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Here's this week's reading material. The first test will be in two weeks. :-)
1) On the Vassiliev Knot Invariants by Dror Bar-Natan, Harvard University ``pre-preprint.''
I went to UC San Diego this week to give a talk, and the timing was nice, because Dror Bar-Natan was there. He is a student of Witten who has started from Witten's ideas relating knot theory and quantum field theory and developed them into a beautiful picture that shows how knot theory, the theory of classical Lie algebras, and abstract Feynman diagrams are three faces of the same thing. To put it boldly, in a deliberately exaggerated form, Bar-Natan has proposed a conjecture saying that knot theory and the theory of classical Lie algebras are one and the same!
This won't seem very exciting if you don't know what a classical Lie algebra is. Let me give a brief and very sketchy introduction, apologizing in advance to all the experts for the terrible sins I will commit, such as failing to distinguish between complex and real Lie algebras.
Well, remember that a Lie algebra is just a vector space equipped with a "bracket" such that the bracket [x,y] of any two vectors x and y is again a vector, and such that the following hold:
a) skew-symmetry: [x,y] = -[y,x]. b) bilinearity: [x,ay] = a[x,y], [x,y+z] = [x,y] + [x,z]. (a is a number.) c) Jacobi identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.
The best known example is good old R^3 with the cross product as the bracket. But the real importance of Lie algebras is that one can get one from any Lie group - roughly speaking, a group that's also a manifold, and such that the group operations are smooth maps. And the importance of Lie groups is that they are what crop up as the groups of symmetries in physics. The Lie algebra is essentially the "infinitesimal version" of the corresponding Lie group, as anyone has seen who has taken physics and seen the relation between the group of rotations in R^3 and the cross product. Here the group is called SO(3) and the Lie algebra is called so(3). (So R^3 with its cross product is called so(3).) One can generalize this to any number of dimensions, letting SO(n) denote the group of rotations in R^n and so(n) the corresponding Lie algebra. (However, so(n) is not isomorphic to R^n except for n = 3, so there is something very special about three dimensions.)
Similarly, if one uses complex numbers instead of real numbers, one gets a group SU(n) and Lie algebra su(n). And if one looks at the symmetries of a 2n-dimensional classical phase space - so-called canonical transformations, or symplectic transformations - one gets the group Sp(n) and Lie algebra sp(n). To be precise, SO(n) consists of all nxn orthogonal real matrices with determinant 1, SU(n) consists of all nxn unitary complex matrices with determinant 1, and Sp(n) consists of all (2n)x(2n) real matrices preserving a nondegenerate skew-symmetric form.
These are all very important in physics. Indeed, all the "gauge groups" of physics are Lie groups of a certain sort, so-called compact Lie groups, and in the standard model all the forces are symmetrical under some gauge group or other. Electromagnetism a la Maxwell is symmetric under the group U(1) of complex numbers of unit magnitude, or "phases". The electroweak force (unified electromagnetism and weak force) is symmetric under U(1) x SU(2), where one uses the fact that one can build up bigger semisimple Lie groups as direct sums (also called products) of smaller ones. The gauge group for the strong force is SU(3). And, finally, the gauge group of the whole standard model is simply U(1) x SU(2) x SU(3), which results from lumping the electroweak and strong gauge groups together. This direct sum business also works for the Lie algebras, so the Lie algebra relevant to the standard model is written u(1) x su(2) x su(3).
There are certain very special Lie algebras called simple Lie algebras which play the role of "elementary building blocks" in the world of Lie algebras. They cannot be written as the direct sum of other Lie algebras (and in fact there is an even stronger sense in which they cannot be decomposed). On the other hand, the Lie algebra of any compact Lie group is a direct sum of simple Lie algebras and copies of u(1) - the one-dimensional Lie algebra with zero Lie bracket which, for technical reasons, people don't call "simple".
These simple Lie algebras were classified by the monumental work of Killing, Cartan and others, and the classification is strikingly simple: there are infinite series of "classical" Lie algebras of type su(n), so(n), and sp(n), and five "exceptional" Lie algebras called G_2, F_4, E_6, E_7, and E_8. Believe it or not, there is a deep connection between the exceptional Lie algebras and the Platonic solids. But that is another story, one I barely know....
Now, Witten showed how one could use quantum field theory to constuct an invariant of knots, or even links, corresponding to any representation of a compact Lie group. (You won't even need to know what a representation is to understand what follows.) This had been done in a different way, in terms of "quantum groups," by Reshetikhin and Turaev (following up on work by many other people). These invariants are polynomials in a variable q (for "quantum"), and if one writes q as e^ħ and expands a power series in ħ, the coefficient of ħ^n is a "Vassiliev invariant of degree n". Recall from last week that given an invariant of oriented knots, one can extend it to knot with arbitrarily many nice crossings by setting the value of the invariant on a knot with a crossing like
\ / \/ /\ / \
to be the invariant of the knot with the crossing changed to
\ / \ / / / \ / \
minus the invariant of the knot with the crossing changed to
\ / \ / \ / \ / \
(Again, the knot has to be oriented for this rule to make sense, and the strands shown in the pictures above should be pointing downwards.) Having made this extension, one says a knot invariant is a Vassiliev invariant of degree n if it vanishes on all knot with n+1 or more double points.
This is where Dror stepped in, roughly. First of all, he showed that the Vassiliev invariant of degree n is just what you get when you do Witten's quantum-field-theoretic calculations perturbatively using Feynman diagrams and look at the terms of order n in Planck's constant, ħ! Secondly, and more surprisingly, he developed a bunch of relationships between Feynman diagrams and pictures of knots! The third and most amazing thing he did takes a bit longer to explain...
Roughly, he showed that any Vassiliev invariant of degree n is determined by some combinatorial data called a "weight system." He showed that any representation of a Lie algebra determines a weight system and hence a Vassiliev invariant. But the really interesting thing he showed is that many of the things one can do for Lie algebras can be done for arbitrary weight systems. This makes it plausible that EVERY weight system, hence every Vassiliev invariant, comes from a representation of a simple Lie algebra. In fact, Dror conjectures that every Vassiliev invariant comes from a representation of a classical simple Lie algebra. Now there is another conjecture floating around these days, namely that Vassiliev invariants almost form a complete set - that is, that if two knots cannot be distinguished by any Vassiliev invariants, they must either be the same or differ simply by reversing the orientation of all the strands. If BOTH these conjectures are true, one has in some sense practically reduced the theory of knots to the theory of the classical Lie algebras! This wouldn't mean that all of sudden we know the answer to every question about knots, but it would certainly help a lot, and more importantly, in my opinion, it would show that the connection between topology and the theory of Lie algebras is far more profound than we really understand. The ramifications for physics, as I hope all my chatting about knots, gauge theories and quantum gravity makes clear, might also be profound.
Well, we certainly don't understand all this stuff yet, since we don't know how to prove these conjectures! But Dror's conjecture - that all weight systems come from representations of simple Lie algebras - is tantalizingly close to being within grasp, since he has reduced it to a fairly elementary combinatorial problem, which I will now state. Note that "elementary" does not mean easy to solve! Just easy to state.
Before I state the combinatorial problem, let me say something about the evidence for the conjecture that all Vassiliev invariants come from representations of classical Lie algebras. In addition to all sorts of "technical" evidence, Dror has shown the conjecture is true for Vassiliev invariants of degree <= 9 by means of many hours of computation using his Sparcstation. In fact, he said in his talk that he felt guilty about having a Sparcstation unless it was always computing something, and that even as he spoke his computer was busily verifying the conjecture for higher degrees. (I suggested that it was the Sparcstation that should feel guilty when it was not working, not him.) He also advertised that his programs, a mixture of C and Mathematica code, are available by anonymous ftp from math.harvard. Use user name "ftp", go to the directory "dror". You folks with Crays should feel VERY guilty if they are just sitting there and not helping Dror verify this important conjecture. (I suggest that you first read his papers and the file README in his directory, then check out his programs, and then ask him where he's at and what would be worth doing. Please don't pester him unless you are a good enough mathematician to discuss this stuff intelligently and have megaflops to burn. If you want to make a fool of yourself, don't say I sent you.)
Okay, with no further ado, here's the conjecture in its elementary combinatorial form. Let B be the vector space spanned by finite graphs with univalent and "oriented" trivalent vertices, modulo some relations... first of all, a trivalent vertex is "oriented" if there is a cyclic ordering of the three incident edges. That is, we "orient" the vertex
\ /
\ /
\ /
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by drawing a little clockwise or counterclockwise-pointing circle at the vertex. (Or, for those of an algebraic bent, label the edges by 1,2,3 but then mod out by cyclic permutations.) The relations are: 1) if we reverse the orientation of a trivalent vertex, that's equivalent to multiplying the graph by -1. (Remember we're in a vector space spanned by graphs.) 2)
------ | | \ / | = |____| - \/ | | | /\ ------ | | / \
(That is, we can make this substitution anywhere we want; these pictures might be part of a bigger graph. Note that the "X" is not a vertex, since there aren't quadrivalent vertices; it's just one edge going over or under another. It doesn't matter whether it goes over or under since these are abstract graphs, not graphs embedded in space.)
Now, let B_m be the vector space spanned by "labelled" finite graphs with univalent and oriented trivalent vertices, modulo some relations... but first I have to say what "labelled" means. It means that each edge is labelled with a 1 or -1. The relations are: 1) if we reverse the orientation of a trivalent vertex, it's the same as multiplying the labellings of all three incident edges by -1. 2)
------ | |
| = |____|
| | |
------ | |
if the internal edge is labelled with a 1. (Here the 4 external edges can have any labellings and we don't mess with that.)
Now, define a linear map from B to B_m by mapping any graph to the signed sum of the 2^{number of edges} ways of labelling the edges with -1 or -1. Symbolically,
1 -1
-------- → --------- - --------- .
Of course, one must work a bit to show this map is well-defined. (This just takes a paragraph - see Proposition 6.5 of Dror's paper.)
Okay, the conjecture is:
THIS MAP IS ONE-TO-ONE.
If you can solve it, you've made great progress in showing that knots and classical Lie groups are just two aspects of the same branch of mathematics. Don't work on it, though, until you get Dror's paper and make sure I stated it exactly right!!!!!
2) Mathematical problems of non-perturbative quantum general relativity, by Abhay Ashtekar, lectures delivered at the 1992 Les Houches summer school on Gravitation and Quantization, December 2, 1992, available as gr-qc/9302024.
This is a good overview of the loop variables approach to quantizing general relativity as it currently stands. It begins with a review of the basic difficulties with quantizing gravity, as viewed from three perspectives: the particle physicist, the mathematical physicist, and the general relativist. Technically, a main problem is that general relativity consists of both evolution equations and constraint equations on the initial data (which are roughly the metric of space at a given time and its first time derivative, or really "extrinsic curvature"). So Ashtekar reviews Dirac's ideas on quantizing constrained systems before sketching how this program is carried out for general relativity.
Then he considers a "toy model" - quantum gravity in 2+1 dimensions. This is a funny theory because classically Einstein's equations in 2+1 dimensions simply say that spacetime is flat (in a vacuum)! No gravitational waves exist as in 3+1 dimensions, and one can say that the information in the gravitational field is "purely global" - locally, everywhere looks the same as everywhere else (like Iowa), but there may be global "twists" that you notice when going around a noncontractible loop. There has been a lot of work on 2+1 gravity recently - in a sense this problem has been solved, by a number of methods - and this allows one to understand some of the conceptual difficulties of honest 3+1-dimensional quantum gravity without getting caught in an endless net of technical complications.
Then Ashtekar jumps back to 3+1 dimensions and gives a more thorough introduction to the loop variables approach. He ends by going through some of the many open problems and possible ways to attack them.
I have worn myself out trying to do justice to Bar-Natan's work, so I will postpone until next week a review of Kapranov and Voevodsky's paper on 2-categories.
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
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