## March 14, 2007

### Quantization and Cohomology (Week 18)

#### Posted by John Baez

Today was the last class on Quantization and Cohomology for the winter quarter! We wrapped up with a summary of what we’d done this quarter, and a sketch of the big picture:

• Week 18 (Mar. 13) - The big picture. Building a Hilbert space from a finite category C equipped with an "amplitude" functor A: C → U(1). Example: discretized version of the free particle on the line. Generalizing from particles to strings by categorifying everything in sight. Building a 2-Hilbert space from a finite 2-category C equipped with a 2-functor A: C → U(1)Tor, where U(1)Tor is the 2-group of U(1)-torsors.

Last week’s notes are here; next week’s notes are here.

Next quarter we’ll dig more deeply into the secret theme of this course: the way cohomology shows up naturally when we repeatedly categorify our work so far. Classes resume at the beginning of April. Ciao!

Posted at March 14, 2007 2:52 AM UTC

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### Re: Quantization and Cohomology (Week 18)

Concerning the issue mentioned around the Good question on slide 50-51: you know I have been thinking about this quite a bit lately.

My impression is that it is of great help to write down the theory for $n=1$ completely in terms of arrows. This answers a lot of subtle questions when it comes to categorification.

For instance, I believe it does answer the question what the “space of functions” on configuration space categorifies to.

Also, the choice of 2-category of 2-vector spaces then answers automatically the question what these “2-functions” should be valued in.

Or so I think. I might of course be wrong. I would love to discuss this stuff in more detail. There are not too many people in the world thinking about this stuff from this point of view.

From your reaction to my last comments I gained the impression that you would maybe rather not spread out everything in public, with your student Alex Hoffnung being supposed to work on this. If that’s so, I would just as well enjoy taking this to private email and keeping top secrecy about any further development.

Given my latest rate of publications, my activity here should hardly be a threat to anyone else’s career. So it would be a pity if we could not exchange more ideas on this.

Posted by: urs on March 19, 2007 2:21 PM | Permalink | Reply to this

### Re: Quantization and Cohomology (Week 18)

I’m not big on secrecy, but I still feel I should make an exception for my grad students’ theses: it might not be good for their career if some idea was ‘well-known’ by the time it appeared in their thesis, or least in some talk or publication of theirs.

Since you’re the only person I can imagine working on these ideas, Urs, and since you’re perfectly trustworthy, there should be some arrangement that will work.

In fact, Urs, given how you develop ideas much faster than you can publish them, you could probably use some grad students. Maybe you should take some of mine!

Or, more seriously, I could share some with you.

Posted by: John Baez on March 19, 2007 7:05 PM | Permalink | Reply to this

### Re: Quantization and Cohomology (Week 18)

Well, maybe there is room for collaboration, proper. I am looking for more collaborators!

With Konrad Waldorf there is great progress writing up the stuff, roughly, which I talked about at the time the $n$-Café got started, on locally trivializable $n$-transport. Jim Stasheff is helping me to compile the material on Lie $n$-algebras. Jens Fjelstad is helping work out some things about how 2dCFT is an example for the general theory of the quantum 2-particle.

All this are aspects of one big picture that is emerging, and which looks like

parallel $n$-transport functor $\stackrel{\mathrm{quantization}}{\rightarrow}$ extended $n$-dimensional QFT $n$-functor

But this picture is really too big for me to draw alone. My least worry is that everything in it will soon be well known!

For instance, writing it this way suggests that this “quantization of the charged $n$-particle” should in fact be an $(n+1)$-functor. This I haven’t even tried to make explicit yet.

My impression is that, concerning the “arrow-theoretic functorial 1-particle” what we did is to some degree complementary (I don’t know how much you already did for the 2-particle along these lines). Maybe we could join forces and at least start drawing a tiny corner of this picture.

Ah, right, and of course also Jeffrey Morton is working on one corner of this picture, that where (in the terminology that I keep advertizing) the 3-particle propagates on the classifying space of a group (modeled as $\mathrm{tar} = \Sigma(G)$) and charged under a Chern-Simons 2-gerbe (or its discrete analog). States of this give correlators of a $(n=2)$-theory, and this holographic thing also needs to be worked out in more detail. In a slightly different language Bruce has a lot to say about this. Incorporating this into the general picture needs to be worked out more clearly. (I wanted to visit Bruce this week, but the gods were against us.)

Posted by: urs on March 19, 2007 10:03 PM | Permalink | Reply to this

### Re: Quantization and Cohomology (Week 18)

Or maybe invite Alex here? :)

Posted by: Eric on March 19, 2007 7:54 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: The Canonical 1-Particle
Weblog: The n-Category Café
Excerpt: On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
Tracked: March 19, 2007 9:05 PM
Read the post Quantization and Cohomology (Week 19)
Weblog: The n-Category Café
Excerpt: Finding critical points of the action in a general context: a smooth category equipped with a smooth 'action' functor.
Tracked: April 20, 2007 8:53 PM
Read the post The n-Café Quantum Conjecture
Weblog: The n-Category Café
Excerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
Tracked: June 8, 2007 11:43 AM
Read the post The n-Café Quantum Conjecture
Weblog: The n-Category Café
Excerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
Tracked: June 8, 2007 11:43 AM
Read the post A Groupoid Approach to Quantization
Weblog: The n-Category Café
Excerpt: On Eli Hawkins' groupoid version of geometric quantization.
Tracked: June 12, 2008 5:50 PM

### Re: Quantization and Cohomology (Week 18)

I was just looking at the figure on page 47 and was wondering if anything is lost be constraining the morphisms to $\gamma:(t,x)\to(t+1,x\pm 1)$?

From a numerical analysis point of view, I can see how the correct continuum limit might emerge in both cases and the difference could amount to a choice of a “forward” or “backward” difference approximation for the time derivative.

Note: With the constrained morphisms to $\pm 1$, I believe the continuum limit $\epsilon\to 0$ should be approached in a such a way the $\Delta t = (\Delta x)^2 = \epsilon$.

Just thinking out loud…

Posted by: Eric on June 12, 2008 6:57 PM | Permalink | Reply to this
Read the post An Exercise in Groupoidification: The Path Integral
Weblog: The n-Category Café
Excerpt: A remark on the path integral in view of groupoidification and Sigma-model quantization.
Tracked: June 13, 2008 6:23 PM

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