Poisson Geometry with a 3-Form Background

Abstract

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.

Poisson Quasi-Nijenhuis Structures with Background

“…In the above results, the bivector π is a true Poisson bivector. So the last structure we obtain is different from a possible compatibility between a Poisson structure with background [13] and a Nijenhuis tensor.…”

On dual pairs in Dirac geometry

“…Courant algebroids may be twisted by closed three-forms. Specifically, let φ ∈ Ω 3 (M ) be closed, and consider the φ-twisted Dorfman bracket Twisted Dirac structures go back, in one form or another, to [27,32,34,35]. A crucial example of such structures is given by Cartan-Dirac structures associated to nondegenerate, invariant inner products on the Lie algebra g of a Lie group G [35, Example 4.2], and whose presymplectic realizations correspond to the quasiHamiltonian g-spaces of [1] (see [10]).…”

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

“…Since the annihilator of λ (π) is the graph of π, these considerations imply the following proposition. This proposition is well-known and can be proved without any appeal to spinors (see [113,111,115]). Here, we emphasize that the modular field appears as the section U such that the ψ-twisted differential of λ (π) satisfies (9.5), which ensures that the graph of π is closed under the ψ-twisted Courant bracket (9.4).…”

“…The nondegenerate 2-form ω is then called twisted symplectic. (See [113,78], and, for a generalization of this correspondence, see [71].) It is easy to construct non-degenerate twisted Poisson structures using this result.…”

“…To each of these theories, there corresponds, more generally, a twisted version. Twisted Poisson manifolds were first considered in some quantization problems in string theory [22,105], then they were defined as geometric objects [62,113], and finally viewed as particular cases of Lie algebroids with twisted Poisson structures [111,72]. Just as a Lie algebra with a triangular r-matrix gives rise to a Lie bialgebra in the sense of Drinfeld [29], a Lie algebroid with a Poisson structure gives rise to a Lie bialgebroid in the sense of Mackenzie and Xu [95,64], while a twisted Poisson structure gives rise to a quasi-Lie-bialgebroid 2 .…”