Classical String Theory and Categorified Symplectic Geometry
Posted by John Baez
As categorification sweeps the land, it hits some areas sooner than others. While it’s had a big impact on fancy forms of mathematical physics like ‘topological quantum field theory’, it hasn’t yet encroached quite so visibly on more basic subjects, like classical mechanics.
However, this is typical of mathematical ideas: they’re often discovered in fancy contexts, but when it becomes clear how simple they are, their realm of application spreads. I believe categorification can be applied to classical mechanics… and then it leads to higherdimensional field theories, including classical string theory!
Chris Rogers, Alex Hoffnung and I are writing a paper on one aspect of this topic:

Chris Roger, Alex Hoffnung and John Baez,
Categorified symplectic geometry and the classical string, draft version. For a more uptodate version try
this.
Abstract: A Lie 2algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures similar to those of a Lie algebra, but where the usual laws hold only up to isomorphism. It is well known that given a manifold equipped with a symplectic 2form, the Poisson bracket gives rise to a Lie algebra of observables. Multisymplectic geometry generalizes the classical mechanics of point particles to $n$dimensional field theories, decribing such a theory in terms of a ‘phase space’ that is a manifold equipped with a closed nondegenerate $(n+1)$form. Here, given a manifold with a closed nondegenerate $3$form, we construct a Lie 2algebra of observables. We then describe how this Lie 2algebra can be used to describe dynamics in classical bosonic string theory.
In fact, Chris just gave a series of 5 lectures on the subject, which you can see here…
Chris Rogers did physical chemistry before switching to math and coming to UCR, so he can think and calculate like a physicist — which comes in handy! His lectures make a nice introduction to our paper, since he was explaining the ideas to some of my other graduate students, which meant that he needed to explain things from scratch, not assuming vast prior knowledge. So, if you want to get started on Hamiltonian mechanics, symplectic and multisymplectic geometry, classical string theory and Lie 2algebras, don’t be scared — here’s a place to start!
 Lectures by Chris Rogers (notes by Alex Hoffnung):
It’s amusing to note that the key idea behind categorified classical mechanics — boosting the symplectic 2form to a multisymplectic 3form — goes back to the work of DeDonder and Weyl in the 1930s. But only much later was it realized that 2forms are to line bundles as 3form are to gerbes! This makes the role of categorification more explicit. By showing that a manifold with a multisymplectic 3form gives a Lie 2algebra of observables, we’re making it so darn explicit that it can no longer be ignored!
Fans of Lie 2algebras will enjoy that we actually get both a ‘semistrict’ Lie 2algebra — where the Jacobi identity holds up to isomorphism but the bracket is skewsymmetric on the nose — and a ‘hemistrict’ one as defined by Roytenberg — where the Jacobi identity holds on the nose but the skewsymmetry holds only up to isomorphism.
But, they’re isomorphic!
I was very surprised when Chris and Alex discovered this.
Re: Classical String Theory and Categorified Symplectic Geometry
Is there a version of geometric quantization in this setting yet?