November 3, 1994

This Week's Finds in Mathematical Physics (Week 42)

John Baez

String theory means different things to different people. The original theory of strings -- at least if I've got my history right -- was a theory of hadrons (particles interacting via the strong force). The strong force wasn't understood too well then, but in 1968 Veneziano cleverly noticed when thumbing through a math book that Euler's beta function had a lot of the properties one would expect of the formula for how hadrons scattered (the so-called S-matrix). Later, around 1970, Nambu and Goto noticed that this function would come out naturally if one thought of hadrons as different vibrational modes of a relativistic string.

This theory had problems, and eventually it was supplanted by the current theory of the strong force, involving quarks and gluons. The gluons are another way of talking about the strong force, which is a gauge field. The biggest puzzle about this approach to hadrons is, "how come we don't see quarks?" This is called the puzzle of confinement. In the late 1970's, one proposed solution was that as you pulled the quark and the antiquark in a meson apart, the strong force effectively formed an elastic "string" with constant tension. This would mean that pulling them apart took energy proportional to how far you pulled them apart. Past a certain point, the energy would be enough to create a new quark-antiquark pair and snap - the string would split into two new strings with quark and antiquark on each end. So here the "string" idea is revived but as an approximation to a theory of gauge fields. One can even try to derive approximate string equations from the equations for the strong force: the Yang-Mills equations. In my paper on strings, loops, knots and gauge fields (see week18), I gave references to some early papers on the subject:

1) QCD and the string model, by Y. Nambu, Phys. Lett. B80 (1979) 372-376.

Gauge fields as rings of glue, A. Polyakov, Nucl. Phys. B164 (1979) 171-188.

The quantum dual string wave functional in Yang-Mills theories, by J. Gervais and A. Neveu, Phys. Lett. B80 (1979), 255-258.

The interaction among dual strings as a manifestation of the gauge group, by F. Gliozzi and M. Virasoro, Nucl. Phys. B164 (1980), 141-151.

Loop-space representation and the large-N behavior of the one-plaquette Kogut-Susskind Hamiltonian, A. Jevicki, Phys. Rev. D22 (1980), 467-471.

Quantum chromodynamics as dynamics of loops, by Y. Makeenko and A. Migdal, Nucl. Phys. B188 (1981) 269-316.

Loop dynamics: asymptotic freedom and quark confinement, by Y. Makeenko and A. Migdal, Sov. J. Nucl. Phys. 33 (1981) 882-893.

These papers make very interesting reading even today. Anyone who knows particle physics will recognize most of these names! Strings were big back then. But then they went out of fashion, because the string models predicted a massless spin-2 particle --- and there's no such thing in particle physics. Later, when people were trying to cook up "theories of everything" including gravity, this flaw was again seen as a plus, since the hypothesized "graviton" meets that description.

The modern, more technical subject of string theory is a lot more fancy than these early papers. In particular, the recognition that conformal invariance was a very good thing when studying strings propagating on fixed background metric (like that of Minkowski space) pushed string theorists into a careful study of 2-dimensional conformal invariant quantum field theories. (Here the 2 dimensions refer to the surface the string traces out as it moves through spacetime.) Conformal field theory then developed a life of its own! By now it's pretty intimidating to the outsider. Mathematicians might find the following summary handy:

2) Conformal field theory, by Krzysztof Gawedzki, Seminaire Bourbaki, Asterisque 177-178 (1989), pp. 95-126.

while physicists might try

3) Introduction to Superstrings, by Michio Kaku, New York, Springer-Verlag, 1988.

String Fields, Conformal Fields, and Topology, by Michio Kaku, New York, Springer-Verlag, 1991.

Kaku's books are a decent overview but rather sketchy in spots, since they cover vast amounts of territory.

Then there is another kind of sophisticated modern string theory, "string field theory", which doesn't assume the strings are moving around on a spacetime with a background geometry. This is clearly more like what one wants to do if one is using strings to explain quantum gravity. I don't understand this nearly as well as I'd like to, but the guru on this subject is Barton Zwiebach, so if one was really gutsy one would, after a suitable warmup with Kaku, plunge in and read something like

4) Quantum background independence of closed string field theory, by Ashoke Sen and Barton Zwiebach, 60 pages, phyzzx.tex, MIT-CTP-2244, available as hep-th/9311009.

Background independent algebraic structures in closed string field theory, by Ashoke Sen and Barton Zwiebach, phyzzx.tex, MIT-CTP-2346, available as hep-th/9408053.

Unfortunately I'm not quite up to it yet....

Then, in a different direction, a bunch of folks from general relativity pursued some ideas about string and loops to the point of developing the "loop representation of quantum gravity." I'm referring to

5) Loop representation for quantum general relativity, by C. Rovelli and L. Smolin, Nucl. Phys. B331 (1990), 80-152.

though it's important to credit some of the people who kept alive the idea that one should study gauge fields as being "loops of string", or more technically, "Wilson loops":

6) Gauge dynamics in the C-representation, by R. Gambini and A. Trias, Nucl. Phys. B278 (1986) 436-448.

Now what's frustrating here is that I understand the loop representation business, but not the "background-free closed string field theory" business, even though they have the same historical roots and are both trying to deal with quantum gravity (among other forces) in a way that assumes that loops are the basic objects. Alas, the two strands speak in different languages! Heavy-duty mathematicians like Getzler, Kapranov and Stasheff know how to think about closed string fields in terms of "operads", and that stuff seems like it should be simple enough to understand, but alas, when I read it I get snowed in detail (so far).

Let me digress to mention what an "operad" is. An "operad" is basically a cool way to handle sets equipped with lots of n-ary operations. These operations might be "parametrized" in various ways. The operad elegantly keeps track of these parametrizations. So, for each n, an operad has a set X(n) which we think of as all the n-ary operations. Think of something in X(n) as a black box that has n "input" tubes and one "output" tube, or a tree-shaped thing

       \  |  /
        \ | /

with n branches and one root (here n = 3). Then suppose we have a bunch of these black boxes. Say we have something in X(n1), something in X(n2), .... and so on up to something in X(nk). Thus we've got a pile of black boxes with a total of n1 + ... + nk input tubes and k output tubes. Now if we also have a guy in X(k), which has k input tubes, we can hook up all the output tubes of all the boxes in our pile to the input tubes of this guy, to get a monstrous machine with n1 + ... + nk input tubes and one output. In short, there is an operation from X(n1) x ... x X(nk) x X(k) to X(n1 + ... + nk). For example, if we take the tree up there, which represents something in X(3), and another thing in X(3), we can hook up their outputs to the inputs of something in X(2), to get something that looks like

       \  |  /   \  |  /
        \ | /     \ | /
         \|/       \|/
          |         |
          |         |
          |         |
          \         /
           \       /
            \     / 
             \   /
              \ /

which is in X(6). The closed string field theorists like operads because there are lots of parametrized ways of gluing together Riemann surfaces with punctures together. It's a handy language, apparently... I am a bit more familiar with operads (though not much) in the context of homotopy theory, where they can be used to elegantly summarize the operations one has floating around in an infinite loop space. Very roughly, an infinite loop space is a space that looks like the space of loops of loops of loops of loops... of loops in some topological space, where you get to make the "dot dot dot" part go on as long as you want! A beautifully unpretentious and utterly readable book on these spaces, operads, and much much more, is:

7) Infinite Loop Spaces, by J. F. Adams, Princeton U. Press, Princeton, NJ, 1978.

Lest "infinite loop spaces" seem abstruse, I should emphasize that the book is really a nice tour of a lot of modern homotopy theory. As he says, "my object has been a more elementary exposition, which I hope may convey the basic ideas of the the subject in a way as nearly painless as I can make it. In this the Princeton audience encouraged me; the more I found means to omit the technical details, the more they seemed to like it." A lot of the general mathematical machinery he discusses, especially in the chapter called "Machinery", is really too nice to be left for only the homotopy theorists!

Anyway, once you have gotten the hang of operads you can try the work of a reformed homotopy theorist, Jim Stasheff, on string field theory:

8) Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli spaces, by Jim Stasheff, available as hep-th/9304061.

Actually Graeme Segal, another string theory guru, also used to do homotopy theory. He's the one who's famous for:

9) Loop groups, by Andrew Pressley and Graeme Segal, Oxford University Press, Oxford, 1986.

So it's possible that these guys didn't really quit homotopy theory, but just figured out how to get physicists interested in it. Notice all those loops! :-)

But where was I... romping through various approaches to string theory, taking a detour to mention loops, but all the while sneaking up on my goal, which is to list a few papers that lend evidence to the thesis of my paper Strings, Loops, Knots and Gauge Fields, namely that a profound "string/gauge field duality" is at work in many physical models, and that the loop representation of quantum gravity, and string theory, may eventually not be seen as so different after all.

Let's see what we've got here:

10) A reformulation of the Ponzano-Regge quantum gravity model in terms of surfaces, Junichi Iwasaki, University of Pittsburgh, 11 pages in LaTeX format available as gr-qc/9410010.

I've discussed the Ponzano-Regge model quite a bit in week16 and week38. It's an approach to quantum gravity that is especially successful in 3 dimensions, and involves chopping spacetime up into simplices. The exact partition function, as they say, can be computed using this combinatorial discrete approximation to the spacetime manifold. (In quantum field theory, when you know enough about the partition function you can compute the expectation values of observables to your heart's content.) Anyway, here Iwasaki does the kind of thing I was pointing towards in my paper, namely, to rewrite the theory, which starts out as a gauge theory, as a theory of surfaces ("string worldsheets") in spacetime.

Meanwhile, more work has been done on the same kind of idea for good old quantum chromodynamics, though here there is a background geometry, and one approximates the spacetime manifold by a discrete lattice not because one expects to get the exact answers out that way, but just because it's a decent approximation that makes things a bit more manageable:

11) Lattice QCD as a theory of interacting surfaces, by B. Rusakov, TAUP-2204-94, 12 pages in LaTeX format available as hep-th/9410004.

U(N) Gauge Theory and Lattice Strings, by Ivan K. Kostov, 26 pages, 8 figures not included, available by mail upon request, T93-079 (talk at the Workshop on string theory, gauge theory and quantum gravity, 28-29 April 1993, Trieste, Italy), available as hep-th/9308158.

Also, if there were any gauge theory that deserved to be a string theory, it's probably Chern-Simons theory, which has so much to do with knots... and indeed something like this seems to be the case, though it's all rather subtle and mysterious so far:

12) Chern-Simons-Witten theory as a topological Fermi liquid, by Michael R. Douglas, Rutgers University preprint RU-94-29, available as hep-th/9403119.

Frequently, when there is a whole lot of frenetic, sophisticated-sounding activity around a certain idea, like this relation between strings and gauge fields, there is a simple truth yearning to be known. Sometimes it takes a while! We'll see.

© 1994 John Baez