N-plectic geometry and Courant algebroids
From CRCG-Wiki
Just as a symplectic manifold is equipped with a closed non-degenerate 2-form, an n-plectic manifold is equipped with a closed non-degenerate (n+1)-form. Any n-plectic manifold gives a Lie n-algebra: the higher analogue of a Lie algebra in which the Jacobi identity holds only up to coherent homotopy. This is one of the ways n-plectic geometry can be seen as an approach to "higher symplectic geometry". When n=1, we are in the context of symplectic geometry and the corresponding Lie 1-algebra is the usual Poisson algebra of smooth functions on a symplectic manifold. In this talk, I will explain how the n=2 case is naturally related, both algebraically and geometrically, to the theory of Courant algebroids. Courant algebroids are, roughly, vector bundles that generalize the structures found in tangent bundles and quadratic Lie algebras, and they too can be interpreted as higher analogs of the structures found in symplectic geometry.
