# Formal Lagrangian Operad

Authors: Alberto S. Cattaneo; Benoit Dherin; Giovanni Felder
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Given a symplectic manifoldM, we may define an operad structure
on the the spaces Ok 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∞(ℝd) is a deformation of the trivial symplectic groupoid
structure of T∗ℝd.


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## Body

## 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 overℝd. 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 ℝd. Our guideline will be the Kontsevich deformation of the usual algebra of functions over ℝd, (C∞(ℝd),·). Namely, the usual pointwise product of functions S02(f,g)=fg generates a suboperad, the product suboperad, 𝒪Sn={S0n}, of the endomorphism operad 𝒪 of C∞(ℝd), where S0n is the n-multilinear map defined by S0n(f1,…,fn)=f1f2⋯fn. For each n one may choose the vector subspace 𝒪defn⊂𝒪n of n-multidifferential operators. The operad structure of 𝒪 induces an operad structure on 𝒪S+𝒪def, which in turn generates an operad structure on 𝒪def which is, however, nonlinear. Then, γ is a deformation of the usual product S02, that is, an element γ∈𝒪def2 such that S02+γ is still an associative product, if γ is a product in the induced deformation operad 𝒪def. We may also consider the formal version by replacing 𝒪def by the formal power series in ϵ, ϵ𝒪def[[ϵ]]. Kontsevich in [1] gives an explicit formal deformation of the product of functions over ℝd, (1.1)Sϵ=S02+∑n=1∞ϵn∑Γ∈Gn,2WΓBΓ,
where the WΓ's are the Kontsevich weights and the BΓ'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 groupoidT*ℝd over ℝd, we see that the multiplication space (1.2)Δ2n:={(p1,x),(p2,x),(p1+p2,x):p1,p2∈Rd*,x∈Rd}
generates an operad 𝒪Δn={Δn}, where (1.3)Δn:={(p1,x),…,(pn,x),(p1+⋯+pn,x):pi∈Rd*,x∈Rd}.Δ2 is a product in this operad. The compositions are given by symplectic reduction as the Δn's are Lagrangian submanifolds of (T*ℝd)n¯×T*ℝd. The main difference with the vector space case is that there is no “true” endomorphism operad where 𝒪Δ would naturally embed into. Thus, the question of finding a deformation operad for 𝒪Δ must be taken with more care. The first remark is that the Δn may be expressed in terms of generating functions (1.4)S0n(p1,…,pn,x)=(p1+⋯+pn)x.
Namely, Δn=graphdS0n. 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 𝒪defn for each n such that 𝒪Δ+𝒪def remains an operad. The formal version of it gives a surprising result. Namely, we may find an explicit deformation of the trivial generating function S02, it is given by the formula (1.5)Sϵ=S02+∑n=1∞ϵn∑Γ∈Tn,2WΓB̂Γ,
where the WΓ are the Kontsevich weights and the B̂Γ are the symbols of the Kontsevich bidifferential operators and the sum is taken over all Kontsevich trees Tn,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 thatU=∑nϵnUn, where (1.6)Un(ξ1,…,ξn)=∑Γ∈GnWΓBΓ(ξ1,…,ξn)
for ξi∈Γ(∧diTM), i=1,…,d is an L∞-morphism from the multivector fields to the multidifferential operators on ℝd. In our perspective, we may still write (1.7)Ũn(ξ1,…,ξn)=∑Γ∈TnWΓB̂Γ(ξ1,…,ξn)
summing over Kontsevich trees instead of Kontsevich graphs and replacing multidifferential operators by their symbols. Exactly, as in Kontsevich case, (1.8)Sϵ=S02+∑n≥1ϵnŨn(α,…,α)
is an associative deformation of the generating function of the trivial symplectic groupoid T*ℝd. However, it is still not completely clear how to define “semi-classical L∞-morphisms”.Organization of the Paper
In Section2, 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 𝒪S(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 𝒪defn such that 𝒪S+𝒪def is still an operad. We describe the deformations of S in terms of products in 𝒪def. 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 ℝd 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 ℝd. 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 Section3, 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 ℝd 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*ℝn.
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*ℝd. 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*ℝd, 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*ℝd 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 overT*ℝd 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 operad𝒪(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 𝒪def(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 ℝd 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 operad𝒪 consists of  (1)
a collection of sets𝒪n, n≥0,(2)
composition laws(2.1)On×Ok1×⋯×Okn⟶Ok1+…+kn(F,G1,…,Gn)⟼F(G1,…,Gn)
satisfying the following associativity relations:
(2.2)F(G1,…,Gn)(H11,…,H1k1,…,Hn1,…,Hnkn)=F(G1(H11,…,H1k1),…,Gn(Hn1,…,Hnkn)),(3)
a unit elementI∈𝒪1 such that F(I,…,I)=FforallF∈𝒪n. 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 category𝒞. 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:
(2.3)(A⊗B)⊗C≃A⊗(B⊗C),e⊗A≃A⊗e≃A
for all A,B,C∈Obj𝒞.
Let𝒞 be a monoidal category and let an object M∈Obj𝒞. We define the endomorphism operad of M in the following way: (1)
𝒪n(M):=Hom(M⊗n,M),𝒪0(M):=Hom(e,M),(2)
F(G1,…,Gn):=F∘(G1⊗⋯⊗Gn),(3)
the unit is given byidM∈𝒪1(M).
The operad axioms follow directly from the bifunctoriality of⊗, that is,
(2.4)(f⊗g)∘(ψ⊗ϕ)=(f∘ψ)⊗(g∘ϕ)idM⊗⋯⊗idM=idM⊗⋯⊗M.
IfM 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 productS∈𝒪2, the associativity of the operad implies that, for any F∈𝒪k, G∈𝒪l and H∈𝒪m we have,
(2.5)S(F,S(G,H))=S(I,S)(F,G,H)=S(S,I)(F,G,H)=S(S(F,G),H).
This notion is the natural generalization of an associative product on a vector space. Namely, ifM 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 productS∈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 operad𝒪(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 productS∈𝒪2(M) generates a suboperad 𝒪S(M), which we call a  product operad, in 𝒪(M) with only one point in each degree:
(2.6)OS0(M):=∅,OS1(M):={I},OS2(M):={S},OS3(M)∶={S(S,I)},OS4(M):={S(S(S,I),I)},…,etc.
To simplify the notation we will denote by S0n the unique element in 𝒪Sn(M).Remark 2.3.
The product operad𝒪S(M) is a suboperad of 𝒪(M) but not a linear suboperad, namely, for each n∈ℕ, 𝒪Sn(M) is not a linear subspace of 𝒪n(M) (it contains only a single point).Definition 2.4.
LetM be an object of an additive monoidal category 𝒞 and let S∈𝒪2(M) be a product. A deformation operad, 𝒪def(M,S), for S is the data, for each n∈ℕ, of a linear subspace 𝒪defn(M,S)⊂𝒪n(M) such that the difference
(2.7)R(γ;γ1,…,γn)∶=(S0n+γ)(S0k1+γ1,…,S0kn+γn)-S0k1+⋯+kn
is in 𝒪defk1+⋯+kn(M,S) for all γ∈𝒪defn(M,S), γi∈𝒪defki(M,S), and i=1,…,n.Remark 2.5.
𝒪S+𝒪def is a suboperad of 𝒪(M) but not a linear one: the spaces 𝒪Sn+𝒪defn(M,S) are not linear subspaces but affine ones.Proposition 2.6.
Let𝒪def(M,S) be a deformation operad for a product S∈𝒪2(M). Then the compositions
(2.8)γ(γ1,…,γn)∶=R(γ;γ1,…,γn),
defined by (2.7) gives 𝒪def(M,S) together with the unit 0∈𝒪def1(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,𝒪def(M,S) is not a linear operad since its compositions, the Rs, are not multilinear.Definition 2.8.
We say that an elementγ∈𝒪def2(M,S) is a deformation of the productS w.r.t. the deformation operad 𝒪def if S+γ is still a product in 𝒪S+𝒪def.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.
LetS∈𝒪2(M) be a product. Take an element γ∈𝒪def2(M,S). Then, γ is a deformation of the product S if and only if γ is a product in 𝒪def(M,S). In particular, 0∈𝒪def2(M,S) is always a product in the deformation operad of S.Proof.
γ is a deformation of S if and only if
(2.9)(S+γ)(S+γ,I)=(S+γ)(I,S+γ),
which is equivalent to
(2.10)S03+R(γ;γ,0)=S03+R(γ;0,γ).From now on, we will write01 for the identity element of the deformation operad which is the zero of 𝒪def1 and 02 for the trivial product of the deformation operad which is the 0 element in 𝒪def2(M,S).Notice that neither𝒪S(M) nor 𝒪S(M)+𝒪def(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 𝒪def(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(2.11)[,]:Odefk(M,S)×Odefl(M,S)⟶Odefk+l-1(M,S)
defined by(2.12)[F,G]=F∘G-(-1)(k-1)(l-1)G∘F,
where (2.13)F∘G=∑i=1k(-1)(i-l)(l-1)R(F;01,…,01,G︸ith,01,…,01).
This bracket is not bilinear. An important fact concerning this bracket is that, (2.14)12[γ,γ]=R(γ;γ,01)-R(γ;01,γ),
which means that γ is a product in the deformation operad if and only if(2.15)12[γ,γ]=0.Moreover, we may define an equivalent of the Hochschild differential(2.16)d:Odefn(M,S)⟶Odefn+1(M,S),(2.17)dF:=[02,F]=R(02;F,01)+(-1)n-1R(02;01,F)-(-1)n-1∑i=1n(-1)i-1R(F;01,…,01,02︸ith,01,…,01).It turn out thatd is still a coboundary operator.Proposition 2.11.
d defined by (2.17) is a coboundary operator, that is, d2=0. Moreover, γ∈𝒪def2(M,S) satisfies product equation (1/2)[γ,γ]=0, in 𝒪def(M,S) if and only if
(2.18)dγ+γ(γ,S01,)-γ(S01,γ)=0.Proof.
Using (2.7) we obtain d in terms of the endomorphism compositions
(2.19)dF=S02(F,S01)+(-1)n-1S02(S01,F)-(-1)n-1∑i=1n(-1)i-1F(S01,…,S02︸ith,…,S01).
The result follows directly from the linearity of the compositions in the endomorphism operad. Using again (2.7) we get
(2.20)12[γ,γ]=R(γ;γ,01)-R(γ;01,γ)=S02(γ,S01)+γ(S02,S01)+γ(γ,S01)-S02(S01,γ)-γ(S01,S02)-γ(S01,γ),
which gives (2.18).A formal deformationSϵ of S is a formal power series (2.21)Sϵ=ϵS1+ϵ2S2+⋯∈Oformn(M,S)∶=ϵOdefn(M,S)⊗k[[ϵ]],n∈N*,
where ϵ is a formal parameter and 𝒪def(M,S) is a deformation operad for S, such that S+Sϵ is a product in 𝒪S(M)+𝒪form(M,S).Equivalently, one may say thatSϵ must satisfy (2.22)[Sϵ,Sϵ]=0,
or, thanks to (2.18) that the Si's satisfy at each order n∈ℕ* the following recursive equation:(2.23)dSn+Hn(Sn-1,…,S1)=0,
where (2.24)Hn(Sn-1,…,S1)=∑n=i+jSi(Sj,S01)-Si(S01,Si).The Kontsevich Product Deformation
Consider the category of real vector spaces. In this category we take the real vector spaceM=C∞(ℝd) of smooth functions on ℝd. The endomorphism operad of C∞(ℝd) is
(2.25)On(M)={n-multilinearmapsfromC∞(Rd)⊗ntoC∞(Rd)}.
The usual product of functions induces a product in 𝒪(M), namely,
(2.26)S02(F,G)(f1,…,fk,g1,…,gl)=F(f1,…,fk)G(g1,…,gl),
for F∈𝒪k(M) and G∈𝒪l(M).
The induced product operad is(2.27)OSn(M)={S0n},
where
(2.28)S0n(f1,…,fn)=f1f2⋯fn.
As deformation operad, we take
(2.29)Odefn(M,S):={n-multidifferentialoperatorsonC∞(Rd)}.
The induced coboundary operator on 𝒪def(M,S) is the Hochschild coboundary operator,
(2.30)dF(f1,…,fn)=F(f1,…,fn)fn+1+(-1)n-1f1F(f2,…,fn+1)-(-1)n-1∑i=1n(-1)(i-1)F(f1,…,fi-1,fifi+1,fi+2,…,fn+1).
and the product equation
(2.31)dγ+γ(γ,S01,)-γ(S01,γ)=0,
is nothing but the usual Maurer-Cartan equation.
Kontsevich in [1] shows that there exists a formal deformation
(2.32)S∈OS2(M)+ϵOdef2(M)[[ϵ]]
of S02. He provides the explicit formula for this deformation
(2.33)S=S02+∑n=1∞ϵn∑Γ∈Gn,2WΓBΓ,
where the Gn,2 are the Kontsevich graphs of type (n,2), WΓ is their associated weight, and BΓ is their associated bidifferential operator (and [1] for more precisions).


### 2.2. Monoidal Structure of𝒮𝒴ℳ

Let us recall that the extended symplectic “category”𝒮𝒴ℳ is given by (2.34)Obj={symplecticmanifolds},Hom(M,N)={L⊂M¯×N:LisLagrangian},
where M¯ denotes the symplectic manifold M with opposite symplectic structure -ω. The identity morphism of Hom(M,M) is the diagonal (2.35)idM∶=ΔM={(m,m)⊂M¯×M}.
The composition of two morphisms L∈Hom(M,N) and L̃∈Hom(N,P) is given by the composition of canonical relations (2.36)L̃∘L:=πM×P((L×L̃)∩(M×ΔN×P))⊂M¯×P.
Everything works fine except the fact that the composition L̃∘L may fail to be a Lagrangian submanifold of M¯×P. It is always the case when L×L̃ intersects M×ΔN×P cleanly (see [10] for more precisions).Let us pretend for a while that𝒮𝒴ℳ 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 objectsM and N of 𝒮𝒴ℳ as the Cartesian product (2.37)M⊗N∶=M×N,
and the tensor product between morphisms as (2.38)L1⊗L2∶={(m,a,n,b):(m,n)∈L1,(a,b)∈L2}∈Hom(M⊗A,N⊗B),
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)
ConsiderL1∈Hom(M,A), L2∈Hom(N,B), L3∈Hom(A,X) and L4∈Hom(B,Y). Then one has the following equality of sets:
(2.39)(L3⊗L4)∘(L1⊗L2)=(L3∘L1)⊗(L4∘L2).(2)
idM⊗idN=idM⊗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)(2.40)I=(L3⊗L4)∘(L1⊗L2)=π(((L1⊗L2)×(L3⊗L4))∩(N×M¯×ΔA×B×X×Y))={(m,n,x̃,ỹ):∃(a,b)∈A×Bs.t.(m,n,a,b)∈L1⊗L2,(a,b,x,y)∈L3⊗L4}={(m,n,x̃,ỹ):∃a∈A,(m,a)∈L1,(a,x)∈L3∃b∈B,(n,b)∈L2(b,y)∈L4}=(L3∘L1)⊗(L4∘L2)
(2)ΔM⊗ΔN={(m,n,m,n):m∈Mandn∈N}=ΔM⊗N.
(3) The associativity between objects is trivial.
(4) For morphisms, we have,(2.41)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}.
(5) is trivial.


### 2.3. Lagrangian Operads

If𝒮𝒴ℳ 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 𝒪restn(M)⊂𝒪n(M) for each n≥0 such that the composition (2.42)Ln(Lk1,…,Lkn):=Ln∘(Lk1⊗⋯⊗Lkn),
yields always a Lagrangian submanifold in 𝒪restk1+⋯+kn(M) for every Ln∈𝒪restn(M) and Lki∈𝒪restki(M), i=1,…,n. For instance, there is always the trivial choice (2.43)Orest1(M)={ΔM},Orestn(M)=∅,n≠1.In this way, we may get a true operad𝒪rest(M).The next natural question to ask is the following.Question 2.3.
What is a product in a Lagrangian operad overM?As a first hint, take the situation where the symplectic manifold is a symplectic groupoidG. In this case, we may generate an operad from the multiplication space Gm∈𝒪2(G) and the base G(0)∈𝒪0(G), the identity being the diagonal ΔG∈𝒪1(G). Remark that Gm is a product in this operad, that is, that Gm(Gm,ΔG)=Gm(ΔG,Gm). 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 isT*ℝd and will try to develop a deformation theory for the product in this case.Local Cotangent Lagrangian Operads
Remember thatT*ℝd has always a structure of a symplectic groupoid over ℝd: the trivial one. The multiplication space is given in this case by
(2.44)Δ2={(p1,x),(p2,x),(p1+p2,x):p1,p2∈Rd*,x∈Rd}.
The base is
(2.45)Δ0={(0,x):x∈Rd}.
If we set further
(2.46)Δn:={(p1,x),…,(pn,x),(p1+…+pn,x):pi∈Rd*,x∈Rd},
it immediate to see that the operad generated by Δ0 and Δ2 is exactly
(2.47)OΔn(T*Rd)={Δn},
and that Δ2 is a product in it.Following [3], we will call this operad the cotangent Lagrangian operad over T*ℝd. 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 𝒪Δ(T*ℝd) into. The idea now is to enlarge the cotangent Lagrangian operad, that is, by considering Lagrangian submanifolds close enough to Δn for each n∈ℕ in order to have still an operad.Notice at this point that theΔn's are given by generating functions. Namely, we may identify (T*ℝd)n¯×T*ℝd with T*Bn, where Bn:=(ℝd*)n×ℝd. Then, (2.48)Δn={((p1,∂S0n∂p1(z)),…,(pn,∂S0n∂pn(z)),(∂S0n∂x(z),x)):z=(p1,…,pn,x)∈Bn},
where S0n is the function on Bn defined by (In the sequels, we will use the shorter notation (p1+⋯+pn)x instead of ∑i=1d(p1i+⋯+pni)xi.)(2.49)S0n(p1,…,pn,x)=∑i=1d(p1i+⋯+pni)xi.
The cotangent Lagrangian operad may then be identified with (2.50)OΔn={S0n},OΔ0={0}.
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 (2.51)F=S0n+F̃,
where F̃∈C∞(Bn). The Lagrangian submanifold associated to F is (2.52)LF∶=graphdF.
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:(2.53)Bn0={0}×Rd⊂Bn,V(Bn0) will stand for the set of all neighborhoods of Bn0 in Bn.Definition 2.13.
We define𝒪locn(T*ℝd) to be the space of germs at Bn0 of smooth functions F̃ (defined on an open neighborhood UF̃⊂Bn of Bn0) which satisfy F̃(0,x)=0 and ∇pF̃(0,x)=0. Note that the composition will always be understood in terms of composition of germs.Proposition 2.14.
LetF∈𝒪Δn+𝒪locn and Gi∈𝒪Δki+𝒪locki for i=1,…,n. Consider the function ϕ defined by the formula
(2.54)ϕ(pG,xF)=G1∪⋯∪Gn(pG,xG)+F(pF,xF)-xGpF,(2.55)pF=∇xG1∪…∪Gn(pG,xG),xG=∇pF(pF,xF),
where
(2.56)G1∪⋯∪Gn(pG,xG)∶=G1(pG1,xG1)+⋯+Gn(pGn,xGn)
and pG=(pG1,…,pGn),pGi∈(ℝd*)ki, xGi∈ℝd and (pGi,xGi)∈UGi, for i=1,…,n.
Then,(2.57)ϕ∈OΔk1+⋯+kn+Oloc⁡k1+⋯+kn,andLϕ=LF(LG1,…,LGn).
In other words, 𝒪Δ+𝒪loc together with the product
(2.58)ϕ=F(G1,…,Gn)
is an operad.
Moreover, the induced operad structure on𝒪loc is given by
(2.59)R(F̃;G̃1,…,G̃n)=H,
where H is the function H∈𝒪lock1+⋯+kn defined by
(2.60)H(pG,xF)=G̃(pG,xG)+F̃(pF,xF)-∇pF̃(pF,xF)∇xG̃(pG,xG),pF=pF0+∇xG̃(pG,xG),pF0∶=(pG1Σ,…,pGnΣ),xG=xG0+∇pF̃(pF,xF),xG0∶=(xF,…,xF).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:
(2.61)∫e(i/ℏ)[F(p1,…,pk,x)+∑i=1k(Gi(πi1,…,πili,yi)-pi⋅yi)]∏i=1kdnpidnyi(2πℏ)n=e(i/ℏ)Φ(π11,…,π1l1,π21,…,π2l2…πk1,…,πklk,x)(C+O(ℏ)),
where C is some constant.Proof of Proposition2.14.
To simplify the computations, we identify(T*ℝd)n with T*(ℝdn) and (T*ℝd)ki with T*(ℝdki). With this identifications the graphs of F and Gi, i=1,…,n may be written as
(2.62)LF={((pF,∇pF(pF,xF)),(∇xF(pF,xF),xF)):(pF,xF)∈UF}⊂T*(Rdn)×T*Rd,LGi={((pGi,∇pGi(pGi,xGi)),(∇xGi(pGi,xGi),xGi)):(pGi,xGi)∈UGi}⊂T*(Rdki)×T*Rd,
where UF∈V(Bn0) and UGi∈V(Bki0) for i=1,…,n.
Consider now the composition,(2.63)LF(LG1,…,LGn)=LF∘(LG1⊗⋯⊗LGn).
First of all, observe that,
(2.64)LG∶=LG1⊗⋯⊗LGn={((pG,∇pG(pG,xG)),(∇xG(pG,xG),xG)):(pGi,xGi)∈UGi}LG⊂T*(Rd(k1+⋯+kn))×T*(Rdn).
Thus,
(2.65)LF∘LG=π((LG×LF)∩(T*Rd(k1+⋯+kn)×ΔT*Rdn×T*Rd))={((pG,∇pG(pG,xG)),(∇xF(pF,xF),xF)):xG=∇pF(pF,xF),pF=∇xG(pG,xG),(pG,xF)∈Ũ}LF∘LG⊂T*(Rd(k1+⋯+kn))×T*Rd,
where Ũ is the subset of (pG,xF)∈Bk1+⋯+kn such that the system,
(2.66)pF=∇xG(pG,xG),xG=∇pF(pF,xF),
has a unique solution (pF,xG) and such that (pGi,xGi)∈UGi, i=1,…,n, and (pF,xF)∈UF. Let us check that Ũ always exists and is a neighborhood of Bk1+⋯+kn0. To begin with, observe that for any (0,xF)∈Bn0 this system has the unique solution (0,∇pF(0,xF)). Set now,
(2.67)H(pG,xF,pF,xG)=(pF-∇xG(pG,xG)xF-∇pF(pF,xF)).
Thanks to the fact that G(0,x)=∑i=1nGi(0,x)=0 we get that the Jacobi matrix
(2.68)DpF,xGH(0,xf,0,∇pF(0,xF))=(id0-∇p∇pF(0,xF)id)
is invertible.
Thus, the implicit function theorem gives us the desired neighborhoodŨ of Bk1+⋯+kn0.
Now, takeϕ as defined in (2.54). The previous considerations tell us that ϕ is exactly defined on Ũ. Let us compute its graphs,
(2.69)Lϕ={((pG,∇pΦ(pG,xF)),(∇xΦ(pG,xF),xF):(pG,xF)∈Ũ}.
We have that
(2.70)∇pϕ(pG,xF)=∇pG(pG,xG)+∇xG(pG,xG)dxGdp+∇pF(pF,xF)dpFdp-pFdxGdp-dpFdpxG=∇pG(pG,xG).
Similarly, ∇xϕ(pG,xF)=∇xF(pF,xF). Thus, Lϕ=LF∘LG.
At last, let us check thatϕ∈𝒪lock1+⋯+kn. First of all, remember that
(2.71)F(pF,xF)=pFΣxF+F̃(pF,xF),G(pG,xF)=∑i=1npGiΣxGi+G̃(pG,xG).
Thus, we obtain immediately that
(2.72)ϕ(pG,xF)=pGΣxF+H(pG,xF),
where H is a function only defined on Ũ by the equations
(2.73)H(pG,xF)=G̃(pG,xG)+F̃(pF,xF)-∇pF̃(pF,xF)∇xG̃(pG,xG),pF=pF0+∇xG̃(pG,xG),pF0:=(pG1Σ,…,pGnΣ),xG=xG0+∇pF̃(pF,xF),xG0:=(xF,…,xF).
But now, if we set pG=0 then pF=0, xG=xG0+∇pF̃(0,xF) and H(0,xF)=0. Similarly, one easily checks that ∇pH(0,xF)=0We will call the operad𝒪Δ+𝒪loc  local cotangent Lagrangian operad over T*ℝd or for short the local Lagrangian operad when no ambiguities arise. The induced operad 𝒪loc will be called the local deformation operad of𝒪Δ.Associative Products in the Local Deformation Operad
We say that a generating functionS∈C∞(B2) satisfies the Symplectic Groupoid Associativity equation if for a point (p1,p2,p3,x)∈B3 sufficiently close to B30 the following implicit system for x¯,p¯,x̃ and p̃:
(2.74)x¯=∇p1S(p¯,p3,x),p¯=∇xS(p1,p2,x¯),x̃=∇p2S(p1,p̃,x),p̃=∇xS(p2,p3,x̃)
has a unique solution and if the following additional equation holds:
(2.75)S(p1,p2,x¯)+S(p¯,p3,x)-x¯p¯=S(p2,p3,x̃)+S(p1,p̃,x)-x̃p̃.
IfS also satisfies the Symplectic Groupoid Structure conditions, that is, if
(2.76)S(p,0,x)=S(0,p,x)=px,S(p,-p,x)=0
then S generates a Poisson structure
(2.77)α(x)=2(∇pk1∇pl2S(0,0,x))k,l=1d
on ℝd together with a local symplectic groupoid integrating it, whose structure maps are given by
(2.78)ϵ(x)=(0,x)unit map,i(p,x)=(-p,x)inverse map,s(p,x)=∇p2S(p,0,x)source map,t(p,x)=∇p1S(0,p,x)target map.
In this case, we callS 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.
S̃∈𝒪loc2 is a product in 𝒪loc if and only if S=S02+S̃ satisfies the Symplectic Groupoid Associativity equation.Proof.
We know thatS̃ is a product in 𝒪loc if and only if S=S02+S̃ is a product in 𝒪Δ+𝒪loc, that is, if and only if S(S,I)=S(I,S). Let us compute.
(2.79)S(S,I)(p1,p2,p3,x)=S∪I(p1,p2,p3,x¯1,x¯2)+S(p¯1,p¯2,x)-x¯1p¯1-p¯2x¯2=S(p1,p2,x¯1)+p3x¯2+S(p¯1,p¯2,x)-p¯1x¯1-p¯2x¯2
with
(2.80)p¯1=∇x1S∪I(p1,p2,p3,x¯1,x¯2)=∇xG(x¯),p¯2=∇x2S∪I(p1,p2,p3,x¯1,x¯2)=p3,x¯1=∇p1S(p¯1,p¯2,x),x¯2=∇p2S(p¯1,p¯2,x).
Then we get
(2.81)S(S,I)=S(p1,p2,x¯)+S(p¯,p3,x)-p¯x¯,x¯=∇p1S(p¯,p3,x),p¯=∇xS(p1,p2,x¯).
Similarly, we get
(2.82)S(I,S)=S(p2,p3,x̃)+S(p1,p̃,x)-p̃x̃,x̃=∇p2S(p1,p̃,x)p̃=∇xS(p2,p3,x̃).
Hence, S̃∈𝒪loc2(T*ℝd) is a product if and only if S02+S̃ 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)[S̃,S̃]=0. We may also still write a formula for the coboundary operator. But, as this time the compositions in 𝒪Δ+𝒪loc are not multilinear, we cannot develop the expression (1/2)[S̃,S̃] 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 eachF∈𝒪Δ1+𝒪loc1, we may associate a symplectomorphism ψF which is defined only on a neighborhood UF of B10 in T*ℝd and which fixes B01. The composition of two such ψG and ψF, which may always be defined on a possibly smaller neighborhood Ũ⊂UG of B10, is exactly ψF(G) where F(G) is the composition of F by G in the local Lagrangian operad.
We denote byF-1∈𝒪Δ1+𝒪loc1 the generating function of the (ψF)-1, that is, the generating function such that F(F-1)=F-1(F)=I. Two associative products S and S̃ will be called equivalent if
(2.83)S̃=F(S)(F-1,F-1)
for a certain F∈𝒪Δ1+𝒪loc1. It is clear that if S∈𝒪Δ1+𝒪loc1 is an associative product, then S̃ also is. The following questions naturally arises.Questions
IfS generates a local symplectic groupoid, does S̃ 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 byψF. As a consequence the induced Poisson structures on the base are the same, that is, (2.84)α(x)=∇p1∇p2S(0,0,x)=∇p1∇p2S̃(0,0,x).The following two Propositions prove these statements.Proposition 2.17.
LetF∈𝒪Δ1+𝒪loc1. The following implicit equations:
(2.85)x1=∇pF(p1,x2),p2=∇xF(p1,x2),
define a symplectomorphism ψF(p1,x1)=(p2,x2) on a neighborhood UF of B10={(0,x):x∈ℝd} in T*ℝd which fixes B10 and which is close to the identity in the sense that F(p,x)=px+F̃(p,x) induces the identity if F̃=0. Consider now ψF and ψG defined, respectively, on UF and UG for F,G∈𝒪Δ1+𝒪loc1. Then one has that ψG∘ψF=ψF(G) on UF(G).Proof.
(1) Let us check that the system (2.85) generates a diffeomorphism around B10. Namely, one verifies that (p¯1,x¯1,p¯2,x¯2):=(0,∇pF(0,x2),0,x2) is a solution of the system. Set now
(2.86)H(p1,x1,p2,x2)∶=(x1-∇pF(p1,x2)p2-∇xF(p1,x2)).
As
(2.87)Dp1,x1H(p¯1,x¯2,p¯2,x¯2)=(-∇p∇pF(0,x¯2)id∇x∇pF(0,x¯2)0),Dp2,x2H(p¯1,x¯2,p¯2,x¯2)=(0∇x∇pF(0,x¯2)id0),
the implicit function theorem gives us the result. Let us call Ũ the neighborhood of B10 where ψF is defined.
(2) We check now thatψF is symplectic. From (2.85) we get the relation
(2.88)∂pl2∂pk1=∂xk1∂xl2,
which directly implies that dψFJ(dψF)*=J where
(2.89)J=(0id-id0).
(3) Let us see thatψF(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+F̃(p,x) with ∇pF̃(0,p)=0 and then ∇x∇pF(0,x2)=x2.
(4) ClearlyF(p,x)=px generates the identity.
(5) Recall that(2.90)LG={(p1,∇pG(p1,x2),∇xG(p1,x2),x2):(p1,x2)∈UG},LF={(p2,∇pF(p2,x3),∇xF(p2,x3),x3):(p2,x3)∈UF}.
Thus, LG=graphψG and LF=graphψF. The composition of these two canonical relations yields that LF∘LG=graphψF∘ψG. On the other hand, LF∘LG=LF(G)=graphψF(G). Taking care on the domain of definitions, we have that ψF∘ψG=ψF(G) on UF(G).Proposition 2.18.
LetS∈𝒪Δ2+𝒪loc2 be a generating function of a symplectic groupoid, that is,
(2.91)S(S,I)=S(I,S),S(p,0,x)=S(0,p,x)=px,S(p,-p,x)=0.
Let F∈𝒪Δ1+𝒪loc1 such that F(-p,x)=-F(p,x). Then,
(2.92)S̃∶=(F(S))(F-1,F-1)
is also a generating function of a symplectic groupoid. The subset of odd function in p forms a subgroup of 𝒪Δ1+𝒪loc1. Moreover, ψF is a groupoid isomorphism between the local symplectic groupoid generated by S and the one generated by S̃. As a consequence S and S̃ induce the same Poisson structure on the base.Proof.
To simplify the notation, we setG=F-1. A straightforward computation gives that
(2.93)F(S)(G,G)(p1,p2,x)=S(p¯,p̃,ẋ)+F(ṗ,x)+G(p1,x¯)+G(p2,x̃)-p¯x¯-p̃x̃-ẋṗ,ẋ=∇pF(ṗ,x),x¯=∇p1S(p¯,p̃,ẋ),x̃=∇p2S(p¯,x̃,ẋ),ṗ=∇xS(p¯,p̃,ẋ),p¯=∇xG(p1,x¯),p̃=∇xG(p2,x̃).(1)
Settingp1=p and p2=0, we have immediately
(2.94)F(S)(G,G)(p,0,x)=G(p,ẋ)+F(ṗ,x)-ẋṗ
with ẋ=∇pF(ṗ,x) and ṗ=∇xG(p,ẋ). We recognize then that
(2.95)F(S)(G,G)(p,0,x)=F(G)(p,x)=I(p,x)=px.
The case p1=0 and p2=p is analog.(2)
One reads directly from the equation(2.96)px=F-1(p,ẋ)+F(ṗ,x)-ẋṗ,
where ẋ=∇pF(ṗ,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 𝒪Δ1+𝒪loc1.(3)
Suppose now thatp1=p and p2=-p. G odd in p implies that p¯=-p̃. As S(p,-p,0)=0, we get immediately that x̃=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 thatψF is also a groupoid isomorphism. Consider the multiplication space of the symplectic groupoid generated by an generating function S, that is,(2.97)G(m)(S)={(p1,∇p1S),(p2,∇p2S),(∇xS,x):p1,p2∈(Rd)*,x∈Rd},
where the partial derivative are evaluated in (p1,p2,x).
We have to show that(ψF×ψF×ψF)(G(m)(S))=G(m)(S̃).
A straightforward computation gives that(2.98)∇p1S̃(p1,p2,x)=∇pG(p1,x¯),∇p2S̃(p1,p2,x)=∇pG(p2,x̃),∇xS̃(p1,p2,x)=∇xF(ṗ,x).
From this, we check immediately that(2.99)ψG((p1,∇p1S̃(p1,p2,x)))=(p¯,∇p1S(p¯,p̃,ẋ)),ψG((p2,∇p2S̃(p1,p2,x)))=(p̃,∇p2S(p¯,p̃,ẋ)),ψF((∇xS(p¯,p̃,ẋ),ẋ))=(∇xS̃(p1,p2,x),x)
which ends the proof.Remark 2.19.
Suppose thatS is a generating function of a local symplectic groupoid. Let F∈𝒪Δ1+𝒪loc1 act on S, that is, S̃=(F(S))(F-1,F-1). Then, the condition S(p,0,x)=S(0,p,x)=px is preserved by any F∈𝒪Δ1+𝒪loc1. 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
(2.100)((-p2,∇p2S(p2,p1,x)),(-p1,∇p1S(p2,p1,x)),(-∇xS(p2,p1,x),x))∈G(m)(S),
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.


## 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 operad𝒪(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 𝒪def(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 ℝd 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 operad𝒪 consists of  (1)
a collection of sets𝒪n, n≥0,(2)
composition laws(2.1)On×Ok1×⋯×Okn⟶Ok1+…+kn(F,G1,…,Gn)⟼F(G1,…,Gn)
satisfying the following associativity relations:
(2.2)F(G1,…,Gn)(H11,…,H1k1,…,Hn1,…,Hnkn)=F(G1(H11,…,H1k1),…,Gn(Hn1,…,Hnkn)),(3)
a unit elementI∈𝒪1 such that F(I,…,I)=FforallF∈𝒪n. 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 category𝒞. 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:
(2.3)(A⊗B)⊗C≃A⊗(B⊗C),e⊗A≃A⊗e≃A
for all A,B,C∈Obj𝒞.
Let𝒞 be a monoidal category and let an object M∈Obj𝒞. We define the endomorphism operad of M in the following way: (1)
𝒪n(M):=Hom(M⊗n,M),𝒪0(M):=Hom(e,M),(2)
F(G1,…,Gn):=F∘(G1⊗⋯⊗Gn),(3)
the unit is given byidM∈𝒪1(M).
The operad axioms follow directly from the bifunctoriality of⊗, that is,
(2.4)(f⊗g)∘(ψ⊗ϕ)=(f∘ψ)⊗(g∘ϕ)idM⊗⋯⊗idM=idM⊗⋯⊗M.
IfM 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 productS∈𝒪2, the associativity of the operad implies that, for any F∈𝒪k, G∈𝒪l and H∈𝒪m we have,
(2.5)S(F,S(G,H))=S(I,S)(F,G,H)=S(S,I)(F,G,H)=S(S(F,G),H).
This notion is the natural generalization of an associative product on a vector space. Namely, ifM 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 productS∈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 operad𝒪(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 productS∈𝒪2(M) generates a suboperad 𝒪S(M), which we call a  product operad, in 𝒪(M) with only one point in each degree:
(2.6)OS0(M):=∅,OS1(M):={I},OS2(M):={S},OS3(M)∶={S(S,I)},OS4(M):={S(S(S,I),I)},…,etc.
To simplify the notation we will denote by S0n the unique element in 𝒪Sn(M).Remark 2.3.
The product operad𝒪S(M) is a suboperad of 𝒪(M) but not a linear suboperad, namely, for each n∈ℕ, 𝒪Sn(M) is not a linear subspace of 𝒪n(M) (it contains only a single point).Definition 2.4.
LetM be an object of an additive monoidal category 𝒞 and let S∈𝒪2(M) be a product. A deformation operad, 𝒪def(M,S), for S is the data, for each n∈ℕ, of a linear subspace 𝒪defn(M,S)⊂𝒪n(M) such that the difference
(2.7)R(γ;γ1,…,γn)∶=(S0n+γ)(S0k1+γ1,…,S0kn+γn)-S0k1+⋯+kn
is in 𝒪defk1+⋯+kn(M,S) for all γ∈𝒪defn(M,S), γi∈𝒪defki(M,S), and i=1,…,n.Remark 2.5.
𝒪S+𝒪def is a suboperad of 𝒪(M) but not a linear one: the spaces 𝒪Sn+𝒪defn(M,S) are not linear subspaces but affine ones.Proposition 2.6.
Let𝒪def(M,S) be a deformation operad for a product S∈𝒪2(M). Then the compositions
(2.8)γ(γ1,…,γn)∶=R(γ;γ1,…,γn),
defined by (2.7) gives 𝒪def(M,S) together with the unit 0∈𝒪def1(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,𝒪def(M,S) is not a linear operad since its compositions, the Rs, are not multilinear.Definition 2.8.
We say that an elementγ∈𝒪def2(M,S) is a deformation of the productS w.r.t. the deformation operad 𝒪def if S+γ is still a product in 𝒪S+𝒪def.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.
LetS∈𝒪2(M) be a product. Take an element γ∈𝒪def2(M,S). Then, γ is a deformation of the product S if and only if γ is a product in 𝒪def(M,S). In particular, 0∈𝒪def2(M,S) is always a product in the deformation operad of S.Proof.
γ is a deformation of S if and only if
(2.9)(S+γ)(S+γ,I)=(S+γ)(I,S+γ),
which is equivalent to
(2.10)S03+R(γ;γ,0)=S03+R(γ;0,γ).From now on, we will write01 for the identity element of the deformation operad which is the zero of 𝒪def1 and 02 for the trivial product of the deformation operad which is the 0 element in 𝒪def2(M,S).Notice that neither𝒪S(M) nor 𝒪S(M)+𝒪def(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 𝒪def(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(2.11)[,]:Odefk(M,S)×Odefl(M,S)⟶Odefk+l-1(M,S)
defined by(2.12)[F,G]=F∘G-(-1)(k-1)(l-1)G∘F,
where (2.13)F∘G=∑i=1k(-1)(i-l)(l-1)R(F;01,…,01,G︸ith,01,…,01).
This bracket is not bilinear. An important fact concerning this bracket is that, (2.14)12[γ,γ]=R(γ;γ,01)-R(γ;01,γ),
which means that γ is a product in the deformation operad if and only if(2.15)12[γ,γ]=0.Moreover, we may define an equivalent of the Hochschild differential(2.16)d:Odefn(M,S)⟶Odefn+1(M,S),(2.17)dF:=[02,F]=R(02;F,01)+(-1)n-1R(02;01,F)-(-1)n-1∑i=1n(-1)i-1R(F;01,…,01,02︸ith,01,…,01).It turn out thatd is still a coboundary operator.Proposition 2.11.
d defined by (2.17) is a coboundary operator, that is, d2=0. Moreover, γ∈𝒪def2(M,S) satisfies product equation (1/2)[γ,γ]=0, in 𝒪def(M,S) if and only if
(2.18)dγ+γ(γ,S01,)-γ(S01,γ)=0.Proof.
Using (2.7) we obtain d in terms of the endomorphism compositions
(2.19)dF=S02(F,S01)+(-1)n-1S02(S01,F)-(-1)n-1∑i=1n(-1)i-1F(S01,…,S02︸ith,…,S01).
The result follows directly from the linearity of the compositions in the endomorphism operad. Using again (2.7) we get
(2.20)12[γ,γ]=R(γ;γ,01)-R(γ;01,γ)=S02(γ,S01)+γ(S02,S01)+γ(γ,S01)-S02(S01,γ)-γ(S01,S02)-γ(S01,γ),
which gives (2.18).A formal deformationSϵ of S is a formal power series (2.21)Sϵ=ϵS1+ϵ2S2+⋯∈Oformn(M,S)∶=ϵOdefn(M,S)⊗k[[ϵ]],n∈N*,
where ϵ is a formal parameter and 𝒪def(M,S) is a deformation operad for S, such that S+Sϵ is a product in 𝒪S(M)+𝒪form(M,S).Equivalently, one may say thatSϵ must satisfy (2.22)[Sϵ,Sϵ]=0,
or, thanks to (2.18) that the Si's satisfy at each order n∈ℕ* the following recursive equation:(2.23)dSn+Hn(Sn-1,…,S1)=0,
where (2.24)Hn(Sn-1,…,S1)=∑n=i+jSi(Sj,S01)-Si(S01,Si).The Kontsevich Product Deformation
Consider the category of real vector spaces. In this category we take the real vector spaceM=C∞(ℝd) of smooth functions on ℝd. The endomorphism operad of C∞(ℝd) is
(2.25)On(M)={n-multilinearmapsfromC∞(Rd)⊗ntoC∞(Rd)}.
The usual product of functions induces a product in 𝒪(M), namely,
(2.26)S02(F,G)(f1,…,fk,g1,…,gl)=F(f1,…,fk)G(g1,…,gl),
for F∈𝒪k(M) and G∈𝒪l(M).
The induced product operad is(2.27)OSn(M)={S0n},
where
(2.28)S0n(f1,…,fn)=f1f2⋯fn.
As deformation operad, we take
(2.29)Odefn(M,S):={n-multidifferentialoperatorsonC∞(Rd)}.
The induced coboundary operator on 𝒪def(M,S) is the Hochschild coboundary operator,
(2.30)dF(f1,…,fn)=F(f1,…,fn)fn+1+(-1)n-1f1F(f2,…,fn+1)-(-1)n-1∑i=1n(-1)(i-1)F(f1,…,fi-1,fifi+1,fi+2,…,fn+1).
and the product equation
(2.31)dγ+γ(γ,S01,)-γ(S01,γ)=0,
is nothing but the usual Maurer-Cartan equation.
Kontsevich in [1] shows that there exists a formal deformation
(2.32)S∈OS2(M)+ϵOdef2(M)[[ϵ]]
of S02. He provides the explicit formula for this deformation
(2.33)S=S02+∑n=1∞ϵn∑Γ∈Gn,2WΓBΓ,
where the Gn,2 are the Kontsevich graphs of type (n,2), WΓ is their associated weight, and BΓ is their associated bidifferential operator (and [1] for more precisions).


## 2.2. Monoidal Structure of𝒮𝒴ℳ

Let us recall that the extended symplectic “category”𝒮𝒴ℳ is given by (2.34)Obj={symplecticmanifolds},Hom(M,N)={L⊂M¯×N:LisLagrangian},
where M¯ denotes the symplectic manifold M with opposite symplectic structure -ω. The identity morphism of Hom(M,M) is the diagonal (2.35)idM∶=ΔM={(m,m)⊂M¯×M}.
The composition of two morphisms L∈Hom(M,N) and L̃∈Hom(N,P) is given by the composition of canonical relations (2.36)L̃∘L:=πM×P((L×L̃)∩(M×ΔN×P))⊂M¯×P.
Everything works fine except the fact that the composition L̃∘L may fail to be a Lagrangian submanifold of M¯×P. It is always the case when L×L̃ intersects M×ΔN×P cleanly (see [10] for more precisions).Let us pretend for a while that𝒮𝒴ℳ 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 objectsM and N of 𝒮𝒴ℳ as the Cartesian product (2.37)M⊗N∶=M×N,
and the tensor product between morphisms as (2.38)L1⊗L2∶={(m,a,n,b):(m,n)∈L1,(a,b)∈L2}∈Hom(M⊗A,N⊗B),
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)
ConsiderL1∈Hom(M,A), L2∈Hom(N,B), L3∈Hom(A,X) and L4∈Hom(B,Y). Then one has the following equality of sets:
(2.39)(L3⊗L4)∘(L1⊗L2)=(L3∘L1)⊗(L4∘L2).(2)
idM⊗idN=idM⊗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)(2.40)I=(L3⊗L4)∘(L1⊗L2)=π(((L1⊗L2)×(L3⊗L4))∩(N×M¯×ΔA×B×X×Y))={(m,n,x̃,ỹ):∃(a,b)∈A×Bs.t.(m,n,a,b)∈L1⊗L2,(a,b,x,y)∈L3⊗L4}={(m,n,x̃,ỹ):∃a∈A,(m,a)∈L1,(a,x)∈L3∃b∈B,(n,b)∈L2(b,y)∈L4}=(L3∘L1)⊗(L4∘L2)
(2)ΔM⊗ΔN={(m,n,m,n):m∈Mandn∈N}=ΔM⊗N.
(3) The associativity between objects is trivial.
(4) For morphisms, we have,(2.41)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}.
(5) is trivial.


## 2.3. Lagrangian Operads

If𝒮𝒴ℳ 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 𝒪restn(M)⊂𝒪n(M) for each n≥0 such that the composition (2.42)Ln(Lk1,…,Lkn):=Ln∘(Lk1⊗⋯⊗Lkn),
yields always a Lagrangian submanifold in 𝒪restk1+⋯+kn(M) for every Ln∈𝒪restn(M) and Lki∈𝒪restki(M), i=1,…,n. For instance, there is always the trivial choice (2.43)Orest1(M)={ΔM},Orestn(M)=∅,n≠1.In this way, we may get a true operad𝒪rest(M).The next natural question to ask is the following.Question 2.3.
What is a product in a Lagrangian operad overM?As a first hint, take the situation where the symplectic manifold is a symplectic groupoidG. In this case, we may generate an operad from the multiplication space Gm∈𝒪2(G) and the base G(0)∈𝒪0(G), the identity being the diagonal ΔG∈𝒪1(G). Remark that Gm is a product in this operad, that is, that Gm(Gm,ΔG)=Gm(ΔG,Gm). 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 isT*ℝd and will try to develop a deformation theory for the product in this case.Local Cotangent Lagrangian Operads
Remember thatT*ℝd has always a structure of a symplectic groupoid over ℝd: the trivial one. The multiplication space is given in this case by
(2.44)Δ2={(p1,x),(p2,x),(p1+p2,x):p1,p2∈Rd*,x∈Rd}.
The base is
(2.45)Δ0={(0,x):x∈Rd}.
If we set further
(2.46)Δn:={(p1,x),…,(pn,x),(p1+…+pn,x):pi∈Rd*,x∈Rd},
it immediate to see that the operad generated by Δ0 and Δ2 is exactly
(2.47)OΔn(T*Rd)={Δn},
and that Δ2 is a product in it.Following [3], we will call this operad the cotangent Lagrangian operad over T*ℝd. 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 𝒪Δ(T*ℝd) into. The idea now is to enlarge the cotangent Lagrangian operad, that is, by considering Lagrangian submanifolds close enough to Δn for each n∈ℕ in order to have still an operad.Notice at this point that theΔn's are given by generating functions. Namely, we may identify (T*ℝd)n¯×T*ℝd with T*Bn, where Bn:=(ℝd*)n×ℝd. Then, (2.48)Δn={((p1,∂S0n∂p1(z)),…,(pn,∂S0n∂pn(z)),(∂S0n∂x(z),x)):z=(p1,…,pn,x)∈Bn},
where S0n is the function on Bn defined by (In the sequels, we will use the shorter notation (p1+⋯+pn)x instead of ∑i=1d(p1i+⋯+pni)xi.)(2.49)S0n(p1,…,pn,x)=∑i=1d(p1i+⋯+pni)xi.
The cotangent Lagrangian operad may then be identified with (2.50)OΔn={S0n},OΔ0={0}.
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 (2.51)F=S0n+F̃,
where F̃∈C∞(Bn). The Lagrangian submanifold associated to F is (2.52)LF∶=graphdF.
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:(2.53)Bn0={0}×Rd⊂Bn,V(Bn0) will stand for the set of all neighborhoods of Bn0 in Bn.Definition 2.13.
We define𝒪locn(T*ℝd) to be the space of germs at Bn0 of smooth functions F̃ (defined on an open neighborhood UF̃⊂Bn of Bn0) which satisfy F̃(0,x)=0 and ∇pF̃(0,x)=0. Note that the composition will always be understood in terms of composition of germs.Proposition 2.14.
LetF∈𝒪Δn+𝒪locn and Gi∈𝒪Δki+𝒪locki for i=1,…,n. Consider the function ϕ defined by the formula
(2.54)ϕ(pG,xF)=G1∪⋯∪Gn(pG,xG)+F(pF,xF)-xGpF,(2.55)pF=∇xG1∪…∪Gn(pG,xG),xG=∇pF(pF,xF),
where
(2.56)G1∪⋯∪Gn(pG,xG)∶=G1(pG1,xG1)+⋯+Gn(pGn,xGn)
and pG=(pG1,…,pGn),pGi∈(ℝd*)ki, xGi∈ℝd and (pGi,xGi)∈UGi, for i=1,…,n.
Then,(2.57)ϕ∈OΔk1+⋯+kn+Oloc⁡k1+⋯+kn,andLϕ=LF(LG1,…,LGn).
In other words, 𝒪Δ+𝒪loc together with the product
(2.58)ϕ=F(G1,…,Gn)
is an operad.
Moreover, the induced operad structure on𝒪loc is given by
(2.59)R(F̃;G̃1,…,G̃n)=H,
where H is the function H∈𝒪lock1+⋯+kn defined by
(2.60)H(pG,xF)=G̃(pG,xG)+F̃(pF,xF)-∇pF̃(pF,xF)∇xG̃(pG,xG),pF=pF0+∇xG̃(pG,xG),pF0∶=(pG1Σ,…,pGnΣ),xG=xG0+∇pF̃(pF,xF),xG0∶=(xF,…,xF).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:
(2.61)∫e(i/ℏ)[F(p1,…,pk,x)+∑i=1k(Gi(πi1,…,πili,yi)-pi⋅yi)]∏i=1kdnpidnyi(2πℏ)n=e(i/ℏ)Φ(π11,…,π1l1,π21,…,π2l2…πk1,…,πklk,x)(C+O(ℏ)),
where C is some constant.Proof of Proposition2.14.
To simplify the computations, we identify(T*ℝd)n with T*(ℝdn) and (T*ℝd)ki with T*(ℝdki). With this identifications the graphs of F and Gi, i=1,…,n may be written as
(2.62)LF={((pF,∇pF(pF,xF)),(∇xF(pF,xF),xF)):(pF,xF)∈UF}⊂T*(Rdn)×T*Rd,LGi={((pGi,∇pGi(pGi,xGi)),(∇xGi(pGi,xGi),xGi)):(pGi,xGi)∈UGi}⊂T*(Rdki)×T*Rd,
where UF∈V(Bn0) and UGi∈V(Bki0) for i=1,…,n.
Consider now the composition,(2.63)LF(LG1,…,LGn)=LF∘(LG1⊗⋯⊗LGn).
First of all, observe that,
(2.64)LG∶=LG1⊗⋯⊗LGn={((pG,∇pG(pG,xG)),(∇xG(pG,xG),xG)):(pGi,xGi)∈UGi}LG⊂T*(Rd(k1+⋯+kn))×T*(Rdn).
Thus,
(2.65)LF∘LG=π((LG×LF)∩(T*Rd(k1+⋯+kn)×ΔT*Rdn×T*Rd))={((pG,∇pG(pG,xG)),(∇xF(pF,xF),xF)):xG=∇pF(pF,xF),pF=∇xG(pG,xG),(pG,xF)∈Ũ}LF∘LG⊂T*(Rd(k1+⋯+kn))×T*Rd,
where Ũ is the subset of (pG,xF)∈Bk1+⋯+kn such that the system,
(2.66)pF=∇xG(pG,xG),xG=∇pF(pF,xF),
has a unique solution (pF,xG) and such that (pGi,xGi)∈UGi, i=1,…,n, and (pF,xF)∈UF. Let us check that Ũ always exists and is a neighborhood of Bk1+⋯+kn0. To begin with, observe that for any (0,xF)∈Bn0 this system has the unique solution (0,∇pF(0,xF)). Set now,
(2.67)H(pG,xF,pF,xG)=(pF-∇xG(pG,xG)xF-∇pF(pF,xF)).
Thanks to the fact that G(0,x)=∑i=1nGi(0,x)=0 we get that the Jacobi matrix
(2.68)DpF,xGH(0,xf,0,∇pF(0,xF))=(id0-∇p∇pF(0,xF)id)
is invertible.
Thus, the implicit function theorem gives us the desired neighborhoodŨ of Bk1+⋯+kn0.
Now, takeϕ as defined in (2.54). The previous considerations tell us that ϕ is exactly defined on Ũ. Let us compute its graphs,
(2.69)Lϕ={((pG,∇pΦ(pG,xF)),(∇xΦ(pG,xF),xF):(pG,xF)∈Ũ}.
We have that
(2.70)∇pϕ(pG,xF)=∇pG(pG,xG)+∇xG(pG,xG)dxGdp+∇pF(pF,xF)dpFdp-pFdxGdp-dpFdpxG=∇pG(pG,xG).
Similarly, ∇xϕ(pG,xF)=∇xF(pF,xF). Thus, Lϕ=LF∘LG.
At last, let us check thatϕ∈𝒪lock1+⋯+kn. First of all, remember that
(2.71)F(pF,xF)=pFΣxF+F̃(pF,xF),G(pG,xF)=∑i=1npGiΣxGi+G̃(pG,xG).
Thus, we obtain immediately that
(2.72)ϕ(pG,xF)=pGΣxF+H(pG,xF),
where H is a function only defined on Ũ by the equations
(2.73)H(pG,xF)=G̃(pG,xG)+F̃(pF,xF)-∇pF̃(pF,xF)∇xG̃(pG,xG),pF=pF0+∇xG̃(pG,xG),pF0:=(pG1Σ,…,pGnΣ),xG=xG0+∇pF̃(pF,xF),xG0:=(xF,…,xF).
But now, if we set pG=0 then pF=0, xG=xG0+∇pF̃(0,xF) and H(0,xF)=0. Similarly, one easily checks that ∇pH(0,xF)=0We will call the operad𝒪Δ+𝒪loc  local cotangent Lagrangian operad over T*ℝd or for short the local Lagrangian operad when no ambiguities arise. The induced operad 𝒪loc will be called the local deformation operad of𝒪Δ.Associative Products in the Local Deformation Operad
We say that a generating functionS∈C∞(B2) satisfies the Symplectic Groupoid Associativity equation if for a point (p1,p2,p3,x)∈B3 sufficiently close to B30 the following implicit system for x¯,p¯,x̃ and p̃:
(2.74)x¯=∇p1S(p¯,p3,x),p¯=∇xS(p1,p2,x¯),x̃=∇p2S(p1,p̃,x),p̃=∇xS(p2,p3,x̃)
has a unique solution and if the following additional equation holds:
(2.75)S(p1,p2,x¯)+S(p¯,p3,x)-x¯p¯=S(p2,p3,x̃)+S(p1,p̃,x)-x̃p̃.
IfS also satisfies the Symplectic Groupoid Structure conditions, that is, if
(2.76)S(p,0,x)=S(0,p,x)=px,S(p,-p,x)=0
then S generates a Poisson structure
(2.77)α(x)=2(∇pk1∇pl2S(0,0,x))k,l=1d
on ℝd together with a local symplectic groupoid integrating it, whose structure maps are given by
(2.78)ϵ(x)=(0,x)unit map,i(p,x)=(-p,x)inverse map,s(p,x)=∇p2S(p,0,x)source map,t(p,x)=∇p1S(0,p,x)target map.
In this case, we callS 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.
S̃∈𝒪loc2 is a product in 𝒪loc if and only if S=S02+S̃ satisfies the Symplectic Groupoid Associativity equation.Proof.
We know thatS̃ is a product in 𝒪loc if and only if S=S02+S̃ is a product in 𝒪Δ+𝒪loc, that is, if and only if S(S,I)=S(I,S). Let us compute.
(2.79)S(S,I)(p1,p2,p3,x)=S∪I(p1,p2,p3,x¯1,x¯2)+S(p¯1,p¯2,x)-x¯1p¯1-p¯2x¯2=S(p1,p2,x¯1)+p3x¯2+S(p¯1,p¯2,x)-p¯1x¯1-p¯2x¯2
with
(2.80)p¯1=∇x1S∪I(p1,p2,p3,x¯1,x¯2)=∇xG(x¯),p¯2=∇x2S∪I(p1,p2,p3,x¯1,x¯2)=p3,x¯1=∇p1S(p¯1,p¯2,x),x¯2=∇p2S(p¯1,p¯2,x).
Then we get
(2.81)S(S,I)=S(p1,p2,x¯)+S(p¯,p3,x)-p¯x¯,x¯=∇p1S(p¯,p3,x),p¯=∇xS(p1,p2,x¯).
Similarly, we get
(2.82)S(I,S)=S(p2,p3,x̃)+S(p1,p̃,x)-p̃x̃,x̃=∇p2S(p1,p̃,x)p̃=∇xS(p2,p3,x̃).
Hence, S̃∈𝒪loc2(T*ℝd) is a product if and only if S02+S̃ 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)[S̃,S̃]=0. We may also still write a formula for the coboundary operator. But, as this time the compositions in 𝒪Δ+𝒪loc are not multilinear, we cannot develop the expression (1/2)[S̃,S̃] 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 eachF∈𝒪Δ1+𝒪loc1, we may associate a symplectomorphism ψF which is defined only on a neighborhood UF of B10 in T*ℝd and which fixes B01. The composition of two such ψG and ψF, which may always be defined on a possibly smaller neighborhood Ũ⊂UG of B10, is exactly ψF(G) where F(G) is the composition of F by G in the local Lagrangian operad.
We denote byF-1∈𝒪Δ1+𝒪loc1 the generating function of the (ψF)-1, that is, the generating function such that F(F-1)=F-1(F)=I. Two associative products S and S̃ will be called equivalent if
(2.83)S̃=F(S)(F-1,F-1)
for a certain F∈𝒪Δ1+𝒪loc1. It is clear that if S∈𝒪Δ1+𝒪loc1 is an associative product, then S̃ also is. The following questions naturally arises.Questions
IfS generates a local symplectic groupoid, does S̃ 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 byψF. As a consequence the induced Poisson structures on the base are the same, that is, (2.84)α(x)=∇p1∇p2S(0,0,x)=∇p1∇p2S̃(0,0,x).The following two Propositions prove these statements.Proposition 2.17.
LetF∈𝒪Δ1+𝒪loc1. The following implicit equations:
(2.85)x1=∇pF(p1,x2),p2=∇xF(p1,x2),
define a symplectomorphism ψF(p1,x1)=(p2,x2) on a neighborhood UF of B10={(0,x):x∈ℝd} in T*ℝd which fixes B10 and which is close to the identity in the sense that F(p,x)=px+F̃(p,x) induces the identity if F̃=0. Consider now ψF and ψG defined, respectively, on UF and UG for F,G∈𝒪Δ1+𝒪loc1. Then one has that ψG∘ψF=ψF(G) on UF(G).Proof.
(1) Let us check that the system (2.85) generates a diffeomorphism around B10. Namely, one verifies that (p¯1,x¯1,p¯2,x¯2):=(0,∇pF(0,x2),0,x2) is a solution of the system. Set now
(2.86)H(p1,x1,p2,x2)∶=(x1-∇pF(p1,x2)p2-∇xF(p1,x2)).
As
(2.87)Dp1,x1H(p¯1,x¯2,p¯2,x¯2)=(-∇p∇pF(0,x¯2)id∇x∇pF(0,x¯2)0),Dp2,x2H(p¯1,x¯2,p¯2,x¯2)=(0∇x∇pF(0,x¯2)id0),
the implicit function theorem gives us the result. Let us call Ũ the neighborhood of B10 where ψF is defined.
(2) We check now thatψF is symplectic. From (2.85) we get the relation
(2.88)∂pl2∂pk1=∂xk1∂xl2,
which directly implies that dψFJ(dψF)*=J where
(2.89)J=(0id-id0).
(3) Let us see thatψF(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+F̃(p,x) with ∇pF̃(0,p)=0 and then ∇x∇pF(0,x2)=x2.
(4) ClearlyF(p,x)=px generates the identity.
(5) Recall that(2.90)LG={(p1,∇pG(p1,x2),∇xG(p1,x2),x2):(p1,x2)∈UG},LF={(p2,∇pF(p2,x3),∇xF(p2,x3),x3):(p2,x3)∈UF}.
Thus, LG=graphψG and LF=graphψF. The composition of these two canonical relations yields that LF∘LG=graphψF∘ψG. On the other hand, LF∘LG=LF(G)=graphψF(G). Taking care on the domain of definitions, we have that ψF∘ψG=ψF(G) on UF(G).Proposition 2.18.
LetS∈𝒪Δ2+𝒪loc2 be a generating function of a symplectic groupoid, that is,
(2.91)S(S,I)=S(I,S),S(p,0,x)=S(0,p,x)=px,S(p,-p,x)=0.
Let F∈𝒪Δ1+𝒪loc1 such that F(-p,x)=-F(p,x). Then,
(2.92)S̃∶=(F(S))(F-1,F-1)
is also a generating function of a symplectic groupoid. The subset of odd function in p forms a subgroup of 𝒪Δ1+𝒪loc1. Moreover, ψF is a groupoid isomorphism between the local symplectic groupoid generated by S and the one generated by S̃. As a consequence S and S̃ induce the same Poisson structure on the base.Proof.
To simplify the notation, we setG=F-1. A straightforward computation gives that
(2.93)F(S)(G,G)(p1,p2,x)=S(p¯,p̃,ẋ)+F(ṗ,x)+G(p1,x¯)+G(p2,x̃)-p¯x¯-p̃x̃-ẋṗ,ẋ=∇pF(ṗ,x),x¯=∇p1S(p¯,p̃,ẋ),x̃=∇p2S(p¯,x̃,ẋ),ṗ=∇xS(p¯,p̃,ẋ),p¯=∇xG(p1,x¯),p̃=∇xG(p2,x̃).(1)
Settingp1=p and p2=0, we have immediately
(2.94)F(S)(G,G)(p,0,x)=G(p,ẋ)+F(ṗ,x)-ẋṗ
with ẋ=∇pF(ṗ,x) and ṗ=∇xG(p,ẋ). We recognize then that
(2.95)F(S)(G,G)(p,0,x)=F(G)(p,x)=I(p,x)=px.
The case p1=0 and p2=p is analog.(2)
One reads directly from the equation(2.96)px=F-1(p,ẋ)+F(ṗ,x)-ẋṗ,
where ẋ=∇pF(ṗ,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 𝒪Δ1+𝒪loc1.(3)
Suppose now thatp1=p and p2=-p. G odd in p implies that p¯=-p̃. As S(p,-p,0)=0, we get immediately that x̃=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 thatψF is also a groupoid isomorphism. Consider the multiplication space of the symplectic groupoid generated by an generating function S, that is,(2.97)G(m)(S)={(p1,∇p1S),(p2,∇p2S),(∇xS,x):p1,p2∈(Rd)*,x∈Rd},
where the partial derivative are evaluated in (p1,p2,x).
We have to show that(ψF×ψF×ψF)(G(m)(S))=G(m)(S̃).
A straightforward computation gives that(2.98)∇p1S̃(p1,p2,x)=∇pG(p1,x¯),∇p2S̃(p1,p2,x)=∇pG(p2,x̃),∇xS̃(p1,p2,x)=∇xF(ṗ,x).
From this, we check immediately that(2.99)ψG((p1,∇p1S̃(p1,p2,x)))=(p¯,∇p1S(p¯,p̃,ẋ)),ψG((p2,∇p2S̃(p1,p2,x)))=(p̃,∇p2S(p¯,p̃,ẋ)),ψF((∇xS(p¯,p̃,ẋ),ẋ))=(∇xS̃(p1,p2,x),x)
which ends the proof.Remark 2.19.
Suppose thatS is a generating function of a local symplectic groupoid. Let F∈𝒪Δ1+𝒪loc1 act on S, that is, S̃=(F(S))(F-1,F-1). Then, the condition S(p,0,x)=S(0,p,x)=px is preserved by any F∈𝒪Δ1+𝒪loc1. 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
(2.100)((-p2,∇p2S(p2,p1,x)),(-p1,∇p1S(p2,p1,x)),(-∇xS(p2,p1,x),x))∈G(m)(S),
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

LetF:ℝn*→ℝ and G:ℝn→ℝ be two smooth functions. Consider the point ϕ∈ℝ defined by(3.1)ϕ∶=G(x¯)+F(p¯)-p¯x¯,
where x¯ and p¯ are defined by the implicit equations(3.2)p¯=∇xG(x¯),x¯=∇pF(p¯).
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 (3.3)G(x)=p0x+∑i=1∞ϵiG(i)(x),F(p)=x0p+∑i=1∞ϵiF(i)(p),
equation (3.2) become (3.4)p¯=p0+∑i=1nϵi∇xG(i)(x¯),x¯=x0+∑i=1nϵi∇pF(i)(p¯),
which are always recursively uniquely solvable.Let us compute the first terms ofp¯, x¯ and ϕ to get a feeling of what is happening: (3.5)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+ϵ(G(1)(x0)+F(1)(p0))+ϵ22∇pF(1)(p0)∇xG(1)(x0)+⋯.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)
Agrapht is given by a set of vertices Vt={1,…,n) and a set of edges Et which is a set of pairs of elements of Vt. 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)}∈Et′ if {v,w}∈Et. 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)
Atree is a graph which has no cycles. Isomorphisms and symmetries are defined the same way as for graphs(3)
Arooted 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)
Abipartite graph is a graph t together with a map ω:Vt→{∘,•} such that ω(v)≠ω(w) if {v,w}∈Et. An isomorphism of bipartite trees is an isomorphism of graphs which respects the coloring, that is, ω(σ(v))=ω(v).(5)
Aweighted graph is a graph t together with a weight map L:Vt→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.Table1 summarizes some notations we will use in the sequel.Table 1TThe set of bipartite treesRTThe set of rooted bipartite treesRT∘The set of elements ofRT with white rootRT•The set of elements ofRT with black rootWe will give the nameCayley 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)
ift1,…,tm∈[RT∘]∞, then the tree [t1,…,tm]•i∈[RT]∞ where [t1,…,tm]•i is defined by connecting the roots of t1,…,tm 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 Section3.1.Definition 3.2.
Given two collections of functionsF={F(i)}i=1∞ and G={G(j)}j=1∞, where Fi:ℝn*→ℝ and Gj:ℝn→ℝ are smooth functions, we may associate to any rooted tree t∈[RT]∞ a vector field on T*ℝd, DCt(F,G)∈Vect(T*ℝd), called the elementary differential and a function on T*ℝd, Ct(F,G)∈C∞(T*ℝd), called the elementary function. (1)
Theelementary differentialDCt(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)
DCt(F,G)=∇x(m+1)G(i)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]∘i;(c)
DCt(F,G)=∇p(m+1)F(j)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]•j.(2)
Theelementary functionCt(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)
Ct(F,G)=∇x(m)G(i)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]∘i;(c)
Ct(F,G)=∇p(m)F(j)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]•j.  The notation ∇x(m) (resp. ∇p(m)) stands for the mth derivative in the direction x (resp. p).Some examples are given in Table2.Table 2DiagramElementary differentialElementary function∇x(2)G(i)∇pF(j)∇xG(i)∇pF(j)∇p(3)F(i)(∇xG(j),∇xG(k))∇p(2)F(i)(∇xG(j),∇xG(k))∇x(3)G(i)(∇pF(j),∇p(2)F(k)∇xG(l))∇x(2)G(i)(∇pF(j),∇p(2)F(k)∇xG(l))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.
Letu=[u1,…,uk],v=[v1,…,vl]∈[RT] (resp. ∈[RT]∞). Following [5], we define the Butcher product as follows:
(3.6)u∘v:=[u1,…,uk,[v1,…,vl]].
We have not written the obvious conditions on the ui's and the vi'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 setA 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 thatCt(F,G)=Ct′(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.
Lett=[t1,…,tm]∈[RT]∞. Consider the list t̃1,…,t̃k of all nonisomorphic trees appearing in t1,…,tm. Define μi as the number of time the tree t̃i appears in t1,…,tm. Then we introduce the symmetry coefficientσ(t) of t by the following recursive definition:
(3.7)σ(t)=μ1!μ2!…σ(t̃1)…σ(t̃k)
and initial condition σ(∘i)=σ(•j)=1.
It is clear thatσ(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 inϵ,
(3.8)G(x)=p0x+∑i=1∞ϵiG(i)(x),F(p)=x0p+∑j=1∞ϵjF(j)(p),
where G(i):ℝn→ℝn* and F(j):ℝn*→ℝn are smooth functions for i,j>0. Define ϕ(p0,x0)∈ℝ[[ϵ]] as
(3.9)ϕ(p0,x0)∶=G(x¯)+F(p¯)-p¯x¯,
where the formal power series x¯(ϵ) and p¯(ϵ) are uniquely determined by the implicit equations
(3.10)p¯=p0+∑i=1∞ϵi∇xG(i)(x¯),x¯=x0+∑j=1∞ϵj∇pF(j)(p¯).
Then, one has that
(3.11)ϕ(p0,x0)=p0x0+∑t∈T∞ϵ‖t‖|t|!Ct(F,G)(p0,x0).The proof of Proposition3.6 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 forp¯(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 forx¯(ϵ) and for p¯(ϵ) which satisfy (3.2). They are given by
(3.12)x¯(ϵ)=x0+∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G),(3.13)p¯(ϵ)=p0+∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G).Proof.
Uniqueness is trivial. Let us check that we have the right formal series. We only check (3.12). The other computation is similar.
(3.14)x¯(ϵ)=x0+∑i≥1ϵi∇pF(i)(p¯)=x0+∑i≥1ϵi∑m≥01m!∇p(m+1)F(i)(∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G),…,∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G))=x0+∑i≥1∑m≥0∑t1∈[RT∘]∞⋯∑tm∈[RT∘]∞ϵi+‖t1‖+⋯+‖tm‖m!σ(t1)…σ(tm)×∇p(m+1)F(i)(DCt1(F,G),⋯,DCtm(F,G))=x0+∑i≥1∑m≥0∑t1…∑tmϵ‖t‖m!σ(t)(μ1!μ2!…)DCt(F,G),witht=[t1,…,tm]•i=x0+∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G).Lemma 3.8.
One has the following expansion forϕ(p0,x0):
(3.15)ϕ(p0,x0)=p0x0+∑t∈[RT]∞ϵ‖t‖σ(t)Ct(F,G)-(∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G))(∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G)).Proof.
We compute the different terms arising inG(x¯)+F(p¯)-p¯x¯ in terms of trees.
(3.16)G(x¯)=p0x¯+∑i≥1ϵi∑m≥01m!∇x(m)G(i)(∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G),…,∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G))=p0x¯+∑i≥1∑m≥0∑t1∈[RT•]∞…∑tm∈[RT•]∞ϵ‖t‖m!σ(t)(μ1!μ2!…)×∇x(m)G(i)(DCt1(F,G),…,DCtm(F,G)),witht=[t1,…,tm]•i=p0x¯+∑t∈[RT∘]∞ϵ‖t‖σ(t)Ct(F,G).
By the same sort of computations we obtain(3.17)F(p¯)=x0p¯+∑t∈[RT•]∞ϵ‖t‖σ(t)Ct(F,G).
Finally, we get the desired result as
(3.18)p0x¯+x0p¯-p¯x¯=p0x0-(∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G))(∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G)).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.
Letu∈[RT∘]∞ and v∈[RT•]∞. Then,
(3.19)DCu(F,G)DCv(F,G)=Cu∘v(F,G)=Cv∘u(F,G).Proof.
Supposeu=[u1,…,um]∘i, v=[v1,…,vl]•j, then we get
(3.20)A=DCu(F,G)DCv(F,G)=∇x(m+1)G(i)(DCu1(F,G),…,DCum(F,G))⋅DCv(F,G)=∇x(m+1)G(i)(DCu1(F,G),…,DCum(F,G),DCv(F,G))=Cu∘v(F,G).Lemma 3.10.
Lett=(Vt,Et)∈T∞. For all v∈Vt let tv be the bipartite rooted tree (Vt,Et,v)∈RT∞. For v∈Vt and e={u,v}∈Et one has
(3.21)|sym(t)||sym(tv)|=|{v′∈Vttv′isisomorphictotv}|,|sym(t)||sym(tu)||sym(tv)|=|{e′∈Ettu′⊔tv′isisomorphictotu⊔tv}|.Proof.
Consider the induced action of the symmetry group of the tree on the set of vertices. Notice that two verticesv and w are in the same orbit if and only if tv is isomorphic to tw. Then the number of vertices of t which lead to rooted tree isomorphic to tv 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(tv)| 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(3.22)ϕ(p0,x0)=p0x0+∑t∈T∞ϵ‖t‖|t|!Ct(F,G).Proof.
Let us perform the last computation.(3.23)ϕ(p0,x0)=p0x0+∑t∈[RT]∞ϵ‖t‖σ(t)Ct(F,G)-∑u∈[RT∘]∞∑v∈[RT•]∞ϵ‖u‖+‖v‖σ(u)σ(v)DCu(F,G)DCv(F,G)=p0x0+∑t¯∈[T]∞ϵ|t¯|Ct¯(F,G){∑t∈t¯1|sym(t)|-∑u∈[RT•]∞,v∈[RT∘]∞u∘v∈t¯1|sym(u)∣∣sym(v)|}=p0x0+∑t∈T∞ϵ‖t‖|t|!Ct(F,G){∑v∈Vt|sym(t)||sym(tv)|1k(t,v),-∑e={u,v)∈Et|sym(t)||sym(tu)∣∣sym(tv)|1l(t,e)}
where k(t,v)=|{v′∈Vt/tv′isisomorphictotv}| and l(t,e)=|{e′∈Et/tu′⊔tv′isisomorphictotu⊔tv)|. 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 thatS is a formal power series we immediately get Proposition 3.6.


## 3.1. The Equation

LetF:ℝn*→ℝ and G:ℝn→ℝ be two smooth functions. Consider the point ϕ∈ℝ defined by(3.1)ϕ∶=G(x¯)+F(p¯)-p¯x¯,
where x¯ and p¯ are defined by the implicit equations(3.2)p¯=∇xG(x¯),x¯=∇pF(p¯).
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 (3.3)G(x)=p0x+∑i=1∞ϵiG(i)(x),F(p)=x0p+∑i=1∞ϵiF(i)(p),
equation (3.2) become (3.4)p¯=p0+∑i=1nϵi∇xG(i)(x¯),x¯=x0+∑i=1nϵi∇pF(i)(p¯),
which are always recursively uniquely solvable.Let us compute the first terms ofp¯, x¯ and ϕ to get a feeling of what is happening: (3.5)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+ϵ(G(1)(x0)+F(1)(p0))+ϵ22∇pF(1)(p0)∇xG(1)(x0)+⋯.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)
Agrapht is given by a set of vertices Vt={1,…,n) and a set of edges Et which is a set of pairs of elements of Vt. 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)}∈Et′ if {v,w}∈Et. 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)
Atree is a graph which has no cycles. Isomorphisms and symmetries are defined the same way as for graphs(3)
Arooted 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)
Abipartite graph is a graph t together with a map ω:Vt→{∘,•} such that ω(v)≠ω(w) if {v,w}∈Et. An isomorphism of bipartite trees is an isomorphism of graphs which respects the coloring, that is, ω(σ(v))=ω(v).(5)
Aweighted graph is a graph t together with a weight map L:Vt→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.Table1 summarizes some notations we will use in the sequel.Table 1TThe set of bipartite treesRTThe set of rooted bipartite treesRT∘The set of elements ofRT with white rootRT•The set of elements ofRT with black rootWe will give the nameCayley 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)
ift1,…,tm∈[RT∘]∞, then the tree [t1,…,tm]•i∈[RT]∞ where [t1,…,tm]•i is defined by connecting the roots of t1,…,tm 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 Section3.1.Definition 3.2.
Given two collections of functionsF={F(i)}i=1∞ and G={G(j)}j=1∞, where Fi:ℝn*→ℝ and Gj:ℝn→ℝ are smooth functions, we may associate to any rooted tree t∈[RT]∞ a vector field on T*ℝd, DCt(F,G)∈Vect(T*ℝd), called the elementary differential and a function on T*ℝd, Ct(F,G)∈C∞(T*ℝd), called the elementary function. (1)
Theelementary differentialDCt(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)
DCt(F,G)=∇x(m+1)G(i)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]∘i;(c)
DCt(F,G)=∇p(m+1)F(j)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]•j.(2)
Theelementary functionCt(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)
Ct(F,G)=∇x(m)G(i)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]∘i;(c)
Ct(F,G)=∇p(m)F(j)(DCt1(F,G),…,DCtm(F,G)) if t=[t1,…,tm]•j.  The notation ∇x(m) (resp. ∇p(m)) stands for the mth derivative in the direction x (resp. p).Some examples are given in Table2.Table 2DiagramElementary differentialElementary function∇x(2)G(i)∇pF(j)∇xG(i)∇pF(j)∇p(3)F(i)(∇xG(j),∇xG(k))∇p(2)F(i)(∇xG(j),∇xG(k))∇x(3)G(i)(∇pF(j),∇p(2)F(k)∇xG(l))∇x(2)G(i)(∇pF(j),∇p(2)F(k)∇xG(l))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.
Letu=[u1,…,uk],v=[v1,…,vl]∈[RT] (resp. ∈[RT]∞). Following [5], we define the Butcher product as follows:
(3.6)u∘v:=[u1,…,uk,[v1,…,vl]].
We have not written the obvious conditions on the ui's and the vi'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 setA 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 thatCt(F,G)=Ct′(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.
Lett=[t1,…,tm]∈[RT]∞. Consider the list t̃1,…,t̃k of all nonisomorphic trees appearing in t1,…,tm. Define μi as the number of time the tree t̃i appears in t1,…,tm. Then we introduce the symmetry coefficientσ(t) of t by the following recursive definition:
(3.7)σ(t)=μ1!μ2!…σ(t̃1)…σ(t̃k)
and initial condition σ(∘i)=σ(•j)=1.
It is clear thatσ(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 inϵ,
(3.8)G(x)=p0x+∑i=1∞ϵiG(i)(x),F(p)=x0p+∑j=1∞ϵjF(j)(p),
where G(i):ℝn→ℝn* and F(j):ℝn*→ℝn are smooth functions for i,j>0. Define ϕ(p0,x0)∈ℝ[[ϵ]] as
(3.9)ϕ(p0,x0)∶=G(x¯)+F(p¯)-p¯x¯,
where the formal power series x¯(ϵ) and p¯(ϵ) are uniquely determined by the implicit equations
(3.10)p¯=p0+∑i=1∞ϵi∇xG(i)(x¯),x¯=x0+∑j=1∞ϵj∇pF(j)(p¯).
Then, one has that
(3.11)ϕ(p0,x0)=p0x0+∑t∈T∞ϵ‖t‖|t|!Ct(F,G)(p0,x0).The proof of Proposition3.6 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 forp¯(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 forx¯(ϵ) and for p¯(ϵ) which satisfy (3.2). They are given by
(3.12)x¯(ϵ)=x0+∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G),(3.13)p¯(ϵ)=p0+∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G).Proof.
Uniqueness is trivial. Let us check that we have the right formal series. We only check (3.12). The other computation is similar.
(3.14)x¯(ϵ)=x0+∑i≥1ϵi∇pF(i)(p¯)=x0+∑i≥1ϵi∑m≥01m!∇p(m+1)F(i)(∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G),…,∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G))=x0+∑i≥1∑m≥0∑t1∈[RT∘]∞⋯∑tm∈[RT∘]∞ϵi+‖t1‖+⋯+‖tm‖m!σ(t1)…σ(tm)×∇p(m+1)F(i)(DCt1(F,G),⋯,DCtm(F,G))=x0+∑i≥1∑m≥0∑t1…∑tmϵ‖t‖m!σ(t)(μ1!μ2!…)DCt(F,G),witht=[t1,…,tm]•i=x0+∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G).Lemma 3.8.
One has the following expansion forϕ(p0,x0):
(3.15)ϕ(p0,x0)=p0x0+∑t∈[RT]∞ϵ‖t‖σ(t)Ct(F,G)-(∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G))(∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G)).Proof.
We compute the different terms arising inG(x¯)+F(p¯)-p¯x¯ in terms of trees.
(3.16)G(x¯)=p0x¯+∑i≥1ϵi∑m≥01m!∇x(m)G(i)(∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G),…,∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G))=p0x¯+∑i≥1∑m≥0∑t1∈[RT•]∞…∑tm∈[RT•]∞ϵ‖t‖m!σ(t)(μ1!μ2!…)×∇x(m)G(i)(DCt1(F,G),…,DCtm(F,G)),witht=[t1,…,tm]•i=p0x¯+∑t∈[RT∘]∞ϵ‖t‖σ(t)Ct(F,G).
By the same sort of computations we obtain(3.17)F(p¯)=x0p¯+∑t∈[RT•]∞ϵ‖t‖σ(t)Ct(F,G).
Finally, we get the desired result as
(3.18)p0x¯+x0p¯-p¯x¯=p0x0-(∑t∈[RT∘]∞ϵ‖t‖σ(t)DCt(F,G))(∑t∈[RT•]∞ϵ‖t‖σ(t)DCt(F,G)).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.
Letu∈[RT∘]∞ and v∈[RT•]∞. Then,
(3.19)DCu(F,G)DCv(F,G)=Cu∘v(F,G)=Cv∘u(F,G).Proof.
Supposeu=[u1,…,um]∘i, v=[v1,…,vl]•j, then we get
(3.20)A=DCu(F,G)DCv(F,G)=∇x(m+1)G(i)(DCu1(F,G),…,DCum(F,G))⋅DCv(F,G)=∇x(m+1)G(i)(DCu1(F,G),…,DCum(F,G),DCv(F,G))=Cu∘v(F,G).Lemma 3.10.
Lett=(Vt,Et)∈T∞. For all v∈Vt let tv be the bipartite rooted tree (Vt,Et,v)∈RT∞. For v∈Vt and e={u,v}∈Et one has
(3.21)|sym(t)||sym(tv)|=|{v′∈Vttv′isisomorphictotv}|,|sym(t)||sym(tu)||sym(tv)|=|{e′∈Ettu′⊔tv′isisomorphictotu⊔tv}|.Proof.
Consider the induced action of the symmetry group of the tree on the set of vertices. Notice that two verticesv and w are in the same orbit if and only if tv is isomorphic to tw. Then the number of vertices of t which lead to rooted tree isomorphic to tv 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(tv)| 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(3.22)ϕ(p0,x0)=p0x0+∑t∈T∞ϵ‖t‖|t|!Ct(F,G).Proof.
Let us perform the last computation.(3.23)ϕ(p0,x0)=p0x0+∑t∈[RT]∞ϵ‖t‖σ(t)Ct(F,G)-∑u∈[RT∘]∞∑v∈[RT•]∞ϵ‖u‖+‖v‖σ(u)σ(v)DCu(F,G)DCv(F,G)=p0x0+∑t¯∈[T]∞ϵ|t¯|Ct¯(F,G){∑t∈t¯1|sym(t)|-∑u∈[RT•]∞,v∈[RT∘]∞u∘v∈t¯1|sym(u)∣∣sym(v)|}=p0x0+∑t∈T∞ϵ‖t‖|t|!Ct(F,G){∑v∈Vt|sym(t)||sym(tv)|1k(t,v),-∑e={u,v)∈Et|sym(t)||sym(tu)∣∣sym(tv)|1l(t,e)}
where k(t,v)=|{v′∈Vt/tv′isisomorphictotv}| and l(t,e)=|{e′∈Et/tu′⊔tv′isisomorphictotu⊔tv)|. 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 thatS 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 onT*ℝd is the perturbative/formal version of the local cotangent operad on T*ℝd. Recall that in the latter the product for F∈𝒪Δn+𝒪locn and Gi∈𝒪Δn+𝒪locki, i=1,…n was expressed as in Proposition 2.14: (4.1)F(G1,…,Gn)(pG,xF)=G1∪⋯∪Gn(pG,xG)+F(pF,xF)-pF⋅xG,pF=∇xG1∪⋯∪Gn(pG,xG),xF=∇pF(pF,xF).
If we consider pG and xF as parameters in the previous equations, we have then that (4.2)G(pG,⋅):Rnd⟶R,F(⋅,xF):(Rnd)*⟶R.
Suppose now that the F and Gi, i=1,…,n, are formal series of the form (4.3)F(pF,xF)=pFΣ⋅xF+∑i=1∞ϵiF(i)(pF,xF),Gl(pGl,xGl)=pGlΣ⋅xGl+∑i=1∞ϵiGl(i)(pGl,xGl),
where (4.4)pΣ:=∑i=1npiforp=(p1,…,pn)∈(Rdn)*.
We may rewrite F and G as (4.5)F(pF,xF)=x0FpF+∑i=1∞ϵiF(i)(pF,xF),G(pG,xG)=p0GxG+∑i=1∞ϵiG(i)(pG,xG),
where x0F=(xF,…,xF)∈ℝdn and p0G=(pG1Σ,…,pGnΣ)∈(ℝnd)* for xG∈ℝdn and pF∈(ℝdn)*.Applying now Proposition3.6, we obtain for the compositions the following expansion:(4.6)F(G1,…,Gn)(pG,xG)=pGΣ⋅xF+∑t∈T∞ϵ‖t‖|t|!Ct(F(⋅,xF),G1∪…∪Gn(pG,⋅))(p0G,x0F).This motivates to define the formal deformation space of the cotangent Lagrangian operad𝒪Δ(T*ℝd) as (4.7)Oformn(T*Rd,Δ)∶={∑i=1∞ϵiF(i):F(i)∈Pin(T*Rd)},wherePin(T*ℝd) stands for the vector space of functions F:Bn→ℝ such that  (1)
F(p,x) is a polynomial in the variables p=(p1,…,pn),(2)
F(μp,x)=μi+1F(p,x).One may think of𝒪loc+𝒪form as the Taylor series of functions in 𝒪Δ+𝒪loc. The compositions are given by formula (4.6), which also tells us that 𝒪Δ+𝒪loc is an operad. The unit is (4.8)I(p,x)=px,I∈OΔ+Oform1.The induced operad structure on𝒪form is then given by(4.9)I∈Oform1,I(p,x)=0,Oformn={∑i=1∞ϵiF(i):F(i)∈Pin(T*Rd)},F(G1,…,Gn)(pG,xF)=∑t∈T∞ϵ‖t‖|t|!Ct(F,G1∪…∪Gn).
This operad will be called the formal deformation operad of the cotangent Lagrangian operad 𝒪Δ.


### 4.2. Product in the Formal Deformation Operad

Exactly as for the local deformation operad,Sϵ is a product in 𝒪form if and only if  S02+Sϵ satisfies formally the SGA equation. Moreover, if S02+Sϵ satisfies the SGS conditions, then S02+Sϵ is the generating function of a formal symplectic groupoid over ℝd.Again, the zero of𝒪form2 is a product in 𝒪form. We will stick to the conventions introduced for 𝒪loc. Namely, 01 will stand for the zero of 𝒪form1, which is also the identity of the operad and 02 will stand for the zero of 𝒪form2, 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 𝒪form 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 in𝒪form as follows: (4.10)[F,G]=F∘G-(-1)(k-1)(l-1)G∘F,
where (4.11)F∘G=∑i=1k(-1)(i-l)(l-1)F{01,…,01,G︸ith,01,…,01),
for F∈𝒪formk and G∈𝒪forml.We are now able to define a true coboundary operator.Proposition 4.1.
Considerd:𝒪formn→𝒪formn+1(4.12)dF∶=[02,F].
Then, d may be written as
(4.13)dF(p1,…,pn+1)=F(p1,…,pn,x)+∑j=1n(-1)n+j-1F(p1,…,pj+pj+1,…,pn,x)+(-1)n-1F(p2,…,pn+1,x).
Moreover, d is linear and d2=0.Proof.
For more clarity, let us break our convention and writeĨ instead of 01 and S̃ instead of 02. We have that [S̃,F]=S̃∘F-(-1)n-1F∘S̃. As S̃=0, only the trees ∘i and •j will contribute to the product. Then we have,
(4.14)I1=S̃∘F(p1,…,pn+1,x)=∑i≥1ϵi(C•i(S̃(⋅,x),F∪Ĩ(p,⋅))((∑1npl,pn+1),(x,x))+(-1)n-1C•i(S̃(⋅,x),Ĩ∪F(p,⋅))((p1,∑2n+1pl),(x,x)))=∑i≥1ϵi(F(i)(p1,…,pn,x)+(-1)n-1F(i)(p2,…,pn+1,x)),I2=F∘S̃(p1,…,pn+1)=∑j=1n(-1)j-1∑i≥1ϵiC∘i(F(⋅,x),(Ĩ∪⋯Ĩ∪S̃︸jth∪Ĩ⋯⋯∪Ĩ)(p,⋅))×((p1,…,pj+pj+1,…,pn+1),(x,…,x))=∑j=1n(-1)j-1∑i≥1ϵiF(i)(p1,…,pj+pj+1,…,pn+1,x),
which gives the desired formula. The check that d2=0 is straightforward.We have then a complex(4.15)(C•=⨁n≥0Oformn,d).
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 (4.16)Hn(C•)≃ϵVn(Rd)[[ϵ]],
where 𝒱n(ℝd) is the space of n-multi-vector fields on ℝd.We come now to the question of finding a productSϵ in the formal deformation operad of 𝒪Δ. This is exactly the same problem as deforming the trivial generating function S02 in 𝒪Δ+𝒪form. We are thus looking for an element Sϵ∈𝒪form2 of the form (4.17)Sϵ=ϵS1+ϵ2S2+⋯
such that(4.18)[Sϵ,Sϵ]=0.
Equation (4.18) becomes, on the level of trees,(4.19)∑t∈T∞ϵ‖t‖|t|!(Ct(Sϵ,Sϵ∪I)-Ct(Sϵ,I∪Sϵ))=0.One sees immediately that this equation is equivalent to the following infinite set of recursive equations:(4.20)dSn+Hn(Sn-1,…,S1)=0,
where (4.21)Hn(Sn-1,…,S1)=∑t∈T∞k,n2≤|k|≤n1|t|!(Ct(Sϵ,Sϵ∪I)-Ct(Sϵ,I∪Sϵ)),
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 structureα on ℝd, one has that
(4.22)Sϵ(α)=∑n=1∞ϵnn!∑Γ∈Tn,2WΓB̂Γ(α)
is a product in the formal deformation operad 𝒪form(T*ℝd,Δ) of the cotangent Lagrangian operad 𝒪Δ(T*ℝd). Moreover, Sϵ(α) is the unique natural product in 𝒪form(T*ℝd,Δ) whose first order is ϵα.In the above theorem, theTn,2 stand for the set of Kontsevich trees of type (n,2), WΓ is the Kontsevich weight of Γ and B̂Γ is the symbol of the bidifferential operator BΓ associated to Γ. We refer the reader to [2] for exact definitions of Kontsevich trees, weights, operators, and naturallity.We calledSϵ(α) the (formal)  symplectic groupoid generating function because, as shown in [2], it generates a “geometric object”, a (formal) symplectic groupoid over ℝd associated to the Poisson structure α whose structure maps are explicitly given by(4.23)ϵϵ(x)=(0,x)unit  map,iϵ(p,x)=(-p,x)inverse  map,sϵ(p,x)=x+∇p2Sϵ(α)(p,0,x)sourcex map,tϵ(p,x)=x+∇p1Sϵ(α)(0,p,x)target  map.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 Sϵ(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.


## 4.1. The Formal Cotangent Lagrangian Operad

The formal cotangent Lagrangian operad onT*ℝd is the perturbative/formal version of the local cotangent operad on T*ℝd. Recall that in the latter the product for F∈𝒪Δn+𝒪locn and Gi∈𝒪Δn+𝒪locki, i=1,…n was expressed as in Proposition 2.14: (4.1)F(G1,…,Gn)(pG,xF)=G1∪⋯∪Gn(pG,xG)+F(pF,xF)-pF⋅xG,pF=∇xG1∪⋯∪Gn(pG,xG),xF=∇pF(pF,xF).
If we consider pG and xF as parameters in the previous equations, we have then that (4.2)G(pG,⋅):Rnd⟶R,F(⋅,xF):(Rnd)*⟶R.
Suppose now that the F and Gi, i=1,…,n, are formal series of the form (4.3)F(pF,xF)=pFΣ⋅xF+∑i=1∞ϵiF(i)(pF,xF),Gl(pGl,xGl)=pGlΣ⋅xGl+∑i=1∞ϵiGl(i)(pGl,xGl),
where (4.4)pΣ:=∑i=1npiforp=(p1,…,pn)∈(Rdn)*.
We may rewrite F and G as (4.5)F(pF,xF)=x0FpF+∑i=1∞ϵiF(i)(pF,xF),G(pG,xG)=p0GxG+∑i=1∞ϵiG(i)(pG,xG),
where x0F=(xF,…,xF)∈ℝdn and p0G=(pG1Σ,…,pGnΣ)∈(ℝnd)* for xG∈ℝdn and pF∈(ℝdn)*.Applying now Proposition3.6, we obtain for the compositions the following expansion:(4.6)F(G1,…,Gn)(pG,xG)=pGΣ⋅xF+∑t∈T∞ϵ‖t‖|t|!Ct(F(⋅,xF),G1∪…∪Gn(pG,⋅))(p0G,x0F).This motivates to define the formal deformation space of the cotangent Lagrangian operad𝒪Δ(T*ℝd) as (4.7)Oformn(T*Rd,Δ)∶={∑i=1∞ϵiF(i):F(i)∈Pin(T*Rd)},wherePin(T*ℝd) stands for the vector space of functions F:Bn→ℝ such that  (1)
F(p,x) is a polynomial in the variables p=(p1,…,pn),(2)
F(μp,x)=μi+1F(p,x).One may think of𝒪loc+𝒪form as the Taylor series of functions in 𝒪Δ+𝒪loc. The compositions are given by formula (4.6), which also tells us that 𝒪Δ+𝒪loc is an operad. The unit is (4.8)I(p,x)=px,I∈OΔ+Oform1.The induced operad structure on𝒪form is then given by(4.9)I∈Oform1,I(p,x)=0,Oformn={∑i=1∞ϵiF(i):F(i)∈Pin(T*Rd)},F(G1,…,Gn)(pG,xF)=∑t∈T∞ϵ‖t‖|t|!Ct(F,G1∪…∪Gn).
This operad will be called the formal deformation operad of the cotangent Lagrangian operad 𝒪Δ.


## 4.2. Product in the Formal Deformation Operad

Exactly as for the local deformation operad,Sϵ is a product in 𝒪form if and only if  S02+Sϵ satisfies formally the SGA equation. Moreover, if S02+Sϵ satisfies the SGS conditions, then S02+Sϵ is the generating function of a formal symplectic groupoid over ℝd.Again, the zero of𝒪form2 is a product in 𝒪form. We will stick to the conventions introduced for 𝒪loc. Namely, 01 will stand for the zero of 𝒪form1, which is also the identity of the operad and 02 will stand for the zero of 𝒪form2, 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 𝒪form 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 in𝒪form as follows: (4.10)[F,G]=F∘G-(-1)(k-1)(l-1)G∘F,
where (4.11)F∘G=∑i=1k(-1)(i-l)(l-1)F{01,…,01,G︸ith,01,…,01),
for F∈𝒪formk and G∈𝒪forml.We are now able to define a true coboundary operator.Proposition 4.1.
Considerd:𝒪formn→𝒪formn+1(4.12)dF∶=[02,F].
Then, d may be written as
(4.13)dF(p1,…,pn+1)=F(p1,…,pn,x)+∑j=1n(-1)n+j-1F(p1,…,pj+pj+1,…,pn,x)+(-1)n-1F(p2,…,pn+1,x).
Moreover, d is linear and d2=0.Proof.
For more clarity, let us break our convention and writeĨ instead of 01 and S̃ instead of 02. We have that [S̃,F]=S̃∘F-(-1)n-1F∘S̃. As S̃=0, only the trees ∘i and •j will contribute to the product. Then we have,
(4.14)I1=S̃∘F(p1,…,pn+1,x)=∑i≥1ϵi(C•i(S̃(⋅,x),F∪Ĩ(p,⋅))((∑1npl,pn+1),(x,x))+(-1)n-1C•i(S̃(⋅,x),Ĩ∪F(p,⋅))((p1,∑2n+1pl),(x,x)))=∑i≥1ϵi(F(i)(p1,…,pn,x)+(-1)n-1F(i)(p2,…,pn+1,x)),I2=F∘S̃(p1,…,pn+1)=∑j=1n(-1)j-1∑i≥1ϵiC∘i(F(⋅,x),(Ĩ∪⋯Ĩ∪S̃︸jth∪Ĩ⋯⋯∪Ĩ)(p,⋅))×((p1,…,pj+pj+1,…,pn+1),(x,…,x))=∑j=1n(-1)j-1∑i≥1ϵiF(i)(p1,…,pj+pj+1,…,pn+1,x),
which gives the desired formula. The check that d2=0 is straightforward.We have then a complex(4.15)(C•=⨁n≥0Oformn,d).
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 (4.16)Hn(C•)≃ϵVn(Rd)[[ϵ]],
where 𝒱n(ℝd) is the space of n-multi-vector fields on ℝd.We come now to the question of finding a productSϵ in the formal deformation operad of 𝒪Δ. This is exactly the same problem as deforming the trivial generating function S02 in 𝒪Δ+𝒪form. We are thus looking for an element Sϵ∈𝒪form2 of the form (4.17)Sϵ=ϵS1+ϵ2S2+⋯
such that(4.18)[Sϵ,Sϵ]=0.
Equation (4.18) becomes, on the level of trees,(4.19)∑t∈T∞ϵ‖t‖|t|!(Ct(Sϵ,Sϵ∪I)-Ct(Sϵ,I∪Sϵ))=0.One sees immediately that this equation is equivalent to the following infinite set of recursive equations:(4.20)dSn+Hn(Sn-1,…,S1)=0,
where (4.21)Hn(Sn-1,…,S1)=∑t∈T∞k,n2≤|k|≤n1|t|!(Ct(Sϵ,Sϵ∪I)-Ct(Sϵ,I∪Sϵ)),
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 structureα on ℝd, one has that
(4.22)Sϵ(α)=∑n=1∞ϵnn!∑Γ∈Tn,2WΓB̂Γ(α)
is a product in the formal deformation operad 𝒪form(T*ℝd,Δ) of the cotangent Lagrangian operad 𝒪Δ(T*ℝd). Moreover, Sϵ(α) is the unique natural product in 𝒪form(T*ℝd,Δ) whose first order is ϵα.In the above theorem, theTn,2 stand for the set of Kontsevich trees of type (n,2), WΓ is the Kontsevich weight of Γ and B̂Γ is the symbol of the bidifferential operator BΓ associated to Γ. We refer the reader to [2] for exact definitions of Kontsevich trees, weights, operators, and naturallity.We calledSϵ(α) the (formal)  symplectic groupoid generating function because, as shown in [2], it generates a “geometric object”, a (formal) symplectic groupoid over ℝd associated to the Poisson structure α whose structure maps are explicitly given by(4.23)ϵϵ(x)=(0,x)unit  map,iϵ(p,x)=(-p,x)inverse  map,sϵ(p,x)=x+∇p2Sϵ(α)(p,0,x)sourcex map,tϵ(p,x)=x+∇p1Sϵ(α)(0,p,x)target  map.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 Sϵ(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.



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*Source: 643605-2011-01-09.xml*