GAUGE EQUIVALENCE OF DIRAC STRUCTURES AND SYMPLECTIC GROUPOIDS

HENRIQUE BURSZTYN AND OLGA RADKO

Abstract. We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gaugeequivalent integrable Poisson structures are Morita equivalent. As an example, we study certain generic sets of Poisson structures on Riemann surfaces: we find complete gauge-equivalence invariants for such structures which, on the 2-sphere, yield a complete invariant of Morita equivalence.

1. Introduction

Dirac structures were introduced in [10, 11] to provide a geometric framework for the study of constrained mechanical systems. Examples of Dirac structures on a manifold M include pre-symplectic forms, Poisson structures and foliations; in general, a Dirac structure determines a singular foliation on M whose leaves carry a pre-symplectic structure.

The notion of gauge equivalence of Dirac structures was introduced in [27] motivated by the study of the geometry of Poisson structures “twisted” by a closed 3-form. (See also [21, 24], where such structures were introduced in connection with Poisson-sigma models.) There is a natural way to modify a Dirac structure on M by a closed 2-form B ∈ Ω^{2}(M): one adds the pull-back of B to the pre-symplectic form on each leaf of the foliation, and the resulting pre-symplectic foliation determines a new Dirac structure. This operation is called a gauge transformation, and it defines an action of the additive group of closed 2-forms on Dirac structures. Two Dirac structures are called gauge equivalent if they lie in the same orbit of this action. As discussed in [27], gauge-equivalent Dirac structures share a lot of important properties; for instance, their corresponding Lie algebroids are always isomorphic.

Gauge transformations of Poisson structures also arise in some quantization problems. Let (M, π) be a Poisson manifold, and let Def(M, π) denote its moduli space of equivalence classes of star products [1]. It follows from Kontsevich’s formality theorem [22] that Def(M, π) is in bijection with equivalence classes of formal Poisson structures on (M, π). As discussed in [4], the classification of deformation quantizations of (M, π) up to Morita equivalence can be expressed in terms of the orbits of a canonical action of the Picard group Pic(M) ≅ H^{2}(M, ℤ) on Def(M, π). Following Kontsevich’s correspondence, there is a Poisson counterpart of algebraic Morita equivalence given by an H^{2}(M, ℤ)-action on formal Poisson structures, and the results in [4, 5, 20] indicate that this action is given by gauge transformations. In this context, a natural question is how gauge equivalence relates to the geometric notion of Morita equivalence of Poisson manifolds [33]. (See [23] for other aspects of the relationship between algebraic and geometric Morita equivalence.)

1991 Mathematics Subject Classification. Primary 53D17, Secondary 58H05.

Key words and phrases. Dirac structures, gauge equivalence, Morita equivalence, symplectic groupoids.

HB thanks MSRI for financial support through NSF grant DMS-9810361.

In this paper we discuss the notion of gauge equivalence in the realm of symplectic dual pairs and symplectic groupoids, and study the relationship between the notions of gauge and Morita equivalence of Poisson structures.

The paper is organized as follows.

In Section 2 we recall some basic facts about Dirac structures and their functorial properties, and establish an equivariance property of Dirac maps with respect to gauge transformations.

In Section 3 we extend the usual notion of a symplectic dual pair [29] to deal with Dirac manifolds. This more general notion, called a pre-dual pair, arises naturally when one considers a gauge transformation of a Poisson manifold which is part of a symplectic dual pair. We show that, under natural regularity conditions, ordinary symplectic dual pairs are obtained as quotients of pre-dual pairs.

In Section 4 we study the effect of gauge transformations of an integrable Poisson manifold on the symplectic structure of its symplectic groupoid. We show that if (G, Ω) is a symplectic groupoid and G_{0} is its identity section, endowed with its natural Poisson structure π, then a gauge transformation of π by B ∈ Ω^{2}(G_{0}) changes the symplectic form on G by

Ω ↦ Ω + α^{∗}B − β^{∗}B,

where α and β are the source and target maps, respectively.

Using this result, we show in Section 5 that two integrable gauge-equivalent Poisson structures are Morita equivalent. The converse is clearly not true, as Morita equivalent Poisson structures need not have the same leaf decomposition. We show that, in fact, the converse is not true even if we consider the possibility of gauge equivalence up to a Poisson diffeomorphism.

Finally, in Section 6, we study gauge and Morita equivalence of a certain generic set of Poisson structures on a compact connected oriented surface Σ. We consider the space 𝒢_{n}(Σ) of Poisson structures vanishing linearly on n smooth disjoint curves on Σ and show that two Poisson structures π, π′ ∈ 𝒢_{n}(Σ) vanishing on the same curves are gauge equivalent if and only if they have the same corresponding modular periods. (A classification of these structures up to isomorphism was obtained in [25].) This result provides a sufficient condition for Morita equivalence in 𝒢_{n}(Σ); for the case Σ = S^{2}, we construct a complete Morita-equivalence invariant for such Poisson structures.

The authors would like to thank Robert Bryant, Marius Crainic, Rui L. Fernandes, André Henriques, Nicholas Proudfoot and Alan Weinstein for valuable discussions and comments. HB thanks the Mathematical Sciences Research Institute for financial support and hospitality while this work was being developed.

2. Dirac structures

2.1. Linear Dirac structures. Let V be a finite-dimensional real vector space. Let us equip V ⊕ V^{∗} with the symmetric pairing

(2.1)  ⟨(x, ω), (y, μ)⟩ = \frac{1}{2}(ω(y) + μ(x)).

A Dirac structure on V is a subspace L ⊂ V ⊕ V^{∗} which is maximally isotropic with respect to ⟨ , ⟩ ^{1}. As we will see later, it is useful to think of (V ⊕ V^{∗}, ⟨ , ⟩) as an “odd” symplectic vector space (in the sense of super geometry), in such a way that its lagrangian subspaces correspond to Dirac structures on V. We denote the set of Dirac structures on V by Dir(V).

Throughout the text, we will identify bilinear forms Ω : V × V ⟶ ℝ (resp. π : V^{∗} × V^{∗} ⟶ ℝ) with linear maps Ω̃ : V ⟶ V^{∗} (resp. π̃ : V^{∗} ⟶ V) by Ω̃(u)(v) = Ω(u, v) (resp. π̃(ν)(η) = π(η, ν)). Whenever the context is clear, we will just write Ω for Ω̃ (resp. π for π̃).

Example 2.1. If Ω is a skew-symmetric bilinear form on V (resp. π is a skewsymmetric bilinear form on V^{∗}, i.e., a bivector on V), then L = graph(Ω) ⊂ V ⊕ V^{∗} (resp. L = graph(π)) is a Dirac structure on V. Conversely, any Dirac structure L satisfying L ∩ V^{∗} = {0} (resp. L ∩ V = {0}) defines a skew-symmetric bilinear form (resp. bivector) on V. Hence pre-symplectic structures (i.e., skew-symmetric bilinear forms) and Poisson structures (i.e., bivectors) on vector spaces are examples of linear Dirac structures.

Consider the natural projections ρ : V ⊕ V^{∗} ⟶ V, and ρ^{∗} : V ⊕ V^{∗} ⟶ V^{∗}, and let L be a Dirac structure on V. It is easy to check that [10]

ρ(L)^{∘} = L ∩ V^{∗}, (2.2) ρ^{∗}(L)^{∘} = L ∩ V, (2.3)

where ^{∘} denotes the annihilator.

Proposition 2.2. A Dirac structure L on V is equivalent to either of the following:

i) A pair (R, Ω), where R ⊆ V is a subspace and Ω is a skew-symmetric bilinear form on R.

ii) A pair (K, π), where K ⊆ V is a subspace and π is a bivector on the quotient

V/K. Moreover, these correspondences are such that R = ρ(L) and K = V ∩ L = ker Ω.

Proof. Let L be a Dirac structure on V. We set R = ρ(L) ⊆ V and define the skew-symmetric bilinear form Ω_{L} : R ⟶ R^{∗} by

(2.4)  Ω_{L}(x) ≐ η|_{R},

where η ∈ V^{∗} is any element such that (x, η) ∈ L. The map Ω_{L} is well defined by (2.2), and ker Ω_{L} = V ∩ L. Similarly, we can define a skew-symmetric form π_{L} on ρ^{∗}(L) and notice that (ρ^{∗}(L))^{∗} = V/ρ^{∗}(L)^{∘} = V/V ∩ L. So, for K ≐ V ∩ L, we have a well-defined bivector π_{L} on V/K with ker π_{L} = L ∩ V^{∗}.

1This is equivalent to L being isotropic and dim L = dim V.

Conversely, given a pair (R, Ω) as in i), we set

(2.5)  L ≐ {(x, η) | x ∈ R, η ∈ V^{∗} such that η|_{R} = Ω(x)}.

A simple dimension count shows that L is a Dirac structure satisfying ρ(L) = R and Ω_{L} = Ω. A similar construction holds for a pair (K, π) as in ii). □

For L ∈ Dir(V), we denote the corresponding skew-symmetric bilinear form on ρ(L) by Ω_{L} and the corresponding bivector on V/V ∩ L by π_{L}.

Example 2.3. Let L = graph(Ω), where Ω is a skew-symmetric bilinear form on V. Then R = ρ(L) = V, Ω_{L} = Ω, K = ker Ω, and π_{L} is the Poisson bivector on the reduced space V/K corresponding to the symplectic form induced by Ω.

Example 2.4. Let L = graph(π), where π is a bivector on V. Then R = π(V^{∗}), and Ω_{L} is the natural symplectic form induced by π. In this case, K = ker Ω_{L} = {0}, and π_{L} = π.

2.2. Functorial properties of linear Dirac structures. Linear Dirac structures have nice functorial properties: they can be both pushed forward and pulled back. This is a consequence of the dual characterizations of Dirac structures (Prop. 2.2) in terms of bilinear forms and bivectors.

We can also describe the functorial properties of Dirac structures in terms of Weinstein’s symplectic category [28], where the objects are symplectic vector spaces and the morphisms are canonical relations. We recall here the main ideas.

Let E, F and H be symplectic vector spaces. A canonical relation between E and F is a lagrangian subspace L ⊂ E × \overline{F}, where \overline{F} is the vector space F with symplectic form multiplied by −1. If L_{1} ⊂ E × \overline{F} and L_{2} ⊂ F × \overline{H}, the usual composition of relations

(2.6) L_{1} ∘ L_{2} = {(x, y) ∈ E × H | ∃z ∈ F such that (x, z) ∈ L_{1} and (z, y) ∈ L_{2}}

defines a lagrangian subspace of E × \overline{H}. If L_{1} = graph(f), L_{2} = graph(g) for symplectomorphisms f : F ⟶ E and g : H ⟶ F, then L_{1} ∘ L_{2} = graph(f ∘ g). In general, the composition of canonical relations defines a map

(2.7)  ∘ : Lag(E × \overline{F}) × Lag(F × \overline{H}) ⟶ Lag(E × \overline{H}),

where Lag(U) denotes the set of lagrangian subspaces of a vector space U. We remark that the same ideas work when symplectic vector spaces are replaced by vector spaces equipped with a nondegenerate symmetric bilinear form with zero signature (thought of as “odd” symplectic spaces); in this case, lagrangian (i.e., maximally isotropic) subspaces still have half the dimension of the total space.

Let V and W be vector spaces, and let ϕ : V ⟶ W be a linear map. Let E = (V ⊕ V^{∗}, ⟨ , ⟩) and F = (W ⊕ W^{∗}, ⟨ , ⟩), regarded as “odd” symplectic vector spaces. We define two canonical relations associated to ϕ:

Fϕ = \{(ϕ(x), η, x, ϕ^{∗}η) | x ∈ V, η ∈ W^{∗}\} ∈ Lag(F × \overline{E}), (2.8)

Bϕ = \{(x, ϕ^{∗}η, ϕ(x), η) | x ∈ V, η ∈ W^{∗}\} ∈ Lag(E × \overline{F}). (2.9)

(As will become clear below, the letters F and B stand for forward and backward.)

Since we have the natural identifications Dir(V) ≅ Lag(E × {0}) and Dir(W) ≅ Lag(F × {0}), the composition of relations (2.7) immediately induces maps

Fϕ : Dir(V) ⟶ Dir(W), (2.10) Bϕ : Dir(W) ⟶ Dir(V). (2.11)

Explicitly, for L_{V} ∈ Dir(V), L_{W} ∈ Dir(W) we have

Fϕ(L_{V}) = {(ϕ(x), η) | x ∈ V, η ∈ W^{∗}, (x, ϕ^{∗}η) ∈ L_{V} }, (2.12) Bϕ(L_{W}) = {(x, ϕ^{∗}η) | x ∈ V, η ∈ W^{∗}, (ϕ(x), η) ∈ L_{W}}. (2.13)

Example 2.5. If L_{V} = graph(π) for a bivector π on V, then Fϕ(L_{V}) = graph(ϕ_{∗}π). Analogously, if L_{W} = graph(Ω) for a skew-symmetric bilinear form Ω in W, then Bϕ(L_{W}) = graph(ϕ^{∗}Ω).

We observe that the maps Fϕ and Bϕ are not inverse to each other in general. A simple computation shows that if ϕ is 1-1, then Bϕ ∘ Fϕ = Id, and if ϕ is onto, then Fϕ ∘ Bϕ = Id.

Proposition 2.6. Let ϕ : V ⟶ W be a linear map, and let L_{V} ∈ Dir(V) and L_{W} ∈ Dir(W).

i) If Fϕ(L_{V}) = L_{W}, then ker Ω_{L_{W}} = ϕ(ker Ω_{L_{V}}) and π_{L_{W}} = ϕ_{∗}(π_{L_{V}}).

ii) If L_{V} = Bϕ(L_{W}), then ρ(L_{V}) = ϕ^{−1}(ρ(L_{W})) and Ω_{L_{V}} = ϕ^{∗}(Ω_{L_{W}}).

Proof. Using (2.12), it is easy to check that if L_{W} = Fϕ(L_{V}), then ker Ω_{L_{W}} = W ∩ L_{W} = {ϕ(x) | x ∈ V, (x, 0) ∈ L_{V} }. Since V ∩ L_{V} = ker Ω_{L_{V}}, it follows that ker Ω_{L_{W}} = ϕ(ker Ω_{L_{V}}).

As η ∈ ϕ(V ∩L_{V})^{∘} implies that ϕ^{∗}η ∈ V ∩L_{V}, we can define ϕ_{∗}π_{L_{V}} on W/ϕ(V ∩L_{V}) by ϕ_{∗}π_{L_{V}} (η) = ϕ(π_{L_{V}} (ϕ^{∗}η)), where η ∈ (W/ϕ(V ∩ L_{V}))^{∗} ≅ W^{∗}/ϕ(V ∩ L_{V})^{∘}.

By definition of π_{L_{V}} (see Prop. 2.2), ϕ(π_{L_{V}} (ϕ^{∗}η)) = ϕ(x), where x is such that (x, ϕ^{∗}η) ∈ L_{V}. On the other hand, if y = π_{L_{W}} (η), then (y, η) ∈ L_{W} = Fϕ(L_{V}), which is the case if and only if y = ϕ(x) and (x, ϕ^{∗}η) ∈ L. Therefore π_{L_{W}} = ϕ_{∗}(π_{L_{V}}) and i) is proven.

The proof of ii) is analogous and is left for the reader. □

Corollary 2.7. Let (V, π_{1}) and (W, π_{2}) be Poisson vector spaces, and let L_{π_{i}} = graph(π_{i}), i = 1, 2. A linear map ϕ : V ⟶ W is Poisson (i.e., φ_{∗}π_{1} = π_{2}) if and only if Fϕ(L_{π_{1}}) = L_{π_{2}}.

Similarly, if (V, Ω_{1}) and (W, Ω_{2}) are pre-symplectic vector spaces, then a linear map ϕ : V ⟶ W satisfies ϕ^{∗}Ω_{2} = Ω_{1} if and only if Bϕ(L_{Ω_{2}}) = L_{Ω_{1}}, where L_{Ω_{i}} = graph(Ω_{i}), i = 1, 2.

This motivates the following definition.

Definition 2.8. Let V and W be vector spaces endowed with Dirac structures L_{V} , L_{W}, respectively. A linear map ϕ : V ⟶ W is called forward Dirac if Fϕ(L_{V}) = L_{W}.

A map satisfying the analogous definition for Bϕ is called backward Dirac. We remark that these definitions are not equivalent. In this paper we will only deal with forward Dirac maps, and we will refer to them simply as Dirac maps.

Example 2.9. Let L ∈ Dir(V). As observed in Prop. 2.2(ii), V/ker Ω_{L} has an induced Poisson structure π_{L}. The projection pr : V ⟶ V/ker Ω_{L} is a Dirac map.

Lemma 2.10. Let L_{V} ∈ Dir(V) and L_{W} ∈ Dir(W). If Φ : V ⟶ W is a Dirac map, then it naturally induces a Poisson map ϕ : V/ker Ω_{L_{V}} ⟶ W/ker Ω_{L_{W}}.

Proof. Let pr_{V} : V ⟶ V/ ker Ω_{L_{V}} and pr_{W} : W ⟶ W/ ker Ω_{L_{W}} be the natural projections. By Prop. 2.2(i), the map ϕ : V/kerΩ_{L_{V}} → W/kerΩ_{L_{W}} given by ϕ(pr_{V} (x)) ≐ pr_{W} (Φ(x)) is well defined. By the observation in Example 2.9, we get

Fϕ(L_{π_{L_{V}}} ) = Fϕ(Fpr_{V} (L_{V})) = Fpr_{W} (FΦ(L_{V})) = Fpr_{W} (L_{W}) = L_{π_{L_{W}}} ,

where L_{π_{L_{V}}} = graph(π_{L_{V}}), L_{π_{L_{W}}} = graph(π_{L_{W}}) for the Poisson structures π_{L_{V}} and π_{L_{W}} on V/kerΩ_{L_{V}} and W/kerΩ_{L_{W}}, respectively. Therefore, ϕ is a Poisson map. □

2.3. Gauge equivalence of linear Dirac structures. Let Bil(W) be the additive group of skew-symmetric bilinear forms on a vector space W. Following Weinstein and Severa [27], we consider the action τ : Bil(W) × Dir(W) ⟶ Dir(W) by gauge transformations,

(2.14)  τ_{B}(L) = {(x, η + B(x)) | (x, η) ∈ L}, B ∈ Bil(W).

Alternatively, using Prop. 2.2, two Dirac structures L_{1}, L_{2} on W are gauge equivalent if

(2.15)  ρ(L_{1}) = ρ(L_{2}) and Ω_{L_{1}} = Ω_{L_{2}} + B|_{ρ(L_{2})},

for B ∈ Bil(W) and L ∈ Dir(W).

A linear map ϕ : V ⟶ W of vector spaces induces an action of Bil(W) on Dir(V), since ϕ^{∗}Bil(W) ⊆ Bil(V).

Lemma 2.11. The map Fϕ : Dir(V) ⟶ Dir(W) is Bil(W)-equivariant.

Proof. We must show that Fϕ(τ_{ϕ^{∗}B}(L)) = τ_{B}(Fϕ(L)) for all L ∈ Dir(V), B ∈ Bil(W). Since τ_{ϕ^{∗}B}(L) = {(x, η + (ϕ^{∗}B)(x)) | (x, η) ∈ L}, we have

Fϕ(τ_{ϕ^{∗}B}(L)) = {(ϕ(x), ξ) | (x, ϕ^{∗}ξ) ∈ τ_{ϕ^{∗}B}(L)} = {(ϕ(x), ξ) | (x, ϕ^{∗}ξ−(ϕ^{∗}B)(x)) ∈ L}.

On the other hand,

τ_{B}(Fϕ(L)) = {(ϕ(x), ζ+B(ϕ(x))) | (x, ϕ^{∗}ζ) ∈ L} = {(ϕ(x), ξ) | (x, ϕ^{∗}ξ−ϕ^{∗}(B(ϕ(x)))) ∈ L}.

Since (ϕ^{∗}B)(x) = ϕ^{∗}(B(ϕ(x))), the result follows. □

A similar equivariance property holds for the map Bϕ.

Let (V, Ω) be a symplectic vector space, and let (W, π) be a Poisson vector space. Let L_{Ω} = graph(Ω) and L_{π} = graph(π) be the corresponding Dirac structures.

Lemma 2.12. Let ϕ : (V, Ω) ⟶ (W, π) be a linear Poisson map, and let B ∈ Bil(W). Then the restricted map ϕ : ker(Ω+ϕ^{∗}B) ⟶ ker Ω_{τ_{B}(L_{π})} is an isomorphism. In particular, the form Ω + ϕ^{∗}B is symplectic if and only if τ_{B}(L_{π}) corresponds to a Poisson structure.

Proof. By Lemma 2.11, Fϕ(τ_{ϕ^{∗}B}(L_{Ω})) = τ_{B}(Fϕ(L_{Ω})) = τ_{B}(L_{π}), since ϕ is Poisson. Hence, by Prop. 2.6 (i),

(2.16)  ker Ω_{τ_{B}(L_{π})} = ϕ(ker(Ω + ϕ^{∗}B)),

and ϕ : ker(Ω + ϕ^{∗}B) ⟶ ker Ω_{τ_{B}(L_{π})} is onto.

On the other hand, since ϕ^{∗}B|_{ker ϕ} = 0, it follows that

ker(ϕ|_{ker(Ω+ϕ^{∗}B)}) = ker ϕ ∩ ker(Ω + ϕ^{∗}B) = ker ϕ ∩ ker Ω = 0.

So ϕ is injective. □

In order to study dual pairs, it will be useful to collect a few results on presymplectic orthogonals in the linear case.

Lemma 2.13. Let L ∈ Dir(V) and ϕ : V ⟶ W be linear. Then

(ker ϕ ∩ ρ(L))^{Ω_{L}} = {x ∈ V | ∃η ∈ W^{∗} such that (x, ϕ^{∗}η) ∈ L}.

Proof. If x ∈ (ker ϕ ∩ ρ(L))^{Ω_{L}}, then Ω_{L}(x)(y) = 0 for all y ∈ ker ϕ ∩ ρ(L). The form η ∈ (ϕ(ρ(L)))^{∗} given by η(ϕ(y)) = Ω_{L}(x)(y) is well defined, ϕ^{∗}η = Ω_{L}(x), and hence (x, ϕ^{∗}η) ∈ L. On the other hand, suppose (x, ϕ^{∗}η) ∈ L for η ∈ W^{∗}. Then Ω_{L}(x) = ϕ^{∗}η|_{ρ(L)}, and if y ∈ ker ϕ ∩ ρ(L), we have Ω_{L}(x)(y) = η(ϕ(y)) = 0. □

Let L ∈ Dir(V), L_{i} ∈ Dir(W_{i}), and let J_{i} : V ⟶ W_{i}, i = 1, 2, be Dirac maps.

Lemma 2.14. Suppose that the following orthogonality condition on the J_{i}-fibers holds:

(2.17)  (ker J_{1} ∩ ρ(L))^{Ω_{L}} = ker J_{2} ∩ ρ(L).

Then for any B ∈ Bil(W_{2}), J_{1} : (V, τ_{J_{2}^{∗}B}(L)) ⟶ (W_{1}, L_{1}) is a Dirac map.

Proof. We must show that FJ_{1}(τ_{J_{2}^{∗}B}(L)) = L_{1}. Since these subspaces have the same dimension, it suffices to prove that FJ_{1}(L) = L_{1} ⊆ FJ_{1}(τ_{J_{2}^{∗}B}(L)). Recall that

FJ_{1}(τ_{J_{2}^{∗}B}(L)) = {(J_{1}(x), η) | x ∈ V, η ∈ W_{1}^{∗}, (x, J_{1}^{∗}η) ∈ τ_{J_{2}^{∗}B}(L)}
= {(J_{1}(x), η) | x ∈ V, η ∈ W_{1}^{∗}, (x, J_{1}^{∗}η − (J_{2}^{∗}B)(x)) ∈ L},

and

FJ_{1}(L) = {(J_{1}(x), η) | x ∈ V, η ∈ W_{1}^{∗}, (x, J_{1}^{∗}η) ∈ L}.

Suppose (J_{1}(x), η) ∈ FJ_{1}(L). By Lemma 2.13, x ∈ (ker J_{1} ∩ ρ(L))^{Ω_{L}} = ker J_{2} ∩ ρ(L). So (J_{2}^{∗}B)(x) = 0, and therefore (x, J_{1}^{∗}η − (J_{2}^{∗}B)(x)) = (x, J_{1}^{∗}η) ∈ L. This implies that (J_{1}(x), η) ∈ FJ_{1}(τ_{J_{2}^{∗}B}(L)), and the result follows. □

Consider a symplectic vector space (V, Ω), Poisson vector spaces (W_{i}, π_{i}), i = 1, 2, and linear Poisson maps J_{i} : V ⟶ W_{i}. We call the diagram

[figure: (V, Ω); (W_{1}, π_{1}); (W_{2}, π_{2}); J_{1}; J_{2}]

a linear dual pair if (ker J_{1})^{Ω} = ker J_{2}. Let us fix such a linear dual pair, and let L_{Ω} = graph(Ω) and L_{π_{i}} = graph(π_{i}). Before we prove the main result of this section, we need the following lemma.

Lemma 2.15. Let B ∈ Bil(W_{2}), and let Ω′ = Ω+J_{2}^{∗}B. Then (ker J_{2})^{Ω′} = (ker J_{2})^{Ω} = ker J_{1} and (ker J_{1})^{Ω′} = (ker J_{1})^{Ω} + ker Ω′.

Proof. Since J_{2}^{∗}B|_{ker J_{2}} = 0, it follows that (ker J_{2})^{Ω′} = (ker J_{2})^{Ω} = ker J_{1}. By taking Ω′-orthogonals, we get (ker J_{1})^{Ω′} = ker J_{2} + ker Ω′. □

We can now prove the main result of this section.

Theorem 2.16. Let B_{i} ∈ Bil(W_{i}), i = 1, 2, and let Ω̂ ≐ Ω + J_{1}^{∗}B_{1} + J_{2}^{∗}B_{2}. Then (i) the maps J_{i} : (V, LΩ̂) ⟶ (W_{i}, τ_{B_{i}}(L_{π_{i}})), i = 1, 2, are Dirac; (ii) the form Ω̂ is symplectic if and only if τ_{B_{i}}(L_{π_{i}}), i = 1, 2, are Poisson; (iii) (ker J_{1})Ω̂ = ker J_{2} + ker Ω̂.

Proof. By Lemma 2.11, J_{2} : (V, L_{Ω+J_{2}^{∗}B_{2}}) ⟶ (W_{2}, τ_{B_{2}}(L_{π_{2}})) is Dirac, and, by Lemma 2.14, J_{1} : (V, L_{Ω+J_{2}^{∗}B_{2}}) ⟶ (W_{1}, π_{1}) is also Dirac. Again by Lemma 2.11, it follows that J_{1} : (V, LΩ̂) ⟶ (W_{1}, τ_{B_{1}}(L_{π_{1}})) is Dirac. By Lemma 2.15, (ker J_{2})^{Ω+J_{2}^{∗}B_{2}} = (ker J_{2})^{Ω} = ker J_{1}. So condition (2.17) is satisfied, and Lemma 2.14 implies that J_{2} : (V, LΩ̂) ⟶ (W_{2}, τ_{B_{2}}(L_{π_{2}})) is a Dirac map. This proves i).

In order to prove ii), let us assume that Ω̂ is symplectic. It follows from i) and Prop. 2.2(i) that τ_{B_{1}}(L_{π_{1}}) and τ_{B_{2}}(L_{π_{2}}) are Poisson. Conversely, if τ_{B_{1}}(L_{π_{1}}) is Poisson, then Ω + J_{1}^{∗}B_{1} is symplectic, by Lemma 2.12. It follows, again from Lemma 2.12, that if τ_{B_{2}}(L_{π_{2}}) is also Poisson, then Ω + J_{1}^{∗}B_{1} + J_{2}^{∗}B_{2} is symplectic.

We now prove iii). Clearly, (ker J_{1})Ω̂ = (ker J_{1})^{Ω+J_{2}^{∗}B_{2}} = ker J_{2} +ker(Ω+J_{2}^{∗}B_{2}), by Lemma 2.15. Thus ker J_{2} + ker Ω̂ ⊆ (ker J_{1})Ω̂. On the other hand, again by Lemma 2.15, ker(Ω + J_{2}^{∗}B_{2}) ⊆ ker J_{1}. So ker Ω̂ ∩ ker J_{1} = ker(Ω + J_{2}^{∗}B_{2}), and therefore ker(Ω + J_{2}^{∗}B_{2}) ⊆ ker Ω̂. Hence ker J_{2} + ker(Ω + J_{2}^{∗}B_{2}) ⊆ ker J_{2} + ker Ω̂, and the result follows. □

2.4. Dirac structures on manifolds. Let M be a smooth manifold. A Dirac structure on M [10] is a subbundle L ⊂ T M ⊕ T^{∗}M which determines linear Dirac structures pointwise and whose sections are closed under the Courant bracket [ , ] : Γ(T M ⊕ T^{∗}M) × Γ(T M ⊕ T^{∗}M) ⟶ Γ(T M ⊕ T^{∗}M),

(2.18)  ((X, ω), (Y, μ)) ↦ ([X, Y], L_{X}μ − L_{Y} ω + \frac{1}{2}d(ω(X) − μ(Y)).

Let (M, L_{M}) and (N, L_{N}) be Dirac manifolds. A smooth map ϕ : M ⟶ N is a (forward) Dirac map if FT_{x}ϕ((L_{M})_{x}) = (L_{N})_{ϕ(x)} for all x ∈ M.

Example 2.17. Let Ω ∈ Ω^{2}(M) (resp. π ∈ χ^{2}(M)). As discussed in Section 2.1, L = graph(Ω) ⊂ T M ⊕ T^{∗}M (resp. L = graph(π)) defines a pointwise linear Dirac structure. In this case, the extra condition involving the Courant bracket in the definition of a Dirac manifold is equivalent to the integrability condition dΩ = 0 (resp. [π, π] = 0, where [ , ] is the Schouten bracket). Hence pre-symplectic and Poisson structures on M are particular cases of Dirac structures.

The Courant bracket (2.18) does not satisfy the Jacobi identity in general. However, the Jacobi identity does hold when this bracket is restricted to sections of a Dirac subbundle L ⊂ T M ⊕ T^{∗}M, and it defines a Lie algebroid structure on L with anchor map ρ|_{L}, where ρ : T M ⊕ T^{∗}M ⟶ T M is the natural projection. When L = graph(π), where π is a Poisson structure, the Lie algebroid structure on L is isomorphic to the natural Lie algebroid structure on T^{∗}M via the projection ρ^{∗} : T M ⊕ T^{∗}M ⟶ T^{∗}M.

A Dirac manifold (M, L) carries a (singular) pre-symplectic foliation: the leaves are the orbits of the corresponding Lie algebroid and the leafwise pre-symplectic structure is defined as in Prop. 2.2(i). This foliation is symplectic if and only if L = graph(π) for a Poisson structure π.

As in the linear case, one can think of a Dirac structure L on M as a “Poisson structure on the leaf space M/K”, where K is the characteristic foliation of the leafwise pre-symplectic form Ω_{L}. More precisely, if (M, L) is a Dirac manifold, we define its set of admissible functions by

(2.19)  A ≐ {f ∈ C^{∞}(M) | df|_{ker(Ω_{L})} = 0} = {f ∈ C^{∞}(M) | df ∈ ρ^{∗}(L)}.

If f ∈ A, there exists X ∈ χ(M) such that (X, df) ∈ L; we call X a hamiltonian vector field of f and denote it by X_{f}. Note that hamiltonian vector fields of admissible functions are defined up to vector fields in ker Ω_{L}. If f, g ∈ A, then the bracket

(2.20)  {f, g} ≐ Ω_{L}(X_{f}, X_{g})

is well defined and makes A into a Poisson algebra. Clearly, if L comes from a Poisson structure, then A is just the Poisson algebra (C^{∞}(M), { , }). If the characteristic foliation K of Ω_{L} is simple, i.e., if M/K is a smooth manifold and pr : M ⟶ M/K is a submersion, then A ≅ C^{∞}(M/K) and there is a naturally defined Poisson structure π_{L} on M/K in such a way that pr is a Dirac map (Example 2.9).

Gauge transformations of Dirac structures are defined analogously to the linear case: if L is a Dirac structure on M and B a closed 2-form, we set

(2.21)  τ_{B}(L) = {(X, η + B(X)), (X, η) ∈ L}.

The closedness of B guarantees that τ_{B}(L) satisfies the integrability condition with respect to the Courant bracket. As in the linear case, two Dirac structures on M in the same τ-orbit are called gauge equivalent.

As observed in [27, Sec. 3], for a Poisson structure π on M, τ_{B}(π) is Poisson if and only if the endomorphism 1 + Bπ : T^{∗}M ⟶ T^{∗}M is invertible. In this case,

τ_{B}(π) = π(1 + Bπ)^{−1}.

Gauge-equivalent Dirac structures share many properties: for instance, they have the same leaf decomposition (though the pre-symplectic forms on the leaves differ by the pullbacks of B (see (2.15))) and their corresponding Lie algebroids are isomorphic [27]. In particular, gauge-equivalent Poisson structures have isomorphic Poisson cohomology.

3. Pre-dual pairs and reduction

A dual pair [29] consists of a symplectic manifold (S, Ω), Poisson manifolds (M_{1}, π_{1}), (M_{2}, π_{2}), and Poisson maps J_{i} : S ⟶ M_{i}, i = 1, 2 with symplectically orthogonal fibers, i.e.,

(3.1)  ker T_{x}J_{1} = (ker T_{x}J_{2})^{Ω}.

A dual pair is called full if the maps J_{1}, J_{2} are surjective submersions, and complete if these Poisson maps are complete (see e.g. [6]).

If M_{1} ←^{J_{1}} S →^{J_{2}} M_{2} is a full dual pair, then

ker T_{x}J_{1} = {X_{J_{2}^{∗}f}, f ∈ C^{∞}(M_{2})}, and (3.2) {J_{1}^{∗}(C^{∞}(M_{1})), J_{2}^{∗}(C^{∞}(M_{2}))} = 0. (3.3)

In order to deal with Dirac structures, we generalize the notion of a dual pair as follows.

Definition 3.1. A pre-dual pair is a pre-symplectic manifold (S, Ω), Dirac manifolds (M_{1}, L_{1}), (M_{2}, L_{2}) and Dirac maps J_{i} : S ⟶ M_{i}, i = 1, 2, such that

(3.4)  (ker T_{x}J_{1})^{Ω} = ker T_{x}J_{2} + ker Ω and (ker T_{x}J_{2})^{Ω} = ker T_{x}J_{1} + ker Ω.

As in the case of dual pairs, we represent pre-dual pairs by a diagram

[figure: (S, Ω); (M_{1}, L_{1}); (M_{2}, L_{2}).; J_{1}; J_{2}]

As before, a pre-dual pair will be called full if each J_{i} is a surjective submersion.

Proposition 3.2. Consider a pre-dual pair M_{1} ←^{J_{1}} S →^{J_{2}} M_{2}. If S is symplectic, then M_{1} and M_{2} are automatically Poisson, and we have a dual pair in the usual sense.

Proof. If ker(Ω) = 0, then ker(Ω_{L_{i}}) = T J_{i}(ker Ω) = 0, and hence L_{i} is automatically Poisson. The maps J_{i},i = 1, 2, are Dirac, and hence Poisson, and the orthogonality property (3.4) reduces to (3.1). □

We now fix a full pre-dual pair M_{1} ←^{J_{1}} S →^{J_{2}} M_{2}.

Proposition 3.3. We have the following generalization of (3.2):

ker T_{x}J_{1} + ker Ω = (ker T_{x}J_{2})^{Ω} = {X_{J_{2}^{∗}f} | f ∈ C^{∞}(M_{2})}.

Proof. Clearly, if v ∈ ker T_{x}J_{2}, Ω(X_{J_{2}^{∗}f}, v) = d_{x}(J_{2}^{∗}f)(v) = df(T_{x}J_{2}(v)) = 0. On the other hand, if u ∈ (ker T_{x}J_{2})^{Ω}, η = Ω_{x}(u) ∈ T_{x}^{∗}S vanishes along ker T_{x}J_{2}. Since J_{2} is a submersion, in a neighborhood of x we can choose an exact 1-form dg with d_{x}g = η and such that dg vanishes on the distribution ker T J_{2}. Hence g is constant along the J_{2}-fibers and can be written in the form J_{2}^{∗}f for f ∈ C^{∞}(M_{2}). Around x, u = X_{J_{2}^{∗}f}. □

The next result generalizes property (3.3).

Proposition 3.4. Let A be the algebra of admissible functions on S, and let A_{i} be the algebra of admissible functions on M_{i}. Then

i) J_{i}^{∗}A_{i} ⊆ A, ii) {J_{1}^{∗}A_{1}, J_{2}^{∗}A_{2}} = 0, iii) {J_{i}^{∗}f, J_{i}^{∗}g} = J_{i}^{∗}{f, g}, for f, g ∈ A_{i}.

Proof. By Prop. 2.6, T J_{i}(X) ∈ ker Ω_{L_{i}}. So, if X ∈ ker Ω and f ∈ A_{i}, then d(J_{i}^{∗}f)(X) = df(T J_{i}(X)) = 0, and i) is proven.

For ii), note that if f_{i} ∈ A_{i}, then {J_{1}^{∗}f_{1}, J_{2}^{∗}f_{2}} = Ω(X_{J_{1}^{∗}f_{1}}, X_{J_{2}^{∗}f_{2}}) = 0 by Prop. 3.4. Finally, since J_{i} is a Dirac map, (X, J_{i}^{∗}dg) ∈ L implies that (T_{x}J_{i}(X), dg) ∈

FT_{x}J_{i}(L) = L_{i}. This means that T_{x}J_{i}(X_{J_{i}^{∗}g}) = X_{g}. Now, if f, g ∈ A_{i}, then

{J_{i}^{∗}f, J_{i}^{∗}g}(x) = d_{x}(J_{i}^{∗}f)(X_{J_{i}^{∗}g}) = d_{J_{i}(x)}f(T_{x}J_{i}(X_{J_{i}^{∗}g})) = d_{J_{i}(x)}f(X_{g}) = {f, g}(J_{i}(x)),

and the result follows. □

Suppose that the characteristic foliations of Ω and Ω_{L_{i}} , K and K_{i}, respectively, are simple. Recall that S/K has an induced symplectic structure Ω_{red} , while M_{i}/K_{i}, i = 1, 2, carry induced Poisson structures π_{i} so that the projections pr : S ⟶ S/K and pr_{i} : S ⟶ M_{i}/K_{i} are Dirac maps.

Theorem 3.5. Let M_{1} ←^{J_{1}} S →^{J_{2}} M_{2} be a full pre-dual pair. The maps J_{i} : S ⟶ M_{i} induce surjective submersions on the quotient, j_{i} : S/K ⟶ M_{i}/K_{i}, in such a way that

[figure: (S/K, Ω_{red}); (M_{1}/K_{1}, π_{1}); (M_{2}/K_{2}, π_{2}); j_{1}; j_{2}]

is a full dual pair.

Proof. By Prop. 2.6(i), T J_{i}(ker Ω) = ker Ω_{L_{i}}. Hence if x and y belong to the same leaf of K, then J_{i}(x) and J_{i}(y) belong to the same leaf of K_{i}. So the map j_{i} : S/K ⟶ M_{i}/K_{i}, j_{i}(pr(x)) = pr_{i}(J_{i}(x)) is well defined and is a surjective submersion. It follows from Lemma 2.10 that j_{i} is a Poisson map. Finally, a simple computation shows that (ker T j_{i})^{Ω_{red}} = pr((ker T J_{i})^{Ω}). Hence (3.4) implies that (ker T j_{1})^{Ω_{red}} = ker T j_{2}. □

We obtain examples of pre-dual pairs from gauge transformations of Poisson structures in dual pairs as follows.

Theorem 3.6. Let (M_{1}, π_{1}) ←^{J_{1}} (S, Ω) →^{J_{2}} (M_{2}, π_{2}) be a full dual pair, and let B_{i} be a closed 2-form on M_{i}, i = 1, 2. Let Ω̂ = Ω + J_{1}^{∗}B_{1} + J_{2}^{∗}B_{2}. Then

[figure: (S, Ω̂); (M_{1}, τ_{B_{1}}(L_{π_{1}})); (M_{2}, τ_{B_{2}}(L_{π_{2}})); J_{1}; J_{2}]

is a full pre-dual pair. Moreover, Ω̂ is symplectic if and only if τ_{B_{i}}(L_{π_{2}}), i = 1, 2, are Poisson, in which case they form a dual pair.

Proof. The result is a consequence of Theorem 2.16. □

4. Gauge equivalence of symplectic groupoids

In this section we will apply the results of Section 3 to dual pairs coming from symplectic groupoids.

A symplectic groupoid [30] is a symplectic manifold (G, Ω) which is a Lie groupoid such that the graph γ_{m} = {(x, y, m(x, y)), (x, y) ∈ G_{2}} of the multiplication is lagrangian in G × G × \overline{G} (as usual, G_{2} denotes the set of composable pairs). We denote the source (resp. target) map of G by α (resp. β), the identity embedding by ϵ : G_{0} ↪ G, and the inversion by i : G ⟶ G. We recall that there exists a unique Poisson structure π on G_{0} making α (resp. β) into a Poisson (resp. anti-Poisson) map.

Let (M, π) be an integrable Poisson manifold, with symplectic groupoid (G, Ω, α, β). Since Lie algebroids corresponding to gauge-equivalent Dirac structures are isomorphic, all the Lie algebroids of Dirac structures in the τ-orbit of π can be integrated to a Lie groupoid isomorphic to (G, α, β). We now discuss the effect of a gauge transformation τ_{B} on the symplectic form Ω.

Let B be a closed 2-form on M, and consider the 2-form Ω_{B} ≐ Ω + α^{∗}B − β^{∗}B on G.

Theorem 4.1. If π_{B} ≐ τ_{B}(π) is Poisson, then G_{B} ≐ (G, Ω_{B}, α, β) is a symplectic groupoid integrating (M, π_{B}).

Proof. By Theorem 2.16, Ω_{B} is symplectic. We must check that the graph

γ_{m} = {(x, y, m(x, y)), (x, y) ∈ G_{2}}

is lagrangian in G_{B} × G_{B} × \overline{G_{B}}. Let (x, y) ∈ G_{2}, and consider a curve (x(t), y(t)) in G_{2} with (x(0), y(0)) = (x, y). Let (u, v) = (x′(0), y′(0)). Then (u, v, T_{(x,y)}m(u, v)) ∈ T_{p}γ_{m}, p = (x, y, m(x, y)) and any element in T_{p}γ_{m} is of this form.

Differentiating the identities α(m(x, y)) = α(x), β(m(x, y)) = β(y) and β(x) = α(y), we get

T αT m(u, v) = T α(u), T βT m(u, v) = T β(v), (4.1) T β(u) = T α(v). (4.2)

Therefore, if (u_{1}, v_{1}, T m(u_{1}, v_{1})), (u_{2}, v_{2}, T m(u_{2}, v_{2})) ∈ T_{p}γ_{m}, we have

(Ω_{B} × Ω_{B} × (−Ω_{B}))((u_{1}, v_{1}, T m(u_{1}, v_{1}), (u_{2}, v_{2}, T m(u_{2}, v_{2}))) = B(T α(u_{1}), T α(u_{2})) − B(T β(u_{1}), T β(u_{2})) + B(T α(v_{1}), T α(v_{2})) − B(T β(v_{1}), T β(v_{2}))− B(T αT m(u_{1}, v_{1}), T αT m(u_{2}, v_{2})) + B(T βT m(u_{1}, v_{1}), T βT m(u_{2}, v_{2})) = 0,

Hence γ_{m} is lagrangian in G_{B} × G_{B} × \overline{G_{B}} and G_{B} is a symplectic groupoid.

By Theorem 3.6, α : (G, Ω_{B}) ⟶ (M, π_{B}) is a Poisson map. Since there is a unique Poisson structure on the identity section of a symplectic groupoid with this property, the Poisson structure induced by Ω_{B} on M is π_{B}. □

Thus, the effect of applying a gauge transformation τ_{B} to the Poisson structure of the identity section of a symplectic groupoid is the following change of the symplectic form on the groupoid:

Ω ⟼^{τ_{B}} Ω_{B} = Ω + α^{∗}B − β^{∗}B.

Note that, in general, τ_{B}(π) is not Poisson, and the form Ω_{B} on G is degenerate. So

[figure: (G, Ω_{B}); (M, π_{B}); (M, −π_{B}); α; β]

is generally just a pre-dual pair. In this context, one is naturally led to consider groupoids equipped with pre-symplectic forms; the associated integration problem is whether a Dirac structure with an integrable Lie algebroid can be integrated to such a “pre-symplectic groupoid” (we believe an analog of the constructions in [7, 13] should clarify this question). Similarly, considering gauge transformations on Poisson groupoids [31], we are led to the more general notion of “Dirac groupoids”. The development of these ideas is the subject of work in progress.

5. Morita equivalence of gauge-equivalent Poisson structures

In this section, we compare the notions of gauge and Morita equivalence for integrable Poisson manifolds.

Two Poisson manifolds (M_{1}, π_{1}), (M_{2}, π_{2}) are called Morita equivalent [33] if there exists a symplectic manifold (S, Ω) and Poisson maps J_{i} : S ⟶ M_{i}, i = 1, 2, so that

[figure: (S, Ω); (M_{1}, π_{1}); (\overline{M_{2}}, π_{2}); J_{1}; J_{2}]

is a complete full dual pair with J_{i}-connected and J_{i}-simply-connected fibers. In this case we call this diagram a Morita equivalence bimodule.

Let (M, π) be an integrable Poisson manifold with α-connected and α-simply-connected symplectic groupoid (G, Ω, α, β). Let B be a closed 2-form on M such that π_{B} = τ_{B}(π) is Poisson.

Theorem 5.1. The Poisson manifolds (M, π), (M, π_{B}) are Morita equivalent, with Morita equivalence bimodule (G, Ω̂, α, β), where Ω̂ = Ω − β^{∗}B.

Proof. By Theorem 3.6, (M, π) ←^{α} (G, Ω̂) →^{β} (M, τ_{−B}(−π)) is a full dual pair with connected and simply connected fibers. Since τ_{−B}(−π)) = −τ_{B}(π), β : (G, Ω̂) ⟶ (M, τ_{B}(π)) is anti-Poisson, and it only remains to show that this dual pair is complete. Let X_{h} denote the hamiltonian vector field of h with respect to Ω. Let X̂_{h} and X_{h}^{B} denote the hamiltonian vector fields with respect to Ω̂ and Ω_{B} = Ω + α^{∗}B − β^{∗}B.

Claim. We have the following relations between the hamiltonian vector fields:

X̂_{α^{∗}f} = X_{α^{∗}f} and X̂_{β^{∗}f} = X_{β^{∗}f}^{B}, for all f ∈ C^{∞}(M).

Proof. Since X_{α^{∗}f} ∈ ker β, β^{∗}B(X_{α^{∗}f}) = 0. Hence Ω̂(X_{α^{∗}f}) = Ω(X_{α^{∗}f}) = d(α^{∗}f). Therefore, X̂_{α^{∗}f} = X_{α^{∗}f}. The other relation can be derived analogously. □

Let now f ∈ C^{∞}(M) be a complete function with respect to π (resp. π_{B}). We must check that X̂_{α^{∗}f} (resp. X̂_{β^{∗}f}) is complete.

By our claim, X̂_{α^{∗}f} = X_{α^{∗}f}, which is complete, since α : (G, Ω) ⟶ (M, π) is the target map of a symplectic groupoid [9, Chp. III]. Analogously, since (G, Ω_{B}, α, β) is a symplectic groupoid for (M, π_{B}), it follows that β : (G, Ω_{B}) ⟶ (M, π_{B}) is complete, and hence X̂_{β^{∗}f} = X_{β^{∗}f}^{B} is complete as well. □

It is clear that Morita equivalent Poisson structures on a manifold M need not be gauge equivalent, as their leaf decompositions are not necessarily the same. More generally, we say that two Poisson structures π and π′ on M are gauge equivalent up to Poisson diffeomorphism if there exists a diffeomorphism f : M ⟶ M such that f_{∗}π and π′ are gauge equivalent. It is clear that integrable Poisson structures which are gauge equivalent up to Poisson diffeomorphism are still Morita equivalent. However, as the next example shows, one can have Morita equivalent Poisson structures on a manifold M which are not gauge equivalent up to Poisson diffeomorphism.

Example 5.2. The example is based on non-cancellation properties of product manifolds (see [19] and references therein). Let F_{1}, F_{2} and B be closed smooth manifolds so that F_{1} and F_{2} have different homotopy types and F_{1}×B is diffeomorphic to F_{2}×B (see [8] for original examples). As discussed in [19], one can take B = S^{q} and F_{i} to be S^{q}-bundles over S^{n}, for q and n suitably chosen (and large). Hence we can assume that B and F_{i} are simply connected.

Let E be the total space of these trivial fibrations, with diffeomorphisms E ⟶^{φ_{i}} B × F_{i}, i = 1, 2. Let M = T^{∗}E. If φ_{i}^{♯} denotes the natural cotangent lift of φ_{i}, we obtain the diffeomorphisms

φ_{i}^{♯} : M ⟶ T^{∗}B × T^{∗}F_{i}, i = 1, 2.

Since the transformations in the structure group of the fiber bundle

[figure: T^{∗}F_{i}; M; T^{∗}B]

are cotangent lifts of diffeomorphisms of F_{i}, they preserve the canonical symplectic forms on T^{∗}F_{i}, i = 1, 2. Thus φ_{i}^{♯} makes M into a bundle of symplectic manifolds in the sense of [18], defining a Poisson structure π_{i} on M. As F_{1} and F_{2} have different homotopy types, so do T^{∗}F_{1} and T^{∗}F_{2}, and hence π_{1} and π_{2} cannot be gauge equivalent up to Poisson diffeomorphism.

Finally note that since F_{1}, F_{2} and B are simply connected, so are T^{∗}F_{1}, T^{∗}F_{2} and T^{∗}B, and [18, Thm. 3] implies that the symplectic structure along the fibers of T^{∗}F_{i} → M → T^{∗}B admits a closed extension. Hence the fundamental class [14] of (M, π_{i}) vanishes, and [33, Thm. 4.3] implies that (M, π_{i}) is Morita equivalent to the base T^{∗}B equipped with the zero Poisson structure. By transitivity, (M, π_{1}) and (M, π_{2}) are Morita equivalent.

6. Gauge and Morita equivalence of topologically stable Poisson structures on surfaces

Let Σ be a compact connected oriented surface. Since for dimensional reasons any bivector field on Σ is Poisson, Poisson structures on Σ form a vector space.

For n ≥ 0, let 𝒢_{n}(Σ) be the set of Poisson structures π on Σ such that • the zero set {p ∈ Σ| π(p) = 0} consists of n smooth disjoint curves γ_{1}(π), ⋯, γ_{n}(π);

• π vanishes linearly on each of the curves γ_{1}(π), ⋯, γ_{n}(π). Let 𝒢 (Σ) ≐ ⨆_{n≥0} 𝒢_{n}(Σ). For n ≥ 1 the symplectic leaves of π ∈ 𝒢 (Σ) are the points in the zero set ⨆_{i=1}^{n} γ_{i} and the connected components of Σ \ ⨆_{i=1}^{n} γ_{i}; for n = 0 the structure is symplectic. We call the Poisson structures in 𝒢 (Σ) topologically stable, since the topology of their zero sets is preserved under small perturbations.

Choosing a non-degenerate Poisson structure π_{0} on Σ, we can identify the space of Poisson structures on Σ with C^{∞}(Σ): any π is represented in the form π = f · π_{0} for f ∈ C^{∞}(Σ). Under this identification, 𝒢 (Σ) corresponds to the space of smooth functions for which 0 is a regular value. This implies that the set of topologically stable Poisson structures 𝒢 (Σ) is generic, i.e., 𝒢 (Σ) is an open dense subset of the space of all Poisson structures on Σ endowed with the Whitney C^{∞} topology.

As shown in [25], the classification of Poisson structures in 𝒢 (Σ) up to Poisson isomorphisms depends on a finite number of invariants. In order to recall what these invariants are, we need a few definitions.

For a Poisson manifold (M, π), let ν be a volume form on M. The modular vector field X^{ν} of π with respect to ν [32] is defined by the formula

X^{ν}h ≐ \frac{L_{X_{h}}ν}{ν} , h ∈ C^{∞}(M).

This vector field measures the degree of invariance of ν under the flows of hamiltonian vector fields; in particular, X^{ν} = 0 if and only if L_{X_{h}}ν = 0 for all h ∈ C^{∞}(M). The modular vector field X^{ν} is Poisson (i.e., its flow preserves π), and, if ν′ is a different volume form, one has

X^{ν′} = X^{ν} + X_{− log k},

where k is the nowhere zero ratio k = |\frac{ν′}{ν} |. Thus the class of X^{ν} in the first Poisson cohomology H_{π}^{1}(M) (i.e., its equivalence class modulo hamiltonian vector fields) is independent of ν. This defines an element in H_{π}^{1}(M) called the modular class and denoted by μ_{(M,π)}.

Suppose that the Poisson tensor π on M vanishes on a closed curve γ ⊂ M, and is nonzero away from γ in a neighborhood of γ. Since the modular vector field X^{ν} preserves the Poisson structure, its flow must take the zero set of π to the zero set of π. Thus the flow of X^{ν} takes γ to γ and so X^{ν} must be tangent to γ. Moreover, for another choice of volume form ν′, we have

X^{ν′}|_{γ} = X^{ν}|_{γ} + (X_{− log | \frac{ν′}{ν} |})|_{γ} = X^{ν}|_{γ},

since all hamiltonian vector fields are zero when restricted to the zero curve γ. It follows that the restriction of the modular vector field X^{ν} to γ is independent of ν, and hence induces an orientation on γ. As was observed in [26], the period of the flow of this vector field around γ is an invariant of the Poisson structure π. We denote this number by T_{γ}(M, π) (or, for short, T_{γ}(π) when it is clear what M is).

For π ∈ 𝒢_{n}(Σ), let Z(π) denote its zero set, consisting on n disjoint curves, taken with the induced orientations. The main result of [25] states that Poisson structures in 𝒢_{n}(Σ) are completely classified up to (orientation-preserving) Poisson isomorphisms by the class of Z(π) modulo orientation-preserving diffeomorphisms of Σ, the n modular periods of π around each connected component of Z(π) and the regularized Liouville volume V (π)(which is a certain regularized sum of symplectic volumes of two-dimensional leaves, taken with appropriate signs).

We shall now consider the questions of gauge and Morita equivalence of Poisson structures in 𝒢 (Σ).

6.1. Gauge equivalence in 𝒢 (Σ). The obvious necessary condition for two Poisson structures π, π′ ∈ 𝒢 (Σ) to be gauge equivalent is Z(π) = Z(π′), i.e. the zero sets of both structures, with the induced orientations, should be the same.

Theorem 6.1. Two Poisson structures π, π′ ∈ 𝒢_{n}(Σ) with Z(π) = Z(π′) = ⨆_{i=1}^{n} γ_{i} are gauge equivalent if and only if their modular periods are the same around all the zero curves, i.e. T_{γ_{i}}(π) = T_{γ_{i}}(π′), for i = 1, …, n.

Proof. Let π = f · π_{0}, π′ = f′ · π_{0}, where f, f′ ∈ C^{∞}(Σ) are functions vanishing linearly on γ_{1}, ⋯, γ_{n} and non-zero elsewhere, and π_{0} is a fixed nondegenerate Poisson structure on Σ.

Suppose that π′ = τ_{B}(π) for a closed 2-form B ∈ Ω^{2}(Σ). Let ω = f^{−1}ω_{0} (resp. ω′ = f′^{−1}ω_{0}) be the symplectic form corresponding to π (resp. π′) on Σ \ Z(π). Then

(6.1)  B|_{Σ\Z(π)} = ω′ − ω = (\frac{1}{f′} − \frac{1}{f} ) ω_{0}.

For each i = 1, …, n, let U_{i} = {(z_{i}, θ_{i})| |z_{i}| < R_{i}, θ_{i} ∈ [0, 2π]} be a small annular neighborhood of the zero curve γ_{i} ∈ Z(π) such that Z(π) ∩ U_{i} = γ_{i} and π|_{U_{i}} = f(z_{i})∂_{z_{i}} ∧ ∂_{θ_{i}}, π′|_{U_{i}} = f′(z_{i})∂_{z_{i}} ∧ ∂_{θ_{i}} with f|_{U_{i}} = c_{i}z_{i} + O(z^{2}), f′|_{U_{i}} = c′_{i}z_{i} + O(z^{2}). An easy computation (see [25, Sec. 2.4]) shows that, in these coordinates, T_{γ_{i}}(π) = \frac{2π}{c_{i}}, T_{γ_{i}}(π′) = \frac{2π}{c′_{i}}. Thus on U_{i} \ γ_{i} we have

\frac{1}{f′} − \frac{1}{f} = \frac{f − f′}{ff′} = \frac{c_{i}z_{i} − c′_{i}z_{i} + O(z_{i}^{2})}{z_{i}^{2}(1 + O(z_{i}))} = \frac{c_{i} − c′_{i}}{z_{i}} · O(1) + O(1), z_{i} ≠ 0.

Since B extends the form in (6.1), we must have c_{i} = c′_{i}, i = 1, ⋯, n.

Conversely, if c_{i} = c′_{i} for i = 1, ⋯, n, the form defined in (6.1) extends to a smooth closed form B ∈ Ω^{2}(Σ). In this case, π′ = τ_{B}(π) and, therefore, the structures π and π′ are gauge equivalent.

□

Let 𝒢_{n}(Σ; γ_{1}, …, γ_{n}; T_{γ_{1}}, …, T_{γ_{n}}) be the space of topologically stable Poisson structures π ∈ 𝒢_{n}(Σ) with fixed Z(π) = ⨆_{i=1}^{n} γ_{i} and fixed modular periods T_{γ_{1}}, …, T_{γ_{n}} around the zero curves.

Corollary 6.2. The additive group of closed 2-forms on Σ acts transitively on the space 𝒢_{n}(Σ; γ_{1}, …, γ_{n}; T_{γ_{1}}, …, T_{γ_{n}}). The regularized Liouville volume changes under this action as follows: V (τ_{B}(π)) = V (π) + Vol(B), where Vol(B) = ∫_{Σ} B is the Liouville volume of B.

For π ∈ 𝒢_{n}(Σ), the anchor π̃ : T^{∗}Σ → T Σ of the corresponding Lie algebroid is injective on the open dense set Σ \ Z(π). According to [15, Thm. 1], a Lie algebroid whose anchor is injective on an open dense set is integrable, so (Σ, π) is an integrable Poisson manifold.

The following result follows from Theorem 5.1.

Theorem 6.3. Two Poisson structures π, π′ ∈ 𝒢_{n}(Σ) with the same zero sets Z(π) = Z(π′) = ⨆_{i=1}^{n} γ_{i} and equal modular periods, T_{γ_{i}}(π) = T_{γ_{i}}(π′), for i = 1, …, n, are Morita equivalent.

We now turn our attention to the study of Morita equivalence in 𝒢 (S^{2}). First, we need to collect a few general results on invariants of Morita equivalence.

6.2. Invariants of Morita equivalence: topology of the leaf space and modular periods. Let (M, π) be a Poisson manifold. Let L(M) be the leaf space of the symplectic foliation of π, endowed with its quotient topology: for a topological space X, a function f : L(M) → X is continuous if and only if f ∘ pr : M → X is continuous, where pr : M → L(M) is the quotient map.

Let (M_{1}, π_{1}) and (M_{2}, π_{2}) be Poisson manifolds, and let (M_{1}, π_{1}) ←^{J_{1}} (S, Ω) →^{J_{2}} (M_{2}, π_{2}) be a Morita equivalence bimodule. It is well-known (see e.g. [2, 6]) that S induces a bijection of sets ϕ_{S} : L(M_{1}) → L(M_{2}) given by

ϕ_{S}(ℒ) = J_{2}(J_{1}^{−1}(ℒ)), for ℒ ∈ L(M_{1}).

The following observation is based on ideas from [12].

Proposition 6.4. The map ϕ_{S} : L(M_{1}) → L(M_{2}) is a homeomorphism of topological spaces.

Proof. Let F_{i} be the subset of T M_{i} consisting of vectors tangent to the symplectic leaves. Let T J_{i} ⊂ T S, i = 1, 2 be the subbundles tangent to the J_{i}-fibers. Then

J_{1}^{∗}F_{1} = J_{2}^{∗}F_{2} = T J_{1} + T J_{2},

where J_{i}^{∗}F_{i} = {v ∈ T S| T J_{i}v ∈ F_{i}} denotes the pull-back of F_{i}. Let F = J_{1}^{∗}F_{1} = J_{2}^{∗}F_{2}.

Since the fibers of J_{i} are connected, i = 1, 2, the natural maps

ψ_{i} : S/F → M_{i}/F_{i}

of leaf spaces are bijections. Moreover, it is not hard to see that ϕ_{S} = ψ_{2} ∘ ψ_{1}^{−1}. So, if we endow S/F with its quotient topology, it is sufficient to prove that ψ_{i}, i = 1, 2, are homeomorphisms.

By the definition of the quotient topology, the map ψ_{i} : S/F → M_{i}/F_{i} is continuous if and only if the map ψ_{i} ∘ pr : S → M_{i}/F_{i} is continuous (here pr : S → S/F is the quotient map). But ψ_{i} = pr_{i} ∘ J_{i}, where pr_{i} : M_{i} → M_{i}/F_{i} is the quotient map. Hence ψ_{i} is continuous.

Similarly, ψ_{i}^{−1} : M_{i}/F_{i} → S/F is continuous if and only if ψ_{i}^{−1} ∘ pr_{i} : M_{i} → S/F is continuous. Since J_{i} is a submersion, this is true if and only if ψ_{i}^{−1}∘pr_{i}∘J_{i} : S → S/F is continuous. But ψ_{i}^{−1} ∘ pr_{i} ∘ J_{i} = pr. Therefore, ψ_{i} is a homeomorphism for i = 1, 2, which implies that ϕ_{S} is a homeomorphism. □

The modular vector field and the modular class are well-behaved under Morita equivalence: the bijection of leaf spaces ϕ_{S} induced by a Morita equivalence bimodule produces an isomorphism of Poisson cohomologies [17, Thm. 3.1]

(6.2)  ϕ_{S}^{∗} : H_{π}^{1}(M_{1}) → H_{π}^{1}(M_{2}),

which preserves the modular class [12, 16],

ϕ_{S}^{∗}(μ_{(M_{1},π_{1})}) = μ_{(M_{2},π_{2})}.

We will need the following remark from the construction of the isomorphism (6.2) in [17].

Remark 6.5. Given volume forms ν_{1} and ν_{2} on M_{1} and M_{2}, respectively, there exists a vector field X on S with the property that (J_{i})_{∗}X = X^{ν_{i}}, i = 1, 2. The vector field X is actually hamiltonian, and its Hamiltonian H is determined by the equation

DJ_{1}^{∗}ν_{1} = ±e^{H}J_{2}^{∗}ν_{2},

where D : Ω^{k}(M) → Ω^{2m−k}(M) is the symplectic ∗-operator (cf. [3]).

We now observe that the modular periods also behave well under Morita equivalence.

Theorem 6.6. Let (M_{i}, π_{i}), i = 1, 2 be Poisson manifolds, and let (M_{1}, π_{1}) ←^{J_{1}} (S, Ω) →^{J_{2}} (M_{2}, π_{2}) be a Morita equivalence bimodule. Assume that Z_{i} ⊂ M_{i} are such that π_{i}|_{Z_{i}} = 0 and the isomorphism of leaf spaces satisfies

ϕ_{S}(Z_{1}) = Z_{2}.

Let Φ_{t}^{i} be the flow of the modular vector field X^{ν_{i}} for some volume form ν_{i} on M_{i}, i = 1, 2. Assume that Φ_{t}^{i} takes Z_{i} to Z_{i} for all t. Then

ϕ_{S} ∘ Φ_{t}^{1} = Φ_{t}^{2} ∘ ϕ_{S}, ∀t ∈ ℝ.

Proof. Let X be a vector field on S such that (J_{i})_{∗}X = X_{i}^{ν_{i}}, i = 1, 2 (see Remark 6.5). By the definition of ϕ_{S}, for each p_{1} ∈ Z_{1} (which forms by itself a symplectic leaf),

ϕ_{S}({p_{1}}) = J_{2}(J_{1}^{−1}({p_{1}})),

which by our assumption on ϕ_{S} is a single point p_{2} ∈ Z_{2}. It follows that J_{1}^{−1}({p_{1}}) ⊂ J_{2}^{−1}({p_{2}}). Reversing the roles of p_{1} and p_{2}, we get J_{1}^{−1}({p_{1}}) = J_{2}^{−1}({p_{2}}). In particular, it follows that

J_{1}^{−1}(Z_{1}) = J_{2}^{−1}(Z_{2}).

Thus for any fixed r ∈ J_{1}^{−1}({p_{1}}) = J_{2}^{−1}({p_{2}}) we obtain

Φ_{t}^{2}(ϕ_{S}({p_{1}})) = J_{2}(Φ_{t}({r})) = J_{2}(J_{1}^{−1}(Φ_{t}^{1}({p_{1}})) = ϕ_{S}(Φ_{t}^{1}({p_{1}})),

where Φ_{t} is the flow of X. Therefore, ϕ_{S} ∘ Φ_{t}^{1} = Φ_{t}^{2} ∘ ϕ_{S}. □

Corollary 6.7. Assume that γ_{i} ⊂ M_{i} are simple closed curves, U_{i} ⊃ γ_{i} are open sets, and that π_{i}|_{γ_{i}} = 0 and π|_{U_{i}\γ_{i}} ≠ 0, i = 1, 2. Assume further that (M, π_{1}) and (M_{2}, π_{2}) are Morita-equivalent via a bimodule S. Assume finally that ϕ_{S}(γ_{1}) = γ_{2}. Then

T_{γ_{1}}(M_{1}, π_{1}) = T_{γ_{2}}(M_{2}, π_{2}).

Proof. Applying the previous theorem with Z_{i} = γ_{i}, i = 1, 2, we obtain that the flows Φ_{t}^{i} of modular vector fields X_{i}^{ν_{i}} are intertwined by ϕ_{S}. Thus, for any p_{1} ∈ γ_{1},

T_{γ_{1}}(M_{1}, π_{1}) = inf{t > 0 : Φ_{t}^{1}(p_{1}) = p_{1}}
= inf{t > 0 : ϕ_{S}(Φ_{t}^{1}(p_{1})) = ϕ_{S}(p_{1})}
= inf{t > 0 : Φ_{t}^{2}(ϕ_{S}(p_{1})) = ϕ_{S}(p_{1})}
= inf{t > 0 : Φ_{t}^{2}(p_{2}) = p_{2}},  p_{2} = ϕ_{S}(p_{1})
= T_{γ_{2}}(M_{2}, π_{2}).
□

6.3. Morita equivalence of topologically stable Poisson structures on S^{2}. We will now show that two Poisson structures in 𝒢_{n}(S^{2}) are Morita equivalent if and only if they are gauge equivalent up to diffeomorphism. As a result, we find a complete Morita-equivalence invariant for topologically stable Poisson structures on S^{2}.

Let Σ = S^{2}, the 2-sphere. Let π ∈ 𝒢 (Σ) and write, as before, π = f ·π_{0}, where π_{0} is a non-degenerate Poisson structure on Σ and f ∈ C^{∞}(Σ). We fix on Σ the orientation induced by (the symplectic form corresponding to) π_{0}. The class of Z(π) modulo (orientation-preserving) diffeomorphisms of Σ can be described combinatorially by a signed tree in the following way [25, Sec. 2.8]. The vertices of the tree correspond to the connected components of Σ \ Z(π) (i.e., the regions bounded by the curves comprising Z(π)). The vertex is assigned a positive sign if f is positive on the corresponding region; otherwise the vertex is assigned a negative sign. Two vertices are connected by an edge if and only if the corresponding regions share a boundary. We denote this signed tree by T ree(π).

Lemma 6.8. If two Poisson structures π, π′ ∈ 𝒢_{n}(S^{2}) are Morita equivalent, then there exists an isomorphism of trees ϕ : T ree(π) → T ree(π′), not necessarily preserving signs, so that T_{γ}(π) = T_{ϕ(γ)}(π′) for every edge γ of T ree(π).

Proof. Assume that (S^{2}, π) and (S^{2}, π′) are Morita equivalent. Let ϕ be the induced homeomorphism of the leaf spaces of (S^{2}, π) and (S^{2}, π′) (see Prop. 6.4).

As a set, the leaf space L of (S^{2}, π) can be identified with the union Z(π) ⊔ {ℓ_{1}, ⋯, ℓ_{n}}, where ℓ_{1}, …, ℓ_{n} are the points corresponding to the 2-dimensional leaves L_{1}, …, L_{n}. The quotient topology of L is easily described: the only open subsets of L have the form U ∪ {ℓ_{i_{1}}} ∪ ⋯ ∪ {ℓ_{i_{k}}}, where i_{1}, …, i_{k} ∈ {1, ⋯, n}, k ≥ 0 and U ⊂ Z(π) is an open subset with the property that if U intersects non-trivially a curve γ ⊂ Z(π), then for both leaves bounding γ the corresponding points of the leaf space occur among {ℓ_{i_{1}}, …, ℓ_{i_{k}}}.

Given L with its topology, consider the collection Y of all subsets Y ⊂ L with the property that L \ Y is Hausdorff. Order Y by inclusion. We claim that X = {ℓ_{1}, …, ℓ_{n}} is a minimal element of Y of smallest cardinality. First, note that X ∈ Y, since the relative topology on L \ X = Z(π) ⊂ L is Hausdorff. Next, assume that Y ∈ Y, and Y ⊅ X. Then ℓ_{i} ∈ L \ Y for some i ∈ {1, ⋯, n}. Now all of the points of the boundary of L_{i} in S^{2} lie in Z(π) and cannot be separated from ℓ_{i} by open sets; thus all of these points must necessarily be in Y. Thus Y must have infinite cardinality.

It follows that ϕ must map X to a subset of L′ with the same minimality property; and hence ϕ must take the complement of X, Z(π), to Z(π′). Thus ϕ induces a map between the set of vertices of T ree(π) and T ree(π′).

Now, two vertices ℓ_{i}, ℓ_{j} ∈ T ree(π) are connected by an edge if and only if the corresponding regions share a boundary in S^{2}. A point x ∈ Z(π) ⊂ L cannot be separated from ℓ_{k} by an open set if and only if x belongs to the boundary of L_{k} in S^{2}. It follows that ℓ_{i}, ℓ_{j} are connected by an edge if and only if there exists a point x ∈ L, such that x ≠ ℓ_{i}, x ≠ ℓ_{j}, but which cannot be separated from either of them by an open set. Since ϕ is a homeomorphism, it must preserve this property, and thus ϕ induces a map of trees from T ree(π) to T ree(π′). The statement about modular periods now follows from Corollary 6.7. □

Theorem 6.9. Two Poisson structures π, π′ ∈ 𝒢_{n}(S^{2}) are Morita equivalent if and only if there exists an isomorphism of trees ϕ : T ree(π) → T ree(π′), not necessarily preserving signs, so that T_{γ}(π) = T_{ϕ(γ)}(π′) for every edge γ of T ree(π).

Proof. The necessity follows from Lemma 6.8.

Assume now that there exists an isomorphism ϕ : T ree(π) → T ree(π′) satisfying the conditions. Let ψ : S^{2} → S^{2} be an orientation-reversing diffeomorphism. By replacing π′ with ψ_{∗}π′ (which is obviously Poisson isomorphic, and hence Morita equivalent to, π′) if necessary, we may assume that ϕ : T ree(π) → T ree(π′) is an isomorphism of signed trees.

Choose a function g ∈ C^{∞}(S^{2}) supported on the interior of one of the twodimensional leaves. Let π′′ = π′ + gπ′. Since π′′ = π′ in a neighborhood of each of the zero curves γ ⊂ Z(π), the modular periods of π′ and π′′ are equal. Therefore, by Theorem 6.1, π′ and π′′ are gauge equivalent, and hence Morita equivalent (Theorem 5.1) for any such choice of g. Also, the isomorphism ϕ induces an isomorphism of trees ϕ′ : T ree(π) → T ree(π′′).

With a suitable choice of g, the regularized Liouville volume of π′′ can be made equal to that of π (see [25] for details). Thus by [25, Thm. 3], π and π′′ are Poisson isomorphic. We conclude that π and π′′ are Morita equivalent, and so are π and π′, by transitivity of Morita equivalence. □

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Henrique Bursztyn, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720

  E-mail address: henrique@math.berkeley.edu

Olga Radko, Department of Mathematics, U. C. Berkeley, Berkeley, CA 94720

  E-mail address: radko@math.berkeley.edu