Given a symplectic manifold , we may define an operad structure on the the spaces of the Lagrangian submanifolds of via symplectic reduction. If 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 () is a deformation of the trivial symplectic groupoid structure of .

1. Introduction

Symplectic groupoids, in the extended symplectic category, may be thought as the analog of associative algebras in the category of vector spaces. For the latter, a deformation theory exists and is well known. In this paper, we will present a conceptual framework as well as an explicit deformation of the trivial symplectic groupoid over . 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 . Our guideline will be the Kontsevich deformation of the usual algebra of functions over , . Namely, the usual pointwise product of functions generates a suboperad, the product suboperad, , of the endomorphism operad of , where is the -multilinear map defined by . For each one may choose the vector subspace of -multidifferential operators. The operad structure of induces an operad structure on , which in turn generates an operad structure on which is, however, nonlinear. Then, is a deformation of the usual product , that is, an element such that is still an associative product, if is a product in the induced deformation operad . We may also consider the formal version by replacing by the formal power series in , . Kontsevich in [1] gives an explicit formal deformation of the product of functions over , where the 's are the Kontsevich weights and the 's are the Kontsevich bidifferential operators associated to the Kontsevich graphs of type (see [2] for a brief introduction).

If we consider the trivial symplectic groupoid over , we see that the multiplication space generates an operad , where is a product in this operad. The compositions are given by symplectic reduction as the 's are Lagrangian submanifolds of . The main difference with the vector space case is that there is no “true” endomorphism operad where would naturally embed into. Thus, the question of finding a deformation operad for must be taken with more care. The first remark is that the may be expressed in terms of generating functions Namely, . 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 for each such that remains an operad. The formal version of it gives a surprising result. Namely, we may find an explicit deformation of the trivial generating function , it is given by the formula where the are the Kontsevich weights and the are the symbols of the Kontsevich bidifferential operators and the sum is taken over all Kontsevich trees . This formula may be seen as the semi-classical part of Kontsevich deformation quantization formula.

As a last comment, note that Kontsevich derives its star product formula from a more general result. In fact, he shows that , where for , is an -morphism from the multivector fields to the multidifferential operators on . In our perspective, we may still write summing over Kontsevich trees instead of Kontsevich graphs and replacing multidifferential operators by their symbols. Exactly, as in Kontsevich case, is an associative deformation of the generating function of the trivial symplectic groupoid . However, it is still not completely clear how to define “semi-classical -morphisms”.

Organization of the Paper
In Section 2, we describe the endomorphism operad associated to any object in a monoidal category. We explain what is an associative product on in a monoidal category and we define the product suboperad of . If the category is further associative, we may choose a deformation operad for , which is a choice, for each of a vector subspace such that is still an operad. We describe the deformations of in terms of products in . 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 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 . We conclude this section by defining equivalence between deformations of the trivial generating function and we show that two equivalent deformations induce the same local symplectic groupoid.
In Section 3, we describe the combinatorial tools needed to give a formal version of the local Lagrangian operad. As the problem consists mainly in taking Taylor's series of some implicit equations we need devices to keep track of all terms to all orders. The crucial point is that these implicit equations, describing the composition in the local Lagrangian operad, have a form extremely close to a special Runge-Kutta method: the partitioned implicit Euler method. We borrow then some techniques form numerical analysis of ODEs to make the expansion at all orders.
In the last section, we describe the formal Lagrangian operad, which is the perturbative version of the local one, in terms of composition of bipartite trees. We give in particular the product equation in the formal deformation operad in terms of these trees. At last, we restate the main theorem of [2] in this language. This tells us that the semi-classical part of Kontsevich star product on is a product in the formal deformation operad of the cotangent Lagrangian operad in 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 .
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 . 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 , 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 in the microsymplectic category contains, as a suboperad, the local lagrangian operad constructed in the present paper (see [8]).
However, there is no formal version of the microsymplectic category to date, and the combinatorics presented here to deal with the compositions in the formal lagrangian operad over have no equivalent in terms of a "formal microsymplectic category"; this is, at the time of writing, still a work in progress.

2. Product in the Extended Symplectic Category

2.1. Basic Constructions and Kontsevich Deformation

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

Definition 2.1. An operad consists of (1)a collection of sets , , (2)composition laws satisfying the following associativity relations: (3)a unit element such that . It usually also requires some equivariant action of the symmetric group. We do not require this here.

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

Product in a Monoidal Category
We consider here a monoidal category . We denote by the product bifunctor and by the neutral object. Let us recall that we have the following canonical isomorphisms: for all .
Let be a monoidal category and let an object . We define the endomorphism operad of in the following way: (1), (2), (3)the unit is given by .
The operad axioms follow directly from the bifunctoriality of , that is,
If is an object of a monoidal category , we may define a product on .

Definition 2.2. An associative product (In [9], Gerstenhaber and Voronov call it a multiplication.) on an operad is an element such that . An associative product on is an associative product in the endomorphism operad . In the sequel, we will constantly use the term product to mean in fact associative product.
Given a product , the associativity of the operad implies that, for any , and we have,
This notion is the natural generalization of an associative product on a vector space. Namely, if is a vector space, is the set of bilinear maps on . As in this case , we have that is an associative product on .

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

Remark 2.3. The product operad is a suboperad of but not a linear suboperad, namely, for each , is not a linear subspace of (it contains only a single point).

Definition 2.4. Let be an object of an additive monoidal category and let be a product. A deformation operad, , for is the data, for each , of a linear subspace such that the difference is in for all , , and .

Remark 2.5. is a suboperad of but not a linear one: the spaces are not linear subspaces but affine ones.

Proposition 2.6. Let be a deformation operad for a product . Then the compositions defined by (2.7) gives together with the unit the structure of an operad.

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

Remark 2.7. Although each of its degrees is a linear subspace, is not a linear operad since its compositions, the s, are not multilinear.

Definition 2.8. We say that an element is a deformation of the product w.r.t. the deformation operad if is still a product in .

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

Proposition 2.10. Let be a product. Take an element . Then, is a deformation of the product if and only if is a product in . In particular, is always a product in the deformation operad of .

Proof. is a deformation of if and only if which is equivalent to

From now on, we will write for the identity element of the deformation operad which is the zero of and for the trivial product of the deformation operad which is the 0 element in .

Notice that neither nor 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 are vector spaces, but the induced operad compositions are not linear in general.

We may however introduce the Gerstenhaber bracket of the deformation operad defined by where This bracket is not bilinear. An important fact concerning this bracket is that, which means that is a product in the deformation operad if and only if

Moreover, we may define an equivalent of the Hochschild differential

It turn out that is still a coboundary operator.

Proposition 2.11. defined by (2.17) is a coboundary operator, that is, . Moreover, satisfies product equation , in if and only if

Proof. Using (2.7) we obtain in terms of the endomorphism compositions The result follows directly from the linearity of the compositions in the endomorphism operad. Using again (2.7) we get which gives (2.18).

A formal deformation of is a formal power series where is a formal parameter and is a deformation operad for , such that is a product in .

Equivalently, one may say that must satisfy or, thanks to (2.18) that the 's satisfy at each order the following recursive equation: where

The Kontsevich Product Deformation
Consider the category of real vector spaces. In this category we take the real vector space of smooth functions on . The endomorphism operad of is The usual product of functions induces a product in , namely, for and .
The induced product operad is where As deformation operad, we take The induced coboundary operator on is the Hochschild coboundary operator, and the product equation is nothing but the usual Maurer-Cartan equation.
Kontsevich in [1] shows that there exists a formal deformation of . He provides the explicit formula for this deformation where the are the Kontsevich graphs of type , is their associated weight, and is their associated bidifferential operator (and [1] for more precisions).

2.2. Monoidal Structure of

Let us recall that the extended symplectic “category” is given by where denotes the symplectic manifold with opposite symplectic structure . The identity morphism of is the diagonal The composition of two morphisms and is given by the composition of canonical relations Everything works fine except the fact that the composition may fail to be a Lagrangian submanifold of . It is always the case when intersects cleanly (see [10] for more precisions).

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

We define the tensor product between two objects and of as the Cartesian product and the tensor product between morphisms as for and .

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

Proposition 2.12. The following statements hold. (1)Consider , , and . Then one has the following equality of sets: (2) for any object and .(3) for any objects , and . (4) for any arrows , and .(5) for all object and for all arrows , where means that the two sets and are in bijection.

Proof. (1)

(3) The associativity between objects is trivial.
(4) For morphisms, we have,
(5) is trivial.

2.3. Lagrangian Operads

If were a true category, we could consider the endomorphism operad of a symplectic manifold . However, we may be able to restrict to a subset of Lagrangian submanifolds for each such that the composition yields always a Lagrangian submanifold in for every and , . For instance, there is always the trivial choice

In this way, we may get a true operad .

The next natural question to ask is the following.

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

As a first hint, take the situation where the symplectic manifold is a symplectic groupoid . In this case, we may generate an operad from the multiplication space and the base , the identity being the diagonal . Remark that is a product in this operad, that is, that . Notice that the inverse of the symplectic groupoid does not play any role in this construction.

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

Local Cotangent Lagrangian Operads
Remember that has always a structure of a symplectic groupoid over : the trivial one. The multiplication space is given in this case by The base is If we set further it immediate to see that the operad generated by and is exactly and that is a product in it.

Following [3], we will call this operad the cotangent Lagrangian operad over . 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 into. The idea now is to enlarge the cotangent Lagrangian operad, that is, by considering Lagrangian submanifolds close enough to for each in order to have still an operad.

Notice at this point that the 's are given by generating functions. Namely, we may identify with , where . Then, where is the function on defined by (In the sequels, we will use the shorter notation instead of .) The cotangent Lagrangian operad may then be identified with In order to define a deformation operad for , a natural idea would be to consider Lagrangian submanifolds whose generating functions are of the form where . The Lagrangian submanifold associated to is 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: will stand for the set of all neighborhoods of in .

Definition 2.13. We define to be the space of germs at of smooth functions (defined on an open neighborhood of ) which satisfy and . Note that the composition will always be understood in terms of composition of germs.

Proposition 2.14. Let and for . Consider the function defined by the formula where and , and , for .
Then, In other words, together with the product is an operad.
Moreover, the induced operad structure on is given by where is the function defined by

Remark 2.15 (Saddle point formula). Formula (2.54) for can be interpreted in terms of saddle point evaluation for of the following integral: where is some constant.

Proof of Proposition 2.14. To simplify the computations, we identify with and with . With this identifications the graphs of and , may be written as where and for .
Consider now the composition, First of all, observe that, Thus, where is the subset of such that the system, has a unique solution and such that , , and . Let us check that always exists and is a neighborhood of . To begin with, observe that for any this system has the unique solution . Set now, Thanks to the fact that we get that the Jacobi matrix is invertible.
Thus, the implicit function theorem gives us the desired neighborhood of .
Now, take as defined in (2.54). The previous considerations tell us that is exactly defined on . Let us compute its graphs, We have that Similarly, . Thus, .
At last, let us check that . First of all, remember that Thus, we obtain immediately that where is a function only defined on by the equations But now, if we set then , and . Similarly, one easily checks that

We will call the operad   local cotangent Lagrangian operad over or for short the local Lagrangian operad when no ambiguities arise. The induced operad will be called the local deformation operad of   .

Associative Products in the Local Deformation Operad
We say that a generating function satisfies the Symplectic Groupoid Associativity equation if for a point sufficiently close to the following implicit system for and : has a unique solution and if the following additional equation holds:
If also satisfies the Symplectic Groupoid Structure conditions, that is, if then generates a Poisson structure on together with a local symplectic groupoid integrating it, whose structure maps are given by
In this case, we call 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. is a product in if and only if satisfies the Symplectic Groupoid Associativity equation.

Proof. We know that is a product in if and only if is a product in , that is, if and only if . Let us compute. with Then we get Similarly, we get Hence, is a product if and only if 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 . We may also still write a formula for the coboundary operator. But, as this time the compositions in are not multilinear, we cannot develop the expression in terms of the coboundary operator. Nevertheless, in Section 4, we will develop the bracket with help of Taylor's expansion and recover a form very close to (2.23) in the additive category case.

Equivalence of Associative Products
To each , we may associate a symplectomorphism which is defined only on a neighborhood of in and which fixes . The composition of two such and , which may always be defined on a possibly smaller neighborhood of , is exactly where is the composition of by in the local Lagrangian operad.
We denote by the generating function of the , that is, the generating function such that . Two associative products and will be called equivalent if for a certain . It is clear that if is an associative product, then also is. The following questions naturally arises.

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

In fact, two equivalent associative products, which are also generating functions of local symplectic groupoids, induce isomorphic local symplectic groupoids. The isomorphism is given explicitly by . As a consequence the induced Poisson structures on the base are the same, that is,

The following two Propositions prove these statements.

Proposition 2.17. Let . The following implicit equations: define a symplectomorphism on a neighborhood of in which fixes and which is close to the identity in the sense that induces the identity if . Consider now and defined, respectively, on and for . Then one has that on .

Proof. (1) Let us check that the system (2.85) generates a diffeomorphism around . Namely, one verifies that is a solution of the system. Set now As the implicit function theorem gives us the result. Let us call the neighborhood of where is defined.
(2) We check now that is symplectic. From (2.85) we get the relation which directly implies that where
(3) Let us see that . We have already noticed that is a solution of the system (2.85). But with and then .
(4) Clearly generates the identity.
(5) Recall that