Graded Generalized Geometry

Abstract

Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular, several integrability conditions can be formulated in terms of a canonical Dorfman bracket, an example of Courant algebroid. On the other hand, smooth manifolds can be generalized to involve functions of $\mathbb{Z}$-graded variables which do not necessarily commute. This leads to a mathematical theory of graded manifolds. It is only natural to combine the two theories by exploring the structures on a generalized tangent bundle associated to a given graded manifold. After recalling elementary graded geometry, graded Courant algebroids on graded vector bundles are introduced. We show that there is a canonical bracket on a generalized tangent bundle associated to a graded manifold. Graded analogues of Dirac structures and generalized complex structures are explored. We introduce differential graded Courant algebroids which can be viewed as a generalization of Q-manifolds. A definition and examples of graded Lie bialgebroids are given.

arXiv commentVersion of the manuscript reuploaded to better match the published version - some sections were shortened

References in corpus (17)

• The Geometry of the Master Equation and Topological Quantum Field Theory
• Geometry of Batalin-Vilkovisky quantization
• Courant algebroids, derived brackets and even symplectic supermanifolds
• Generalized complex geometry
• Torsion-free generalized connections and Heterotic Supergravity
• Introduction to supergeometry
• $Q$-manifolds and Higher Analogs of Lie Algebroids
• Poisson-Lie T-duality and Courant algebroids
• Generalized complex geometry and T-duality
• Integration of Exact Courant Algebroids
• Poisson-Lie T-duality as a boundary phenomenon of Chern-Simons theory
• Poisson-Lie T-duality of String Effective Actions: A New Approach to the Dilaton Puzzle
• Introduction to graded geometry
• Heterotic Reduction of Courant Algebroid Connections and Einstein-Hilbert Actions
• Kaluza-Klein Reduction of Low-Energy Effective Actions: Geometrical Approach
• Effective Actions for $\sigma$-Models of Poisson-Lie Type
• Supergravity Fluxes and Generalised Geometry

Cited by in corpus (1)

• Threefold Nature of Graded Vector Bundles