On the Integration of Poisson Manifolds, Lie Algebroids, and Coisotropic Submanifolds

Cattaneo, Alberto S.

doi:10.1023/b:math.0000027690.76935.f3

Cited by 35 publications

(61 citation statements)

References 16 publications

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“…More precisely, conormal bundles of coisotropic submanifolds are all possible Lagrangian Lie subalgebroids of T * M with its canonical symplectic structure. If M is integrable, coisotropic submanifolds are also in correspondence with Lagrangian subgroupoids of the symplectic groupoid of M. (see [3]). EXAMPLE 2.1.…”

Section: Coisotropic Submanifoldsmentioning

confidence: 99%

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

Cattaneo

Felder

2004

Letters in Mathematical Physics

210

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Abstract. General boundary conditions ('branes') for the Poisson sigma model are studied. They turn out to be labeled by coisotropic submanifolds of the given Poisson manifold. The role played by these boundary conditions both at the classical and at the perturbative quantum level is discussed. It turns out to be related at the classical level to the category of Poisson manifolds with dual pairs as morphisms and at the perturbative quantum level to the category of associative algebras (deforming algebras of functions on Poisson manifolds) with bimodules as morphisms. Possibly singular Poisson manifolds arising from reduction enter naturally into the picture and, in particular, the construction yields (under certain assumptions) their deformation quantization. Mathematics Subject Classifications (2000). 81T45 (primary); 22A22, 53D17, 53D20, 53D55, 81T70 (secondary).

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Section: Coisotropic Submanifoldsmentioning

confidence: 99%

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

Cattaneo

Felder

2004

Letters in Mathematical Physics

210

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“…6) whereX : T D → T * M is the algebroid homotopy betweenγ 0 andγ 1 seen as an algebroid morphism from the disk, thanks to the boundary conditions. Because of (4.5), this definition of equivalence does not depend on the concrete choice of (D,X).…”

Section: I) the Ssc Symplectic Groupoid G(m ) Is Prequantizable If Anmentioning

confidence: 99%

“…Consider a surface Σ with n boundary components, ∂Σ = [6]. As we said, the boundary conditions are defined by a choice of C, i.e., X : …”

Section: On-shell Gauge Transformations With Branesmentioning

confidence: 99%

Geometric quantization and non-perturbative Poisson sigma model

Bonechi¹,

Cattaneo²,

Zabzine³

2006

Advances in Theoretical and Mathematical Physics

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In this note, we point out the striking relation between the conditions arising within geometric quantization and the non-perturbative Poisson sigma model. Starting from the Poisson sigma model, we analyze necessary requirements on the path integral measure which imply a e-print archive: http://arXiv.org/abs/math.sg/0507223 FRANCESCO BONECHI ET AL.certain integrality condition for the Poisson cohomology class [α]. The same condition was considered before by Crainic and Zhu but in a different context. In the case when [α] is in the image of the sharp map, we reproduce the Vaisman's condition for prequantizable Poisson manifolds. For integrable Poisson manifolds, we show, with a different procedure than in Crainic and Zhu, that our integrality condition implies the prequantizability of the symplectic groupoid. Using the relation between prequantization and symplectic reduction, we construct the explicit prequantum line bundle for a symplectic groupoid. This picture supports the program of quantization of Poisson manifold via symplectic groupoid. At the end, we discuss the case of a generic coisotropic D-brane.

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“…That groupoid was obtained by symplectic reduction of an infinite-dimensional manifold. That method may in fact be used for any Lie algebroid, as shown by Cattaneo [6]. A complete solution of the integration problem for Lie algebroids was obtained by M. Crainic and R. L. Fernandes [9].…”

Section: The Anchor ρ Of That Lie Algebroid Is the Map T S Restrictedmentioning

confidence: 96%

Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

Marle

2008

Dissertationes Math.

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We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator d E similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds, whose links with Lie algebroids are very close.

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On the Integration of Poisson Manifolds, Lie Algebroids, and Coisotropic Submanifolds

Cited by 35 publications

References 16 publications

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

Geometric quantization and non-perturbative Poisson sigma model

Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

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