We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable Courant algebroids. The additive group of 2-forms acts on twisted Poisson structures and permits them to be seen as glued from ordinary Poisson structures defined on local patches. We conclude with remarks on deformation quantization and twisted symplectic groupoids.The ideas presented in this note grew out of an attempt to understand how Poisson geometry on a manifold is affected by the presence of a closed 3-form "field". Such forms are playing an important role in contemporary string theory. We refer, for example, to Park 14) as well as to Cornalba and Sciappa 5) and Klimčík and Ströbl 9) . Our aim here is to show that the notions of Courant algebroid and Dirac structure provide a framework in which one can easily carry out computations in Poisson geometry in the presence of a background 3-form. It seems clear that a proper understanding of the global effect of such a 3-form involves gerbes (see for example Brylinski 3) ); our work here should at least partially substantiate the claim that Courant algebroids are appropriate infinitesimal objects to associate with gerbes.Our work was stimulated in part by the many talks at the Workshop on Deformation Quantization and String Theory at Keio University (March, 2001) in which such 3-forms played an essential role. It is essentially an application of some of the ideas contained in a series of letters fromŠevera to Weinstein written in 1998.
Given a Lie groupoid G over a manifold M , we show that multiplicative 2forms on G relatively closed with respect to a closed 3-form φ on M correspond to maps from the Lie algebroid of G into T * M satisfying an algebraic condition and a differential condition with respect to the φ-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to φ-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-hamiltonian spaces and group-valued momentum maps.Motivated by the relationship between symplectic realizations of Poisson manifolds and hamiltonian actions [9], we study presymplectic realizations of twisted Dirac structures. Just as in usual Poisson geometry, these presymplectic realizations carry natural actions of presymplectic groupoids. In fact, it is this property that determines our definition of presymplectic realizations. An important example of twisted Dirac structure is described in [27, Example 4.2]: any nondegenerate invariant inner product on the Lie algebra h of a Lie group H induces a natural Dirac structure on H, twisted by the invariant Cartan 3-form; we call such structures Cartan-Dirac structures. We show that presymplectic realizations of Cartan-Dirac structures are equivalent to the quasi-hamiltonian h-spaces of Alekseev, Malkin and Meinrenken [1] in such a way that the realization maps are the associated groupvalued momentum maps. It also follows from our results that the transformation groupoid H ⋉ H corresponding to the conjugation action carries a canonical twisted presymplectic structure, which we obtain explicitly by "integrating" the Cartan-Dirac structure. As a result, we recover the 2-form on the "double" D(H) of [1] and the AMM groupoid of [4]. (Closely related forms were introduced earlier in [18,30].) A unifying approach to momentum map theories based on Morita equivalence of presymplectic groupoids has been developed by Xu in [34]; much of our motivation for considering quasi-hamiltonian spaces comes from his work. Our results indicate that Dirac structures provide a natural framework for the common description of various notions of momentum maps (as e.g. in [1,22], see also [31]).We illustrate our results on multiplicative 2-forms and Dirac structures in many examples. In the case of action groupoids, we obtain an explicit formula for the natural map from the cohomology of the Cartan model of an H-manifold [2, 16] to (Borel) equivariant cohomology [2,3] in degree three; for the monodromy groupoid of a foliation F, we show that multiplicative 2-forms are closely related to the usual cohomology and spectral sequence of F [19].The entire discussion of relatively closed multiplicative 2-forms on groupoids may be embedded in the more general context of a van Est theorem for the "bar-de Rham" double complex of forms on the simplicial space of composable sequences in a groupoid G, whose total complex computes the cohomology of the classifying space...
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincaré-BirkhoffWitt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A
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