Poisson fibrations and fibered symplectic groupoids

Brahic, Olivier; Fernandes, Reynaldo

doi:10.1090/conm/450/08733

Cited by 12 publications

(129 citation statements)

References 16 publications

(29 reference statements)

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“…General coupling Dirac structures can be seen as infinite dimensional Yang-Mills-Higgs phase spaces. This was observed in [3] and often offers guidance on how to extend constructions which work for a Yang-Mills-Higgs phase space to a general coupling Dirac structure. We will use this principle later, so we will recall here this Poisson gauge theory.…”

Section: 4mentioning

confidence: 82%

“…The notion of a coupling was first introduced in the context of Dirac geometry in [30] and [3], but their origins lie in the theory of symplectic and hamiltonian fibrations (see, e.g., [23]). In this section we will recall the definition of a coupling Dirac structure and study its first properties.…”

Section: Coupling Dirac Structuresmentioning

confidence: 99%

“…We discuss the first two integration steps above. For this, we recall briefly from [3] a few notions about fibered symplectic groupoids. 4.1.1.…”

Section: Integration Of the Yang-mills-higgs Phase Spacementioning

confidence: 99%

“…In fact, standard facts about integration of Lie algebroids yield the following (see [3] for details): Theorem 4.6. Let p : E → B be a Poisson fibration with fiber type (F, π F ) an integrable Poisson manifold.…”

Section: Definition 43mentioning

confidence: 99%

“…We are interested in the more general situation of a Poisson fibration: now one looks for a coupling Dirac structure on the total space of the fibration which glues the Poisson structures on the fibers. This idea of a coupling is only a rough approximation: Dirac structures are very flexible and extra care must be taken in formulating precisely what a coupling Dirac structure is ( [3,30]). …”

Section: Introductionmentioning

confidence: 99%

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Integration of coupling Dirac structures

Brahic

Fernandes

2015

Pacific J. Math.

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Abstract. Coupling Dirac structures are Dirac structures defined on the total space of a fibration, generalizing hamiltonian fibrations from symplectic geometry, where one replaces the symplectic structure on the fibers by a Poisson structure. We study the associated Poisson gauge theory, in order to describe the presymplectic groupoid integrating coupling Dirac structures. We find the obstructions to integrability and we give explicit geometric descriptions of the integration.

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Section: 4mentioning

confidence: 82%

Section: Coupling Dirac Structuresmentioning

confidence: 99%

“…We discuss the first two integration steps above. For this, we recall briefly from [3] a few notions about fibered symplectic groupoids. 4.1.1.…”

Section: Integration Of the Yang-mills-higgs Phase Spacementioning

confidence: 99%

Section: Definition 43mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integration of coupling Dirac structures

Brahic

Fernandes

2015

Pacific J. Math.

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Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

2019

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We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinite singularities we can provide these wrinkled fibrations in dimension 6 with near-symplectic structures.2010 Mathematics Subject Classification. 57R17, 53D17. after combining the definition of generalized bLf [19] together with the results of [6] and [10] on Poisson structures. Theorem 1.1. Let M and X be closed oriented smooth manifolds with dim(M ) = 2n, dim(X) = 2n − 2, and f : M → X a generalized broken Lefschetz fibration. Then there is a complete singular Poisson structure of rank 2 whose associated bivector vanishes on the singularity set of f . If none of its symplectic leaves are, or contain, 2-spheres, then this Poisson structure is integrable.The proof of this theorem is a direct application of Theorem 2.11 and the definition of completeness. The integrability condition is verified in the relevant cases, as explained in 2.4.In section 3 we focus on the Poisson structure on the total space of a generalized wrinkled fibration in dimension 6. We give the general steps for buidling Poisson bivectors around all types of singularities of corank 1, which can be applied on any given dimension. This idea allows us to show the following. Theorem 1.2. Let M be a closed, orientable, smooth 6-manifold equipped with a generalized wrinkled fibration f : M → X over a smooth 4-manifold X. Then there exists a complete Poisson structure whose symplectic leaves correspond to the fibres of the given fibration structure, and the singularities of both the fibration and the Poisson structures coincide. Moreover, for each singularity, the Poisson bivector and induced symplectic form on the leaves are given by the following equations:Folds: 13) If none of its symplectic leaves are, or contain, 2-spheres, then this Poisson structure is integrable.The existence of a Poisson structure with the stated properties follows from Theorem 2.11, previously shown by the first and third named authors together with García-Naranjo [10]. The proof of this theorem follows from an application of Theorem 2.11 and the definition of completeness.These results allow us to present in section 2.5.1 countably many examples of Poisson structures on the same underlying smooth manifold that are Morita inequivalent. In our examples the leaves of the symplectic foliations change topology, as the fibrations involved undergo deformations.In section 3, the local models for the bivectors are shown to hold true. Then in section 4 we prove that the local models for the symplectic forms on the leaves are also the claimed ones.Finally, in section 5 we turn to near-symplectic geometry. Explicit models of near-symplectic forms have appeared in previous work [11,1,14,19]. Here we show a further construction of local near-symplectic forms that follows a...

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Compatible poisson structures on fibered 5-manifolds

Flores-Espinoza¹,

Ruíz-Pantaleón²,

Vorobiev³

2018

Bol. Soc. Mat. Mex.

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We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a 5-dimensional orientable fibered manifolds with 2-dimensional bases, we describe a global behavior of almost coupling Poisson tensors and their singularities by using a bigraded factorization of the Jacobi identity. In particular, we present some unimodularity criteria and describe a class of gauge type transformations preserving the almost coupling property.

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Poisson fibrations and fibered symplectic groupoids

Abstract: Abstract. We show that Poisson fibrations integrate to a special kind of symplectic fibrations, called fibered symplectic groupoids.

Cited by 12 publications

References 16 publications

Integration of coupling Dirac structures

Integration of coupling Dirac structures

Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

Compatible poisson structures on fibered 5-manifolds

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