INTEGRATION OF TWISTED DIRAC BRACKETS

HENRIQUE BURSZTYN, MARIUS CRAINIC, ALAN WEINSTEIN, AND CHENCHANG ZHU

Abstract. Given a Lie groupoid G over a manifold M, we show that multiplicative 2forms on G relatively closed with respect to a closed 3-form ϕ on M correspond to maps from the Lie algebroid of G into T^{∗}M satisfying an algebraic condition and a differential condition with respect to the ϕ-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to ϕ-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-hamiltonian spaces and group-valued momentum maps.

Contents

1. Introduction 1 2. Basic definitions and the main results 3 2.1. Twisted Dirac structures 3 2.2. Groupoids and algebroids 5 2.3. Dirac ↔ presymplectic 6 2.4. The main results 7 3. Multiplicative 2-forms on groupoids 8 4. Multiplicative 2-forms and induced Dirac structures 12 5. Reconstructing multiplicative forms 18 6. Examples 22 6.1. Multiplicative 2-forms: first examples 22 6.2. Examples related to Poisson manifolds 23 6.3. Over-presymplectic groupoids and singular presymplectic groupoids 25 6.4. Lie group actions and equivariant cohomology 27 7. Presymplectic realizations of Dirac structures 30 7.1. Presymplectic realizations 30 7.2. Realizations of Cartan-Dirac structures and quasi-hamiltonian spaces 31 8. Multiplicative 2-forms, foliations and regular Dirac structures 34 8.1. Foliations 34 8.2. Multiplicative 2-forms on monodromy groupoids 35 8.3. Dirac structures associated to foliations 36 8.4. Presymplectic groupoids of regular Dirac structures 37 References 41

1. Introduction

The correspondence between Poisson structures and symplectic groupoids [7, 9, 14], analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of hamiltonian and Poisson actions (see e.g. [31]). In this paper, we extend this correspondence to the context of Dirac structures twisted by a closed 3-form.

2000 Mathematics Subject Classification. 58H05 (primary), 53C12, 53D17, 53D20 (secondary).

Dirac structures were introduced by Courant in [10, 11], motivated by the study of constrained mechanical systems. Examples of Dirac structures include Poisson structures, presymplectic forms and regular foliations. Connections between Poisson geometry and topological sigma models [20, 26] have led to the notion of Poisson structures twisted by a closed 3-form, which were described by Weinstein and Severa in [27] as special cases of twisted Dirac structures. The diagram below describes these various generalizations of Poisson structures and the corresponding global objects.

Poisson structures

[figure]

[figure]

[figure] [figure]

[figure] [figure]

symplectic groupoids

[figure]

[figure] [figure]

[figure]

Dirac structures

[figure]

[figure]

[figure]

?

[figure]

twisted Dirac structures

[figure] [figure]

twisted ?

twisted Poisson structures

[figure] [figure] [figure]

twisted symplectic groupoids

[figure]

In this paper, we fill the gaps in the diagram above by introducing the notion of twisted presymplectic groupoid relative to a closed 3-form ϕ on a manifold M and establishing a bijective correspondence (up to natural isomorphisms) between source-simply-connected ϕ-twisted presymplectic groupoids over M and ϕ-twisted Dirac structures. Our results complete, and were inspired by, earlier work of Bursztyn and Radko [5] on gauge transformations of Poisson structures, and of Cattaneo and Xu [8], who used the method of sigma models [7] to construct global objects attached to twisted Poisson structures. We remark that the degeneracies of presymplectic forms force our techniques to be different in nature from those used to establish the correspondence of Poisson structures and symplectic groupoids (as in [9, 29, 8]). In our search for the right notion of nondegeneracy that 2-forms on presymplectic groupoids must satisfy (Definition 2.1), the examples provided by [5, 14] were essential. (In fact, our presymplectic groupoids are closely related to those introduced in [14]).

Twisted presymplectic groupoids turn out to be best approached through a general study of multiplicative 2-forms on groupoids, which turn out to be extremely rigid. (On groups, they are all zero.) Given a closed 3-form ϕ on the base M of a groupoid G, we call a 2-form ω on G relatively ϕ-closed if dω = s^{∗}ϕ − t^{∗}ϕ, where s and t are the source and target maps of G. If G is source-simply-connected, we identify the infinitesimal counterparts of relatively ϕ-closed multiplicative 2-forms on G as those maps to T^{∗}M from the Lie algebroid A of G which satisfy an algebraic condition and a differential condition related to the ϕ-twisted Courant bracket [27]. In fact, the [twisted] Courant bracket itself could be rediscovered from the properties of [relatively] closed 2-forms on groupoids. The reconstruction of multiplicative 2-forms out of the infinitesimal data is based on the constructions of [13, Sec. 4.2].

Motivated by the relationship between symplectic realizations of Poisson manifolds and hamiltonian actions [9], we study presymplectic realizations of twisted Dirac structures. Just as in usual Poisson geometry, these presymplectic realizations carry natural actions of presymplectic groupoids. In fact, it is this property that determines our definition of presymplectic realizations. An important example of twisted Dirac structure is described in [27, Example 4.2]: any nondegenerate invariant inner product on the Lie algebra 𝔥 of a Lie group H induces a natural Dirac structure on H, twisted by the invariant Cartan 3-form; we call such structures Cartan-Dirac structures. We show that presymplectic realizations of Cartan-Dirac structures are equivalent to the quasi-hamiltonian 𝔥-spaces of Alekseev, Malkin and Meinrenken [1] in such a way that the realization maps are the associated groupvalued momentum maps. It also follows from our results that the transformation groupoid H ⋉ H corresponding to the conjugation action carries a canonical twisted presymplectic structure, which we obtain explicitly by “integrating” the Cartan-Dirac structure. As a result, we recover the 2-form on the “double” D(H) of [1] and the AMM groupoid of [4]. (Closely related forms were introduced earlier in [18, 30].) A unifying approach to momentum map theories based on Morita equivalence of presymplectic groupoids has been developed by Xu in [34]; much of our motivation for considering quasi-hamiltonian spaces comes from his work. Our results indicate that Dirac structures provide a natural framework for the common description of various notions of momentum maps (as e.g. in [1, 22], see also [31]).

We illustrate our results on multiplicative 2-forms and Dirac structures in many examples. In the case of action groupoids, we obtain an explicit formula for the natural map from the cohomology of the Cartan model of an H-manifold [2, 16] to (Borel) equivariant cohomology [2, 3] in degree three; for the monodromy groupoid of a foliation F, we show that multiplicative 2-forms are closely related to the usual cohomology and spectral sequence of F [19].

The entire discussion of relatively closed multiplicative 2-forms on groupoids may be embedded in the more general context of a van Est theorem for the “bar-de Rham” double complex of forms on the simplicial space of composable sequences in a groupoid G, whose total complex computes the cohomology of the classifying space BG. We reserve this more general discussion for a future paper.

Outline of the paper: In Section 2, we review basic definitions concerning Dirac structures and groupoids, we introduce various notions of “adapted” 2-forms on groupoids, and we state our two main results, which allow one to go back and forth between [twisted] presymplectic groupoids and their infinitesimal counterparts, [twisted] Dirac manifolds. Section 3 begins the work of proving the theorems with a study of the global objects, namely multiplicative 2-forms on groupoids. In Section 4, we pass from the groupoids to the infinitesimal objects, and in Section 5, we go in the opposite direction, completing the proofs of the main theorems. Section 6 begins with simple examples. Then, after a discussion of 2-forms on groupoids which become presymplectic only after one passes to the quotient by a foliation, we show how twisted-multiplicative forms on action groupoids are related to equivariant cohomology. This leads us to Section 7, where we study the AMM-groupoid and apply our results to quasi-hamiltonian actions. Finally, Section 8 is devoted to multiplicative 2-forms on foliation groupoids, with applications to Dirac structures whose presymplectic leaves all have the same dimension.

Acknowledgments: We would like to thank many people for discussions which were essential to our work, including Anton Alekseev, Alberto Cattaneo, Rui Loja Fernandes, Johannes Huebschmann, Yvette Kosmann-Schwarzbach, Eckhard Meinrenken, Pavol Ševera, Marco Zambon, and especially Ping Xu, who should be considered as a “virtual co-author” of this paper. Many institutions were our hosts during part of the work, including MSRI and Université Libre de Bruxelles (Bursztyn), Institut Mathématique de Jussieu (Weinstein), the Erwin Schrödinger Institute (Crainic). Our collaboration was facilitated by invitations to the conferences Poisson 2002 (Institute Superior Tecnico, Lisbon) and Groupoidfest 2002 (University of Nevada, Reno). For financial support, we would like to thank the National Science Foundation (Bursztyn, grant DMS-9810361, Weinstein and Zhu, grant DMS-0204100) and the Miller Institute for Basic Research in Science and the Dutch Royal Academy (Crainic).

2. Basic definitions and the main results

2.1. Twisted Dirac structures. We recall some basic concepts in Dirac geometry [10]. Let V be a finite dimensional vector space, and equip V ⊕ V^{∗} with the symmetric pairing

(2.1)  ⟨(x, ξ), (y, η)⟩_{+} = ξ(y) + η(x).

A linear Dirac structure on V is a subspace L ⊂ V ⊕ V^{∗} which is maximally isotropic with respect to ⟨ , ⟩_{+}. The set of all Dirac structures on V is a smooth submanifold of a Grassmann manifold; we denote it by Dir (V).

Natural examples of vector spaces carrying linear Dirac structures are presymplectic and Poisson vector spaces (i.e, spaces equipped with a skew-symmetric bilinear form and a bivector, respectively). More precisely, a skew-symmetric bilinear form θ (resp. bivector π) corresponds to the Dirac structure L_{θ} (resp. L_{π}) given by the graph of the map θ̃: V ⟶ V^{∗}, θ̃(x)(y) = θ(x, y) (resp. π̃ : V^{∗} ⟶ V, π̃(α)(β) = π(β, α)).

General Dirac structures can be described either in terms of bilinear forms or bivectors. Let pr_{1} : V ⊕ V^{∗} ⟶ V and pr_{2} : V ⊕ V^{∗} ⟶ V^{∗} be the natural projections. A linear Dirac structure L has an associated range,

R(L) = pr_{1}(L) = {v ∈ V : (v, ξ) ∈ L for some ξ ∈ V^{∗}} ⊂ V,

and a skew-symmetric bilinear form θ_{L} on R(L),

(2.2)  θ_{L}(v_{1}, v_{2}) = ξ_{1}(v_{2}), where ξ_{1} ∈ V^{∗} is such that (v_{1}, ξ_{1}) ∈ L.

For the description of L in terms of a bivector, let us define the kernel of L as the kernel of θ_{L}:

Ker (L) := Ker (θ_{L}) = pr_{2}(L)^{∘} = {v ∈ V : (v, 0) ∈ L} ⊂ V.

(Here ^{∘} stands for the annihilator.) The bivector π_{L}, defined on V/Ker (L), is the one induced by a form analogous to (2.2) on pr_{2}(L) ⊂ V^{∗}. It is not difficult to see that L is completely characterized by the pair (R(L), θ_{L}), or, analogously, by the pair (Ker (L), π_{L}). We observe that

(i) R(L) = V if and only if L = L_{θ} for some skew-symmetric bilinear form θ on V; (ii) Ker (L) = 0 if and only if L = L_{π} for some bivector π on V. If V and W are vector spaces, any linear map ψ : V ⟶ W induces a push-forward map

𝔉ψ : Dir (V) ⟶ Dir (W) by

(2.3)  𝔉ψ(L_{V}) = {(ψ(x), η) | x ∈ V, η ∈ W^{∗}, (x, ψ^{∗}(η)) ∈ L_{V} },

where L_{V} ∈ Dir (V). We note that the map 𝔉ψ is not continuous at every L_{V}.

Given L_{V} ∈ Dir (V) and L_{W} ∈ Dir (W), we call a linear map ψ : V ⟶ W forward Dirac if 𝔉ψ(L_{V}) = L_{W}. There is a corresponding concept of a backward Dirac map (see e.g. [5]), but we will not deal with it in this paper. Hence, for simplicity, we will refer to forward Dirac maps just as Dirac maps. As an example, we recall that a Dirac map between Poisson vector spaces is just a Poisson map.

An almost Dirac structure on a smooth manifold M is a subbundle L ⊂ T M ⊕ T^{∗}M defining a linear Dirac structure on each fiber. Note that the dimensions of the range R(L) and the kernel Ker (L) (defined fiberwise) may vary from a point to another.

A Dirac structure is an almost Dirac structure whose sections are closed under the Courant bracket^{1} [ , ] : Γ(T M ⊕ T^{∗}M) × Γ(T M ⊕ T^{∗}M) ⟶ Γ(T M ⊕ T^{∗}M),

(2.4)  [(X, ξ), (Y, η)] = ([X, Y], L_{X}η − i_{Y} dξ).

For example, a bivector field π on M corresponds to a Dirac structure if and only if it defines a Poisson structure.

As observed in [27], one can use a closed 3-form ϕ on M to modify the standard Courant bracket as follows:

(2.5)  [(X, ξ), (Y, η)]_{ϕ} = ([X, Y], L_{X}η − i_{Y} dξ + ϕ(X, Y, ·)).

A Dirac structure twisted by ϕ, or simply a ϕ-twisted Dirac structure, is an almost Dirac structure whose sections are closed under [·, ·]_{ϕ}. A bivector π defines a ϕ-twisted Dirac structure if and only if

[π, π] = 2(∧^{3})π̃(ϕ),

where [ , ] is the Schouten bracket. Similarly, a 2-form ω defines a ϕ-twisted Dirac structure if and only if dω+ϕ = 0. These two kinds of Dirac structures are called ϕ-twisted Poisson structures and ϕ-twisted presymplectic structures.

Let (M, L_{M}, ϕ_{M}) and (N, L_{N}, ϕ_{N}) be twisted Dirac manifolds. A smooth map ψ : M ⟶ N is called a (forward) Dirac map if 𝔉(dψ)_{x}((L_{M})_{x}) = (L_{N})_{ψ(x)}, for all x ∈ M. (Here, and throughout this paper, we denote by (df)_{x} : T_{x}M ⟶ T_{f(x)}N the differential of a smooth function f : M ⟶ N at the point x ∈ M.)

We observe that the push-forward operation between Dirac structures defined in the linear case (2.3) is not well defined for manifolds in general. For instance, if ψ : M ⟶ N is a surjective submersion and L_{M} is a Dirac structure on M, the pointwise push-forward structures 𝔉d_{x}ψ((L_{M})_{x}) may differ for points x along the same ψ-fiber; even if they coincide, the resulting family of vector spaces might not be a subbundle, because of discontinuities of the map 𝔉.

2.2. Groupoids and algebroids. Throughout the text, G will denote a Lie groupoid. We denote the unit map by ε : M ⟶ G, the inversion by i : G ⟶ G and the source (resp. target) map by s : G ⟶ M (resp. t : G ⟶ M). We denote the set of composable pairs of groupoid elements by G_{2} (adopting the convention that (g, h) ∈ G × G is composable if s(g) = t(h)), and we write m : G_{2} ⟶ G for the multiplication operation. We will often identify M with the submanifold of G of identity arrows . In particular, given x ∈ M and v ∈ T_{x}M, ϵ(x) = 1_{x} ∈ G will be identified with x, and the tangent vector (dϵ)_{x}(v) ∈ T_{x}G with v ∈ T_{x}M.

1This is the non-skew-symmetric version of Courant’s [10] original bracket, as introduced in [21] and used in [27].

We emphasize that the Lie groupoids we consider may be non-Hausdorff. Basic important examples come from foliation theory and from integration of bundles of Lie algebras. However, M and the fibers of s are always assumed to be Hausdorff manifolds, and s-fibers will be assumed to be connected. We say that G is s-simply connected if the s-fibers are simply connected. We will use the notation G(−, x) = s^{−1}(x) (arrows starting at x ∈ M), and, for y ←^{g} x, we write the corresponding right multiplication map as R_{g} : G(−, y) ⟶ G(−, x), R_{g}(a) = ag.

The infinitesimal version of a Lie groupoid is a Lie algebroid. To fix our notation, we recall that a Lie algebroid A over M is a vector bundle A over M together with a Lie bracket [·, ·] on the space of sections Γ(A), and a bundle map ρ : A ⟶ T M satisfying the Leibniz rule

[α, fβ] = f[α, β] + L_{α}(f)β.

Here and elsewhere in this paper, we use the notation L_{α} for the Lie derivative with respect to the vector field ρ(α).

Given a Lie groupoid G, the associated Lie algebroid A = Lie(G) has fibers A_{x} = Ker (ds)_{x} = T_{x}(G(−, x)), for x ∈ M. Any α ∈ Γ(A) extends to a unique right-invariant vector field on G, which will be denoted by the same letter α. This correspondence identifies Γ(A) with the space X_{inv}^{s}(G) of vector fields on G which are tangent to the s-fibers and right invariant. The usual Lie bracket on vector fields induces the bracket on Γ(A), and the anchor is given by ρ = dt : A ⟶ T M.

Given a Lie algebroid A, an integration of A is a Lie groupoid G together with an isomorphism A ≅ Lie(G). By abuse of language, we will often call G alone an integration of A. If such a G exists, we say that A is integrable. In contrast with the case of Lie algebras, not all Lie algebroids are integrable (see [13] and references therein). However, as with Lie algebras, integrability implies the existence of a canonical s-simply connected integration G(A) of A, and any other s-simply connected integration of A will be isomorphic to G(A). Roughly speaking, G(A) consists of A-homotopy classes of A-paths, where an A-path consists of a path γ : I ⟶ M together with an “A-derivative of γ”, i.e. a path a : I ⟶ A above γ with the property that ρ(a) is the usual derivative \frac{dγ}{dt}. In general, G(A) is a topological groupoid carrying an additional “smooth structure” as the leaf space of a foliation, called in [13] the Weinstein groupoid of A, and its being a manifold is equivalent to the integrability of A. Further details and the precise obstructions to integrability can be found in [13], though some facts about G(A) will be recalled in Section 5.

A ϕ-twisted Dirac structure L has an induced Lie algebroid structure: the bracket on Γ(L) is the restriction of the Courant bracket [ ·, ·]_{ϕ}, and the anchor is the restriction of the projection pr_{1} : T M ⊕ T^{∗}M ⟶ T M. We denote by G(L) the groupoid associated to this Lie algebroid structure. The main theme in this paper is the description of the extra structure on G(L) induced by the Dirac structure. For instance, if L = L_{π} is the Dirac structure coming from a Poisson tensor π ∈ Γ(Λ^{2}T M), then G(L_{π}) carries a canonical symplectic structure making it into a symplectic groupoid (also interpreted as the phase space of an associated Poisson sigma-model, see [7, 14]). If π is a twisted Poisson structure, G(L_{π}) becomes what Cattaneo and Xu [8] call a non-degenerate quasi-symplectic groupoid (and we call a twisted symplectic groupoid).

2.3. Dirac ↔ presymplectic. Symplectic structures appear in several ways in connection with Poisson manifolds, e.g., as symplectic leaves, symplectic realizations and symplectic groupoids. With the relationship (Poisson manifolds) ↔ (symplectic structures) in mind, we briefly discuss in the remainder of this section the analogous correspondence for (twisted) Dirac manifolds.

Recall that a presymplectic manifold is a manifold M equipped with a closed 2-form ω. When Ker (ω) has constant rank, it defines a foliation on M; if this foliation is simple (i.e., the space of leaves M/Ker (ω) is smooth and the quotient map is a submersion), then M/Ker (ω) becomes a symplectic manifold, with symplectic form induced by ω. Hence, modulo global regularity issues, presymplectic manifolds can be reduced to symplectic manifolds. In the case of a ϕ-twisted presymplectic manifold, the reduction mentioned above works only when Ker (ω) ⊂ Ker (ϕ).

Just as Poisson structures can be viewed as singular foliations whose leaves are symplectic manifolds, ϕ-twisted Dirac structures L are singular foliations whose leaves are ϕ-twisted presymplectic manifolds. Given L, the foliation is defined at each x ∈ M by its range R(L_{x}). Note that this (singular) distribution coincides with the image of the anchor map of the Lie algebroid structure of L, and hence is necessarily integrable. In particular, the leaves of the foliation are the orbits of this Lie algebroid. For each such leaf S, the ϕ|_{S}- presymplectic form θ_{S} ∈ Ω^{2}(S) is, at each point, just the 2-form θ_{L_{x}} associated to the linear Dirac space L_{x}. As above, under certain regularity conditions (e.g. if Ker (ω) ⊂ Ker (ϕ) and Ker (L) is simple) one can quotient out M by Ker (L) and reduce (M, L) to a twisted Poisson manifold M/Ker (L) whose symplectic leaves are precisely the reductions of the presymplectic leaves of L. This suggests that Dirac structures could very well be called “pre-Poisson structures”.

The notion of a presymplectic realization of a ϕ-twisted Dirac structure L on M is more subtle: it is a Dirac map

μ : (P, η) ⟶ (M, L),

where η ∈ Ω^{2}(P) is a μ^{∗}ϕ-twisted presymplectic form (dη + μ^{∗}ϕ = 0), with the extra property that Ker (dμ) ∩ Ker (η) = {0}. This “non-degeneracy condition” for η will be explained in more detail in Section 6. As we will see, presymplectic realizations are the infinitesimal counterpart of the actions studied in [34], from where much of our motivation comes.

The only thing still to be explained in the correspondence between Dirac structures and presymplectic structures are presymplectic groupoids, and this leads us to the main results of the paper.

2.4. The main results. A 2-form ω on a Lie groupoid G is called multiplicative if the graph of m : G_{2} ⟶ G is an isotropic submanifold of (G, ω) × (G, ω) × (G, −ω), or, equivalently, if

(2.6)  m^{∗}ω = pr_{1}^{∗}ω + pr_{2}^{∗}ω,

where pr_{i} : G_{2} ⟶ G, i = 1, 2, are the natural projections.

Let G be a Lie groupoid over M, ϕ a closed 3-form on M, and ω a multiplicative 2-form on G. We call ω relatively ϕ-closed if dω = s^{∗}ϕ − t^{∗}ϕ.

Definition 2.1. We call (G, ω, ϕ) a presymplectic groupoid twisted by ϕ, or a ϕ-twisted presymplectic groupoid, if ω is relatively ϕ-closed, dim(G) = 2 dim(M), and if

(2.7)  Ker (ω_{x}) ∩ Ker (ds)_{x} ∩ Ker (dt)_{x} = {0},

for all x ∈ M.

Condition (2.7) provides a restriction on how degenerate ω can be; we will discuss it further in Section 4. If ϕ = 0 and ω is nondegenerate, the groupoid in Definition 2.1 is just a symplectic groupoid in the usual sense [28].

We find that, modulo integrability issues, (twisted) Dirac structures on M are basically the same thing as (twisted) presymplectic groupoids over M. This extends the relationship between integrable Poisson manifolds and symplectic groupoids (see e.g. [9]). More precisely, we have two main results on this correspondence. The first starts with the groupoid.

Theorem 2.2. Let (G, ω, ϕ) be a ϕ-twisted presymplectic groupoid. Then

(i) There is a (canonical, and unique) ϕ-twisted Dirac structure L on M, such that t : G ⟶ M is a Dirac map, while s is anti-Dirac;

(ii) There is an induced isomorphism between the Lie algebroid Lie(G) of G and the Lie algebroid of L.

We will also see that known properties of symplectic groupoids naturally extend to our setting. For instance, (Ker (dt)_{g})^{⊥} = Ker (ds)_{g} + Ker (ω_{g}) for all g ∈ G, and (T M)^{⊥} = T M + Ker (ω) (where ⊥ denotes the orthogonal with respect to ω).

Remark 2.3. The alternative convention of identifying the Lie algebroid of G with leftinvariant vector fields would lead to s being Dirac and t being anti-Dirac. This is the convention adopted in [9, 28].

In the situation of the theorem, we say that (G, ω) is an integration of the ϕ-twisted Dirac structure L. Note that such an integration immediately gives rise to an integration of the Lie algebroid associated with L (in the sense of subsection 2.2), namely the Lie groupoid G together with the isomorphism insured by Theorem 2.2, part (ii) (which does depend on ω!).

Our second result starts with a Dirac structure.

Theorem 2.4. Let L be a ϕ-twisted Dirac structure on M whose associated Lie algebroid is integrable, and let G(L) be its s-simply connected integration. Then there exists a unique 2-form ω_{L} such that (G(L), ω_{L}) is an integration of the ϕ-twisted Dirac structure L.

Hence we obtain a one-to-one correspondence between integrable ϕ-twisted Dirac structures on M and ϕ-twisted presymplectic groupoids over M by

L ↔ (G(L), ω_{L}).

In order to prove these two theorems, we have to understand the intricacies of multiplicative 2-forms, starting with their infinitesimal counterpart. We will prove the following result, which we expect to be useful in other settings as well.

Theorem 2.5. Let G be an s-simply connected Lie groupoid over M, with Lie algebroid A, and let ϕ ∈ Ω^{3}(M) be a closed 3-form. Then there is a one-to-one correspondence between

(i) multiplicative 2-forms ω ∈ Ω^{2}(G), with dω = s^{∗}ϕ − t^{∗}ϕ.

(ii) bundle maps ρ^{∗} : A ⟶ T^{∗}M with the properties:

\begin{matrix}⟨ρ^{∗}(α), ρ(β)⟩ & = −⟨ρ^{∗}(β), ρ(α)⟩; \\ ρ^{∗}([α, β]) = L_{α}(ρ^{∗}(β))− & L_{β}(ρ^{∗}(α)) + d⟨ρ^{∗}(α), ρ(β)⟩ + i_{ρ(α)∧ρ(β)}(ϕ). \end{matrix}

In fact, for a given ω, the corresponding ρ^{∗} is ρ_{ω}^{∗}(α)(X) = ω(α, X).

3. Multiplicative 2-forms on groupoids

In this section we discuss general properties of multiplicative 2-forms on Lie groupoids.

Lemma 3.1. If ω ∈ Ω^{2}(G) is multiplicative, then (i) ε^{∗}ω = 0, and i^{∗}ω = −ω; (ii) Ker (ds)_{g} + Ker (ω_{g}) ⊂ (Ker (dt)_{g})^{⊥} for all g ∈ G; (iii) For all arrows g : y ⟵ x, the map (di)_{g} induces an isomorphism

Ker (ω_{g}) ⟶ Ker (ω_{g^{−1}}),

and (dR_{g})_{y} induces isomorphisms

Ker (ω_{y}) ∩ Ker (ds)_{y} ⟶ Ker (ω_{g}) ∩ Ker (ds)_{g},

Ker (ω_{y}) ∩ Ker (ds)_{y} ∩ Ker (dt)_{y} ⟶ Ker (ω_{g}) ∩ Ker (ds)_{g} ∩ Ker (dt)_{g},

Ker (ds)_{y} ∩ (Ker (ds)_{y})^{⊥} ⟶ Ker (ds)_{g} ∩ (Ker (ds)_{g})^{⊥}.

(iv) On each orbit S of G, there is an induced 2-form θ_{S}, uniquely determined by the formula ω|_{G_{S}} = t^{∗}θ_{S} − s^{∗}θ_{S}, where G_{S} = s^{−1}(S) = t^{−1}(S) is the restriction of G to S. Moreover, if ω is relatively ϕ-closed, then dθ = −ϕ|_{S}; hence each orbit of G becomes a (ϕ|_{S})-twisted presymplectic manifold.

Note that, in (iii), Ker (ω_{y}) and Ker (ω_{g}) are not isomorphic in general.

Remark 3.2. The statements in (ii) and (iii) clearly hold with the roles of s and t interchanged (and right multiplication replaced by left multiplication).

Proof. We will need the following simple identities:

(3.1)  v_{g} = (dm)_{y,g}((dt)_{g}(v_{g}), v_{g}) = (dm)_{g,x}(v_{g}, (ds)_{g}(v_{g})), for all v_{g} ∈ T_{g}G,
(3.2)  (dR_{g})_{y}(α_{y}) = (dm)_{y,g}(α_{y}, 0), for all α_{y} ∈ Ker (ds)_{y}.
(3.3)  (dt)_{g}(v_{g}) = (dm)_{g,g^{−1}}(v_{g}, (di)_{g}(v_{g})), for all v_{g} ∈ T_{g}G.

The identity in (3.1) is obtained by differentiating id_{G} = m ∘ (t, id_{G}) = m ∘ (id_{G}, s). We verify the other two formulas similarly: for (3.2), we write R_{g} : G(−, y) ⟶ G as a ↦ (a, g) ↦^{m} ag, while for (3.3) we write the target map t as the composition of G ⟶ G ×_{M} G, g ↦ (g, g^{−1}) with m.

We now prove the lemma. Consider the map (id × i) : G ⟶ G ×_{M} G, g ↦ (g, g^{−1}). Applying (id × i)^{∗} to (2.6) and using (3.3), we deduce that t^{∗}ε^{∗}ω = ω + i^{∗}ω. Now applying ε^{∗} to this equation, we get ε^{∗}ω = ε^{∗}ω + ε^{∗}ω = 0, and therefore ω + i^{∗}ω = 0. This proves (i).

Using (3.1), (3.2), and the multiplicativity of m, we get

(3.4)  ω_{g}((dR_{g})_{y}(α_{y}), v_{g}) = ω_{y}(α_{y}, (dt)_{g}(v_{g}))

for all α_{y} ∈ Ker (ds)_{y} and v_{g} ∈ T_{g}G. When v_{g} ∈ Ker (dt)_{g}, since (dR_{g})_{y} maps Ker (ds)_{y} isomorphically into Ker (ds)_{g}, it follows that Ker (ds)_{g} ⊂ (Ker (dt)_{g})^{⊥}. Since Ker (ω) is inside all orthogonals, (ii) follows.

The first isomorphism in (iii) follows from i^{∗}ω = −ω. In order to check the following two, note that (dR_{g})_{y} maps Ker (ds)_{y} isomorphically onto Ker (ds)_{g}, and Ker (ds)_{y} ∩ Ker (dt)_{y} isomorphically onto Ker (ds)_{g} ∩ Ker (dt)_{g}. So it suffices to prove the first isomorphism induced by (dR_{g})_{y} (as the second follows from it). Let α_{y} ∈ Ker (ds)_{y}. Using (3.2) and the multiplicativity of ω, we have

ω_{g}((dR_{g})_{y}(α_{y}), (dm)_{y,g}(v_{y}, w_{g})) = ω_{y}(α_{y}, v_{y})

for all (v_{y}, w_{g}) tangent to the graph of the multiplication, which shows that α_{y} ∈ Ker (ds)_{y} ∩ Ker (ω_{y}) if and only if (dR_{g})_{y}(α_{y}) ∈ Ker (ω_{g}). The last isomorphism in (iii) is implied by

ω_{g}((dR_{g})_{y}(α_{y}), (dR_{g})_{y}(u_{y})) = ω_{y}(α_{y}, u_{y})

for all α_{y}, u_{y} ∈ Ker (ds)_{y}, which follows from (3.2) and the multiplicativity of ω.

Part (iv) is a statement about transitive groupoids (namely G|_{S}), so we may assume that S = M and G is transitive. If G = G(M) is the pair groupoid M × M over M, the proof is straightforward. In general, a transitive groupoid can be written as G = (P × P)/K, the quotient of the pair groupoid G(P) = P × P by the action of a Lie group K, where P ⟶ M is a principal K-bundle (fix x ∈ M and take P as the set of arrows starting at x). Let p_{1} : G(P) ⟶ G and p_{2} : P ⟶ M be the natural projections. (By abuse of language, we denote source and target maps on either G(P) or G by s and t, since the context should avoid any confusion.) If ω ∈ Ω^{2}(G) is multiplicative, then so is p_{1}^{∗}ω, and we can write p_{1}^{∗}ω = t^{∗}θ_{0} − s^{∗}θ_{0}, for some θ_{0} ∈ Ω^{2}(P). Since the correspondence θ_{0} ↦ t^{∗}θ_{0} − s^{∗}θ_{0} is injective, the fact that p_{1}^{∗}ω is basic implies that θ_{0} is basic. Hence θ_{0} = p_{2}^{∗}θ for some θ ∈ Ω^{2}(M). Since p_{1} is a submersion, it follows that ω = t^{∗}θ − s^{∗}θ. Similarly, s^{∗}ϕ − t^{∗}ϕ = dω = t^{∗}dθ − s^{∗}dθ implies that dθ = −ϕ. □

We now look at what happens at points x ∈ M. The following is a first sign of the rigidity of multiplicative 2-forms.

Lemma 3.3. If ω ∈ Ω^{2}(G) is multiplicative, then, at points x ∈ M,

Ker (ds)_{x} + Ker (ω_{x}) = (Ker (dt)_{x})^{⊥}
T_{x}M + Ker (ω_{x}) = (T_{x}M)^{⊥}
Ker (ω_{x}) = Ker (ω_{x}) ∩ Ker (ds)_{x} ⊕ Ker (ω_{x}) ∩ T_{x}M

(The first and third identities also hold for s and t interchanged.)

Furthermore, the following identities hold:

\{ \begin{matrix}dim(Ker (ω_{x}) ∩ T_{x}M) = \frac{1}{2}(dim(Ker (ω_{x})) + 2 dim(M) − dim(G)) \\ dim(Ker (ω_{x}) ∩ Ker (ds)_{x}) = \frac{1}{2}(dim(Ker (ω_{x})) − 2 dim(M) + dim(G)) \end{matrix}

Proof. Let us first prove the third equality. Let u_{x} ∈ Ker (ω_{x}), and write

u_{x} = (u_{x} − (ds)_{x}(u_{x})) + (ds)_{x}(u_{x}).

It then suffices to show that (ds)_{x}(u_{x}) ∈ Ker (ω_{x}). Using (3.1) (the one involving ds) and the multiplicativity of ω, we get

ω_{x}(u_{x}, (dm)_{x,x}(v_{x}, w_{x})) = ω_{x}(u_{x}, v_{x}) + ω_{x}((ds)_{x}(u_{x}), w_{x})

for all (v_{x}, w_{x}) tangent to the graph of m at (x, x). This shows that, indeed, ω_{x}((ds)_{x}(u_{x}), w_{x}) = 0 for all w_{x} ∈ T_{x}G. Now, for the first two equalities, note that the direct inclusions are consequences of Lemma 3.1. We now compute the dimensions of the spaces involved. First recall that for any subspace W of a linear presymplectic space (V, ω), we have

(3.5)  dim(W^{⊥}) = dim(V) − dim(W) + dim(W ∩ Ker (ω)).

Then, comparing dimensions in Ker (ds) + Ker (ω) ⊂ (Ker (dt))^{⊥}, we get

dim(Ker (ω)) ≤ dim(Ker (ω) ∩ Ker (ds)) + dim(Ker (ω) ∩ Ker (dt)) +
+2 dim(M) − dim(G)

at all g ∈ G. At a point x ∈ M,

dim(Ker (ω) ∩ Ker (ds)) = dim(Ker (ω) ∩ Ker (dt))

(since (di)_{x} is an isomorphism between these spaces), so

(3.6)  dim(Ker (ω_{x})) ≤ 2 dim(Ker (ω_{x}) ∩ Ker (ds)_{x}) + 2 dim(M) − dim(G).

Comparing dimensions in T_{x}M + Ker (ω_{x}) ⊂ (T_{x}M)^{⊥}, we get

dim(Ker (ω_{x})) ≤ dim(Ker (ω_{x}) ∩ T_{x}M) + dim(G) − 2 dim(M).

Since we already proved the third equality in the statement, we do know that

(3.7)  dim(Ker (ω_{x}) ∩ T_{x}M) = dim(Ker (ω_{x})) − dim(Ker (ω_{x}) ∩ Ker (ds)_{x}

Plugging this into the last inequality, we get precisely the opposite of (3.6). This shows that the inequality (3.6), as well as the direct inclusions for the first two relations in the statement, must become equalities. This immediately implies the dimension identities in the statement of the lemma and completes the proof. □

Note that the first and third identities in the lemma immediately imply that

(3.8)  (Ker (dt)_{x})^{⊥} = Ker (ds)_{x} + Ker (ω_{x}) ∩ T_{x}M, x ∈ M.

Corollary 3.4. Two multiplicative forms ω, ω′ ∈ Ω^{2}(G) which have the same differential dω = dω′, and which coincide at all x ∈ M, must coincide globally.

Proof. We may clearly assume that ω′ = 0. Equation (3.4) implies that i_{v}(ω) = 0 for all v tangent to the s-fibers. Since ω is closed, it follows that L_{v}(ω) = 0 for all such v’s, and therefore ω = s^{∗}η for some 2-form η on M. Since ω|_{T M} = 0, we must have η = 0, hence ω = 0. □

We now discuss the infinitesimal counterpart of multiplicative forms. A multiplicative 2-form ω ∈ Ω^{2}(G) induces a bundle map ρ_{ω}^{∗} : A ⟶ T^{∗}M characterized by the equation

(3.9)  ρ_{ω}^{∗}(α) ∘ (dt) = i_{α}(ω).

This equation is in fact equivalent to ρ_{ω}^{∗}(α) = i_{α}(ω)|_{M}, as a consequence of Lemma 3.1, part (ii).

Proposition 3.5. The bundle map ρ_{ω} associated to a multiplicative 2-form ω ∈ Ω^{2}(G) has the following properties:

(i) for α, β ∈ Γ(A), we have ⟨ρ_{ω}^{∗}(α), ρ(β)⟩ = −⟨ρ_{ω}^{∗}(β), ρ(α)⟩; (ii) if ϕ is a closed 3-form on M and dω = s^{∗}ϕ − t^{∗}ϕ, then

ρ_{ω}^{∗}([α, β]) = L_{α}(ρ_{ω}^{∗}(β)) − L_{β}(ρ_{ω}^{∗}(α)) + d⟨ρ_{ω}^{∗}(α), ρ(β)⟩ + i_{ρ(α)∧ρ(β)}(ϕ),

for all α, β ∈ Γ(A); (iii) Two multiplicative forms ω_{1} and ω_{2} coincide if and only if ρ_{ω_{1}}^{∗} = ρ_{ω_{2}}^{∗}, and dω_{1} =

dω_{2}.

Proof. By the comment preceding the proposition, (i) is clear, and (iii) is just Corollary 3.4. For part (ii), we fix v a vector field on M, and we prove that the right and left hand sides of (ii) applied to v are the same. Let α, β ∈ Γ(A), and denote the vector fields on G tangent to the s-fibers induced by them by the same letters. We consider

(3.10)  dω(α, β, ṽ) = −ω([α, β], ṽ) + ω([α, ṽ], β) − ω([β, ṽ], α) +
+L_{α}(ω(β, ṽ)) − L_{β}(ω(α, ṽ)) + Lṽ(ω(α, β))

at points x ∈ M, where ṽ is a vector field on G to be chosen. We will pick ṽ so that it agrees with v on M, and so that it is t-projectable, i.e.

(dt)_{g}(ṽ_{g}) = ṽ_{t(g)} = v_{t(g)}.

Note that, at points x ∈ M,

dω(α, β, ṽ) = (s^{∗}ϕ − t^{∗}ϕ)(α, β, ṽ) = −ϕ(ρ(α), ρ(β), v).

We claim that this observation and (3.10) imply part (ii).

In order to check that, we need some remarks on t-projectable vector fields and tprojectable functions on G (i.e. functions f with f(g) = f(t(g))):

1) any vector field v on M admits (locally if G is non-Hausdorff) a t-projectable extension ṽ to G;

2) if X is a vector field on G, and f is a t-projectable function, then L_{X}(f)(x) = L_{dt(X_{x})}(f|_{M})(x) for all x ∈ M;

3) if α, β ∈ Γ(A), then ω(α, β), as a function on G, is t-projectable;

4) if α ∈ Γ(A) and ṽ is t-projectable, then ω(α, ṽ) is t-projectable;

5) if α ∈ Γ(A) and ṽ is t-projectable, then (dt)_{x}[α, ṽ]_{x} = [ρ(α), v]_{x} for all x ∈ M. Note that, if ṽ extends v, then this implies that ω([α, ṽ], β) = ω([ρ(α), v], β) at all x ∈ M, and for all β ∈ Γ(A) (as a result of s and t-fibers being orthogonal with respect to ω).

Taking these remarks into account in (3.10), we immediately obtain

ρ_{ω}^{∗}([α, β])(v) = i_{[ρ(β),v]}ρ_{ω}^{∗}(α) − i_{[ρ(α),v]}ρ_{ω}^{∗}(β) + L_{ρ(α)}i_{v}ρ_{ω}^{∗}(β) −
L_{ρ(β)}i_{v}ρ_{ω}^{∗}(α) + i_{v}d(⟨ρ_{ω}^{∗}(β), ρ(α)⟩) + ϕ(ρ(α), ρ(β), v).

(Note that, in the statement of (ii), we follow the convention that L_{α} means L_{ρ(α)}, and the same for L_{β}). Now, using the identity i_{[X,Y]} = L_{X}i_{Y} − i_{Y} L_{X} on the first two terms of the right hand side, we obtain (ii).

Let us briefly discuss the remarks above:

In order to prove 1), let σ be a splitting of the bundle map dt : T G ⟶ t^{∗}T M. (In general, if G is non-Hausdorff, we can only find such a splitting locally. Since the formula to be proven is local, this is enough.) Then u_{x} = v_{x} − σ_{x}(v_{x}) is in Ker (dt)_{x}. Now we extend it to a vector field on G tangent to the t-fibers by left translations, and set ṽ_{g} = σ_{g}(v_{t(g)}) + u_{g};

Remark 2) is clear, while 3) and 4) follow from equation (3.4);

To check 5), we first look at (dt)_{x}[α, ṽ]_{x}. We see that

(3.11)  (dt)_{x} \frac{d}{dϵ}|_{ϵ=0} (dΦ_{α}^{ϵ})_{Φ_{α}^{−ϵ}(x)}(ṽ_{Φ_{α}^{−ϵ}(x)}) = \frac{d}{dϵ}|_{ϵ=0} d(t ∘ Φ_{α}^{ϵ})_{Φ_{α}^{−ϵ}(x)}(ṽ_{Φ_{α}^{−ϵ}(x)}),

where Φ_{α}^{ϵ} is the flow of α viewed as a vector field on G (extended by right translation). We have t ∘ Φ_{α}^{ϵ} = Φ_{ρ(α)}^{ϵ} ∘ t, and, since ṽ is t-projectable, (dt)(ṽ_{Φ_{α}^{−ϵ}(x)}) = v_{Φ_{ρ(α)}^{−ϵ}(x)}. Hence the last term in (3.11) equals to

\frac{d}{dϵ}|_{ϵ=0} d(t ∘ Φ_{α}^{ϵ})_{Φ_{α}^{−ϵ}(x)}(v_{Φ_{ρ(α)}^{−ϵ}(x)}),

and this gives us [ρ(α), v]_{x}. □

Remark 3.6. Note that the bundle map ρ_{ω}^{∗} (3.9) carries all the information about ω at units: at x ∈ M one has a canonical decomposition

T_{x}G ≅ T_{x}M ⊕ A_{x}, v_{x} ↦ (ds(v_{x}), v_{x} − (ds)_{x}(v_{x})),

and, with respect to this decomposition, it is easy to see that

(3.12)  ω((X, α), (Y, β)) = ρ_{ω}^{∗}(α)(Y) − ρ_{ω}^{∗}(β)(X) + ⟨ρ_{ω}^{∗}(α), ρ(β)⟩.

4. Multiplicative 2-forms and induced Dirac structures

In this section we discuss the relationship between multiplicative 2-forms and Dirac structures, proving in particular Theorem 2.2.

Let G be a Lie groupoid over M, and let ϕ ∈ Ω^{3}(M) be a closed 3-form. Given ω ∈ Ω^{2}(G) multiplicative, we first look at when the Dirac structure L_{ω} associated to ω can be linearly pushed forward by the target map t. For g ∈ G, let t_{∗}(L_{ω,g}) denote the push-forward of L_{ω,g} by (dt)_{g}:

(4.1)  t_{∗}(L_{ω,g}) = {((dt)_{g}(v_{g}), ξ_{x}) : i_{v_{g}}(ω) = ξ_{x} ∘ (dt)_{g}} ⊂ T_{x}M ⊕ T_{x}^{∗}M,

where x = t(g). In particular, restricting (4.1) to points in M, we get a (possibly nonsmooth, but of constant rank) bundle of linear Dirac structures L_{M} on M:

L_{M,x} = {((dt)_{x}(v_{x}), ξ_{x}) : i_{v_{x}}(ω) = ξ_{x} ∘ (dt)_{x}} ⊂ T_{x}M ⊕ T_{x}^{∗}(M).

The problem is to understand when this bundle agrees with (4.1) for all g ∈ G.

Definition 4.1. We say that a multiplicative 2-form ω on G is of Dirac type if t_{∗}(L_{ω,g}) = L_{M,t(g)} for all g ∈ G.

The next result provides alternative characterizations of 2-forms of Dirac type.

Lemma 4.2. Given a multiplicative 2-form ω on G, one has

(4.2)  L_{M} = {(ρ(α) + u, ρ_{ω}^{∗}(α)) : α ∈ A, u ∈ Ker (ω) ∩ T M},

and the following are equivalent:

(i) ω is of Dirac type; (ii) Ker (ds)_{g}^{⊥} = Ker (dt)_{g} + Ker (ω_{g}) for all g ∈ G ; (iii) (dt)_{g} : Ker (ω_{g}) ⟶ Ker (ω_{t(g)}) ∩ T M is surjective for all g ∈ G; (iv) dim(Ker (ω_{g})) = \frac{1}{2}(dim(Ker (ω_{x})) + dim(Ker (ω_{y}))) for all g ∈ G.

Proof. Let x ∈ M. Using the first and the third equalities in Lemma 3.3, we see that Ker (dt)_{x}^{⊥} = Ker (ds)_{x}+Ker (ω_{x})∩T M. Since the equation i_{v}(ω) = ξ∘(dt)_{x} in the definition of L_{M,x} implies that v ∈ Ker (dt)_{x}^{⊥}, the elements of L_{M,x} are pairs ((dt)_{x}(α) + u, ξ), where α ∈ Ker (ds)_{x} = A_{x}, u ∈ Ker (ω) ∩ T M, and i_{α}(ω) = ξ_{x} ∘ (dt)_{x}. Since this last equation is exactly the one defining ρ_{ω}^{∗}(α), (4.2) is proven.

Let g ∈ G, with s(g) = x, t(g) = y. First, let us compute the codimension of Ker (dt)_{g} + Ker (ω_{g}) in Ker (ds)_{g}^{⊥}. Using (3.5), and the fact that Ker (ds)_{g} ∩ Ker (ω_{g}) ≅ Ker (ds)_{y} ∩ Ker (ω_{y}), and similarly for t (cf. Lemma 3.1), we find that the codimension equals to

2 dim(M) − dim(G) + dim(Ker (ω_{x}) ∩ Ker (dt)_{x}) +
dim(Ker (ω_{y}) ∩ Ker (ds)_{y}) − dim(Ker (ω_{g})).

Using the last two formulas of Lemma 3.3, we get

(4.3) dim(Ker (ds)_{g}^{⊥}) − dim(Ker (dt)_{g} + Ker (ω_{g})) = − dim(Ker (ω_{g})) +

\frac{1}{2}(dim(Ker (ω_{y})) + dim(Ker (ω_{x}))).

Next, we compute the corank of the map in (iii) (note that (dt)_{g}(Ker (ω_{g})) ⊆ Ker (ω_{y})∩T M by (3.4)):

dim(Ker (ω_{y} ∩ T_{y}M)) − dim(Ker (ω_{g})) + dim(Ker (ω_{g}) ∩ Ker (dt)_{g}),

where, as above, we may replace g by x in the last term; replacing the first and last terms by the last two formulas of Lemma 3.3, we obtain

(4.4) dim(Ker (ω_{y} ∩ T_{x}M)) − dim((dt)_{g}(Ker (ω_{g})) = − dim(Ker(ω_{g})) + \frac{1}{2}(dim(Ker (ω_{y})) + dim(Ker (ω_{x}))).

Equations (4.3) and (4.4) immediately imply the equivalence of (ii), (iii), and (iv).

To see that (i) implies (iii), assume that v ∈ Ker (ω_{y}) ∩ T M. Since (v, 0) ∈ t_{∗}(L_{ω,y}), we must have (v, 0) ∈ t_{∗}(L_{ω,g}), i.e. v = (dt)_{g}(w), with w ∈ Ker (ω_{g}), and this proves (iii). For the converse, since L_{M,y} and t_{∗}(L_{ω,g}) have the same dimension, it suffices to prove that L_{M,y} ⊂ t_{∗}(L_{ω,g}). Let (ρ(α) + u, ρ_{ω}^{∗}(α)) be an element in L_{M,y}. Write ρ(α) = (dt)_{g}(α̃), and u = (dt)_{g}(ũ), where α̃ = (dR_{g})_{y}(α), and ũ ∈ Ker (ω_{g}). By formula (3.4) we have iα̃(ω) = i_{α}(ω)∘(dt)_{g} = ρ^{∗}(α)∘(dt)_{g}, and we now see that (ρ(α)+u, ρ^{∗}(α)) is in t_{∗}(L_{ω,g}). □

Note that, in contrast to the Poisson bivectors on Poisson groupoids [29], which always satisfy a condition like that in Definition 4.1, a form ω can be multiplicative and closed without being of Dirac type. (See Example 6.4.)

It can also happen that ω is multiplicative and of Dirac type, but “the leaves of L_{M}” do not coincide with the orbits of G (take, e.g., ω to be zero on a non-transitive groupoid). When they do coincide, the situation becomes much more rigid, as we now discuss. Recall that the orbits S of G are (twisted) presymplectic manifolds (see Lemma 3.1, (iii)); we denote the corresponding family of 2-forms by θ = {θ_{S}}, and we consider the bundle L_{θ} of linear Dirac structures on M induced by θ,

L_{θ,x} := {(v_{x}, ξ_{x}) : v_{x} ∈ T_{x}S, ξ_{x}|_{S} = i_{v_{x}}(θ)} ⊂ T_{x}M ⊕ T_{x}^{∗}M.

Lemma 4.3. Given a multiplicative 2-form ω ∈ Ω^{2}(G), and x ∈ M, the following are equivalent:

(i) the range R(L_{M,x}) equals to the tangent space to the orbit of G through x; (ii) the conormal bundle to the orbit through x, (Im (ρ_{x}))^{∘}, sits inside Im (ρ_{ω}^{∗}); (iii) Ker (ω_{x}) ∩ T_{x}M ⊂ Im (ρ); (iv) Ker (ω_{x}) ∩ T_{x}M = Ker (θ_{x});

(v) L_{M,x} = L_{θ,x}. Moreover, if these hold at all x ∈ M, then ω is of Dirac type.

Proof. Let us first show that

(4.5)  Ker (θ_{x}) = Ker (ω_{x}) ∩ Im (ρ), and Ker (ω_{x}) ∩ T_{x}M = Im (ρ_{ω}^{∗})^{∘}.

Given v ∈ T_{x}S = Im (ρ), we write v = ρ(α) with α ∈ Ker (ds)_{x}. From the defining property for θ = θ_{S}, we have θ(v, w) = θ(ρ(α), ρ(β)) = ω(α, β) for all w = ρ(β) ∈ T_{x}S. Hence a vector v is in Ker (θ_{x}) if and only if v = ρ(α) has the property that ω(α, β) = 0 for all β ∈ Ker (ds)_{x}, i.e. α ∈ Ker (ds)_{x} ∩(Ker (ds)_{x})^{⊥} = Ker (ds)_{x} ∩(Ker (dt)_{x} +Ker (ω_{x})∩T_{x}M), where for the last equality we used (3.8). Hence

Ker (θ_{x}) = (dt)_{x}(Ker (ds)_{x} ∩ (Ker (ds)_{x})^{⊥}
= (dt)_{x}(Ker (ds)_{x} ∩ (Ker (dt)_{x} + Ker (ω_{x}) ∩ T_{x}M)),

which is easily seen to be Ker (ω_{x})∩(dt_{x})(Ker (ds)_{x}) = Ker (ω_{x})∩Im (ρ). The other identity in (4.5) easily follows from the explicit formula for ω_{x} in terms of ρ and ρ_{ω}^{∗} mentioned in Remark 3.6. Now note that (4.5) proves the equivalence of (iii) and (iv), and also shows that (ii) and (iii) are just dual to each other. Hence we are left with proving that (i) is equivalent with (iii). But this is immediate since R(L_{M,x}) = Im (ρ_{x}) + Ker (ω_{x}) ∩ T M (e.g., cf. (4.2)).

We now prove the last assertion of the lemma. We will do that by showing that Ker (ds)_{g}^{⊥} = Ker (dt)_{g} + Ker (ω_{g}) for all g ∈ G (and using Lemma 4.2, (ii)). So let g : y ← x be in G.

Claim 1: Given v ∈ T_{g}(G), one has (dt)_{g}(v) ∈ Ker (ω_{y}) if and only if v ∈ Ker (ds)_{g}^{⊥}. This follows immediately from (3.4) and Lemma (3.1), part (i). Claim 2: If (i)-(v) hold at all x ∈ M, then

(dt)_{g}(Ker (dt)_{g}^{⊥}) = Im (ρ_{y}).

Clearly, the left hand side contains the right hand side, and we will compare the dimensions of the two spaces. Recalling that dim(M) = dim(G) − dim(Ker (dt)_{g}) and using (3.5), we find that the dimension of the left hand side is

dim(Ker (dt)_{g}^{⊥}) − dim(Ker (dt)_{g}^{⊥} ∩ Ker (dt)_{g}) = dim(Ker (dt)_{g} ∩ Ker (ω_{g}))
− dim(Ker (dt)_{g}^{⊥} ∩ Ker (dt)_{g})
+ dim(M).

Note that, by Lemma (3.1), part (iii) (see Remark (3.2)), we can replace g in the right hand side of the last formula by x = s(g). Hence the dimension we are interested in equals the one of (dt)_{x}(Ker (dt)_{x}^{⊥}). Using again that Ker (dt)_{x}^{⊥} = Ker (ds)_{x} +Ker (ω_{x})∩T_{x}M, together with (iii), we obtain (dt)_{x}(Ker (dt)_{x}^{⊥}) = Im (ρ_{x}). But dim(Im (ρ_{x})) = dim(Im (ρ_{y})) because x and y are on the same orbit, and this concludes the proof of the claim.

Claim 3: If (i)-(v) hold at all x ∈ M, then the following is a short exact sequence:

(4.6) 0 ⟶ Ker (dt)_{g} ∩ Ker (dt)_{g}^{⊥} ⟶ Ker (ds)_{g}^{⊥} ∩ Ker (dt)_{g}^{⊥} ⟶^{(dt)_{g}} Ker (ω_{y}) ∩ T M ⟶ 0.

The claim easily follows if one checks the surjectivity of the last map. To see that, note that if u ∈ Ker (ω_{y}) ∩ T M, then u ∈ Im (ρ) by (iii); so Claim 2 implies that u = dt_{g}(v), for some v ∈ Ker (dt)_{g}^{⊥}. But by Claim 1, v ∈ Ker (ds)_{g}^{⊥}. Hence the last map is surjective.

As a result of Claim 3, we get

(4.7) dim(Ker (ds)_{g}^{⊥} ∩ Ker (dt)_{g}^{⊥}) = dim(Ker (dt)_{g} ∩ Ker (dt)_{g}^{⊥}) + dim(Ker (ω_{y}) ∩ T M).

Let us recall again that

dim(Ker (dt)_{g}^{⊥} ∩ Ker (dt)_{g}) = dim(Ker (dt)_{x}^{⊥} ∩ Ker (dt)_{x}).

Using the short exact sequence

0 ⟶ Ker (dt)_{x}^{⊥} ∩ Ker (dt)_{x} ⟶ Ker (dt)_{x}^{⊥} ⟶^{(dt)_{x}} Im (ρ_{x}) ⟶ 0,

and (3.5) to compute dim(Ker (dt)_{x}^{⊥}), we find

(4.8) dim(Ker (dt)_{g}^{⊥} ∩ Ker (dt)_{g}) = dim(M) + dim(Ker (dt)_{x} ∩ Ker (ω_{x})) − dim(Im (ρ_{x})).

Let us now compute the dimension of Ker (ds)_{g}^{⊥} ∩ Ker (dt)_{g}^{⊥} = (Ker (ds)_{g} + Ker (dt)_{g})^{⊥}. Using (3.5) and replacing dim(Ker (ds)_{g} + Ker (dt)_{g}) by dim(Im (ρ_{x}) + dim(G) − dim(M), which is possible due to the exact sequence

0 ⟶ Ker (ds)_{g} ⟶ Ker (ds)_{g} + Ker (dt)_{g} ⟶^{(ds)_{g}} Im (ρ_{x}) ⟶ 0,

we find

(4.9) dim(Ker (ds)_{g}^{⊥} ∩ Ker (dt)_{g}^{⊥}) = dim(M) − dim(Im (ρ_{x})) +
dim((Ker (ds)_{g} + Ker (dt)_{g}) ∩ Ker (ω_{g})).

Plugging (4.8) and (4.9) into (4.7), we find

dim((Ker (ds)_{g} + Ker (dt)_{g}) ∩ Ker (ω_{g})) = dim(Ker (dt)_{x} ∩ Ker (ω_{x})) +
dim(Ker (ω_{y}) ∩ T M)
= \frac{1}{2}(dim(Ker (ω_{x})) + dim(Ker (ω_{y}))),

where the last identity follows from the last two formulas in Lemma 3.3. In particular,

\frac{1}{2}(dim(Ker (ω_{x})) + dim(Ker (ω_{y})) − dim(Ker (ω_{g})) ≤ 0.

On the other hand, we have seen in the proof of Lemma 4.2 that this number is the codimension of Ker (dt)_{g} + Ker (ω_{g}) in Ker (ds)_{g}^{⊥}, hence positive. So it must be zero, and this completes the proof. □

Note that, if L_{θ} is smooth, then it is automatically a ϕ-twisted Dirac structure since the 2-forms θ_{S} satisfy dθ_{S} = −ϕ|_{S}. In particular, we obtain

Corollary 4.4. Let ω be a relatively ϕ-closed, multiplicative 2-form. Then the following are equivalent:

(i) the bundle L_{M} is smooth and Ker (ω) ∩ T M ⊂ Im (ρ) (or any of the equivalent conditions in Lemma 4.3);

(ii) there is a ϕ-twisted Dirac structure L on M so that t : G ⟶ M is a Dirac map,

and the presymplectic leaves of L coincide with the orbits of G. If these conditions hold, then L coincides with

L_{M} = {(ρ(α) + u, ρ_{ω}^{∗}(α)) : α ∈ A, u ∈ (Im (ρ_{ω}^{∗}))^{∘}},

and L_{M} is a ϕ-twisted Dirac structure with presymplectic leaves (S, θ_{S}) and kernel Ker (ω)∩ T M = (Im (ρ_{ω}^{∗}))^{∘}.

Another interesting consequence is the following:

Proposition/Definition 4.5. Given a relatively ϕ-closed, multiplicative 2-form ω on G, the following are equivalent:

(i) there exists a ϕ-twisted Poisson structure on M so that t : G ⟶ M is a Dirac map;

(ii) dim(Ker (ω_{x})) = dim(G) − 2 dim(M) for all x ∈ M; (iii) Ker (dt_{g})^{⊥} = Ker (ds_{g}); (iv) Ker (ω_{x}) ⊂ Ker (ds)_{x}∩Ker (dt)_{x} for all x ∈ M (or, equivalently, Ker (ρ_{ω}^{∗}) ⊂ Ker (ρ)).

In this case we say that (G, ω) is a twisted over-symplectic groupoid.

Proof. Condition (i) is equivalent to L_{M} being smooth and having zero kernel, i.e. Ker (ω)∩ T M = {0}. Hence, by (3.8), (Ker (dt)_{x})^{⊥} = Ker (ds)_{x}, for all x ∈ M. A simple dimension counting, using (3.5) and Ker (ω_{x}) = Ker (ω_{x}) ∩ Ker (dt)_{x}, directly shows (ii). Using once again (3.5) to compute (dim(dt)_{g})^{⊥}, and recalling that dim(Ker (ω_{g}) ∩ Ker (dt)_{g}) = dim(Ker (ω_{x}) ∩ Ker (dt)_{x}), for x = s(g), another dimension counting shows that (ii) implies (iii). Note that (iii) implies that Ker (ω_{x}) ⊂ Ker (ds)_{x}, and since (iii) also holds for t and s interchanged, we get Ker (ω_{x}) ⊂ Ker (ds)_{x} ∩ Ker (dt)_{x} and (iv) follows. Finally, if (iv) holds, then (4.2) implies that (ρ, ρ_{ω}^{∗}) : A ⟶ L_{M} is surjective; thus L_{M} is smooth. Since Ker (ω_{x}) ∩ T_{x}M = Ker (ds)_{x} ∩ Ker (dt)_{x} ∩ T_{x}M = {0}, ω is of Dirac type and the Dirac structure induced on M is Poisson. □

Definition 4.6. Let G be a Lie groupoid over M, and let ϕ be a closed 3-form on M. A multiplicative 2-form ω on G is called robust if the isotropy bundle 𝔤(ω), defined by

𝔤_{x}(ω) = Ker (ω_{x}) ∩ Ker (ds)_{x} ∩ Ker (dt)_{x}, x ∈ M,

has constant rank equal to dim(G) − 2 dim(M). If ω is also relatively ϕ-closed, we say that (G, ω) is a ϕ-twisted over-presymplectic groupoid.

We will explain this “over” terminology in Remark 4.9 below.

A ϕ-twisted presymplectic groupoid (Def. 2.1) is a ϕ-twisted over-presymplectic groupoid (G, ω) with dim(G) = 2 dim(M). If ω is nondegenerate, then (G, ω) is called a ϕ-twisted symplectic groupoid. (This coincides with the notion of quasi-symplectic groupoid of Cattaneo and Xu [8].)

Note that, if (G, ω) is a ϕ-twisted over-presymplectic groupoid, then 𝔤(ω) is a smooth bundle of Lie algebras.

Lemma 4.7. Let ω be a multiplicative 2-form on G. The following statements are equivalent:

(i) ω is robust;

(ii) (ρ, ρ_{ω}^{∗}) : A ⟶ L_{M} is surjective;

(iii) Ker (dt)_{g}^{⊥} = Ker (ds)_{g} + Ker (ω_{g}) ∩ Ker (dt)_{g}; (iv) (dt)_{g} : Ker (ω_{g}) ∩ Ker (ds)_{g} ⟶ Ker (ω_{t(g)}) ∩ T M is onto;

(v) Ker (ω_{g}) = Ker (ω_{g}) ∩ Ker (ds)_{g} + Ker (ω_{g}) ∩ Ker (dt)_{g}. Note that (iii), (iv) and (v) are required to hold for all g ∈ G or, equivalently, for all g = x ∈ M. In this case, ω is of Dirac type and L_{M} is a ϕ-twisted Dirac structure on M.

Proof. The equivalence of (i) and (ii) is immediate by a dimension counting. Using (3.5) and noticing that (iii) holds if and only if the dimensions of the right and left hand sides coincide (due to the inclusion in Lemma 3.1, part (ii)), we conclude that (iii) is equivalent to

(4.10)  dim(Ker (ds)_{g}) − dim(M) = dim(𝔤_{g}(ω))

Also, (ρ, ρ_{ω}^{∗}) is surjective if and only if the dimension of its image (which equals dim(Ker (ds))− dim(Ker (ω) ∩ Ker (ds) ∩ Ker (dt))) coincides with the dimension of L_{M} (which equals dim(M)), and this is precisely (4.10) at points in M. On the other hand, Lemma 3.1, part (iii), shows that (4.10) is equivalent to the same relation at the point t(g) ∈ M. So we conclude that (i), (ii) and (iii) are equivalent to each other. A direct dimension counting, allied with Lemma 3.1, and then the last two formulas of Lemma 3.3, shows that (iv) is also equivalent to (i).

Finally, a similar argument (i.e. dimension counting together with Lemma 3.1 and the last formula of Lemma 3.3) shows that the equality in (v) is equivalent to

(4.11)  dim(Ker ω_{g}) = \frac{1}{2}(dim(Ker (ω_{x}) + dim(Ker (ω_{y})) +
dim(G) − 2 dim(M) − dim(𝔤_{x}(ω))

at all g ∈ G (where x = s(g), y = t(g)). Evaluating this expression at g = x ∈ M immediately implies that ω is robust (i.e., (i)). Conversely, if (iii) holds, then the equivalence of (ii) and (iv) of Lemma 4.2 implies (4.11).

□

Corollary 4.8. Let G be a Lie groupoid over M, and let ω ∈ Ω^{2}(G) be multiplicative. The following statements are equivalent:

(i) dim(G) = 2 dim(M), and ω is robust;

(ii) (ρ, ρ_{ω}^{∗}) : A ⟶ L_{M} is an isomorphism;

(iii) Ker (dt)_{g}^{⊥} = Ker (ds)_{g} ⊕ Ker (ω_{g}) ∩ Ker (dt)_{g}; (iv) Ker (ω_{g}) = Ker (ω_{g}) ∩ Ker (ds)_{g} ⊕ Ker (ω_{g}) ∩ Ker (dt)_{g};

(v) (dt)_{g} : Ker (ω_{g}) ∩ Ker (ds)_{g} ⟶ Ker (ω_{t(g)}) ∩ T M is an isomorphism, where (iii), (iv) and (v) are required to hold for all g ∈ G or, equivalently, for all g = x ∈ M.

Moreover, if ω is relatively ϕ-closed (i.e. (G, ω) is a ϕ-twisted presymplectic groupoid), then L_{M} is a ϕ-twisted Dirac structure on M, characterized by the properties that t is a Dirac map and that (ρ, ρ_{ω}^{∗}) in (ii) above is an isomorphism of algebroids.

Proof. The equivalence of (i) and (ii) follows immediately from (4.10). The rest is a direct consequence of Lemma 4.7. □

Clearly, the last corollary proves Theorem 2.2.

Remark 4.9.

(i) For any ϕ-twisted over-presymplectic groupoid (G, ω), the isotropy bundle 𝔤(ω) is a smooth bundle of Lie algebras, and a subbundle of the (possibly non-smooth) isotropy Lie algebra bundle of G, 𝔤(G) = Ker (ρ). Integrating 𝔤(ω) to a bundle of simply connected Lie groups G(ω), assuming that the quotient G/G(ω) is smooth, we see that ω reduces to a multiplicative 2-form \overline{ω} on G/G(ω), and (G/G(ω), \overline{ω}) becomes a ϕ-twisted presymplectic groupoid, over the same manifold M, which induces the same ϕ-twisted Dirac structure on M. (Of course, if (G, ω) is oversymplectic, then (G/G(ω), \overline{ω}) is a symplectic groupoid.) Although this does not always work (i.e. G/G(ω) may be non-smooth), this explains our “over” terminology, and it shows that 𝔤(ω) can be viewed as an obstruction to our final goal of obtaining a one-to-one correspondence between ϕ-twisted Dirac structures on M and groupoids over M equipped with a certain extra structure. Examples of overpresymplectic groupoids which cannot be reduced to presymplectic groupoids will be given in Section 6.3.

(ii) Similarly, if (G, ω) is a ϕ-twisted presymplectic groupoid over M then, again modulo global regularity issues, one can quotient it out by Ker (ω) to obtain a ϕ-twisted syplectic groupoid over the ϕ-twisted Poisson manifold M/Ker (L). Details of this construction will be given elsewhere.

5. Reconstructing multiplicative forms

In this section we explain how multiplicative forms can be reconstructed from their infinitesimal counterpart. In particular, we will complete the proof of Theorem 2.4.

Recall that (cf. Proposition 3.5) for ω ∈ Ω^{2}(G) multiplicative and relatively ϕ-closed, the associated bundle map ρ_{ω}^{∗} satisfies

(5.1)  ⟨ρ_{ω}^{∗}(β), ρ(α)⟩ = −⟨ρ_{ω}^{∗}(α), ρ(β)⟩ for all α, β ∈ Γ(A);
(5.2)  ρ_{ω}^{∗}([α, β]) = L_{α}(ρ_{ω}^{∗}(β)) − L_{β}(ρ_{ω}^{∗}(α)) − d⟨ρ_{ω}^{∗}(β), ρ(α)⟩ + i_{ρ(α)∧ρ(β)}(ϕ).

We will use the notation

(d_{A}ρ_{ω}^{∗})(α, β) := ρ_{ω}^{∗}([α, β]) − L_{α}(ρ_{ω}^{∗}(β)) + L_{β}(ρ_{ω}^{∗}(α)) + d⟨ρ_{ω}^{∗}(β), ρ(α)⟩.

The rest of this section is devoted to the proof of the next result (see also [8]).

Theorem 5.1. If G is an s-simply connected Lie groupoid, and ϕ ∈ Ω^{3}(M) is closed, then the correspondence ω ↦ ρ_{ω}^{∗} induces a bijection between the space of relatively ϕ-closed multiplicative 2-forms on G, and bundle maps ρ^{∗} : A ⟶ T^{∗}M satisfying conditions (5.1) and (5.2) above.

By Prop. 3.5, part (iii), we only have to check that the map ω ↦ ρ_{ω}^{∗} is surjective, i.e., given ρ^{∗} : A ⟶ T^{∗}M satisfying (5.1) and (5.2), we must produce a relatively ϕ-closed multiplicative 2-form ω on G. By Corollary 4.8, if Ker (ρ^{∗}) ∩ Ker (ρ) = {0}, then the resulting 2-form makes G into a twisted presymplectic groupoid; note that if L is a Dirac structure on M, then Theorem 5.1, applied to ρ^{∗} = pr_{2} : L ⟶ T^{∗}M, together with Corollary 4.8 imply the correspondence in Theorem 2.4.

Let us recall [13] how G can be reconstructed from A. Let I = [0, 1] and p : A ⟶ M be the natural projection (we will also denote other bundle projections by p whenever the context is clear). Consider the Banach manifold P̃(A) consisting of paths a : I ⟶ A of class C^{1}, whose base path γ = p ∘ a : I ⟶ M is of class C^{2}, and the submanifold P (A) defined by the equation ρ ∘ a = \frac{d}{dt}γ (i.e. a is an A-path). The manifold P (A) comes endowed with an infinitesimal action of the infinite dimensional Lie algebra 𝔤 consisting of time dependent sections η_{t} (t ∈ [0, 1])^{2} of A, with η_{0} = η_{1} = 0. To define the Lie algebra map

𝔤 ∋ η ↦ X_{η} ∈ X (P (A))

describing the action, it will be more convenient to introduce the flows of the vector fields X_{η}. One advantage of this approach is that X_{η} will be defined on the entire P̃(A). Given a_{0} ∈ P (A), we construct the flow a_{ϵ} = Φ_{X_{η}}^{ϵ}(a_{0}) in such a way that a_{ϵ} are paths above γ_{ϵ}(t) = Φ_{ρ(η_{t})}^{ϵ}γ_{0}(t), where γ_{0} is the base path of a_{0}, and Φ_{ρ(η_{t})}^{ϵ} is the flow of the vector field ρ(η_{t}). We choose a time dependent section ξ_{0} of A with ξ_{0}(t, γ_{0}(t)) = a_{0}(t), and consider the (ϵ, t)-dependent section of A, ξ = ξ(ϵ, t), solution of

(5.3)  \frac{dξ}{dϵ} − \frac{dη}{dt} = [ξ, η], ξ(0, t) = ξ_{0}(t).

2In this section, t will denote a “time” parameter.

Then a_{ϵ}(t) = ξ_{ϵ}(t, γ_{ϵ}(t)). This defines the desired vector fields X_{η}, the action of 𝔤, and the foliation on P (A). It is clear from the definition that

(5.4)  (dp)(X_{η}) = ρ(η) ∘ p and p ∘ Φ_{X_{η}}^{ϵ}(a) = Φ_{ρ(η)}^{ϵ} ∘ p(a).

Now, G(A) = P (A)/ ∼ is a topological groupoid for any A: the source (resp. target) map is obtained by taking the starting (resp. ending) point of the base paths, and the multiplication is defined by concatenation of paths. Moreover, G must be isomorphic to G(A).

We will construct forms on G(A) by constructing forms on P (A) which are basic with respect to the the action (i.e. L_{X_{η}}ω = 0 and i_{X_{η}}ω = 0 for all η ∈ 𝔤). First, the 3-form ϕ on M induces a 2-form on T M that, at X ∈ T_{x}M, is i_{X}(p_{M}^{∗}ϕ_{x}), where p_{M} : T M ⟶ M is the projection. We pull it back by ρ to A and lift it to P̃(A) to get a 2-form ω_{ϕ} on P̃(A):

ω_{ϕ,a}(V, W) = ∫_{0}^{1} ϕ(ρ(a(t)), (dp)_{a(t)}(V (a(t)), (dp)_{a(t)}(W (a(t))))dt.

In order to produce basic forms, we must understand the behavior of ω_{ϕ,a} when we take derivatives along, or interior products by, vector fields of the form X_{η}.

Lemma 5.2. For any closed 3-form ϕ on M,

ω_{ϕ}(X_{η,a}, X_{a}) = ∫_{0}^{1} ϕ(ρa(t), ρη(t, γ(t)), (dp)_{a(t)}(X_{a}(t)))dt,
dω_{ϕ} = t^{∗}ϕ − s^{∗}ϕ,

where η ∈ 𝔤, X_{a} is a vector tangent at a to P̃(A), and s, t : P̃(A) ⟶ M take the start/end points of the base path.

Proof. The first formula is immediate from the definition of ω_{ϕ} and the the first equation in (5.4), while the second formula follows from Stokes’ Theorem. □

Let us now consider the Liouville one-form σ^{c} on T^{∗}M, and the associated canonical symplectic form ω^{c}. Recall that, on T^{∗}M,

σ_{ξ_{x}}^{c}(X_{ξ_{x}}) = ⟨ξ_{x}, (dp)_{ξ_{x}}(X_{ξ_{x}})⟩, (ξ_{x} ∈ T_{x}^{∗}M, X_{x} ∈ T_{ξ_{x}}T^{∗}M)

and ω^{c} = −dσ^{c}. Using ρ^{∗} : A ⟶ T^{∗}M, we pull σ^{c} and ω^{c} back to A, and we denote by σ̃ and ω̃ the resulting forms on P̃(A). Hence ω̃ = −dσ̃, and

σ̃_{a}(X_{a}) = ∫_{0}^{1} ⟨ρ^{∗}(a(t)), (dp)_{a(t)}(X_{a}(t))⟩dt.

Lemma 5.3. If ρ^{∗} satisfies (5.1), then, for any A-path a and any vector field X on P̃(A),

i_{X_{η,a}}(σ̃) = − ∫_{0}^{1} ⟨ρ^{∗}η(t, γ(t)), ρ(a(t))⟩dt, and

L_{X_{η}}(σ̃)(X_{a}) = ∫_{0}^{1} ⟨(d_{A}ρ^{∗})(a(t), η(t, γ(t)), (dp)(X_{a}(t))⟩dt − d(∫_{0}^{1} ⟨ρ^{∗}η(t, γ(t)), \frac{dγ}{dt} ⟩dt),

where the last term is the differential of the function a ↦ ∫_{0}^{1} ⟨ρ^{∗}η(t, γ(t)), \frac{dγ}{dt} ⟩dt, γ is the base path of a and η ∈ 𝔤.

Proof. For the first formula, we use the definition of σ̃:

σ̃(X_{η,a}) = σ_{ρ^{∗}(a)}^{c}((dρ^{∗})_{a}(X_{η,a})) = ∫_{0}^{1} ⟨ρ^{∗}(a), (dp)(X_{η,a})⟩,

the first formula in (5.4), and (5.1). To prove the second formula, we use the definition of σ̃ and the second formula in (5.4) to rewrite the Lie derivative

\begin{aligned}L_{X_{η}}(σ̃)(X_{a}) & = \frac{d}{dϵ}|_{ϵ=0} σ̃(dΦ_{X_{η}}^{ϵ})_{a}(X_{a}) = \\ & = ∫_{0}^{1} \frac{d}{dϵ}|_{ϵ=0} ⟨ρ^{∗}ξ_{ϵ}(t, γ_{ϵ}(t)), (dΦ_{ρ(η_{t})}^{ϵ})_{γ(t)}(X′(t))⟩dt \end{aligned}

where γ_{ϵ}(t) and ξ_{ϵ} are as in the construction above of the vector fields X_{η}, and X′(t) = (dp)_{a(t)}(X_{a}(t)). To compute this expression, we use a connection ∇ on M. The expression in the last integral has the following two terms:

(5.5)  ⟨ρ^{∗}ξ_{ϵ}(t, γ_{ϵ}(t)), ∂_{ϵ}(dΦ_{ρ(η_{t})}^{ϵ})_{γ(t)}(X′(t))⟩
(5.6)  ⟨∂_{ϵ}ρ^{∗}ξ_{ϵ}(t, γ_{ϵ}(t)), X′(t)⟩

where ∂_{ϵ} is the derivation of paths in T M and T^{∗}M induced by the connection. On the other hand, for any vector fields V and W on M, ∂_{ϵ}(dΦ_{W}^{ϵ})_{x}(V_{x}) = \overline{∇}_{V_{x}}(W), where \overline{∇}_{V} (W) = ∇_{W} (V) + [V, W] is the conjugated connection (this is a simple check in local coordinates). Hence (5.5) equals to

⟨ρ^{∗}(ξ), [X′, ρ(η)] + ∇_{ρ(η)}(X′)⟩_{γ(t)} = ⟨L_{ρ(η)}(ρ^{∗}(ξ)) − ∇_{ρ(η)}(ρ^{∗}(ξ)), X′(t)⟩_{γ(t)},

where, for a moment, we have made X′ into a vector field extending X′(t) (for each fixed t). On the other hand, (5.6) equals to

⟨∇_{\frac{dγ}{dt}} (ρ^{∗}(ξ)) + \frac{dρ^{∗}(ξ_{ϵ})}{dϵ}, X′(t)⟩ = ⟨∇_{ρ(η)}(ρ^{∗}(ξ)) + ρ^{∗}([ξ, η]) + \frac{dρ^{∗}(η_{t})}{dt}, X′(t)⟩,

(at the point γ(t)), where we have used the defining equation (5.3) for ξ. Adding the two expressions we obtained for (5.5) and (5.6) (at ϵ = 0), we get

\begin{aligned}L_{X_{η}}(σ̃)(X_{a}) & = ∫_{0}^{1} ⟨L_{ρ(η)}(ρ^{∗}(ξ_{0})) + ρ^{∗}([ξ_{0}, η]) + \frac{dρ^{∗}(η_{t})}{dt}, X′(t)⟩_{γ(t)}dt \\ & = ∫_{0}^{1} ⟨(d_{A}ρ^{∗})(ξ_{0}, η) + i_{ρ(ξ_{0})}(dρ^{∗}(η)) + \frac{dρ^{∗}(η_{t})}{dt}, X′(t)⟩dt \end{aligned}

where, in the last equality, we used the definition of d_{A}(ρ^{∗}). At this point, the computation is transfered to M, since the expression

L_{X_{η}}(σ̃)(X_{a}) − ∫_{0}^{1} ⟨(d_{A}ρ^{∗})(a(t), η(t, γ(t))), (dp)_{a(t)}(X_{a}(t))⟩dt

equals to

(5.7)  ∫_{0}^{1} ⟨i_{\frac{dγ}{dt}} (du^{t}) + \frac{du^{t}}{dt} , X′(t)⟩_{γ(t)}dt

where u^{t} = ρ^{∗}(η^{t}). To finish the proof, we will use the next lemma. □

Lemma 5.4. For any path γ on M, any path X′ : I ⟶ T M above γ, and any timedependent 1-form u^{t} on M, we have

L_{X′} (∫_{0}^{1} ⟨u(t, γ(t)), \frac{dγ}{dt} ⟩dt) + ∫_{0}^{1} ⟨i_{\frac{dγ}{dt}} (du^{t}) + \frac{du^{t}}{dt} , X′(t)⟩_{γ(t)}dt = ⟨u(t, γ(t)), X′(t)⟩|_{0}^{1}

(The function on which L_{X′} acts is defined as in Lemma 5.3.)

Proof. We may assume that there is a vector field Z such that Z(γ(t)) = X′(t) (otherwise one just brakes γ into smaller paths, and note that the formula to be proven is additive with respect to concatenation of paths). We first compute the first integral using, as above, a connection ∇ and the formula ∂_{ϵ}(dΦ_{W}^{ϵ})_{x}(V_{x}) = \overline{∇}_{V_{x}}(W):

\frac{d}{dϵ}|_{ϵ=0} ∫_{0}^{1} ⟨u(t, Φ_{Z}^{ϵ}(γ(t))), \frac{d}{dt}(Φ_{Z}^{ϵ}(γ(t)))⟩dt =

= ∫_{0}^{1} ⟨u^{t}, ∂_{ϵ}(dΦ_{Z}^{ϵ})_{γ(t)}(\frac{dγ}{dt})⟩dt + ⟨∂_{ϵ}(u(t, Φ_{Z}^{ϵ}(γ(t))), \frac{dγ}{dt} ⟩ =

= ∫_{0}^{1}(⟨u^{t}, \overline{∇}_{\frac{dγ}{dt}} (X′)⟩ + ⟨u^{t}, \frac{dγ}{dt} ⟩)dt

Now, it is easy to see that the sum of the term in the last integral with the term appearing in the second integral in the statement is precisely

⟨u, ∇_{\frac{dγ}{dt}} (X′)⟩_{γ(t)} + ⟨∇_{\frac{dγ}{dt}} (u) + \frac{du^{t}}{dt} , X′⟩_{γ(t)} = \frac{d}{dt}⟨u(t, γ(t)), X′(t)⟩

□

Using Cartan’s formula L_{X} = di_{X} + i_{X}d, the next result follows directly from Lemma 5.2.

Lemma 5.5. If ρ^{∗} satisfies (5.1), then, for any A-path a and any vector field X on P̃(A), we have

i_{X_{η,a}}(ω̃)(X_{a}) = d(∫_{0}^{1} ⟨ρ^{∗}(η), \frac{dγ}{dt} − ρ(a)⟩dt) − ∫_{0}^{1} ⟨(d_{A}ρ^{∗})(a, η), (dp)(X_{a})⟩dt.

We can now complete the proof of Theorem 5.1, i.e., reconstruct the 2-form ω out of ρ^{∗}. Let us assume that ρ^{∗} satisfies both conditions (5.1) and (5.2), and put ω̃_{ϕ} = ω̃ + ω_{ϕ}. Then the first equation in Lemma 5.2 and the equation in Lemma 5.5 show that i_{X_{η}}(ω̃_{ϕ}) = 0 on P (A). On the other hand, the second equation in Lemma 5.2 and the fact that ds and dt vanish on X_{η}’s (since η vanishes at end-points), imply that

i_{X_{η}}dω̃_{ϕ} = i_{X_{η}}(t^{∗}ϕ − s^{∗}ϕ) = 0.

Hence ω̃_{ϕ} is basic, and induces a 2-form ω_{0} on G(A). The multiplicativity of ω_{0} follows from the additivity of integration. To compute the associated ρ_{ω_{0}}^{∗}(α)(X) = ω(α, X), one has to look at the identification of the Lie algebroid of G(A) with A (see [13]). After straightforward computations, we find that

ρ_{ω_{0}}^{∗}(α)(X) = ω_{can}((ρ(α), ρ^{∗}(α)), (X, 0)).

Here, ω_{can} is the linear version of the symplectic form,

ω_{can}((X_{1}, η_{1}), (X_{2}, η_{2})) = η_{2}(X_{1}) − η_{1}(X_{2}).

We see that ρ_{ω_{0}}^{∗} = −ρ^{∗}. On the other hand dω_{0} = t^{∗}ϕ − s^{∗}ϕ, as it follows from the similar formula satisfied by ω_{ϕ} (cf. Lemma 5.2). Hence ω = −ω_{0} will have the desired properties.

6. Examples

We discuss in this section some examples of multiplicative 2-forms, presymplectic groupoids and their corresponding Dirac structures.

6.1. Multiplicative 2-forms: first examples. We now discuss some basic examples of multiplicative 2-forms on Lie groupoids.

Example 6.1. (Lie groups)

If H is a Lie group (so the base M is just the one-point space, consisting of the identity in H), then the zero form is the only multiplicative form on H. This follows from Lemma 3.1 (part (ii), or part (iv)).

Example 6.2. (Lie groupoids integrating tangent bundles)

Let M be a ϕ-twisted presymplectic manifold. Hence ϕ ∈ Ω^{3}(M) is closed, and M is equipped with a 2-form ω_{M} with dω_{M} +ϕ = 0. Consider the pair groupoid G = M ×M with the product (x, y) ∘ (y, z) = (x, z) (hence s = pr_{2}, t = pr_{1}). A simple computation shows that M × M equipped with the 2-form ω = pr_{1}^{∗}ω_{M} − pr_{2}^{∗}ω_{M} is a ϕ-twisted presymplectic groupoid, and that the ϕ-twisted Dirac structure induced on M (identified with the diagonal in M ×M) is just the one associated with ω_{M}. As usual, one obtains the s-simply connected ϕ-twisted presymplectic groupoid corresponding to ω_{M} by pulling ω ∈ Ω^{2}(M × M) back to Π(M), the fundamental groupoid of M, using the natural covering map Π(M) ⟶ M × M (which is also a groupoid morphism).

Example 6.3. (Pull-backs)

Let L be a ϕ-twisted Dirac structure on M, and let (G(L), ω_{L}) be a presymplectic groupoid integrating it. If f : P ⟶ M is a submersion (we will see that weaker conditions are possible), we can form the pull-back groupoid f^{∗}G(L) := P ×_{M} G(L)×_{M} P consisting of triples (p, g, q) with g : f(p) ← f(q), s = pr_{3}, t = pr_{1}, and (p, g, q) · (q, h, r) = (p, gh, r). It is simple to check that dim(f^{∗}G(L)) = 2 dim(P), and that the form pr_{2}^{∗}ω_{L} ∈ Ω^{2}(f^{∗}G(L)) is multiplicative, robust and relatively f^{∗}ϕ-closed. So (f^{∗}G(L), pr_{2}^{∗}ω_{L}) is a f^{∗}ϕ-twisted presymplectic groupoid over P. Infinitesimally, it corresponds to the pull-back Dirac structure (see e.g. [5])

f^{∗}L = {(X, f^{∗}(ξ)) : ((df)(X), ξ) ∈ L}.

We remark that the construction of the pull-backs f^{∗}L and f^{∗}G(L) is also possible in situations where f is not a submersion. In this case, (f^{∗}G(L), pr_{2}^{∗}ω_{L}) is often just a (twisted) over-presymplectic groupoid, but not presymplectic. Such examples arise for instance when one considers inclusions of submanifolds (see Example 6.7).

Example 6.4. (Multiplicative 2-forms of non-Dirac type)

There are closed multiplicative 2-forms which are not of Dirac type. In order to provide an explicit example, we start with a general observation. Let G be a groupoid over M, and let θ be a closed 2-form on M.

Claim: If the multiplicative 2-form ω = t^{∗}θ − s^{∗}θ on G is of Dirac type, then

Im (ρ_{x}) + Im (ρ_{x})^{⊥_{θ}}

has constant dimension along points x in a fixed orbit of G. (Here “⊥_{θ}” is the orthogonal with respect to θ.)

Let us prove the claim, following the notation of Section 4.

If ω is of Dirac type, then the ranges of t_{∗}(L_{ω,g}) and L_{M,t(g)} must coincide. Let us denote s(g) = x, t(g) = y. Since ω = t^{∗}θ − s^{∗}θ ∈ Ω^{2}(G), we can write ρ_{ω}^{∗}(α) = i_{ρ(α)}(θ). Now the second formula in (4.5) implies that Ker (ω) ∩ T_{y}M = Im (ρ_{y})^{⊥_{θ}} for all y ∈ M, and, using (4.2), we see that

dim(R(L_{M,y})) = dim(Im (ρ_{y}) + Im (ρ_{y})^{⊥_{θ}}).

On the other hand, it is easy to see from the definition of t_{∗}(L_{ω,g}) that R(t_{∗}(L_{ω,g})) = (dt)_{g}(Ker (dt_{g})^{⊥}) which, as shown in the proof of Claim 2, Lemma 4.3, has dimension equal to (dt)_{x}(Ker (dt_{x})^{⊥}) = (dt)_{x}(Ker (ds_{x}) + Ker (ω_{x})). Hence

dim(R(t_{∗}(L_{ω,g})) = dim(Im (ρ_{x}) + Ker (ω_{x}) ∩ T_{x}M) = dim(Im (ρ_{x}) + Im (ρ_{x})^{⊥_{θ}}),

and this proves the claim.

To find our example, let v be a vector field on M, and let Φ_{v} denote its flow. The domain G(v) ⊂ ℝ×M of Φ_{v} is a groupoid over M (the source is the second projection, the target is Φ_{v}, and the multiplication is defined by (t_{1}, y)(t_{2}, x) = (t_{1}+t_{2}, x)) integrating the action Lie algebroid defined by v (the underlying vector bundle is the trivial line bundle, the anchor is multiplication by v, and the bracket is [f, g] = fv(g) − v(f)g). On such a groupoid, the image of ρ is either zero or one-dimensional, so Im (ρ_{x}) + Im (ρ_{x})^{⊥_{θ}} = Im (ρ_{x})^{⊥_{θ}} for any closed θ ∈ Ω^{2}(M), and this need not be constant along orbits of v. For instance, one can take

M = ℝ^{2}, v = x\frac{∂}{∂y} − y \frac{∂}{∂x}, θ = ydxdy.

Then the circle S^{1} is an integral curve, and Im (ρ_{x})^{⊥} is one-dimensional everywhere on S^{1}, except for the points (1, 0) and (−1, 0).

6.2. Examples related to Poisson manifolds.

Example 6.5. (Symplectic groupoids)

Let (G, ω) be a presymplectic groupoid over M, and let L be the corresponding Dirac structure on M. Recall that L comes from a Poisson structure if and only if Ker (L) = Ker (ω) ∩ T M = {0}. But this condition is equivalent to Ker (ω_{x}) = 0 for all x ∈ M (by Corollary 4.8, part (v), and Lemma 3.3), which is in turn equivalent to Ker (ω_{g}) = 0 for all g ∈ G (by Lemma 4.2, (iv)). Hence L comes from a Poisson structure if and only if ω is nondegenerate. We see in this way that our main result, restricted to Poisson structures, recovers the well-known correspondence between Poisson manifolds and symplectic groupoids. In the presence of a closed 3-form, we recover the twisted version of this correspondence, which was conjectured in [27] and proved in [8].

Example 6.6. (Gauge transformations of Poisson manifolds)

Following [5], we now explain how to produce twisted presymplectic groupoids out of symplectic groupoids through gauge transformations associated to 2-forms.

Let M be a smooth manifold equipped with a ϕ-twisted Poisson structure π. We denote the corresponding ϕ-twisted Dirac structure by L_{π} = graph(π̃). The gauge transformation of L_{π} associated to a 2-form B on M is given by

L_{π} ↦ τ_{B}(L_{π}) = {(π̃(η), η + B̃(π̃(η))), η ∈ T^{∗}M}

As explained in [27], τ_{B}(L_{π}) is a (ϕ − dB)-twisted Dirac structure which may fail to be Poisson.

Let (G, ω) be a ϕ-twisted symplectic groupoid integrating π. Since L_{π} and τ_{B}(L_{π}) have isomorphic Lie algebroids (see [27]), τ_{B}(L_{π}) is integrable as an algebroid to a groupoid isomorphic to G. The 2-form

(6.1)  τ_{B}(ω) := ω + t^{∗}B − s^{∗}B ∈ Ω^{2}(G)

is easily seen to be multiplicative, robust and relatively (ϕ − dB)-closed, and, as remarked in [5, Thm 2.16], it induces the Dirac structure τ_{B}(L_{π}) on M. So (G, τ_{B}(ω)) is the presymplectic groupoid corresponding to τ_{B}(L_{π}).

Our results also show that this is true more generally: If L is a (twisted) Dirac structure on M associated with a presymplectic groupoid (G(L), ω_{L}), then ρ_{τ_{B}(ω_{L})}^{∗} = ρ_{ω}^{∗} + i_{ρ(α)}B; hence the image of (ρ, ρ_{τ_{B}(ω_{L})}^{∗}) is τ_{B}(L), and (G(L), τ_{B}(ω_{L})) is a presymplectic groupoid integrating τ_{B}(L).

Example 6.7. (Dirac submanifolds of Poisson manifolds)

In this example, we relate our results to those in [14, Sec. 9]. We describe how certain submanifolds of Dirac manifolds carrying an induced Dirac structure (such submanifolds of Poisson manifolds were studied in [14, 33]) give rise to over-presymplectic groupoids, whose reduction (in the sense of Remark 4.9, (i)) produce presymplectic groupoids of the submanifolds. For simplicity, we restrict the discussion to the untwisted case.

Let L_{M} be a Dirac structure on M, let N ↪ M be a submanifold, and suppose that the pull-back Dirac structure induced at each point by inclusion,

L_{N} := {(X, ξ|_{T N}) : X ∈ T_{x}N, (X, ξ) ∈ L_{M}} ⊂ T N ⊕ T^{∗}N,

is a smooth bundle; it is not difficult to check that L_{N} defines a Dirac structure on N. In the particular case of L_{M} coming from a Poisson structure π_{M} on M, and L_{N} coming from a Poisson structure π_{N} on N, N is called a Poisson-Dirac submanifold^{3} of (M, π_{M}) [14, Sec. 9].

Let us consider the vector bundle

𝔤_{N}(M) := T N^{∘} ∩ (L_{M} ∩ T^{∗}M) = (R(L_{M}) + T N)^{∘}

over N, which is in fact a bundle of Lie algebras, and assume that it has constant rank. (Here ^{∘} denotes the annihilator.) In this case, the restriction of the Lie algebroid L_{M} to N [17] is well-defined and determines a Lie subalgebroid L_{N}(M) whose underlying vector bundle is

{(X, ξ) : X ∈ T N, (X, ξ) ∈ L_{M}}.

This Lie algebroid fits into the following exact sequence of Lie algebroids:

0 ⟶ 𝔤_{N}(M) ⟶ L_{N}(M) ⟶ L_{N} ⟶ 0.

Let us assume that L_{M} is integrable, and let (G(L_{M}), ω_{M}) be the associated presymplectic groupoid. Then L_{N}(M) is also integrable (as it sits inside L_{M} as a Lie subalgebroid), and the associated groupoid G(L_{N}(M)) is a subgroupoid of G(L_{M}). Moreover, the restriction ω_{N,M} of ω_{M} to G(L_{N}(M)) makes G(L_{N}(M)) into an over-presymplectic groupoid over N, corresponding to the pull-back Dirac structure L_{N}.

We observe, however, that the reduction procedure of Remark 4.9, (i), produces a smooth presymplectic groupoid if and only if L_{N} is also integrable. In this case, the reduced groupoid will be precisely the pull-back (see Example 6.3) presymplectic groupoid (G(L_{N}), ω_{N}) of L_{N}. Hence, the presymplectic groupoids of a Dirac structure and of a Dirac submanifold are related by reduction of an intermediary over-presymplectic groupoid, as illustrated below:

3A Poisson submanifold is a Poisson-Dirac submanifold for which the inclusion (N, π_{N}) ↪ (M, π_{M}) is a Poisson map; this is equivalent to Im (π̃_{N}) = Im (π̃_{M}).

\xymatrix{(G(L_{N}(M)), ω_{N,M}) \ar@{^{(}->}[r] \ar[dr] & (G(L_{M}), ω_{M}) \\  & (G(L_{N}), ω_{N})}

In general, this quotient space may not be a manifold, but it is the same as the Weinstein groupoid introduced in [13] (see also [7]).

If L_{M} comes from a Poisson structure π_{M}, the discussion above shows that N is a Poisson-Dirac submanifold of (M, π_{M}) if and only if G(L_{N}(M)) is an over-symplectic groupoid, and the reduction procedure just described recovers Proposition 9.13 in [14].

6.3. Over-presymplectic groupoids and singular presymplectic groupoids. In this subsection we discuss examples of over-presymplectic groupoids that cannot be reduced to presymplectic groupoids, as already mentioned in Example 6.7. In particular, we will provide concrete examples of non-integrable Poisson structures admitting over-symplectic groupoids inducing them.

Let us consider the following particular case of the construction in Example 6.7. Let (M, π) be a Poisson manifold, with s-simply connected symplectic groupoid (G, ω). Let C be a Casimir function on M, and let a be a regular value of C. Then the level manifold N = C^{−1}(a) ↪ M is a Poisson submanifold of M. As we saw in the previous example, the restriction G_{N,M} = t^{∗}(N) = s^{∗}(N) of G to N is an s-simply connected groupoid over N, and the pull-back ω_{N,M} of ω to G_{N,M} makes it into an over-symplectic groupoid over N, inducing on N its Poisson structure. Indeed, the kernel of ω_{N,M} is spanned by the hamiltonian vector field of t^{∗}C = s^{∗}C, and since this vector field projects to zero on M, it is easy to check that condition (iv) of Lemma 4.5 is satisfied. To pass to a symplectic groupoid for N, we simply form the quotient of G_{N,M} by the hamiltonian flow.

Example 6.8. (Over-symplectic groupoids of nonintegrable Poisson submanifolds)

To obtain an interesting class of examples of the above construction, take G to be the cotangent bundle T^{∗}H ≅ H ⋉ 𝔥^{∗} of a simply-connected Lie group H, M = 𝔥^{∗}, and

C : 𝔥^{∗} → ℝ, u ↦ \frac{1}{2}(u, u)_{𝔥^{∗}}

the kinetic energy function of a bi-invariant (possibly indefinite) metric (·, ·)_{𝔥^{∗}} on H. The hamiltonian flow of t^{∗}(C) is then the geodesic flow on T^{∗}H. If N is the unit “sphere” 𝔥_{1/2}^{∗} = C^{−1}(1/2), then the unit co“sphere” bundle G_{N,M} = (T^{∗}H)_{1/2} is an oversymplectic groupoid over it. The quotient Q of (T^{∗}H)_{1/2} by the geodesic flow is then the canonical symplectic groupoid for the Poisson manifold 𝔥_{1/2}^{∗}. The elements of Q are geodesics in H considered as oriented submanifolds. One can view them as cosets of an open subset of the connected oriented one-dimensional subgroups of H. From this point of view, the groupoid structure of Q is just the one induced from the Baer groupoid (see [31]) consisting of all the cosets in H.

A specific case of the construction in the previous paragraph recovers the “pathology” of symplectic groupoids discussed in Section 6 of [7]. We let H be the product of a “space manifold” SU(2) = S^{3} with its usual riemannian metric and a “time manifold” ℝ carrying the negative of its usual metric. The unit “sphere” in 𝔥^{∗} for this Lorentz metric may then be identified with the product S^{2} × ℝ, with the Poisson structure for which the S^{2} slice over each τ ∈ ℝ is a symplectic leaf with symplectic structure equal to 1 + τ^{2} times the standard symplectic structure. The critical point of 1 + τ^{2} at τ = 0 is responsible for a singularity in the space of unit-speed (equivalently, space-like) geodesics. Most of these geodesics are diffeomorphic to ℝ, but the ones which are perpendicular to the τ direction are circles. As a result, the 6-dimensional quotient groupoid, i.e. the space of geodesics in S^{3} × ℝ, is a singular fiber bundle over S^{2} × S^{2}—the smooth 4-dimensional manifold of oriented geodesics^{4} in S^{3}. The fiber, which is neither Hausdorff nor locally Euclidean, is the quotient space ℝ^{2}/(x, y) ∼ (x, x + y) It can be obtained from the standard cone in ℝ^{3} by removing the vertex and replacing it with a line, with a topology such that any sequence of points on the cone converging to where the vertex used to be now converges to every point on the line.

An alternative way of obtaining over-presymplectic groupoids (which may not be reducible) inducing a given Dirac structure L is by means of extensions of the Lie algebroid of L by 2-cocycles (see e.g. [23]).

Let L be a Dirac structure on M, and let u ∈ Γ(Λ^{2}L^{∗}) be a 2-cocycle of the Lie algebroid of L. Let us assume, for simplicity, that L is not twisted. Then there is an associated algebroid L ⋉_{u} ℝ = L ⊕ ℝ which fits into an exact sequence

(6.2)  0 → ℝ → L ⋉_{u} ℝ → L → 0.

The anchor of L ⋉_{u} ℝ is just (X, a) ↦ ρ(X), where ρ is the anchor of L, while the Lie bracket on Γ(L) is

[(X, a), (Y, b)] = ([X, Y], L_{X}b − L_{Y} a + u(X, Y)).

The interesting point of this construction is that L ⋉_{u} ℝ may be integrable even when L is not. Moreover, the associated groupoid G(L ⋉_{u} ℝ) admits a canonical 1-form σ_{u} (whose construction is similar to the construction of the form σ in Proposition 8.1 (iii)), and (G(L ⋉_{u} ℝ), dσ_{u}) will be an over-presymplectic groupoid over M inducing L on the base. In the case of a Poisson manifold (M, π) (L = L_{π}), the groupoid corresponding to the extension of the Lie algebroid T^{∗}M by u = π, together with the 1-form σ_{u}, is an example of a contact groupoid.

Example 6.9. (Contact groupoids)

Contact groupoids are groupoids associated with Jacobi manifolds, as we now briefly explain (see [15] and references therein for details).

A Jacobi manifold is a manifold equipped with a bivector Λ and a vector field E satisfying [Λ, Λ] = 2Λ ∧ E, and [Λ, E] = 0. A particular example is given by Poisson manifolds (M, π), in which case Λ = π and E = 0; more generally, if g ∈ C^{∞}(M), then gπ may fail to be Poisson, but Λ = gπ and E = X_{g} define a Jacobi structure on M.

Any Jacobi manifold (M, Λ, E) determines a Lie algebroid structure on T^{∗}M ⊕ ℝ, which in the case of a Poisson manifold (M, π) is precisely T^{∗}M ⋉_{π} ℝ defined in (6.2). Just as Poisson structures correspond to symplectic groupoids (or Dirac structures correspond to presymplectic groupoids), Jacobi manifolds are associated with contact groupoids. These are groupoids G endowed with a contact 1-form σ and a smooth function f ∈ C^{∞}(G) such that σ is f-multiplicative:

m^{∗}σ = pr_{2}^{∗}f · pr_{1}^{∗}σ + pr_{2}^{∗}σ.

(Here pr_{j} : G×G → G is the natural projection onto the j-th factor.) When M is a Poisson manifold, the function f is constant equal to 1, so σ is multiplicative. One can therefore associate two groupoids to a Poisson manifold (M, π), its symplectic groupoid (G_{s}, ω) and the contact groupoid (G_{c}, σ) obtained by regarding (M, π) as a Jacobi manifold. Since σ is multiplicative, so is the 2-form dσ, and one can check that (G_{c}, dσ) is an over-symplectic groupoid over M inducing on M its Poisson structure. As a result, one can see (G_{s}, ω) as a reduction of (G_{c}, dσ).

4The projection T^{∗}H → T^{∗}S^{3} is equivariant with respect to the geodesic flows.

So, whenever a Poisson manifold (M, π) is integrable as a Jacobi manifold, it automatically admits an over-symplectic groupoid inducing π. As explained in [15], (M, π) can be a nonintegrable Poisson manifold, and yet be integrable as a Jacobi manifold (and viceversa). For example, the co“sphere” bundle (T^{∗}H)_{1/2} in Example 6.8 is actually a contact groupoid with contact form given by the restriction of the canonical 1-form on T^{∗}H. For more concrete examples and a detailed comparison of the obstructions to Poisson and Jacobi integrability, we refer the reader to [15].

6.4. Lie group actions and equivariant cohomology. Let H be a connected Lie group acting on a manifold M. We consider the action groupoid H ⋉ M over M, with

s(g, x) = x, t(g, x) = gx, ∀(g, x) ∈ H × M,

and multiplication of composable pairs given by

m((g_{1}, x_{1}), (g_{2}, x_{2})) = (g_{1}g_{2}, x_{2}).

It was pointed out in [4] that the space of twisted multiplicative 2-forms on H ⋉M is closely related to the equivariant cohomology of M in degree three. Our main results provide the following description of this relationship at the infinitesimal level.

The Lie algebroid of H ⋉ M (the action Lie algebroid 𝔥 ⋉ H) is the trivial bundle 𝔥_{M} := 𝔥 × M → M; the anchor is defined by the infinitesimal action, and the bracket is uniquely determined by the Leibniz rule and the Lie bracket on 𝔥. Following Theorem 2.5, the infinitesimal counterpart of twisted multiplicative 2-forms on H ⋉ M are pairs (ρ^{∗}, ϕ), where ϕ is a closed 3-form on M, and ρ^{∗} : 𝔥_{M} ⟶ T^{∗}M is a bundle map satisfying conditions (5.1), (5.2). We denote by ω_{ρ^{∗},ϕ} ∈ Ω^{2}(H ⋉ M) the relatively ϕ-closed, multiplicative 2-form associated with (ρ^{∗}, ϕ).

When H is a compact Lie group, the equivariant cohomology of M can be computed by Cartan’s complex of equivariant differential forms on M, denoted Ω_{H}^{∗}(M) (see e.g. [16]). This complex consists of H-invariant, Ω^{∗}(M)-valued polynomials on 𝔥, with degree twice the polynomial degree plus the form-degree:

Ω_{H}^{k}(M) = ( ⨁_{2i+j=k} S^{i}(𝔥^{∗}) ⊗ Ω^{j}(M))^{H}.

The differential is d_{H} := d_{1} − d_{2} where d_{1}(P)(v) = d(P (v)), d_{2}(P)(v) = i_{v}(P (v)); if α is an Ω^{∗}(M)-valued polynomial on 𝔥, invariance means that

(6.3)  g^{∗}(α(Ad_{g}(v))) = α(v),

for all v ∈ 𝔥 and g ∈ H.

Note that equivariantly closed 3-forms can be written as

ρ^{∗} + ϕ ∈ Ω_{H}^{3}(M),

where ϕ ∈ Ω^{3}(M) is closed, and ρ^{∗} ∈ 𝔥^{∗} ⊗ Ω^{1}(M), which are both invariant (as in (6.3)) and satisfy

\{ \begin{matrix}i_{v}(ρ^{∗}(v)) = 0 \\ i_{v}(ϕ) − d(ρ^{∗}(v)) = 0 \end{matrix}

for all v ∈ 𝔥. Infinitesimally, the invariance condition for ρ^{∗} reads

ρ^{∗}([v, w]) = L_{v}(ρ^{∗}(w)),

for all v, w ∈ 𝔥. Using this equation, one can easily check that conditions (5.1), (5.2) are satisfied, and hence there is a corresponding multiplicative 2-form ω_{ρ^{∗},ϕ}. Assuming that ρ^{∗} + ϕ ∈ Ω_{H}^{3}(M), we will now describe how one can obtain a simple explicit formula for ω_{ρ^{∗},ϕ} just using general properties of multiplicative forms.

Let pr_{g} and pr_{x} be the natural projections of H × M onto H and M, respectively, and let λ, \overline{λ} denote the left and right-invariant Maurer-Cartan forms on H (i.e., λ_{g}(V) = (dL_{g^{−1}})_{g}(V), \overline{λ}_{g}(V) = (dR_{g^{−1}})_{g}(V)).

Proposition 6.10. Suppose that ρ^{∗} and ϕ satisfy conditions (5.1), (5.2), and let ω = ω_{ρ^{∗},ϕ} ∈ Ω^{2}(H ⋉ M) be the corresponding 2-form. The following are equivalent:

(i) ρ^{∗} + ϕ ∈ Ω_{H}^{3}(M); (ii) the restriction of ω to all slices {g} × M vanishes, for all g ∈ H; (iii) ω is given by the formula

ω_{g,x} = ⟨ρ_{x}^{∗}pr_{g}^{∗}λ, pr_{x}^{∗} + \frac{1}{2}ρ_{x}pr_{g}^{∗}λ⟩,

or, explicitly,

(6.4) ω_{g,x}((V, X), (V′, X′)) = ⟨ρ_{x}^{∗}(λ_{g}(V)), ρ_{x}(λ_{g}(V′))⟩+⟨ρ_{x}^{∗}(λ_{g}(V)), X′⟩−⟨ρ_{x}^{∗}(λ_{g}(V′)), X⟩

Proof. We first observe a few facts. The defining formula for ρ^{∗} implies that

(6.5)  ω((v, 0), (0, X)) = ⟨ρ^{∗}(v), X⟩,

for v ∈ 𝔥, X ∈ T M. Using (3.4), we can write

ω_{g,x}((V_{g}, 0), (V′_{g}, X′_{x})) = ω_{gx}(((dR_{g}^{−1})_{g}(V_{g}), 0), dt_{g,x}(V′_{g}, X′_{x}))
= ω_{gx}(((dR_{g}^{−1})_{g}(V_{g}), 0), dt_{gx}((dR_{g}^{−1})_{g}V′_{g}, gX′_{x}))
= ⟨ρ_{gx}^{∗}(dR_{g}^{−1})_{g}(V_{g}), ρ_{gx}((dR_{g}^{−1})_{g}(V′_{g})) + gX′_{x}⟩

for all V_{g}, V′_{g} ∈ T_{g}H, X′_{x} ∈ T_{x}M. (Here gX denotes the infinitesimal action of H on T M.) Hence, for general pairs ((V_{g}, X_{x}), (V′_{g}, X′_{x})), we have

(6.6)  ω_{g,x}((V_{g}, X_{x}), (V′_{g}, X′_{x})) = ω_{g,x}^{0}((V_{g}, X_{x}), (V′_{g}, X′_{x})) + ω_{g,x}((0_{g}, X_{x}), (0_{g}, X′_{x})),

where

ω_{g,x}^{0}((V_{g}, X_{x}), (V′_{g}, X′_{x})) = ⟨ρ_{gx}^{∗}dR_{g^{−1}}(V_{g}), ρ_{gx}dR_{g^{−1}}(V′_{g})⟩ + ⟨ρ_{gx}^{∗}dR_{g^{−1}}(V_{g}), gX′_{x}⟩ −

⟨ρ_{gx}^{∗}dR_{g^{−1}}(V′_{g}), gX_{x}⟩ (6.7)

Let ω^{1} = ω − ω^{0}. We make two simple remarks: First, ω^{1} encodes precisely the restrictions of ω to the slices {g} × M (see (6.6)); Second, if ρ^{∗} is invariant, then (6.7) coincides with the formula (6.4) in the statement. Hence it suffices to show that ω = ω^{0} if and only if (ρ^{∗}, ϕ) is an equivariant form. One possible route to prove that is as follows: one can show that ω^{1} (or, equivalently, ω^{0}) is multiplicative if and only if ρ^{∗} is invariant, and ω^{1} is closed if and only if i_{v}(ϕ) − d(ρ^{∗}(v)) = 0. Since ρ_{ω^{1}}^{∗} = 0, the uniqueness of Corollary 3.4 implies the proposition. We will present an alternative argument instead.

Let us rewrite ω^{1} = ω−ω^{0} as ⟨c(g), X_{x} ∧ X′_{x}⟩, defining a smooth function c ∈ C^{∞}(H; Ω^{2}(M)). The multiplicativity of ω, applied on vectors ((0, X), (0, X′)), reads

(6.8)  ω_{hg,x}((0, X_{x}), (0, X′_{x})) = ω_{h,gx}((0, gX_{x}), (0, gX′_{x})) + ω_{g,x}((0, X_{x}), (0, X′_{x})),

which precisely means that

(6.9)  c(hg) = g^{∗}c(h) + c(g), ∀ g, h, ∈ H

(i.e., c is an Ω^{2}(M)-valued 1-cocycle on H). Note that, in order to prove that c = 0, it suffices to show that L_{v}(c) = 0 for all v ∈ 𝔥. Indeed, by differentiating (6.9), we obtain L_{V_{g}}(c) = 0 for all V_{g} = dR_{g}(v) ∈ T_{g}H and all g ∈ H, and c must be constant. Since, again by (6.9), c(1) is clearly zero, c must vanish (see also Remark 6.11).

We now claim that

(6.10)  L_{v}(c) = d_{H}(ρ^{∗} + ϕ)(v) = d(ρ^{∗}(v)) − i_{v}(ϕ)

for all v ∈ 𝔥. In order to prove (6.10), let V be a vector field on H extending v, and let X and X′ be vector fields on M. We evaluate dω = s^{∗}ϕ − t^{∗}ϕ on (V, 0), (0, X), (0, X′):

dω((V, 0), (0, X), (0, X′)) = L_{(V,0)}(ω((0, X), (0, X′))) − L_{(0,X)}(ω((V, 0), (0, X′))) +
L_{(0,X′)}(ω((V, 0), (0, X))) + ω((V, 0), (0, [X, X′]))
= −ϕ(ρ(λ̄(V)), X, X′),

where λ̄(V)_{g} = (dR_{g}^{−1})_{g}(V_{g}). Using (6.5) and evaluating the previous formula at g = 1 ∈ H, we find

L_{v}(c)(X, X′) = L_{X}(ρ^{∗}(v)(X′)) − L_{X′}(ρ^{∗}(v)(X)) − ρ^{∗}(v)([X, Y]) − ϕ(ρ(v), X, Y),

which is just (6.10). This proves the proposition. □

Remark 6.11. We observe that the proof of Proposition 6.10 indicates how to express ω_{ρ^{∗},ϕ} for general pairs (ρ^{∗}, ϕ) (i.e. which only satisfy the conditions (5.1), (5.2)). More precisely, the cocycle condition (6.9) for c is equivalent to saying that g ↦ (g, c(g)) is a group homomorphism from H into the group H ⋉ Ω^{2}(M) defined by (h, a)(g, b) = (hg, g^{∗}a + b). The proof of Proposition 6.10 then shows that the induced Lie algebra map 𝔥 ⟶ 𝔥⋉Ω^{2}(M) is

(6.11)  v ↦ (v, d(ρ^{∗}(v)) − i_{v}(ϕ)).

So, if H is simply connected, the Lie algebra cocycle (6.11) integrates uniquely to a group cocycle c, and ω will be given by (6.7) plus c(g)(X_{x}, Y_{x}).

Remark 6.12. In general, there is a natural map

H^{∗}(Ω_{H}^{∗}(M)) ⟶ H_{H}^{∗}(M)

from the cohomology of the Cartan complex into the equivariant cohomology of M, which is an isomorphism if H is compact [2]. The equivariant cohomology groups can be obtained from a double complex Ω^{p}(H^{q} ×M), with de Rham differential increasing the degree p, and a group-cohomology differential increasing q; see e.g. [3]. Our result gives both an explicit description of this map in degree three,

ρ^{∗} + ϕ ↦ ω_{ρ^{∗},ϕ} + ϕ,

as well as an interpretation of this map in terms of multiplicative forms.

If ρ^{∗} + ϕ ∈ Ω_{H}^{3}(M) satisfies the non-degeneracy condition dim(Ker (ρ) ∩ Ker (ρ^{∗})) = dim(H) − dim(M), then (H ⋉ M, ω_{ρ^{∗},ϕ}) becomes an over-presymplectic groupoid, which is presymplectic if dim(M) = dim(H). The associated ϕ-twisted Dirac structure can be described directly as

L = {(ρ(v), ρ^{∗}(v)) : v ∈ 𝔥} ⊂ T M ⊕ T^{∗}M.

A simple example is M = 𝔥^{∗} with the coadjoint action of H, ϕ = 0, and ρ^{∗} given by ρ_{ξ}^{∗}(v) = v. The associated groupoid will be H ⋉ 𝔥^{∗} ≅ T^{∗}H, with the canonical symplectic form. A more interesting example will be the AMM groupoid of Subsection 7.2.

7. Presymplectic realizations of Dirac structures

Let (M, π) be a Poisson manifold. Recall that a symplectic realization of M is a Poisson map from a symplectic manifold (P, η) to M (see e.g. [6]). The following important property of symplectic realizations brings them close to the theory of hamiltonian actions: any symplectic realization μ : P → M induces a canonical action of the Lie algebroid T^{∗}M, induced by π, on P by assigning to each α ∈ Ω^{1}(M) the vector field X ∈ X (P) defined by

i_{X}η = μ^{∗}α.

When μ complete (i.e., the hamiltonian vector field X_{μ^{∗}f} is complete whenever f ∈ C^{∞}(M) has compact support), M is integrable [14] and this action extends to a symplectic action of G, the s-simply connected symplectic groupoid of M (see [9, 14]), with moment map μ [24]. In this way, we get a natural correspondence between symplectic actions of G and complete symplectic realizations of M. In particular, if M = 𝔥^{∗} is the dual of a Lie algebra, the action of the associated symplectic groupoid T^{∗}H = H ⋉ 𝔥^{∗} factors through an H-action, and complete symplectic realizations of 𝔥^{∗} become hamiltonian H-spaces. In this section, we will extend this picture to twisted Dirac manifolds.

7.1. Presymplectic realizations. We recall the definition introduced in Section 2.3.

Definition 7.1. A presymplectic realization of a ϕ-twisted Dirac manifold (M, L) is a Dirac map μ : (P, η) ⟶ (M, L), where η is a μ^{∗}ϕ-closed 2-form (i.e., dη + μ^{∗}ϕ = 0), such that Ker (dμ) ∩ Ker (η) = {0}.

The following results explain this definition.

Lemma 7.2. Let (M, L) be a ϕ-twisted Dirac manifold, let μ : P ⟶ M be a smooth map, and let P be equipped with a 2-form η satisfying dη +μ^{∗}ϕ = 0. The following are equivalent:

(i) The map μ is a presymplectic realization of L; (ii) For all p ∈ P, (w, ξ) ∈ L_{μ(p)}, there exists a unique X ∈ T_{p}P satisfying the equations:

\{ \begin{matrix}w = dμ(X) \\ μ^{∗}(ξ) = i_{X}(η) \end{matrix} ,

(iii) The map μ is Dirac and dμ maps Ker (η) isomorphically onto Ker (L).

Proof. Note that μ being a Dirac map is equivalent to the equations in (ii) having a solution for X. The uniqueness of the solutions is equivalent to Ker (dμ) ∩ Ker (η) = {0}, so (i) and (ii) are equivalent. Note that Ker (L) = {w = dμ(X) | i_{X}η = 0} = dμ(Ker (η)), so dμ : Ker (η) → Ker (L) is an isomorphism if and only Ker (dμ) ∩ Ker (η) = {0}. Hence all the conditions are equivalent. □

Note that if the conditions in Lemma 7.2 hold, then (ii) defines a map ρ_{P} : L_{μ(p)} ⟶ T_{p}P, (w, ξ) ↦ X. A direct computation shows that

(1) the induced map ρ_{P} : Γ(L) ⟶ X (P) is a map of Lie algebras (Γ(L) is equipped with twisted Courant bracket);

(2) dμ(ρ_{P} (l)) = ρ(l) for all l ∈ L, which precisely means that ρ_{P} is an infinitesimal action of the Lie algebroid L on P (and this is what we were after!).

Corollary 7.3. Any presymplectic realization μ : (P, η) ⟶ (M, L) of a ϕ-twisted Dirac structure is canonically equipped with an infinitesimal action of the Lie algebroid of L.

We call a presymplectic realization μ : P → M complete if ρ_{P} (l) is a complete vector field whenever l ∈ Γ(L) has compact support. As with symplectic realizations of Poisson manifolds, a complete realization defines a complete Lie algebroid action [25], which can be integrated to a global action of the groupoid G(L) associated with L on P: indeed, an algebroid action of L defines a map ∇ : Γ(L) ⊗ C^{∞}(P) ⟶ C^{∞}(P) which behaves like a flat (L-) connection; then parallel transport defines the desired action of G(L) on P (see [14, pp.26–27] for details). For the integration of general Lie algebroid actions, see [25].

For complete symplectic realizations of Poisson manifolds, the induced action of the symplectic groupoid is symplectic [24]. The following property generalizes this fact. Let (M, L) be a ϕ-twisted Dirac structure, and let (G(L), ω_{L}) be the associated groupoid.

Corollary 7.4. If the realization μ : P → M is complete (e.g., if P is compact), then there is an induced action of G(L) on P, m_{P} : G(L) ×_{M} P ⟶ P. Moreover, if G(L) is smooth (i.e., if L is integrable), then the action is smooth and

(7.1)  m_{P}^{∗} η = pr_{G}^{∗}ω_{L} + pr_{P}^{∗} η

(where pr_{G} : G(L)×_{M} P ⟶ G(L) and pr_{P} : G(L)×_{M} P ⟶ P are the natural projections).

Proof. In order to check (7.1), note that the 2-forms ω_{1} := m_{P}^{∗} η and ω_{2} := pr_{G}^{∗}ω_{L}+pr_{P}^{∗} η are both multiplicative in the semi-direct product groupoid G(L) ⋉ P. A direct computation shows that ρ_{ω_{1}}^{∗} = ρ_{ω_{2}}^{∗}, so it follows from Theorem 5.1 that the forms must coincide. □

Remark 7.5. By the same arguments as in [14, Thm. 8.2], the existence of a complete symplectic realization μ : P → M which is a surjective submersion implies the integrability of L. Note also that such realizations μ can be used to compute (G(L), ω_{L}) (though we do not know how to use this to give a direct proof of the integrability of L): First, we note that the groupoid G(L) ⋉ P over P is isomorphic to the monodromy groupoid G(Im (ρ_{P})) of the (regular) foliation Im (ρ_{P}), so that G(L) is a quotient of G(Im (ρ_{P})); second, (7.1) says that the form t^{∗}η − s^{∗}η on G(Im (ρ_{P})) descends to ω_{L} on G(L).

7.2. Realizations of Cartan-Dirac structures and quasi-hamiltonian spaces. As observed in [27, Example 4.2], any Lie group with a bi-invariant metric carries a ϕ-twisted Dirac structure, where ϕ is the associated bi-invariant Cartan form. We call it a Cartan-Dirac structure. In this section, we will discuss presymplectic realizations and groupoids of Cartan-Dirac structures. We recover, in this framework, quasi-hamiltonian spaces [1] and the AMM-groupoid of [4], proving the following result.

Theorem 7.6. Let H be a connected Lie group, and let (·, ·)_{𝔥} be an invariant inner product on its Lie algebra 𝔥. Let L denote the associated Cartan-Dirac structure on H. Then

(i) There is a one-to-one correspondence between presymplectic realizations of (H, L) and quasi-hamiltonian 𝔥-spaces (which are infinitesimal versions of quasi-hamiltonian H-spaces introduced in [1]);

(ii) The AMM-groupoid of [4] is a presymplectic groupoid inducing the Cartan-Dirac structure L on H.

Before we prove this theorem, let us recall some definitions and fix our notation.

A quasi-hamiltonian H-space [1] is a manifold P endowed with a smooth action of H, an invariant 2-form η ∈ Ω^{2}(P), and an equivariant map μ : P ⟶ H (the moment map), such that (1) the differential of η is given by

dη = −μ^{∗}ϕ;

(2) the map μ satisfies

i_{ρ_{P} (v)}(η) = \frac{1}{2}μ^{∗}(λ + λ̄, v)_{𝔥};

(3) at each p ∈ P, the kernel of η_{p} is given by

Ker (η_{p}) = {ρ_{P,p}(v) : v ∈ Ker (Ad_{μ(p)} + 1)}.

Here, ρ_{P} : 𝔥 ⟶ T P is the induced infinitesimal action of 𝔥 on P, λ (resp. λ̄) is the left- (resp. right-) invariant Maurer-Cartan form on H, and ϕ ∈ Ω^{3}(H) is the bi-invariant Cartan form:

ϕ = \frac{1}{12}(λ, [λ, λ])_{𝔥} = \frac{1}{12}(λ̄, [λ̄, λ̄])_{𝔥}.

On the Lie algebra, we have ϕ(u, v, w) = \frac{1}{2}(u, [v, w])_{𝔥}. The equivariance of μ is with respect to the action of H on itself by conjugation. Infinitesimally, equivariance becomes

(7.2)  (dμ)_{p}(ρ_{P} (v)) = ρ_{H}(v)

for all v ∈ 𝔥, where ρ_{H} : 𝔥 ⟶ T H is the infinitesimal conjugation action (explicitly, ρ_{H}(v) = v_{r} − v_{l}, where v_{l} and v_{r} are the vector fields obtained from v ∈ 𝔥 by left and right translations).

Definition 7.7. A quasi-hamiltonian 𝔥-space is a manifold P carrying an 𝔥-action ρ_{P} : 𝔥 ⟶ T P, together with an 𝔥-invariant 2-form η ∈ Ω^{2}(P) and and equivariant μ : P → H (as in (7.2)), satisfying conditions (1), (2) and (3).

Conditions (1),(2) and (3) in the definition of quasi-hamiltonian spaces strongly resemble the conditions we used to define presymplectic realizations, see Lemma 7.2, (ii). In order to find the underlying Dirac structure L on H making quasi-hamiltonian 𝔥-spaces into presymplectic realizations, recall that an 𝔥-action on P together with an equivariant map μ : P → H is equivalent to an action of the action Lie algebroid 𝔥 ⋉ H, with moment μ [24]. Hence the Lie algebroid of L is isomorphic to 𝔥 ⋉ H, with anchor ρ_{H}; in other words, there is a map ρ^{∗} : 𝔥 → T^{∗}H such that (ρ_{H}, ρ^{∗}) : 𝔥 ⋉ H → L is an isomorphism. To find ρ^{∗}, we compare condition (2) for quasi-hamiltonian spaces and the second equation in Lemma 7.2, (ii), and obtain

ρ^{∗}(v) = \frac{1}{2}(λ + λ̄, v)_{𝔥} = \frac{1}{2}(v_{r} + v_{l}),

where in the last equality we use the metric to identify T^{∗}H with T H. More explicitly, the Dirac structure we obtain on H is

L = \{(v_{r} − v_{l}, \frac{1}{2}(v_{r} + v_{l})) : v ∈ 𝔥\} ⊂ T H ⊕ T H,

which is precisely the ϕ-twisted Dirac structure discussed in [27, Example 4.2]. We call L the Cartan-Dirac structure on H associated with (·, ·)_{𝔥}.

We now proceed to the proof of Theorem 7.6.

Proof. Suppose that μ : P → H is a presymplectic realization of the Cartan-Dirac structure L on H. Let ρ_{P}^{L} be the induced infinitesimal action of L on P, with moment μ. Since the Lie algebroid of L is isomorphic to 𝔥 ⋉ H, it immediately follows that ρ_{P}^{L} determines an 𝔥-action ρ_{P} on P, for which μ is 𝔥-equivariant. Explicitly,

(7.3)  ρ_{P} (v) = ρ_{P}^{L}(v_{r} − v_{l}, \frac{1}{2}(λ + λ̄, v)_{𝔥}), v ∈ 𝔥.

Since dη + μ^{∗}ϕ = 0, (1) in Definition 7.7 holds. Condition (2) is just the second equation in Lemma 7.2, (ii). Since dμ : Ker (η) → Ker (L) is an isomorphism, and Ker (L_{g}) = {ρ_{H}(v) : v ∈ Ker (Ad_{g}+1)}, the equivariance of μ, dμ(ρ_{P} (v)) = ρ_{H}(v) implies condition (3). Finally, we note that η is 𝔥-invariant:

L_{ρ_{P} (v)}(η) = di_{ρ_{P} (v)}η + i_{ρ_{P} (v)}dη = \frac{1}{2}d(μ^{∗}(λ + λ̄, v)_{𝔥})) − i_{ρ_{P} (v)}μ^{∗}ϕ = 0,

where the last equality follows from the Maurer-Cartan equations for λ and λ̄. So P is a quasi-hamiltonian 𝔥-space.

Conversely, if P is a quasi-hamiltonian 𝔥-space, then dη + μ^{∗}ϕ = 0, and we must check that condition (ii) in Lemma 7.2 holds. If (w, ξ) = (v_{r} − v_{l}, \frac{1}{2}(λ + λ̄, v)_{𝔥}) ∈ L, then w = ρ_{H}(v) = dμ(ρ_{P} (v)), so the first equation in (ii) has a solution X = ρ_{P} (v). The second equation in (ii) is just (2) in Definition 7.7. The uniqueness of this solution follows from (3): if i_{X}η = 0, then X = ρ_{P} (v) for some v with v_{l} +v_{r} = 0; since dμ(X) = ρ_{H}(v) = v_{r} −v_{l} = 0, we must have v = 0. So μ : P → H is a presymplectic realization. This proves part (i) of the theorem.

In order to prove part (ii), we describe the presymplectic groupoids associated with the Cartan-Dirac structure L. Since L, as a Lie algebroid, is isomorphic to the action Lie algebroid 𝔥⋉H, the action groupoid H⋉H (with action given by conjugation, g·x = gxg^{−1}) integrates it. (On H ⋉ H, s(g, x) = x, t(g, x) = gxg^{−1}, (g_{1}, x_{1}) · (g_{2}, x_{2}) = (g_{1}g_{2}, x_{2}).) We are exactly in the situation of Example 6.4. As observed in [1], ρ^{∗} + ϕ ∈ Ω_{H}^{3}(H). (In particular, it follows from this observation that L is indeed a ϕ-twisted Dirac structure.) Applying Proposition 6.10, we immediately obtain the formula for the multiplicative 2-form on H × H corresponding to L:

ω_{(g,x)} = \frac{1}{2} ((Ad_{x}p_{g}^{∗}λ, p_{g}^{∗}λ)_{𝔥} + (p_{g}^{∗}λ, p_{x}^{∗}(λ + λ̄))_{𝔥}) .

As in Proposition 6.10, p_{g} and p_{x} denote the projections onto the first and second components of H × H. This is precisely the 2-form in the “double” D(H) = H × H introduced in [1]; the groupoid (H ⋉ H, ω) also appears in [4], where it is called the AMM groupoid. So the AMM groupoid is a presymplectic groupoid associated with the Cartan-Dirac structure on H, though it is not necessarily s-simply connected. If H is simply connected, then (G(L), ω_{L}) = (H ⋉ H, ω), while, in general, one must pull-back ω to H̃ ⋉ H, where H̃ is the universal cover of H. □

Finally, if the infinitesimal action can be integrated to a global action, then the notion of quasi-hamiltonian H-space coincides with that of quasi-hamiltonian 𝔥-space. For instance, if H is simply connected, there is a one-to-one correspondence between complete quasihamiltonian H-space (“complete” meaning that the action of 𝔥 is by complete vector fields), and complete presymplectic realizations of L ( “complete” in the sense of the Lie-algebroid action of L on P). In particular,

Corollary 7.8. If H is simply connected, then there is a one-to-one correspondence between compact quasi-hamiltonian H-spaces and compact presymplectic realizations of the Cartan-Dirac structure L on H.

8. Multiplicative 2-forms, foliations and regular Dirac structures

In this section, we explain connections between our results and some aspects of foliation theory: we will see that multiplicative 2-forms on monodromy groupoids of foliations are directly related to foliated cohomology, and they are relevant for the explicit description of presymplectic groupoids associated to regular Dirac structures.

8.1. Foliations. Let us recall some basic facts of foliations theory. The reader is referred to [19] and references therein for details.

By the Frobenius theorem, a foliation on M can be viewed as a subbundle F of T M (of vectors tangent to the leaves) for which [F, F] ⊂ F; alternatively, foliations are the same thing as algebroids with injective anchor map. The monodromy groupoid of F consists of leafwise homotopy classes of leafwise paths in M (i.e., each s-fiber s^{−1}(x) is the universal cover of the leaf through x, constructed with x as base point). This groupoid is the same as the one described in Section 5, i.e., it is the unique s-simply connected Lie groupoid integrating F viewed as an algebroid. We denote it by G(F). The space of foliated forms on M, Ω^{•}(F) = Γ(∧^{•}F^{∗}), carries a foliated de Rham operator

(8.1)  d_{F}ω(X_{1}, …, X_{p+1}) = ∑_{i}(−1)^{i}L_{X_{i}}(ω(X_{1}, …, X̂_{i}, …, X_{p+1}))
+ ∑_{i<j}(−1)^{i+j−1}ω([X_{i}, X_{j}], X_{1}, …, X̂_{i}, …, X̂_{j}, …, X_{p+1})),

and we denote by H^{•}(F) the resulting cohomology (which is just the cohomology of F as an algebroid). One defines, in a similar way, the foliated cohomology H^{•}(F; E) with coefficients in a foliated bundle E, i.e., a bundle E over M endowed with a flat F-connection ∇ : Γ(F) × Γ(E) → Γ(E). The corresponding complex is now Ω^{•}(F; E) = Γ(∧^{•}F^{∗} ⊗ E), and the differential is given just as in (8.1), with L_{X} replaced by ∇_{X}. The basic example of a foliated bundle is the normal bundle ν = T M/F, with F-connection given by the well-known Bott connection, ∇ : Γ(F) × Γ(ν) → Γ(ν),

∇_{V} \overline{X} = \overline{[V, X]},

where X ↦ \overline{X} is the projection from T M onto ν. As usual, the Bott connection induces connections on the dual ν^{∗} and on the associated tensor bundles.

Here we will deal with the cohomology spaces H^{•}(F) and H^{•}(F; ν^{∗}) (in degrees one and two). These spaces are relate by a transversal de Rham operator

(8.2)  d_{ν} : H^{•}(F) ⟶ H^{•}(F; ν^{∗})

(which can be extended to higher exterior powers of ν^{∗}). Let us give a direct description of this map in degree two, since higher degrees can be treated analogously. Given a class [θ] ∈ H^{2}(F) represented by a foliated 2-form θ, let θ̃ be a 2-form on M with θ = θ̃|_{F}. Since dθ̃|_{F} = 0, it follows that the map Γ(∧^{2}F) → Γ(ν^{∗}) defined by

(V, W) ↦ dθ̃(V, W, −),

gives a closed foliated 2-form with coefficients in ν^{∗}; we set d_{ν}([θ]) to be its class in H^{2}(F; ν^{∗}).

8.2. Multiplicative 2-forms on monodromy groupoids. In this example we relate the space of closed multiplicative 2-forms on monodromy groupoids to cohomology spaces which are well known in foliation theory.

Let Mult^{2}(G) denote the space of closed multiplicative 2-forms on a Lie groupoid G.

Proposition 8.1. Let F be a foliation on M, and let G = G(F) be the monodromy groupoid of F. Then (i) any ω ∈ Mult^{2}(G) induces a cohomology class c(ω) ∈ H^{2}(F). (ii) a foliated cohomology class c ∈ H^{2}(F) is of type c(ω) if and only if d_{ν}(c) = 0. (iii) c(ω) = 0 if and only if ω is multiplicatively exact, i.e. ω = dσ with σ ∈ Ω^{1}(G)

multiplicative.

Before we prove this proposition, let us recall a more conceptual way to describe the operator (8.2). There is a spectral sequence associated to the foliation F (see e.g [19]), converging to H^{•}(M), with

(8.3)  E_{1}^{p,q} = H^{p}(F; Λ^{q}ν^{∗}),

and so that d_{ν} is just the boundary map d_{1}^{p,q} : E_{1}^{p,q} ⟶ E_{1}^{p,q+1}. The spectral sequence is associated to the filtration F_{p}Ω^{•}(M) of Ω^{•}(M) with

F_{q}Ω^{n}(M) = {η ∈ Ω^{n}(M) : i_{V_{1}} … i_{V_{n−q+1}}η = 0, for all V_{i} ∈ Γ(F)}

(F_{0}Ω^{•}(M) = Ω^{•}(M), and F_{q}Ω^{n}(M) = 0 for q > n). It is easy to see that

E_{0}^{p,q} = F_{q}Ω^{p+q}(M)/F_{q+1}Ω^{p+q+1}(M) ≅ Ω^{p}(F; Λ^{q}ν^{∗})

and that the boundary d_{0}^{p,q} is precisely the leafwise de Rham operator d_{F}. This shows that the E_{1}-terms are indeed given by (8.3), and a standard computation shows that d_{1}^{p,q} has the explicit description mentioned above.

Proof. Note that we have an isomorphism

\frac{F_{0}Ω^{2}(M)}{F_{2}Ω^{2}(M)} ⟶^{∼} \{ρ^{∗} : F ⟶ T^{∗}M : ρ^{∗} satisfies (5.1)\},

sending [η] to ρ^{∗}, defined by ⟨ρ^{∗}(V), X⟩ = η(V, X). Moreover, the closedness of [η] in the complex F_{0}Ω^{•}(M)/F_{2}Ω^{•}(M) corresponds to (5.2) for ρ^{∗}. On the other hand, ρ^{∗} corresponds to an exact [η] if and only if ⟨ρ^{∗}(V), X⟩ = dσ(V, X) for some σ ∈ F_{0}Ω^{1}(M)/F_{2}Ω^{1}(M) = Ω^{1}(M). But then the closed multiplicative 2-form ω associated with ρ^{∗} is ω_{0} = d(t^{∗}σ −s^{∗}σ) (note that ω_{0} is multiplicative and closed, and it is easy to see that ρ_{ω_{0}}^{∗} = ρ^{∗}). As a result, we get an isomorphism

H^{2} (\frac{F_{0}Ω^{•}(M)}{F_{2}Ω^{•}(M)}) ⟶^{∼} \frac{Mult^{2}(G)}{\{d(t^{∗}σ − s^{∗}σ) : σ ∈ Ω^{1}(M)\}}.

Now, using the short exact sequence of complexes

0 ⟶ \frac{F_{1}Ω^{•}(M)}{F_{2}Ω^{•}(M)} ⟶ \frac{F_{0}Ω^{•}(M)}{F_{2}Ω^{•}(M)} ⟶ \frac{F_{0}Ω^{•}(M)}{F_{1}Ω^{•}(M)} ⟶ 0,

we get an exact sequence in cohomology

(8.4) H^{1}(F) ⟶^{d_{ν}} H^{1}(F; ν^{∗}) ⟶ \frac{Mult^{2}(G(F))}{\{d(t^{∗}σ − s^{∗}σ) : σ ∈ Ω^{1}(M)\}} ⟶^{c} H^{2}(F) ⟶^{d_{ν}} H^{2}(F; ν^{∗}).

This immediately implies statements (i) and (ii). Note that the map c in (8.4) is given by c(ω) = [c_{ω}], where c_{ω} ∈ Ω^{2}(F) is defined by c_{ω}(V, W) = ⟨ρ_{ω}^{∗}(V), W ⟩.

We now prove (iii). The fact that c_{ω} = 0 for multiplicative 2-forms of type ω = dσ, with σ multiplicative, follows by showing that the restriction of σ to F (a foliated 1-form) gives, after differentiation, precisely the foliated 2-form c_{ω} induced by ω. This can be checked by an argument similar to the one used to prove the formula of Proposition 3.5, part (ii) (but the argument is simpler, following from Remarks 2) and an analogue of 3) in that proof). For the converse, we fix ω with [c_{ω}] = 0. From the exact sequence (8.4), we may assume that c_{ω} = 0. But then Lemma 5.3 shows that the 1-form σ̃ = −ρ^{∗}σ^{c} is basic, so it descends to a multiplicative 1-form σ on G(F) = G. Since ω is induced by ω̃ = dσ̃, we conclude that ω = dσ. □

8.3. Dirac structures associated to foliations. Any regular foliation F on M defines a Dirac structure L_{F} whose presymplectic leaves are precisely the leaves of F, with the zero form. In other words,

L_{F} = F ⊕ ν^{∗} ⊂ T M ⊕ T^{∗}M.

The Dirac structure L_{F} is always integrable, and we now describe the associated presymplectic groupoid. As above, we denote by G(F) the monodromy groupoid of F, and by ν the normal bundle. The parallel transport with respect to the Bott connection ∇ (see Section 8.1) is well defined along leafwise paths (because ∇ is an F-connection), and is invariant under leafwise homotopy (because ∇ is flat). Hence it defines an action of G(F) on ν, which we dualize to an action on ν^{∗}. We form the semi-direct product groupoid

G(F) ⋉ ν^{∗}

consisting of pairs (g, v) with v ∈ ν_{s(g)}^{∗}, source and target maps induced by those of G(F), and multiplication

(g, v)(h, w) = (gh, h^{−1}v + w).

Restricting the canonical symplectic form ω_{can} on T^{∗}M to ν^{∗}, its pull-back

ω_{F} = pr_{2}^{∗}ω_{can}

by the second projection is a multiplicative 2-form on G(F) ⋉ ν^{∗}. It is not difficult to check the following

Lemma 8.2. (G(F) ⋉ ν^{∗}, ω_{F}) is the presymplectic groupoid associated with L_{F}.

A slight “twist” of this result will yield more example of multiplicative 2-forms which are not of Dirac type.

Let us consider a closed 3-form ϕ on M with the property that

i_{V} i_{W} ϕ = 0, ∀ V, W ∈ Γ(F).

Using the filtration of Section 8.1, this condition means that ϕ ∈ F_{2}Ω^{3}(M). By Theorem 5.1, there exists a unique multiplicative 2-form ω_{ϕ} on G(F) such that

dω_{ϕ} = s^{∗}ϕ − t^{∗}ϕ, ω_{ϕ,x} = 0 ∀ x ∈ M.

Proposition 8.3. The following are equivalent:

(i) ω_{ϕ} = 0; (ii) ω_{ϕ} is of Dirac type; (iii) ϕ ∈ F_{3}Ω^{3}(M) (or, equivalently, ϕ is basic).

Proof. By Lemma 3.1, both Ker (ds) and Ker (dt) sit inside Ker (ω_{ϕ}) at all points g ∈ G(F). This implies that Ker (ds)_{g}^{⊥} = T_{g}G and Ker (dt)_{g} + Ker (ω_{ϕ,g}) = Ker (ω_{ϕ,g}), and the equivalence of (ii) and (i) follows from condition Lemma 4.2, part (ii).

Next, note that, while (i) is equivalent to s^{∗}ϕ − t^{∗}ϕ = 0 at all g ∈ G, (iii) is equivalent to the same condition at all x ∈ M (by Corollary 3.4). Since s^{∗}ϕ − t^{∗}ϕ is a multiplicative 3-form which has zero differential, the equivalence of (i) and (iii) follows from a degree three version of Corollary 3.4 (proven in the same way). □

Let us point out that, although ω_{ϕ} is not of Dirac type in general, this form is still relevant for the construction of forms of Dirac type and of presymplectic groupoids. In order to see that, note that the Dirac structure L_{F} is a ϕ-twisted Dirac structure, since ϕ vanishes along the leaves of the foliation. By the properties of ω_{ϕ} in the proof above, adding ω_{ϕ} does not affect the the isotropy bundle closed 2-forms. Using the notation of the previous example, we get

Corollary 8.4. Viewing L_{F} as a ϕ-twisted Dirac structure, the associated ϕ-twisted presymplectic groupoid is G(F) ⋉ ν^{∗} with the 2-form ω_{F} + pr_{1}^{∗}ω_{ϕ}.

8.4. Presymplectic groupoids of regular Dirac structures. We will call a Dirac structure regular if its presymplectic leaves have constant dimension. To begin, we will restrict ourselves to the untwisted case, with ϕ = 0. If L is regular, then it determines

(1) a regular foliation F (whose leaves are the presymplectic leaves of L);

(2) a closed foliated 2-form θ ∈ Ω^{2}(F) (defined by the leafwise presymplectic forms of

L). Conversely, we can recover L from F and θ:

L = {(X, ξ) : X ∈ F, ξ|_{F} = i_{X}(θ)}.

In this section, we discuss examples of regular Dirac structures for which G(L) admits a simplified description in terms of this data, F and θ. (Note that the case θ = 0 has been treated in Section 8.3.)

A simplified description of G(L) depends on the classifying class of L, denoted by c(L), that we now discuss. Using the transversal de Rham operator d_{ν} defined in (8.2), c(L) is defined as

c(L) = d_{ν}(θ) ∈ H^{2}(F; ν^{∗}).

While θ carries all the information of L as a Dirac structure, c(L) characterizes L as a Lie algebroid. As suggested by the exact sequence

0 ⟶ ν^{∗} ⟶ L ⟶ F ⟶ 0,

the relationship between L and c(L) is the same as the one of extensions and 2-cocycles, as briefly discussed in Section 6.3. Let us make it more explicit. First of all, any closed u ∈ Ω^{2}(F; ν^{∗}) defines an algebroid F ⋉_{u}ν^{∗}, with underlying vector bundle F ⊕ν^{∗}, projection on the first factor as anchor map, and bracket

[(X, v), (Y, w)] = ([X, Y], ∇_{X}(w) − ∇_{Y} (v) + u(X, Y)).

When u = 0, we simplify the notation to F ⋉ ν^{∗}. (Note that this is the Lie algebroid underlying the Dirac structure L_{F} of the Section 8.3.) The isomorphism class of the Lie algebroid F ⋉_{u} ν^{∗} depends only on the cohomology class of u: if u′ = u + dv with v ∈ Ω^{1}(F; ν^{∗}), then

(X, ξ) ↦ (X, ξ + v(X))

is an isomorphism between F ⋉_{u′} ν^{∗} and F ⋉_{u} ν^{∗}.

In order to see that c(L) is the class corresponding to L, we choose a linear splitting σ of the map L ⟶ F. On one hand,

(8.5)  u_{σ}(X, Y) = [σ(X), σ(Y)] − σ([X, Y])

is a representative of d_{ν}(θ) (this follows from the explicit description of d_{ν} given in Section 8.2); on the other hand, σ induces a linear isomorphism L ≅ F ⊕ν^{∗} which maps the brackets on L into the brackets of F ⋉_{u} ν^{∗}.

Example 8.5. (The case c(L) = 0) We now describe the presymplectic groupoid of L when c(L) = 0. This case is closely related to our discussion in Section 8.2, which we now extend. Any multiplicative 2-form ω on the monodromy groupoid G(F) defines a foliated form c_{ω} = ω|_{F} (where we view F ⊂ T M ⊂ T G(F)), whose cohomology class is precisely the c(ω) defined in Section 8.2. In particular, we have an induced regular Dirac structure L (namely the one defined by F and c_{ω}). In this case, we say that L comes from ω, and we write L = L(ω).

Corollary 8.6. For a regular Dirac structure L on M the following are equivalent: (i) c(L) = 0; (ii) L comes from a closed multiplicative 2-form on the monodromy groupoid of F; (iii) the underlying algebroid of L is isomorphic to F ⋉ ν^{∗}.

In this case L is integrable. Moreover, if one chooses ω as in (ii) (in which case L = L(ω)), then

(G(L), ω_{L}) ≅ (G(F) ⋉ ν^{∗}, ω_{F} − pr_{1}^{∗}ω)

(Here G(F) ⋉ ν^{∗} and ω_{F} = pr_{2}^{∗}ω_{can} are as in Section 8.3).

Proof. Using the map ρ_{ω}^{∗} : F ⟶ T^{∗}M induced from ω, we have

L ≅ F ⋉ ν^{∗}, (v, ξ) ↦ (v, ξ − ρ_{ω}^{∗}(v)),

which is an isomorphism of Lie algebroids. Hence G(L) ≅ G(F) ⋉ ν^{∗}.

To find the 2-form ω_{L} on G(F)⋉ν^{∗}, we look at its infinitesimal counterpart ρ^{∗} : F⋉ν^{∗} ⟶ T^{∗}M. This is obtained by transporting pr_{2} : L ⟶ T^{∗}M (which defines ω_{L} on G(L)) by the isomorphism above. Hence ρ^{∗}(v, ξ) = ξ − ρ_{ω}^{∗}(v). Now, ρ_{0}^{∗}(v, ξ) = ξ is precisely the infinitesimal counterpart of the multiplicative 2-form ω_{F}, while ρ_{1}^{∗}(v, ξ) = ρ_{ω}^{∗}(v) comes from the multiplicative form ω on G(F) and the projection on the first factor. Hence the form induced by ρ^{∗} is ω_{F} − pr_{1}^{∗}ω. □

Another case in which we can make G(L) more explicit is when c(L) is integrable as a foliated cohomology class; as we will see, this is similar to Van Est’s approach to Lie’s third theorem for Lie algebras (see [12] and references therein).

Example 8.7. (The case of integrable c(L)) Recall that, if a Lie groupoid G acts on a vector bundle E, we can define differentiable cohomology groups H_{diff}^{∗}(G; E), and the Van Est map maps these cohomology groups into Lie algebroid cohomology with coefficients in E. We refer the reader to [32, 12] for a general discussion. Here we only deal with the Van Est map for G(F), with coefficients in ν^{∗}, and in degree two:

Φ : H_{diff}^{2}(G(F); ν^{∗}) ⟶ H^{2}(F; ν^{∗}).

We now recall its definition. A differentiable 2-cocycle on G(F) with coefficients in ν^{∗} is a smooth function c which associates to any composable pair (g, h) an element c(g, h) ∈ ν_{t(g)}^{∗}, which vanishes whenever g or h is a unit. We say that c is closed if

gc(h, k) − c(gh, k) + c(g, hk) − c(g, h) = 0,

for all triples (g, h, k) of composable arrows in G(F). Two cocycles c and c′ are said to be cohomologous if their difference is of type (g, h) ↦ gd(h) − d(gh) + d(g) for some section d ∈ Γ(G; t^{∗}ν^{∗}). This defines H_{diff}^{2}(G(F)).

Any closed c defines a foliated form Φ(c) ∈ Ω^{2}(F; ν^{∗}): roughly speaking, Φ(c) is obtained from c by taking derivatives along leafwise vector fields (for the precise formulas, see [12, 32]). For our purpose, it will be useful to give a more abstract description of Φ(c) using extensions. ; Analogously to Lie algebroid extensions by algebroid 2-cocycles, differentiable 2-cocycles induce groupoid structures on G(F) × ν^{∗} with the multiplication extending the one in G(F) ⋉ ν^{∗}:

(g, v)(h, w) = (gh, h^{−1}v + w + (gh)^{−1}c(g, h)).

The resulting groupoid is denoted by G(F) ⋉_{c} ν^{∗}. Still as in the infinitesimal case, the isomorphism class of G(F) ⋉_{c} ν^{∗} depends only on the cohomology class of c, and this groupoid fits into an exact sequence of groupoids

(8.6)  1 ⟶ ν^{∗} ⟶ G(F) ⋉_{c} ν^{∗} ⟶ G(F) ⟶ 1.

Passing to Lie algebroids, this induces an extension of F by ν^{∗}, hence a cohomology class in H^{2}(F; ν^{∗}). This defines Φ([c]), and determines Φ at the cohomology level. Note that this cohomology class has a canonical representative, and that will define Φ(c), i.e, the map Φ at the chain level. The exact sequence (8.6) has a canonical splitting, which induces a linear splitting σ at the algebroid level; the associated foliated form (8.5) defines Φ(c).

Corollary 8.8. If the characteristic class c(L) comes from a differentiable cocycle c (i.e. if c(L) is integrable), then L is integrable and G(L) ≅ G(F) ⋉_{c} ν^{∗}.

At first sight, this corollary is just the definition of the integrability of c(L). This is due to our definition of Φ in terms of extensions. However, [12] gives us a precise description of when c(L) is integrable, related to the monodromy groups of L. In this way the result becomes meaningful.

More precisely, by [12] we know that u ∈ H^{2}(F; ν^{∗}) is integrable if and only if all its leafwise periods vanish. This means that, for each leaf S and any 2-sphere γ in S, ∫_{γ} u|_{S} = 0. On the other hand, by the very definition of the monodromy groups N_{x}(L) [13], and by the description of c(L) above in terms of a splitting σ, we have

N_{x}(L) = {∫_{γ} c(L)|_{S} : γ ∈ π_{2}(S, x)},

where S is the leaf through x. As in [14], this groups can be interpreted (or defined) as the groups defined by the variations of the presymplectic areas. And, still completely analogous to the Poisson case treated in [14], we state the conclusion without further details.

Corollary 8.9. The following are equivalent (i) c(L) is integrable; (ii) all the leafwise periods of c(L) vanish; (iii) all the monodromy groups N_{x}(L) vanish.

The discussion of general regular Dirac structures (i.e., when c(L) is not necessarily integrable) can be treated, again, exactly as in the Poisson case [14].

So far, we have dealt with the case of ϕ = 0. However, much of the discussion in this section extends to ϕ-twisted regular Dirac structures. For instance, one should use the ϕ-twisted Courant bracket in the construction of the foliated form (8.5). This defines the classifying class of L, c(L) ∈ H^{2}(F; ν^{∗}), and its integrals over leafwise 2-loops defines the monodromy groups N_{x}(L).

We finish the section with remarks on the particular twisted case discussed in Section 8.3.

Example 8.10. (Twisted regular Dirac structures)

Let ϕ ∈ F_{1}Ω^{3}(M), i.e.

i_{U}i_{V} i_{W} ϕ = 0, ∀ U, V, W ∈ F.

Let L be a regular Dirac structure L (with presymplectic foliation F). Then L is automatically a ϕ-twisted regular presymplectic Dirac structure. To distinguish these two structures, we write (L, ϕ) for L viewed as a twisted Dirac structure. We also write c(L, ϕ) for its class, G(L, ϕ) for the associated groupoid, etc.

As in Section 8.3, if ϕ ∈ F_{2}Ω^{3}(M), then G(L, ϕ) = G(L); if moreover ϕ ∈ F_{3}Ω^{3}(M), then the corresponding multiplicative 2-forms coincide.

Let us now consider the class induced by ϕ, ϕ̄ ∈ F_{1}Ω^{3}(M)/F_{2}Ω^{3}(M) ≅ Ω^{2}(F; ν^{∗}), i.e.

ϕ̄(V, W) = i_{V} i_{W} (ϕ).

Note that, if ϕ̄ = dψ for some ψ ∈ Ω^{1}(F; ν^{∗}), then Theorem 5.1 applied to G(F), with the twist by ϕ, and to ρ^{∗} : F ⟶ T^{∗}M which is just ψ viewed as a bundle map. In this way we get an induced ϕ-twisted multiplicative 2-form

ω_{ψ} ∈ Ω^{2}(G(F)).

Let us denote by pr : G(L) ⟶ G(F) the projection induced by the first projection pr_{1} : L ⟶ F.

Corollary 8.11. If [ϕ̄] ∈ H^{2}(F; ν^{∗}) vanishes, then G(L, ϕ) = G(L).

More precisely, any choice of ψ ∈ Ω^{1}(F; ν^{∗}) such that ϕ̄ = dψ induces an isomorphism G(L, ϕ) ≅ G(L) which maps the multiplicative 2-form on G(L, ϕ) to the form ω_{L} + pr^{∗}ω_{ψ}.

Proof. First of all, by the description of the classifying classes in terms of splittings σ and of formula (8.5), it immediately follows that the forms u_{σ} representing c(L), and u_{σ,ϕ} representing c(L, ϕ), satisfy u_{σ,ϕ} = u_{σ} + ϕ̄. In particular,

c(L, ϕ) = c(L) + [ϕ̄].

Hence, by the classifying properties of c(L), the first assertion follows.

The second assertion follows from the general properties mentioned above: σ induces algebroid isomorphisms L ≅ F ⋉_{u_{σ}} ν^{∗}, (L, ϕ) ≅ F ⋉_{u_{σ,ϕ}} ν^{∗}, while ψ induces an isomorphism between the algebroids associated to the two cocycles. Actually the resulting isomorphism does not depend on σ, and is just (X, ξ) ↦ (X, ψ̄(X) + ξ). It is clear now that the infinitesimal counterpart of the twisted multiplicative form is, on the untwisted L, just the sum of pr_{2} : L ⟶ T^{∗}M (defining ω_{L}), and the composition of pr_{1} : L ⟶ F with ϕ̄ (which is the infinitesimal counterpart of pr^{∗}ω_{ψ}). This concludes the proof. □

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Department of Mathematics, University of Toronto, Toronto, Ontario M5S3G3, Canada

Department of Mathematics, University of California, Berkeley, CA 94720, USA, and Utrecht University, P.O. Box 80.010, 3508 TA, Utrecht, The Netherlands

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Department of Mathematics, University of California, Berkeley, CA 94720, USA