So, last "week", I said a bit about how supersymmetry could be handy for constructing topological quantum field theories, and this week I want to say a bit more about what that has to do with getting a purely combinatorial description of Donaldson theory.
But first, I want to lighten things up a bit by mentioning a good science fiction novel!
1) Permutation City, by Greg Egan, published in Britain by Millenium (should be available in the U.S. by autumn).
There is a lot of popular interest these days in the anthropic principle. Roughly, this claims to explain certain features of the universe by noting that if the universe didn't have those features, there would be no intelligent life. So, presumably, the very fact that we are here and asking certain questions guarantees that the questions will have certain answers.
Of course, the anthropic principle is controversial. Suppose one could really show that if the universe didn't have property X, there would be no intelligent life. Does this really count as an "explanation" of property X? People like arguing about this. But this question is much too subtle for a simple-minded soul such as myself. I'm still stuck on more basic things!
For example, are there any examples where we can really show that if the universe didn't have property X, there would be no intelligent life? It seems that to answer this, we need to have some idea about what we're counting as "all possible universes", and what counts as "intelligent life". So far we only know ONE example of a universe and ONE example of intelligent life, so it is difficult to become an expert on these subjects! It'd be all to easy for us to unthinkingly assume that all intelligent life is carbon-based, metabolizes using oxidation, and eats pizza, just because folks around here do.
Our unthinking parochialism is probably all the worse as far as different universes are concerned! What counts as a possible universe, anyway? Rather depressingly, we must admit we don't even know the laws of this universe, so we don't really know what it takes for a universe to be possible, in the strong sense of capable of actually existing as a universe. We are a little bit better off if we consider all logically possible universes, but not a whole lot better. Certainly every axiom system counts as a logically possible set of laws of a universe - every set of axioms in every possible formal system. But who is to say that universes must have laws of this form? We don't even know for sure that ours does!
So this whole topic will remain a hopeless quagmire until one takes a small, carefully limited piece of it and studies that. People studying artificial life are addressing one of these bite-sized pieces, and getting some interesting results. I hope everyone has heard about Thomas Ray's program Tierra, for example: he created an artificial ecosystem - one could call it a "possible universe" - and found, after seeding it with one self-reproducing program, a rapid evolution of parasites, etc., following many of the patterns of ecology here. But so far, perhaps merely due to time and memory limitations, no intelligence!
One of the cool things about "Permutation City" is an imagined cellular automaton, the "Autoverse", which is complicated enough to allow life. But something much cooler is the main theme of the book. Egan calls it the "Dust Theory". It's an absolutely outrageous theory, but if you think about it carefully, you'll see that it's rather hard to spot a flaw. It depends on the tricky puzzles concealed in the issue of "isomorphism".
Being a mathematician, one thing that always puzzled me about the notions of "intelligent life" and "all possible universes" was the question of isomorphisms between universes. Certainly we all agree that, say, the Heisenberg "matrix mechanics" and Schrodinger "wave mechanics" formulations of quantum mechanics are isomorphic. In both of them, the space of states is a Hilbert space, but in one the states are described as sequences of numbers, while in the other they are described as wavefunctions. At first they look like quite different theories. But in a while people realized that there was a unitary operator from Heisenberg's space of states to Schrodinger's, and that via this correspondence all of matrix mechanics is equivalent to wave mechanics.
So does Heisenberg's universe count as the same one as Schrodinger's, or a different one? It seems clear that they're the same. But say we had two quantum-mechanical systems whose Hamiltonians have the same eigenvalues (or spectrum); does that mean they are the "same" system, really? Is that all there is to a physical system, a list of eigenvalues??? If we are going to go around talking about "all possible universes", it would probably pay to think a little about this sort of thing!
Say we had two candidates for "laws of the universe", written down as axioms in different formal systems. How would we decide if these were describing different universes, or were simply different ways of talking about the same universe? Pretty soon it becomes clear that the issue is not a black-and-white one of "same" versus "different" universes. Instead, laws of physics, or universes satisfying these laws, can turn out to be isomorphic or not depending on how much structure you want the isomorphism to preserve. And even if they are isomorphic, there may not be a "unique" isomorphism or a "canonical" isomorphism. (Very roughly speaking, a canonical isomorphism is a "God-given best one", but one can use some category theory to make this precise.) If you think about this carefully you'll see that our universe could be isomorphic to some very different-seeming ones, or could have some very different-seeming ones `embedded' in it.
Greg Egan takes this issue and runs with it -- in a very interesting direction. Everyone interested in cellular automata, artificial life, virtual reality, or other issues of simulation should read this, as well as anyone who likes philosophy or just a good story.
Okay, back to business here...
2) Alberto Cattaneo, Teorie topologiche di tipo BF ed invarianti dei nodi, doctoral thesis, department of physics, University of Milan.
Alberto Cattaneo, Paolo Cotta-Ramusino, Juerg Froehlich, and Maurizio Martellini, Topological BF theories in 3 and 4 dimensions, preprint available as hep-th/9505027.
So, last week I said a wee bit about Blau and Thompson's paper on supersymmetry and the Casson invariant. All I said was that suitably concocted supersymmetric field theories could be used to compute the Euler characteristics of your favorite spaces, and that Blau and Thompson talked about one which computed the Casson invariant, which is (in a rather subtle sense) the Euler characteristic of the moduli space of flat connections on a trivial SU(2) bundle over a 3-manifold. Traditionally one requires that the 3-manifold be a homology 3-sphere, but Kevin Walker showed how to do it for rational homology spheres, and Blau and Thompson mention other work in which the Casson invariant is generalized still further.
But I didn't say which supersymmetric field theory computes the Casson invariant for you. The answer is, N = 2 supersymmetric BF theory with gauge group SU(2). So now I should say a little about BF theory. Actually I have already mentioned it here and there, especially in "week36". But I should say a bit more. This is going to be pretty technical, though, so fasten your seatbelts.
The people I know who are the most excited about BF theory are the folks I was visiting at Milan, namely Cotta-Ramusino, Martellini and his student Cattaneo. They are working on BF theory in 3 and 4 dimensions. Let me talk about BF theory in 3 dimensions, which is what's most directly relevant here. Well, in any dimension, say n, the fields in BF theory are a connection A on a trivial bundle (take your favorite gauge group G), whose curvature F we'll think of as a 2-form taking values in the Lie algebra of G, and Lie-algebra-valued (n-2)-form B. Then the Lagrangian of the theory is
L(B,F) = tr(B ^ F)
where in the "trace" we're using something like the Killing form to get an honest n-form which we can integrate over spacetime.
But in 3 dimensions, since B is a 1-form, you can add an extra "cosmological constant" term and take as your Lagrangian
L(B,F,c) = tr(B ^ F + (c2/3) B ^ B ^ B)
where I have put in "c2/3" as my "cosmological constant" for insidious reasons to become clear momentarily. Now what the article by Cattaneo, Cotta-Ramusino, Froehlich and Martellini makes really clear is how BF theory is related to Chern-Simons theory. This is implicit in Witten's work on 3d gravity (see "week16"), which is just the special case where G is SO(2,1) or SO(3), and where the cosmological constant really is the usual cosmological constant. But I'd never noticed it. Recall that the Chern-Simons action is
L(A) = tr(A ^ dA + (2/3)A ^ A ^ A)
Thus if we have 1-form B around as well, we can set
A' = A + cB, A'' = A - cB
so we get two different Chern-Simons theories with actions L(A') and L(A''), respectively. OR, we can form a theory whose action is the difference of these two, and, lo and behold:
L(A') - L(A'') = 4cL(B,F,c)
In other words, BF theory with cosmological constant is just a "difference of two Chern-Simons theories". Fans of topological quantum field theory may perhaps be more familiar with this if I point out that the Turaev-Viro theory is just BF theory with gauge group SU(2), and the fact that the partition function for this theory is the absolute value squared of that for Chern-Simons theory is a special case of what I'm talking about. The nice thing about all this is that the funny phases coming from framings in Chern-Simons theory precisely cancel out when you form this "difference of two Chern-Simons theories".
Now the Casson invariant is related to BF theory in 3 dimensions without cosmological constant, i.e., taking c = 0. We might get worried by the equation above, which we can't solve for L(B,F) when c = 0, but as Cattaneo and company point out,
L(B,F) = lim L(A') - L(A'') c → 0 ----------------- 4c
so BF theory without cosmological constant is just a limiting case, actually a kind of derivative of Chern-Simons theory. They use this to make clearer the relation between the vacuum expectation values of Wilson loops in Chern-Simons theory -- which give you the HOMFLY polynomial for G = SU(N) -- and the corresponding vacuum expectation values in BF theory without cosmological constant -- which give you the Alexander polynomial! Very pretty stuff.
Now back to the Casson invariant and some flagrant speculation on my part concerning Crane and Frenkel's ideas on Donaldson theory. (I said last week that this is where I was heading, and now I'm almost there!) Okay: we know how to define Chern-Simons theory in a purely combinatorial way using quantum groups. I.e., we can compute the partition function of Chern-Simons theory with gauge group G using the quantum version of the group G... let me just call it "quantum G". If we take c to be imaginary above, one can show that BF theory with cosmological constant can be computed in a very similar way starting with the quantum group corresponding to the complexification of G, i.e. "quantum CG". The point is that A+cB can then be thought of as a connection on a bundle with gauge group CG. So far this is not flagrant speculation. Slightly more flagrantly, but not really very much at all, the formulas above hint that BF theory without cosmological constant can be computed in a similar way starting with the quantum group corresponding to the tangent bundle of G, or "quantum TG". (The tangent bundle of a Lie group is again a Lie group, and as we let c → 0 what we are really doing is taking a limit in which CG approaches TG; folks call this a "contraction", and in the SU(2) case many of the details appear in Witten's paper on 3d quantum gravity; the tangent bundle of SO(2,1) being just the Poincare group in 3 dimensions.) If anyone knows whether folks have worked out the quantization of these tangent bundle groups, let me know! I think some examples have been worked out.
Okay, but Blau and Thompson say that to compute the Casson invariant you need to use, not BF theory with gauge group SU(2), but supersymmetric BF theory with gauge group SU(2). Well, no problemo -- just compute it with "quantum super-T(SU(2))"! Here I'm getting a bit flagrant; there are theories of quantum supergroups, but I don't know much about them, especially "quantum super-TG" for G compact semisimple. Again, if anybody does, please let me know! (Actually Blau told me to check out a paper by Saleur and somebody on this, but I never did....)
Okay, but now let's get seriously flagrant. Recall that the Casson invariant is really the Euler characteristic of something, just a number, but this number is just the superdimension of a super-vector-space, namely the Floer cohomology. From numbers to vector spaces: this is a typical sort of "categorification" process that one would expect as one goes from 3d to 4d TQFTs. And indeed, folks suspect that the Floer cohomology is the space of states for a 4d TQFT, or something like a 4d TQFT, namely Donaldson theory. ("Something like it" because of many quirky twists that one wouldn't expect of a full-fledged TQFT satisfying the Atiyah axioms.) So, just as the Casson invariant is associated to a certain Hopf algebra, namely "quantum super-T(SU(2))", we'd expect Donaldson theory to be associated to a certain Hopf category, the "categorification of quantum super-T(SU(2))". So all we need to do is figure out how to categorify quantum super-T(SU(2)) and we've got a purely combinatorial definition of Donaldson theory!
Well, that's not quite so easy, of course. And I may have made, not only the inevitable errors involved in painting a simplified sketch of what is bound to be a rather big task, but also other worse errors. Still, this business should clarify, if only a wee bit, what Crane and Frenkel are up to when they are categorifying SU(2). In fact, it's likely that working with SU(2) rather than T(SU(2)) will remove some of the divergences from the state sum, since, being compact, SU(2) has a discrete set of representations (and quantum SU(2) has finitely many interesting ones, at roots of unity). So they may get a theory that's allied to but not exactly the same as Donaldson theory, yet better-behaved as far as the TQFT axioms go.
If anyone actually does anything interesting with these ideas I'd very much appreciate hearing about it.
© 1995 John Baez