154 research outputs found
Integrating morphisms of Lie 2-algebras
We show how to integrate a weak morphism of Lie algebra crossed-modules to a
weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for
2-term L_infty algebras. In particular, we obtain a new description of the
bicategory of 2-term L_infty algebras. We use butterflies to give a functorial
construction of connected covers of Lie 2-groups. We also discuss the notion of
homotopy fiber of a morphism of 2-term L_infty algebras.Comment: discussion of connected covers of Lie 2-groups expanded, 26 page
Omni-Lie 2-algebras and their Dirac structures
We introduce the notion of omni-Lie 2-algebra, which is a categorification of
Weinstein's omni-Lie algebras. We prove that there is a one-to-one
correspondence between strict Lie 2-algebra structures on 2-sub-vector spaces
of a 2-vector space \V and Dirac structures on the omni-Lie 2-algebra
\gl(\V)\oplus \V . In particular, strict Lie 2-algebra structures on \V
itself one-to-one correspond to Dirac structures of the form of graphs.
Finally, we introduce the notion of twisted omni-Lie 2-algebra to describe
(non-strict) Lie 2-algebra structures. Dirac structures of a twisted omni-Lie
2-algebra correspond to certain (non-strict) Lie 2-algebra structures, which
include string Lie 2-algebra structures.Comment: 23 page
Twisted Courant algebroids and coisotropic Cartan geometries
In this paper, we show that associated to any coisotropic Cartan geometry
there is a twisted Courant algebroid. This includes in particular parabolic
geometries. Using this twisted Courant structure, we give some new results
about the Cartan curvature and the Weyl structure of a parabolic geometry. As
more direct applications, we have Lie 2-algebra and 3D AKSZ sigma model with
background associated to any coisotropic Cartan geometry
AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories
We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz-
Zaboronsky superfield formalism using the language of graded manifolds. As a
main illustarting example, to every Courant algebroid structure we associate
canonically a three-dimensional topological sigma-model. Using the AKSZ
formalism, we construct the Batalin-Vilkovisky master action for the model.Comment: 13 pages, based on lectures at Rencontres mathematiques de Glanon
200
Integration of twisted Poisson structures
Poisson manifolds may be regarded as the infinitesimal form of symplectic
groupoids. Twisted Poisson manifolds considered by Severa and Weinstein
[math.SG/0107133] are a natural generalization of the former which also arises
in string theory. In this note it is proved that twisted Poisson manifolds are
in bijection with a (possibly singular) twisted version of symplectic
groupoids.Comment: 12 pages; minor corrections (especially in terminology: "twisted
symplectic" replaces "quasi-symplectic"), references updated; to appear in J.
Geom. Phy
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