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Gauge transformations of Jacobi structures and contact groupoids

We define gauge transformations of Jacobi structures on a manifold. This is related to gauge transformations of Poisson structures via the Poissonization. Finally, we discuss how the contact structure of a contact groupoid is effected by a gauge transformation of the Jacobi structure on its base.


Introduction
The notion of gauge transformations of Poisson structures associated with certain closed 2-forms was introduced byŠevera and Weinstein [23] in connection with Poisson-sigma models. Gauge transformations of Poisson structures also arise in some quantization problems [13]. Roughly, a gauge transformation of a given Poisson structure modify its leafwise symplectic forms by means of the pullback of a globally defined 2-form. Gauge equivalent Poisson structures share many important properties, namely, they gives rise to same singular foliation on the manifold, and corresponds to isomorphic Lie algebroid structures on the cotangent bundle. Gauge transformations of Poisson structures was further studied by Bursztyn and Radko [2,3] from the perspective of symplectic groupoids. They also provide a relationship between gauge transformations and Xu's Morita equivalence of Poisson manifolds.
The most natural framework to study gauge transformations of Poisson structures is that of Dirac structure. Gauge transformations have also been studied in the context of multiplicative Poisson and Dirac structures on a Lie groupoid [2,21]. Recently, the present author introduced gauge transformations of Nambu-Poisson structures and showed that these transformations commute with the reduction procedure [7].
Our main objective of this paper is the notion of Jacobi manifold, introduced by Lichnerowicz [17]. A Jacobi structure on a smooth manifold M consists of a pair (π, E) of a bivector field π ∈ Γ(∧ 2 T M ) and a vector field E ∈ Γ(T M ) satisfying certain conditions. Jacobi structures include symplectic, Poisson, contact and locally conformal symplectic (l.c.s.) structures [16]. A Jacobi structure (π, E) on M defines a bundle map (π, E) ♯ : T * M × R → T M × R, (α, g) → (π ♯ α + gE, − α, E ), for (α, g) ∈ Γ(T * M × R). One might expect that the natural framework to study gauge transformations of Jacobi structures is the notion of Dirac-Jacobi structure (also called E 1 (M )-Dirac structure) introduced by Wade [27]. A Dirac-Jacobi structure on M consists of a subbundle L ⊂ E 1 (M ) = (T M × R) ⊕ (T * M × R) satisfying certain maximally isotropic and integrability condition (Definition 2.6). Then L inherits the structure of a Lie algebroid and there is a distinguised 1-cocycle of this Lie algebroid. The graph of the bundle map (π, E) ♯ associated to a Jacobi structure defines a Dirac-Jacobi structure. Hence, the 1-jet bundle T * M × R of a Jacobi manifold M carries a Lie algebroid structure by identifying this bundle with the graph of (π, E) ♯ .
In section 3, we define an action τ : Ω 1 (M ) × DJ(M ) → DJ(M ), (B, L) → τ B (L) of the abelian group Ω 1 (M ) on the space DJ(M ) of all Dirac-Jacobi structures on M . When the Dirac-Jacobi structure L comes from the graph of a Jacobi structure (π, E), then for any B, the Dirac-Jacobi structure τ B (L) need not be the graph of another Jacobi structure. This amounts to the invertibility of a certain map and in this case, the new Jacobi structure (denoted by (π B , E B ) or τ B (π, E)) on M is called the gauge transformation of (π, E) associated to the 1-form B. We prove that gauge equivalent Jacobi structures on M gives rise to isomorphic Lie algebroid structures on T * M × R (cf. Proposition 3.4). As a remark, we get that gauge equivalent Jacobi structures gives rise to isomorphic Lichnerowicz-Jacobi cohomology. We show that gauge transformations of contact structures are contact and gauge transformations of l.c.s. structures are l.c.s. (cf. Remarks 3.6, 3.7). Moreover, any two contact structures on a manifold are gauge equivalent (cf. Remark 3.8).
In section 4, we show that our gauge transformations of Dirac-Jacobi structures is related to the gauge transformations of Dirac structures via the Diracization process (cf. Proposition 4.1). In the particular case, it shows the relation between gauge transformations of Jacobi structures and gauge transformations of Poisson structures (cf. Proposition 4.3).
A contact groupoid is a Lie groupoid G ⇒ M together with a contact 1-form η ∈ Ω 1 (G) and a multiplicative function σ ∈ C ∞ (G) that satisfies certain multiplicativity condition. Contact groupoids correspond to Jacobi structures on its base. In section 5, we discuss how the contact structure of a contact groupoid is effected by a gauge transformation of the Jacobi structure on its base (cf. Theorem 5.6).
In section 6, we deal with gauge transformations of multiplicative Jacobi structures on a Lie groupoid. Namely, we define gauge transformations of Jacobi groupoids and generalized Lie bialgebroids.
Generalized contact structures are odd analouge of generalized complex structures and generalization of contact structures [11] (see also [22,26]). Motivated from the notion of B-field transformation of generalized complex structures, in section 7, we define a similar notion for generalized contact structures. We also noticed that B-field transformation of contact structures (considered as generalized contact structures) need not be contact. Hence, this notion is different than gauge transformations of contact structures (considered as Jacobi structures).
Conclusions. We remark that the line bundle approach of Jacobi structures was introduced by Kirillov to study locally conformal Jacobi structures [15] (see also [1,19]). More precisely, a local Lie algebra structure on M consists of a line bundle L over M together with a Lie bracket {−, −} : ΓL × ΓL → ΓL on the space of sections of L, which is local in the sense that, for u, v ∈ ΓL supported in some open set U ⊂ M , the bracket {u, v} is supported in U as well. Kirillov's local Lie algebras are same as Jacobi structures when the line bundle L is trivial. We refer [4,22,25,26] for more details on the line bundle approach of contact structures, contact groupoids, Dirac-Jacobi structures and generalized contact structures. Finally, we remark that the contents of the present paper can also be discussed in the line bundle framework.
We assume that the reader is familiar with some basics of Lie groupoids and Lie algebroids. See [18] for details. Given a Lie groupoid G ⇒ M , the source map and the target map are denoted by α and β, respectively. The space of composable arrows is defined by We denote the de Rham differential of a manifold by d.
A Jacobi manifold is a manifold equipped with a Jacobi structure as above. If E = 0 then a Jacobi structure is nothing but a Poisson structure. Moreover, given a Jacobi structure (π, E) on M the product manifold M × R carries a Poisson structure whose Poisson bivector is given by This is called the Poissonization of the Jacobi structure (π, E) on M .
It is important to note that given a Jacobi structure (π, E) on M , there is a bundle map (π, E) ♯ : The set of all hamiltonian vector fields X h := pr 1 • (π, E) ♯ (dh, h) = π ♯ (dh) + hE, h ∈ C ∞ (M ), generates a distribution D on M , called the characteristic distribution. Here pr 1 : T M × R → T M denotes the projection onto the first factor.
Given a contact manifold (M, η) there exists an isomorphism of C ∞ (M )-modules The corresponding Jacobi structure (π, E) is given by In this case, the induced bundle map (π, E) ♯ : Conversely, a Jacobi structure (π, E) on M is induced from a contact structure on M if the bundle map (π, E) ♯ is invertible.

Example.
A locally conformal symplectic (l.c.s.) manifold is a smooth manifold M 2n together with a non-degenerate 2-form ω ∈ Ω 2 (M ) with the property that for each x ∈ M there is an open neighbourhood U x of x and a function f ∈ C ∞ (U x ) such that (U x , e −f ω) is a symplectic manifold. Alternatively, a l.c.s. manifold is a manifold M 2n together with a non-degenerate 2-form ω ∈ Ω 2 (M ) and a closed 1-form θ ∈ Ω 1 (M ) such that dω = θ ∧ ω (see [24] for more details). Given a l.c.s. manifold (M 2n , ω, θ), one can define a Jacobi structure (π, E) on M by π(α, β) = ω(♭ −1 (α), ♭ −1 (β)) and 2.4. Remark. The Poissonization of a contact structure η on M is given by a symplectic structure η = e t (pr * 1 dη + dt ∧ pr * 1 η) on M × R. If (ω, θ) defines a l.c.s. structure on M 2n , then around every point x in M there is a local chart (U x ; q 1 , . . . , q n , p 1 , . . . , p n ) and a function f on U x such that Therefore, the induced Jacobi structure (π, E) on M is given by Hence, around the local chart (q 1 , . . . , q n , p 1 , . . . , p n , t) on M × R, the Poissonization is given by the formula (1) where (π, E) is defined above.
A Jacobi structure (π, E) on M is called 'transitive' if the map is surjective. The only transitive Jacobi structures are given by contact structures on odd dimensional manifolds and locally conformal symplectic (l.c.s.) structures on even dimensional manifolds [15]. More generally, if (π, E) is an arbitrary Jacobi structure, the leaves of its characteristic distribution D carries transitive Jacobi structures. A conformal change of a Jacobi structure (π, E) on M by a nowhere vanishing smooth function σ is given by (π σ , E σ ) where π σ = σπ and E σ = π ♯ (dσ) + σE.
2.5. Definition. A Jacobi map between two Jacobi manifolds ( M , π, E) and (M, π, E) is a smooth map φ : M → M which preserves the corresponding bivector fields and vector fields, that is, φ * π = π and φ * E = E. The map φ is called a conformal Jacobi map with respect to a nowhere vanishing function σ ∈ C ∞ ( M ) if φ : M → M is a Jacobi map when M is equipped with the conformal Jacobi structure ( π σ , E σ ). We denote a conformal Jacobi map simply by (φ, σ) : M → M .
Next we recall Dirac-Jacobi structure (or E 1 (M )-Dirac structure) on a manifold studied by Wade [27]. A line bundle approach of this notion was further studied by Vitagliano [25]. First observe that for any smooth manifold M , the bundle T M × R → M has a Lie algebroid structure whose bracket and anchor are given by Then one can verify that the following identity holds Hence, we can define a non-degenerate pairing −, − on the space of sections of the bundle and a generalized Dorfman bracket −, − on the space of sections of E 1 (M ) by for (X, f ) ⊕ (α, g), (Y, h) ⊕ (β, k) ∈ Γ(E 1 (M )).
2.6. Definition. A Dirac-Jacobi structure on a manifold M is a subbundle which is maximally isotropic with respect to the pairing −, − and such that ΓL is closed under the generalized Dorfman bracket −, − .
The kernel of a Dirac-Jacobi structure L is defined as ker (L) := L ∩ ((T M × R) ⊕ {0}). Given a Dirac-Jacobi structure L, its opposite Dirac-Jacobi structure is given by

2.7.
Remark. Note that if L is a Dirac-Jacobi subbundle then it is equipped with a Lie algebroid structure over M . The Lie bracket on ΓL is given by the restriction of the generalized Dorfman bracket −, − and the anchor is given by the projection on T M . Moreover, there is a distinguised Lie algebroid 1-cocycle given by Given a Dirac-Jacobi structure L ⊂ (T M × R) ⊕ (T * M × R) on M , one can correspond a Dirac structure L on M × R given by following Diracization [12] process A characterization of Jacobi structures is given by the following [20,27].
is a Dirac-Jacobi structure on M .
With this Lie algebroid structure on T * M ×R, the bundle map (π, E) ♯ : T * M ×R → T M ×R defined in (2) is a Lie algebroid morphism. Moreover, we remark that the corresponding Diracization L (π,E) on M × R is given by the graph of the Poissonization. In other words, L (π,E) = L (π,E) .

2.10.
Remark. Let L be a Dirac-Jacobi structure on M such that its Diracization L is given by a Poisson structure on M × R. Then it follows from the definition of L that it must be of the following form for some bivector field π on M and a vector field E on M . The Dirac-Jacobi structure L is then given by L = L (π,E) . Moreover, by Proposition 2.8 defines a Jacobi structure on M .
Another example of Dirac-Jacobi structure is given by a precontact 1-form. The term 'precontact' is just a terminology and this is nothing but a usual 1-form. Given a 1-form η on M , one can define a Dirac-Jacobi structure on M given by The next theorem suggests when a Dirac-Jacobi structure L comes from a contact 1-form η [11].
2.11. Theorem. A Dirac-Jacobi structure L on an odd dimensinal manifold M is given by a contact 1-form η if and only if it satisfies for all x ∈ M.

Gauge transformations
In this section, we introduce gauge transformations of Dirac-Jacobi and Jacobi structures on a manifold. Let be a Dirac-Jacobi structure on M and take (B 1 , B) ∈ Ω 2 (M ) × Ω 1 (M ) be a pair of a 2-form and a 1-form. Consider the subbundle It is easy to see that the bundle is also maximally isotropic with respect to the pairing , . Moreover, we have the following.
Therefore, for any 1-form B ∈ Ω 1 (M ), the transformation τ (dB,B) (L) defines a Dirac-Jacobi structure on M . We denote this Dirac-Jacobi structure simply by τ B (L) and is called the gauge transformation of L associated to the 1-form B. The Dirac-Jacobi structure L and τ B (L) are called gauge equivalent. We remark that when L = L η , the gauge transformation τ B (L) is given by L η+B .
The proof of the following is obvious.

3.2.
Proposition. Gauge transformations of Dirac-Jacobi structures satisfy the following properties.
defines an isomorphism between Lie algebroid structures and under this isomorphism, the distinguised 1-cocycles are same.
Next, we consider Dirac-Jacobi structures on M which are graph of Jacobi structures. Let (π, E) be a Jacobi structure on M . Take any B ∈ Ω 1 (M ). Consider the Dirac-Jacobi structure τ B (L (π,E) ) gauge equivalent to L (π,E) ,

If the bundle map
is invertible, then τ B (L (π,E) ) is the graph of the map In this case, the 1-form B is called (π, E)-admissible. Moreover, the map defined in (7) is skewsymmetric, thus, given by a pair (π B , E B ) of a bivector field π B and a vector field E B on M . The pair (π B , E B ) is completely determined by and, in this case, Therefore, it follows from Proposition 2.8 that (π B , E B ) is a Jacobi structure on M . The Jacobi structure (π B , E B ) which is also denoted by τ B (π, E), is called the gauge transformation of (π, E) associated with the 1-form B. The Jacobi structures (π, E), (π B , E B ) are called gauge equivalent.
3.3. Remark. Since the map (6) is an isomorphism, it follows that Im ( . Therefore, gauge equivalent Jacobi structures gives rise to same characteristic distribution.
More generally, gauge equivalent Jacobi structures on M correspond to isomorphic Lie algebroids on T * M × R.

3.4.
Proposition. Let (π, E) be a Jacobi structure on M , and (π B , E B ) be a gauge equivalent Jacobi structure associated with the 1-form B. Then the Lie algebroid structures on T * M × R associated to (π, E) and (π B , E B ) are isomorphic.
Proof. Consider the bundle isomorphism Φ : This map commute with the corresponding anchors, as Hence the proof.
Note that the isomorphism Φ pulls back the cocycle (−E B , 0) to the cocycle (−E, 0). This can be shown by considering the distinguised cocycles of the corresponding Dirac-Jacobi structures.
3.5. Remark. Gauge equivalent Jacobi structures on M gives rise to isomorphic Lie algebroid cohomology of T * M × R. In other words, they have isomorphic Lichnerowicz-Jacobi cohomology.
3.6. Remark. Let (π, E) be a transitive Jacobi structure on M . Let B ∈ Ω 1 (M ) be such that the gauge transformation τ B (L (π,E) ) defines a Jacobi structure (π B , E B ) on M . Then we have Therefore, Since the map Id+ (dB, B)•(π, E) ♯ is invertible and pr 1 •(π, E) ♯ : T * M ×R → T M is surjective, it follows that pr 1 • (π B , E B ) ♯ is also surjective. Therefore, gauge transformations of transitive Jacobi structures are transitive. Thus, gauge transformations of contact structures are contact and gauge transformations of l.c.s. structures are l.c.s. In the next remark we give an alternative argument of this fact for contact structures. Since a gauge transformation of a Jacobi structure preserves the characteristic distribution, it only transform the contact or l.c.s. structures on the characteristic leaves by the pullback of the 1-form B.

3.7.
Remark. Let η be a contact 1-form on M with associated Jacobi structure (π, E). Let (π B , E B ) be its gauge transformation associated with the (π, E)-admissible 1-form B. As we have the map (π B , E B ) ♯ is also invertible and the inverse is given by Therefore, the gauge transformation of contact structures are contact.
If η B denotes the gauge transformation of η associated with B, then it follows from (3) and (8) 3.8. Remark. Let η and η ′ be any two contact structures on M with associated Jacobi structures (π, E) and (π ′ , E ′ ), respectively. Define a map θ : Next, we discuss the effect of gauge transformations on l.c.s. structures. Let (ω, θ) be a l.c.s. structure on M with associated Jacobi structure (π, E). See Example 2.3 for details. The corresponding Dirac-Jacobi structure is given by For any 1-form B ∈ Ω 1 (M ), we have from Equation (9) that If B is (π, E)-admissible, that is, the map (Id + (dB, B) • (π, E) ♯ ) is invertible, then it follows from (10) that the two form (ω − dB − B ∧ θ) is non-degenerate. In this case, the pair (ω − dB − B ∧ θ, θ) defines a l.c.s. structure on M . Moreover, is the Dirac-Jacobi structure induced from the l.c.s. structure (ω − dB − B ∧ θ, θ). Hence, the gauge transformation of the l.c.s. structure (ω, θ) is given by (ω − dB − B ∧ θ, θ).

Gauge transformations commute with the Poissonization
In this section, we prove that gauge transformations of Jacobi structures on a manifold commute with the Poissonization process. We first observe that gauge transformations of Dirac-Jacobi structures commute with the Diracization. Let Proof. We have Hence, On the other hand, from the definition of L it follows that To prove that gauge transformations of Jacobi structures commute with the Poissonization, we need the following lemma.

4.2.
Lemma. Let (π, E) be a Jacobi structure on M and B ∈ Ω 1 (M ) be a 1-form. Then B is (π, E)-admissible if and only if B is (π, E)-admissible.
Since τ B (π, E) defines a Poisson structure on M × R, it follows from Remark 2.10 that the Dirac-Jacobi structure τ B (L (π,E) ) is given by a Jacobi structure. In other words, B is (π, E)-admissible.
By Proposition 4.1 and Lemma 11 we have the following.
Therefore, gauge transformations of Jacobi structures commute with the Poissonization.

4.4.
Remark. In the particular case of contact structure, the proof is more simple. This follows from the fact that the Poissonization of a contact structure η on M is given by a symplectic structure η = e t (pr * 1 dη + dt ∧ pr * 1 η) on M × R. Hence, in this case,

Gauge transformations and contact groupoids
The notion of contact groupoid was first introduced in [14]. See [4,5] for recent developments on contact groupoids and integrable Jacobi structures. In this section, we describe how the contact structure of a contact groupoid is effected by a gauge transformation of the Jacobi structure on its base.

Definition.
A contact groupoid is a Lie groupoid G ⇒ M together with a contact 1-form η ∈ Ω 1 (G) and a function σ ∈ C ∞ (G) such that where ⊕ T G denotes the groupoid (partial) multiplication on the tangent Lie groupoid T G ⇒ T M .
It follows from the above condition that σ is a multiplicative function on G, that is, σ(gh) = σ(g) + σ(h), for all (g, h) ∈ G (2) . Given a contact groupoid (G ⇒ M, η, σ), the manifold M carries a unique Jacobi structure such that (α, e σ ) is a conformal Jacobi map and β is an anti-Jacobi map (see [5,10] for more details). In this case, the contact groupoid (G ⇒ M, η, σ) is said to integrate the base Jacobi structure. Alternatively, a Jacobi structure (π, E) on M is integrable if and only if the corresponding Lie algebroid structure on T * M × R → M is integrable [5]. In this case, the source-connected, simply connected Lie groupoid integrating T * M ×R → M carries a unique contact form and a multiplicative function which makes it a contact groupoid. Under this correspondence, the multiplicative function on the groupoid differentiates to the distinguised 1-cocycle (−E, 0) of the Lie algebroid.
Let (M, π, E) be an integrable Jacobi manifold, with contact groupoid (G ⇒ M, η, σ). Let B ∈ Ω 1 (M ) be a (π, E)-admissible 1-form on M . Since the Jacobi structures (π, E) and τ B (π, E) corresponds to isomorphic Lie algebroid structures on T * M × R and also the distinguised 1-cocycles are same, the Jacobi structure τ B (π, E) and the corresponding distinguised 1-cocycle can be integrated by a Lie groupoid isomorphic to G ⇒ M and by the multiplicative function σ (Proposition 3.4). Here we discuss the effect of the gauge transformation τ B on the base Jacobi manifold to the contact 1-form of the Lie groupoid G (Theorem 5.6).
We recall the following definition from [12].
We show that a smooth Jacobi map φ : M → M between two Jacobi manifolds is same as forward Dirac-Jacobi map when the manifolds are equipped with corresponding Dirac-Jacobi structures. This follows from the following observation.
This shows that φ is a forward Dirac-Jacobi map. It is also easy to verify that if φ is a Dirac-Jacobi map then φ is a Jacobi map.
The next lemma shows the relation between gauge transformations and push-forward of Dirac-Jacobi structures.
Proof. We have Therefore, On the other hand,
Proof. From the definition of L −η , we have τ φ * B L −η = L −η+φ * B . Hence, The second equality follows from Lemma 5.3 and the last equality follows since φ is a Jacobi map.

5.5.
Remark. Similarly, one can prove the followings.

5.7.
Remark. Let η be a contact 1-form on M considered as a Jacobi structure. Then it is integrable and the corresponding contact groupoid is given by (M × R × M ⇒ M, η = e σ pr * 3 η − pr * 1 η, σ). The source, target and the partial multiplication of the groupoid structure is given by α(x, t, y) = pr 3 (x, t, y) = y, β(x, t, y) = pr 1 (x, t, y) = x, (x, t, y)(y, s, z) = (x, t + s, z), for (x, t, y), (y, s, z) ∈ M × R × M . The multiplicative function σ on this groupoid is just the projection onto the second factor.
If B ∈ Ω 1 (M ) is a 1-form on the base M such that the gauge transformation defines a Jacobi structure (this is infact a contact structure), then we have τ B (η) = η − B. By Theorem 5.6 the corresponding contact 1-form on the groupoid is given by This is precisely the global contact 1-form η − B on the groupoid associated to the contact structure τ B (η) = η − B on the base.
In the next section we give an alternative proof of Theorem 5.6 using gauge transformations of multiplicative Jacobi structures (see Theorem 6.5 and Remark 6.7).

Gauge transformations of Jacobi groupoids
In this section, we study gauge transformations of Jacobi groupoids and of generalized Lie bialgebroids.
Conversely, the base of a generalized Lie bialgebroid carries a Jacobi structure. Let ((A, φ 0 ), (A * , X 0 )) be a generalized Lie bialgebroid over M . Suppose the bracket and anchor of the Lie algebroids A and A * are given by ([−, −], a) and ([−, −] * , a * ) , respectively. Then the induced Jacobi structure (π, E) on M is given by where the bundle maps (a(−), φ 0 (−)) : A −→ T M × R and (a * (−), X 0 (−)) : A * −→ T M × R are respectively given by X → (a(X), φ 0 (X)) and α → (a * (α), X 0 (α)), for X ∈ ΓA, α ∈ ΓA * [9]. Let B ∈ Ω 1 (M ) be a (π, E)-admissible 1-form on M . One can easily check that this condition is being equivalent to the invertability of the map Hence, one can define a new Lie algebroid structure on A * whose bracket and anchor are given by We denote this Lie algebroid structure on A * by (A * ) B . Moreover, X B 0 (−) := X 0 • ψ −1 B (−) defines a 1-cocycle of this Lie algebroid. Actually, we have (a B * (−), X B 0 (−)) = (a * (−), X 0 (−)) • ψ −1 B (−). We denote the Jacobi algebroid ((A * ) B , X B 0 ) simply by (A * , X 0 ) B . Moreover, the pair ((A, φ 0 ), (A * , X 0 ) B ) is a generalized Lie bialgebroid. We call this generalized Lie bialgebroid as a gauge transformation of the given one. Note that the bundle map associated to the Jacobi structure on M induced from the generalized Lie bialgebroid ((A, φ 0 ), (A * , X 0 ) B ) is given by It is easy to show that this map coincides with (π, E) ♯ • Id + (dB, B) • (π, E) ♯ −1 which is same as (τ B (π, E)) ♯ . Hence, the Jacobi structure on M induced from the generalized Lie bialgebroid ((A, φ 0 ), (A * , X 0 ) B ) is simply given by τ B (π, E). Next, we recall multiplicative Jacobi structures on Lie groupoids and study gauge transformations of them in the multiplicative sense. First, let us recall few things. Let G ⇒ M be a Lie groupoid and σ ∈ C ∞ (G) be a multiplicative function. Then the tangent Lie groupoid T G ⇒ T M can be twisted by σ to define a new Lie groupoid T G × R ⇒ T M × R. The source, target and partial multiplication are given by Let A → M be the Lie algebroid of G ⇒ M . One can also twist the usual cotangent groupoid T * G ⇒ A * by the multiplicative function σ to define a new groupoid T * G × R ⇒ A * with structure maps (α) σ (ω g , γ) = e −σ(g)α (ω g ) . This Lie groupoid is called the 1-jet Lie groupoid of G twisted by σ [10]. 6.4. Definition. [10] A Jacobi groupoid is a Lie groupoid G ⇒ M together with a multiplicative function σ ∈ C ∞ (G) and a Jacobi structure (π G , E G ) on G such that the induced map (π G , E G ) ♯ : from the 1-jet Lie groupoid to the twisted tangent Lie groupoid defined above.

B-field transformations of generalized contact structures
Generalized contact structures are odd analouge of generalized complex structures and generalization of contact structures [11]. A line bundle approach of this notion was studied in [22,26]. In this final section, we study some symmetries of generalized contact structures, namely, B-field transformations. Here I * denote the adjoint of I with respect to non-degenerate pairing −, − defined in (4) and N I denote the Nijenhuis torsion of I with respect to the generalized Dorfman bracket −, − . A manifold equipped with a generalized contact structure is called a generalized contact manifold.

7.2.
Example. Any contact manifold is a generalized contact manifold. In particular, if η is a contact 1-form on M with Jacobi structure (π, E), then as a matrix block Obviously, this is not given by any contact form. Therefore, B-field transformations of a contact form (considered as a generalized contact structure) need not be contact. This holds if and only if B = 0.