|
|
|
|
For people in theoretical physics, Trieste is a kind of mecca. It's an Italian town on the Adriatic quite near the border with Slovenia, and it's quite charming, especially the castle of Maximilian near the coast, built when parts of northern Italy were under Hapsburg rule. Maximilian later took his architect with him to Mexico when he became Emperor there, who built another castle for him in Mexico City. (The Mexicans, apparently unimpressed, overthrew and killed Maximilian.) These days, physicists visit Trieste partially for the charm of the area, but mainly to go to the ICTP and SISSA, two physics institutes, the latter of which has grad students, the former of which is purely for research. There are lots of conferences and workshops at Trieste, and I was lucky enough to be invited to Trieste while one I found interesting was going on.
As I described to some extent in "week44" and "week45", Seiberg and Witten have recently shaken up the subject of Donaldson theory by using some physical reasoning to radically simplify the computations involved. Donaldson theory has always had a lot to do with physics, since it uses the special features of of gauge theory in 4 dimensions to obtain invariants of 4-dimensional manifolds. So perhaps it is not surprising that physicists have had a lot to say about Donaldson theory all along, even before the recent Seiberg-Witten revolution. And indeed, at Trieste lots of mathematicians and physicists were busy talking to each other about Donaldson theory, trying to catch up with the latest stuff and trying to see what to do next.
Now I don't know much about Donaldson theory, but I have a vague interest in it for various reasons. First, it's supposed to be a 4-dimensional topological quantum field theory, or TQFT. Indeed, the very first paper on TQFTs was about Donaldson theory in 4 dimensions:
1) Topological quantum field theory, by Edward Witten, Comm. Math. Phys. 117 (1988) 353.
Only later did Witten turn to the comparatively easier case of Chern-Simons theory, which is a 3-dimensional TQFT:
2) Quantum field theory and the Jones polynomial, by Edward Witten, Comm. Math. Phys. 121 (1989) 351.
However, when mathematicians talk about TQFTs they usually mean something satisfying Atiyah's axioms for a TQFT (which are nicely presented in his book - see "week39"). Now it turns out that Chern-Simons theory can be rigorously constructed as a TQFT satisfying these axioms most efficiently using braided monoidal categories, which play a big role in 3d topology. So it makes quite a bit of sense in a general sort of way that Crane and Frenkel are trying to construct Donaldson theory using braided monoidal 2-categories, which seem to play a comparable role in 4d topology. In the paper which I cite in "week50", they begin to construct a braided monoidal 2-category related to the group SU(2), which they conjecture gives a TQFT related to Donaldson theory. That also makes some general sense, because Donaldson theory, at least "old" Donaldson theory, is closely related to gauge theory with gauge group SU(2). Still, I've always wanted to see a more specific reason why Donaldson theory should be related to the Crane-Frenkel ideas, not necessarily a proof, but at least a good heuristic argument.
Luckily George Thompson, who invited me to Trieste, knows a bunch about TQFTs. Unluckily he was sick and I never really got to talk to him very much! But luckily his collaborator Matthias Blau was also there, so I took the opportunity to pester him with questions. I learned a bit, most of which is in their paper:
3) N = 2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant, by Matthias Blau and George Thompson, Comm. Math. Phys. 152 (1993), 41-71.
This paper helped me a lot in understanding Crane and Frenkel's ideas. But so that this "week" doesn't get too long, I'll just focus on one basic aspect of the paper, which is the importance of supersymmetric quantum theory for TQFTs. Then next week I'll say a bit more about the Donaldson theory business.
If you look at Witten's paper on Donaldson theory above, you'll see he writes down the Lagrangian for a "supersymmetric" field theory, which is supposed to be a TQFT, namely, Donaldson theory. Supersymmetric field theories treat bosons and fermions in an even-handed way. But why does supersymmetry show up here? The connection with TQFTs is actually pretty simple and beautiful, at least in essence.
Suppose we are doing quantum field theory, and "space" (as opposed to "spacetime") is some manifold M. Then we have some Hilbert space of states Z(M) and some Hamiltonian H, which is a self-adjoint operator on Z(M). To evolve a state (a vector in Z(M)) in time, we hit it with the unitary operator exp(-itH), where t is the amount of time we want to evolve by, and the minus sign is just a convention designed to confuse you.
We can think of this geometrically as follows. We are taking spacetime to be [0,t] x M. You can visualize spacetime as a kind of pipe, if you want, and then imagine sticking in the state ψ at one end and having exp(-itH)ψ pop out at the other end.
Now say we bend the pipe around and connect the input end to the output end! Then we get the spacetime S1 x M, where S1 is the circle of circumference t, formed by gluing the two ends of the interval [0,t] together. For this kind of "closed" spacetime, or compact manifold, a quantum field theory should give us not an operator like exp(-itH), but a number, the "partition function", which in this case is just the trace tr(exp(-itH)).
The deep reason for this is that taking the trace of an operator - remember, that means taking the sum of the diagonal entries, when you think of it as a matrix - is really very much like as taking a pipe and bending it around, connecting the input end to the output end, forming a closed loop. This may seem bizarre, but observe that taking the sum of the diagonal entries really is just a quantitative measure of how much the "output constructively interferes with the input". (And a very nice one, since it winds up not depending on the basis in which we write the matrix!) This sort of idea is basic in the Bohm-Aharonov effect, where we take a particle in an electomagnetic field around a loop and see how much it interferes with itself, and it is also the basic idea of a "Wilson loop", where we do the same thing for a particle in a gauge field. In other words, the trace measures the amount of "positive feedback". If this still seems bizarre, or just vague, take a look at:
4) Knots and Physics, by Louis Kauffman, World Scientific Press, Singapore, 1991.
You will see that the same idea shows up in knot theory, where taking a trace corresponds to taking something (like a braid or tangle) and folding it over to connect the input and output. In a later "week" I'll talk a bit about a new paper by Joyal, Street and Verity that studies the notion of "trace", "feedback" and "folding over" in a really general context, the context of category theory.
Anyway, the partition function tr(exp(-itH)) typically depends on t, or in other words, it depends on the circumference of our circle S1, not just on the topology of the manifold S1 x M. In a TQFT, the partition function is only supposed to depend on the topology of spacetime! So, how can we get tr(exp(-itH)) to be independent of t?
There is a banal answer and a clever answer. The banal answer is to take H = 0! Then tr(exp(-itH)) = tr(1) is just the dimension of the Hilbert space:
tr(exp(-itH)) = dim(Z(M)).
Actually this isn't quite as banal as it may sound; indeed, the basic equation of quantum gravity is the Wheeler-DeWitt equation,
H ψ = 0,
which must hold for all physical states. In other words, in quantum gravity there is a big space of "kinematical states" on which H is an operator, but the really meaningful "physical states" are just those in the subspace
Z(M) = {ψ: H ψ = 0}.
Read "week11" for more on this.
But there is a clever answer involving supersymmetry! You might hope that there were some more interesting self-adjoint operators H such that tr(exp(-itH)) is time-independent, but there aren't. So we seem stuck. This reminds me of a course I took from Raoul Bott. He said "so, we think about the problem... and still we are stuck, so what should we do? SUPERTHINK!"
Recall that the Hamiltonian of a free particle in quantum mechanics is - up to boring constants - just minus the Laplacian on configuration space which is some Riemannian manifold that the particle roams around on. For this Hamiltonian, tr(exp(-itH)) doesn't quite make sense, since the Hilbert space is infinite-dimensional and the sum of the diagonal matrix entries diverges. But tr(exp(-tH)) often does converge. This is why folks often replace t by -it in formulas, which is called "going to imaginary time" or a "Wick transform"; it really amounts to replacing Schrodinger's equation by the heat equation: i.e., instead of a quantum particle, we have a particle undergoing Brownian motion! In any event, tr(exp(-tH)) certainly depends on t in these situations, but there is something very similar that does NOT.
Namely, let's replace the Laplacian on functions by the Laplacian on differential forms. I won't try to remind you what these are; I'll simply note that functions are 0-forms, but there are also 1-forms, 2-forms, and so on - tensor fields of various sorts - and the Laplacian of a j-form is another j-form. So for each j we get a kind of Hamiltonian Hj, which is just minus the Laplacian on j-forms. We can also consider the space of all forms, never mind the j, and on this space there is a Hamiltonian H, which is just minus the Laplacian on all forms. Now, we could try to take the trace of exp(-tH), but it's more interesting to take the "supertrace":
str(exp(-tH)) = tr(exp(-tH0)) - tr(exp(-tH1)) + tr(exp(-tH2)) - ...
in other words, the trace of exp(-tH) acting on even forms, minus the trace on odd forms.
Why?? Well, odd forms are sort of "fermionic" in nature, while even forms are sort of "bosonic". The idea of supersymmetry is to throw in minus signs when you've got "odd things", because they are like fermions, and physicists know that lots formulas for fermions are just like formulas for bosons, which are "even things", except for these signs. That's the rough idea. It's all related to how, when you interchange two identical bosons, their wavefunction remains unchanged, while for fermions it picks up a phase of -1.
Now the amazing cool thing is that str(exp(-tH)) is independent of t. This follows from some stuff called Hodge theory, or, if you want to really show off, index theory. Basically it works like this. If you have an operator A with eigenvalues λ_i, then
tr(exp(-tA)) = ∑i exp(-t λi)
if the sum makes sense. We can use this formula to write out str(exp(-tH)) in terms of eigenvalues of the Laplacians Hj, and it turns out that all the terms coming from nonzero eigenvalues exactly cancel! So all that's left is the part coming from the zero eigenvalues, which is independent of t. If you believe this for a second, it means we can compute the supertrace by taking the limit as t → ∞. The eigenvalues are all nonnegative, so all the quantities exp(-t λi) go to zero except for the zero eigenvalues, and we're left with str(exp(-tH)) being equal to the alternating sum of the dimensions of the spaces
{ψ : Hj ψ = 0}
Now in fact, Hodge theory tells us that these spaces are really just the "cohomology groups" of our configuration space, so the answer we get for str(exp(-tH)) is what folks call the "Euler characteristic" of our configuration space... an important topological invariant.
So, generalizing the heck out of this idea, we can hope to get TQFTs from supersymmetric quantum field theories as follows. Start with some recipe for associating to each choice of "space" M a "configuration space" C(M)... some space of fields on M, typically. Let Z(M) be the space of all forms on C(M), and let H be the minus the Laplacian, as an operator on Z(M). Then we expect that the partition function str(exp(-tH)) will be independent of t. This is just what one wants in a TQFT. Moreover, the partition function will be the Euler characteristic of the configuration space C(M).
But what if we want to get a TQFT out of this trick, and avoid reference to the Laplacian? Then we can just do the following equivalent thing (at least it's morally equivalent: there will usually be things to check). Let Z+(M) be the direct sum of all the even cohomology groups of C(M), and let Z-(M) be the direct sum of all the odd ones. Then
str(exp(-tH)) = dim(Z+(M)) - dim(Z-(M))
so what we expect is, not quite a TQFT in the Atiyah sense, but a "superTQFT" whose space of states has an "even" part equal to Z+(M) and an "odd" part equal to Z-(M); the right hand side is then the "superdimension" of the space of states this "superTQFT" assigns to M.
Now actually in real life things get tricky because the configuration space C(M) might be infinite-dimensional, or a singular variety. If C(M) is too weird, it gets hard to say what its Euler characteristic should be! But as Blau and Thompson's paper and the references in it point out, one can often still make it make sense, with enough work. In particular, when we are dealing with Donaldson theory, C(M) is just the moduli space of flat SU(2) connections on M. This means that the partition function of S1 x M should be the Euler characteristic of moduli space, better known as the Casson invariant. And what is the vector space our superTQFT assigns to M? Well, it's called Floer homology. Now actually there are a lot of subtleties here I'm deliberately sloughing over. Read Blau and Thompson's paper for some of them -- and read the references for more!
© 1995 John Baez
baez@math.removethis.ucr.andthis.edu
|
|
|
|