Categorified Symplectic Geometry and the Classical String
Posted by John Baez
After weeks of polishing, maybe this is ready for the arXiv:
- John Baez, Alex Hoffnung and Chris Rogers, Categorified symplectic geometry and the classical string.
We’ve already talked about this paper. Besides the title, what’s new?
There are two main changes:
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First, in Section 2 we describe a general recipe for getting an -plectic manifold from an -dimensional field theory. This is already known to people familiar with multisymplectic geometry — but there aren’t many such people. So, it deserves explanation.
An -form is nondegenerate if the -form we get by plugging in one tangent vector is zero only if that vector is zero. An -plectic manifold is a manifold equipped with a closed nondegenerate -form. When , an -plectic manifold is usually called a symplectic manifold. Symplectic manifolds serve as phase spaces in the classical mechanics of point particles. The path or ‘worldline’ of a point particle is 1-dimensional, so we say . But the idea generalizes to higher !
Suppose we’re studying a field theory where fields are maps where the ‘parameter space’ is an -dimensional manifold and the ‘target space’ is a manifold of any dimension. Then there’s a standard way to build an ‘extended phase space’ for this theory which is an -plectic manifold. We describe how this works.
In the case , a map describes the worldline of a particle moving in the spacetime . In this case, the ‘extended phase space’ we’re talking about is just the cotangent bundle . This becomes a symplectic manifold in a well-known way.
We’re mainly interested in the case , where a map describes the worldsheet of a string moving in the spacetime . In this case the extended phase space is a bit more tricky to explain. In our previous draft we just blurted out the answer… but it probably looked quite ad hoc. Now we derive the answer from an already known framework that works for all .
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Second, in Section 5 we describe how the presence of a field affects the 2-plectic structure for a string. This is just like how an electromagnetic field affects the symplectic structure for a particle!
I remember being amazed when I first read that just by modifying the symplectic structure in an obvious way, the equations of motion for a free particle become the equations for a charged particle in an electromagnetic field. No need to change the Hamiltonian! The same thing works for a string in a field.
We also fixed millions of mistakes, some caught by people here.
But the main idea of the paper is unchanged: just as a symplectic manifold gives a Lie algebra of observables, a 2-plectic manifold gives a Lie 2-algebra of observables.
It’s well-known that we can describe the dynamics of a particle using its Lie algebra of observables. For example, bracketing with an observable called the Hamiltonian says how other observables change with time. Similarly, we can describe the dynamics of a string using its Lie 2-algebra of observables.
This material is based upon work supported by the National Science Foundation under Grant No. 0653646.
Re: Categorified Symplectic Geometry and the Classical String
Concerning the extended phase space being a bundle of forms over something that at least locally is a product of parameter space with target space :
this is something apparently of deep significance which has not generally been appreciated or even fully understood, it seems.
I once mentioned an observation Witten makes about this starting on the bottom of page 29 here.
He covers with open sets and then looks at those sheaves of vertex operator algebras over it.