### Courant Algebroids From Categorified Symplectic Geometry

#### Posted by John Baez

*guest post by Chris Rogers*

‘Higher symplectic geometry’ is a topic that has come up recently in posts here in the $n$-Café and in some ideas presented by Urs and others in the $n$Lab. This is a subject I’ve been thinking a lot about lately, and I just finished a (rough!) draft of a paper that attempts to establish some connections between two different approaches to generalizing, extending, and/or categorifying symplectic geometry. So of course I happily accepted John’s invitation to write a guest post about some of this work. I hope by doing so I can contribute a little something to the ongoing discussion and learn more about the different ideas people have on the subject.

There are many approaches in the literature that attempt to develop a more generalized notion of symplectic geometry. Some are cast in the language of ‘oidization’ (e.g. Courant algebroids, Lie $n$-algebroids ), or in terms of supergeometry (e.g. NQ-manifolds), while others use more traditional geometric ingredients to invent interesting new structures (e.g. multisymplectic geometry, and $k$-symplectic geometry). Some of these approaches had higher-algebraic or higher-geometric structures explicitly built in from the beginning. But many of them did not, and it is only rather recently that they have been reinterpreted in some kind of categorified way.

I’d like to focus on two approaches here. I’ll just sketch the main
ideas but you can get the full story by reading the
draft
available for now on my web page. The
first approach is what John and I call ‘categorified symplectic
geometry’ or ‘$n$-plectic geometry’. Here one studies the
categorified algebraic and geometric structures that naturally arise
on manifolds equipped with a closed non-degenerate $n+1$-form. Such manifolds show up as phase spaces in $n$-dimensional classical field theory. The case relevant to the classical bosonic string is when $n=2$ and
is called ‘2-plectic geometry’. So, just as the phase space of the
classical particle is a manifold equipped with a closed,
non-degenerate 2-form, the phase space of the classical string is a
(finite-dimensional) manifold equipped with a closed
non-degenerate **3-form**.

And there is also an algebraic side to this correspondence. The symplectic form gives the space of smooth functions the structure of a Poisson algebra. Analogously, the 2-plectic form gives a Lie 2-algebra of observables associated to any 2-plectic manifold. And when the 2-plectic manifold is a compact simple Lie group $G$, this Lie 2-algebra is closely related to the string Lie 2-algebra. (See this paper for more about Hamiltonian observables and classical strings, and this paper for more on how the string Lie 2-algebra gets involved.)

String theory, closed 3-forms and Lie 2-algebras also play important roles in the theory of Courant algebroids. Roughly, Courant algebroids are vector bundles which simultaneously combine and generalize the structures found in the tangent bundles of manifolds and quadratic Lie algebras. In particular, the underlying vector bundle of a Courant algebroid comes equipped with an antisymmetric bracket on the space of global sections. However, unlike the Lie bracket of vector fields, the bracket need not satisfy the Jacobi identity.

In a fascinating series of letters to Alan Weinstein, Pavol Ševera showed (among many other things) that so-called ‘exact Courant algebroids’ naturally arise in the context of string theory. He developed the notions of connection and curvature on exact Courant algebroids and then showed that these algebroids are classified up to isomorphism by the third de Rham cohomology of the base space. Around the same time Dmitry Roytenberg and Alan Weinstein showed that the bracket on the Courant algebroid induces an $L_{\infty}$ structure on the global sections. Roytenberg later reinterpreted this $L_{\infty}$-algebra as a Lie 2-algebra.

So there are some very strong similarities between these two approaches! And perhaps the punchline to the story is now obvious: On any 2-plectic manifold there is an exact Courant algebroid whose Ševera class is specified by the 2-plectic form. Algebraically, this corresponds to a nice embedding of the Lie 2-algebra of observables on the 2-plectic manifold into the Lie 2-algebra induced by the bracket on the Courant algebroid’s space of global sections.

I think this is just the beginning of some deeper relationships between these two approaches, and I am optimistic that they will mesh well with the ideas that Urs presented here. Thank you for reading! Comments, questions, corrections, and suggestions are most welcome.

## Re: Courant Algebroids From Categorified Symplectic Geometry

Thanks for this post!

You mention:

Your link here points to $n$-symplectic manifold. I want to emphasize that the ideas mentioned at this entry – up to just slight idiosyncracies on my part (discussed in the discussion part) – are those of Dmitry Roytenberg and Pavol Ševera, as indicated in the list of references.

A few additional ideas of mine on this topic (linked to from there) is instead at schreiber:symplectic $\infty$-Lie algebroid.