Quantization, deformations, and new homological and categorical methods in mathematical physics

Courant algebroids are a natural generalisation of Lie algebras with an invariant inner product to the world of vector bundles. They provide a natural framework for describing integrability of various geometric structures (such as distributions, bivector fields and 2-forms), and for two-dimensional variational problems. Other examples include doubles of Lie bialgebroids, BRST algebra, the commutative and non-commutative Weil algebra. It turns out that Courant algebroids are most conveniently described using the notion of a graded symplectic supermanifold (specifically, the symplectic form must be of degree two). We will first describe the structure of such supermanifolds, showing that they are in 1-1 correspondence with vector bundles with a fiberwise inner product. Then we will show that Courant algebroid structures correspond to cubic Hamiltonians H satisfying the structure equation {H,H}=0. We thus obtain the standard and deformation complexes for Courant algebroids.