to [4, Prop. 2.8] every L ∞ -algebra L is isomorphic to the product of a linear con-tractible one and a minimal one ( L, Q ) ∼ = ( V ⊕ W, e Q ) . This means L ∼ = V ⊕ W as vector spaces,such that V is an acyclic cochain complex with differential d V and W is an L ∞ -algebra with cod-ifferential Q W with Q W, = 0 . The codifferential e Q on S(( V ⊕ W )[1]) is given on v ∨ · · · ∨ v m with v , . . . , v k ∈ V and v k +1 , . . . , v m ∈ W by e Q ( v ∨ · · · ∨ v m ) = − d V ( v ) , for k = m = 1 Q W ( v ∨ · · · ∨ v m ) , for k = 00 , else.This implies in particular that the canonical maps I W : W −→ V ⊕ W and P W : V ⊕ W −→ W are L ∞ -morphisms. We want to show now that I W ◦ P W ∼ id . Choose a contracting homotopy h V : V → V [ − with h V d V + d V h V = id V and define the maps P ( t ) : V ⊕ W ∋ ( v, w ) ( tv, w ) ∈ V ⊕ W and H ( t ) : V ⊕ W ∋ ( v, w ) ( − h V ( v ) , ∈ V ⊕ W. Note that P ( t ) is a path of L ∞ -morphisms by the explicit form of the codifferential. We clearlyhave dd t P ( t ) = pr V = e Q ◦ H ( t ) + H ( t ) ◦ e Q = b Q ( H ( t )) since h V is a contracting homotopy. This implies dd t P ( t ) = b Q ( H ( t ) ∨ exp( P ( t )) since im( H ( t )) ⊆ V and as the higher brackets of e Q vanish on V . Since P (0) = I W ◦ P W and P (1) = id we conclude that I W ◦ P W ∼ id . We choose a similar splitting for a L ′ = V ′ ⊕ W ′ withthe same properties and consider an L ∞ -quasi-isomorphism F : L → L ′ . Since I W , I W ′ , P W and P W ′ are L ∞ -quasi-isomorphisms and we have that F W = P W ′ ◦ F ◦ I W : W −→ W ′ an L ∞ -isomorphism. Hence it invertible and we denote the inverse G W ′ . We define now G = I W ◦ G W ′ ◦ P W ′ : L ′ −→ L. Since by Proposition 3.5 and Proposition 3.6 compositions of homotopic L ∞ -morphisms with an L ∞ -morphism are again homotopic, we get F ◦ G = F ◦ I W ◦ G W ′ ◦ P W ′ ∼ I W ′ ◦ P W ′ ◦ F ◦ I W ◦ G W ′ ◦ P W ′ = I W ′ ◦ F W ◦ G W ′ ◦ P W ′ = I W ′ ◦ P W ′ ∼ id and similarly G ◦ F ∼ id .The other direction follows from Proposition 3.4. Suppose F ◦ G ∼ id and G ◦ F ∼ id , thenwe know that F ◦ G and G ◦ F are both chain homotopic to the identity. Therefore, F and G are L ∞ -quasi-isomorphisms. (cid:3) orollary 3.9 Let F : ( L, Q ) → ( L ′ , Q ′ ) be a an L ∞ -quasi-isomorphism with two given quasi-inverses G, G ′ : ( L ′ , Q ′ ) → ( L, Q ) in the sense of Definition 3.7. Then one has G ∼ G ′ . Proof: One has G ∼ G ◦ ( F ◦ G ′ ) = ( G ◦ F ) ◦ G ′ ∼ G ′ . (cid:3) Let now F : ( g , Q ) → ( g ′ , Q ′ ) be an L ∞ -morphism between DGLAs with complete descending andexhaustive filtrations. Instead of comparing the twisted morphisms F π and F π ′ with respect totwo equivalent Maurer-Cartan elements π and π ′ , we consider for simplicity just a Maurer-Cartanelement π ∈ F g equivalent to zero via π = exp([ g, · ]) ⊲ , i.e. λ ( t ) = g = ˙ A ( t ) ∈ F d g [ t ] . Thenwe know that and S = F (exp( π )) ∈ F ( g ′ ) are equivalend Maurer-Cartan elements in ( g ′ , d ′ ) .Let the equivalence be implemented by an A ′ ( t ) ∈ F \ ( g ′ ) [ t ] as in Proposition 2.13. Then we havethe diagram ( g ′ , d ′ )( g , d) ( g ′ , d ′ + [ S, · ])( g , d + [ π, · ]) e [ A ′ (1) , · ] Fe [ A (1) , · ] F π (3.6)where e [ A (1) , · ] and e [ A ′ (1) , · ] are well-defined by the completeness of the filtrations. In the followingwe show that it commutes up to homotopy, which is indicated by the vertical arrow. Proposition 3.10 The L ∞ -morphisms F and e [ − A ′ (1) , · ] ◦ F π ◦ e [ A (1) , · ] are homotopic, i.e. gaugeequivalent Maurer-Cartan elements in Hom(S( g [1]) , g ′ ) . The candidate for the path between F and e [ − A ′ (1) , · ] ◦ F π ◦ e [ A (1) , · ] is F ( t ) = e [ − A ′ ( t ) , · ] ◦ F π ( t ) ◦ e [ A ( t ) , · ] . However, F ( t ) is not necessarily in the completion \ (Hom(S( g [1]) , g ′ ) [ t ]) with respect to the filtra-tion from (3.5) since for example F ( t ) mod F Hom(S( g [1]) , g ′ )[[ t ]] = e [ − A ′ ( t ) , · ] ◦ F π ( t )1 ◦ e [ A ( t ) , · ] is in general no polynomial in t . To solve this problem we introduce a new filtration on theconvolution DGLA h = Hom(S( g [1]) , g ′ ) that takes into account the filtrations on S( g [1]) and g ′ : h = F h ⊃ F h ⊃ · · · ⊃ F k h ⊃ · · · F k h = X n + m = k n f ∈ Hom(S( g [1]) , g ′ ) | f (cid:12)(cid:12) S The above filtration (3.7) is a complete descending filtration on the convolutionDGLA Hom(S( g [1]) , g ′ ) . Proof: The filtration is obviously descending and h = F h since we consider in the convolutionDGLA only maps that are compatible with respect to the filtration. It is compatible with theconvolution DGLA structure and complete since g ′ is complete. (cid:3) hus we can finally prove Proposition 3.10. Proof (of Prop. 3.10): The path F ( t ) = e [ − A ′ ( t ) , · ] ◦ F π ( t ) ◦ e [ A ( t ) , · ] is an element in thecompletion \ (Hom(S( g [1]) , g ′ )[1]) [ t ] with respect to the filtration from (3.7). This is clear since A ( t ) ∈ F d g [ t ] , A ′ ( t ) ∈ F \ ( g ′ ) [ t ] and π ( t ) ∈ F d g [ t ] imply that n − X i =1 e [ − A ′ ( t ) , · ] ◦ F π ( t ) i ◦ e [ A ( t ) , · ] mod F n (Hom(S( g [1]) , g ′ )[1])[[ t ]] is polynomial in t . Moreover, F ( t ) satisfies by (2.10) d F ( t )d t = − exp([ − A ′ ( t ) , · ]) ◦ [ λ ′ ( t ) , · ] ◦ F π ( t ) ◦ e [ A ( t ) , · ] + e [ − A ′ ( t ) , · ] ◦ F π ( t ) ◦ [ λ ( t ) , · ] ◦ e [ A ( t ) , · ] + e [ − A ′ ( t ) , · ] ◦ d F π ( t ) d t ◦ e [ A ( t ) , · ] . But we have d F π ( t ) k d t ( X ∨ · · · ∨ X k ) = F π ( t ) k +1 ( Q π ( t ) , ( λ ( t )) ∨ X ∨ · · · ∨ X k )= F π ( t ) k +1 ( Q π ( t ) ,k +1 k +1 ( λ ( t ) ∨ X ∨ · · · ∨ X k )) + F π ( t ) k +1 ( λ ( t ) ∨ Q π ( t ) ,kk ( X ∨ · · · ∨ X k )))= Q S ( t ) , F π ( t ) , k +1 ( λ ( t ) ∨ X ∨ · · · ∨ X k ) + Q S ( t ) , F π ( t ) , k +1 ( λ ( t ) ∨ X ∨ · · · ∨ X k ) − F π ( t ) , k ◦ Q π ( t ) ,kk +1 ( λ ( t ) ∨ X ∨ · · · ∨ X k ) + F π ( t ) k +1 ( λ ( t ) ∨ Q π ( t ) ,kk ( X ∨ · · · ∨ X k )) . Setting now λ Fk ( t )( · · · ) = F π ( t ) k +1 ( λ ( t ) ∨ · · · ) we get d F π ( t ) k d t = b Q t, ( λ Fk ( t )) + b Q t, ( λ F ( t ) ∨ F π ( t ) ) − F π ( t ) k ◦ [ λ ( t ) , · ] + [ λ ′ ( t ) , · ] ◦ F π ( t ) k . Thus we get d F ( t )d t = e [ − A ′ ( t ) , · ] ◦ (cid:16) b Q t, ( λ F ( t )) + b Q t, ( λ F ( t ) ∨ F π ( t ) ) (cid:17) ◦ e [ A ( t ) , · ] = b Q ( e [ − A ′ ( t ) , · ] λ F ( t ) e [ A ( t ) , · ] ) + b Q ( e [ − A ′ ( t ) , · ] λ F ( t ) e [ A ( t ) , · ] ∨ F ( t )) since exp([ A ( t ) , · ]) and exp([ A ′ ( t ) , · ]) commute with the brackets and intertwine the differentials.Thus F (0) = F and F (1) are homotopy equivalent and by Theorem 2.12 also gauge equivalent. (cid:3) Now we want to apply the above general results to the globalization of the Kontsevich formalityto smooth real manifolds M . More precisely, the globalization procedure proved by Dolgushevin [5, 6] depends on the choice of a torsion-free covariant derivative on M and we show that theglobalizations with respect to two different covariant derivatives are homotopic. Let us briefly recall the globalization procedure from from [5, 6] to establish the notation.• T kpoly denotes the bundle of formal fiberwise polyvector fields of degree k over M . Its sectionsare C ∞ ( M ) -linear operators v : Λ k +1 Γ ∞ ( SM ) → Γ ∞ ( SM ) of the local form v = ∞ X p =0 v j ...j k i ...i p ( x ) y i · · · y i p ∂∂y j ∧ · · · ∧ ∂∂y j k . Analogously, the sections of formal fiberwise differential operators D kpoly are C ∞ ( M ) -linearoperators X : N k +1 Γ ∞ ( SM ) → Γ ∞ ( SM ) of the local form X = X α ,...,α k ∞ X p =0 X α ...α k i ...i p ( x ) y i · · · y i p ∂∂y α ⊗ · · · ⊗ ∂∂y α k . Here X α ...α k i ...i p are symmetric in the indices i , . . . , i p and α are multi-indices α = ( j , . . . , j k ) .Moreover, the sum in the orders of the derivatives is finite.• D = − δ + ∇ + [ A, · ] = d + [ B, · ] is the Fedosov differential, where δ = [d x i ∂∂y i , · ] , ∇ =d x i ∂∂x i − [d x i Γ kij ( x ) y j ∂y k , · ] with Christoffel symbols Γ kij of a torsion-free connection on M with curvature R = − d x i d x j ( R ij ) kl ( x ) y l ∂∂y k , and A ∈ Ω ( M, T poly ) ⊆ Ω ( M, D poly ) is theunique solution of δ ( A ) = R + ∇ A + [ A, A ] ,δ − ( A ) = r,σ ( A ) = 0 . Here r ∈ Ω ( M, T poly ) is arbitrary but fixed and has vanishing constant and linear term withrespect to the y -variables. We refer to ( ∇ , r ) as globalization data.• τ : Γ ∞ δ ( T poly ) → Z (Ω( M, T poly ) , D ) denotes the Fedosov Taylor series, given by τ ( a ) = a + δ − ( ∇ τ ( a ) + [ A, τ ( a )]) . Here one has Γ ∞ δ ( T poly ) = { v = P k v j ...j k ( x ) ∂∂y j ∧ · · · ∧ ∂∂y jk } , analogously for the polydif-ferential operators. In addition, ∂ M denotes the fiberwise Hochschild differential.• ν : Γ ∞ δ ( T poly ) → T poly ( M ) is given by ν ( w )( f , . . . , f k ) = σw ( τ ( f ) , . . . , τ ( f k )) for f , . . . , f k ∈ C ∞ ( M ) , where σ sets the d x i and y j coordinates to zero, analogously for the polydifferential operators.• U B is the fiberwise formality of Kontsevich U twisted by B = D − d = − d x i ∂∂y i − d x i Γ kij ( x ) y j ∂∂y k + X p ≥ d x i A kij ...j p ( x ) y j · · · y j p ∂∂y k . By the properties of the Kontsevich formality the first two summands do not contribute, i.e. U B = U A .One obtains the diagram T poly ( M ) τ ◦ ν − −→ (Ω( M, T poly ) , D ) U B −→ (Ω( M, D poly ) , D + ∂ M ) τ ◦ ν − ←− D poly ( M ) , where τ ◦ ν − are quasi-isomorphisms of DGLAs and where U B is an L ∞ -quasi-isomorphism. Ina next step, the morphism U B ◦ τ ◦ ν − is modified to a quasi-isomorphism U : T poly ( M ) −→ ( Z D (Ω( M, D poly )) , ∂ M , [ · , · ] G ) , see [6, Prop. 5]. By [8, Lemma 1] we know that U B ◦ τ ◦ ν − and U are homotopic. The desiredquasi-isomorphism U ( ∇ ,r ) = ν ◦ σ ◦ U : T poly ( M ) → D poly ( M ) is then the composition of U withthe DGLA isomorphism ν ◦ σ : ( Z D (Ω( M, D poly )) , ∂ M , [ · , · ] G ) −→ ( D poly ( M ) , ∂, [ · , · ] G ) . Corollary 4.1 The formality U ( ∇ ,r ) induces a one-to-one correspondence between equivalent for-mal Poisson structures on M and equivalent differential star products on C ∞ ( M )[[ ~ ]] , i.e. abijection U ( ∇ ,r ) : Def ( T poly ( M )[[ ~ ]]) −→ Def ( D poly ( M )[[ ~ ]]) . (4.1) .2 Explicit Construction of the Projection L ∞ -morphism As an alternative to the modification of the formality in [6, Prop. 5] we want to construct the L ∞ -quasi-inverse of τ ◦ ν − . We want to use the construction from [12, Prop. 3.2] that gives aformula for the L ∞ -quasi-inverse of an inclusion of DGLAs, see also [17] for the existence in moregeneral cases. In our setting we have the contraction ( D poly ( M ) , ∂ ) (Ω( M, D poly ) , ∂ M + D ) , τ ◦ ν − ν ◦ σ h (4.2)where the homotopy h with respect to ∂ M + D is constructed as follows: As in the Fedosovconstruction in the symplectic setting one has a homotopy D − for the differential D , see also [6,Thm. 3]: Proposition 4.2 The map D − = − δ − − [ δ − , ∇ + [ A, · ]] = − − [ δ − , ∇ + [ A, · ]] δ − (4.3) is a homotopy for D on Ω( M, D poly ) , i.e. one has X = DD − X + D − DX + τ σ ( X ) . (4.4) Proof: The proof is the same as in the symplectic setting, see e.g. [20, Prop. 6.4.17]. (cid:3) If this homotopy is also compatible with the Hochschild differential ∂ M , then we can indeedapply [12, Prop. 3.2] to describe the L ∞ -morphism extending ν ◦ σ . Let us denote by ( D − ) k +1 the extended homotopy on S k +1 (Ω( M, D poly )[1]) and let us write Q D poly , Q D poly for the inducedcodifferentials on the symmetric algebras. Then we get: Proposition 4.3 The homotopy D − anticommutes with ∂ M , whence it is also a homotopy for ∂ M + D . Therefore, one obtains an L ∞ -quasi-isomorphism P : S(Ω( M, D poly )[1]) → S( D poly ( M )[1]) with recursively defined structure maps P = ν ◦ σ and P k +1 = ( Q D poly , ◦ P k +1 − P k ◦ Q k D poly ,k +1 ) ◦ ( D − ) k +1 . (4.5) Proof: The fact that D − anticommutes with ∂ M is clear as ∇ + [ A, · ] and δ − anticommutewith ∂ , and the rest follows directly from [12, Prop. 3.2]. (cid:3) Summarizing, we obtain another global formality: Corollary 4.4 Given the globalization data ( ∇ , r ) there exists an L ∞ -quasi-isomorphism F ( ∇ ,r ) = P ◦ U B ◦ τ ◦ ν − : T poly ( M ) −→ D poly ( M ) (4.6) with F being the Hochschild-Kostant-Rosenberg map. Proof: We immediately get F ( ∇ ,r ) , = P ◦ ( U B ) ◦ τ ◦ ν − = ν ◦ σ ◦ U ◦ τ ◦ ν − and the statementfollows since U is the fiberwise Hochschild-Kostant-Rosenberg map. (cid:3) The higher structure maps of P k +1 of the L ∞ -projection contain copies of the homotopy D − that decrease the antisymmetric form degree. Therefore, they vanish on Ω ( M, D poly ) andare needed to get rid of the form degrees arising from the twisting with B , analogously to themodifying of the formality from U B ◦ τ ◦ ν − to U .As a last point, we want to remark that we can use the L ∞ -projection P to obtain a splittingof Ω( M, D poly ) similar to the one used in the proof of Lemma 3.8: Instead of splitting into theproduct of the cohomology as minimal L ∞ -algebra and a linear contractible one, we can prove inour setting: emma 4.5 One has an L ∞ -isomorphism L : Ω( M, D poly ) −→ D poly ( M ) ⊕ im[ D, D − ] . (4.7) Here the L ∞ -structure on D poly ( M ) is the usual one consisting of Gerstenhaber bracket andHochschild differential ∂ and on im[ D, D − ] the L ∞ -structure is just given by the differential ∂ M + D . The L ∞ -structure on D poly ( M ) ⊕ im[ D, D − ] is the product L ∞ -structure. Proof: By Proposition 4.3 we already have an L ∞ -morphism P : Ω( M, D poly ) → D poly ( M ) withfirst structure map ν ◦ σ . Now we construct an L ∞ -morphism F : Ω( M, D poly ) → im[ D, D − ] . Weset F = DD − + D − D and F n = − D − ◦ F n − ◦ ( Q D poly ) n − n for n ≥ and note that in particular F k = 0 for k ≥ by D − D − = 0 . In the following, we denote by Q D poly the L ∞ -structure on Ω( M, D poly ) and by e Q the L ∞ -structure on im[ D, D − ] with e Q = − ( ∂ M + D ) as only vanishingcomponent. We have F n = D − ◦ L ∞ ,n with L ∞ ,n = − F n − ◦ ( Q D poly ) n − n . By [12, Lemma 3.1]we know that if F is an L ∞ -morphism up to order n , i.e. if ( e QF ) k = ( F Q D poly ) k for all k ≤ n ,then one has e Q ◦ L ∞ ,n +1 = − L ∞ ,n +1 ◦ ( Q D poly ) n +1 n +1 . By Proposition 4.3 we know that F is an L ∞ -morphism up to order one. Assuming it defines an L ∞ -morphism up to order n , then we getwith (4.4) e Q ◦ F n +1 = − ( ∂ M + D ) ◦ D − ◦ L ∞ ,n +1 = D − ◦ ∂ M ◦ L ∞ ,n +1 − L ∞ ,n +1 + D − ◦ D ◦ L ∞ ,n +1 + τ ◦ σ ◦ L ∞ ,n +1 = − L ∞ ,n +1 − D − ◦ e Q ◦ L ∞ ,n +1 = − L ∞ ,n +1 + F n +1 ◦ ( Q D poly ) n +1 n +1 . Thus F is an L ∞ -morphism up to order n + 1 and therefore an L ∞ -morphism.The universal property of the product gives the desired L ∞ -morphism L = P ⊕ F which iseven an isomorphism since its first structure map ( ν ◦ σ ) ⊕ ( DD − + D − D ) is an isomorphismwith inverse ( τ ◦ ν − ) ⊕ id , see e.g. [4, Prop. 2.2]. (cid:3) The above globalization of the Kontsevich Formality depends on the choice of a covariant derivative.We want to show that globalizations with respect to different covariant derivatives are homotopicin the sense of Definition 3.3. The ideas are similar to those in the proof of [2, Theorem 2.6]:observe that changing the covariant derivative corresponds to twisting with a Maurer-Cartanelement which is equivalent to zero, and apply Proposition 3.10. Remark 4.6 (Filtrations on Fedosov resolutions) In order to apply Proposition 3.10 weneed complete descending and exhaustive filtrations on the Fedosov resolutions. As in [2, Ap-pendix C] we assign to d x i and y i the degree and to ∂∂y i the degree − and consider the induceddescending filtration. The filtration on Ω( M, T poly ) is complete and bounded below since Ω( M, T poly ) ∼ = lim ←− Ω( M, T poly ) / F k Ω( M, T poly ) and Ω( M, T poly ) = F − d Ω( M, T poly ) , where d is the dimension of M . In the case of the differential operators the filtration is unboundedin both directions. Instead of D poly we consider from now on its y -adic completion withoutchanging the notation. This is the completion with respect to the filtration induced by assigning y i the degree and ∂∂y i the degree − . The globalization of the formality works just the sameand one obtains the desired properties Ω( M, D poly ) ∼ = lim ←− Ω( M, D poly ) / F k Ω( M, D poly ) and Ω( M, D poly ) = [ k F k Ω( M, D poly ) . Let ( ∇ ′ , r ′ ) be a second pair of globalization data, then there is a second diagram T poly ( M ) τ ′ ◦ ν − −→ (Ω( M, T poly ) , D ′ ) U B ′ −→ (Ω( M, D poly ) , D ′ + ∂ M ) τ ′ ◦ ( ν ′ ) − ←− D poly ( M ) , here D ′ = − δ + ∇ ′ + [ A ′ , · ] = d + [ B ′ , · ] , and A ′ is the unique solution of δ ( A ′ ) = R ′ + ∇ ′ A ′ + [ A ′ , A ′ ] ,δ − ( A ′ ) = r ′ ,σ ( A ′ ) = 0 . In the case of polyvector fields one easily sees that ν = ν ′ . Note ∇ ′ − ∇ = (cid:20) − d x i y j (Γ ′ kij − Γ kij ) ∂∂y k , · (cid:21) = [ δs, · ] for s = − y i y j (Γ ′ kij − Γ kij ) ∂∂y k . Thus we get D ′ = − δ + ∇ + [ A ′ + δs, · ] = − δ + ∇ + [ e A, · ] (4.8)and since R ′ = R + ∇ δs + [ δs, δs ] we know that e A = A ′ + δs is the unique solution of δ ( e A ) = R + ∇ e A + [ e A, e A ] ,δ − ( e A ) = r ′ + s,σ ( e A ) = 0 . As in [2, Appendix C] one can now show that B and B ′ can be interpreted as equivalentMaurer-Cartan elements: Proposition 4.7 There exists an element h ∈ F Ω ( M, T poly ) (4.9) that is at least quadratic in the fiber coordinates y such that one has B ′ − B = e A − A = − exp([ h, · ]) − id[ h, · ] Dh ∈ F Ω ( M, T poly ) (4.10) and exp([ h, · ]) ◦ D ◦ exp([ − h, · ]) = D ′ . (4.11) Thus B ′ − B is gauge equivalent to zero in in (Ω( M, T poly ) , D ) and (Ω( M, D poly ) , D + ∂ M ) , where h implements the gauge equivalences. Proof: For the existence of the element h ∈ F Ω ( M, T poly ) encoding the gauge equivalence inthe polyvector fields see [2, Appendix C]. Thus we have a path B ( t ) = − exp([ th, · ]) − id[ th, · ] D ( th ) ∈ F \ Ω ( M, T poly )[ t ] that satisfies B (0) = 0 , B (1) = B ′ − B and d B ( t )d t = Q ( λ ( t ) ∨ exp( B ( t ))) with λ ( t ) = h. The formality U B satisfies in the notation of Proposition 2.15 e B ( t ) = U B, (exp( B ( t )) = B ( t ) and e λ ( t ) = U B, ( h ∨ exp( B ( t ))) = h since the higher orders of the Kontsevich formality vanish if one only inserts vector fields. (cid:3) Now it follows directly from Proposition 3.10 and ( U B ) B ′ − B = U B ′ that the twisted formalitiesare homotopic. orollary 4.8 The L ∞ -morphisms U B and e − [ h, · ] ◦ U B ′ ◦ e [ h, · ] are homotopic. Moreover, the Fedosov Taylor series is compatible in the following sense: Corollary 4.9 For all X ∈ T poly ( M ) one has e [ h, · ] ◦ τ ◦ ν − ( X ) = τ ′ ◦ ( ν ′ ) − ( X ) . (4.12) Proof: By the above proposition exp([ h, · ]) maps the kernel of D into the kernel of D ′ . There-fore, e [ h, · ] ◦ τ ◦ ν − ( X ) = τ ′ ◦ σ ◦ e [ h, · ] ◦ τ ◦ ν − ( X ) = τ ′ ◦ ν − ( X ) since h is at least quadratic in the y coordinates. (cid:3) Similarly, one has on the differential operator side the following identity: Lemma 4.10 For all X ∈ Z D ′ (Ω( M, D poly )) one has ν ◦ σ ◦ e − [ h, · ] ( X ) = ν ′ ◦ σ ( X ) . (4.13) Proof: Using the definition of ν we compute for X ∈ Z D ′ (Ω( M, D n − poly )) and f , . . . , f n ∈ C ∞ ( M )( ν ◦ σ ◦ e − [ h, · ] X )( f , . . . , f n ) = σ (( σ ◦ e − [ h, · ] X )( τ ( f ) , . . . , τ ( f n )))= σ ( e − h ( X ( e h τ ( f ) , . . . , e h τ ( f n ))))= σ (( σ ◦ X )( τ ′ ( f ) , . . . , τ ′ ( f n )))= ( ν ′ ◦ σX )( f , . . . , f n ) and the statement is shown. (cid:3) As a last preparation we want to compare the two different L ∞ -projections P ′ and P ◦ e − [ h, · ] from (Ω( M, D poly ) , ∂ M + D ′ ) to ( D poly ( M ) , ∂ ) . Lemma 4.11 The L ∞ -projections P ′ and P ◦ e − [ h, · ] are homotopic. Proof: Since the higher structure maps of P and P ′ vanish on the zero forms, we have byLemma 4.10 P ◦ e − [ h, · ] ◦ τ ′ ◦ ( ν ′ ) − = ν ◦ σ ◦ e − [ h, · ] ◦ τ ′ ◦ ( ν ′ ) − = ν ′ ◦ σ ◦ τ ′ ◦ ( ν ′ ) − = id D poly ( M ) . Instead of directly using Lemma 3.8, we recall the splitting from Lemma 4.5 and adapt the proofof Lemma 3.8. Define M ( t ) : D poly ( M ) ⊕ im[ D, D − ] ∋ ( D , D ) ( D , tD ) ∈ D poly ( M ) ⊕ im[ D, D − ] which is an L ∞ -morphism with respect to the product L ∞ -structure. Setting H ( t ) : D poly ( M ) ⊕ im[ D, D − ] ∋ ( D , D ) (0 , − D − D ) ∈ D poly ( M ) ⊕ im[ D, D − ] we obtain again dd t M ( t ) = pr im[ D,D − ] = 0 ⊕ ( DD − + D − D )= − ∂ ⊕ ( ∂ M + D ) ◦ H ( t ) − H ( t ) ◦ ( ∂ ⊕ ( ∂ M + D ))= b Q ( H ( t ) ∨ exp( M ( t ))) . Therefore, it follows that L ( t ) = L − ◦ M ( t ) ◦ L : Ω( M, D poly ) −→ Ω( M, D poly ) encodes the homotopy between L (0) = τ ◦ ν − ◦ P and L (1) = id . But this implies with Proposition 3.5 P ◦ e − [ h, · ] ∼ P ◦ e − [ h, · ] ◦ τ ′ ◦ ( ν ′ ) − ◦ P ′ = P ′ and the statement is shown. (cid:3) ombining all the above statements we can show that the globalizations with respect to dif-ferent covariant derivatives are homotopy equivalent. Theorem 4.12 Let ( ∇ , r ) and ( ∇ ′ , r ′ ) be two pairs of globalization data. Then the formalitiesconstructed via Dolgushev’s globalization and the globalized formalities via the L ∞ -projection areall are homotopic, i.e. one has U ( ∇ ,r ) ∼ F ( ∇ ,r ) ∼ F ( ∇ ′ ,r ′ ) ∼ U ( ∇ ′ ,r ′ ) . (4.14) Proof: By Proposition 3.4 we already know that compositions of homotopic L ∞ -morphisms withDGLA morphisms are homotopic, which yields U ∼ U B ◦ τ ◦ ν − ∼ e − [ h, · ] ◦ U B ′ ◦ e [ h, · ] ◦ τ ◦ ν − = e − [ h, · ] ◦ U B ′ ◦ τ ′ ◦ ( ν ′ ) − ∼ e − [ h, · ] ◦ U ′ . It follows with Lemma 4.11, Proposition 3.5 and Proposition 3.6 U ( ∇ ,r ) = ν ◦ σ ◦ U = P ◦ U ∼ P ◦ U B ◦ τ ◦ ν − ∼ P ◦ e − [ h, · ] ◦ U B ′ ◦ τ ′ ◦ ( ν ′ ) − ∼ P ′ ◦ U B ′ ◦ τ ′ ◦ ( ν ′ ) − ∼ P ′ ◦ U ′ = ν ′ ◦ σ ◦ U ′ = U ( ∇ ′ ,r ′ ) and the theorem is shown. (cid:3) Corollary 4.13 Let M be a smooth manifold and let ( ∇ , r ) be a globalization data. For everycoordinate patch ( U, x ) F ( ∇ ,r ) (cid:12)(cid:12) U ∼ K (cid:12)(cid:12) U , holds, where K denotes the Kontsevich formality on R d , and where d is the dimension of M . Proof: The formalities themselves are differential operators and can therefore be restricted toopen neighbourhoods. Moreover, the Kontsevich formality coincides with the Dolgushev formalityon R d for the choice of the canonical flat covariant derivative and r = 0 . (cid:3) This allows us to recover [2, Theorem 2.6], i.e. that the induced maps on equivalence classesof Maurer-Cartan elements are independent of the choice of the covariant derivative. It impliesin particular that globalizations with respect to different covariant derivatives lead to equivalentstar products. Corollary 4.14 The induced map Def ( T poly ( M )[[ ~ ]]) −→ Def ( D poly ( M )[[ ~ ]]) does not depend onthe choice of a covariant derivative. Proof: The statement follows directly from Theorem 4.12 and Proposition 3.4. (cid:3) Finally, note that Theorem 4.12 also holds in the equivariant setting of an action of a Liegroup G on M with G -invariant torsion-free covariant derivatives ∇ and ∇ ′ . Proposition 4.15 Let G act on M and consider two pairs of globalization data ( ∇ , r ) and ( ∇ ′ , r ′ ) ,where ∇ and ∇ ′ are two G -invariant torsion-free covariant derivatives and where r and r ′ are G -invariant. Then the formalities are equivariant and equivariantly homotopic U ( ∇ ,r ) ∼ G F ( ∇ ,r ) ∼ G F ( ∇ ′ ,r ′ ) ∼ G U ( ∇ ′ ,r ′ ) , (4.15) i.e. all paths encoding the equivalence relation from (2.9) are G -equivariant. Proof: The formalities are equivariant since all involved maps are [6, Theorem 5]. Moreover, U B and e − [ h, · ] ◦ U B ′ ◦ e [ h, · ] are equivariantly homotopic by the explicit form of the homotopyfrom Proposition 3.10. Moreover, again by [6, Theorem 5] we know that U and U B ◦ τ ◦ ν − are equivariantly homotopic, the same holds for the ( ∇ ′ , r ′ ) case. Thus by Theorem 4.12 it onlyremains to show that P ◦ e − [ h, · ] and P ′ are equivariantly homotopic. But this follows directlyfrom Lemma 4.11 since all involved maps are equivariant. (cid:3) In the case of proper Lie group actions one knows that invariant covariant derivatives alwaysexist and one has even an invariant Hochschild-Kostant-Rosenberg theorem, compare [19, Theo-rem 5.11]. Thus the L ∞ -morphisms from (4.15) restrict to the invariant DGLAs and one obtainshomotopic formalities from ( T poly ( M )) G to ( D poly ( M )) G . Final Remarks In [7, Thm. 6] it is proven that the construction of the formality U ( ∇ , is functorial for diffeo-morphisms of pairs ( M, ∇ ) . More explicitly, the objects of the source category are pairs ( M, ∇ ) of manifolds with torsion-free covariant derivatives, and a morphism from ( M, ∇ ) to ( M ′ , ∇ ′ ) isa diffeomorphism φ : M → M ′ such that φ ∗ ( ∇ X Y ) = ∇ ′ φ ∗ X φ ∗ Y for all X, Y ∈ Γ ∞ ( T M ) . The target category is the category of triples ( A, B, F ) , where A, B are L ∞ -algebras and where F : A → B is an L ∞ -quasi-isomorphism. A morphism is a pair ( f, g ) : ( A, B, F ) → ( A ′ , B ′ , F ′ ) consisting of two L ∞ -morphisms f : A → A ′ and g : B → B ′ suchthat A BA ′ B ′ Ff gF ′ commutes. The functor from [7] is hence given by ( M, ∇ ) −→ ( T poly ( M ) , D poly ( M ) , U ( ∇ , ) , mapping diffeomorphisms to push-forwards of vectorfields resp. differential operators. Our inves-tigations from above lead now to a functor from the category of manifolds with diffeomorphismsinto a category as above but with morphisms being homotopy classes of L ∞ -quasi-isomorphimsand L ∞ -morphisms, respectively. It is given by M −→ (( T poly ( M ) , D poly ( M ) , [ U ( ∇ , ]) , where [ · ] indicates the homotopy class and where ∇ is an arbitrary torsion-free connection.Moreover, for a Lie group G , we can consider the source category of G -manifolds with proper G -actions and with equivariant diffeomorphisms to get the functor M → ( T poly ( M ) G , D poly ( M ) G , [ U ( ∇ , ]) . Here ∇ is an arbitrary G -invariant torsion-free connection. Bibliography [1] Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D.: Deforma-tion Theory and Quantization . Ann. Phys. (1978), 61–151. 2[2] Burzstyn, H., Dolgushev, V., Waldmann, S.: Morita equivalence and characteristicclasses of star products . Crelle’s J. reine angew. Math. (2012), 95–163. 2, 5, 8, 9, 16, 17,19[3] Calaque, D.: Formality for Lie Algebroids . Commun. Math. Phys. (2005), 563–578. 2[4] Canonaco, A.: L ∞ -algebras and quasi-isomorphisms . In: Seminari di Geometria Algebrica1998-1999 , 67–86. Scuola Normale Superiore, Pisa, 1999. 2, 4, 5, 6, 7, 9, 10, 11, 16[5] Dolgushev, V.: A formality theorem for Hochschild chains . Adv. Math. (2006), 51–101.2, 7, 13[6] Dolgushev, V. A.: Covariant and equivariant formality theorems . Adv. Math. (2005),147–177. 2, 3, 4, 13, 14, 15, 19[7] Dolgushev, V. A.: A Proof of Tsygan’s Formality Conjecture for an Arbitrary SmoothManifold . PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, 2005.math.QA/0504420. 2, 3, 4, 7, 20[8] Dolgushev, V. A.: Erratum to: "A Proof of Tsygan’s Formality Conjecture for an ArbitrarySmooth Manifold" . PhD thesis, 2007. arXiv:math/0703113. 2, 7, 8, 14 Dotsenko, V., Poncin, N.: A tale of three homotopies . Applied Categorical Structures (2016), 845–873. 2, 5, 7[10] Dotsenko, V., Shadrin, S., Vallette, B.: The twisting procedure . Preprint arXiv:1810.02941 (2018), 93 pages. 2, 12[11] Esposito, C., de Kleijn, N.: L ∞ -resolutions and twisting in the curved context . Preprint arXiv:1801.08472 (2018), 16 pages. 2, 4[12] Esposito, C., Kraft, A., Schnitzer, J.: The Strong Homotopy Structure of PoissonReduction . Preprint arXiv:2004.10662 (2020), 30 pages. 15, 16[13] Getzler, E.: Lie theory for nilpotent L ∞ -algebras . Ann. of Math. .1 (2009), 271–301. 6[14] Goldman, W. M., Millson, J. J.: The deformation theory of representations of funda-mental groups of compact Kähler manifolds . Inst. Hautes Études Sci. Publ. Math. (1988),43–96. 4[15] Kontsevich, M.: Deformation Quantization of Poisson manifolds . Lett. Math. Phys. (2003), 157–216. 2, 3, 10[16] Liao, H.-Y., Stiénon, M., Xu, P.: Formality theorem for differential graded manifolds .Comptes Rendus Mathematique .1 (2018), 27– 43. 2[17] Loday, J.-L., Vallette, B.: Algebraic Operads , vol. 346 in Grundlehren der mathematis-chen Wissenschaften . Springer-Verlag, Berlin, Heidelberg, New York, 2012. 15[18] Manetti, M.: Deformation theory via differential graded Lie algebras , 2005. 2, 4, 6[19] Miaskiwskyi, L.: Invariant Hochschild cohomology of smooth functions . Preprint arXiv:1808.08096 (2018), 16 pages. 19[20] Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung .Springer-Verlag, Heidelberg, Berlin, New York, 2007. 15.Springer-Verlag, Heidelberg, Berlin, New York, 2007. 15