### 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, $n$-plectic geometry is good for classical field theory on $n$-dimensional spacetime. The case $n = 1$ is symplectic geometry; the new stuff starts with $n = 2$.

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 $3$-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 $G$ 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 $G$. He was right! The Lie 2-algebra associated to the 2-plectic manifold $G$ comes equipped with an action of $G$ 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

globalorintegratedviewpoint 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 $\mathfrak{g}$ `extended’ by $\mathbb{R}$.

In particular, I didn’t know of a simple direct geometric way to understand that ‘$\mathfrak{g}$’ and the ‘$\mathbb{R}$’ appearing in the classification of Lie 2-algebras.

This paper gives us such a geometric intuition. It says that we can think of that ‘$\mathfrak{g}$’ as really being the

left invariant Hamiltonian 1-formson the group $G$ (in the sense of 2-plectic geometry), and we can think of that ‘$\mathbb{R}$’ as really being theleft invariant smooth functions on $G$. Something like that, anyhow. It’s always nicer to have a direct geometric understanding of a particular incarnation of $\mathbb{R}$, rather than just saying ‘$\mathbb{R}$’.