
Last weekend we had a conference on Knots and Quantum Gravity here at Riverside. I will briefly describe the talks, many of which will eventually appear in a conference proceedings volume. I think that to be nice I will list these talks in order of how technical my descriptions will be, rather than chronologically.
Oleg Viro spoke on "simplicial topological quantum field theories". There has been a lot of interest recently in constructing topological quantum field theories using triangulations of manifolds. The most famous of these is due to Turaev and Viro. Witten showed that this one is the same as quantum gravity in 2+1 dimensions. The nice thing is that this gives an alternate description of the TuraevViro theory in terms of more traditional ideas from field theory, so the same theory has a "discrete" and a "continuum" formulation  some evidence for my notion that quantum gravity will resolve the old "is space continuous or discrete" argument by saying "both, and neither," just as quantum mechanics resolved the old "is light a wave or a particle" dispute! (Hegel would've loved it.)
When constructing simplicial topological quantum field theories, one has to prove that the answer you get is independent of the triangulation. Viro reviewed a couple sets of "moves" whereby one can get between any two triangulations of the same manifold  the Alexander moves, and the Pachner moves. He also discussed an alternate, and more convenient, way of describing manifolds by "special spines". Here the idea is as follows. Pick a bunch of points in the manifold. From each one, start blowing a little bubble, which grows bigger and bigger until it bumps into the other bubbles. The result is something very much like a foam of soap bubbles, which generically have polyhedral faces, with edges and vertices of a special sort. Look at a mess of foam sometime if you don't know what I mean: in 3 dimensions, three bubbles meet at an edge, and four at a vertex. One can describe this situation purely combinatorially, and it contains all the information about the manifold. There are a certain set of moves, the Matveev moves, relating any two such "special spines" for the same manifold. One can figure out what these moves might be by staring some foam and watching how the bubbles move.
Louis Kaufmann changed his talk from the announced subject and instead talked about Vassiliev invariants of knots and their relation to perturbative ChernSimons theory. Let me just recall what all this is about. ChernSimons theory is a TQFT (topological quantum field theory) in 3 dimensions in which the field is a connection. In physical terms, a connection is just a generalization of the vector potential in electromagnetism. Recall that in 3 dimensional space, the vector potential is a vector field A whose curl is the magnetic field. In quantum theory, the significance of the vector potential is as follows. If we take a particle and carry it around a loop, its wavefunction gets multiplied by a phase, that is, a complex number of absolute value 1. These "phases" form a group, since the product of two phases is a phase. This group is called U(1), since we can think of phases as 1x1 unitary matrices. A key idea in modern physics is to generalize the heck out of electromagnetism by allowing other groups to play the role of phases. The group we choose is called the "gauge group." The second simplest choice after U(1) is SU(2), the 2x2 unitary matrices with determinant = 1.
You can do ChernSimons theory with any gauge group but it's especially simple with gauge group SU(2). An SU(2) connection is just a kind of field that lets one do "parallel translation" around a loop in space and get an element of SU(2). Mathematicians call this the holonomy of the connection around the loop. Physicists typically take the trace of the group element (in this case, just the sum of the diagonal entries of the 2x2 matrix), and call that the "Wilson loop observable," a function of the connection that depends on the loop.
Now the great thing about ChernSimons theory is that the theory is independent of any choice of coordinates or background metric on spacetime. This is part of what we mean by saying the theory is a TQFT. Another aspect of a TQFT is that there is a "vacuum state" and we can calculate the expectation value of a Wilson loop in the vacuum state. The idea one should have is that the connection is undergoing all sorts of "quantum fluctuations" in the vacuum state, but that we can ask for the average value for the trace of the connection of the holonomy around a loop in the vacuum state. Given a knot K, we write this expectation value as <K>. Now the great thing about ChernSimons theory is that the vacuum state does not care what coordinates you use to describe it. Thus <K> does not depend on the geometry of the knot K (which would take coordinates or a metric to describe), but only on its topology. In other words, <K> is a knot invariant. In fact we can define <K> not just for knots, but also for links (bunches of knots, possibly intertangled), by taking the expectation value of the product of the Wilson loops, one for each knot. So ChernSimons theory really gives a link invariant. Witten showed that this link invariant is just the Kauffman bracket, which is an invariant easily calculated using the rules:
Rule 1: If K is the ``empty link,'' the link with NO components whatsoever  i.e., just the empty set  we have
<K> = 1.
This is sort of a normalization rule.
Rule 2: If K' is obtained from K by adding an unlinked copy of the unknot (an unknotted circle) to K,
<K'> = (a^2 + a^{2})<K>.
Here a is an adjustable parameter that appears in ChernSimons theory  a function of the coupling constant.
Rule 3: Suppose K, L, and L' are 3 knots or links differing at just one crossing (we're supposing them to be drawn as pictures in 2 dimensions). And suppose at this crossing they look as follows.
K looks like:
\ / \ / / / \ / \
L looks like:
       
and L' looks like
\ / \_/
_ / \ / \
Any rotated version of this picture is fine too.
Then
<K> = a<L> + a^{1}<L'>.
That's it! I leave as an exercise for the reader to calculate <K> for the trefoil knot,
________________________ / _______________ \ \ / \  \ /   /   / \   / \   \ /   \ /   /   / \   / \   \ /   \ /   /   / \   / \________________/  \_________________________/
and get A^5  A^{3}  A^{7}. Then try the mirrorimage trefoil, or "lefthanded trefoil," and see what you get.
Now in quantum field theory people like doing perturbative calculations, and that's interesting here even though we know the exact answer. Namely, there is a coupling constant c in ChernSimons theory such that A = exp(c), and if one uses Feynman diagrams and the rest of the usual machinery for quantum field theory and does a perturbative calculation of the vacuum expectation value of Wilson loops, one gets the same answer, but as a power series in c. The coefficient of the c^n is a special sort of link invariant called a "Vassiliev invariant" of degree n. I discussed these a lot in "week3" (see below for how to get that article), so I won't repeat myself here. In any event, Kauffman gave a nice discussion of this sort of thing.
Viktor Ginzburg talked about his work with Milgram on Vassiliev invariants. He had hoped to show that these were an almost complete set of knot invariants, able to distinguish between any two knots that weren't just orientationreversed versions of each other (here we equip the knots with an orientation, or field of arrows running along the knot). He came to the conference, as he said, sadder and wiser. He presented a nice result on Vassiliev invariants that might be a step towards proving completeness.
Dana Fine spoke on "ChernSimons theory and the WessZuminoWitten model". There is a very interesting "ladder of field theories" that contains "topological quantum gravity" in 4 dimensions, ChernSimons theory in 3 dimensions, and the WessZuminowitten model in 2 dimensions. Dana Fine spoke on the bottom 2 rungs of this ladder. he described a very explicit, although still formal, reduction of the ChernSimons path integral (the integral one does to compute the expectation values I mentioned above) to the path integral in the WessZuminoWitten model. The relation between CS theory and the WZW model is what witten used in his original argument that CS theory gives interesting link invariants, so this is of interest in knot theory as well as physics.
On Saturday morning we had a nice trio of talks from the Syracuse gang, Syracuse University being a hotbed of new work on quantum gravity. Ashtekar and Smolin are there, as are a bunch of good grad students (Bernd Bruegmann was there until very recently) and postdocs, including Jerzy Lewandowski and Renate Loll. The whole gang is moving down to Penn State this summer, and they will be hiring Jorge Pullin, now at Utah State. There are not many people working on quantum gravity  and not many jobs in the field  so Penn State will become arguably the center, at least in the US (let us not forget Penrose, Hawking, Isham, et al!), for work on this subject.
Renate Loll spoke on the "Loop representation of gauge theory and gravity." This was an introduction to the ideas of Gambini, Trias, Rovelli, Smolin, et al on doing quantum field theory solely in terms of Wilson loops. It is this approach that makes the connection between quantum gravity and knot theory.
Abhay Ashtekar spoke on "Loop transforms." The process of encoding a connection in terms of all its Wilson loops is called the loop transform, and it can be regarded as a nonlinear generalization of the Fourier transform. Ashtekar has led an effort to make this transform thoroughly rigorous and mathematically respectable, and he discussed this work.
Jorge Pullin spoke on "The quantum Einstein equations and the Jones polynomial". He outlined his work with Gambini and Bruegmann in which they show perturbatively that the Kaufmann bracket (or, alternatively, ChernSimons theory) gives a state of 4d quantum gravity. This is perhaps the most exciting aspect of the "ladder of field theories" mentioned above, since it touches upon the real world  or at least comes darn close.
On Sunday, Gerald Johnson started things off with an introduction to his work on making the Feynman path integral rigorous. This is relevant because a main problem with Witten's otherwise marvelous work is that the path integral in ChernSimons theory has not been made rigorous. Dana Fine's talk offered one approach, and my own talk offered another (based on my recent paper).
Perhaps the most novel talk was by Paolo CottaRamusino ("4d quantum gravity and knot theory") describing his work with Maurizio Martellini on 4dimensional TQFTs and invariants of 2knots, that is, embedded surfaces in R^4 (or more general 4manifolds). This is an attempt to push the Wilson loop story up one dimension, in an effort to make it applicable to theories similar to quantum gravity. These theories are the socalled "BF theories," whose Lagrangian is of the form tr(B wedge F), where B is a Lie algebra valued 2form and F is the curvature of a connection. Martellini and CottaRamusino's work on this is still in a preliminary stage but it seems rather promising.
Perhaps the most controversial talk was by Louis Crane, entitle "Quantum gravity, spin geometry and categorical physics." This was about his ideas on using category theory to construct 4dimensional TQFTs. He also emphasized the importance of TQFTs that use triangulations but wind up being independent of the triangulation, thus slipping through the discrete/continuous distinction. Many of his assertions provoked violent reactions from the physicists present.
Finally, I spoke on "Strings, tangles and gauge fields," beginning by pointing out some relationships between closed string field theory and the loop representation of quantum gravity, and then retreating to safer ground and describing my work on trying to make the ChernSimons path integral and the loop representation more mathematically rigorous. I will write a paper on this subject this summer and try to further build up my case for the conjecture that string theory and gauge field theory are in a sense dual descriptions of certain TQFTs.
A rather technical introduction to currently interesting topics in closed string field theory has just appeared:
1) Closed string field theory  an introduction, by Barton Zwiebach, preprint available as hepth/9305026 (requires the phyzzx macros to print; these macros are also available from hepth; see below).
Unfortunately for me, he mainly treats theories of strings moving around on a manifold with a background metric, while it seems that the loop representation of quantum gravity is very like a "backgroundfree" string field theory. A paper that recently came out and appears to support my notions is the following:
2) Twodimensional YangMills Theories Are String Theories, by S.G. Naculich, H.A. Riggs, and H.J. Schnitzer, 14 pages, preprint available in latex form as hepth/9305097.
Apparently this builds on work by Gross, Taylor, and Minahan which treated SU(n) YangMills theories in 2 dimensions as string theories, and does something similar for the gauge groups SO(n) and Sp(n).
I have a pack of interesting papers to describe but I am already worn out, so I will put that off until next week, except for the following paper by Smolin that I seem to have neglected:
3) What can we learn from the study of nonperturbative quantum general relativity? by Lee Smolin, 37 pages, preprint available in latex form as grqc/9211019.
This is a nice introduction to current issues associated to the loop representation of quantum gravity and nonperturbative quantum gravity in general. As should be evident from my weekly reviews, the subject seems to be moving faster and faster, and it is best to read some of the review papers like this one by Smolin if one wants to figure out where things are now and where they might be heading.
© 1993 John Baez
baez@math.removethis.ucr.andthis.edu
