February 15, 1999

This Week's Finds in Mathematical Physics (Week 129)

John Baez

For the last 38 years the Austrians have been having winter workshops on nuclear and particle physics in a little Alpine ski resort town called Schladming. This year it was organized by Helmut Gausterer and Hermann Grosse, and the theme was "Geometry and Quantum Physics":

1) Geometry and Quantum Physics lectures, 38th Internationale Universitaetswochen fuer Kern- und Teilchenphysik, http://physik.kfunigraz.ac.at/utp/iukt/iukt_99/iukt99-lect.html

I was invited to give some talks about spin foam models, and the other talks looked interesting, so I decided to leave my warm and sunny home for the chilly north. I flew out to Salzburg in early January and took a train to Schladming from there. Jet-lagged and exhausted, I almost slept through my train stop, but I made it and soon collapsed into my hotel bed.

The next day I alternately slept and prepared my talks. The workshop began that evening with a speech by Helmut Grosse, a speech by the town mayor, and a reception featuring music by a brass band. The last two struck me as a bit unusual - there's something peculiarly Austrian about drinking beer and discussing quantum gravity over loud oompah music! This was also the first conference I've been to that featured skiing and bowling competitions.

Anyway, there were a number of 4-hour minicourses on different subjects, which should eventually appear as articles in this book:

2) Geometry and Quantum Physics, proceedings of the 38th Int. Universitaetswochen fuer Kern- und Teilchenphysik, Schladming, Austria, Jan. 9-16, 1999, eds. H. Gausterer, H. Grosse and L. Pittner, to appear in Lecture Notes in Physics, Springer-Verlag, Berlin.

Right now they exist in the form of lecture notes:

All these talks were about different ways of combining quantum theory and geometry. Quantum theory is so strange that ever since its invention there has been a huge struggle to come to terms with it at all levels. It took a while for it to make its full impact in pure mathematics, but now you can see it happening all over: there are lots of papers on quantum topology, quantum geometry, quantum cohomology, quantum groups, quantum logic... even quantum set theory! There are even some fascinating attempts to apply quantum mechanics to unsolved problems in number theory like the Riemann hypothesis... will they bear fruit? And if so, what does this mean about the world? Nobody really knows yet; we're in a period of experimentation - a bit of a muddle.

I don't have the energy to summarize all these talks so I'll concentrate on part of Alekseev's - just a tiny smidgen of it, actually! But first, let me just quickly say a word about each speaker's topic.

Alekseev talked about some ideas related to the stationary phase approximation. This is one of the main tools linking classical mechanics to quantum mechanics. It's a trick for approximately computing the integral of a function of the form exp(iS(x)) knowing only S(x) and its 2nd derivative at points where its first derivative vanishes. In physics, people use it to compute path integrals in the semiclassical limit where what matters most is paths near the classical trajectories. Alekseev discussed problems where the stationary phase approximation gives the exact answer. There's a wonderful thing called the Duistermaat-Heckman formula which says that this happens in certain situations with circular symmetry. There are also generalizations to more complicated symmetry groups. These are related to `equivariant cohomology' - more on that later.

I talked about the spin foam approach to quantum gravity. I've already discussed this in "week113", "week114", "week120", and "week128", so there's no need to say more here.

Cesar Gomez gave a wonderful introduction to string theory, starting from scratch and rapidly working up to T-duality and D-branes. The idea behind T-duality is very simple and pretty. Basically, if you have closed strings living in a space with one dimension curled up into a circle of radius R, there is a symmetry that involves replacing R by 1/R and switching two degrees of freedom of the string, namely the number of times it winds around the curled-up direction and its momentum in the curled-up direction. Both these numbers are integers. D-branes are something that shows up when you consider the consequences of this symmetry for open strings.

String theory is rather conservative in that, at least until recently, it usually treated spacetime as a manifold with a fixed geometry and only applied quantum mechanics to the description of the strings wiggling around in spacetime. In spin foam models, by contrast, spacetime itself is modelled quantum-mechanically as a kind of higher-dimensional version of a Feynman diagram. There are also other ideas about how to treat spacetime quantum-mechanically. One of them is to treat the coordinates on spacetime as noncommuting variables. In this approach, called noncommutative geometry, the uncertainty principle limits our ability to simultaneously know all the coordinates of a particle's position, giving spacetime a kind of quantum "fuzziness". Personally I don't find noncommutative geometry convincing as a theory of physical spacetime, because there are no clues that spacetime actually has this sort of fuzziness. But I find it quite interesting as mathematics.

Daniel Kastler talked about Alain Connes' theories of physics based on noncommutative geometry. He discussed both the original Connes-Lott version of the Standard Model and newer theories that include gravity. Kastler is a real character! As usual, his talks lauded Connes to the heavens and digressed all over the map in a frustrating but entertaining manner. Throughout the conference, he kept us well-fed with anecdotes, bringing back the aura of heroic bygone days. A random example: Pauli liked to work long into the night - so when a student asked "Could I meet you at your office at 9 a.m.?" he replied "No, I can't possibly stay that late".

One nice idea mentioned by Kastler came from this paper:

3) Alain Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995), 6194.

The idea is to equip spacetime with extra curled-up dimensions shaped like the quantum group SUq(2) where q is a 3rd root of unity. A quantum group is actually a kind of noncommutative algebra, but using Connes' ideas you can think of it as a kind of "space". If you mod out this particular algebra by its nilradical, you get the algebra M1(C) + M2(C) + M3(C), where Mn(C) is the algebra of n x n complex matrices. This has a tantalizing relation to the gauge group of the Standard Model, namely U(1) x SU(2) x SU(3).

John Madore also spoke about noncommutative geometry, but more on the general theory and less on the applications to physics. He concentrated on the notion of a "differential calculus" - a structure you can equip an algebra with in order to do differential geometry thinking of it as a kind of "space".

Julius Wess also spoke on noncommutative geometry, focussing on a q-deformed version of quantum mechanics. The process of "q-deformation" is something you can do not only to groups like SU(2) but also other spaces. You get noncommutative algebras, and these often have nice differential calculi that let you go ahead and do noncommutative geometry. Wess had a nice humorous way of defusing tense situations. When one questioner pointedly asked him whether the material he was presenting was useful in physics or merely a pleasant game, he replied "That's a very good question. I will try to answer that later. For now you're just like students in calculus: you don't know why you're learning all this stuff...." And when Kastler and other mathematicians kept hassling him over whether an operator was self-adjoint or merely hermitian, he begged for mercy by saying "I would like to be a physicist. That was my dream from the beginning."

Anyway, I hope that from these vague descriptions you get some sense of the ferment going on in mathematical physics these days. Everyone agrees that quantum theory should change our ideas about geometry. Nobody agrees on how.

Now let me turn to Alekseev's talk. In addition to describing his own work, he explained many things I'd already heard about. But he did it so well that I finally understood them! Let me talk about one of these things: equivariant deRham cohomology. For this, I'll assume you know about deRham cohomology, principal bundles, connections and curvature. So I assume you know that given a manifold M, we can learn a lot about its topology by looking at differential forms on M and figuring out the space of closed p-forms modulo exact ones - the so-called pth deRham cohomology of M. But now suppose that some Lie group G acts on M in a smooth way. What can differential forms tell us about the topology of this group action?

All sorts of things! First suppose that G acts freely on M - meaning that gx is different from x for any point x of M and any element g of G other than the identity. Then the quotient space M/G is a manifold. Even better, the map M → M/G gives us a principal G-bundle with total space M and base space M/G.

Can we figure out the deRham cohomology of M/G? Of course if we were smart enough we could do it by working out M/G and then computing its cohomology. But there's a sneakier way to do it using the differential forms on M. The map M → M/G lets us pull back any form on M/G to get a form on M. This lets us think of forms on M/G as forms on M satisfying certain equations - people call them "basic" differential forms because they come from the base space M/G.

What are these equations? Well, note that each element v of the Lie algebra of G gives a vector field on M, which I'll also call v. This give two operations on the differential forms on M: the Lie derivative Lv and the interior product iv. It's easy to see that any basic differential form is annihilated by these operations for all v. The converse is true too! So we have some nice equations describing the basic forms.

If we now take the space of closed basic p-forms modulo the exact basic p-forms, we get the deRham cohomology of M/G! This lets us study the topology of M/G using differential forms on M. It's very convenient.

If the action of G on M isn't free, the quotient space M/G might not be a manifold. This doesn't stop us from defining "basic" differential forms on M just as before. We can also define some cohomology groups by taking the closed basic p-forms modulo the exact ones. But topologists know from long experience that another approach is often more useful. Group actions that aren't free are touchy, sensitive creatures - a real nuisance to work with. Luckily, when you have an action that's not free, you can tweak it slightly to make it free. This involves "puffing up" the space that the group acts on - replacing it by a bigger space that the group acts on freely.

For example, suppose you have a group G acting on a one-point space. Unless G is trivial, this action isn't free. In fact, it's about as far from free as you can get! But we can "puff it up" and get a space called EG. Like the one-point space, EG is contractible, but G acts freely on it. Actually there are various spaces with these two properties, and it doesn't much matter which one we use - people call them all EG. People call the quotient space EG/G the "classifying space" of G, and they denote it by BG.

More generally, suppose we have any action of G on a manifold M. How can we puff up M to get a space on which G acts freely? Simple: just take its product with EG. Since G acts on M and EG, it acts on the product M x EG in an obvious way. Since G acts freely on EG, its action on M x EG is free. And since EG is contractible, the space M x EG is a lot like M, at least as far as topology goes. More precisely, it has the same homotopy type!

Actually the last 2 paragraphs can be massively generalized at no extra cost. There's no need for G to be a Lie group or for M to be a manifold. G can be any topological group and M can be any topological space! But since I want to talk about deRham cohomology, I don't need this extra generality here.

Anyway, now we know the right substitute for the quotient space M/G when the action of G on M isn't free: it's the quotient space (M x EG)/G.

So now let's figure out how to compute the pth deRham cohomology of (M x EG)/G. Since G acts freely on M x EG, this should be just the closed basic p-forms on M x EG modulo the exact ones, where "basic" is defined as before. In fact this is true. We call the resulting space the pth "equivariant deRham cohomology" of the space M. It's a kind of well-behaved substitute for the deRham cohomology of M/G in the case when M/G isn't a manifold.

There's only one slight problem: the space EG is very big, so it's not easy to deal with differential forms on M x EG!

You'll note that I didn't say much about what EG looks like. All I said is that it's some contractible space on which G acts freely. I didn't even say it was a manifold, so it's not even obvious that "differential forms on EG" makes sense! If you are smart you can choose your space EG so that it's a manifold. However, you'll usually need it to be infinite-dimensional.

Differential forms make perfect sense on infinite-dimensional manifolds, but they can be a bit tiresome when we're trying to do explicit calculations. Luckily there is a small subalgebra of the differential forms on EG that's sufficient for the purpose of computing equivariant cohomology! This is called the "Weil algebra", WG.

To guess what this algebra is, let's just list all the obvious differential forms on EG that we can think of. Well, I guess none of them are obvious unless we know a few more facts! First of all, since the action of G on EG is free, the quotient map EG → BG gives us a principal G-bundle with total space EG and base space BG. This bundle is very interesting. It's called the "universal" principal G-bundle. The reason is that any other principal G-bundle is a pullback of this one.

(I guess I'm upping the sophistication level again here: I'm assuming you know how to pull back bundles!)

Even better, if we choose our space EG so that it's a manifold, then there is a god-given connection on the bundle EG → BG, and any other principal G-bundle with connection is a pullback of this one.

(And now I'm assuming you know how to pull back connections! However, this pullback stuff is not necessary in what follows, so just ignore it if you like.)

Okay, so how can we get a bunch of differential forms on EG just using the fact that it's the total space of a G-bundle equipped with a connection?

Well, whenever we have a G-bundle E → B, we can think of a connection on it as a 1-form on E taking values in the Lie algebra of G. Let's see what differential forms on E this gives us! Let's call the connection A. If we pick a basis of the Lie algebra, we can take the components of A in this basis, and we get a bunch of 1-forms Ai on E. We also get a bunch of 2-forms dAi. We also get a bunch of 2-forms Ai ^ Aj. And so on.

In general, we can form all possible linear combinations of wedge products of the Ai's and the dAi's. We get a big fat algebra. In the case when our bundle is EG → BG, equipped with its god-given connection, we define this algebra to be the Weil algebra, WG!

Great. But let's try to define WG in a purely algebraic way, so we can do computations with it more easily. We're starting out with the 1-forms Ai and taking all linear combinations of wedge products of them and their exterior derivatives. There are in fact no relations except the obvious ones, so WG is just "the supercommutative differential graded algebra freely generated by the variables Ai". Note: all the mumbo-jumbo about supercommutative differential graded algebras is a way of mentioning the obvious relations.

Warning: people don't usually describe the Weil algebra quite this way. They usually seem describe it in terms of the connection 1-forms and curvature 2-forms. However, the curvature is related to the connection by the formula F = dA + A ^ A, and if you use this you can go from the usual description of the Weil algebra to mine - I think.

(Actually, people often describe the Weil algebra as an algebra generated by a bunch of things of degree 1 and a bunch of things of degree 2, without telling you that the things of degree 1 are secretly components of a connection 1-form and the things of degree 2 are secretly components of a curvature 2-form! That's why I'm telling you all this stuff - so that if you ever study this stuff you'll have a better chance of seeing what's going on behind all the murk.)

Okay, so here is the upshot. Say we want to compute the equivariant deRham cohomology of some manifold M on which G acts. In other words, we want to compute the deRham cohomology of (M x EG)/G. On the one hand, we can start with the differential forms on M x EG, figure out the "basic" p-forms, and take the space of closed basic p-forms modulo exact ones. But remember: up to details of analysis, the algebra of differential forms on M x EG is just the tensor product of the algebra of forms on M and the algebra of forms on EG. And we have this nice small "substitute" for the algebra of forms on EG, namely the Weil algebra WG. So let's take the algebra of differential forms on M and just tensor it with WG. We get a differential graded algebra with Lie derivative operations Lv and interior product operations iv defined on it. We then proceed as before: we take the space of closed basic elements of degree p modulo exact ones. Voila! This is something one can actually compute, with sufficient persistence. And it gives the same answer, at least when G is connected and simply connected.

There are all sorts of other things to say. For example, if we take the simplest posssible case, namely when M is a single point, this gives a nice trick for computing the deRham cohomology of EG/G = BG. Guys in this cohomology ring are called "characteristic classes", and they're really important in physics. Since any principal G-bundle is a pullback of EG → BG, and cohomology classes pull back, these characteristic classes give us cohomology classes in the base space of any principal G-bundle - thus helping us classify G-bundles. But if I started explaining this now, we'd be here all night.

Also sometime I should say more about how to construct EG.

© 1999 John Baez