Poisson Geometry with a 3-Form Background

By P. Ševera, A. Weinstein. 2001

471 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5d0b598fdff49933a9feb5337f16dc562df49ab6

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Citations in the corpus

Generalized Complex Geometry (2004, background): “As observed in [35], one may replace the exterior derivative in (2.”
Linear and Multiplicative 2-Forms (2009, background): “If L ⊂ TM ⊕ T M is a φ-twisted Dirac structure [7, 17], then the projection L −→ T M is naturally an IM 2-form, so the correspondence above includes the integration of twisted Dirac structures as a special case.”
New aspects of the ddc-lemma (2005, unknown): “∂ B and ∂ B, are given by ∂ B = eB∂e−B; ∂ B = eB∂e−B. 3. Twisted Generalized Complex Structures The idea of a twisted generalized complex structure comes from the observation by ˇSevera and Weinstein [84] that the Courant bracket can be twisted by a closed 3-form H(also called an NS-ﬂux): [X+ξ,Y+η] ”
Multiplicative forms at the infinitesimal level (2010, background): “Dirac structures [10, 28] fit into a similar picture, being the infinitesimal versions of certain multiplicative 2-forms, see [5].”
Membrane sigma-models and quantization of non-geometric flux backgrounds (2012, unknown): “The structure maps of C comprise the skew-symmetrization of the H-twisted Courant–Dorfman bracket given by [79] [ (Y1, α1) , (Y2, α2) ] H := ( [Y1, Y2]TM , LY1α2 − LY2α1 (2.”
Integration of twisted Dirac brackets (2003, methodology): “A φ-twisted Dirac structure L has an induced Lie algebroid structure (see [10], [21], [27]): the bracket on 0(L) is the restriction of the Courant bracket [ ·, ·]φ , and the anchor is the restriction of the projection pr1 : T M ⊕ T M −→ T M .”
Generalized Kähler Geometry (2010, background): “These twisted structures are therefore classified by a characteristic class in H1(Ω2,cl(M)) and are called exact Courant algebroids [13, 18, 19].”
Universal lifting theorem and quasi-Poisson groupoids (2005, background): “As explained in [24], a twisted Poisson structure induces a Lie algebroid structure on T ∗M with anchor map π and Lie bracket of sections σ and τ defined by [σ, τ ] := Lπ♯(σ)τ − Lπ♯(τ)σ − dπ(σ, τ) + φ(π (σ), π(τ), ·).”
Quasi-Lie Bialgebroids and Twisted Poisson Manifolds (2001, methodology): “In general, the derived bracket is not skew-symmetric; the resulting structure on E coincides (after skew-symmetrization) with the Courant algebroid introduced by Liu, Weinstein and Xu [11] as a candidate for the double of (A,A∗); the Courant algebroid axioms follow easily from (2.6) (see [16]) for ”
Derived Brackets (2003, methodology): “Courant algebroids were introduced by Liu, Weinstein and Xu in [38], as a generalization of the bracket defined by Courant [11] on the sections of TM ⊕ T ∗M .”
Maximally homogeneous para-CR manifolds of semisimple type (2006, unknown): “also [34]), i.”
New aspects of the dd c -lemma (2004, background): “Twisted Generalized Complex Structures The idea of a twisted generalized complex structure comes from the observation by Ševera and Weinstein [82] that the Courant bracket can be twisted by a closed 3-form H (also called an NS-flux): [X + ξ, Y + η]H = [X + ξ, Y + η] + Y b(XbH).”
Reduction and submanifolds of generalized complex manifolds (2005, result): “Remark 1.5. In [16] it is shown that supersymmetry is also related with generalized, almost complex structures that are integrable (i.e., have a vanishing tensor (1.4)) with respect to the ySevera-Weinstein Courant bracket [ 21 ]”
Reduction of Courant algebroids and generalized complex structures (2005, unknown): “When the Courant algebroid is split, with curvature H ∈ Ω3 cl(M), these are usually referred to as H -twisted Dirac structures [26].”
Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, methodology): “They are defined in terms of a bivector field π and a closed 3-form, denoted by φ in [52], but which we shall denote by ψ to avoid confusion with the above.”
Poisson geometry and Morita equivalence (2004, methodology): “The bracket (2.7) is the non-skew-symmetric version, introduced in [56] (see also [ 79 ]), of”