September 11, 1993

This Week's Finds in Mathematical Physics (Week 18)

John Baez

I will be resuming this series of articles this fall, though perhaps not at a rate of one "Week" per week, as I'll be pretty busy. For those of you who haven't seen this series before, let me explain. It's meant to be a guide to some papers, mostly in preprint form, that I have found interesting. I should emphasize that it's an utterly personal and biased selection - if more people did this sort of thing, we might get a fairer sample, but I'll be unashamed in focussing on my own obsessions, which these days lean towards quantum gravity, topological quantum field theories, knot theory, and the like.

Quite a pile of papers has built up over the summer, but I will start by describing what I did over my summer vacation:

1) Strings, loops, knots, and gauge fields, by John Baez, preprint available in LaTeX form as hep-th/9309067, 34 pages.

When I tell layfolk that I'm working on the loop representation of quantum gravity, and try to describe its relation to knot theory, I usually say that in this approach one thinks of space, not as a smoothly curved manifold (well, I try not to say "manifold"), but as a bunch of knots linked up with each other. If thy have been exposed to physics popularizations they will usually ask me at this point if I'm talking about superstring theory. To which I used to respond, somewhat annoyed, that no, it was quite different. Superstring theory, I explained, is a grandiose "theory of everything" that tries to describe all known forces and particles, and lots more, too, as being vibrating loops of string hurling around in 349-dimensional space. (Well, maybe just 10, or 26.) It is a complicated mishmash of all previous failed approaches to unifying gravity with the other forces: Yang-Mills theory, Kaluza-Klein models, strings, and supersymmetry. (The last is a symmetry principle that postulates for every particle another one, a mysterious "superpartner," despite the fact that no such superpartners have been seen.) And it has made no testable predictions as of yet. The loop representation of quantum gravity, on the other hand, is a much more conservative project. It simply attempts to use some new mathematics to reconcile two theories which both seem true, but up to now have been as immiscible as oil and water: quantum field theory, and general relativity. If it works, it will still be only the first step towards uniftying gravity with the other forces. If the questioner has the gall to ask if it has made any testable predictions, I say that so far it is essentially a mathematics project. On the one hand, here are Einstein's equations; on the other hand, here are the rules of thumb for "quantizing" some equations. Is there a consistent and elegant way of applying those rules to those equations? People have tried for 40 years or so without real success, but quite possibly they just weren't being clever enough, since the rules of thumb leave a lot of scope for creativity. Then a physicist named Ashtekar came along and reformulated Einstein's equations using some new variables (usually known by experts as the "new variables"). This made the equations look much more like those that describe the other forces in physics. This led to renewed hope that Einstein's equations might be consistently quantized after all. Then physicists named Rovelli and Smolin , working with Ashtekar, made yet another change of variables, based on the new variables. Rovelli and Smolin's variables were labelled by loops in space, so they are called the loop variables. These loops are quite unlike strings, since they are merely mathematical artifacts for playing with Einstein's equations, not actual little objects whizzing about. But using them, Rovelli and Smolin were able to quantize Einstein's equations and actually find a lot of solutions! However, they were making up a lot of new mathematics as they went along, and, as usual in theoretical physics, it wasn't 100% rigorous (which, as we know, is like the the woman who could trace her descent from William the Conqueror with only two gaps). So I, as a mathematician, got interested in this and am trying to help out and see how much of this apparently wonderful development is for real....

The odd thing is that there are a lot of mathematical connections between string theory and the loop representation. Gradually, as time went on, I became more and more convinced that maybe the layfolk were right - maybe the loop representation of quantum gravity really WAS string theory in disguise, or vice versa. This made a little embarassed by how much I had been making fun of string theory. Still, it could be a very good thing. On the one hand, the loop representation of quantum gravity is much more well-motivated from basic physical principles than string theory - it's not as baroque - a point I still adhere to. So maybe one could use it to understand string theory a lot more clearly. On the other hand, string theory really attempts to explain, not just gravity, but a whole lot more - so maybe it might help people see what the loop representation of quantum gravity has to do with the other forces and particles (if in fact it actually works).

I decided to write a paper about this, and as I did some research I was intrigued to find more and more connections between the two approaches, to the point where it is clear that while they are presently very distinct, they come from the same root, historically speaking.

Here's what I wound up saying:

The notion of a deep relationship between string theories and gauge theories is far from new. String theory first arose as a model of hadron interactions. Unfortunately this theory had a number of undesirable features; in particular, it predicted massless spin-2 particles. It was soon supplanted by quantum chromodynamics (QCD), which models the strong force by an SU(3) Yang-Mills field. However, string models continued to be popular as an approximation of the confining phase of QCD. Two quarks in a meson, for example, can be thought of as connected by a string-like flux tube in which the gauge field is concentrated, while an excitation of the gauge field alone can be thought of as a looped flux tube. This is essentially a modern reincarnation of Faraday's notion of ``field lines,'' but it can be formalized using the notion of Wilson loops. If A denotes a classical gauge field, or connection, a Wilson loop is simply the trace of the holonomy of A around a loop in space. If instead A denotes a quantized gauge field, the Wilson loop may be reinterpreted as an operator on the Hilbert space of states, and applying this operator to the vacuum state one obtains a state in which the Yang-Mills analog of the electric field flows around the loop.

In the late 1970's, Makeenko and Migdal, Nambu, Polyakov, and others attempted to derive equations of string dynamics as an approximation to the Yang-Mills equation, using Wilson loops. More recently, D. Gross and others have been able to exactly reformulate Yang-Mills theory in 2-dimensional spacetime as a string theory by writing an asymptotic series for the vacuum expectation values of Wilson loops as a sum over maps from surfaces (the string worldsheet) to spacetime. This development raises the hope that other gauge theories might also be isomorphic to string theories. For example, recent work by Witten and Periwal suggests that Chern-Simons theory in 3 dimensions is also equivalent to a string theory.

String theory eventually became popular as a theory of everything because the massless spin-2 particles it predicted could be interpreted as the gravitons one obtains by quantizing the spacetime metric perturbatively about a fixed ``background'' metric. Since string theory appears to avoid the renormalization problems in perturbative quantum gravity, it is a strong candidate for a theory unifying gravity with the other forces. However, while classical general relativity is an elegant geometrical theory relying on no background structure for its formulation, it has proved difficult to describe string theory along these lines. Typically one begins with a fixed background structure and writes down a string field theory in terms of this; only afterwards can one investigate its background independence. The clarity of a manifestly background-free approach to string theory would be highly desirable.

On the other hand, attempts to formulate Yang-Mills theory in terms of Wilson loops eventually led to a full-fledged ``loop representation'' of gauge theories, thanks to the work of Gambini, Trias, and others. After Ashtekar formulated quantum gravity as a sort of gauge theory using the ``new variables,'' Rovelli and Smolin were able to use the loop representation to study quantum gravity nonperturbatively in a manifestly background-free formalism. While superficially quite different from modern string theory, this approach to quantum gravity has many points of similarity, thanks to its common origin. In particular, it uses the device of Wilson loops to construct a space of states consisting of ``multiloop invariants,'' which assign an amplitude to any collection of loops in space. The resemblance of these states to wavefunctions of a string field theory is striking. It is natural, therefore, to ask whether the loop representation of quantum gravity might be a string theory in disguise - or vice versa.

The present paper does not attempt a definitive answer to this question. Rather, we begin by describing a general framework relating gauge theories and string theories, and then consider a variety of examples. Our treatment of examples is also meant to serve as a review of Yang-Mills theory in 2 dimensions and quantum gravity in 3 and 4 dimensions.

I should add that the sort of string theory I talk about in this paper is fairly crude compared to that which afficionados of the subject usually concern themselves with. It treats strings only as maps from a surface (the string worldsheet) into spacetime, and only cares about such maps up to diffeomorphism, i.e., smooth change of coordinates. In most modern string theory the string worldsheet is equipped with more geometrical structure (a conformal structure) - it looks locally like the complex plane, so one can talk about holomorphic functions on it and the like. This is why string theorists are always muttering about conformal field theory. But the sort of string theory that Gross and others (Taylor, Minahan, and Polychronakos, particularly) have been using to describe 2d Yang-Mills theory does not require a conformal structure on the string worldsheet, so it's at least possible that more interesting theories like 4d quantum gravity can be formulated as string theories without reference to conformal structures. (Of course, if one integrates over all conformal structures, that's a way of referring to conformal structures without actually picking one.) I guess I'm rambling on here a bit, but this is really the most mysterious point as far as I'm concerned.

One hint of what might be going on is as follows. And here, I'm afraid, I will be quite technical. As noted by Witten and formalized by Moore, Seiberg, and Crane, a rational conformal field theory gives rise to a particularly beautiful sort of category called a modular tensor category. This contains, as it were, the barest essence of the theory. Any modular tensor category gives rise in turn to a 3d topological quantum field theory - examples of which are Chern-Simons theory and quantum gravity in 3 dimensions. And Crane and Frenkel have shown (or perhaps it's fairer to say that if they ever finish their paper they will have shown) that the nicest modular tensor categories give rise to braided tensor 2-categories, which should, if there be justice, give 4d topological quantum field theories. (For more information on all these wonderful things - which no doubt seem utterly intimidating to the uninitiated - check out previous "This Week's Finds.") Quantum gravity in 4 dimensions is presumably something roughly of this sort, if it exists. So what I'm hinting at, in brief, is that a bunch of category theory may provide the links between modern string theory with its conformal fields and the loop representation of quantum gravity. This is not as outre as it may appear. The categories being discussed here are really just ways of talking about symmetries (see my stuff on categories and symmetries for more on this). As usual in physics, the clearest way to grasp the connection between two seemingly disparate problems is often by recognizing that they have the same symmetries.

© 1993 John Baez