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January 26, 2009

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:

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, nn-plectic geometry is good for classical field theory on nn-dimensional spacetime. The case n=1n = 1 is symplectic geometry; the new stuff starts with n=2n = 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 33-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 GG 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 GG. He was right! The Lie 2-algebra associated to the 2-plectic manifold GG comes equipped with an action of GG via left translations. The translation-invariant elements form a Lie 2-algebra in their own right… and this is the string Lie 2-algebra!

Posted at January 26, 2009 11:39 PM UTC

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

In particular, I didn’t know of a simple direct geometric way to understand that ‘𝔤\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 ‘𝔤\mathfrak{g}’ as really being the left invariant Hamiltonian 1-forms on the group GG (in the sense of 2-plectic geometry), and we can think of that ‘\mathbb{R}’ as really being the left invariant smooth functions on GG. 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}’.

Posted by: Bruce Bartlett on January 27, 2009 11:27 AM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the String Lie 2-Algebra

Nice result.

A similar question I once talked about with Danny was if/how one can think of “the universal enveloping algebra of the String Lie 2-algebra” as a quantum group. I was once hoping that one could “explain” quantum groups as universal enveloping algebras of Lie 2-algebras. But I couldn’t see how to.

Posted by: Urs Schreiber on January 27, 2009 12:09 PM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the String Lie 2-Algebra

I like the idea of relating quantum groups to universal enveloping algebras of string Lie 2-algebras. After all, in both cases we’re deforming a simple Lie algebra 𝔤\mathfrak{g} using the god-given element of H 3(𝔤,)H^3(\mathfrak{g},\mathbb{R}). We also know that quantum groups are closely related to central extensions of loop groups, which are closely related to string 2-groups.

So, there has to be some relation between quantum groups and string Lie 2-algebras. The question is just how to find an elegant one.

Posted by: John Baez on January 27, 2009 4:26 PM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the String Lie 2-Algebra

Chris Rogers is giving a talk on our paper in Göttingen next week. You can see the talk here.

Posted by: John Baez on January 29, 2009 5:31 AM | Permalink | Reply to this

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