The n-Category Café

I guess by now it’s okay to give away the answer to John Armstrong’s dilemma.

We’re not considering a flexible but inelastic rope hanging between two posts. That would trace out a catenary.

Instead, we’re considering a flexible and elastic spring hanging between posts, idealized so its energy of stretching is proportional to

$$\int \mid \frac{dq}{ds}{\mid }^{2}ds$$

where $q(s)$ is the position of the spring as a function of the parameter $s$ that “counts atoms” as we move along the spring. Such a spring would have the least energy when it shrinks down to zero length.

And, such a spring traces out a parabola when you hang it between two posts!

In fact, that paper was how Louis Crane got excited about the “ladder of dimensions” which relates field theories in neighboring dimensions through a process of categorification. And he, in turn, got me interested in n-categories. So, maybe it’s finally all fitting together.

Interesting. I had not known this part of the story.

I am wondering how many other people in the world seriously think about the step WZW to CS in terms of categorification. Willerton and Bartlett, inspired by Freed, do #. According to the first slide of Gukov’s talk at Strings 06, possibly also Witten does. But it’s not clear. There is still a difference between using a category here and there and taking things seriously.

In Vienna both Anton Kapustin as well as Tony Pantev, after their brilliant talks on Langlands (Pan., Kap. ) found little appreciation for the fact that the eigenbrane condition is the condition for an eigen-2-vector in categorified linear algebra - a very simple and obvious statement that makes the entire field appear in a different light. To me, it’s the difference between happening to use a category and taking things seriously.

Anyway. Above # I wrote:

actually, I don’t yet talk there explicitly about the quantum 2-particle

I was hoping to find the time to work more today. But the blissful period between semesters is coming to an end, causing lots of distractions (and I guess I was also being distracted by thinking about Euler characteristics).

But I should maybe indicate how using the puzzle pieces that I talked about a coherent picture seems to emerge.

Let the 3-particle be the 2-category

with two objects, two nontrivial 1-morphisms and 1 nontrivial 2-morphism.

Its configuration space is the space of objects of the functor category

(To be compared with the first couple of definitions here.)

For $C_2$ a braided monoidal category, let

be the 3-category of its endomorphisms.

According to this we want to look at lax 2-functors

into that, which assign algebras internal to $C_2$ to points, left-right induced bimodules to edges.

By the general logic # a state of the 3-particle coupled to the 2-bundle with connection represented by $\mathrm{tra}$ is a section $e$ of that 3-bundle, which again is a morphism

where ${\mathrm{tra}}^0$ sends everything to the idendity on the tensor unit.

Meditating over that for a while reveals that such a morphism is more or less determined by the same form of disk diagrams that are featured at the end of that text on disk holonomy #.

In fact, I believe the resulting disk diagram should be essentially that which appears in section 4.3 of my FRS notes #, except that the $U$- and $V$- ribbons which were inserted by hand there now appear automatically due to # the nature of $\mathrm{End}(C_2)$. Very nice.

So this story starts as a description of a state for the 3-particle and ends as a description of the disk correlator of a 2-particle.

Now I need the time to work this out in detail.

You could say that a lot of the first quarter of this course is an attempt to understand the relation between Hamiltonian and Lagrangian mechanics. Of course people understand the relation already. But, some aspects of this relation still feel like ‘tricks’, and they shouldn’t — everything should be very clear.

In particular, it seems a bit ‘tricky’ how Hamiltonian mechanics is usually formalized in terms of a 2-form on phase space, the symplectic structure $\omega$. What’s the physical meaning of this 2-form?

A 2-form is something that you integrate over a surface… so what does it mean when you integrate $\omega$ over a surface in phase space?

The quantity you get has units of action, since the symplectic structure looks like $$\omega =d{p}_i\wedge d{q}^i,$$ and momentum times position has units of action.

But, why should a surface in phase space have an action associated to it?

In the Lagrangian approach, what gets an action is a path, not a surface.

So, it’s all a bit mysterious.

However, we can imagine a system tracing out a periodic orbit. We get a path in phase space that’s a loop. This loop might be the boundary of a surface. If the loop is the boundary of some surface, we could try to define the action of that surface to be the action of the loop!

This is definitely a good idea, and Bohr and Sommerfeld thought about it a lot when quantizing the hydrogen atom. They realized that the hydrogen atom seemed to enjoy orbits that go around surfaces where the integral of $\omega$ is an integer times Planck’s constant. This is the Bohr-Sommerfeld quantization condition.

So, there’s something important about surfaces in phase space where the integral of $\omega$ is an integer. Maybe that’s why integrals are called integrals: Nature likes them best when they’re integral!

But, there are a bunch of things that don’t seem to gell at first. For example, action is really the integral of $$Hdt-p_{i}d{q}^i$$ along a path. What does this have to do with the integral of $\omega =d{p}_i\wedge d{q}^i$ over a surface? How do time and energy get unified with position and momentum, exactly? Clearly this has something do with relativity… but what?

In the notes I try to answer these questions.

However, I should improve all my explanations, so people can see my motivations better! Your questions are already helping me do this.

By the way, the notes from my course on classical mechanics could be helpful. I stick to more standard subjects, so everything is better organized… less of a personal quest into unknown territory.