Camille Laurent-Gengoux
email: camille.laurent-gengoux@univ-lorraine.fr

Friedrich Wagemann
email: wagemann@math.univ-nantes.fr

Lie rackoids integrating Courant algebroids

We construct an infinite dimensional Lie rackoid Y which hosts an integration of the standard Courant algebroid. As a set, Y = C ∞ ([0, 1], T * M ) for a compact manifold M . The rackoid product is by automorphisms of the Dorfman bracket. The first part of the article is a study of the Lie rackoid Y and its tangent Leibniz algebroid a quotient of which is the standard Courant algebroid. In a second part, we study the equivalence relation related to the quotient on the rackoid level and restrict then to an integrable Dirac structure. We show how our integrating object contains the corresponding integrating Weinstein Lie groupoid in the case where the Dirac structure is given by a Poisson structure.

Introduction

Courant algebroids, introduced by Courant [START_REF] Courant | Dirac manifolds[END_REF] in 1990, permit to treat Poisson structures and presymplectic structures on an equal footing. Courant algebroids are not Lie algebroids, but contain lots of Lie algebroids, namely the Lie algebroids associated to Dirac structures. Courant and Dirac geometry gained much interest during the last 20 years and have become central subjects in Poisson geometry.

Courant algebroids are not a classical differential geometric structure, as they are not related to Lie groups or Lie algebras, but rather to Leibniz algebras. A Courant algebroid carries a bracket which can be either expressed as an antisymmetric bracket which does not satisfy the Jacobi identity (Courant bracket [START_REF] Courant | Dirac manifolds[END_REF]), or as a non-antisymmetric bracket satisfying the Leibniz identity (Dorfman bracket [START_REF] Dorfman | Dirac structures of integrable evolution equations[END_REF]). Therefore classical integration theory for Lie algebras or Lie algebroids does not apply, and the search for a global differential geometric object integrating a given Courant algebroid was wide open until lately.

Mehta-Tang [START_REF] Mehta | Xiang From double Lie groupoids to local Lie 2-groupoids[END_REF], Li Bland-Severa [START_REF] Li-Bland | Pavol Integration of exact Courant algebroids[END_REF] and Sheng-Zhu [START_REF] Sheng | Chenchang Higher Extensions of Lie Algebroids and Application to Courant Algebroids[END_REF] attacked the problem of integrating Courant algebroids and found solutions in the realm of graded differential geometry. It is not easy to work with these solutions and in particular, the authors did not give hints about how to extract from their global objects the Lie groupoids integrating integrable Dirac structures in the Courant algebroid -concerning the latest progress in this direction, see [START_REF] Mehta | Xiang Constant symplectic 2-groupoids[END_REF]. We believe that the integrating object which we propose in the present article remedies this and reveals the Lie groupoids corresponding to integrable Dirac structures more easily, to the cost of working with an unusual, but natural differential geometric structure.

In the present article, we introduce an object integrating the standard Courant algebroid on TM = T M ⊕ T * M on a compact manifold M . As it comes from an integration of the underlying Leibniz algebroid TM and in some sense Leibniz algebras can be integrated into Lie racks in the same way as Lie algebras can be integrated into Lie groups, we dubbed it a Lie rackoid, see [10] where this notion appears for the first time.

In order to prepare the basic facts about Lie rackoids, compare our article [10]. We give there the basic definitions concerning Lie rackoids and show that the tangent object of a Lie rackoid is a Leibniz algebroid and every Lie groupoid has an underlying Lie rackoid. The intuition which goes with Lie rackoids is that in the same way as a rack retains from the structure of a group only the conjugation (forgetting the group product), a Lie rackoid can be viewed as a Lie groupoid where one retains only the conjugation. It is thus natural that the product structure in a Lie rackoid brings bisections of a (pre)category into play.

In a nutshell, a Lie rackoid is a smooth precategory Y t / / s / / M with a rack product on its bisections and a compatible rack action of bisections on elements of Y . The bisections of a precategory over a compact manifold M form a Fréchet manifold (see [START_REF] Schmeding | Christoph The Lie group of bisections of a Lie groupoid[END_REF], [10] Proposition 3.3) and the rack product on bisections is supposed to be smooth with respect to this manifold structure and the rack action is supposed to be by diffeomorphisms of Y . Our basic object in the present article is the Lie rackoid of (smooth) cotangent maps Y = {a : [0, 1] → T * M }. The precategory Y has as its source and target the projections to the starting-and end-point of the path a. The bisections of Y are of the form Σ = (φ, η) where φ 0 = id M and φ 1 is some diffeomorphism of M , and η is a path of cotangent vectors. The matching between attachment points is such that φ * s η s is a 1-form on M for each s ∈ [0, 1]. We denote

β Σ = 1 0
φ * s η s ds the 1-form on M naturally attached to the bisection Σ. The rack product on bisections reads then for T = (ψ, ζ)

Σ ⊲ T = (φ -1 1 • ψ t • φ 1 , φ * 1 (ζ t -i ψt (φ - 1
1 ) * dβ Σ )), while the rack action of a bisection on an element a = (γ, θ) of Y is given by Σ ⊲ a = (φ -1 1 (γ), φ * 1 (θ -i γt (φ -1 1 ) * dβ Σ )).

Note that the group of automorphisms of the Dorfman bracket of the standard Courant algebroid on TM is the semidirect product of diffeomorphisms and closed 2-forms [START_REF] Bursztyn | Marco Reduction of Courant algebroids and generalized complex structures[END_REF], and such an automorphism (φ 1 , dβ Σ ) acts exactly in the way displayed above. The Lie rackoid Y is an infinite-dimensional Lie rackoid (first main theorem: Theorem 2.12) whose infinitesimal Leibniz algebroid A admits a quotient isomorphic to the standard Courant algebroid (Proposition 3.7). In Section 5, we define an equivalence relation giving rise to a Fréchet foliation (Corollary 5.5) which corresponds to the passage to the quotient on the global level. In our second main theorem, we show that the Lie rack of bisections Bis(Y ) factors to a quotient rack Bis(Y )/ ∼ which has a manifold structure and the correct tangent space in (id, 0) ∈ Bis(Y ), see Theorem 5.7.

We will call a subbundle D ⊂ TM a Dirac structure if it is maximally isotropic and its sections are closed under the Dorfman bracket. In this case, the Dorfman bracket reduces to a Lie bracket on the section of the subbundle D and D becomes a Lie algebroid. We call a Dirac structure D integrable if the underlying Lie algebroid is integrable, i.e. integrates into a Lie groupoid. The identification of the integrable Dirac structures (inside A and its quotient) and the corresponding Lie groupoids (inside Y and its quotient) takes the following form, see Section 5.2. Given a Dirac structure D ⊂ TM , there are subobjects

A(D) ⊂ A and Y D ⊂ Y defined by A(D) m := {(X t , α t ) ∈ A m | ( Ẋt , α t ) ∈ D m ∀t ∈ [0, 1]} and Y D := {(γ t , α t ) ∈ Y | ( γ, α) ∈ D γ(t) ∀t ∈ [0, 1]}.
We strongly believe that the identification of integrable Dirac structures inside our Leibniz algebroid A and of their associated Lie groupoids inside our Lie rackoid work for a general integrable Dirac structure D, but for the moment, we can show these only for Dirac structures coming from integrable Poisson structures on M . For this class of Dirac structures, we show that Y D is a subrackoid (Proposition 5.12) and that with respect to the above equivalence relation, Y D / ∼ is a Lie rackoid isomorphic to the Lie rackoid underlying the symplectic Lie groupoid (smooth Weinstein groupoid) integrating the Dirac structure D (Corollary 5.15 and Proposition 5.19).

The link between D and Y is facilitated by the intrinsic symplectic geometry of Y , see Section 4. In terms of the symplectic 2-form ω on Y , inherited from T * M , the 2-form dβ Σ is just the pull-back of ω by the bisection Σ, viewed as a section of the source map. thank furthermore Christoph Wockel who has been part of this project for some time for his insights and contributions.

The path rackoid

We refer to [10], Section 3 for the basic definitions concerning Lie rackoids. The goal of the present section is to define the path Lie rackoid which is part of the Lie rackoid of cotangent paths integrating the standard Courant algebroid. The path rackoid serves here as an introduction to the rackoid of cotangent paths.

Let M be a finite dimensional compact manifold. Let X be the set of smooth paths γ : [0, 1] → M .Then X is a smooth Fréchet manifold with respect to the C ∞ -topology (with left resp. right derivatives of all orders in 0 ∈ [0, 1] resp. 1 ∈ [0, 1]), see [START_REF] Michor | Manifolds of differentiable mappings[END_REF] Theorem 10.4, p. 91. It becomes a smooth precategory introducing source and target maps s, t : X → M which send a path γ to its starting point γ(0) and its end point γ(1) respectively. The unit map associates to m ∈ M the constant path in m.

A Lie rackoid structure (see [10] for definitions) is a structure on the bisections of a smooth pre-category. Therefore we determine in the following lemma the set of bisections Bis(X). Note that Bis(X) carries a smooth Fréchet manifold structure obtained as in Proposition 3.3 of [10]. Indeed, by topologizing path spaces putting systematically the factor [0, 1] to the left (using the bijection from the exponential law

C ∞ (A × B, C) = C ∞ (A, C ∞ (B, C)))
, one deals only with finite-dimensional compact manifolds.

Lemma 1.1. The Fréchet manifold of smooth bisections Bis(X) of the precategory s, t :

X → M is Bis(X) = {γ : [0, 1] → C ∞ (M, M ) | γ 0 = id M and γ 1 diffeomorphism }. Proof. A bisection of the precategory Y is by definition a map σ : M → Y such that s•σ = id M and t•σ is a diffeomorphism of M . Let us view maps from M to C ∞ ([0, 1], M ) = X as maps from [0, 1] to C ∞ (M, M ), i.e. as paths in C ∞ (M, M )
under this identification. The requirements on such a path γ in C ∞ (M, M ) to be a bisection are clearly that γ 0 = id M and that γ 1 is a diffeomorphism of M .

A Lie rackoid is, roughly speaking (see Definition 3.5 in [10]) a smooth precategory X together with a smooth rack product on its manifold of bisections and a smooth rack action on X itself.

In the case at hand, we define on Bis(X) a rack product for two bisections ϕ and ψ by (ψ ⊲ ϕ) t,m = ψ -1 1 ϕ t,ψ1(m) , for all t ∈ [0, 1] and all m ∈ M , and a rack action for a bisection ψ and an element γ ∈ X by

(ψ ⊲ γ)(t) = (t → ψ -1 1 (γ(t))).
Let us check the self-distributivity relation for three bisections ϕ, ψ and χ:

(χ ⊲ (ψ ⊲ ϕ)) t,m = χ -1 1 ψ -1 1 ϕ t,ψ1(χ1(m)) = χ -1 1 ψ -1 1 χ 1 χ -1 1 ϕ t,χ1χ -1 1 ψ1χ1(m) = ((χ ⊲ ψ) ⊲ (χ ⊲ ϕ)) t,m .
This shows that X becomes a (unital, infinite dimensional) Lie rackoid.

Proposition 1.2. There is a surjective morphism of rackoids from X to the rackoid underlying the fundamental groupoid associating to a path its homotopy class.

The Lie rackoid of cotangent paths

In this section, we define the central object of this article, namely the Lie rackoid of cotangent paths. This Lie rackoid and a quotient of it are the object we propose for an integration of the standard Courant algebroid. The Lie rackoid which we have constructed in the previous section constitues the first component of the Lie rackoid of cotangent paths.

Definition of the Lie rackoid of cotangent paths

Let M be a finite dimensional compact manifold and let π : T * M → M denote its cotangent bundle. Consider the set of smooth cotangent paths in M , i.e.

Y = {(γ, η) : [0, 1] → T * M | η smooth}.
Here the actual cotangent path is η and η contains the base path γ = π • η. For better readability of the upcoming formulae, the splitting into base path and (total) cotangent path maybe helpful. The set Y is a smooth Fréchet manifold with respect to the C ∞ -topology and a smooth precategory over M with respect to the compositions π • ev 0 and π • ev 1 of the evaluation in 0 and 1 with the bundle projection π.

Again, in order to define a Lie rackoid structure, we determine the manifold of bisections Bis(Y ) of the pre-category Y .

Lemma 2.1. The bisections Bis(Y ) are the pairs Σ = (φ, η) where the first component φ is an element of the path Lie rackoid. The second component s → η s is a section of the bundle φ * t T * M to which we can associate a path of 1-forms s → ηs on M by setting

T m M ∋ X → ηs (X) := η s,m (T m φ s (X)).
Proof. The first part is very similar to the proof of Lemma 1.1, the second part is immediate from the definitions. Remark 2.2. We denote the 1-form ηs by φ * s η s . Observe that this is an abuse of notation. In fact, the value of η s at m ∈ M , i.e. η s,m , is already a cotangent vector at φ s (m), but the tangent vector X at m has to be taken to φ s (m) by T m φ s . Thus η s is somehow the halfway pullback φ * s η s . This is a ntural operation on the bundle φ * t T * M . In order to define the Lie rackoid, we now define a rack action for a bisection Σ acting on a cotangent path a ∈ Y . Composing a with the bundle projection π : T * M → M gives an underlying path γ := π • a, [0, 1] ∋ s → γ s , on M with cotangent vectors θ s attached to γ s for all s ∈ [0, 1]. We let the bisection Σ act on the given cotangent path (γ, θ) according to the following formula:

Σ ⊲ a = (φ -1 1 (γ), φ * 1 (θ -i γt (φ -1 1 ) * d 1 0 φ * s η s ds )). ( 1
)
Remark 2.3. Let us comment on the algebraic form of the bracket operation. The bracket in Equation ( 1) is composed of the natural action by diffeomorphisms on the base path and on the cotangent path θ. Added to this is the gauge transformation using a certain exact 2-form d 1 0 φ * s η s ds. To understand this better, observe that there is a map ϕ :

Y → C ∞ ([0, 1], TM ) which sends el- ements of (γ, θ) ∈ Y to ( γ, θ) ∈ C ∞ ([0, 1], TM ), where TM = T M ⊕ T * M
is the extended tangent bundle of M . Recall the fact (see [START_REF] Bursztyn | Marco Reduction of Courant algebroids and generalized complex structures[END_REF], Proposition 2.3) that the automorphism group of the standard Courant algebroid on the extended tangent bundle TM is the semidirect product Ω 2 cl (M ) ⋊ Diff(M ) of Diff(M ) and Ω 2 cl (M ) where diffeomorphism and closed 2-form act exactly in this way. We therefore conclude that the action in Equation ( 1) is sent under ϕ to the natural action of an automorphism of the standard Courant algebroid.

Definition 2.4. The 1-form 1 0 φ * s η s ds on M is canonically associated to the bisection Σ = (φ, η). We denote it by definition as

β Σ := 1 0 φ * s η s ds.
Lemma 2.5. Formula (1) defines an element of Y , i.e. a cotangent path.

Remark 2.6. There is a similar rackoid product on tangent paths given by φ ⊲ a = (φ 1 (γ), (φ -1 1 ) * (θ)). This is in some sense the opposite of the above rack product without the cocycle term. It is easy to see that the associated Leibniz algebroid (see Theorem 5.3 in [10]) to this Lie rackoid is the vector bundle with typical fiber

A m = {a : [0, 1] → T m M | a(0) = 0}
and whose ancre map is the evaluation in 1. The associated Leibniz bracket on the sections reads

[a(t), b(t)] = [a(1), b(t)].
This is closely related to the generalized Poisson bracket on the dual of a Leibniz algebra of Dhérin-Wagemann [START_REF] Dhérin | Friedrich Deformation quantization of Leibniz algebras[END_REF].

For the Lie rackoid structure on Y , we will also need a rack product on Bis(Y ). In fact, it can be derived from the above rack action in Equation ( 1) by acting with a bisection on the bisection associated to the corresponding element of Y . This means that Formula (1) induces a formula of the bisection acting on another bisection by viewing a bisection as a subset of Y and applying the first formula point-by-point to the elements of Y constituing it. This procedure permits in general to derive from some kind of action on a precategory the corresponding kind of product on the bisections.

Lemma 2.7. The formula for the action of a bisection on a bisection associated to Formula ( 1) is

Σ ⊲ T = (φ -1 1 • ψ t • φ 1 , φ * 1 (ζ t -i ψt (φ -1 1 ) * dβ Σ )), (2)
where Σ = (φ, η), T = (ψ, ζ) and β Σ the associated 1-form of Σ (cf Definition 2.4).

Proof. Let us discuss separately first and second component of the bisection.

Concerning the first component, we have to show that when we view

ψ t in φ -1 1 • ψ t • φ 1 not as a family of maps in C ∞ (M, M ), but rather as a subset of π • Y = C ∞ ([0, 1], M
) and we apply to each point individually the formula

(φ ⊲ γ)(t) = (t → φ -1 1 (γ(t))), then we obtain φ -1 1 • ψ t • φ 1 . To do this, look at ψ ∈ C ∞ ([0, 1], C ∞ (M, M )) = C ∞ (M, C ∞ ([0, 1], M )) as a subset {ψ m } m∈M ⊂ Y = C ∞ ([0, 1], M
). An individual ψ m may be viewed as a path γ in the above formula which we may therefore apply. Observe that in order to compare the result, we must take in

φ ⊲ {ψ m } m∈M = {φ -1 1 (ψ m (t))} m∈M,t∈[0,1]
the path starting in m ∈ M , i.e. φ -1 1 (ψ φ1(m) )(t), and this is exactly the path in

C ∞ (M, M ) which is described by φ -1 1 • ψ t • φ 1 .
In the same way, we also treat the second component. It is clear that the cotangent vectors ζ t , brought back to the path φ -1 1) along the path. Concerning the second factor of the second component, the terms are similar, with the exception of i ψt which requires interpretation. Indeed, ψt is seen as associating to each t ∈ [0, 1] a tangent vector to C ∞ (M, M ) at ψ(t), i.e. as a family of tangent vectors to the path ψ t . The operation φ * 1 (i ψt (φ -1 1 ) * (-)) renders it then the insertion of a tangent vector to the path φ

1 • ψ t • φ 1 via φ * 1 , correspond to the cotangent vectors φ * 1 θ in Equation (
β Σ⊲T = φ * 1 β T -φ * 1 1 0 ψ * s i ψs (φ -1 1 ) * dβ Σ ds.
Proof. This is straight forward:

1 0 (φ -1 1 • ψ s • φ 1 ) * φ * 1 (ζ s -i ψs (φ -1 1 ) * dβ Σ )ds = φ * 1 β T -φ * 1 1 0 ψ * s i ψs (φ -1 1 ) * dβ Σ ds.

Selfdistributivity of the rackoid product

Now we show that the rackoid product proposed in Formula (1) satisfies the self-distributivity equation

Σ ⊲ (T ⊲ a) = (Σ ⊲ T ) ⊲ (Σ ⊲ a).
This implies then that it defines a Lie rackoid structure on the set of cotangent paths. For this, we need the following lemma:

Lemma 2.9. Let ψ t be a smooth path of maps in C ∞ (M, M ) with ψ 0 and ψ 1 diffeomorphisms and µ ∈ Ω 1 (M ). Then we have

d 1 0 ψ * s i ψs dµ ds = ψ * 1 dµ -ψ * 0 dµ.
Proof . We have

d 1 0 ψ * s i ψs dµ ds = 1 0 d(ψ * s i ψs dµ) ds.
Let us show that dψ * s i ψs dµ = d ds ψ * s dµ.

For this, we may suppose that µ = f dg and therefore that dµ = df ∧ dg. With this expression, we obtain

ψ * s dµ = d(f • ψ s ) ∧ d(g • ψ s ),
and then

d ds ψ * s dµ = d ds (d(f • ψ s ) ∧ d(g • ψ s )) = d(T ψs f ( ψs )) ∧ d(g • ψ s ) + d(f • ψ s ) ∧ d(T ψs g( ψs )) = dψ * s (df ( ψs ))dg + (dg( ψs ))df = dψ * s i ψs dµ.
This implies then

d 1 0 ψ * s i ψs dµ ds = 1 0 d ds ψ * s dµ ds. = ψ * 1 dµ -ψ * 0 dµ.
This lemma permits us to find an easy expression for the de Rahm differential of the canonical 1-form β Σ⊲T of Σ ⊲ T described in Lemma 2.8 in terms of the 1-forms β Σ of Σ and β T of T :

Corollary 2.10. The 2-form dβ Σ⊲T corresponding to the rack product of Σ and T has the following expression:

dβ Σ⊲T = φ * 1 dβ T -φ * 1 ψ * 1 (φ -1 1 ) * dβ Σ + dβ Σ .
Proof. By Lemmas 2.8 and 2.9, we have with µ

= (φ -1 1 ) * β Σ and ψ 0 = id M dβ Σ⊲T = φ * 1 dβ T -φ * 1 d 1 0 ψ * s i ψs (φ -1 1 ) * dβ Σ ds. = φ * 1 dβ T -φ * 1 ψ * 1 ((φ -1 1 ) * dβ Σ ) -ψ * 0 ((φ -1 1 ) * dβ Σ ) = φ * 1 dβ T -φ * 1 ψ * 1 (φ -1 1 ) * dβ Σ + dβ Σ .
Remark 2.11. Observe that the expression for the 2-form dβ Σ⊲T in Corollary 2.10 is the transformation of a closed 2-form under conjugation in the semidirect product Ω 2 cl (M ) ⋊ Diff(M ), cf Remark 2.3. The following theorem is the first main results of this article.

Theorem 2.12. Formula (1) defines a rack action of Bis(Y ) on Y , i.e. for all bisections Σ and T in Bis(Y ) and all a ∈ Y , we have Recall also that the expression φ * s η s is a 1-form and the notations

Σ ⊲ (T ⊲ a) = (Σ ⊲ T ) ⊲ (Σ ⊲ a).
β Σ = 1 0 φ * s η s ds, and β T = 1 0 ψ * s ζ s ds.
With these notations, we have:

T ⊲ a = (ψ -1 1 (γ), ψ * 1 (α -i γt (ψ -1 1 ) * dβ T )), Σ ⊲ a = (φ -1 1 (γ), φ * 1 (α -i γt (φ -1 1 ) * dβ Σ )), and Σ ⊲ T = (φ -1 1 • ψ t • φ 1 , φ * 1 (ζ t -i ψt (φ -1 1 ) * dβ Σ )).

The Leibniz algebroid of cotangent paths

Here we will explore the infinitesimal/tangent Leibniz algebroid of Y , the Leibniz algebroid of cotangent paths. The general procedure associating to a Lie rackoid a tangent Leibniz algebroid can be found in Section 5.2 of [10].

Definition of the Leibniz algebroid

The Lie rackoid Y = {γ : [0, 1] → T * M } has the underlying precategory the set of cotangent paths with the source s and target t maps to M induced by the starting-and endpoint projections. Its tangent Leibniz rackoid is by definition the kernel of T s, i.e.

A := Γ * ([0, 1], T M ⊕ T * M ).
A has as its underlying vector bundle the bundle of (pointed) paths over

I = [0, 1] in the extended tangent bundle TM = T M ⊕ T * M . It is an infinite- dimensional vector bundle over M . The fiber at m ∈ M is Γ * ([0, 1], T m M ⊕ T * m M
), the space of (pointed) paths in the vector space T m M ⊕ T * m M . The anchor in the Leibniz algebroid A is by definition the restriction of T t, which is just the evaluation map (X t , α t ) → X 1 (up to a sign).

Let us compute the infinitesimal bracket with respect to the above Lie rackoid product given in Formula (1). It is a bracket on the space of sections of A, namely on

Γ(A) =: Y = Γ * ([0, 1], Vect(M ) ⊕ Ω 1 (M )).
Thus elements of Y are pairs (X t , α t ) of a 1-parameter family of vector fields X t with X 0 = 0 and a 1-parameter family of 1-forms α t . The restriction on X t comes from the fact that the family of diffeomorphisms φ s satisfies φ 0 = id M .

Lemma 3.1. The infinitesimal bracket corresponding to Formula (1) reads

[(X t , α t ), (Y t , β t )] = ([X 1 , Y t ], L X1 β t -i Ẏt d 1 0 α s ds). (3)
Proof. We apply the following procedure. Replace φ and ψ in Formula (2) by paths φ s and ψ s depending smoothly on a new parameter s ∈ [-1, 1] such that at s = 0, both are the constant path in id M , and ∂ ∂s | s=0 φ s = X and

| s=0 ∂ ∂u | u=0 (φ u 1 ) -1 • ψ s t • φ u 1 = [X 1 , Y t ].
In the same vein, the first summand is by definition the Lie derivative with respect to X 1 . In the second summand of the second component, observe that by the product rule only the term with i Ẏt survives, and in this term, φ 0 1 = id M .

Lemma 3.2. The bracket in Equation ( 3) is a Leibniz bracket.

Proof . In view of Lemma 3.1, the claim does not need a proof. However, we will give a direct proof for the sake of transparency.

The (left) Leibniz identity is clear for the first component. In the second component, we have for [(X t , α t ), [(Y t , β t ), (Z t , γ t )]],

L X1 L Y1 γ t -i Żt d 1 0 β s ds -i [Y1, Żt] d 1 0 α s ds, for [[(X t , α t ), (Y t , β t )], (Z t , γ t )], L [X1,Y1] γ t -i Żt d 1 0 (L X1 β s + i Ẏs d 1 0 α u du)ds, and for [(Y t , β t ), [(X t , α t ), (Z t , γ t )], L Y1 L X1 γ t -i Żt d 1 0 α s ds -i [X1, Żt] d 1 0 β s ds.
We recollect the terms corresponding to α, β and γ which must cancel separately.

The fact that the Lie derivative is a representation takes care of the first terms in these three expressions:

L X1 L Y1 γ t = L [X1,Y1] γ t + L Y1 L X1 γ t .
The compatibility of the Lie derivative and the insertion operator (i.e. [L U , i V ] = i [U,V ] ) and the identity dL X = L X d show that

L X1 i Żt d 1 0 β s ds = i Żt d 1 0 L X1 β s ds + i [X1, Żt] d 1 0 β s ds. Therefore, it remains to show i [Y1, Żt] d 1 0 α s ds = i Żt d 1 0 i Ẏs d 1 0 α u du ds + L Y1 i Żt d 1 0 α s ds.

This last equality boils down (using again [L

U , i V ] = i [U,V ] and L U d = di U d) to 1 0 i Ẏs d 1 0 α u du ds = i Y1 d 1 0 α u du, which follows from 1 0 Ẏs ds = Y 1 -Y 0 with the restriction Y 0 = 0.

The exponential map

The link of the infinitesimal picture with the global picture can be performed via the exponential map. The following lemma records formulae for the iterates of the Leibniz bracket: Lemma 3.3. We have the following formulae for the iteration of the bracket introduced in Lemma (3.1):

(a) ad n+1 (X,α) (Y, β) = ad n+1 X (Y ), L n+1 x β - n k=0 n + 1 k i L k X Y dL n-k X α , (b)
e ad (X,α) (Y, β) = e adX (Y ), e LX β - ∞ n=-1 n k=0 i L X k! Y d L n-k X (n -k + 1)! α , = e adX (Y ), e LX β -i e L X Y d e LX -1 L X α . Remark 3.4. Recall the formula (Φ X t ) * (f ) = n≥0 t n X n (f ) n!
relating the flow Φ X t of a real analytic vector field X and the exponential of its action on a real analytic function f which is also valid for the action on other real analytic geometric objects on M . This formula follows from showing that the right hand side satisfies the same initial value problem

(Φ X t ) ′ = X • Φ X t , Φ X 0 = id M
as the flow of X. For this, one inverts sum and derivative thanks to normal convergence, which is true on a compact manifold M . This formula shows that the first component of e ad (X,α) (Y, β) is in fact (Φ X t ) * Y , as expected.

The quotient standard Courant algebroid

In this section, we pass to a quotient of the Leibniz algebroid A = Γ * ([0, 1], T M ⊕ T * M ) with respect to a subalgebroid of ideals such that the quotient is isomorphic as Leibniz algebroids to the standard Courant algebroid. Define the following subalgebroid in A:

I := (X t , α t ) | X 1 = 0 and 1 0 α s ds = 0 .
Observe that I is certainly a subbundle of the vector bundle A, as all requirements are C ∞ (M )-linear. That it is a subalgebroid will follow from the next lemma:

Lemma 3.5. The space of sections I := Γ(I) is a (two-sided) Leibniz ideal contained in the left center of Y = Γ(A).

Proof. It is obvious from the formula for the Leibniz bracket on the space of sections of A that have [I, Y] = 0, thus I is contained in the left center of the Leibniz algebra Y.

On the other hand, for (X t , α t ) ∈ Y and (Y t , β t ) ∈ I, we have

[(X t , α t ), (Y t , β t )] = ([X 1 , Y t ], L X1 β t -i Ẏt d 1 0 α s ds) with ([X 1 , Y t ])| t=1 = 0 and 1 0 (L X1 β t -i Ẏt d 1 0 α s ds) dt = 0.
Indeed, the first factor in the integral is zero, because 1 0 β t dt = 0, and the second factor in the integral is zero, because ω := 1 0 α s ds does not depend on t, and thus

1 0 i Ẏt dω dt = i Y1 dω -i Y0 dω = 0.
Remark 3.6. Observe that the squares of the Leibniz algebra Y are not included in I. Therefore the quotient is not necessarily a Lie algebra (as expected). Indeed, it is easily computed that the squares are of the form

([X 1 , X t ], L X1 α t -i Ẋt d 1 0 α s ds),
and, while [X 1 , X t ]| t=1 = 0, the integral of the form-part gives Proof. We claim that the map of vector bundles

1 0 (L X1 α t -i Ẋt d 1 0 α s ds) dt = L X1 ( 1 0 α s ds) -i X1 d( 1 0 α s ds) = d i X1 1 0 α s ds .
ϕ = (ev 1 , 1 0 (-) dt) : A → TM, (X t , α t ) → (X 1 , 1 0 α t dt)
is a morphism of Leibniz algebroids. Observe that the first component of ϕ is the anchor of A -this shows the compatibility of ϕ with the anchors.

The compatibility with the bracket is also immediate. Let (X t , α t ) and (Y t , β t ) be elements of Y and denote by ω :=

([(X t , α t ), (Y t , β t )]) = ([X 1 , Y 1 ], L X1 ζ -i X1 dω) = [ϕ(X t , α t ), ϕ(Y t , β t )].
It is clear that the kernel of ϕ is the subalgebroid I from Lemma 3.5.

Remark 3.8. We have also a section for the map ϕ := (ev 1 ,

1 0 (-)dt) : A → TM . It consists simply of the map (X, α) → (X t : t → tX, α) for t ∈ [0, 1].
It is easily computed that this section is compatible with the Leibniz bracket.

4 Symplectic geometry of the cotangent rackoid

Canonical 1-form and symplectic form

In this section, we interpret the form dβ Σ = d 1 0 φ * s η s ds associated to a bisection Σ = (φ, η) in terms of a symplectic form. This is not mysterious as it is clear that the manifold of cotangent paths carries a symplectic structure inherited from the cotangent bundle.

Observe that there are the following three geometric objects defined on

Y = C ∞ ([0, 1], T * M ):
(a) the function c : (γ, α) → The 2-form ω is a symplectic form, thus the symplectic manifold Y is exact. Indeed, for any symplectic manifold S, the Fréchet manifold C ∞ ([0, 1], S) is symplectic for the form obtained by integrating the pointwise form along [0, 1], and in case S is exact, the resulting symplectic manifold C ∞ ([0, 1], S) is exact as well, cf [START_REF] Cattaneo | Giovanni Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model[END_REF], [START_REF] Kotov | Thomas Dirac sigma models[END_REF].

The above three objects induce similar objects on the manifold of bisections Bis(Y ), and the 1-form λ on a bisection Σ = (φ, η) is the 1-form β Σ which we introduced before. Denote by s Σ : Σ → M the restriction of the source map s : Y → M to the bisection Σ ⊂ Y and similarly for the target map t : Y → M . View Σ as a section of the source map denoted s -1 Σ : M → Y .

(t -1 Σ ) * λ is (φ -1 1 ) * β Σ .
Proof. Recall that we view bisections of Y , i. e. elements in

C ∞ (M, C ∞ ([0, 1], T * M )),
rather as elements of C ∞ ([0, 1], C ∞ (M, T * M )), i.e. as paths in C ∞ (M, T * M ). This is expressed in the notation (φ t , η t ). In order to perform the pullback, however, we have to view them as families of elements of Y , i.e. for a fixed m ∈ M , we obtain an element of Y which we denote accordingly (φ t (m), η t (m)) =: (γ t , α t ). We have to compute the expression of λ at such a point. In fact, it is immediate that the expression λ (γ,α) =

1 0 α γ(t) (δγ t ) dt com- puted at a point (φ t (m), η t (m)) =: (γ t , α t ) gives β Σ = 1 0 φ * t η t dt.
The only thing to note is that for all tangent vectors X ∈ T m M , we have

(s -1 Σ ) * (α γ(t) (δγ t ))(X) = η t,m (T m φ t (X)).
The other claims of the lemma follow immediately from (s -1

Σ ) * λ = β Σ .
Lemma 4.2. The three objects c, β Σ and dβ Σ on Y and Bis(Y ) respectively satisfy the following equivariance relations with respect to the rackoid product:

(a) the function c is invariant: The condition that a Lie groupoid is symplectic is much more restrictive than the condition that a Lie rackoid is symplectic. The former condition means that for the symplectic form ω on the symplectic groupoid Γ → M , we have

c (γ,α) = c Σ⊲(γ,α) , (b)
β Σ⊲T = φ * 1 β T -φ * 1 1 0 ψ * s i ψs (φ -1 1 ) * dβ Σ ds, (c) dβ Σ⊲T = φ * 1 dβ T -φ * 1 ψ * 1 (φ -1 1 ) * dβ Σ + dβ Σ . (4
0 = pr * 1 ω -comp * ω + pr * 2 ω,
where for i = 1, 2, pr i : Γ × M Γ → Γ are the two projections and comp : Γ × M Γ → Γ is the composition. The fact that the relation contains three terms implies for example that the identity bisection ǫ : M → Γ is isotropic. For a Lie rackoid, we have the following less restrictive property:

Lemma 4.5. In a symplectic Lie rackoid Y , the isotropic bisections form a subrack.

Proof. This follows from the property (c) above (resp. Lemma 2.10) together with dβ Σ = 0 and dβ T = 0.

Proposition 4.6. The Lie rackoid associated to a symplectic Lie groupoid Γ → M is a symplectic Lie rackoid.

Proof. We refer to Proposition 5.1 in [10] for the Lie rackoid underlying a Lie groupoid.

The multiplicativity of the symplectic form in a symplectic Lie groupoid Γ → M can be expressed as

ω γ1γ2 (a • b, c • d) = ω γ1 (a, c) + ω γ2 (b, d),
where γ 1 , γ 2 ∈ Γ are composable, a, b, c, d are tangent vectors to Γ in γ 1 resp. γ 2 , and (a, c) → a • c is the product on the tangent groupoid associated to Γ.

Denoting by * the group product on bisections of Γ, we obtain thus

ω(Σ * T (u), Σ * T (v)) = ω(Σ(u) * T ((φ 1 ) * (u)), Σ(v) * T ((φ 1 ) * (v))) = ω(Σ(u), Σ(v)) + ω(T ((φ 1 ) * (u)), T ((φ 1 ) * (v))),
for two tangent vectors u, v. Thanks to the fact that dβ Σ is the pullback of the symplectic form on Y , this relation for all u, v translates into

dβ Σ * T = dβ Σ + φ * 1 dβ T .
From this, we deduce first of all for the inverse bisection Σ

-1 of Σ 0 = dβ Σ * Σ -1 = dβ Σ + φ * 1 dβ Σ -1 , which implies then dβ Σ -1 = -(φ -1 1 ) * dβ Σ . With this, we compute dβ Σ * T * Σ -1 = dβ Σ + φ * 1 dβ T * Σ -1 = dβ Σ + φ * 1 (ψ * 1 dβ Σ -1 + dβ T ) = dβ Σ -φ * 1 ψ * 1 (φ -1 1 ) * dβ Σ + φ * 1 dβ T
This gives for the Lie groupoid, viewed as Lie rackoid: Proof. For this, we first notice that the bracket of two elements in Γ(A(D)) may be written as an integral over [0, 1]: We can also define the corresponding notion on the global level. Let

dβ Σ⊲T = dβ Σ + φ * 1 dβ T -φ * 1 ψ * 1 (φ -1 1 ) * dβ Σ .

Identifying Dirac structures inside

[(X s , α s ), (Y t , β t )] = ([X 1 , Y t ], L X1 β t -i Ẏt d 1 0 α s ds) = ([ 1 0 Ẋs ds, Y t ], 1 0 L Ẋs β t ds -i Ẏt d 1 0 α s ds) = 1 0 ([ Ẋs , Y t ], L Ẋs β t -i Ẏt dα s ) ds. Next, the integrand ([ Ẋs , Y t ], L Ẋs β t -i Ẏt dα s ) is in D m at each point,
Y D := {(γ t , α t ) ∈ Y | ( γ, α) ∈ D γ(t) ∀t ∈ [0, 1]}.
It is then clear that Y D is a subprecategory of Y . The properties of Y D will occupy a large space in the second half of Section 5.

Integration of the standard Courant algebroid

To integrate the standard Courant algebroid, we introduce an equivalence relation on the level of bisections Bis(Y ) and on the total space Y . The equivalence relation is rather difficult to handle, but well behaved at the point (id, 0) ∈ Bis(Y ). The corresponding infinitesimal Leibniz algebroid at (id, 0) is isomorphic to the standard Courant algebroid. In the second half of this section, we study the behaviour of the equivalence relation on Y D for Dirac structures D coming from integrable Poisson structures. We show that the corresponding quotient gives its integrating symplectic Lie groupoid.

The equivalence relation on Bis(Y ) and Y

The equivalence relation is the following. In order to formulate it, recall that we had shown for a bisection Σ that β Σ = (s -1 Σ ) * λ and dβ Σ = (s -1 Σ ) * ω. In the following, we will deal with families of bisections {Σ t } t∈[0,1] . For such a family, we introduce the notation β = Σ * λ = (s -1

Σ ) * λ ∈ Ω 1 ([0, 1] × M ) and ω = Σ * ω = d β ∈ Ω 2 ([0, 1] × M ), where Σ : [0, 1] × M → Y is the map (ǫ, m) → (s → (φ Σǫ s (m), η Σǫ s (m)).
On the other hand, we will view Σ ǫ either as the member at ǫ of the family

{Σ t } t∈[0,1] or as the map Σ ǫ : [0, 1] → Bis(Y ). Denote by u = ∂ ∂t . Let us define i ǫ : M → M × [0, 1] by m → (m, ǫ). We shall study the relation Σ ∼ Σ ′ :⇔ φ 1 = φ ′ 1 and (5)
∃ {Σ t } t∈[0,1] with Σ 0 = Σ, Σ 1 = Σ ′ and i * ǫ i u ω = 0.
With loss of generality, we can assume that Σ t is constant in some neighborhoods of 0 and 1 which allows to glue such families and thus implies that ∼ is an equivalence relation.

Lemma 5.1. (a) The condition i * ǫ i u ω = 0 is equivalent to i * ǫ d β(u) = ∂βΣ ǫ ∂ǫ . (b) The condition i * ǫ i u ω = 0 is equivalent to ω Y ∂Σ t ∂t (m), T Σ t (δm) = 0
for every tangent vector δm ∈ T m M .

(c) The condition i * ǫ i u ω = 0 is equivalent to the condition that ∂Σt ∂t is in the kernel of the map

T Σ C ∞ (M, Y ) ⊃ T Σ Bis(Y ) → Ω 1 (M ), δΣ → T β(δΣ) -i * ǫ d λ, δΣ . Proof. (a) Recall the map i ǫ : M → M × [0, 1]
given by m → (m, ǫ). We have

d • i * ǫ = i * ǫ • d where one de Rham differential d is on M and the other on M × [0, 1]. We claim that ∂β Σǫ ∂ǫ = i * ǫ L u β. Indeed, if β = i a i (x, t)dx i + b(x, t)dt,
we have on the one hand

i * ǫ di u β = i ∂b ∂x i (x, ǫ)dx i ,
and on the other hand

i * ǫ i u d β = i ∂a i ∂t (x, ǫ)dx i - i ∂b ∂x i (x, ǫ)dx i . But ∂β Σǫ ∂ǫ = i ∂a i ∂t (x, ǫ)dx i , because β Σǫ = i a i (x, ǫ)dx i is the restriction β| M×{ǫ} of β to M × {ǫ}, whence the claim.
We obtain therefore

∂β Σǫ ∂ǫ = i * ǫ L u β = i * ǫ i u d β + i * ǫ di u β = i * ǫ i u ω + i * ǫ di u β. Thus the condition i * ǫ d β(u) = ∂β Σǫ ∂ǫ is equivalent to i * ǫ i u ω = 0. (b),(c) Recall that the tangent space T Σ C ∞ (M, Y ) to C ∞ (M, Y ) at a map Σ : M → Y consists of the maps f : M → T Y such that p • f = Σ where p : T Y → Y is the canonical projection. Let us view β as a map β : Bis(Y ) → Ω 1 (M ), Σ → β Σ .
As Bis(Y ) inherits its manifold structure from C ∞ (M, Y ), its tangent space at Σ is a subspace of T Σ C ∞ (M, Y ). The map T β Σ0 , the value of the bundle map T β at the bisection Σ 0 , sends a tangent vector δΣ to Bis(Y ) at Σ 0 to a 1-form on M .

Therefore the condition

i * ǫ d β(u) =
∂β Σǫ ∂ǫ in the congruence relation ( 5) is equivalent to the condition

i * ǫ dλ ∂Σ ǫ ∂ǫ = T β Σǫ ∂Σ ǫ ∂ǫ . (6)
This follows on the one hand from

T β Σǫ ∂Σ ǫ ∂ǫ = ∂β Σǫ ∂ǫ ,
and on the other hand from

β(u) = Σ * λ, u = λ, T Σ(u) = λ, ∂Σ ǫ ∂ǫ .
Note that Equation (6) implies i * ǫ dλ(δΣ) = T β Σǫ (δΣ) for each tangent vector δΣ to Bis(Y ) as all of them may be described as being the derivative ∂Σt ∂t of a curve. In case a family of bisections {Σ t } t∈[0,1] satisfies the equivalence relation, we have

ω Y ∂Σ t ∂t (m), T Σ t (δm) = 0
for every tangent vector δm to M . Indeed, as ω = Σ * ω, we have

i * ǫ i u ω = i * ǫ ω ∂ ∂t = i * ǫ ( Σ * ω) ∂ ∂t = i * ǫ ω(T Σ * ∂ ∂t , T Σ * (-)) = i * ǫ ω(T Σ * ∂ ∂t , T Σ * (-)) = ω ∂Σ t ∂t , T Σ t (-) .
Remark 5.2. Note that the property

i * ǫ d β(u) =
∂β Σǫ ∂ǫ may be stated as the fact that the family of 1-forms β Σǫ on M is constant up to a well-determined exact form, namely di

* ǫ β(u ǫ ) = i * ǫ d β(u ǫ ). In particular, the exact 2-form dβ Σǫ is constant. Therefore if Σ ∼ Σ ′ , then dβ Σ = dβ Σ ′ . Lemma 5.3. The equivalence relation is a congruence relation, i.e. if Σ ∼ Σ ′ , then we have T ⊲ Σ ∼ T ⊲ Σ ′ and Σ ⊲ T ∼ Σ ′ ⊲ T for every bisection T . Proof. First note that Σ ⊲ T = Σ ′ ⊲ T , because Σ ∼ Σ ′ implies φ 1 = φ ′ 1 and dβ Σ = dβ Σ ′ , cf Remark 5.2. In particular, Σ ⊲ T ∼ Σ ′ ⊲ T .
Now for the equality T ⊲ Σ ∼ T ⊲ Σ ′ , consider three bisections Σ, Σ ′ and T with Σ ∼ Σ ′ . The goal is to show that

dβ T ⊲Σǫ (u ǫ )| {ǫ}×M = ∂β T ⊲Σǫ ∂ǫ , (7)
because this implies then T ⊲ Σ ∼ T ⊲ Σ ′ . Using Formula (2) for T ⊲ Σ ǫ (with φ ǫ,s = φ Σǫ s (resp. ψ s ) the first component of Σ ǫ (resp. T )), we have for the right hand side of Formula ( 7)

∂β T ⊲Σǫ ∂ǫ = ∂ ∂ǫ ψ * 1 β Σǫ -ψ * 1 1 0 ∂ ∂ǫ φ * ǫ,s i ∂φǫ,s ∂s (ψ -1 1 ) * dβ T ds = ψ * 1 d β(u ǫ ) -ψ * 1 1 0 ∂ ∂ǫ φ * ǫ,s i ∂φǫ,s ∂s (ψ -1 1 ) * dβ T ds.
The left hand side of Formula (7) becomes

di uǫ βT ⊲Σ = d 1 0 η T ⊲Σǫ s ∂φ T ⊲Σǫ s ∂ǫ ds = d 1 0 ψ * 1 η Σǫ s -ψ * 1 i ∂φ Σǫ s ∂s (ψ -1 1 ) * dβ T ∂(ψ -1 1 φ Σǫ s ψ 1 ) ∂ǫ ds = d 1 0 ψ * 1 η Σǫ s -ψ * 1 i ∂φ Σǫ s ∂s (ψ -1 1 ) * dβ T (T ψ -1 1 ) * ∂φ Σǫ s ∂ǫ ds = ψ * 1 d 1 0 η Σǫ s ∂ ∂ǫ φ Σǫ s ds -ψ * 1 d 1 0 i ∂φ Σǫ s ∂ǫ i ∂φ Σǫ s ∂s (ψ -1 1 ) * dβ T ds = ψ * 1 d β(u ǫ ) -ψ * 1 d 1 0 i ∂φ Σǫ s ∂ǫ i ∂φ Σǫ s ∂s (ψ -1 1 ) * dβ T ds
In order to show Formula (7), it therefore remains to show

1 0 ∂ ∂ǫ φ * ǫ,s i ∂φǫ,s ∂s (ψ -1 1 ) * dβ T ds = d 1 0 i ∂φ Σǫ s ∂ǫ i ∂φ Σǫ s ∂s (ψ -1 1 ) * dβ T ds. (8)
We will show this by restricting to a path γ(t) in T * M . Without loss of generality, we can assume the 2-form (ψ -1 1 ) * dβ T to be df ∧ dg. Observing that the de Rahm differential d becomes on the path γ(t) the derivative ∂ ∂t | t=0 , we obtain for the right hand side of Formula (8)

1 0 ∂ ∂t t=0 ∂ ∂ǫ f (φ Σǫ s (γ(t))) ∂ ∂s g(φ Σǫ s (γ(t))) -same with (f ↔ g) ds.
In the same vein, we obtain for the left hand side

1 0 ds ∂ ∂ǫ ∂ ∂s g(φ Σǫ s (γ(t))) ∂ ∂t t=0 f (φ Σǫ s (γ(t))) -same with (f ↔ g).
It is clear that integration par parts transforms one side of (the restriction to γ(t) of) Formula ( 8) into the other. The boundary terms of the integration by parts involve derivatives with respect to ǫ for s = 0 or s = 1 of expressions in φ Σǫ s . But φ Σǫ 0 = id M and φ Σǫ 1 = φ 1 is a fixed diffeomorphism, both being independent of ǫ. Therefore the derivatives, and as a consequence the boundary terms, are zero.

As the two sides of Formula ( 8) are thus equal when restricted to an arbitrary path γ(t), they must be equal.

We now show the following differential geometric lemma concerning the manifold structure of the quotient with respect to the congruence relation: Lemma 5.4. The map

T Σ C ∞ (M, Y ) ⊃ T Σ Bis(Y ) → Ω 1 (M ), δΣ → T β(δΣ) -i * ǫ d λ, δΣ
is split surjective.

Proof. A section to the map T Σ Bis(Y ) → Ω 1 (M ) is constructed as follows. Let Σ = (φ s , η s ) ∈ Bis(Y ) be given. There exists a neighborhood V of M × {0} in M × [0, 1] such that for all (m, t) ∈ V , m → φ t (m) is a diffeomorphism, because for t = 0, we have φ 0 = id M . Furthermore, there exists a real valued smooth function χ such that χ ≡ 0 on the complement V c of V in M × [0, 1] and 1 0 χ(m, t)dt = 1 for each m which arises as projection of an element of V onto its first component. Now construct the section

ω : Ω 1 (M ) → T Σ Bis(Y ), α → ω(α)
as ω(α) = δη s where for a tangent vector δm ∈ T φs(m) M we put δη s (δm) := χ(m, t) α m , T φ -1 s (δm) .

Up to a matching of attachment points using T φ -1 s , the section may be expressed as being the 1-form α, seen as a tangent vector to Bis(Y ) ⊂ C ∞ (M, Y ) which is constant on [0, 1] and has zero component in the T * M -direction, multiplied with the cut-off function χ. Observe that this implies that i * ǫ d λ, δη s = 0. We therefore have T β(δη s )(δm) = showing that ω is a section to the above map T β -i * ǫ d λ, -.

Corollary 5.5. The foliation given by the congruence relation is a regular Fréchet foliation.

Lemma 5.6. In the point (id, 0) ∈ Bis(Y ), the infinitesimal relation in T (id,0) Bis(M ) corresponding to the congruence relation ( 5) is (X, α) ∈ T (id,0) F :⇔ X 1 = 0 and

1 0 α s ds = 0,
where F is the leaf of the foliation corresponding to the congruence relation.

Proof. As a first condition, we get clearly X 1 = 0 from φ 1 = id M . Using the expression of the map in Lemma 5.1 (c), the congruence relation reads in Σ = (id, 0) as T β (id,0) = 0, because λ (id,0) = 0.

Hence the tangent map T β (id,0) of β : The set of equivalence classes is a quotient of a Fréchet manifold by a regular Fréchet foliation whose associated distribution of tangent spaces gives the correct infinitesimal relation in (id, 0), i.e. the quotient Leibniz algebra is isomorphic to the Leibniz algebra of sections Γ(TM ) of the standard Courant algebroid.

C ∞ (M, Y ) → Ω 1 (M ), Σ → β Σ has a simple description, namely T β (id,0) (X, α) =
Remark 5.8. Unfortunately, the quotient Bis(Y ) is not a Lie rack. We were not even able to show that it is a local Lie rack. The rack operation is clearly smooth, but the foliation may be ill-behaved off (id, 0).

Integration of Dirac structures inside TM

In this section, we will identify the Lie groupoids corresponding to integrable Dirac structures in Y . We believe once again that this works for a general integrable Dirac structure D, but for the moment, we can show it only for integrable Dirac structures coming from Poisson structures π on M .

Let D be an integrable Dirac structure. We will consider Y D , given by The main idea is to perform symplectic reduction: Given a symplectic category X with source and target maps to the manifold M and symplectic form ω ∈ Ω 2 (X), consider a coisotropic subcategory Y ⊂ X. The distribution of subspaces of the tangent spaces given by ker(ω) defines an integrable regular distribution on Y thanks to the fact that Y is coisotropic. Thus the quotient Y / ∼ carries an induced non-degenerate closed 2-form, but the quotient is not a manifold in general (but rather some orbifold). In our case, an identification with the Weinstein groupoid which has been proven to be a manifold for an integrable Lie algebroid by Crainic-Fernandes [START_REF] Crainic | Rui Loja Integrability of Lie brackets[END_REF] will let us conclude that Y D / ∼ is a manifold in the integrable case.

Y D = {(γ t , α t ) ∈ Y | ( γ, α) ∈ D γ(t) ∀t ∈ [0, 1]}.
In order to proceed (we will take X := Y , the manifold of cotangent paths, and Y := Y D the subset of cotangent paths (γ t , η t ) with ( γt , η t ) ∈ D γ(t) for all t ∈ [0, 1]), we have to show that our Y D associated to a Dirac structure D is coisotropic with respect to the symplectic structure ω on Y which is just the symplectic form on T * M taken pointwise on the path. Lemma 5.9. For any Dirac structure D, Y D is a cosiotropic subset of Y .

Proof. The symplectic form on T * M is n i=1 dq i ∧ dp i , where the tangent space to T * M has the usual fourfold coordinates (q i , p i , ∂ ∂qi , ∂ ∂pi ) with i = 1, . . . , n in case dim(M ) = n. The symplectic form is zero on a subset of Y if for all t ∈ [0, 1], there is no (t-dependent) tangent vector with vector part having a non-zero coefficient in some ( ∂ ∂qi , ∂ ∂pi ) for some i = 1, . . . , n. Recall that a Dirac structure is a Lagrangian subbundle of TM which is involutive, i.e. its sections are closed under the Dorfman bracket. In particular, local section of D are v + α ∈ TM with α(v) = 0.

Let us consider the tangent directions to elements (γ t , η t ) ∈ Y D . For this, let us fix paths (γ t , η t ) ∈ Y D with values in a fixed algebroid leaf S 0 of the Dirac structure. The tangent direction to the η t -component is just the direction given by η t itself as the section η t is linear in the fiber direction. The constraint ( γt , η t ) ∈ D γ(t) means thus that the tangent directions in direction of the path γ(t) at t and in direction of η t pair to zero for all t ∈ [0, 1]. In particular, the coefficient in ( ∂ ∂qi , ∂ ∂pi ) is zero for all i = 1, . . . , n. As a consequence, the symplectic form inherited from T * M must be zero on paths in Y D with values in the fixed leaf S 0 . But being coisotropic means for Y D exactly that the paths with values in a fixed leaf are Lagrangian, thus we have shown that Y D is coisotropic.

In order to show that Y D is a subrackoid, we need two propositions. . This map can also be understood as the quotient map with respect to the homotopy relation, see [START_REF] Crainic | Rui Loja Integrability of Lie brackets[END_REF].

We state the next proposition for Dirac structures induced by the graph of a Poisson structure, because we could not locate in the literature the corresponding statement for general Dirac structures.

We are convinced, however, that this extension is true, except that G D is not obtained by symplectic reduction. Instead, it is obtained by dividing Y D by the kernel of ω with the condition that the end points of the paths in Y D are fixed. In particular ω D may be degenerate. Proposition 5.11. Let ω be the symplectic form on Y and ω D be the symplectic form on G D . We have the following situation:

Y D iD / / I (Y, ω) (G D , ω D ) Then the pullback I * ω D of ω D to Y D is equal to the restriction i * D ω of ω to Y D .
Proof. This follows from [START_REF] Cattaneo | Giovanni Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model[END_REF] using the fact that G D is the symplectic reduction of the coisotropic submanifold Y D ⊂ Y .

After these preparations, we can come to the announced result. Here we mean by "the bisection Σ includes the path γ at m 0 " that for Σ, viewed as a map in C ∞ (M, C ∞ ([0, 1], M )), there exists m 0 such that Σ m0 = γ. Observe that the difference between the equivalence relations ∼ and ≈ on Y D is that while i * ǫ i u ω = 0 is true at all m ∈ M for ≈, this is true only in m 0 for ∼. The equivalence relation ≈ induces in turn an equivalence relation ≈ on bisections, namely Σ 0 ≈ Σ 1 if there exists Σ t from Σ 0 to Σ 1 such that ∂Σt ∂t ∈ ker(ω D ) for all t ∈ [0, 1]. Observe that Σ 0 ≈ Σ 1 for two bisections Σ 0 , Σ 1 ∈ Bis(Y D ) implies Σ 0 ∼ Σ 1 and Σ 0 ⊲ -= Σ 1 ⊲ -. Proof. Include the homotopy γ between y and y ′ into a family of bisections {Σ t } t∈[0,1] . Call the limiting bisections at 0 and 1 of {Σ t } t∈[0,1] Σ and Σ ′ respectively. The bisections Σ and Σ ′ are equivalent in m 0 ∈ M . We have to show that T ⊲ Σ and T ⊲ Σ ′ are still equivalent in some m 1 ∈ M . Denote T = (ψ t , ζ t ), Σ = (φ, η) and Σ ′ = (φ ′ , η ′ ). Our claim follows from the formula (see Corollary 2.10)

dβ T ⊲Σt = ψ * 1 dβ Σt -ψ * 1 φ * 1,t (ψ -1 1 ) * dβ T + dβ T .
Indeed, when computing the condition i * ǫ i u ω = 0 with this formula, i u kills all terms which do not contain dt, thus we are left with the term in dt coming from ψ * 1 dβ Σt . One concludes by commuting ψ * 1 with i * ǫ and i u . Proof. Cotangent paths Y π for a bivector π form a coisotropic subset if and only if π is a Poisson bivector. This statement appears in different forms in the literature, cf Equation (10) in [START_REF] Schaller | Thomas Poisson structure induced (topological) field theories[END_REF], Equation [START_REF] Dorfman | Dirac structures of integrable evolution equations[END_REF] in [START_REF] Klimčík | Thomas WZW-Poisson manifolds[END_REF], Theorem 1.1 in [START_REF] Cattaneo | Coisotropic submanifolds and dual pairs[END_REF] and Theorem 3.1 in [START_REF] Cattaneo | Giovanni Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model[END_REF]. The fact that the symplectic reduction in the Poisson case gives the Weinstein groupoid is shown in [START_REF] Cattaneo | Giovanni Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model[END_REF].

Remark 5.17. We believe that this is valid for a general integrable Dirac structure. A hint in this direction is Theorem 2 (p. 22) in [START_REF] Kotov | Thomas Dirac sigma models[END_REF]. Thus in our framework, Y D / ∼ is the Weinstein groupoid. In order to link this approach to Crainic-Fernandes framework, we prove the following:

-1 1 •

 1 ψ t • φ 1 . Let us compute the canonical 1-form associated to the bisection Σ ⊲ T . Recall that we denote by β Σ = 1 0 φ * s η s ds the 1-form associated to Σ = (φ, η) and denote by β T = 1 0 ψ * s ζ s ds the 1-form associated to T = (ψ, ζ). Lemma 2.8. The 1-form associated to Σ ⊲ T is

  Proof . Let Σ = (φ, η) and T = (ψ, ζ) be bisections in Bis(Y ) and let a = (γ, α) ∈ Y . Recall that φ = φ s and ψ = ψ s are paths of maps in C ∞ (M, M ) with φ 0 = ψ 0 = id M and φ 1 , ψ 1 diffeomorphisms of M . Let us display mainly only the second components of the elements in Bis(Y ) or Y . The selfdistributivity in the first component has been checked in Section 1.

∂

  ∂s | s=0 ψ s = Y are the corresponding paths of vector fields. The paths of 1forms do not undergo much change. By definition of the Lie bracket of vector fields, ∂ ∂s

Proposition 3 . 7 . 1 0

 371 The map ϕ := (ev 1 , (-)dt) : A → TM induces an isomorphism of Leibniz algebroids of A/I with the standard Courant algebroid TM .

1 0 1 0

 11 α s ds, ζ := β s ds the integration of the corresponding forms.

  ϕ

1 0 1 0

 11 α γ(t) ( γ(t)) dt, (b) the 1-form λ with λ (γ,α) (δγ t ) = α γ(t) (δγ t ) dt and (c) the exact 2-form ω = dλ.

Lemma 4 . 1 .

 41 The pullback (s -1 Σ ) * λ is the 1-form β Σ = 1 0 φ * s η s ds. Thus dβ Σ becomes the pullback of the symplectic form ω on Y and the pullback

)

  Proof. (a) Straight forward computation. (b) This has been shown in Lemma 2.8. (c) This has been shown in Lemma 2.10. Definition 4.3. A Lie rackoid with a closed 2-form on the underlying manifold (total space of the submersions) is called a symplectic Lie rackoid if the form satisfies the equivariance property displayed in Equation (4).Definition 4.4. A bisection Σ of a Lie rackoid Y is called isotropic in case dβ Σ = 0.

A 1 0 1 = 1 0 1 0(Proposition 4 . 8 .

 111148 Recall our Leibniz algebroid A, obtained by passing to the tangent Leibniz algebroid of the Lie rackoid of cotangent paths Y . Let D ⊂ TM denote a Dirac structure, i.e. a maximally isotropic subbundle of TM whose sections form a Leibniz subalgebra under the Dorfman bracket. This implies that the Dorfman bracket becomes a Lie bracket for the sections of the subbundle D and D is thus in a natural way a Lie algebroid. On top of this, we will always suppose that our Dirac structures are integrable, i.e. the Lie algebroid corresponding to D integrates into a Lie groupoid. This will play a role in Section 5 where we identify the Lie groupoids integrating Dirac structures inside Y .To D ⊂ TM , we associate a Leibniz subalgebroid A(D) of A byA(D) m := {(X t , α t ) ∈ A m | ( Ẋt , α t ) ∈ D m ∀t ∈ [0, 1]} at each point m ∈ M . Observe that this defines a vector subbundle A(D) → M of A → M .Proposition 4.7. The restriction of the map ϕ = (ev 1 , (-)dt) : A → TM to A(D) has values in D ⊂ TM . Proof. We have X Ẋt dt, because X 0 = 0, and thus ϕ(X t , α t ) = Ẋt , α t ) dt ∈ D m , because ( Ẋt , α t ) ∈ D m for all m ∈ M . The bracket on Γ(A) = Y restricts to a bracket on Γ(A(D)): [Γ(A(D)), Γ(A(D))] ⊂ Γ(A(D)).

  because the Dirac structure is closed under the Courant (Dorfman) bracket. Thus [(X s , α s ), (Y t , β t )] ∈ D m for all t, s ∈ [0, 1].

Corollary 4 . 9 .

 49 The induced map ϕ| A(D) : A(D) → D is a morphism of Leibniz algebroids. In particular, the kernel of (the induced map on the level of sections of) ϕ| A(D) contains (the intersection of A(D) with) the ideal of squares.Remark 4.10. Observe that we have used in this proof the fact that the Dirac structure is closed under the Courant (Dorfman) bracket.

  s) α m , δm ds = α, δm ,

1 0 1 0Theorem 5 . 7 .

 1157 α s ds. Thus we obtain as a second condition α s ds = 0.In conclusion, we have shown the second main theorem of this article: The set of equivalence classes Bis(Y ) := Bis(Y ) / ∼ of bisections of Y with respect to the equivalence relation (5) becomes a rack with the rack product induced from the rack product of bisections.

  The goal is to pass from an equivalence relation on Bis(Y D ) to an interesting equivalence relation on Y D itself such that the quotient Y D / ∼ of Y D identifies with the (smooth) Weinstein groupoid associated to the integrable Dirac structure D.

1 )

 1 Let G D / / / / M be a source connected Lie groupoid integrating D and let ω D be the corresponding 2-form on G D . In other words, ω D is the symplectic form of the symplectic realization G D / / / / M of the Poisson manifold (M, π). Given a bisection Σ = (φ s , η s ) of G D , there is a map Σ : TM ((φ1) * ,φ * defined as the composition of ((φ 1 ) * , φ * 1 ) : TM → TM pushing forward (resp. pulling back) tangent (resp. cotangent) vectors to M via the diffeomorphism φ 1 and then Σ * ω D : TM → TM sending (X, α) to (X, α -i X Σ * ω D ). The composition of these maps covers the infinitesimal automorphisms of the Dorfman bracket described in Proposition 2.3 of [1]. Proposition 5.10. Let G D / / / / M be a source connected Lie groupoid integrating D with 2-form ω D . Let φ 1 be a diffeomorphism of M and Σ be a bisection of G D over φ 1 , i.e. Σ = (φ s , η s ) with φ 1 the given diffeomorphism of M . Then the map Σ : TM ((φ1) * ,φ * 1 ) -→ TM Σ * ωD -→ TM sends D to D. Proof. The map Σ covers an infinitesimal automorphism of the Dorfman bracket. As Σ ∈ Bis(Y D ), this infinitesimal automorphism is inner, i.e. given by the Dorfman bracket with an element of D ⊂ TM . As D is closed under the Dorfman bracket, the infinitesimal automorphism sends D to D. Thus the global automorphism which covers it must also send D to D. This last step follows from the fact that the exponential images in a Lie groupoid cover the identity-component subgroupoid, see Proposition 3.6.3, p. 134, in [12]. Disposing of a Lie groupoid G D / / / / M integrating D makes it possible to integrate paths in the Lie algebroid D ⊂ T * M into paths in the groupoid G D , i.e. we obtain an integration map I : Y D → G D , obtained by evaluating the path in G D in 1 ∈ [0, 1]

Proposition 5 . 12 .

 512 Let D be an integrable Dirac structure induced by the graph of a Poisson structure π on M . Then Y D is a subrackoid of Y , i.e. for all y ∈ Y D and all bisections T ∈ Bis(Y D ), we have T ⊲ y ∈ Y D , and for all bisections Σ, T ∈ Bis(Y D ), we have Σ ⊲ T ∈ Bis(Y D ).Proof. Observe that the bisections of Y D are simplyBis(Y D ) = {Σ = (φ, η) ∈ Bis(Y ) | ∀m ∈ M, ( φ m t , η (m,t) ) ∈ D φ m (t) },where we recall that for the path φ ∈ C ∞ (M, M ), the corresponding family of paths in M is φ m (cf proof of Lemma 2.7).Notice that the 2-form in Proposition 5.10 is ω D , while in our rack product, it is the 2-form of Y , i.e. ω, which takes its place. Thanks to Proposition 5.11, both forms coincide on Y D , and therefore dβ Σ = Σ * ω D which implies then Σ ⊲ T ∈ Bis(Y D ) for Σ, T ∈ Bis(Y D ). Thus Y D is a subrackoid of Y . Now we will induce an equivalence relation (in fact even a congruence relation) on the cosiotropic subset Y D ⊂ Y . The congruence relation on the bisections Bis(Y D ) is the same as for Bis(Y ) (see Equation (5)) with the difference that the family {Σ t } t∈[0,1] must have values in Y D : Definition 5.13. (a) For Σ, Σ ′ ∈ Bis(Y D ), we define that Σ ∼ Σ ′ in case there exists {Σ t } t∈[0,1] where Σ t ∈ Y D for all t ∈ [0, 1], Σ 0 = Σ, Σ 1 = Σ ′ and i * ǫ i u ω = 0. (b) Call a path γ in Y D a homotopy from γ(0) to γ(1) in case all families of bisections {Σ t } t∈[0,1] which include γ at m 0 satisfy i * ǫ i u ω| m0 = 0. For y, y ′ ∈ Y D , define y ∼ y ′ if there exists a homotopy γ from y to y ′ . (c) For y, y ′ ∈ Y D , define y ≈ y ′ if there exists y t with y 0 = y and y 1 = y ′ and ẏt ∈ ker(ω D ) for all t ∈ [0, 1].

Lemma 5 . 14 .

 514 The relation (b) on Y D is a congruence relation, i.e. for all y, y ′ ∈ Y D with y ∼ y ′ and all T ∈ Bis(Y D ), we have T ⊲ y ∼ T ⊲ y ′ .

Corollary 5 . 15 .

 515 The quotient Y D / ∼ is a rackoid.Proof. By Proposition 5.12, Y D is a subrackoid of Y , and by Lemma 5.14, the equivalence relation is a congruence relation and respects thus the classes.On the other hand, the symplectic reduction gives the Weinstein groupoid: Theorem 5.16(Cattaneo-Felder, Strobl). Let π be an integrable Poisson bivector on the manifold M . Then the set of cotangent paths Y π is a coisotropic subset of Y and the corresponding symplectic reduction gives the symplectic Weinstein groupoid integrating π.

Proposition 5 . 18 .

 518 The equivalence relation ∼ on Y D is the one given by the coisotropic foliation, i.e. relations (b) et (c) of Definition 5.13 coincide.Proof. This assertion follows from the comparison of the two relations discussed above: On the one hand, the tangency to the leaves means for families of bisections{Σ t } t∈[0,1] in Y D ω YD ( ∂Σ t ∂t (m), δΣ) = 0

Acknowledgements: This work has been done during the last five years and has been the subject of many visits of FW to Metz and Paris, as well as visits of CLG to Max Planck Institut in Bonn. We thank all the involved institutions for their kind hospitality and excellent working conditions. We

Putting these together, we obtain for the left hand side of the assertion of the theorem:

In the same way, we obtain (the same first component and for the second component of) (for) the right hand side of the assertion of the theorem, using Corollary 2.10 for an expression of dβ Σ⊲T :

Here we have used that for a diffeomorphism f : N → M , a vector field X ∈ X (N ) and a 2-form ω ∈ Ω 2 (M ):

and we used that (φ 1 ) * ( φ-1 1 (γ) t ) = γt as follows easily from the chain rule.

Observe that the terms involving α and dβ T coincide in the expressions of Σ ⊲ (T ⊲ a) and (Σ ⊲ T ) ⊲ (Σ ⊲ a). The remaining terms in (Σ ⊲ T ) ⊲ (Σ ⊲ a) are

1 ) * dβ Σ , and thus both expressions are equal. Remark 2.13. In the light of Remarks 2.3 and 2.11, we can interprete Theorem 2.12 in a more conceptual way.

Indeed, recall the notion of an augmented rack, see e.g. [10]: Given a group G which acts on a set X such that there exists an equivariant map p : X → G (with respect to the conjugation action on G), X inherits the structure of a rack defined by x ⊲ y := p(x) • y. In our context, the map

) is in fact an augmented rack for the action of (θ, ψ) ∈ Aut(TM ) on Σ ∈ Bis(Y ) given by (θ, ψ)

The computations are very similar to the above proof. Note that the above discussion implies in particular that p is a morphism of racks (with values in the conjugation rack underlying Aut(TM )).

for every tangent vector δΣ to Y D . We claim that this property is equivalent to the property (see Lemma 5.1) which satisfies the family of bisections {Σ t } t∈[0,1] in Y D in our congruence relation, namely

for every tangent vector δm to M . In order to show this, we have to construct for a given tangent vector δΣ to Y D a family {Σ t } t∈[0,1] in Y D and a tangent vector δm to M such that T Σ t (δm) = δΣ. This follows from a contractibility property of Y D . As elements of Y D are sections of a vector bundle, one can contract the fiber component to zero and then the resulting path to the trivial path. This means that Y D permits the extension of sections and bisections, and we can thus find a bisection passing through a given point or even passing through a given tangent vector. Proof. The proof goes as follows. Let us describe the action of a bisection σ of a source simply connected Lie groupoid G on itself. Choose some g ∈ G and let a be an A-path which represents g, with A the Lie algebroid of G. Then Ad σ (a) is again an A-path and it represents Ad σ (g). (Recall from [START_REF] Crainic | Rui Loja Integrability of Lie brackets[END_REF], Section 1, that the set of homotopy classes of G-paths is homeomorphic to the set of homotopy classes of A-paths in the sense of [START_REF] Crainic | Rui Loja Integrability of Lie brackets[END_REF]). Here Ad σ stands for the adjoint action of the bisection σ on the Lie algebroid A.

For the Lie groupoid G D integrating a Dirac structure D arising from a Poisson structure, the adjoint action of bisections goes as follows. To a bisection Σ of G D , one associates a diffeomorphism φ 1 of the base and a closed 2-form: the pull-back Σ * ω GD of the symplectic 2-form ω GD on G D by the source map. The adjoint action of Σ on the Lie algebroid D is then the transformation described in Proposition 5.10.

Let us compare this action with the rackoid product ⊲. For any bisection Σ = (φ t , η t ) of Y , and for any a = (γ, θ) ∈ Y , the rackoid product Σ ⊲ a is obtained by associating to a the path (γ ′ , θ) in TM and then acting on TM with the help of the transformation Σ defined as in Proposition 5.10. By this procedure, we obtain a path in TM lying over the path φ 1 • γ. Its component in T * M is an element of Y which is equal to Σ ⊲ a. Again, the rackoid product Σ ⊲ -depends only on the pair (φ 1 , dβ Σ ), with φ 1 , β Σ as in Definition 2.4. In particular, for a bisection Σ of Y D , the rackoid product Σ ⊲ -depends only on the image of Σ in the set of bisections of G D and, in view of Proposition 5.11, the closed 2-form dβ Σ coincides with the closed 2-form Σ * ω GD described above.

In conclusion, the rack product Σ ⊲ a, with Σ a bisection of Y D and a ∈ Y D is equal to Ad σ (a), with σ the image of Σ in the set of bisection of Y D . In view the first paragraph, this completes the proof of the proposition.


Dr Florin-Constantin Mihai
email: mihai.florinconstantin@gmail.com

Dr Corneliu Iatu

Dr Adrian Grozavu

RURAL WASTE DISPOSAL ISSUES WITHIN URBAN BORDERS

Keywords: waste management, illegal waste dumping, peri-urban areas, pollution

The paper examines the waste management issues in the villages annexed to administrative-territorial units of the Romanian cities which have been frequently neglected by urban waste operators. The lack of waste collection services in such periurban communities favored the illegal waste disposal practices particularly prior to EU accession. The extension of waste collection services from main cities to such areas is compulsory in order to mitigate the environmental risks and the public health threats. The paper estimates the amounts of household waste susceptible to be uncontrolled disposed of by peri-urban villages in different geographical areas of North-East region with a particular focus on Neamt county. The paper points out that these rural settlements should receive the same attention concerning the municipal waste management services as the main urban areas. Traditional recovery of waste fractions at the household level (e.g., home composting) should be further promoted in such areas in order to avoid illegal dumping issue and to prevent the landfill of biodegradable waste as requested by EU regulations.

INTRODUCTION

Municipal waste management systems pose serious challenges across the globe and they must be continually improved in order to avoid pollution and threatening public health [START_REF] Hoornweg | What a waste: a global review of solid waste management[END_REF]. Population access to sound waste management services is still limited in developing and transition countries, particularly in their rural areas [START_REF] Han | Influencing factors of domestic waste characteristics in rural areas of developing countries[END_REF]. Villages are dealing with serious improper waste disposal practices such as open dumping or open burning of household waste [START_REF] Taboada-González | Household Solid Waste Characteristics and Management in Rural Communities[END_REF]. Local administrative units (LAU) of cities may include one or several villages. These areas are susceptible to unsound waste management practices because there are usually neglected by urban waste operators and receive less attention from urban authorities [START_REF]Solid waste management in the world"s cities[END_REF]. This situation prevailed in Romanian urban areas, especially prior to EU accession. The expansion of waste collection coverage across the cities of North-East region is significant compared to 2003, but gaps in waste collection systems are detected in the case of villages annexed to urban areas [START_REF] Mihai | Territorial disparities in municipal waste management[END_REF]. The paper reveals the amounts of household waste susceptible to uncontrolled disposal in such areas during 2004-2015.

MATERIALS AND METHODS

The North-East Region of Romania comprises six counties (Suceava, Neamt, Bacau, Botosani, Iasi, and Vaslui) with a total population of 3,302,217 according to the last population census (2011). There are 46 urban areas in the region, which include 119 villages in their administrative area. However, several major cities have no rural localities under their administrative area (Iasi, Suceava, Bacau, Botosani, Barlad, Roman, Husi, Campulung Moldovenesc, Radauti), therefore, they are excluded from the analysis. The number of such villages varies between one (eg. Parcovaci villageas part of Harlau city in Iasi county) and nine (eg. Haleasa, Lungeni, Neagra, Cotargasi, Darmoxa, Frasin, Holda, Holdita, and Pietroasaas part of Brosteni town in Suceava county). The paper estimates the amounts of household waste uncontrolled disposed (Q ud ) for each village included in the local administrative area of the city (LAU 2 level) within the North-East Region. The data are mapped at an urban LAU1 level which is equivalent to NUTS4 regions (http://ec.europa.eu/eurostat/web/nuts/localadministrative-units).

The time scale analysis is broken down into two main different periods, such as: 1) The period 2004-2009, in which smaller urban areas lack of formal waste collection services, and the population of the main cities is partially covered by such services while the rural localities included in their LAU are neglected by waste operators and household waste are improperly disposed on the surroundings. Several communes had been declared towns during 2003-2004 (eg. Roznov, Podu Iloaiei, Murgeni, Stefanesti, Flamanzi, Vivocu de Sus) despite the lack of basic utilities such as adequate sanitation facilities and waste management services. In this context, an evaluation for 2004 is performed where the ratio of inhabitants in such villages of total LAU population is determined. 2) The period 2010-2015, which is characterized by the expansion of urban waste collection coverage at regional level following the closure of local dumpsites from smaller urban areas and rural municipalities, but with a low collection efficiency outside the core-urban area. The indicator mentioned above was calculated using the following relation if there are no waste collection services across villages as in the first stage [START_REF] Mihai | Territorial disparities in municipal waste management[END_REF]:

Q ud = {Q wu -[( Q wu -0.4 * Q bw ) + ( Q wu -0.1 * Q r )]} Q wu =
waste uncollected by formal waste management services (waste operators), which is calculated according to the formula: Q wu = P * W g *365 /1000, Ppopulation of the village (noWCS scenario) W gper capita waste generation rate = 0.3 kg.inhab.yr -1 (Population Census 2002 data for time series: 2004-2009) and 0.33 kg.inhab.day -1 (Population Census 2011 data for time-series 2010-2015). The per-capita waste generation rate is below the national flat rate (0.4 kg.inhab.day -1 ) as stipulated by National Waste Management Plan (NWMP) due to poorer socioeconomic features of the study area [6]. Q bw -biodegradable fraction of household waste Q r -recyclables (metals, paper and cardboard, plastics, glass, wood)

The data for these fractions are extracted from the amounts of waste uncollected (Q wu ) using a rural municipal waste composition provided by the regional waste management plan [START_REF]Regional waste management plan of North East Region[END_REF] applied for the 2004-2009 period, and data determined by Ingleziakis et al. for the period 2010-2015 [START_REF] Inglezakis | Municipal solid waste experimental studies in Romania and Bulgaria[END_REF]. Waste collection efficiency may differ from one county to another and there may be disparities between cities within the same region [START_REF] Wilson | Wasteaware" benchmark indicators for integrated sustainable waste management in cities[END_REF].

Recycling and recovery practices at individual household level prevent or mitigate the uncontrolled waste disposal practices. In China, some villages used biogas ecological cycle technology to dispose of organic waste (80% of household waste) as a source of bionergy for rural residents [START_REF] Wang | Rural Solid Waste Management in China: Status, Problems, and Challenges[END_REF]. The national waste management plan from Poland stipulates that part of the volume of biodegradable waste generated, particularly in villages and smaller towns, is utilized by the inhabitants on their own [11]. Home composting and animal feed account 15% of total biowaste generated in smaller towns [11]. In Romania, the ratio of biodegradable waste is higher of total municipal waste fraction, particularly in smaller urban areas and rural municipalities (communes). In this context, a factor of 0.4 is used to calculate the biowaste generated and uncollected across rural localities included in the urban LAU and 0.1 for recyclables. The magnitude of home composting and animal feed practices are considered lower in such villages compared to other distant rural municipalities where agricultural activity is more intense.

Currently, all urban areas of North-East Region are served by public or private waste operators. However, field observations, sanitation campaigns, and mass-media points out that such villages are still being exposed to illegal waste disposal practices. There is no waste management data at such level and the model use a low collection efficiency (Cef) of 40% (WCS40) to calculate the amounts of waste uncontrolled disposed in the second stage (2010-2015) as follows: Q wu = Q wu (noWCS) -Q wu (noWCS) *Cef. The data combined from the main two periods reveal the amounts of household waste uncontrolled disposed of by such villages across 12 years.

The paper performs a comparative analysis of the amounts of household waste uncontrolled disposed by such villages and those volumes reported to be eliminated in open dumps. The volume data are transformed into tons using a specific density of 400 kg/m 3 in case of rural dumps. The data are provided by local environmental authorities, but there are available only for 20 out 119 villages included under urban LAU (see table 1).

RESULTS AND DISCUSSION

Several rural municipalities (communes) were declared towns during 2003-2004 despite the lack of proper infrastructure, specifically in urban areas including a sound waste management system. Such localities cover several villages in their administrative area which may count over 40% of the total urban LAU population. Statistically, they are included as urban population (LAU 1 level) and only population censuses provide demographic data at the village level (LAU 2 equivalent to NUTS 5 regions). There are 118 rural localities included in administrative areas of cities or towns which had counted 116992 inhabitants based on Population Census data from 2002. The left map of figure 1 shows that most of "urban population" of several towns (LAU) lived basically in such villages without access to basic utilities. There are towns where over 60% of the population live in the villages annexed to their administrative areas such as Stefanesti (77%, Botosani county), Dolhasca and Salcea towns (67%, Suceava county), Brosteni (65%, Suceava county), Flamanzi (61%, Botosani county).

More than 3 villages are found under the administrative area of cities and towns as follows: Brosteni (9 villages), Dolhasca (7 villages), Bicaz, Murgeni, Negresti (6 villages) Vaslui, Pascani, Darmanesti, Saveni, Liteni (5 villages), Flamanzi, Stefanesti, Salcea, Podu Iloaiei (4 villages). The poor waste management facilities prevail outside the main urban area, particularly in the first stage (2004)(2005)(2006)(2007)(2008)(2009). This fact is also supported by the dumpsites counted by local environmental authorities which should have been closed until 16 July 2009. In 2004, the amount of household waste uncontrolled disposed of by villages under the urban administrative area was 9332.54 t which means 55995.24 t during 2004-2009 (6 years).

The maximum value is 769 t.yr -1 (Pascani city) followed by Dolhasca (596.28 t.yr -1 ) and Flamanzi (575 t.yr -1 ). Significant amounts of household waste uncontrolled disposed (300-500 t.yr) were estimated in case of Brosteni, Liteni, Salcea (Suceava county), Darmanesti (Bacau county), Stefanesti (Botosani county), Vaslui, Murgeni (Vaslui county). Such urban areas must expand the waste collection services to surrounding villages to avoid serious pollution threats. Most of the urban areas are responsible for 100 t of solid waste dumped into the environment each year with the exception of Gura Humorului, Cajvana, or Buhusi with less population in such villages (< 60 t per yr -1 ). North-East of Romania is a poor and a peripheral EU region, which needs access to a proper road infrastructure (better roads maintenance, highways, high-speed railways) to improve connection to West (EU) and South (Bucharest capital city) and to stimulate massive investments in the region. Significant socioeconomic disparities are found across North-East region between the largest cities (Iasi, Suceava, Bacau, Piatra Neamt) and smaller towns with strong rural features (Flamanzi, Stefanesti, Roznov, Salcea, Milisauti, Liteni, Dolhasca, Murgeni) [START_REF]ADR Nord-Est 2013 Regional Development Strategy of North-East 2014-2020[END_REF]. In the latter case, full coverage of population to waste collection services is problematic and illegal disposal practices are still visible on the field.

Eu funds have improved waste management infrastructure across several counties, but delays in the construction process (waste collection platforms, sanitary landfills, sorting and composting stations, transfer stations) lead to illegal waste disposal practices on peripheral urban areas. In fact, the waste management service may be available to 100 percent of the city, but perhaps only 80 percent actually use that service [START_REF]Solid waste management in the world"s cities[END_REF]. The rural localities within urban borders are at risk of being neglected by sound waste management services compared to the main urban area. Same discrepancies are revealed in Varna (Bulgaria) where full coverage ends at the city limit and city"s 5 villages (included in the urban administrative area) are each served with a single container that is collected once per month if at all [START_REF]Solid waste management in the world"s cities[END_REF]. Thus, such villages receive less attention from urban authorities in terms of waste collection schemes which further expose the local population to illegal dumping practices.

For the second period (2010-2015), the paper estimates 24648.5 t to be uncontrolled across rural localities included in the urban local administrative units or 4108 t per yr with a collection efficiency set up to 40%. The right map of figure 1 reveals the critical period during 2004-2009 compared to 2010-2015 when waste collection services have begun to emerge. The total amount of household waste uncontrolled disposed of by the all 119 villages under the urban administration is 80643.74 t during 12 years (2004-2015). Such values vary from 6622 t (Pascani urban LAU) towards less than 1000 t and highlight the fact that such villages may be a significant pollution source at local level. This fact is demonstrated by illegal dumping practices observed across Neamt County as shown in figure 2.

The larger cities (Piatra-Neamt, Roman, Targu Neamt) provide a proper waste collection within the main urban area and the wastes collected were disposed into urban landfills, while neighboring villages have problems with the waste management sector. The worse situation characterized the smaller urban areas in the first stage. Roznov town was not served by a formal waste collection services until 2010 and Bicaz city had an obsolete waste management infrastructure. In the latter case, seven dispersed mountain villages make more difficult to provide a coherent waste collection system without proper funds. The poor coverage of such villages (which are under the urban administration) to reliable waste management services suggests a poor urban governance across the North-East region.

Despite the lack or poor waste management facilities during 2004-2010, only 20 villages (out of 119) have data concerning the surface (ha) and volumes of dumpsites which are ultimately transformed into tons. These dumpsites cover almost 10 ha being disposed of 28928 t compared to 8098 t estimated to be uncontrolled disposed in the first stage which represented almost 28%. Data concerning the rural dumpsites have a poor geographical coverage and their reliability depends on local authority involvement and honesty. As an example, there is no data concerning dumpsites in the villages of urban LAU"s across Neamt county despite clearly evidences shown in figure 2. Also, some dumpsites are temporary (exposed to floods or they are burnt) which are not comprised by statistics. The data broken down per village reveal several situations as shown in table 1: (i) the largest dumpsite is reported in Bobulesti village (1.6 ha -3400 t) with a direct estimated contribution of 680.61 t (ii) the largest amount disposed is 8000 t in Bohoghina village (1ha) despite a low contribution of village (85.67 t) (iii) the smallest dumpsite is reported in Nicolae Balcescu village (0.01 ha with 80 t) despite a direct contribution of 1912.3 t (iv) the lowest amount of waste disposed -40 t in Carja village despite a direct contribution of 579 t (v) similar values between amount of waste disposed (400 t) and direct contribution (439.86 t) in case of village Valea Mare (vi) same area reported (0.4 ha) for all villages of Negresti LAU (vii) dumps with same area may have different volumes and viceversa. The above situations point out that assessment of illegal waste disposal practices is a complex task and it is poorly covered by official statistics. Generally, the amounts waste disposed in dumps are expected to be larger than those estimated as direct contributions by villages because there may be other inputs to take into consideration besides local residents such as economic agents, illegal traffic of waste from the main urban area, leisure activities etc. Dumpsites data for six villages are significantly below than those estimated and the other hand, in some cases, there are huge differences between these data. Economic agents (local or regional) that do not sign contracts with waste operators (although they are obliged by law) may significantly contribute to the illegal dumping practices (household and packaging waste, demolition and construction) on improper sites in the outskirts of cities. Furthermore, the expansion of waste collection services towards rural areas attached to urban LAU"s cities have been partially achieved, especially in recent years.

CONCLUSION

Full population coverage throughout the urban local administrative units is a basic requirement for the implementation of a proper and efficient waste management system which significantly decrease the exposure of villages to illegal dumping practices. The paper reveals that all 119 villages included in the urban LAU"s are responsible for dumping 80643.74 t during 12 years (2004-2015) with a critical situation during 2004-2009 characterized by the lack of waste collection services and widespread improper disposal practices. Good urban governance requires equitable access to basic utilities across all settlements under administration. Municipal waste management system must be continually adapted to the factors that favor urban LAU development by implementing selective collection facilities, better home composting practices in order to support sustainable programs for waste recovery and waste diversion from landfills. The responsibility and receptivity of citizens to such programs will play an important role in the sustainability of this process which finally will improve the quality of life.

Figure 1 .

 1 Figure 1. Estimation of household waste uncontrolled disposed in villages annexed to urban administrative areas (LAU 1) across the North-East Region of Romania In 2011, the number of inhabitants which lived in the villages under the urban administration was 104393 according to the last Population Census with 12599 inhabitants less than the previous census (2002). This demographic decline is explained by living the country by people looking for work abroad, mainly in EU (e.g. Spain, Italy).

Figure 2 .

 2 Figure 2. Uncontrolled waste disposal practices across rural localities of urban LAU"s in Neamt county

Table 1 .

 1 Rural dumps vs amounts of household waste uncontrolled disposed

	Village _ part of urban	S (ha)	Q (t)	Qud_2004_2009_
	LAU_county			noWCS (t)
	MURGENI (LAU)_VS
	Carja	0.02	40	578.19
	Floreni	0.05	100	306.80
	Latesti	0.04	80	192.89
	Raiu	0.5	1000	344.61
	Sarateni	0.04	80	213.95
	Schineni	0.03	60	294.84
	NEGRESTI (LAU)_VS
	Cazanesti	0.4	800	219.21
	Glodeni	0.4	…	143.59
	Parpanita	0.4	800	289.57
	Poiana	0.4	400	132.10
	Valea mare	0.4	400	439.86
	PODU ILOAIEI (LAU)_IS
	Budai	0.5	1600	448.96
	Cositeni	0.2	48	154.60
	Scobalteni	1	4000	496.82
	STEFANESTI (LAU)_BT
	Badiuti	1.2	3040	425.50
	Bobulesti	1.6	3400	680.61
	Stanca	1.3	3000	444.17
	FLAMANZI (LAU)_BT
	Nicolae balcescu	0.01	80	1912.13
	Chitoveni	0.5	2000	293.88
	BUCECEA (LAU)_BT
	Bohoghina	1	8000	85.67

Source: Local Environmental Protection Agencies and author calculations

ACKNOWLEDGEMENTS

This work was supported by a grant of the "Alexandru Ioan Cuza" University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2017-06