Formal Lagrangian Operad

Abstract

Given a symplectic manifold 𝑀, we may define an operad structure on the the spaces 𝑂^𝑘 of the Lagrangian submanifolds of (𝑀)^𝑘×𝑀 via symplectic reduction. If 𝑀 is also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semiclassical part of Kontsevich's deformation of 𝐶^∞(ℝ^𝑑) is a deformation of the trivial symplectic groupoid structure of 𝑇^∗ℝ^𝑑.

1. Introduction

Symplectic groupoids, in the extended symplectic category, may be thought as the analog of associative algebras in the category of vector spaces. For the latter, a deformation theory exists and is well known. In this paper, we will present a conceptual framework as well as an explicit deformation of the trivial symplectic groupoid over ℝ^𝑑. In fact, rephrased appropriately, most constructions of the deformation theory of algebras can be extended to symplectic groupoids, at least for the trivial one over ℝ^𝑑. Our guideline will be the Kontsevich deformation of the usual algebra of functions over ℝ^𝑑, (𝐶^∞(ℝ^𝑑),⋅). Namely, the usual pointwise product of functions 𝑆²₀(𝑓,𝑔)=𝑓𝑔 generates a suboperad, the product suboperad, 𝒪^𝑛_𝑆={𝑆^𝑛₀}, of the endomorphism operad 𝒪 of 𝐶^∞(ℝ^𝑑), where 𝑆^𝑛₀ is the 𝑛-multilinear map defined by 𝑆^𝑛₀(𝑓₁,…,𝑓_𝑛)=𝑓₁𝑓₂⋯𝑓_𝑛. For each 𝑛 one may choose the vector subspace 𝒪^𝑛_def⊂𝒪^𝑛 of 𝑛-multidifferential operators. The operad structure of 𝒪 induces an operad structure on 𝒪_𝑆+𝒪_def, which in turn generates an operad structure on 𝒪_def which is, however, nonlinear. Then, 𝛾 is a deformation of the usual product 𝑆²₀, that is, an element 𝛾∈𝒪²_def such that 𝑆²₀+𝛾 is still an associative product, if 𝛾 is a product in the induced deformation operad 𝒪_def. We may also consider the formal version by replacing 𝒪_def by the formal power series in 𝜖, 𝜖𝒪_def[[𝜖]]. Kontsevich in [1] gives an explicit formal deformation of the product of functions over ℝ^𝑑, 𝑆_𝜖=𝑆²₀+^∞_𝑛=1𝜖^𝑛_{Γ∈𝐺𝑛,2}𝑊_Γ𝐵_Γ, (1.1) where the 𝑊_Γ's are the Kontsevich weights and the 𝐵_Γ's are the Kontsevich bidifferential operators associated to the Kontsevich graphs of type (𝑛,2) (see [2] for a brief introduction).

If we consider the trivial symplectic groupoid T^∗ℝ^𝑑 over ℝ^𝑑, we see that the multiplication space Δ^𝑛₂∶=𝑝₁,𝑥,𝑝₂,𝑥,𝑝₁+𝑝₂,𝑥∶𝑝₁,𝑝₂∈ℝ^𝑑∗,𝑥∈ℝ^𝑑 (1.2) generates an operad 𝒪^𝑛_Δ={Δ_𝑛}, where Δ_𝑛∶=𝑝₁,𝑥,…,𝑝_𝑛,𝑥,𝑝₁+⋯+𝑝_𝑛,𝑥∶𝑝_𝑖∈ℝ^𝑑∗,𝑥∈ℝ^𝑑. (1.3)Δ₂ is a product in this operad. The compositions are given by symplectic reduction as the Δ_𝑛's are Lagrangian submanifolds of (𝑇^∗ℝ^𝑑)^𝑛×𝑇^∗ℝ^𝑑. The main difference with the vector space case is that there is no “true” endomorphism operad where 𝒪_Δ would naturally embed into. Thus, the question of finding a deformation operad for 𝒪_Δ must be taken with more care. The first remark is that the Δ_𝑛 may be expressed in terms of generating functions 𝑆^𝑛₀𝑝₁,…,𝑝_𝑛,𝑥=𝑝₁+⋯+𝑝_𝑛𝑥. (1.4) Namely, Δ_𝑛=graph𝑑𝑆^𝑛₀. The idea is to look at the operad structure induced on the generating functions by symplectic reduction. In fact it is possible to find a vector space of special functions 𝒪^𝑛_def for each 𝑛 such that 𝒪_Δ+𝒪_def remains an operad. The formal version of it gives a surprising result. Namely, we may find an explicit deformation of the trivial generating function 𝑆²₀, it is given by the formula 𝑆_𝜖=𝑆²₀+^∞_𝑛=1𝜖^𝑛_{Γ∈𝑇𝑛,2}𝑊_Γ𝐵_Γ, (1.5) where the 𝑊_Γ are the Kontsevich weights and the 𝐵_Γ are the symbols of the Kontsevich bidifferential operators and the sum is taken over all Kontsevich trees 𝑇_𝑛,2. This formula may be seen as the semi-classical part of Kontsevich deformation quantization formula.

As a last comment, note that Kontsevich derives its star product formula from a more general result. In fact, he shows that 𝑈=∑_𝑛𝜖^𝑛𝑈_𝑛, where 𝑈_𝑛𝜉₁,…,𝜉_𝑛=_{Γ∈𝐺𝑛}𝑊_Γ𝐵_Γ𝜉₁,…,𝜉_𝑛 (1.6) for 𝜉_𝑖∈Γ(∧^𝑑𝑖𝑇𝑀), 𝑖=1,…,𝑑 is an 𝐿_∞-morphism from the multivector fields to the multidifferential operators on ℝ^𝑑. In our perspective, we may still write 𝑈_𝑛𝜉₁,…,𝜉_𝑛=_{Γ∈𝑇𝑛}𝑊_Γ𝐵_Γ𝜉₁,…,𝜉_𝑛 (1.7) summing over Kontsevich trees instead of Kontsevich graphs and replacing multidifferential operators by their symbols. Exactly, as in Kontsevich case, 𝑆_𝜖=𝑆²₀+_𝑛≥1𝜖^𝑛𝑈_𝑛(𝛼,…,𝛼) (1.8) is an associative deformation of the generating function of the trivial symplectic groupoid 𝑇^∗ℝ^𝑑. However, it is still not completely clear how to define “semi-classical 𝐿_∞-morphisms”.

Organization of the Paper
In Section 2, we describe the endomorphism operad 𝒪(𝑀)=Hom(𝑀^⊗𝑛,𝑀) associated to any object 𝑀 in a monoidal category. We explain what is an associative product 𝑆 on 𝑀 in a monoidal category and we define the product suboperad 𝒪_𝑆(𝑀) of 𝒪(𝑀). If the category is further associative, we may choose a deformation operad for 𝑆, which is a choice, for each 𝑛∈ℕ of a vector subspace 𝒪^𝑛_def such that 𝒪_𝑆+𝒪_def is still an operad. We describe the deformations of 𝑆 in terms of products in 𝒪_def. As an example of this construction, we expose Kontsevich product deformation in this language. At last, we show that the extended symplectic category, although not being a true category, exhibits monoidal properties allowing us to carry the precedent construction up to a certain point. Then, we focus on the trivial symplectic groupoid over ℝ^𝑑 case and define the product operad associated to its multiplications space. We give a deformation operad on a local form, the local deformation operad. In particular, we show that any local deformation of the trivial product gives rise to a local symplectic groupoid over ℝ^𝑑. We conclude this section by defining equivalence between deformations of the trivial generating function and we show that two equivalent deformations induce the same local symplectic groupoid.
In Section 3, we describe the combinatorial tools needed to give a formal version of the local Lagrangian operad. As the problem consists mainly in taking Taylor's series of some implicit equations we need devices to keep track of all terms to all orders. The crucial point is that these implicit equations, describing the composition in the local Lagrangian operad, have a form extremely close to a special Runge-Kutta method: the partitioned implicit Euler method. We borrow then some techniques form numerical analysis of ODEs to make the expansion at all orders.
In the last section, we describe the formal Lagrangian operad, which is the perturbative version of the local one, in terms of composition of bipartite trees. We give in particular the product equation in the formal deformation operad in terms of these trees. At last, we restate the main theorem of [2] in this language. This tells us that the semi-classical part of Kontsevich star product on ℝ^𝑑 is a product in the formal deformation operad of the cotangent Lagrangian operad in 𝑑 dimensions.

Paper Genesis and Subsequent Works
This paper was inspired in large part by the unpublished note [3], in which the notion of lagrangian operads first appeared, and from the Ph.D. thesis [4]. It was originally conceived as a development of [2], providing a framework (the theory of operads), in which the results and computations of the latter article could be understood in a cleaner and more conceptual manner: each Taylor series expansion arising in [2] can be seen as a certain composition in the formal lagrangian operad over 𝑇^∗ℝ^𝑛.
The combinatorics of bicolored Runge-Kutta trees was borrowed from the numerical analysis of ODE (see [5]). We used it first in [2] to expand the structure equation (also called the “SGA equation”) for symplectic groupoid generating functions in formal power series. Actually, this combinatorics happens to control the compositions in the formal lagrangian operad over 𝑇^∗ℝ^𝑑. It is very reminiscent of the one used, in the context of bicolored operads, to define versions of operad morphisms “up to homotopy” (see [6] and also [7]). However, in the case of the formal lagrangian operad over 𝑇^∗ℝ^𝑑, we are not dealing with weak structures or weak maps of any kind, at least in a direct way. The actual nature of the relationship between these two formally similar but contextually different combinatorics, if any, is unknown to the authors' best knowledge.
As far as geometric quantization of Poisson manifolds using symplectic groupoid techniques is concerned, recent works seem to indicate that the language of symmetric monoidal categories is better suited than the one of operads, namely, the microsymplectic category developed in [8] is a better fit than the notion of lagrangian operads for understanding functorial aspects of geometric quantization. At any rate, the endomorphism operad of 𝑇^∗ℝ^𝑑 in the microsymplectic category contains, as a suboperad, the local lagrangian operad constructed in the present paper (see [8]).
However, there is no formal version of the microsymplectic category to date, and the combinatorics presented here to deal with the compositions in the formal lagrangian operad over 𝑇^∗ℝ^𝑑 have no equivalent in terms of a "formal microsymplectic category"; this is, at the time of writing, still a work in progress.

2. Product in the Extended Symplectic Category

2.1. Basic Constructions and Kontsevich Deformation

In this section, we describe, in any monoidal category, a natural generalization of an associative algebra structure over a vector space. It is the notion of product in the endomorphism operad 𝒪(𝑀) of an object 𝑀 in the category. If the category is further additive, we explain what is a deformation of a product 𝑆∈𝒪²(𝑀) and construct a non-linear operad, the deformation operad 𝒪_def(𝑀,𝑆) associated to 𝑆 in which any product is equivalent to a deformation of 𝑆. We present the well-known Kontsevich deformation of the usual product of functions over ℝ^𝑑 in this language. At last, we see that most parts of this construction, can be applied to the extended symplectic category, leading to the notion of Lagrangian operad.

Definition 2.1. An operad 𝒪 consists of (1)a collection of sets 𝒪^𝑛, 𝑛≥0, (2)composition laws 𝒪^𝑛×𝒪^𝑘1×⋯×𝒪^𝑘𝑛⟶𝒪^{𝑘1+…+𝑘𝑛}𝐹,𝐺₁,…,𝐺_𝑛⟼𝐹𝐺₁,…,𝐺_𝑛 (2.1) satisfying the following associativity relations: 𝐹𝐺₁,…,𝐺_𝑛𝐻₁₁,…,𝐻_1𝑘1,…,𝐻_𝑛1,…,𝐻_𝑛𝑘𝑛=𝐹𝐺₁𝐻₁₁,…,𝐻_1𝑘1,…,𝐺_𝑛𝐻_𝑛1,…,𝐻_𝑛𝑘𝑛, (2.2)(3)a unit element 𝐼∈𝒪¹ such that 𝐹(𝐼,…,𝐼)=𝐹forall𝐹∈𝒪^𝑛. It usually also requires some equivariant action of the symmetric group. We do not require this here.

The structure we have just defined should then be called more correctly “nonsymmetric operad”. However, we will simply keep using the term “operad” instead of “non symmetric operad” in the sequels.

Product in a Monoidal Category
We consider here a monoidal category 𝒞. We denote by ⊗∶𝒞×𝒞→𝒞 the product bifunctor and by 𝐞∈𝒞 the neutral object. Let us recall that we have the following canonical isomorphisms: (𝐴⊗𝐵)⊗𝐶≃𝐴⊗(𝐵⊗𝐶),𝐞⊗𝐴≃𝐴⊗𝐞≃𝐴 (2.3) for all 𝐴,𝐵,𝐶∈Obj𝒞.
Let 𝒞 be a monoidal category and let an object 𝑀∈Obj𝒞. We define the endomorphism operad of 𝑀 in the following way: (1)𝒪^𝑛(𝑀)∶=Hom(𝑀^⊗𝑛,𝑀),𝒪⁰(𝑀)∶=Hom(𝑒,𝑀), (2)𝐹(𝐺₁,…,𝐺_𝑛)∶=𝐹∘(𝐺₁⊗⋯⊗𝐺_𝑛), (3)the unit is given by id_𝑀∈𝒪¹(𝑀).
The operad axioms follow directly from the bifunctoriality of ⊗, that is, 𝑓⊗𝑔∘𝜓⊗𝜙=𝑓∘𝜓⊗𝑔∘𝜙id_𝑀⊗⋯⊗id_𝑀=id_{𝑀⊗⋯⊗𝑀}. (2.4)
If 𝑀 is an object of a monoidal category 𝒞, we may define a product on 𝑀.

Definition 2.2. An associative product (In [9], Gerstenhaber and Voronov call it a multiplication.) on an operad 𝒪 is an element 𝑆∈𝒪² such that 𝑆(𝐼,𝑆)=𝑆(𝑆,𝐼). An associative product on 𝑀 is an associative product in the endomorphism operad 𝒪(𝑀). In the sequel, we will constantly use the term product to mean in fact associative product.
Given a product 𝑆∈𝒪², the associativity of the operad implies that, for any 𝐹∈𝒪^𝑘, 𝐺∈𝒪^𝑙 and 𝐻∈𝒪^𝑚 we have, 𝑆((()()(𝑆()).𝐹,𝑆𝐺,𝐻))=𝑆𝐼,𝑆)(𝐹,𝐺,𝐻=𝑆𝑆,𝐼)(𝐹,𝐺,𝐻=𝑆𝐹,𝐺,𝐻 (2.5)
This notion is the natural generalization of an associative product on a vector space. Namely, if 𝑀 is a vector space, 𝒪²(𝑀) is the set of bilinear maps on 𝑀. As in this case 𝒪⁰(𝑀)=Hom(ℂ,𝑀)=𝑀, we have that 𝑆∶𝒪⁰(𝑀)×𝒪⁰(𝑀)→𝒪⁰(𝑀) is an associative product on 𝑀.

Product Deformation in a Monoidal Additive Category
Suppose we have a product 𝑆∈𝒪²(𝑀), where 𝑀 is an object of a monoidal category 𝒞. If the category 𝒞 is further additive, we may try to deform 𝑆, that is, to find an element 𝛾∈𝒪²(𝑀) such that 𝑆+𝛾 is still a product.
At this point, the standard way is to introduce the Hochschild complex of the linear operad 𝒪(𝑀), to define the bilinear Gerstenhaber bracket and the Hochschild differential associated with the product 𝑆. A deformation of 𝑆 would then be a solution of the Maurer-Cartan equation written in the Hochschild differential graded Lie algebra controlling the deformations of 𝑆.
We will however rephrase slightly this deformation theory in a way that will allow us to deal with categories whose hom-sets are still linear spaces but with a morphism composition that does not respect this linear structure, as it will be the case in the next sections.
The first step is to notice that a product 𝑆∈𝒪²(𝑀) generates a suboperad 𝒪_𝑆(𝑀), which we call a product operad, in 𝒪(𝑀) with only one point in each degree: 𝒪⁰_𝑆(𝑀)∶=∅,𝒪¹_𝑆(𝑀)∶={𝐼},𝒪²_𝑆(𝑀){𝑆},𝒪∶=_3𝑆(𝑀)∶={𝑆(𝑆,𝐼)},𝒪⁴_𝑆(𝑀)∶={𝑆(𝑆(𝑆,𝐼),𝐼)},…,etc. (2.6) To simplify the notation we will denote by 𝑆^𝑛₀ the unique element in 𝒪^𝑛_𝑆(𝑀).

Remark 2.3. The product operad 𝒪_𝑆(𝑀) is a suboperad of 𝒪(𝑀) but not a linear suboperad, namely, for each 𝑛∈ℕ, 𝒪^𝑛_𝑆(𝑀) is not a linear subspace of 𝒪^𝑛(𝑀) (it contains only a single point).

Definition 2.4. Let 𝑀 be an object of an additive monoidal category 𝒞 and let 𝑆∈𝒪²(𝑀) be a product. A deformation operad, 𝒪_def(𝑀,𝑆), for 𝑆 is the data, for each 𝑛∈ℕ, of a linear subspace 𝒪^𝑛_def(𝑀,𝑆)⊂𝒪^𝑛(𝑀) such that the difference 𝑅𝛾;𝛾¹,…,𝛾^𝑛∶=𝑆^𝑛₀+𝛾𝑆^𝑘1₀+𝛾¹,…,𝑆^𝑘𝑛₀+𝛾^𝑛−𝑆^{𝑘1+⋯+𝑘𝑛}₀ (2.7) is in 𝒪^{𝑘1+⋯+𝑘𝑛}_def(𝑀,𝑆) for all 𝛾∈𝒪^𝑛_def(𝑀,𝑆), 𝛾^𝑖∈𝒪^𝑘𝑖_def(𝑀,𝑆), and 𝑖=1,…,𝑛.

Remark 2.5. 𝒪_𝑆+𝒪_def is a suboperad of 𝒪(𝑀) but not a linear one: the spaces 𝒪^𝑛_𝑆+𝒪^𝑛_def(𝑀,𝑆) are not linear subspaces but affine ones.

Proposition 2.6. Let 𝒪_def(𝑀,𝑆) be a deformation operad for a product 𝑆∈𝒪²(𝑀). Then the compositions 𝛾𝛾¹,…,𝛾^𝑛∶=𝑅𝛾;𝛾¹,…,𝛾^𝑛, (2.8) defined by (2.7) gives 𝒪_def(𝑀,𝑆) together with the unit 0∈𝒪¹_def(𝑀,𝑆) the structure of an operad.

Proof. The proof is direct using only (2.7) and the operad structure of the endomorphism operad 𝒪(𝑀).

Remark 2.7. Although each of its degrees is a linear subspace, 𝒪_def(𝑀,𝑆) is not a linear operad since its compositions, the 𝑅s, are not multilinear.

Definition 2.8. We say that an element 𝛾∈𝒪²_def(𝑀,𝑆) is a deformation of the product 𝑆 w.r.t. the deformation operad 𝒪_def if 𝑆+𝛾 is still a product in 𝒪_𝑆+𝒪_def.

Remark 2.9. All what we have said still applies if we start with any linear operad instead of the endomorphism operad of an object in an additive monoidal category. This allows us to define a notion of product deformations in a specific class of deformations (which is given by the data of the deformation operad) in general linear operads.

Proposition 2.10. Let 𝑆∈𝒪²(𝑀) be a product. Take an element 𝛾∈𝒪²_def(𝑀,𝑆). Then, 𝛾 is a deformation of the product 𝑆 if and only if 𝛾 is a product in 𝒪_def(𝑀,𝑆). In particular, 0∈𝒪²_def(𝑀,𝑆) is always a product in the deformation operad of 𝑆.

Proof. 𝛾 is a deformation of 𝑆 if and only if 𝑆+𝛾𝑆+𝛾,𝐼=𝑆+𝛾𝐼,𝑆+𝛾, (2.9) which is equivalent to 𝑆³₀+𝑅𝛾;𝛾,0=𝑆³₀+𝑅𝛾;0,𝛾. (2.10)

From now on, we will write 0₁ for the identity element of the deformation operad which is the zero of 𝒪¹_def and 0₂ for the trivial product of the deformation operad which is the 0 element in 𝒪²_def(𝑀,𝑆).

Notice that neither 𝒪_𝑆(𝑀) nor 𝒪_𝑆(𝑀)+𝒪_def(𝑀,𝑆) is a linear operad in the sense that, although the compositions are multilinear, the spaces for each degree are not vector spaces but affine spaces. On the other hand, the spaces for each degrees of the deformation operad 𝒪_def(𝑀,𝑆) are vector spaces, but the induced operad compositions are not linear in general.

We may however introduce the Gerstenhaber bracket of the deformation operad [,]∶𝒪^𝑘_def(𝑀,𝑆)×𝒪^𝑙_def(𝑀,𝑆)⟶𝒪^{𝑘+𝑙−1}_def(𝑀,𝑆) (2.11) defined by[𝐹,𝐺]=𝐹∘𝐺−(−1)^{(𝑘−1)(𝑙−1)}𝐺∘𝐹, (2.12) where 𝐹∘𝐺=^𝑘_𝑖=1(−1)^{(𝑖−𝑙)(𝑙−1)}𝑅𝐹;0₁,…,0₁,𝐺_}𝑖th,0₁,…,0₁. (2.13) This bracket is not bilinear. An important fact concerning this bracket is that, 12𝛾,𝛾=𝑅𝛾;𝛾,0₁−𝑅𝛾;0₁,𝛾, (2.14) which means that 𝛾 is a product in the deformation operad if and only if12𝛾,𝛾=0. (2.15)

Moreover, we may define an equivalent of the Hochschild differential 𝑑∶𝒪^𝑛_def(𝑀,𝑆)⟶𝒪^𝑛+1_def(),0𝑀,𝑆𝑑𝐹∶=₂,𝐹=𝑅0₂;𝐹,0₁+(−1)^𝑛−1𝑅0₂;0₁−(),𝐹−1^𝑛𝑛−1_𝑖=1(−1)^𝑖−1𝑅𝐹;0₁,…,0₁,0_2𝑖th,0₁,…,0₁. (2.16)(2.17)

It turn out that 𝑑 is still a coboundary operator.

Proposition 2.11. 𝑑 defined by (2.17) is a coboundary operator, that is, 𝑑²=0. Moreover, 𝛾∈𝒪²_def(𝑀,𝑆) satisfies product equation (1/2)[𝛾,𝛾]=0, in 𝒪_def(𝑀,𝑆) if and only if 𝑑𝛾+𝛾𝛾,𝑆¹₀,−𝛾𝑆¹₀,𝛾=0. (2.18)

Proof. Using (2.7) we obtain 𝑑 in terms of the endomorphism compositions 𝑑𝐹=𝑆²₀𝐹,𝑆¹₀+(−1)^𝑛−1𝑆²₀𝑆¹₀−(),𝐹−1^𝑛𝑛−1_𝑖=1(−1)^𝑖−1𝐹𝑆¹₀,…,𝑆²_0𝑖th,…,𝑆¹₀. (2.19) The result follows directly from the linearity of the compositions in the endomorphism operad. Using again (2.7) we get 12𝛾,𝛾=𝑅𝛾;𝛾,0₁−𝑅𝛾;0₁,𝛾=𝑆²₀𝛾,𝑆¹₀+𝛾𝑆²₀,𝑆¹₀+𝛾𝛾,𝑆¹₀−𝑆²₀𝑆¹₀,𝛾−𝛾𝑆¹₀,𝑆²₀−𝛾𝑆¹₀,𝛾, (2.20) which gives (2.18).

A formal deformation 𝑆_𝜖 of 𝑆 is a formal power series 𝑆_𝜖=𝜖𝑆₁+𝜖²𝑆₂+⋯∈𝒪^𝑛_form(𝑀,𝑆)∶=𝜖𝒪^𝑛_def(𝑀,𝑆)⊗𝑘[[𝜖]],𝑛∈ℕ_∗, (2.21) where 𝜖 is a formal parameter and 𝒪_def(𝑀,𝑆) is a deformation operad for 𝑆, such that 𝑆+𝑆_𝜖 is a product in 𝒪_𝑆(𝑀)+𝒪_form(𝑀,𝑆).

Equivalently, one may say that 𝑆_𝜖 must satisfy 𝑆_𝜖,𝑆_𝜖=0, (2.22) or, thanks to (2.18) that the 𝑆_𝑖's satisfy at each order 𝑛∈ℕ_∗ the following recursive equation:𝑑𝑆_𝑛+𝐻_𝑛𝑆_𝑛−1,…,𝑆₁=0, (2.23) where 𝐻_𝑛𝑆_𝑛−1,…,𝑆₁=_{𝑛=𝑖+𝑗}𝑆_𝑖𝑆_𝑗,𝑆¹₀−𝑆_𝑖𝑆¹₀,𝑆_𝑖. (2.24)

The Kontsevich Product Deformation
Consider the category of real vector spaces. In this category we take the real vector space 𝑀=𝐶^∞(ℝ^𝑑) of smooth functions on ℝ^𝑑. The endomorphism operad of 𝐶^∞(ℝ^𝑑) is 𝒪^𝑛(𝑀)=𝑛-multilinearmapsfromC^∞ℝ^d⊗𝑛toC^∞ℝ^𝑑. (2.25) The usual product of functions induces a product in 𝒪(𝑀), namely, 𝑆²₀(𝐹,𝐺)𝑓₁,…,𝑓_𝑘,𝑔₁,…,𝑔_𝑙=𝐹𝑓₁,…,𝑓_𝑘𝐺𝑔₁,…,𝑔_𝑙, (2.26) for 𝐹∈𝒪^𝑘(𝑀) and 𝐺∈𝒪^𝑙(𝑀).
The induced product operad is 𝒪^𝑛_𝑆(𝑀)=𝑆^𝑛₀, (2.27) where 𝑆^𝑛₀𝑓₁,…,𝑓_𝑛=𝑓₁𝑓₂⋯𝑓_𝑛. (2.28) As deformation operad, we take 𝒪^𝑛_def(𝑀,𝑆)∶=𝑛-multidifferentialoperatorsonC^∞ℝ^𝑑. (2.29) The induced coboundary operator on 𝒪_def(𝑀,𝑆) is the Hochschild coboundary operator, 𝑑𝐹𝑓₁,…,𝑓_𝑛=𝐹𝑓₁,…,𝑓_𝑛𝑓_𝑛+1+(−1)^𝑛−1𝑓₁𝐹𝑓₂,…,𝑓_𝑛+1−(−1)^𝑛𝑛−1_𝑖=1(−1)^(𝑖−1)𝐹𝑓₁,…,𝑓_𝑖−1,𝑓_𝑖𝑓_𝑖+1,𝑓_𝑖+2,…,𝑓_𝑛+1. (2.30) and the product equation 𝑑𝛾+𝛾𝛾,𝑆¹₀,−𝛾𝑆¹₀,𝛾=0, (2.31) is nothing but the usual Maurer-Cartan equation.
Kontsevich in [1] shows that there exists a formal deformation 𝑆∈𝒪²_𝑆(𝑀)+𝜖𝒪²_def(𝑀)[[𝜖]] (2.32) of 𝑆²₀. He provides the explicit formula for this deformation 𝑆=𝑆²₀+^∞_𝑛=1𝜖^𝑛_{Γ∈𝐺𝑛,2}𝑊_Γ𝐵_Γ, (2.33) where the 𝐺_𝑛,2 are the Kontsevich graphs of type (𝑛,2), 𝑊_Γ is their associated weight, and 𝐵_Γ is their associated bidifferential operator (and [1] for more precisions).

2.2. Monoidal Structure of 𝒮𝒴ℳ

Let us recall that the extended symplectic “category” 𝒮𝒴ℳ is given by ,()=Obj=symplecticmanifoldsHom𝑀,𝑁𝐿⊂𝑀×𝑁∶𝐿isLagrangian, (2.34) where 𝑀 denotes the symplectic manifold 𝑀 with opposite symplectic structure −𝜔. The identity morphism of Hom(𝑀,𝑀) is the diagonal id_𝑀∶=Δ_𝑀=(𝑚,𝑚)⊂𝑀×𝑀. (2.35) The composition of two morphisms 𝐿∈Hom(𝑀,𝑁) and 𝐿∈Hom(𝑁,𝑃) is given by the composition of canonical relations 𝐿∘𝐿∶=𝜋_𝑀×𝑃𝐿×𝐿∩𝑀×Δ_𝑁×𝑃⊂𝑀×𝑃. (2.36) Everything works fine except the fact that the composition 𝐿∘𝐿 may fail to be a Lagrangian submanifold of 𝑀×𝑃. It is always the case when 𝐿×𝐿 intersects 𝑀×Δ_𝑁×𝑃 cleanly (see [10] for more precisions).

Let us pretend for a while that 𝒮𝒴ℳ is a true category or, better, that we have selected special symplectic manifolds and special arrows between them such that the composition is always well-defined.

We define the tensor product between two objects 𝑀 and 𝑁 of 𝒮𝒴ℳ as the Cartesian product 𝑀⊗𝑁∶=𝑀×𝑁, (2.37) and the tensor product between morphisms as 𝐿₁⊗𝐿₂∶=(𝑚,𝑎,𝑛,𝑏)∶(𝑚,𝑛)∈𝐿₁,(𝑎,𝑏)∈𝐿₂∈Hom(𝑀⊗𝐴,𝑁⊗𝐵), (2.38) for 𝐿₁∈Hom(𝑀,𝑁) and 𝐿₂∈Hom(𝐴,𝐵).

The neutral object is {∗}, the one-point symplectic manifold. The following proposition tells us that 𝒮𝒴ℳ would be a monoidal category if it were a true category.

Proposition 2.12. The following statements hold. (1)Consider 𝐿₁∈Hom(𝑀,𝐴), 𝐿₂∈Hom(𝑁,𝐵), 𝐿₃∈Hom(𝐴,𝑋) and 𝐿₄∈Hom(𝐵,𝑌). Then one has the following equality of sets: 𝐿₃⊗𝐿₄∘𝐿₁⊗𝐿₂=𝐿₃∘𝐿₁⊗𝐿₄∘𝐿₂. (2.39)(2)id_𝑀⊗id_𝑁=id_𝑀⊗𝑁 for any object 𝑀 and 𝑁.(3)(𝑀⊗𝐴)⊗𝑋=𝑀⊗(𝐴⊗𝑋) for any objects 𝑀, 𝐴 and 𝑋. (4)(𝐿₁⊗𝐿₂)⊗𝐿₃=𝐿₁⊗(𝐿₂⊗𝐿₃) for any arrows 𝐿₁∈Hom(𝑀,𝐴), 𝐿₂∈Hom(𝑁,𝐵) and 𝐿₃∈Hom(𝑃,𝐶).(5){∗}⊗𝐴≃𝐴≃𝐴⊗{∗} for all object 𝐴 and 𝑖𝑑_{∗}⊗𝐿≃𝐿≃𝐿⊗𝑖𝑑_{∗} for all arrows 𝐿, where 𝐴≃𝐵 means that the two sets 𝐴 and 𝐵 are in bijection.

Proof. (1) 𝐼=𝐿₃⊗𝐿₄∘𝐿₁⊗𝐿₂=𝜋𝐿₁⊗𝐿₂×𝐿₃⊗𝐿₄∩𝑁×𝑀×Δ_𝐴×𝐵=𝑦()()×𝑋×𝑌𝑚,𝑛,𝑥,∶∃𝑎,𝑏∈𝐴×𝐵s.t.𝑚,𝑛,𝑎,𝑏∈𝐿₁⊗𝐿₂,𝑎,𝑏,𝑥,𝑦∈𝐿₃⊗𝐿₄=𝑚,𝑛,𝑥,𝑦∶∃𝑎∈𝐴,(𝑚,𝑎)∈𝐿₁,(𝑎,𝑥)∈𝐿₃∃𝑏∈𝐵,(𝑛,𝑏)∈𝐿₂𝑏,𝑦∈𝐿₄=𝐿₃∘𝐿₁⊗𝐿₄∘𝐿₂ (2.40)

(2)Δ_𝑀⊗Δ_𝑁={(𝑚,𝑛,𝑚,𝑛)∶𝑚∈𝑀and𝑛∈𝑁}=Δ_𝑀⊗𝑁.
(3) The associativity between objects is trivial.
(4) For morphisms, we have, 𝐿₁⊗𝐿₂=(𝑚,𝑛,𝑎,𝑏)∶(𝑚,𝑎)∈𝐿₁,(𝑛,𝑏)∈𝐿₂,𝐿₁⊗𝐿₂⊗𝐿₃=𝑚,𝑛,𝑝,𝑎,𝑏,𝑐∶(𝑚,𝑎)∈𝐿₁,(𝑛,𝑏)∈𝐿₂,𝑝,𝑐∈𝐿₃,𝐿₂⊗𝐿₃=𝑛,𝑝,𝑏,𝑐∶(𝑛,𝑏)∈𝐿₂,𝑝,𝑐∈𝐿₃,𝐿₁⊗𝐿₂⊗𝐿₃=𝑚,𝑛,𝑝,𝑎,𝑏,𝑐∶(𝑚,𝑎)∈𝐿₁,(𝑛,𝑏)∈𝐿₂,𝑝,𝑐∈𝐿₃. (2.41)
(5) is trivial.

2.3. Lagrangian Operads

If 𝒮𝒴ℳ were a true category, we could consider the endomorphism operad of a symplectic manifold 𝑀. However, we may be able to restrict to a subset of Lagrangian submanifolds 𝒪^𝑛_rest(𝑀)⊂𝒪^𝑛(𝑀) for each 𝑛≥0 such that the composition 𝐿_𝑛𝐿_𝑘1,…,𝐿_𝑘𝑛∶=𝐿_𝑛∘𝐿_𝑘1⊗⋯⊗𝐿_𝑘𝑛, (2.42) yields always a Lagrangian submanifold in 𝒪^{𝑘1+⋯+𝑘𝑛}_rest(𝑀) for every 𝐿_𝑛∈𝒪^𝑛_rest(𝑀) and 𝐿_𝑘𝑖∈𝒪^𝑘𝑖_rest(𝑀), 𝑖=1,…,𝑛. For instance, there is always the trivial choice 𝒪¹_rest(𝑀)=Δ_𝑀,𝒪^𝑛_rest(𝑀)=∅,𝑛≠1. (2.43)

In this way, we may get a true operad 𝒪_rest(𝑀).

The next natural question to ask is the following.

Question 2.3. What is a product in a Lagrangian operad over 𝑀?

As a first hint, take the situation where the symplectic manifold is a symplectic groupoid 𝐺. In this case, we may generate an operad from the multiplication space 𝐺^𝑚∈𝒪²(𝐺) and the base 𝐺⁽⁰⁾∈𝒪⁰(𝐺), the identity being the diagonal Δ_𝐺∈𝒪¹(𝐺). Remark that 𝐺^𝑚 is a product in this operad, that is, that 𝐺^𝑚(𝐺^𝑚,Δ_𝐺)=𝐺^𝑚(Δ_𝐺,𝐺^𝑚). Notice that the inverse of the symplectic groupoid does not play any role in this construction.

We will answer this question completely for the case were the symplectic manifold is 𝑇^∗ℝ^𝑑 and will try to develop a deformation theory for the product in this case.

Local Cotangent Lagrangian Operads
Remember that 𝑇^∗ℝ^𝑑 has always a structure of a symplectic groupoid over ℝ^𝑑: the trivial one. The multiplication space is given in this case by Δ₂=𝑝₁,𝑥,𝑝₂,𝑥,𝑝₁+𝑝₂,𝑥∶𝑝₁,𝑝₂∈ℝ^𝑑∗,𝑥∈ℝ^𝑑. (2.44) The base is Δ₀=(0,𝑥)∶𝑥∈ℝ^𝑑. (2.45) If we set further Δ_𝑛∶=𝑝₁,𝑥,…,𝑝_𝑛,𝑥,𝑝₁+…+𝑝_𝑛,𝑥∶𝑝_𝑖∈ℝ^𝑑∗,𝑥∈ℝ^𝑑, (2.46) it immediate to see that the operad generated by Δ₀ and Δ₂ is exactly 𝒪^𝑛_Δ𝑇^∗ℝ^𝑑=Δ_𝑛, (2.47) and that Δ₂ is a product in it.

Following [3], we will call this operad the cotangent Lagrangian operad over 𝑇^∗ℝ^𝑑. It is the exact analog of the product operad in a monoidal category, the only difference is that there is no true endomorphism operad to embed 𝒪_Δ(𝑇^∗ℝ^𝑑) into. The idea now is to enlarge the cotangent Lagrangian operad, that is, by considering Lagrangian submanifolds close enough to Δ_𝑛 for each 𝑛∈ℕ in order to have still an operad.

Notice at this point that the Δ_𝑛's are given by generating functions. Namely, we may identify (𝑇^∗ℝ^𝑑)^𝑛×𝑇^∗ℝ^𝑑 with 𝑇^∗𝐵_𝑛, where 𝐵_𝑛∶=(ℝ^𝑑∗)^𝑛×ℝ^𝑑. Then, Δ_𝑛=𝑝₁,𝜕𝑆^𝑛₀𝜕𝑝₁(𝑧),…,𝑝_𝑛,𝜕𝑆^𝑛₀𝜕𝑝_𝑛(𝑧),𝜕𝑆^𝑛₀(𝑧)𝑝𝜕𝑥,𝑥∶𝑧=₁,…,𝑝_𝑛,𝑥∈𝐵_𝑛, (2.48) where 𝑆^𝑛₀ is the function on 𝐵_𝑛 defined by (In the sequels, we will use the shorter notation (𝑝₁+⋯+𝑝_𝑛)𝑥 instead of ∑^𝑑_𝑖=1(𝑝^𝑖₁+⋯+𝑝^𝑖_𝑛)𝑥_𝑖.)𝑆^𝑛₀𝑝₁,…,𝑝_𝑛,𝑥=^𝑑_𝑖=1𝑝^𝑖₁+⋯+𝑝^𝑖_𝑛𝑥_𝑖. (2.49) The cotangent Lagrangian operad may then be identified with 𝒪^𝑛_Δ=𝑆^𝑛₀,𝒪⁰_Δ={0}. (2.50) In order to define a deformation operad for 𝑆, a natural idea would be to consider Lagrangian submanifolds whose generating functions are of the form 𝐹=𝑆^𝑛₀+𝐹, (2.51) where 𝐹∈𝐶^∞(𝐵_𝑛). The Lagrangian submanifold associated to 𝐹 is 𝐿_𝐹∶=graph𝑑𝐹. (2.52) As such, the idea does not work in general. In fact, we have to consider generating functions only defined in some neighborhood. Let us be more precise.

We introduce the following notation: 𝐵⁰_𝑛={0}×ℝ^𝑑⊂𝐵_𝑛, (2.53)𝑉(𝐵⁰_𝑛) will stand for the set of all neighborhoods of 𝐵⁰_𝑛 in 𝐵_𝑛.

Definition 2.13. We define 𝒪^𝑛_loc(𝑇^∗ℝ^𝑑) to be the space of germs at 𝐵⁰_𝑛 of smooth functions 𝐹 (defined on an open neighborhood 𝑈_𝐹⊂𝐵_𝑛 of 𝐵⁰_𝑛) which satisfy 𝐹(0,𝑥)=0 and ∇_𝑝𝐹(0,𝑥)=0. Note that the composition will always be understood in terms of composition of germs.

Proposition 2.14. Let 𝐹∈𝒪^𝑛_Δ+𝒪^𝑛_loc and 𝐺_𝑖∈𝒪^𝑘𝑖_Δ+𝒪^𝑘𝑖_loc for 𝑖=1,…,𝑛. Consider the function 𝜙 defined by the formula 𝜙𝑝_𝐺,𝑥_𝐹=𝐺₁∪⋯∪𝐺_𝑛𝑝_𝐺,𝑥_𝐺+𝐹𝑝_𝐹,𝑥_𝐹−𝑥_𝐺𝑝_𝐹,𝑝_𝐹=∇_𝑥𝐺₁∪…∪𝐺_𝑛𝑝_𝐺,𝑥_𝐺,𝑥_𝐺=∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹, (2.54)(2.55) where 𝐺₁∪⋯∪𝐺_𝑛𝑝_𝐺,𝑥_𝐺∶=𝐺₁𝑝_𝐺1,𝑥_𝐺1+⋯+𝐺_𝑛𝑝_𝐺𝑛,𝑥_𝐺𝑛 (2.56) and 𝑝_𝐺=(𝑝_𝐺1,…,𝑝_𝐺𝑛),𝑝_𝐺𝑖∈(ℝ^𝑑∗)^𝑘𝑖, 𝑥_𝐺𝑖∈ℝ^𝑑 and (𝑝_𝐺𝑖,𝑥_𝐺𝑖)∈𝑈_𝐺𝑖, for 𝑖=1,…,𝑛.
Then, 𝜙∈𝒪^{𝑘1+⋯+𝑘𝑛}_Δ+𝒪^{𝑘1+⋯+𝑘𝑛}_loc,and𝐿_𝜙=𝐿_𝐹𝐿_𝐺1,…,𝐿_𝐺𝑛. (2.57) In other words, 𝒪_Δ+𝒪_loc together with the product 𝜙=𝐹𝐺₁,…,𝐺_𝑛 (2.58) is an operad.
Moreover, the induced operad structure on 𝒪_loc is given by 𝑅𝐹;𝐺₁,…,𝐺_𝑛=𝐻, (2.59) where 𝐻 is the function 𝐻∈𝒪^{𝑘1+⋯+𝑘𝑛}_loc defined by 𝐻𝑝_𝐺,𝑥_𝐹=𝐺𝑝_𝐺,𝑥_𝐺+𝐹𝑝_𝐹,𝑥_𝐹−∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺,𝑝_𝐹=𝑝⁰_𝐹+∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺,𝑝⁰_𝐹∶=𝑝^Σ_𝐺1,…,𝑝^Σ_𝐺𝑛,𝑥_𝐺=𝑥⁰_𝐺+∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹,𝑥⁰_𝐺∶=𝑥_𝐹,…,𝑥_𝐹. (2.60)

Remark 2.15 (Saddle point formula). Formula (2.54) for Φ can be interpreted in terms of saddle point evaluation for ℏ→0 of the following integral: e^{(𝑖/ℏ)[𝐹(𝑝1,…,𝑝𝑘,𝑥)+∑𝑘𝑖=1(𝐺𝑖(𝜋𝑖1,…,𝜋𝑖𝑖𝑙,𝑦𝑖)−𝑝𝑖⋅𝑦𝑖𝑘)]}_𝑖=1d^𝑛𝑝^𝑖d^𝑛𝑦_𝑖(2𝜋ℏ)^𝑛=e^{(i/ℏ)Φ(𝜋11,…,𝜋11𝑙,𝜋21,…,𝜋22𝑙…𝜋𝑘1,…,𝜋𝑘𝑘𝑙,𝑥)}(𝐶+𝑂(ℏ)), (2.61) where 𝐶 is some constant.

Proof of Proposition 2.14. To simplify the computations, we identify (𝑇^∗ℝ^𝑑)^𝑛 with 𝑇^∗(ℝ^𝑑𝑛) and (𝑇^∗ℝ^𝑑)^𝑘𝑖 with 𝑇^∗(ℝ^𝑑𝑘𝑖). With this identifications the graphs of 𝐹 and 𝐺_𝑖, 𝑖=1,…,𝑛 may be written as 𝐿_𝐹=𝑝_𝐹,∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹,∇_𝑥𝐹𝑝_𝐹,𝑥_𝐹,x_𝐹∶𝑝_𝐹,𝑥_𝐹∈𝑈_𝐹⊂𝑇^∗ℝ^𝑑𝑛×𝑇^∗ℝ^𝑑,𝐿_𝐺𝑖=𝑝_𝐺𝑖,∇_𝑝𝐺_𝑖𝑝_𝐺𝑖,𝑥_𝐺𝑖,∇_𝑥𝐺_𝑖𝑝_𝐺𝑖,𝑥_𝐺𝑖,𝑥_𝐺𝑖∶𝑝_𝐺𝑖,𝑥_𝐺𝑖∈𝑈_𝐺𝑖⊂𝑇^∗ℝ^𝑑𝑘𝑖×𝑇^∗ℝ^𝑑, (2.62) where 𝑈_𝐹∈𝑉(𝐵⁰_𝑛) and 𝑈_𝐺𝑖∈𝑉(𝐵⁰_𝑘𝑖) for 𝑖=1,…,𝑛.
Consider now the composition, 𝐿_𝐹𝐿_𝐺1,…,𝐿_𝐺𝑛=𝐿_𝐹∘𝐿_𝐺1⊗⋯⊗𝐿_𝐺𝑛. (2.63) First of all, observe that, 𝐿_𝐺∶=𝐿_𝐺1⊗⋯⊗𝐿_𝐺𝑛=𝑝_𝐺,∇_𝑝𝐺𝑝_𝐺,𝑥_𝐺,∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺,𝑥_𝐺∶𝑝_𝐺𝑖,𝑥_𝐺𝑖∈𝑈_𝐺𝑖𝐿_𝐺⊂𝑇^∗ℝ^{𝑑(𝑘1+⋯+𝑘𝑛)}×𝑇^∗ℝ^𝑑𝑛. (2.64) Thus, 𝐿_𝐹∘𝐿_𝐺=𝜋𝐿_𝐺×𝐿_𝐹∩𝑇^∗ℝ^{𝑑(𝑘1+⋯+𝑘𝑛)}×Δ_𝑇^∗_ℝ^𝑑𝑛×𝑇^∗ℝ^𝑑=𝑝_𝐺,∇_𝑝𝐺𝑝_𝐺,𝑥_𝐺,∇_𝑥𝐹𝑝_𝐹,𝑥_𝐹,𝑥_𝐹∶𝑥_𝐺=∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹,𝑝_𝐹=∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺,𝑝_𝐺,𝑥_𝐹∈𝑈𝐿_𝐹∘𝐿_𝐺⊂𝑇^∗ℝ^{𝑑(𝑘1+⋯+𝑘𝑛)}×𝑇^∗ℝ^𝑑, (2.65) where 𝑈 is the subset of (𝑝_𝐺,𝑥_𝐹)∈𝐵_{𝑘1+⋯+𝑘𝑛} such that the system, 𝑝_𝐹=∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺,𝑥_𝐺=∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹, (2.66) has a unique solution (𝑝_𝐹,𝑥_𝐺) and such that (𝑝_𝐺𝑖,𝑥_𝐺𝑖)∈𝑈_𝐺𝑖, 𝑖=1,…,𝑛, and (𝑝_𝐹,𝑥_𝐹)∈𝑈_𝐹. Let us check that 𝑈 always exists and is a neighborhood of 𝐵⁰_{𝑘1+⋯+𝑘𝑛}. To begin with, observe that for any (0,𝑥_𝐹)∈𝐵⁰_𝑛 this system has the unique solution (0,∇_𝑝𝐹(0,𝑥_𝐹)). Set now, 𝐻𝑝_𝐺,𝑥_𝐹,𝑝_𝐹,𝑥_𝐺=𝑝𝐹−∇_𝑥𝐺𝑝𝐺,𝑥𝐺𝑥𝐹−∇_𝑝𝐹𝑝𝐹,𝑥𝐹. (2.67) Thanks to the fact that 𝐺(0,𝑥)=∑^𝑛_𝑖=1𝐺_𝑖(0,𝑥)=0 we get that the Jacobi matrix 𝐷_{𝑝𝐹,𝑥𝐺}𝐻0,𝑥_𝑓,0,∇_𝑝𝐹0,𝑥_𝐹=id0−∇_𝑝∇_𝑝𝐹0,𝑥_𝐹id (2.68) is invertible.
Thus, the implicit function theorem gives us the desired neighborhood 𝑈 of 𝐵⁰_{𝑘1+⋯+𝑘𝑛}.
Now, take 𝜙 as defined in (2.54). The previous considerations tell us that 𝜙 is exactly defined on 𝑈. Let us compute its graphs, 𝐿_𝜙=𝑝_𝐺,∇_𝑝Φ𝑝_𝐺,𝑥_𝐹,∇_𝑥Φ𝑝_𝐺,𝑥_𝐹,𝑥_𝐹∶𝑝_𝐺,𝑥_𝐹∈𝑈. (2.69) We have that ∇_𝑝𝜙𝑝_𝐺,𝑥_𝐹=∇_𝑝𝐺𝑝_𝐺,𝑥_𝐺+∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺𝑑𝑥_𝐺𝑑𝑝+∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹𝑑𝑝_𝐹𝑑𝑝−𝑝_𝐹𝑑𝑥_𝐺−𝑑𝑝𝑑𝑝_𝐹𝑥𝑑𝑝^G=∇_𝑝𝐺𝑝_𝐺,𝑥_𝐺. (2.70) Similarly, ∇_𝑥𝜙(𝑝_𝐺,𝑥_𝐹)=∇_𝑥𝐹(𝑝_𝐹,𝑥_𝐹). Thus, 𝐿_𝜙=𝐿_𝐹∘𝐿_𝐺.
At last, let us check that 𝜙∈𝒪^{𝑘1+⋯+𝑘𝑛}_loc. First of all, remember that 𝐹𝑝_𝐹,𝑥_𝐹=𝑝^Σ_𝐹𝑥_𝐹+𝐹𝑝_𝐹,𝑥_𝐹,𝐺𝑝_𝐺,𝑥_𝐹=^𝑛_𝑖=1𝑝^Σ_𝐺𝑖𝑥_𝐺𝑖+𝐺𝑝_G,𝑥_𝐺. (2.71) Thus, we obtain immediately that 𝜙𝑝_𝐺,𝑥_𝐹=𝑝^Σ_𝐺𝑥_𝐹+𝐻𝑝_𝐺,𝑥_𝐹, (2.72) where 𝐻 is a function only defined on 𝑈 by the equations 𝐻𝑝_𝐺,𝑥_𝐹=𝐺𝑝_𝐺,𝑥_𝐺+𝐹𝑝_𝐹,𝑥_𝐹−∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺,𝑝_𝐹=𝑝⁰_𝐹+∇_𝑥𝐺𝑝_𝐺,𝑥_𝐺,𝑝⁰_𝐹∶=𝑝^Σ_𝐺1,…,𝑝^Σ_𝐺𝑛,𝑥_𝐺=𝑥⁰_𝐺+∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹,𝑥⁰_𝐺∶=𝑥_𝐹,…,𝑥_𝐹. (2.73) But now, if we set 𝑝_𝐺=0 then 𝑝_𝐹=0, 𝑥_𝐺=𝑥⁰_𝐺+∇_𝑝𝐹(0,𝑥_𝐹) and 𝐻(0,𝑥_𝐹)=0. Similarly, one easily checks that ∇_𝑝𝐻(0,𝑥_𝐹)=0

We will call the operad 𝒪_Δ+𝒪_loc  local cotangent Lagrangian operad over T^∗ℝ^𝑑 or for short the local Lagrangian operad when no ambiguities arise. The induced operad 𝒪_loc will be called the local deformation operad of   𝒪_Δ.

Associative Products in the Local Deformation Operad
We say that a generating function 𝑆∈𝐶^∞(𝐵₂) satisfies the Symplectic Groupoid Associativity equation if for a point (𝑝₁,𝑝₂,𝑝₃,𝑥)∈𝐵₃ sufficiently close to 𝐵⁰₃ the following implicit system for 𝑥,𝑝,𝑥 and 𝑝: 𝑥=∇_𝑝1𝑆𝑝,𝑝₃,𝑥,𝑝=∇_𝑥𝑆𝑝₁,𝑝₂,𝑥,𝑥=∇_𝑝2𝑆𝑝₁,𝑝,𝑥,𝑝=∇_𝑥𝑆𝑝₂,𝑝₃,𝑥 (2.74) has a unique solution and if the following additional equation holds: 𝑆𝑝₁,𝑝₂,𝑥+𝑆𝑝,𝑝₃,𝑥−𝑥𝑝=𝑆𝑝₂,𝑝₃,𝑥+𝑆𝑝₁,𝑝,𝑥−𝑥𝑝. (2.75)
If 𝑆 also satisfies the Symplectic Groupoid Structure conditions, that is, if 𝑆𝑝,0,𝑥=𝑆0,𝑝,𝑥=𝑝𝑥,𝑆𝑝,−𝑝,𝑥=0 (2.76) then 𝑆 generates a Poisson structure 𝛼(𝑥)=2∇_𝑝¹_𝑘∇_𝑝²_𝑙𝑆(0,0,𝑥)^𝑑_𝑘,𝑙=1 (2.77) on ℝ^𝑑 together with a local symplectic groupoid integrating it, whose structure maps are given by 𝜖(𝑥)=()𝑖=𝑠0,𝑥unitmap,𝑝,𝑥−𝑝,𝑥inversemap,𝑝,𝑥=∇_𝑝2𝑆𝑡𝑝,0,𝑥sourcemap,𝑝,𝑥=∇_𝑝1𝑆0,𝑝,𝑥targetmap. (2.78)
In this case, we call 𝑆 a generating function of the Poisson structure α or a generating function of the local symplectic groupoid. See [2, 4, 10] for proofs and explanations about generating functions of Poisson structures.

The following proposition explains what is a product in the local cotangent Lagrangian operad.

Proposition 2.16. 𝑆∈𝒪²_loc is a product in 𝒪_loc if and only if 𝑆=𝑆²₀+𝑆 satisfies the Symplectic Groupoid Associativity equation.

Proof. We know that 𝑆 is a product in 𝒪_loc if and only if 𝑆=𝑆²₀+𝑆 is a product in 𝒪_Δ+𝒪_loc, that is, if and only if 𝑆(𝑆,𝐼)=𝑆(𝐼,𝑆). Let us compute. 𝑆(𝑆,𝐼)𝑝₁,𝑝₂,𝑝₃,𝑥=𝑆∪𝐼𝑝₁,𝑝₂,𝑝₃,𝑥₁,𝑥₂+𝑆𝑝₁,𝑝₂,𝑥−𝑥₁𝑝₁−𝑝₂𝑥₂=𝑆𝑝₁,𝑝₂,𝑥₁+𝑝₃𝑥₂+𝑆𝑝₁,𝑝₂,𝑥−𝑝₁𝑥₁−𝑝₂𝑥₂ (2.79) with 𝑝₁=∇_𝑥1𝑆∪𝐼𝑝₁,𝑝₂,𝑝₃,𝑥₁,𝑥₂=∇_𝑥𝐺(𝑥),𝑝₂=∇_𝑥2𝑆∪𝐼𝑝₁,𝑝₂,𝑝₃,𝑥₁,𝑥₂=𝑝₃,𝑥₁=∇_𝑝1𝑆𝑝₁,𝑝₂,𝑥,𝑥₂=∇_𝑝2𝑆𝑝₁,𝑝₂,𝑥. (2.80) Then we get 𝑆(𝑆,𝐼)=𝑆𝑝₁,𝑝₂,𝑥+𝑆𝑝,𝑝₃,𝑥−𝑝𝑥,𝑥=∇_𝑝1𝑆𝑝,𝑝₃,𝑥,𝑝=∇_𝑥𝑆𝑝₁,𝑝₂,𝑥. (2.81) Similarly, we get 𝑆(𝐼,𝑆)=𝑆𝑝₂,𝑝₃,𝑥+𝑆𝑝₁,𝑝,𝑥−𝑝𝑥,𝑥=∇_𝑝2𝑆𝑝₁,𝑝,𝑥𝑝=∇_𝑥𝑆𝑝₂,𝑝₃,𝑥. (2.82) Hence, 𝑆∈𝒪²_loc(𝑇^∗ℝ^𝑑) is a product if and only if 𝑆²₀+𝑆 satisfies the SGA equation.

At this point, we may still introduce the Gerstenhaber bracket as in (2.12) and the product equation in terms of the bracket would still be (1/2)[S,𝑆]=0. We may also still write a formula for the coboundary operator. But, as this time the compositions in 𝒪_Δ+𝒪_loc are not multilinear, we cannot develop the expression (1/2)[𝑆,𝑆] in terms of the coboundary operator. Nevertheless, in Section 4, we will develop the bracket with help of Taylor's expansion and recover a form very close to (2.23) in the additive category case.

Equivalence of Associative Products
To each 𝐹∈𝒪¹_Δ+𝒪¹_loc, we may associate a symplectomorphism 𝜓_𝐹 which is defined only on a neighborhood 𝑈_𝐹 of 𝐵⁰₁ in 𝑇^∗ℝ^𝑑 and which fixes 𝐵¹₀. The composition of two such 𝜓_𝐺 and 𝜓_𝐹, which may always be defined on a possibly smaller neighborhood 𝑈⊂𝑈_𝐺 of 𝐵⁰₁, is exactly 𝜓_𝐹(𝐺) where 𝐹(𝐺) is the composition of 𝐹 by 𝐺 in the local Lagrangian operad.
We denote by 𝐹⁻¹∈𝒪¹_Δ+𝒪¹_loc the generating function of the (𝜓_𝐹)⁻¹, that is, the generating function such that 𝐹(𝐹⁻¹)=𝐹⁻¹(𝐹)=𝐼. Two associative products 𝑆 and 𝑆 will be called equivalent if 𝑆=𝐹(𝑆)𝐹⁻¹,𝐹⁻¹ (2.83) for a certain 𝐹∈𝒪¹_Δ+𝒪¹_loc. It is clear that if 𝑆∈𝒪¹_Δ+𝒪¹_loc is an associative product, then 𝑆 also is. The following questions naturally arises.

Questions
If 𝑆 generates a local symplectic groupoid, does 𝑆 also generate one? Are these two local groupoids isomorphic?

In fact, two equivalent associative products, which are also generating functions of local symplectic groupoids, induce isomorphic local symplectic groupoids. The isomorphism is given explicitly by 𝜓_𝐹. As a consequence the induced Poisson structures on the base are the same, that is, 𝛼(𝑥)=∇_𝑝1∇_𝑝2𝑆(0,0,𝑥)=∇_𝑝1∇_𝑝2𝑆(0,0,𝑥). (2.84)

The following two Propositions prove these statements.

Proposition 2.17. Let 𝐹∈𝒪¹_Δ+𝒪¹_loc. The following implicit equations: 𝑥₁=∇_𝑝𝐹𝑝₁,𝑥₂,𝑝₂=∇_𝑥𝐹𝑝₁,𝑥₂, (2.85) define a symplectomorphism 𝜓_𝐹(𝑝₁,𝑥₁)=(𝑝₂,𝑥₂) on a neighborhood 𝑈_𝐹 of 𝐵⁰₁={(0,𝑥)∶𝑥∈ℝ^𝑑} in 𝑇^∗ℝ^𝑑 which fixes 𝐵⁰₁ and which is close to the identity in the sense that 𝐹(𝑝,𝑥)=𝑝𝑥+𝐹(𝑝,x) induces the identity if 𝐹=0. Consider now 𝜓_𝐹 and 𝜓_𝐺 defined, respectively, on 𝑈_𝐹 and 𝑈_𝐺 for 𝐹,𝐺∈𝒪¹_Δ+𝒪¹_loc. Then one has that 𝜓_𝐺∘𝜓_𝐹=𝜓_𝐹(𝐺) on 𝑈_𝐹(𝐺).

Proof. (1) Let us check that the system (2.85) generates a diffeomorphism around 𝐵⁰₁. Namely, one verifies that (𝑝₁,𝑥₁,𝑝₂,𝑥₂)∶=(0,∇_𝑝𝐹(0,𝑥₂),0,𝑥₂) is a solution of the system. Set now 𝐻𝑝₁,𝑥₁,𝑝₂,𝑥₂𝑥∶=¹−∇_𝑝𝐹𝑝₁,𝑥₂𝑝₂−∇_𝑥𝐹𝑝₁,𝑥₂. (2.86) As 𝐷_𝑝1,𝑥1𝐻𝑝₁,𝑥₂,𝑝₂,𝑥₂=−∇_𝑝∇_𝑝𝐹0,𝑥₂∇id_𝑥∇_𝑝𝐹0,𝑥₂0,𝐷_𝑝2,𝑥2𝐻𝑝₁,𝑥₂,𝑝₂,𝑥₂=0∇_𝑥∇_𝑝𝐹0,𝑥₂,id0 (2.87) the implicit function theorem gives us the result. Let us call 𝑈 the neighborhood of 𝐵⁰₁ where 𝜓_𝐹 is defined.
(2) We check now that 𝜓_𝐹 is symplectic. From (2.85) we get the relation 𝜕𝑝²_𝑙𝜕𝑝¹_𝑘=𝜕𝑥¹_𝑘𝜕𝑥²_𝑙, (2.88) which directly implies that 𝑑𝜓_𝐹𝐽(𝑑𝜓_𝐹)^∗=𝐽 where 𝐽=0id−id0. (2.89)
(3) Let us see that 𝜓_𝐹(0,𝑥)=(0,𝑥). We have already noticed that (0,∇_𝑝𝐹(0,𝑥₂),0,𝑥₂) is a solution of the system (2.85). But 𝐹(𝑝,𝑥)=𝑝𝑥+𝐹(𝑝,𝑥) with ∇_𝑝𝐹(0,𝑝)=0 and then ∇_𝑥∇_𝑝𝐹(0,𝑥₂)=𝑥₂.
(4) Clearly 𝐹(𝑝,𝑥)=𝑝𝑥 generates the identity.
(5) Recall that𝐿_𝐺=𝑝₁,∇_𝑝𝐺𝑝₁,𝑥₂,∇_𝑥𝐺𝑝₁,𝑥₂,𝑥₂∶𝑝₁,𝑥₂∈𝑈_𝐺,𝐿_𝐹=𝑝₂,∇_𝑝𝐹𝑝₂,𝑥₃,∇_𝑥𝐹𝑝₂,𝑥₃,𝑥₃∶𝑝₂,𝑥₃∈𝑈_𝐹. (2.90) Thus, 𝐿_𝐺=graph𝜓_𝐺 and 𝐿_𝐹=graph𝜓_𝐹. The composition of these two canonical relations yields that 𝐿_𝐹∘𝐿_𝐺=graph𝜓_𝐹∘𝜓_𝐺. On the other hand, 𝐿_𝐹∘𝐿_𝐺=𝐿_𝐹(𝐺)=graph𝜓_𝐹(𝐺). Taking care on the domain of definitions, we have that 𝜓_𝐹∘𝜓_𝐺=𝜓_𝐹(𝐺) on 𝑈_𝐹(𝐺).

Proposition 2.18. Let 𝑆∈𝒪²_Δ+𝒪²_loc be a generating function of a symplectic groupoid, that is, 𝑆(𝑆,𝐼)=𝑆(𝐼,𝑆),𝑆𝑝,0,𝑥=𝑆0,𝑝,𝑥=𝑝𝑥,𝑆𝑝,−𝑝,𝑥=0. (2.91) Let 𝐹∈𝒪¹_Δ+𝒪¹_loc such that 𝐹(−𝑝,𝑥)=−𝐹(𝑝,𝑥). Then, 𝑆∶=(𝐹(𝑆))𝐹⁻¹,𝐹⁻¹ (2.92) is also a generating function of a symplectic groupoid. The subset of odd function in 𝑝 forms a subgroup of 𝒪¹_Δ+𝒪¹_loc. Moreover, 𝜓_𝐹 is a groupoid isomorphism between the local symplectic groupoid generated by 𝑆 and the one generated by 𝑆. As a consequence 𝑆 and 𝑆 induce the same Poisson structure on the base.

Proof. To simplify the notation, we set 𝐺=𝐹⁻¹. A straightforward computation gives that 𝐹(𝑆)(𝐺,𝐺)𝑝₁,𝑝₂,𝑥=𝑆𝑝,𝑝,𝑥̇+𝐹𝑝,̇𝑥+𝐺𝑝₁,𝑥+𝐺𝑝₂,𝑥−𝑝𝑝𝑥̇̇̇𝑥−𝑥−𝑝,𝑥=∇_𝑝𝐹𝑝,̇𝑥,𝑥=∇_𝑝1𝑆𝑝,𝑝,𝑥̇,𝑥=∇_𝑝2𝑆𝑥̇,̇𝑝,𝑥,𝑝=∇_𝑥𝑆𝑝,𝑝,𝑥̇,𝑝=∇_𝑥𝐺𝑝₁,𝑥,𝑝=∇_𝑥𝐺𝑝₂,𝑥. (2.93)(1)Setting 𝑝₁=𝑝 and 𝑝₂=0, we have immediately 𝐹(𝑆)(𝐺,𝐺)𝑝,0,𝑥=𝐺𝑝,𝑥̇+𝐹𝑝,̇𝑥−𝑥̇𝑝̇ (2.94) with 𝑥̇=∇_𝑝𝐹(𝑝,̇𝑥) and 𝑝̇=∇_𝑥𝐺(𝑝,𝑥̇). We recognize then that 𝐹(𝑆)(𝐺,𝐺)𝑝,0,𝑥=𝐹(𝐺)𝑝,𝑥=𝐼𝑝,𝑥=𝑝𝑥. (2.95) The case 𝑝₁=0 and 𝑝₂=𝑝 is analog.(2)One reads directly from the equation 𝑝𝑥=𝐹⁻¹𝑝,𝑥̇+𝐹𝑝,̇𝑥−𝑥̇𝑝,̇ (2.96) where 𝑥̇=∇_𝑝𝐹(𝑝,̇𝑥) and 𝑝̇=∇_𝑥𝐹⁻¹(𝑝,𝑥̇), that if 𝐹 is odd in 𝑝 then is also 𝐹⁻¹ and reciprocally. Similarly, we check directly from the composition formula that 𝐹(𝐺) is odd in 𝑝 if 𝐹 and 𝐺 both are. Thus, the odd functions form a subgroup of 𝒪¹_Δ+𝒪¹_loc.(3)Suppose now that 𝑝₁=𝑝 and 𝑝₂=−𝑝. 𝐺 odd in 𝑝 implies that 𝑝=−𝑝. As 𝑆(𝑝,−𝑝,0)=0, we get immediately that 𝑥=𝑥 and 𝑝̇=0 which in turns implies that 𝑥̇=𝑥. Putting everything together, we get that (𝐹(𝑆))(𝐺,𝐺)(𝑝,−𝑝,𝑥)=0(4)Let us prove now that 𝜓_𝐹 is also a groupoid isomorphism. Consider the multiplication space of the symplectic groupoid generated by an generating function 𝑆, that is, 𝐺^(𝑚)(𝑆)=𝑝₁,∇_𝑝1𝑆,𝑝₂,∇_𝑝2𝑆,∇_𝑥𝑆,𝑥∶𝑝₁,𝑝₂∈ℝ^𝑑∗,𝑥∈ℝ^𝑑, (2.97) where the partial derivative are evaluated in (𝑝₁,𝑝₂,𝑥).
We have to show that (𝜓_𝐹×𝜓_𝐹×𝜓_𝐹)(𝐺^(𝑚)(𝑆))=𝐺^(𝑚)(𝑆).
A straightforward computation gives that ∇_𝑝1𝑆𝑝₁,𝑝₂,𝑥=∇_𝑝𝐺𝑝₁,𝑥,∇_𝑝2𝑆𝑝₁,𝑝₂,𝑥=∇_𝑝𝐺𝑝₂,𝑥,∇_𝑥𝑆𝑝₁,𝑝₂,𝑥=∇_𝑥𝐹𝑝,̇𝑥. (2.98)
From this, we check immediately that 𝜓_𝐺𝑝₁,∇_𝑝1𝑆𝑝₁,𝑝₂,𝑥=𝑝,∇_𝑝1𝑆𝑥̇,𝜓𝑝,𝑝,_𝐺𝑝₂,∇_𝑝2𝑆𝑝₁,𝑝₂,𝑥=𝑝,∇_𝑝2𝑆𝑥̇,𝜓𝑝,𝑝,_𝐹∇_𝑥𝑆𝑝,𝑝,𝑥̇,𝑥̇=∇_𝑥𝑆𝑝₁,𝑝₂,𝑥,𝑥 (2.99) which ends the proof.

Remark 2.19. Suppose that 𝑆 is a generating function of a local symplectic groupoid. Let 𝐹∈𝒪¹_Δ+𝒪¹_loc act on 𝑆, that is, 𝑆=(𝐹(𝑆))(𝐹⁻¹,𝐹⁻¹). Then, the condition 𝑆(𝑝,0,𝑥)=𝑆(0,𝑝,𝑥)=𝑝𝑥 is preserved by any 𝐹∈𝒪¹_Δ+𝒪¹_loc. However, the condition 𝑆(𝑝,−𝑝,0) is only preserved by the odd 𝐹𝑠. Observe now that we have imposed the inverse map to be 𝑖(𝑝,𝑥)=(−𝑝,𝑥). This implies that −𝑝₂,∇_𝑝2𝑆𝑝₂,𝑝₁,𝑥,−𝑝₁,∇_𝑝1𝑆𝑝₂,𝑝₁,𝑥,−∇_𝑥𝑆𝑝₂,𝑝₁,𝑥,𝑥∈𝐺^(𝑚)(𝑆), (2.100) and thus, that 𝑆(𝑝₁,𝑝₂,𝑥)=−𝑆(−𝑝₁,−𝑝₂,𝑥). From this last equation, we get that 𝑆 must satisfy 𝑆(𝑝,−𝑝,𝑥)=0 and that the induced local symplectic groupoid is a symmetric one, that is, 𝑡(𝑝,𝑥)=𝑠(−𝑝,𝑥). Thus, odd transformations map symmetric groupoids to symmetric groupoids. However, they are not the only ones.

3. The Combinatorics

In this section, we present some tools which will allow us to write down at all orders the perturbative version of the composition, (2.54), in the local cotangent operad. All these compositions have essentially the same form. We will first give an abstract version of the equations describing the compositions, then we will introduce some trees which will help us to keep track of the terms involved in the computations and, at last, we will perform the expansion in the general case.

The tools and methods presented here are essentially the same as those used in the Runge–Kutta theory of ODEs to determine the order conditions of a particular numeric method. We follow approximatively the notations of [5].

3.1. The Equation

Let 𝐹∶ℝ^𝑛∗→ℝ and 𝐺∶ℝ^𝑛→ℝ be two smooth functions. Consider the point 𝜙∈ℝ defined by𝜙∶=𝐺(𝑥)+𝐹𝑝−𝑝𝑥, (3.1) where 𝑥 and 𝑝 are defined by the implicit equations𝑝=∇_𝑥𝐺(𝑥),𝑥=∇_𝑝𝐹𝑝. (3.2) Without any assumptions on 𝐹 and 𝐺, (3.2) may not have a solution at all or the solution may be not unique. Hence, the value 𝜙 is not always defined. However, if we assume that 𝐹 and 𝐺 are formal power series of the form 𝐺(𝑥)=𝑝₀𝑥+^∞_𝑖=1𝜖^𝑖𝐺^(𝑖)(𝑥),𝐹𝑝=𝑥₀𝑝+^∞_𝑖=1𝜖^𝑖𝐹^(𝑖)𝑝, (3.3) equation (3.2) become 𝑝=𝑝₀+^𝑛_𝑖=1𝜖^𝑖∇_𝑥𝐺^(𝑖)(𝑥),𝑥=𝑥₀+^𝑛_𝑖=1𝜖^𝑖∇_𝑝𝐹^(𝑖)𝑝, (3.4) which are always recursively uniquely solvable.

Let us compute the first terms of 𝑝, 𝑥 and 𝜙 to get a feeling of what is happening: 𝑝=𝑝₀+𝜖∇_𝑥𝐺⁽¹⁾𝑥₀+𝜖²∇_𝑥⁽²⁾𝐺⁽¹⁾𝑥₀∇_𝑝𝐹⁽¹⁾𝑝₀+⋯,𝑥=𝑥₀+𝜖∇_𝑝𝐹⁽¹⁾𝑥₀+𝜖²∇_𝑝⁽²⁾𝐹⁽¹⁾𝑥₀∇_𝑥𝐺⁽¹⁾𝑥₀+⋯,𝜙=𝑝₀𝑥₀+𝜖𝐺⁽¹⁾𝑥₀+𝐹⁽¹⁾𝑝₀+𝜖²2∇_𝑝𝐹⁽¹⁾𝑝₀∇_𝑥𝐺⁽¹⁾𝑥₀+⋯. (3.5)

As we continue the expansion, the terms get more and more involved and, very soon, expressions as such become intractable. One common strategy in physics as in numeric analysis is to introduce some graphs to keep track of the fast growing terms. Let us present these graphs. We mainly take our inspiration from the book [5].

3.2. The Trees

Definition 3.1. We have the following (1)A graph 𝑡 is given by a set of vertices 𝑉_𝑡={1,…,𝑛) and a set of edges 𝐸_𝑡 which is a set of pairs of elements of 𝑉_𝑡. We denote the number of vertices by |𝑡|. An isomorphism between two graphs 𝑡 and 𝑡′ having the same number of vertices is a permutation 𝜎∈𝑆_|𝑡| such that {𝜎(𝑣),𝜎(𝑤)}∈𝐸_𝑡′ if {𝑣,𝑤}∈𝐸_𝑡. Two graphs are called equivalent if there is an isomorphism between them. The symmetries of a graph are the automorphisms of the graph. We denote the group of symmetries of a graph 𝑡 by 𝑠𝑦𝑚(𝑡).(2)A tree is a graph which has no cycles. Isomorphisms and symmetries are defined the same way as for graphs (3)A rooted tree is a tree with one distinguished vertex called root. An isomorphism of rooted trees is an isomorphism of graphs which sends the root to the root. Symmetries and equivalence are defined correspondingly.(4)A bipartite graph is a graph 𝑡 together with a map 𝜔∶𝑉_𝑡→{∘,•} such that 𝜔(𝑣)≠𝜔(𝑤) if {𝑣,𝑤}∈𝐸_𝑡. An isomorphism of bipartite trees is an isomorphism of graphs which respects the coloring, that is, 𝜔(𝜎(𝑣))=𝜔(𝑣). (5)A weighted graph is a graph 𝑡 together with a weight map 𝐿∶𝑉_𝑡→ℕ⧵{0}. An isomorphism of weighted graph is an isomorphism of graph 𝜎 which respects the weights, that is, 𝜎(𝐿(𝑣))=𝐿(𝜎(𝑣)). We denote by ‖𝑡‖ the sum of the weights on all vertices of 𝑡.

Table 1 summarizes some notations we will use in the sequel.

Table 1 

We will give the name Cayley trees to trees in 𝑇.

We denote by [𝐴] the set of equivalence classes of graphs in 𝐴 (ex: [𝑅𝑇]). They are called topological “𝐴” trees. Moreover, we denote by 𝐴_∞ the weighted version of graphs in 𝐴. Notice that we will use the notation [𝐴]_∞ instead of the more correct [𝐴_∞].

The elements of [𝑅𝑇]_∞ can be described recursively as follows: (1)∘_𝑖,•_𝑗∈[𝑅𝑇]_∞ where 𝑖=𝐿(∘_𝑖) and 𝑗=𝐿(•_𝑗);(2)if 𝑡₁,…,𝑡_𝑚∈[𝑅𝑇_∘]_∞, then the tree [𝑡₁,…,𝑡_𝑚]_•𝑖∈[𝑅𝑇]_∞ where [𝑡₁,…,𝑡_𝑚]_•𝑖 is defined by connecting the roots of 𝑡₁,…,𝑡_𝑚 with the weighted vertex •_𝑖 and declaring that •_𝑖 is the new root. And the same if we interchange ∘ and •.

Now, let us describe in terms of trees the expressions arising in the expansions of Section 3.1.

Definition 3.2. Given two collections of functions 𝐹={𝐹^(𝑖)}^∞_𝑖=1 and 𝐺={𝐺^(𝑗)}^∞_𝑗=1, where 𝐹_𝑖∶ℝ^𝑛∗→ℝ and 𝐺_𝑗∶ℝ^𝑛→ℝ are smooth functions, we may associate to any rooted tree 𝑡∈[𝑅𝑇]_∞ a vector field on 𝑇^∗ℝ^𝑑, 𝐷𝐶_𝑡(𝐹,𝐺)∈Vect(𝑇^∗ℝ^𝑑), called the elementary differential and a function on 𝑇^∗ℝ^𝑑, 𝐶_𝑡(𝐹,𝐺)∈𝐶^∞(𝑇^∗ℝ^𝑑), called the elementary function. (1)The elementary differential 𝐷𝐶_𝑡(𝐹,𝐺) is recursively defined as follows: (a)𝐷𝐶_∘𝑖(𝐹,𝐺)(𝑝,𝑥)=∇_𝑥𝐺^(𝑖)(𝑥), 𝐷𝐶_•𝑗(𝐹,𝐺)(𝑝,𝑥)=∇_𝑝𝐹^(𝑗)(𝑝); (b)𝐷𝐶_𝑡(𝐹,𝐺)=∇_𝑥^(𝑚+1)𝐺^(𝑖)(𝐷𝐶_𝑡1(𝐹,𝐺),…,𝐷𝐶_𝑡𝑚(𝐹,𝐺)) if 𝑡=[𝑡₁,…,𝑡_𝑚]_∘𝑖; (c)𝐷𝐶_𝑡(𝐹,𝐺)=∇_𝑝^(𝑚+1)𝐹^(𝑗)(𝐷𝐶_𝑡1(𝐹,𝐺),…,𝐷𝐶_𝑡𝑚(𝐹,𝐺)) if 𝑡=[𝑡₁,…,𝑡_𝑚]_•𝑗. (2)The elementary function 𝐶_𝑡(𝐹,𝐺), are recursively defined as follows: (a)𝐶_∘𝑖(𝐹,𝐺)(𝑝,𝑥)=𝐺^(𝑖)(𝑥), 𝐶_•𝑗(𝐹,𝐺)(𝑝,𝑥)=𝐹^(𝑗)(𝑝); (b)𝐶_𝑡(𝐹,𝐺)=∇_𝑥^(𝑚)𝐺^(𝑖)(𝐷𝐶_𝑡1(𝐹,𝐺),…,𝐷𝐶_𝑡𝑚(𝐹,𝐺)) if 𝑡=[𝑡₁,…,𝑡_𝑚]_∘𝑖; (c)𝐶_𝑡(𝐹,𝐺)=∇_𝑝^(𝑚)𝐹^(𝑗)(𝐷𝐶_𝑡1(𝐹,𝐺),…,𝐷𝐶_𝑡𝑚(𝐹,𝐺)) if 𝑡=[𝑡₁,…,𝑡_𝑚]_•𝑗. The notation ∇_𝑥^(𝑚) (resp. ∇_𝑝^(𝑚)) stands for the 𝑚th derivative in the direction 𝑥 (resp. 𝑝).

Some examples are given in Table 2.

Table 2 

Remark that for elementary functions it is not important which vertex is the root. This is not the case for elementary differentials.

Definition 3.3. Let 𝑢=[𝑢₁,…,𝑢_𝑘],𝑣=[𝑣₁,…,𝑣_𝑙]∈[𝑅𝑇] (resp. ∈[𝑅𝑇]_∞). Following [5], we define the Butcher product as follows: 𝑢∘𝑣∶=𝑢₁,…,𝑢_𝑘,𝑣₁,…,𝑣_𝑙. (3.6) We have not written the obvious conditions on the 𝑢_𝑖's and the 𝑣_𝑖's so that the product remains bipartite (resp., weighted bipartite).

Definition 3.4 (equivalence relation on (weighted) rooted topological trees). Recall that an equivalence relation on a set 𝐴 is a special subset 𝑅 of 𝐴×𝐴. The equivalence relations on 𝐴 are moreover ordered by inclusion. It makes then sense to consider the minimal equivalence on 𝐴 containing a certain subset 𝑈⊂𝐴.
We consider here the minimal equivalence relation on [𝑅𝑇] (resp. on [𝑅𝑇]_∞)) such that 𝑢∘𝑣∼𝑣∘𝑢.

Properties of this Relation
The following is clear that.(1)Two topological rooted trees are equivalent if it is possible to pass from one to the other by changing the root. More precisely: 𝑡,𝑡^′∈[𝑅𝑇]_(∞), 𝑡∼𝑡^′ if and only if there exists a representative (𝐸,𝑉,𝑟) of 𝑡 and a representative (𝐸^′,𝑉^′,𝑟^′) of 𝑡^′ and a vertex 𝑟^′′∈𝑉 such that (𝐸,𝑉,𝑟^′′) and (𝐸^′,𝑉^′,𝑟^′) are isomorphic (weighted) rooted trees. (2)The quotient of [𝑅𝑇]_(∞) by this equivalence relation is exactly [𝑇]_(∞). (3)It follows immediately from the definition that 𝐶_𝑡(𝐹,𝐺)=𝐶_𝑡′(𝐹,𝐺) if 𝑡∼𝑡^′ for 𝑖=1,2.
Then, it makes sense to define the elementary functions on bipartite trees.
At last, we introduce some important functions on trees: the symmetry coefficients.

Definition 3.5. Let 𝑡=[𝑡₁,…,𝑡_𝑚]∈[𝑅𝑇]_∞. Consider the list 𝑡₁̃,…,𝑡_𝑘̃ of all nonisomorphic trees appearing in 𝑡₁,…,𝑡_𝑚. Define 𝜇_𝑖 as the number of time the tree 𝑡_𝑖̃ appears in 𝑡₁,…,𝑡_𝑚. Then we introduce the symmetry coefficient 𝜎(𝑡) of 𝑡 by the following recursive definition: 𝜎(𝑡)=𝜇₁!𝜇₂!…𝜎𝑡₁̃…𝜎𝑡_𝑘̃ (3.7) and initial condition 𝜎(∘_𝑖)=𝜎(•_𝑗)=1.
It is clear that 𝜎(𝑡) is the number of symmetries for each representative of 𝑡 (i.e. 𝜎(𝑡)=|𝑆𝑦𝑚(𝑡′)| for all 𝑡′∈𝑡).

3.3. The Expansion

We give now a power series expansion for (3.1).

Proposition 3.6. Suppose that we are given the following formal power series in 𝜖, 𝐺(𝑥)=𝑝₀𝑥+^∞_𝑖=1𝜖^𝑖𝐺^(𝑖)(𝑥),𝐹𝑝=𝑥₀𝑝+^∞_𝑗=1𝜖^𝑗𝐹^(𝑗)𝑝, (3.8) where 𝐺^(𝑖)∶ℝ^𝑛→ℝ^𝑛∗ and 𝐹^(𝑗)∶ℝ^𝑛∗→ℝ^𝑛 are smooth functions for 𝑖,𝑗>0. Define 𝜙(𝑝₀,𝑥₀)∈ℝ[[𝜖]] as 𝜙𝑝₀,𝑥₀∶=𝐺(𝑥)+𝐹𝑝−𝑝𝑥, (3.9) where the formal power series 𝑥(𝜖) and 𝑝(𝜖) are uniquely determined by the implicit equations 𝑝=𝑝₀+^∞_𝑖=1𝜖^𝑖∇_𝑥𝐺^(𝑖)(𝑥),𝑥=𝑥₀+^∞_𝑗=1𝜖^𝑗∇_𝑝𝐹^(𝑗)𝑝. (3.10) Then, one has that 𝜙𝑝₀,𝑥₀=𝑝₀𝑥₀+_{𝑡∈𝑇∞}𝜖^‖𝑡‖|𝑡|!𝐶_𝑡(𝐹,𝐺)𝑝₀,𝑥₀. (3.11)

The proof of Proposition 3.6 is broken into several lemmas.

The method used is essentially the same as in numerical analysis when one wants to express the Taylor series of the numerical flow of a Runge–Kutta method. Namely, the defining equations for 𝑝(𝜖) and 𝑥(𝜖) have a form very close to the partitioned implicit Euler method (see [5]).

Lemma 3.7. There exist unique formal power series for 𝑥(𝜖) and for 𝑝(𝜖) which satisfy (3.2). They are given by 𝑥(𝜖)=𝑥₀+_{𝑡∈𝑅𝑇•∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺),𝑝(𝜖)=𝑝₀+_{𝑡∈𝑅𝑇∘∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺). (3.12)(3.13)

Proof. Uniqueness is trivial. Let us check that we have the right formal series. We only check (3.12). The other computation is similar. 𝑥(𝜖)=𝑥₀+_𝑖≥1𝜖^𝑖∇_𝑝𝐹^(𝑖)𝑝=𝑥₀+_𝑖≥1𝜖^𝑖_𝑚≥01∇𝑚!^𝑝(𝑚+1)𝐹^(𝑖)_{𝑡∈𝑅𝑇∘∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺),…,_{𝑡∈[𝑅𝑇∘]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺)=𝑥₀+_{𝑖≥1𝑚≥0𝑡1∈𝑅𝑇∘∞}⋯_{𝑡𝑚∈𝑅𝑇∘∞}𝜖^{𝑖+‖𝑡1‖+⋯+‖𝑡𝑚‖}𝑚!𝜎𝑡₁…𝜎𝑡_𝑚×∇_𝑝^(𝑚+1)𝐹^(𝑖)𝐷𝐶_𝑡1(𝐹,𝐺),⋯,𝐷𝐶_𝑡𝑚(𝐹,𝐺)=𝑥₀+_{𝑖≥1𝑚≥0𝑡1}…_𝑡𝑚𝜖^‖𝑡‖(𝑡)𝜇𝑚!𝜎¹!𝜇₂!…𝐷𝐶_𝑡(𝐹,𝐺),with𝑡=𝑡₁,…,𝑡_𝑚•𝑖=𝑥₀+_{𝑡∈𝑅𝑇•∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺). (3.14)

Lemma 3.8. One has the following expansion for 𝜙(𝑝₀,𝑥₀): 𝜙𝑝₀,𝑥₀=𝑝₀𝑥₀+_{[]𝑡∈𝑅𝑇∞}𝜖^‖𝑡‖𝜎(𝑡)𝐶_𝑡(𝐹,𝐺)−_{𝑡∈[𝑅𝑇∘]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺)_{𝑡∈[𝑅𝑇•]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺). (3.15)

Proof. We compute the different terms arising in 𝐺(𝑥)+𝐹(𝑝)−𝑝𝑥 in terms of trees. 𝐺(𝑥)=𝑝₀𝑥+_𝑖≥1𝜖^𝑖_𝑚≥01∇𝑚!^𝑥(𝑚)𝐺^(𝑖)_{𝑡∈[𝑅𝑇•]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺),…,_{𝑡∈[𝑅𝑇•]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺)=𝑝₀𝑥+_{𝑖≥1𝑚≥0𝑡1∈[𝑅𝑇•]∞}…_{𝑡𝑚∈[𝑅𝑇•]∞}𝜖^‖𝑡‖(𝑡)𝜇𝑚!𝜎¹!𝜇₂!…×∇_𝑥^(𝑚)𝐺^(𝑖)𝐷𝐶_𝑡1(𝐹,𝐺),…,𝐷𝐶_𝑡𝑚(𝐹,𝐺),with𝑡=𝑡₁,…,𝑡_𝑚•𝑖=𝑝₀𝑥+_{𝑡∈[𝑅𝑇∘]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐶_𝑡(𝐹,𝐺). (3.16)
By the same sort of computations we obtain 𝐹𝑝=𝑥₀𝑝+_{𝑡∈𝑅𝑇•∞}𝜖^‖𝑡‖𝜎(𝑡)𝐶_𝑡(𝐹,𝐺). (3.17) Finally, we get the desired result as 𝑝₀𝑥+𝑥₀𝑝−𝑝𝑥=𝑝₀𝑥₀−_{𝑡∈[𝑅𝑇∘]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺)_{𝑡∈[𝑅𝑇•]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐷𝐶_𝑡(𝐹,𝐺). (3.18)

Thus, 𝜙(𝑝₀,𝑥₀) is expressed as sums over topological weighted rooted bipartite trees. We would like now to regroup the terms of the formula in the previous lemma. To do so, we express all terms in terms of topological trees (no longer rooted).

Lemma 3.9. Let 𝑢∈[𝑅𝑇_∘]_∞ and 𝑣∈[𝑅𝑇_•]_∞. Then, 𝐷𝐶_𝑢(𝐹,𝐺)𝐷𝐶_𝑣(𝐹,𝐺)=𝐶_𝑢∘𝑣(𝐹,𝐺)=𝐶_𝑣∘𝑢(𝐹,𝐺). (3.19)

Proof. Suppose 𝑢=[𝑢₁,…,𝑢_𝑚]_∘𝑖, 𝑣=[𝑣₁,…,𝑣_𝑙]_•𝑗, then we get 𝐴=𝐷𝐶_𝑢(𝐹,𝐺)𝐷𝐶_𝑣(𝐹,𝐺)=∇_𝑥^(𝑚+1)𝐺^(𝑖)𝐷𝐶_𝑢1(𝐹,𝐺),…,𝐷𝐶_𝑢𝑚(𝐹,𝐺)⋅𝐷𝐶_𝑣(𝐹,𝐺)=∇_𝑥^(𝑚+1)𝐺^(𝑖)𝐷𝐶_𝑢1(𝐹,𝐺),…,𝐷𝐶_𝑢𝑚(𝐹,𝐺),𝐷𝐶_𝑣(𝐹,𝐺)=𝐶_𝑢∘𝑣(𝐹,𝐺). (3.20)

Lemma 3.10. Let 𝑡=(𝑉_𝑡,𝐸_𝑡)∈𝑇_∞. For all 𝑣∈𝑉_𝑡 let 𝑡_𝑣 be the bipartite rooted tree (𝑉_𝑡,𝐸_𝑡,𝑣)∈𝑅𝑇_∞. For 𝑣∈𝑉_𝑡 and 𝑒={𝑢,𝑣}∈𝐸_𝑡 one has 𝑠𝑦𝑚(𝑡)𝑠𝑦𝑚𝑡_𝑣=𝑣^′∈𝑉_𝑡𝑡_𝑣′isisomorphicto𝑡_𝑣,𝑠𝑦𝑚(𝑡)𝑠𝑦𝑚𝑡_𝑢𝑠𝑦𝑚𝑡_v=𝑒^′∈𝐸_𝑡𝑡_𝑢′⊔𝑡_𝑣′isisomorphicto𝑡_𝑢⊔𝑡_𝑣. (3.21)

Proof. Consider the induced action of the symmetry group of the tree on the set of vertices. Notice that two vertices 𝑣 and 𝑤 are in the same orbit if and only if 𝑡_𝑣 is isomorphic to 𝑡_𝑤. Then the number of vertices of 𝑡 which lead to rooted tree isomorphic to 𝑡_𝑣 is exactly the cardinality of the orbit of 𝑣, which is exactly |𝑠𝑦𝑚(𝑡)| divided by the cardinality of the isotropy subgroup which fixes 𝑣. But the latter is |𝑠𝑦𝑚(𝑡_𝑣)| by definition. We then get the first statement.
For the second statement we have to consider the induced action on the edges and apply the same type of argument.

Lemma 3.11. We get 𝜙𝑝₀,𝑥₀=p₀𝑥₀+_{𝑡∈𝑇∞}𝜖^‖𝑡‖|𝑡|!𝐶_𝑡(𝐹,𝐺). (3.22)

Proof. Let us perform the last computation. 𝜙𝑝₀,𝑥₀=𝑝₀𝑥₀+_{𝑡∈[𝑅𝑇]∞}𝜖^‖𝑡‖𝜎(𝑡)𝐶_𝑡(𝐹,𝐺)−_{𝑢∈[𝑅𝑇∘]∞𝑣∈[𝑅𝑇•]∞}𝜖^{‖𝑢‖+‖𝑣‖}𝜎(𝑢)𝜎(𝑣)𝐷𝐶_𝑢(𝐹,𝐺)𝐷𝐶_𝑣(𝐹,𝐺)=𝑝₀𝑥₀+_{[𝑇]𝑡∈∞}𝜖^|𝑡|𝐶_𝑡(𝐹,𝐺)_𝑡∈𝑡1𝑠𝑦𝑚(𝑡)−_{𝑢∈𝑅𝑇•∞,𝑣∈𝑅𝑇∘∞𝑢∘𝑣∈𝑡}1𝑠𝑦𝑚(𝑢)∣∣𝑠𝑦𝑚(𝑣)=𝑝₀𝑥₀+_{𝑡∈𝑇∞}𝜖^‖𝑡‖|𝑡|!𝐶_𝑡(𝐹,𝐺)_{𝑣∈𝑉𝑡}𝑠𝑦𝑚(𝑡)𝑠𝑦𝑚𝑡_𝑣1𝑘(),𝑡,𝑣−_{{)𝑒=𝑢,𝑣∈𝐸𝑡}𝑠𝑦𝑚(𝑡)𝑠𝑦𝑚𝑡_𝑢∣∣𝑠𝑦𝑚𝑡_𝑣1𝑙(𝑡,𝑒) (3.23) where 𝑘(𝑡,𝑣)=|{𝑣^′∈𝑉_𝑡/𝑡_𝑣′isisomorphicto𝑡_𝑣}| and 𝑙(𝑡,𝑒)=|{𝑒^′∈𝐸_𝑡/𝑡_𝑢′⊔𝑡_𝑣′isisomorphicto𝑡_𝑢⊔𝑡_𝑣)|. Using Lemma 3.10 and the fact that for a tree the difference between the number of vertices and the number of edges is equal to 1 we get the desired result.

Using now the fact that 𝑆 is a formal power series we immediately get Proposition 3.6.

4. Deformation of a Nonlinear Structure

4.1. The Formal Cotangent Lagrangian Operad

The formal cotangent Lagrangian operad on 𝑇^∗ℝ^𝑑 is the perturbative/formal version of the local cotangent operad on 𝑇^∗ℝ^𝑑. Recall that in the latter the product for 𝐹∈𝒪^𝑛_Δ+𝒪^𝑛_loc and 𝐺_𝑖∈𝒪^𝑛_Δ+𝒪^𝑘𝑖_loc, 𝑖=1,…𝑛 was expressed as in Proposition 2.14: 𝐹𝐺₁,…,𝐺_𝑛𝑝_𝐺,𝑥_𝐹=𝐺₁∪⋯∪𝐺_𝑛𝑝_𝐺,𝑥_𝐺+𝐹𝑝_𝐹,𝑥_𝐹−𝑝_𝐹⋅𝑥_𝐺,𝑝_𝐹=∇_𝑥𝐺₁∪⋯∪𝐺_𝑛𝑝_𝐺,𝑥_𝐺,𝑥_𝐹=∇_𝑝𝐹𝑝_𝐹,𝑥_𝐹. (4.1) If we consider 𝑝_𝐺 and 𝑥_𝐹 as parameters in the previous equations, we have then that 𝐺𝑝_𝐺,⋅∶ℝ^𝑛𝑑⟶ℝ,𝐹⋅,𝑥_𝐹∶ℝ^𝑛𝑑∗⟶ℝ. (4.2) Suppose now that the 𝐹 and 𝐺_𝑖, 𝑖=1,…,𝑛, are formal series of the form 𝐹𝑝_𝐹,𝑥_𝐹=𝑝^Σ_𝐹⋅𝑥_𝐹+^∞_𝑖=1𝜖^𝑖𝐹^(𝑖)𝑝_𝐹,𝑥_𝐹,𝐺_𝑙𝑝_𝐺𝑙,𝑥_𝐺𝑙=𝑝^Σ_𝐺𝑙⋅𝑥_𝐺𝑙+^∞_𝑖=1𝜖^𝑖𝐺_𝑙^(𝑖)𝑝_𝐺𝑙,𝑥_𝐺𝑙, (4.3) where 𝑝^Σ∶=^𝑛_𝑖=1𝑝_𝑖for𝑝=𝑝₁,…,𝑝_𝑛∈ℝ^𝑑𝑛∗. (4.4) We may rewrite 𝐹 and 𝐺 as 𝐹𝑝_𝐹,𝑥_𝐹=𝑥^𝐹₀𝑝_𝐹+^∞_𝑖=1𝜖^𝑖𝐹^(𝑖)𝑝_𝐹,𝑥_𝐹,𝐺𝑝_𝐺,𝑥_𝐺=𝑝^𝐺₀𝑥_𝐺+^∞_𝑖=1𝜖^𝑖𝐺^(𝑖)𝑝_𝐺,𝑥_𝐺, (4.5) where 𝑥^𝐹₀=(𝑥_𝐹,…,𝑥_𝐹)∈ℝ^𝑑𝑛 and 𝑝^𝐺₀=(𝑝^Σ_𝐺1,…,𝑝^Σ_𝐺𝑛)∈(ℝ^𝑛𝑑)^∗ for 𝑥_𝐺∈ℝ^𝑑𝑛 and 𝑝_𝐹∈(ℝ^𝑑𝑛)^∗.

Applying now Proposition 3.6, we obtain for the compositions the following expansion:𝐹𝐺₁,…,𝐺_𝑛𝑝_𝐺,𝑥_𝐺=𝑝^Σ_𝐺⋅𝑥_𝐹+_{𝑡∈𝑇∞}𝜖^‖𝑡‖|𝑡|!𝐶_𝑡𝐹⋅,𝑥_𝐹,𝐺₁∪…∪𝐺_𝑛𝑝_𝐺,⋅𝑝^𝐺₀,𝑥^𝐹₀. (4.6)

This motivates to define the formal deformation space of the cotangent Lagrangian operad 𝒪_Δ(𝑇^∗ℝ^𝑑) as 𝒪^𝑛_form𝑇^∗ℝ^𝑑,Δ∶=^∞_𝑖=1𝜖^𝑖𝐹^(𝑖)∶𝐹^(𝑖)∈𝑃^𝑛_𝑖𝑇^∗ℝ^𝑑, (4.7)

where 𝑃^𝑛_𝑖(𝑇^∗ℝ^𝑑) stands for the vector space of functions 𝐹∶𝐵_𝑛→ℝ such that (1)𝐹(𝑝,𝑥) is a polynomial in the variables 𝑝=(𝑝₁,…,𝑝_𝑛), (2)𝐹(𝜇𝑝,𝑥)=𝜇^𝑖+1𝐹(𝑝,𝑥).

One may think of 𝒪_loc+𝒪_form as the Taylor series of functions in 𝒪_Δ+𝒪_loc. The compositions are given by formula (4.6), which also tells us that 𝒪_Δ+𝒪_loc is an operad. The unit is 𝐼𝑝,𝑥=𝑝𝑥,𝐼∈𝒪_Δ+𝒪¹_form. (4.8)

The induced operad structure on 𝒪_form is then given by𝐼∈𝒪¹_form𝒪,𝐼𝑝,𝑥=0,_𝑛form=^∞_𝑖=1𝜖^𝑖𝐹^(𝑖)∶𝐹^(𝑖)∈𝑃^𝑛_𝑖𝑇^∗ℝ^𝑑,𝐹𝐺₁,…,𝐺_𝑛𝑝_𝐺,𝑥_𝐹=_{𝑡∈𝑇∞}𝜖^‖𝑡‖|𝑡|!𝐶_𝑡𝐹,𝐺₁∪…∪𝐺_𝑛. (4.9) This operad will be called the formal deformation operad of the cotangent Lagrangian operad 𝒪_Δ.

4.2. Product in the Formal Deformation Operad

Exactly as for the local deformation operad, 𝑆_𝜖 is a product in 𝒪_form if and only if 𝑆²₀+𝑆_𝜖 satisfies formally the SGA equation. Moreover, if 𝑆²₀+𝑆_𝜖 satisfies the SGS conditions, then 𝑆²₀+𝑆_𝜖 is the generating function of a formal symplectic groupoid over ℝ^𝑑.

Again, the zero of 𝒪²_form is a product in 𝒪_form. We will stick to the conventions introduced for 𝒪_loc. Namely, 0₁ will stand for the zero of 𝒪¹_form, which is also the identity of the operad and 0₂ will stand for the zero of 𝒪²_form, which is the trivial product of the operad.

Thanks to the composition formula (4.6), we are now able to rewrite the product equation in 𝒪_form as a cohomological equation, exactly as the deformation equation of a product in an additive category. Note that the Taylor expansion plays the same role as the linear expansion played in the additive case.

Let us define the Gerstenhaber bracket in 𝒪_form as follows: [𝐹,𝐺]=𝐹∘𝐺−(−1)^{(𝑘−1)(𝑙−1)}𝐺∘𝐹, (4.10) where 𝐹∘𝐺=^𝑘_𝑖=1(−1)^{(𝑖−𝑙)(𝑙−1)}𝐹0₁,…,0₁,𝐺_}𝑖th,0₁,…,0₁, (4.11) for 𝐹∈𝒪^𝑘_form and 𝐺∈𝒪^𝑙_form.

We are now able to define a true coboundary operator.

Proposition 4.1. Consider 𝑑∶𝒪^𝑛_form→𝒪^𝑛+1_form𝑑𝐹∶=0₂,𝐹. (4.12) Then, 𝑑 may be written as 𝑑𝐹𝑝₁,…,𝑝_𝑛+1=𝐹𝑝₁,…,𝑝_𝑛+,𝑥_𝑛𝑗=1(−1)^{𝑛+𝑗−1}𝐹𝑝₁,…,𝑝_𝑗+𝑝_𝑗+1,…,𝑝_𝑛,𝑥+(−1)^𝑛−1𝐹𝑝₂,…,𝑝_𝑛+1,𝑥. (4.13) Moreover, 𝑑 is linear and 𝑑²=0.

Proof. For more clarity, let us break our convention and write 𝐼̃ instead of 0₁ and 𝑆 instead of 0₂. We have that [𝑆,𝐹]=𝑆∘𝐹−(−1)^𝑛−1𝐹∘𝑆. As 𝑆=0, only the trees ∘_𝑖 and •_𝑗 will contribute to the product. Then we have, 𝐼₁=𝑆∘𝐹𝑝₁,…,𝑝_𝑛+1,𝑥=_𝑖≥1𝜖^𝑖𝐶_•𝑖𝑆(⋅,𝑥),𝐹∪𝐼̃𝑝,⋅^𝑛₁𝑝_𝑙,𝑝_𝑛+1,()+()𝑥,𝑥−1^𝑛−1𝐶_•𝑖𝑆(⋅,𝑥),𝐼̃∪𝐹𝑝,⋅𝑝₁,^𝑛+1₂𝑝_𝑙,(𝑥,𝑥)=_𝑖≥1𝜖^𝑖𝐹^(𝑖)𝑝₁,…,𝑝_𝑛,𝑥+(−1)^𝑛−1𝐹^(𝑖)𝑝₂,…,𝑝_𝑛+1,𝐼,𝑥₂=𝐹∘𝑆𝑝₁,…,𝑝_𝑛+1=^𝑛_𝑗=1(−1)^𝑗−1_𝑖≥1𝜖^𝑖𝐶_∘𝑖𝐹(⋅,𝑥),𝐼̃∪⋯𝐼̃∪𝑆^}_𝑗th∪̃𝐼̃×𝑝𝐼⋯⋯∪𝑝,⋅₁,…,𝑝_𝑗+𝑝_𝑗+1,…,𝑝_𝑛+1,()=𝑥,…,𝑥_𝑛𝑗=1(−1)^𝑗−1_𝑖≥1𝜖^𝑖𝐹^(𝑖)𝑝₁,…,𝑝_𝑗+𝑝_𝑗+1,…,𝑝_𝑛+1,𝑥, (4.14) which gives the desired formula. The check that 𝑑²=0 is straightforward.

We have then a complex 𝐶^•=_𝑛≥0𝒪^𝑛_form,𝑑. (4.15) This complex is exactly the Hochschild complex of (formal) multidifferential operators lifted on the level of symbols (see for instance [11]). This remark gives us the cohomology of the complex 𝐻^𝑛𝐶^•≃𝜖𝒱^𝑛ℝ^𝑑[[𝜖]], (4.16) where 𝒱^𝑛(ℝ^𝑑) is the space of 𝑛-multi-vector fields on ℝ^𝑑.

We come now to the question of finding a product 𝑆_𝜖 in the formal deformation operad of 𝒪_Δ. This is exactly the same problem as deforming the trivial generating function 𝑆²₀ in 𝒪_Δ+𝒪_form. We are thus looking for an element 𝑆_𝜖∈𝒪²_form of the form 𝑆_𝜖=𝜖𝑆₁+𝜖²𝑆₂+⋯ (4.17) such that𝑆_𝜖,𝑆_𝜖=0. (4.18) Equation (4.18) becomes, on the level of trees,_{𝑡∈𝑇∞}𝜖^‖𝑡‖|𝑡|!𝐶_𝑡𝑆_𝜖,𝑆_𝜖∪𝐼−𝐶_𝑡𝑆_𝜖,𝐼∪𝑆_𝜖=0. (4.19)

One sees immediately that this equation is equivalent to the following infinite set of recursive equations: 𝑑𝑆_𝑛+𝐻_𝑛𝑆_𝑛−1,…,𝑆₁=0, (4.20) where 𝐻_𝑛𝑆_𝑛−1,…,𝑆₁=_{𝑡∈𝑇∞𝑘,𝑛|𝑘|2≤≤𝑛}1|𝑡|!𝐶_𝑡𝑆_𝜖,𝑆_𝜖∪𝐼−𝐶_𝑡𝑆_𝜖,𝐼∪𝑆_𝜖, (4.21) where 𝑇_∞^𝑘,𝑛 is the subset of trees in 𝑇_∞^𝑘,𝑛 with 𝑘 vertices and such that ‖𝑡‖=𝑛. These recursive equations are the exact analog of (2.23).

4.3. Formal Symplectic Groupoid Generating Function

We restate now the main theorem of [2], Theorem 4.2, in terms of the new structures defined in this paper.

Theorem 4.2. For each Poisson structure α on ℝ^𝑑, one has that 𝑆_𝜖(𝛼)=^∞_𝑛=1𝜖^𝑛𝑛!_{Γ∈𝑇𝑛,2}𝑊_Γ𝐵_Γ(𝛼) (4.22) is a product in the formal deformation operad 𝒪_form(T^∗ℝ^𝑑,Δ) of the cotangent Lagrangian operad 𝒪_Δ(T^∗ℝ^𝑑). Moreover, 𝑆_𝜖(𝛼) is the unique natural product in 𝒪_form(T^∗ℝ^𝑑,Δ) whose first order is 𝜖𝛼.

In the above theorem, the 𝑇_𝑛,2 stand for the set of Kontsevich trees of type (𝑛,2), 𝑊_Γ is the Kontsevich weight of Γ and 𝐵_Γ is the symbol of the bidifferential operator 𝐵_Γ associated to Γ. We refer the reader to [2] for exact definitions of Kontsevich trees, weights, operators, and naturallity.

We called 𝑆_𝜖(𝛼) the (formal) symplectic groupoid generating function because, as shown in [2], it generates a “geometric object”, a (formal) symplectic groupoid over ℝ^𝑑 associated to the Poisson structure α whose structure maps are explicitly given by𝜖_𝜖(𝑥)=()𝑖0,𝑥unitmap,_𝜖=𝑠𝑝,𝑥−𝑝,𝑥inversemap,_𝜖𝑝,𝑥=𝑥+∇_𝑝2𝑆_𝜖(𝛼)𝑡𝑝,0,𝑥sourcexmap,_𝜖𝑝,𝑥=𝑥+∇_𝑝1𝑆_𝜖(𝛼)0,𝑝,𝑥targetmap. (4.23)

This exhibits a strong relationship between star products and symplectic groupoids already foreseen by Costes, et al., Karasëv and Zakrzewski in respectively [12–14]. Recently and from a completely different point of view, Karabegov in [15] went still a step further by showing how to associate a kind of “formal symplectic groupoid” to any star product.

In [4, 10], we prove that the product 𝑆_𝜖(𝛼) has a nonzero convergence radius provided that the Poisson structure α is analytic. In this case, the generated formal symplectic groupoid is the local one. We also compared this local symplectic groupoid with the one constructed by Karasëv and Maslov in [13], and we proved that this two local symplectic groupoids are not only isomorphic as they should but exactly identical.

Acknowledgments

The authors thank Domenico Fiorenza and Jim Stasheff for useful comments and suggestions. A. S. Cattaneo acknowledges partial support of SNF Grant no. 200020-107444/1 and the IHES for hospitality. B. Dherin acknowledges partial support from SNF Grant PA002-113136 and from the Netherlands Organisation for Scientific Research (NWO) Grant 613.000.602. B. Dherin and G. Felder acknowledge partial support of SNF Grant no. 21-65213.01.