Categorified Symplectic Geometry and the Classical String

August 2, 2008

Categorified Symplectic Geometry and the Classical String

Posted by John Baez

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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?

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There are two main changes:

First, in Section 2 we describe a general recipe for getting an $n$ -plectic manifold from an $n$ -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 $(n + 1)$ -form is nondegenerate if the $n$ -form we get by plugging in one tangent vector is zero only if that vector is zero. An $n$ -plectic manifold is a manifold equipped with a closed nondegenerate $(n + 1)$ -form. When $n = 1$ , an $n$ -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 $n = 1$ . But the idea generalizes to higher $n$ !

Suppose we’re studying a field theory where fields are maps $ϕ : Σ \to M$ where the ‘parameter space’ $Σ$ is an $n$ -dimensional manifold and the ‘target space’ $M$ is a manifold of any dimension. Then there’s a standard way to build an ‘extended phase space’ for this theory which is an $n$ -plectic manifold. We describe how this works.

In the case $n = 1$ , a map $ϕ : Σ \to M$ describes the worldline of a particle moving in the spacetime $M$ . In this case, the ‘extended phase space’ we’re talking about is just the cotangent bundle $T^{*} (Σ \times M)$ . This becomes a symplectic manifold in a well-known way.

We’re mainly interested in the case $n = 2$ , where a map $ϕ : Σ \to M$ describes the worldsheet of a string moving in the spacetime $M$ . 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 $n$ .
Second, in Section 5 we describe how the presence of a $B$ 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 $B$ 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.

Posted at August 2, 2008 1:56 PM UTC

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11 Comments & 3 Trackbacks

Re: Categorified Symplectic Geometry and the Classical String

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Concerning the extended phase space being a bundle of forms over something that at least locally is a product $Σ \times X$ of parameter space $Σ$ with target space $X$ :

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 $Σ \times X$ with open sets and then looks at those sheaves of vertex operator algebras over it.

Posted by: Urs Schreiber on August 2, 2008 3:45 PM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

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Thanks for the clue!

I guess Witten is talking about the case where $Σ$ is 2-dimensional? That would be about right if you want to get vertex operator algebras.

To understand this, maybe we should drop down a dimension and consider the case where $Σ$ is 1-dimensional. Then our extended phase space is $T^{*} (Σ \times X)$ . We can imagine trying to do deformation quantization of this phase space and get an algebra. It’s not hard, but suppose you foolishly start with the case where $Σ$ and $X$ are contractible. Then you’re doing deformation quantization to $T^{*} (ℝ^{1} \times ℝ^{d})$ , which is easy. You get a Weyl algebra. But if you try to ‘glue together’ these algebras to get a deformation quantization of the whole extended phase space, you’re dealing with a problem which reminds me of ‘sheaves of algebras’. I don’t quite see a sheaf of algebras here, since we only have algebras for contractible open sets — but still, there’s a lot of old work on deformation quantization that starts locally and then uses sheaves, and this reminds me of that.

So, we might imagine something similar is happening one dimension up. We could try to assemble a vertex operator algebra for some interesting target space $X$ starting from a sheaf of ‘local’ ones. Or something like that.

I’ll have to read the paper a bit more to see if this vague idea is on some right track.

By the way, I think there’s a nice definition of vertex operator algebras using operads. This might allow us to define something analogous to a VOA for other values of $n$ (the dimension of the parameter space), and then see how to quantize $n$ -plectic manifolds using sheaves of these ‘ $n$ -algebras’. The case $n = 1$ should be some familiar stuff about sheaves of Weyl algebras.

Posted by: John Baez on August 2, 2008 4:31 PM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

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I guess Witten is talking about the case where $Σ$ is 2-dimensional?

Yes, I should have said that. He is looking at 2-dimensional conformal field theory and/or possibly some flavor of topological twist of it.

[…] starts locally and then uses sheaves, and this reminds me of that.

Good point, possibly there is a close relation. I wish I had 36 hours a day to go and check that right now.

Just in case it matters for anything: part of the point here is that in 2d there is that famous chiral deRham complex, locally a certain vertex operator algebra, which arranges itself to a sheaf on target space. The crucial point is that there may be non-free 2d CFTs whose non-freedom comes only from global effects. Locally – locally in target space! – they behave like free field theories. More geometrically: if we think of a $σ$ -model of maps from parameter space to target space and then restrict those maps to land only in contractible open subsets of the target, then all these restrictions are free field theories. Gluing them all together on target space may yield a non-free field theory (or it may fail altogether, if the relevant anomaly=obstruction does not vanish).

Somehow that’s a deep statement, because more naively it would seem that for a sigma-model a local description makes sense only with respect to parameter space. But at least in some cases there is more.

By the way, I think there’s a nice definition of vertex operator algebras using operads.

Yes! By Huang and his student Kong.

The vertex operator algebra can be seen to pretty directly encode the properties of functors from the multicategory whose morphisms are Riemann spheres with $n$ incoming and 1 outgoing puncture to certain graded vector spaces, which are required to depend holomorphically on the insertion points. The crucial ingredient of the VOA, that operation “ $Y$ ” which eats a complex number $z$ and a vector $u$ and spits out a linear map $Y (u, z) : V \to V$ is the image under such a functor of the sphere with three punctures at $0$ , $z$ and $∞$ , respectively.

So the VOA is pretty directly a “holomorphic genus 0” CFT in the sense of Segal, only difference being that Segal has proper circle-boundaries where here one uses parameterized punctures. It’s an imprtant difference but still just a technical one.

Liang Kong taught me a bit about all this here in Bonn. He has a whole series of articles where he takes this observation and runs with it, describing open, closed, oplen-closed CFT at genus 0, genus 1 and potentially higher, relating it to Frobenius algebras and “Cardy algebras” and whatnot in the modular tensor category of reps of the VOA.

Posted by: Urs Schreiber on August 4, 2008 9:56 AM | Permalink | Reply to this

Read the post Classical String Theory and Categorified Symplectic Geometry
Weblog: The n-Category Café
Excerpt: Just as any symplectic manifold gives a Lie algebra of observables, any 2-plectic manifold gives a Lie 2-algebra of observables. This shows up in string theory!
Tracked: August 3, 2008 12:49 PM

Re: Categorified Symplectic Geometry and the Classical String

This all looks fascinating. I look forward to getting involved with this stuff.

Posted by: Bruce Bartlett on August 4, 2008 10:57 AM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

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I look forward to getting involved with this stuff.

Any news on connecting your work on 2-[geometric quantization] with the general Baez-Hoffnung-Rogers framework?

(You should at least make them cite your article in the outlook section, as something that should be a cool application once the connection is understood in more detail.)

Posted by: Urs Schreiber on August 4, 2008 11:47 AM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

No, nothing to report at this time.

Posted by: Bruce Bartlett on August 5, 2008 11:13 AM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

Good point — we should cite Bruce’s work as an example of ‘categorified geometric quantization’. Now the trick will be going from finite groups to compact simple Lie groups, which are blessed with very interesting 2-plectic structures.

Posted by: John Baez on August 7, 2008 12:36 PM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

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In a discussion I had I was pointed to Michael Forger’s work on multisymplectic geometry, such as

Michael Forger, Cornelius Paufler, Hartmann Römer, Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory.

I was just wondering if you are aware of this work and maybe looked at this article? (I haven’t yet.)

Posted by: Urs Schreiber on November 12, 2008 7:25 PM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

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We haven’t looked at that stuff. We should! Thanks!

Posted by: John Baez on November 13, 2008 9:12 PM | Permalink | Reply to this

Re: Categorified Symplectic Geometry and the Classical String

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By the way, Bruce Bartlett kindly pointed out to me

H. M. Khudaverdian and T. Voronov, Higher Poisson Brackets and Differential Forms.

There Theodore Voronov is using his powerful – by now probably seminal – method of derived brackets to obtain $∞$ -Poisson algebras. In his setup these come from inhomogeneous multivectorfields which are even graded, and induce on the algebra of functions the structure of an $L_{∞}$ -algebra.

I was chatting with Bruce a bit about how this might be related to the Lie 2-algebra on 1-forms which you obtain from 2-plectic structures. Superficially it looks like both constructions are two different kinds of higher versions of ordinary Poisson algebras. But of course they might well be related somehow. Maybe simply by allowing odd multivectorfields in Voronov’s setup. Or maybe it’s more subtle.

Posted by: Urs Schreiber on November 13, 2008 10:05 PM | Permalink | Reply to this

Read the post Super Version of 2-Plectic Geometry for Classical Superstrings?
Weblog: The n-Category Café
Excerpt: Here are some basic questions about describing the classical superstring using the 'super' analogue of multisymplectic geometry, if such a thing exists.
Tracked: December 16, 2008 7:44 PM

Read the post Categorified Symplectic Geometry and the String Lie 2-Algebra
Weblog: The n-Category Café
Excerpt: A new paper shows how to build the string Lie 2-algebra by taking a compact Lie group with its canonical closed 3-form and then using ideas from multisymplectic geometry.
Tracked: January 27, 2009 12:03 AM

Re: Categorified Symplectic Geometry and the Classical String

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This paper was just accepted for publication in Communications in Mathematical Physics. I’m glad it got accepted now, since Alex Hoffnung is applying for jobs this fall!

Posted by: John Baez on August 19, 2009 7:18 AM | Permalink | Reply to this

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