The n-Category Café

Week 1 (Oct. 3) - How the dynamics of p-branes resembles the statics of (p+1)-branes.

Cool. That touches upon many things that I have been thinking about lately.

Let me propose this general way of looking at the situation:

Let $P_n(X)$ be our geometric $n$-category of $n$-paths in target space $X$.

Let $d^{p-1}$ be a $p$-particle. This means: let $d^{p-1}$ be the $(p-1)$-category which encodes the internal structure of a $p$-particle.

For $p=1$ we usually choose

to be the one-object 0-category.

For $p=2$ we usually choose

to be the poset of the oriented interval $[0,\pi]$ for the “open string”, or, similarly for the “closed string”.

And so on.

The configuration space of the $p$-particle with target space $P_n(X)$ is the functor $n$-category

Objects in here are images of the $p$-particle in target space.

Morphisms in here are worldvolumes traced out by the $p$-particle.

By the nature of pseudonatural transformations, this automatically ensures that the endpoints of the $p$-particle need not be nailed down. In fact, these worldvolumes are automatically (by the logic of $n$-functor categories) cobordisms cobounding the source and target $p$-particle image.

There are also higher morphisms in $[d^{p-1},P_n(X)]$, which encode various gauge invariances (reparameterizations and generalizations thereof) of the worldvolume of a $p$-particle.

Next, we want to associate phases to morphisms in $[d^{p-1},P_n(X)]$.

To get that, let

be an $n$-bundle with connection on $X$, under which our $p$-particle shall be charged.

(Take the trivial bundle with trivial connection if in your application the $p$-particle is not charged.)

Composing with $\mathrm{tra}$ provides us with an $n$-functor

This functor reads in a $p$-particle in target space and spits out a “fiber” over it.

It reads in a worldvolume of a $p$-particle and spits out the “phase” associated to this due to the charge of the $p$-particle.

Now let $\mathrm{triv}$ be a transport functor which factors through the category $\{\bullet\}$ with a single object and no nontrivial morphisms.

The space of states of our $p$-particle is the $n$-category

(Recall that $\mathrm{tra}_*$ was the functor from configuration space $[d^{p-1},P_n(X)]$ to phases $[d^{p-1},T]$.)

An object in

is a section of the bundle of “fibers” over the configuration space of the $p$-particle.

In order to be able to study the quantum mechanics of our $p$-particle charged under $\mathrm{tra}$, we need to assume that the category of generalized phases

is a category with duals.

If that is the case, we can form

(Here $\tilde T$ is whatever the $\mathrm{Hom}$ takes values in, depending on what $[d^{n-1},T]$ is enriched over.)

Similarly for $\mathrm{triv}$.

The point of this is that given any two sections

we get a morphism

On objects, this encodes the scalar product on the space of sections.

On morphisms, this encodes a scalar product on covariant derivatives

Notice that this is the $e_1$, $e_2$ matrix element of the Hamiltonian

of the charged $p$-particle.

I have more details of this discussion scattered on notes flying around on a couple of tables of the $n$-Café, for instance here and here.

I think I have checked that for ordinary charged 1-particles, the above prescription indeed reproduces the ordinary quantization of the particle.

I am in the process of working out what the above says for strings charged under an abelian gerbe. I think everything works as expected, but this is work in progress.

One further aspect one should be able to discuss along these lines is interaction of $p$-particles for $p \gt 1$, by passing from $d^{p-1}$ to suitable interaction diagrams. For instance for the triangle one gets the multiplicative structure on the space of sections over the configuration space of the open string, as described above.

John wrote in part:

Here’s a little refresher course on classical mechanics for mathematicians, if you want it:

• John Baez, Classical Mechanics.

You’ll need to fix the permissions so people can read the errata.

Also, I notice that you have some homeworks from the 2002 course, including 2 solutions written by me. I have all 5 solutions, if you want them.

Urs wrote:

Maybe I am exaggerating here. But in as far as we are interested in quantizing and categorifying the charged point, it seems to me that the central message of geometric quantization useful for categorification is:

• the quantum Hilbert space is that of (square integrable) sections of the bundle that the point is charged under;
• the Hamiltonian is the covariant (with respect to the connection on the bundle) Laplace operator acting on these sections.

These two statement lend themselves to categorification very nicely, as I tried to indicate above #.

I could be wrong, but it seems you’re not talking about full-fledged geometric quantization - just a special case that’s a bit more general than Schrödinger quantization.

For readers not up on their quantization techniques, I should say a bit more. Take a little nap, Urs…

In Schrödinger quantization we take a “configuration space” - a space of possible positions for our particle, say a Riemannian manifold $Q$ - and build a Hilbert space $L^2(Q)$ which serves to describe states of the quantum version of the particle.

In geometric quantization we take a “phase space” - a space of possible positions and momenta for our particle, say a symplectic manifold $X$ - and build a Hilbert space which serves to describe states of the quantum version of the particle.

What is this Hilbert space? It’s not so easy to describe. Our first guess, $L^2(X)$, is wrong in several ways.

The tricky thing to realize is that when the symplectic structure on $X$ is “integral”, it determines a line bundle

$$E \to X$$

with a connection on it (at least up to isomorphism). Instead of thinking about $L^2(X)$, we need to think about $L^2(X,E)$ - the Hilbert space of square-integrable sections of this line bundle. I guess the easiest way to see this is to think very hard about the old quantum mechanics - Bohr and Sommerfeld’s approach to quantization before Schrödinger came along.

But, $L^2(X,E)$ is still not right; it doesn’t match the Schrödinger prescription in the case when both strategies apply, namely when

$$X = T^* Q.$$

So, we call $L^2(X,E)$ the prequantum Hilbert space. It’s too big: we need to chop it down to something smaller to get the right answer. To do this, we need to choose an extra structure on $X$, called a “polarization”, which picks out certain allowed sections of $E$, giving a subspace of $L^2(X,E)$, the quantum Hilbert space $H(X,E)$.

This matches Schrödinger’s prescription when $X = T^* Q$: we have a nice isomorphism

$$H(T^*Q, E) \cong L^2(Q).$$

Anyway, that’s a short version of a long story - for more, try this.

Okay, Urs, you can wake up!

In your discussion of the point particle, you seem to be going a bit beyond Schrödinger quantization, but not all the way to general geometric quantization, since your Hilbert space is $L^2(Q,E)$ - you’re still using configuration space, not phase space - but you’ve got a nontrivial line bundle over it.

This seems like something I could categorify. And, I guess you already have, to some extent. Do you follow Dan Freed’s strategy, and try to build a 2-Hilbert space of sections of a 2-line bundle associated to a gerbe?

(I’m sorry, I should read your notes, but I’m a lazy guy; I like to just sit around and chat.)

This sort of thing is already fascinating, but I can’t help wanting to categorify full-fledged geometric quantization, something like this:

We start with a phase space $X$ with a closed integral 3-form $\omega$ on it. This determines a gerbe $E \to X$ connection on it, at least up to equivalence.

Following Freed, $E$ this should have some 2-Hilbert space of sections $L^2(X,E)$. I don’t actually know how to do this rigorously, and I don’t think Freed did either, but I’ve learned a lot about infinite-dimensional 2-Hilbert spaces recently, in my work with Laurent Freidel, so I feel optimistic.

But then, I want some concept of “2-polarization” - some geometrical structure I can put on $X$, which determines a sub-2-space of $L^2(X,E)$, the “quantum 2-Hilbert space”.

To figure out this concept, we need to look at examples, starting with the Schrödinger quantization of the string - as before, geometric quantization should be backwards-compatible with Schrödinger quantization!

But, one wants to go quite a bit further… I guess the trick is finding more examples: that’s how they invented geometric quantization in the first place.

PS - the above link about the old quantum mechanics doesn’t say anything about the Bohr-Sommerfeld quantization condition and its relation to the crucial “integrality” condition on symplectic structures - it’s all implicit in the charmingly quaint pictures of electron orbitals. But, it leads to a nice story involving Sommerfeld and a young student named Heisenberg, who was defending his dissertation on turbulent fluid flow:

Trouble began when acceptance of the dissertation brought admission of the candidate to the final orals. The examining committee consisted of Sommerfeld and Wien in physics, along with representatives in Heisenberg’s two minor subjects, mathematics and astronomy. Much was at stake, for the only grades a candidate received for his studies were those based on the dissertation and final oral: one grade for each subject and one for overall performance. The grades ranged from I (equivalent to an A) to V (an F).

As the 21-year-old Heisenberg appeared before the four professors on July 23, 1923, he easily handled Sommerfeld’s questions and those in mathematics, but he began to stumble on astronomy and fell flat on his face on experimental physics. In his laboratory work Heisenberg had to use a Fabry-Perot interferometer, a device for observing the interference of light waves, which the class had studied extensively. But Heisenberg had no idea how to derive the resolving power of the interferometer nor, to Wien’s surprise, could he derive the resolution (ability to distinguish objects) of such common instruments as the telescope or the microscope. When an angry Wien asked him how a storage battery works, the candidate was still lost. Wien saw no reason to pass the young man, no matter how brilliant he might be in other branches of physics. An argument arose between Sommerfeld and Wien over the relative importance of theoretical physics in relation to experimental physics. The result was that Heisenberg received a III, equivalent to a C, in physics and for the overall grade for his doctorate. Both of these grades were probably averages between Sommerfeld’s grade (an A) and Wien’s grade (an F).

Sommerfeld was shocked. Heisenberg was mortified. Accustomed to being always at the top of his class, Heisenberg found it hard to accept a mediocre grade for his doctorate. Sommerfeld held a small party at his home later that evening for the new Dr. Heisenberg, but Heisenberg excused himself early, packed his bag, and took the midnight train to Göttingen, showing up in Max Born’s office the next morning. Born had already hired Heisenberg as his teaching assistant for the coming school year. After informing Born of the debacle of his orals, Heisenberg asked sheepishly, “I wonder if you still want to have me.”

Born did not answer until he had gone over the questions Heisenberg had missed. Convincing himself that the questions were “rather tricky,” Born let his employment offer stand. But that fall Heisenberg’s worried father wrote to the famed Göttingen experimentalist James Franck, asking Franck to teach his boy some experimental physics. Franck did his best, but he could not overcome Heisenberg’s complete lack of interest and gave up the effort. If Heisenberg was going to survive at all in physics it would be only as a theorist.

Great Week 1 notes on quantization and cohomology. I had never fully appreciated before the great unity of the subject: that the dynamics of one dimension is the statics of the next, and so on, and that you can think of the action as a one form on “path space” just like the force is a one form on configuration space.

This latter idea ties in perfectly with another great set of notes I’ve recently seen:

• Graeme Segal, Notes on symplectic manifolds and quantization, Kavli Institute webpage, 1999.

Of course, his talks on A mathematician looks at quantum field theory are just as inspiring, but slightly off the topic for now. But in lecture 1 he does make some fascinating comments on qft from a calculus of variations point of view - relating this to the “renormalizability” of the theory.

Dear John Baez,

I am going through all your QC courses again, now in more detail. I hope to be posting my comments and questions here in your blog as time allows. I have very simple questions and suggestions for now, here they are:

Questions for qc.pdf - week 1
====================

- First of all, I think it would be nice to state what the pre-requisites for the course are. It would also be interesting to include references to some terms which are introduced without definition. For instance, on page 5, references to cohomology and homotopy classes. I know these are quite basic terms in topology, but it would be a plus to have a pointer to them (an appendix with some basic definitions or a list of references). Or something like, for instance, Srednicki has done in his QFT book - see his “Preface for Students”, in which he lists a few equations that the students should recognise and be comfortable with them before proceeding. I thought that was very nice!

- Second paragraph of page 5:

Thus, we have a “configuration space” X (…)

I think this is a bit confusing. The term “configuration space X^n” fits nicely with what is usually defined in classical mechanics, for instance, applied to a N-body system: the whole system is represented in the *configuration space* - this space is 6N-dimensional - as a single point, and in the *phase space* - which is 6-dimensional - as N points, each point representing a single particle of the system, and so on. But here you say that the space X (and the path space PX, and so on) is a configuration space as well right? This is just a terminology issue of course, but I found it confusing at first, because for me it is like that the configuration space only makes sense when you include all degrees of freedom of the system. The other spaces are then just projections…

- page 6, first paragraph: please define what \Delta_X is.

- page 7, last line: include a point - “function. In short…”

- page 8: “not exactly isomorphic, but close to it at any rate.”

This sounds very interesting but mysterious to me. Close in what sense?

Thanks,
Christine