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I've been travelling around a lot lately. For a couple of weeks I was in Turkey, resisting the lure of the many internet cafes. I urge you all to visit Istanbul when you get a chance! Fascinating music fills the streets. There are a lot of nice bookstore-cafes on Istiklal Caddesi near Taksim Square, and a huge number of musical instrument shops at the other end of this street, down near Tunel Square. I bought a nice doumbek at one of these shops, and looked at lots of ouzes, sazes and neys, none of which I can play. It's also imperative to check out the Grand Bazaar, the mosques, and the Topkapi Palace - the harem there has most beautiful geometric tiling patterns I've ever seen. I'm not sure why that's true; perhaps this is where the sultans spent most of their time.
The mathematics of tilings is a fascinating subject, but that's not what I'm going to talk about. After my trip to Turkey, I went to a conference at Stanford:
1) Conference on Algebraic Topological Methods in Computer Science, Stanford University, http://math.stanford.edu/atmcs/index.htm
There were lots of fun talks, but I'll just mention two.
The talk most related to physics was the one by my friend Dan Christensen, who spoke on "Spin Networks, Spin Foams and Quantum Gravity", describing a paper he is writing with Greg Egan on efficient algorithms for computing Riemannian 10j symbols. Dan is a homotopy theorist at the University of Western Ontario, and Greg is my favorite science fiction writer. They're both interested in quantum gravity, and they're both good at programming. Together with some undergraduate students of Dan's, the three of us are starting to study the Riemannian and Lorentzian Barrett-Crane models of quantum gravity with the help of computer simulations. But to get anywhere with this, we need to get good at computing "10j symbols".
Huh? "10j symbols"??
Well, as with any quantum field theory, the key to the Barrett-Crane model is the partition function. In the Riemannian version of this theory, you compute the partition function as follows. First you take your 4-dimensional manifold representing spacetime and triangulate it. Then you label all the triangles by spins j = 0, 1/2, 1, 3/2, etcetera. Following certain specific formulas you then calculate a number for each 4-simplex, a number for each tetrahedron, and a number for each triangle, using the spin labellings. Then you multiply all these together. Finally you sum over all labellings to get the partition function. The only tricky part is the convergence of this sum, which was proved by Perez:
2) Alejandro Perez, Finiteness of a spin foam model for euclidean quantum general relativity, Nucl. Phys. B599 (2001) 427-434. Also available as gr-qc/0011058.
The most interesting aspect of all this is the formula giving numbers for 4-simplices. A 4-simplex has 10 triangular faces all of which get labelled by spins, and the formula says how to compute a number from these 10 spins - the so-called "10j symbol".
How do you compute 10j symbols? One approach involves representation theory, or in lowbrow terms, multiplying a bunch of matrices. Unfortunately, if you go about this in the most simple-minded obvious fashion, when the spins labelling your triangles are all about equal to j, you wind up needing to work with matrices that are as big as N x N, where
N = (2j+1)12.
If you do this, already for j = 1/2 you are dealing with square matrices that are 212 by 212. This is too big to be practical!
In computer science lingo, this algorithm sucks because it uses O(j12) time and also O(j12) space. You might think it was O(j24), but it's not that bad... however, it's still very bad!
Luckily, Dan and Greg have figured out a much more efficient algorithm, which uses only O(j6) time and O(j) space. Alternatively, with more caching of data, they can get O(j5) time and O(j3) space, or maybe even better. Using an algorithm of this sort, Dan can compute the 10j symbol for spins up to 55. For all spins equal to 55, the calculation took about 10 hours on a normal desktop computer. However, for computing partition functions it appears that small spins are much more important, and then the computation takes milliseconds.
(Actually, for computing partition functions, Dan is not using a desktop: he is using a Beowulf cluster, which is a kind of supercomputer built out of lots of PCs. This works well for partition functions because the computation is highly parallelizable.)
John Barrett has also figured out a very different approach to computing 10j symbols:
3) John W. Barrett, The classical evaluation of relativistic spin networks, Adv. Theor. Math. Phys. 2 (1998), 593-600. Also available as math.QA/9803063.
In this approach one computes the 10j symbols by doing an integral over the space of geometries of a 4-simplex - or more precisely, over a product of 5 copies of the 3-sphere, where a point on one of these 3-spheres describes the normal vector to one of the 5 tetrahedral faces of the 4-simplex.
Dan and Greg have also written programs that calculate the 10j symbols by doing these integrals. The answers agree with their other approach.
We've already been getting some new physical insights from these calculations. If you write down the integral formula for the Riemannian 10j symbols, a stationary phase argument due to John Barrett and Ruth Williams suggests that, at least in the limit of large spins, the dominant contribution to the integral for the 10j symbol comes from 4-simplices whose face areas are the 10 spins in your 10j symbols:
4) John W. Barrett and Ruth M. Williams, The asymptotics of an amplitude for the 4-simplex, Adv. Theor. Math. Phys. 3 (1999), 209-215. Also available as gr-qc/9809032.
However, Dan and Greg's calculations suggest instead that the dominant contribution comes from certain "degenerate" configurations. Some of these correspond to points on the product of 5 copies of the 3-sphere that are close to points of the form (v,v,v,v,v) - or roughly speaking, 4-simplices whose 5 normal vectors are all pointing the same way. Others come from sprinkling minus signs in this list of vectors. Heuristically, we can think of these degenerate configurations as extremely flattened-out 4-simplices.
For simplicity, we have concentrated so far on studying the 10j symbols in the case when all 10 spins are equal. In this case we can show that the only nondegenerate 4-simplex with these spins as face areas is the regular 4-simplex (all of whose faces are congruent equilateral triangles). Greg used stationary phase to compute the contribution of this regular 4-simplex to Barrett's integral formula for the 10j symbols, and it turned out that asymptotically, for large j, this contribution decays like j-9/2. On the other hand, Dan's numerical computations of the 10j symbol suggests that it goes like j-2. This suggests that for large j, the contribution of the regular 4-simplex is dwarfed by that of the degenerate 4-simplices.
Greg has gotten more evidence for this by studying the integral formula for the 10j symbols and estimating the contribution due to degenerate 4-simplices. This estimate indeed goes like j-2 for large j.
There is a lot more to be understood here, but plunging ahead recklessly, we can ask what all this means for the physics of the Barrett-Crane model. For example: is the dominant contribution to the partition function going to come from spacetime geometries with lots of degenerate 4-simplices?
I think that's a premature conclusion, because we already have evidence that 4 -simplices with large face areas are not contributing that much compared to those with small face areas when we compute the partition function as a sum over spin foams. In other words, it seems that in the Riemannian Barrett-Crane model, spacetime is mostly made of lots of small 4-simplices, rather than a few giant ones. If so, the tendency for the giant ones to flatten out may not be so bad.
Of course the really important thing will be to study these questions for the Lorentzian theory, but it's good to look at the Riemannian theory too.
Another talk on a subject close to my heart was given by Noson Yanofsky. It was based on these papers of his, especially the last:
5) Noson S. Yanofsky, Obstructions to coherence: natural noncoherent associativity, Jour. Pure Appl. Alg. 147 (2000), 175-213. Also available at math.QA/9804106.
The syntax of coherence. To appear in Cahiers Top. Geom. Diff.. Also available at math.CT/9910006.
Coherence, homotopy and 2-theories. To appear in K-Theory. Also available at math.CT/0007033.
One of the cool things Yanofsky has done is to study what happens when we categorify Lawvere's concept of an "algebraic theory". I've already explained this idea of "algebraic theory" in "week53" and "week136", so I'll just quickly recap it here:
The notion of "algebraic theory" is just a slick way to study sets equipped with extra algebraic structure. We call a category C with finite products an "algebraic theory" if its objects are all of the form 1, X, X2, X3, ... for some particular object X. We call a product-preserving functor F: C → Set a "model" of the theory. And we call a natural transformation between such functors a "homomorphism" between models. This gives us a category Mod(C) consisting of models and homomorphisms between them, and it turns out that many categories of algebraic gadgets are of this form: the category of monoids, the category of groups, the category of abelian groups, and so on.
Since algebraic theories are good for studying sets with extra algebraic structure, we might hope that by categorifying, we could obtain a concept of "algebraic 2-theories" which is good for studying categories with extra algebraic structure. And it's true! In 1974, John Gray defined an "algebraic 2-theory" to be a 2-category C with finite products, all of whose objects are of the form 1, X, X2, X3, ... for some particular object X. Define a "model" of this 2-theory to be a product-preserving 2-functor F: C → Cat. And define a "homomorphism" between models to be a pseudonatural transformation between such 2-functors.
Huh? "Pseudonatural"??
Sorry, now things are getting a bit technical: the right thing going between 2-functors is not a natural transformation but something a bit weaker called a "pseudonatural transformation", where the usual commuting squares in the definition of a natural transformation are required to commute only up to certain specified 2-isomorphisms, which in turn satisfy some coherence laws described here:
6) G. Maxwell Kelly and Ross Street, Review of the elements of 2-categories, Springer Lecture Notes in Mathematics 420, Berlin, 1974, pp. 75-103.
However, you don't need to understand the details right now. There is also something going between pseudonatural transformations called a "modification", and this gives us "2-homomorphisms" between homomorphisms between models of our algebraic theory. Thanks to these there is a 2-category Mod(C) consisting of models of our 2-theory homomorphisms between those, and 2-homomorphisms between those.
Some examples might help! For example, there's a 2-theory C called the "theory of weak monoidal categories". Models of C are weak monoidal categories, homomorphisms are monoidal functors, and 2-homomorphisms are natural transformations, so Mod(C) is the usual 2-category of monoidal 2-categories. There's a similar 2-theory C' called "the theory of strict monoidal categories", for which Mod(C') is the usual 2-category of strict monoidal categories.
(Hyper-technical note for n-category mavens only: in both examples here, monoidal functors are required to preserve unit and tensor product only up to coherent natural isomorphism. This nuance is what we get from working with pseudonatural rather than natural transformations. Without this nuance, some of the stuff I'm about to say would be false.)
Now, whenever we have a product-preserving 2-functor between 2-theories, say F: C → C', we get an induced 2-functor going the other way,
F*: Mod(C') → Mod(C).
For example, there's a product-preserving 2-functor from the theory of weak monoidal categories to the theory of strict monoidal categories, and this lets us turn any strict monoidal category into a weak one.
Now in this particular example, F* is a biequivalence, which is the nice way to say that the 2-categories Mod(C) and Mod(C') are "the same" for all practical purposes. And in fact, saying that this particular F* is a biequivalence is really just an ultra-slick version of Mac Lane's theorem - the theorem we use to turn weak monoidal categories into strict ones.
Now, Mac Lane's theorem is the primordial example of a "strictification theorem" - a theorem that lets us turn "weak" algebraic structures on categories into "strict" ones, where lots of isomorphisms, like the associators in the monoidal category example, are assumed to be equations. This suggests that lots of coherence theorems can be stated by saying that 2-functors of the form F* are biequivalences.
So: is there a super-general strictification theorem where we can start from any 2-theory C and get a "strictified" version C' together with an F: C → C' such that F* is a biequivalence?
As a step in this direction, Yanofsky has cooked up a model category of algebraic 2-theories, in which F: C → C' is a weak equivalence precisely when F* is a biequivalence.
Huh? "Model category"??
Well, if you don't know what a "model category" is, you're in serious trouble now! They're a concept invented by Quillen for generalizing the heck out of homotopy theory. Try reading his book:
7) Daniel G. Quillen, Homotopical Algebra, Springer Lecture Notes in Mathematics, vol. 43, Springer, Berlin, 1967.
or for something newer:
8) Mark Hovey, Model Categories, American Mathematical Society Mathematical Surveys and Monographs, vol 63., Providence, Rhode Island, 1999.
or else:
9) Paul G. Goerss and John F. Jardine, Simplicial Homotopy Theory, Birkhauser, Boston, 1999.
(By the way, Jardine was one of the organizers of this Stanford conference, along with Gunnar Carlsson. He told me he had created a hypertext version of this book, but has not been able to get the publisher interested in it. Sad!)
Anyway, in the framework of model categories, the problem of "strictifying" an algebraic structure on categories then amounts to finding a "minimal model" of a given 2-theory C - roughly speaking, a weakly equivalent 2-theory with as little flab as possible. The concept of "minimal model" is important in homotopy theory, but apparently Yanofsky is the first to have given a general definition of this concept applicable to any model category. Yanofsky has not shown that every algebraic 2-theory admits a minimal model, but this seems like a fun and interesting question.
© 2001 John Baez
baez@math.removethis.ucr.andthis.edu
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