Graded Generalized Geometry

TL;DR

This paper extends generalized geometry to graded manifolds, introducing graded Courant algebroids, Dirac structures, and generalized complex structures, thereby enriching the mathematical framework for string theory applications.

Contribution

It develops the theory of graded Courant algebroids and structures on graded tangent bundles, combining graded geometry with generalized geometry for the first time.

Findings

Introduces a canonical bracket on graded tangent bundles.

Defines graded Dirac and generalized complex structures.

Presents graded Courant algebroids and their relation to Q-manifolds.

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.