Categorified Symplectic Geometry and the String Lie 2-Algebra
Posted by John Baez
Just in time for the workshop in Göttingen, my student Chris Rogers and I have finished a paper that uses 2-plectic geometry to give a new construction of the string Lie 2-algebra:
- John Baez and Chris Rogers, Categorified symplectic geometry and the string Lie 2-algebra.
If you have comments or corrections, I’d love to hear ‘em.
What’s the idea?
Well, symplectic geometry is part of a more general subject called multisymplectic geometry, invented by DeDonder and Weyl in the 1930s. Just as symplectic geometry is good for the classical mechanics of point particles, -plectic geometry is good for classical field theory on -dimensional spacetime. The case is symplectic geometry; the new stuff starts with .
In particular, just as the phase space of a classical point particle is a symplectic manifold, a classical string may be described using a finite-dimensional ‘2-plectic’ manifold. Here the nondegenerate closed 2-form familiar from symplectic geometry is replaced by a nondegenerate closed -form.
Just as the smooth functions on a symplectic manifold form a Lie algebra under the Poisson bracket operation, any 2-plectic manifold gives rise to a ‘Lie 2-algebra’. This is a categorified version of a Lie algebra: that is, a category equipped with a bracket operation obeying the usual Lie algebra laws up to isomorphism. Alternatively, we may think of a Lie 2-algebra as a 2-term chain complex equipped with a bracket satisfying the Lie algebra laws up to chain homotopy.
Now, every compact simple Lie group has a god-given 2-plectic structure, built from the Killing form and the Lie bracket. It is natural to wonder what Lie 2-algebra this example gives. Danny Stevenson suggested that it should be related to the already known ‘string Lie 2-algebra’ of . He was right! The Lie 2-algebra associated to the 2-plectic manifold comes equipped with an action of via left translations. The translation-invariant elements form a Lie 2-algebra in their own right… and this is the string Lie 2-algebra!
Re: Categorified Symplectic Geometry and the String Lie 2-Algebra
This is great. Shooting from the hip, the thing that appears most attractive to me is that we now have an alternative global or integrated viewpoint on the string Lie 2-algebra. As I understood things, before this paper the only global/geometric take on the string Lie 2-algebra was that it was the infinitesimal version of the string 2-group (the string 2-group has objects which are smooth paths in the group starting at the unit element).
But strictly speaking, the infinitesimal version of the string 2-group is really the `path Lie 2-algebra’ (which is infinite dimensional). It was then an interesting step to see that this is equivalent (as a Lie 2-algebra) to the finite-dimensional one consisting of a Lie algebra `extended’ by .
In particular, I didn’t know of a simple direct geometric way to understand that ‘’ and the ‘’ appearing in the classification of Lie 2-algebras.
This paper gives us such a geometric intuition. It says that we can think of that ‘’ as really being the left invariant Hamiltonian 1-forms on the group (in the sense of 2-plectic geometry), and we can think of that ‘’ as really being the left invariant smooth functions on . Something like that, anyhow. It’s always nicer to have a direct geometric understanding of a particular incarnation of , rather than just saying ‘’.