Pre-Courant Algebroids
arXiv:1608.01585 · doi:10.1016/j.geomphys.2019.04.007
Abstract
Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive quasi-cochain complex. In particular we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non-negative grading. VB-Courant algebroids are natural examples of what we call weighted pre-Courant algebroids, and our approach drastically simplifies working with them. We remark that examples of pre-Courant algebroids are plentiful - natural examples include the cotangent bundle of any almost Lie algebroid.
Dedicated to the memory of James Alfred Bruce
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- Geometric BV for twisted Courant sigma models and the BRST power finesse
- Metric Algebroid and Poisson-Lie T-duality in DFT
- The Algebroid Structure of Double Field Theory
- Homological sections of Lie algebroids
- 'Anti-Commutable' Pre-Leibniz Algebroids and Admissible Connections
- Instances of Higher Geometry in Field Theory