Integration of twisted Dirac brackets

By H. Bursztyn, M. Crainic, A. Weinstein, Chenchang Zhu. 2003

185 citations recorded by the source corpus.

Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form phi 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 phi-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to phi-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.

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

View Paper: https://www.semanticscholar.org/paper/f3bbc2aa1db98d693ed874f562f4d7b156a7673e

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Citations in the corpus

Linear and Multiplicative 2-Forms (2009, background): “42) were introduced in [3] and are referred to as IM 2-forms on A (relative to φ).”
Topics in Differential Geometry (2008, background): “Dirac structures — a common generalization of symplectic and Poisson structures ([37], [29], [28]).”
Multiplicative forms and Spencer operators (2012, methodology): “Motivated by Dirac geometry and the theory of Lie group-valued moment maps, the case of closed multiplicative 2-forms was analyzed in [9].”
Representations up to homotopy of Lie algebroids (2009, methodology): “Sometimes it is not clear what the relevant cohomology theory is -for example when looking at the infinitesimal equations underlying multiplicative two forms (see the equations in the main theorem of [7]), or the equations satisfied by the k-differentials of [18].”
PROPER GROUPOIDS AND MOMENTUM MAPS: LINEARIZATION, AFFINITY, AND CONVEXITY GROUPOPROPRES ET APPLICATIONS MOMENT: LIN ´ EARISATION, CARACT ` ERE AFFINE, ET CONVEXIT ´ E (2004, background): “Consider now a quasi-symplectic groupoid in the sense of Xu [32] (also known as twisted presymplectic groupoid [3]).”
Supergroupoids, double structures, and equivariant cohomology (2006, background): “ to “integrate” various structures from the level of algebroids to that of groupoids. One example is the integration of (twisted) Dirac structures to (twisted) presymplectic groupoids, carried out in [13]. 94 Appendix A Simplicial structures This appendix provides a brief review of some results from”
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].”
Integrability of Poisson Brackets (2002, background): “on-smooth subgroupoid of Σ(M). We will still have a diagram as above relating Σ(M), Σ(N) and ΣN(M). So, morally, Poisson-Dirac submanifolds are the base manifolds of presymplectic groupoids (see also [2]), with Poisson submanifolds corresponding to base manifolds of coisotropic subgroupoids. One can”