Lie 2-algebras from 2-plectic geometry
Just as symplectic geometry is a natural setting for the classical mechanics of point particles, 2-plectic geometry can be used to describe classical strings. Just as a symplectic manifold is equipped with a closed non-degenerate 2-form, a "2-plectic manifold" is equipped with a closed non-degenerate 3-form.
The Poisson bracket makes the smooth functions on a symplectic manifold form a Lie algebra. Similarly, any 2-plectic manifold gives a "Lie 2-algebra": the categorified analogue of a Lie algebra, where the usual laws hold only up to isomorphism. We explain these ideas and use them to give a new construction of the "string Lie 2-algebra" associated to a simple Lie group.
Slides are available here.
For more details, see:
- John Baez, Alexander Hoffnung and Chris Rogers, Categorified Symplectic Geometry and the Classical String
- John Baez and Chris Rogers, Categorified Symplectic Geometry and the String Lie 2-Algebra