15 research outputs found
Integration of twisted Dirac brackets
The correspondence between Poisson structures and symplectic groupoids,
analogous to the one of Lie algebras and Lie groups, plays an important role in
Poisson geometry; it offers, in particular, a unifying framework for the study
of hamiltonian and Poisson actions. In this paper, we extend this
correspondence to the context of Dirac structures twisted by a closed 3-form.
More generally, given a Lie groupoid over a manifold , we show that
multiplicative 2-forms on relatively closed with respect to a closed 3-form
on correspond to maps from the Lie algebroid of into the
cotangent bundle of , satisfying an algebraic condition and a
differential condition with respect to the -twisted Courant bracket. This
correspondence describes, as a special case, the global objects associated to
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.Comment: 42 pages. Minor changes, typos corrected. Revised version to appear
in Duke Math.
Integration of Dirac-Jacobi structures
We study precontact groupoids whose infinitesimal counterparts are
Dirac-Jacobi structures. These geometric objects generalize contact groupoids.
We also explain the relationship between precontact groupoids and homogeneous
presymplectic groupoids. Finally, we present some examples of precontact
groupoids.Comment: 10 pages. Brief changes in the introduction. References update
Integration of Lie 2-algebras and their morphisms
Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by
integrating the corresponding Lie algebra crossed module. On the other hand,
the integration procedure of Getzler and Henriques will also produce a 2-group.
In this paper, we show that these two integration results are Morita
equivalent. As an application, we integrate a non-strict morphism between Lie
algebra crossed modules to a generalized morphism between their corresponding
Lie group crossed modules.Comment: 19 pages, Lett. Math. Phys. 102 (2), (2012.11), 223-24