October 23, 1998

This Week's Finds in Mathematical Physics (Week 124)

John Baez

I'm just back from Tucson, where I talked a lot with my friend Minhyong Kim, who teaches at the math department of the University of Arizona. I met Minhyong in 1986 when I was a postdoc and he was a grad student at Yale. At the time, strings were all the rage. Having recently found 5 consistent superstring theories, many physicists were giddy with optimism, some even suggesting that the Theory of Everything would be completed before the turn of the century. A lot of mathematicians were going along for the ride, delighted by the beautiful and intricate mathematical infrastructure: conformal field theory, vertex operator algebras, and so on. Minhyong was considering doing his thesis on one of these topics, so we spent a lot of time talking about mathematical physics.

However, he eventually decided to work with Serge Lang on arithmetic geometry. This is a branch of algebraic geometry where you work over the integers instead of a field - especially important for Diophantine equations. Personally, I was a bit disappointed. Perhaps it was because I thought physics was more important than the decadent pleasures of pure mathematics - or perhaps it was because it made it much less likely that we'd ever collaborate on a paper.

However, a lot of the math Minhyong learned when studying string theory is also important in arithmetic geometry. An example is the theory of elliptic curves. Roughly speaking, an elliptic curve is a torus formed taking a parallelogram in the complex plane and identifying opposite edges.

You might wonder why something basically doughnut-shaped is called an elliptic curve! Let's clear that up right away. The "elliptic" part comes from a relationship to elliptic functions, which generalize the familiar trig functions from circles to ellipses. The "curve" part comes from the fact that it takes one complex number z = x+iy to describe your location on a surface with two real coordinates (x,y), so showoffs like to say that a torus is one-dimensional - one complex dimension, that is! - hence a "curve". In short, you have to already understand elliptic curves to know why the heck they're called elliptic curves.

Anyway, why are elliptic curves important? On the one hand, they show up all over number theory, like in Wiles' proof of Fermat's last theorem. On the other hand, in string theory, a string traces out a surface in spacetime called the string worldsheet, and points on this surface are conveniently described using a single complex number, so it's what those showoffs call a "curve" - and among the simplest examples are elliptic curves!

If you're interested to see how Fermat's last theorem was reduced to a problem about elliptic curves - the so-called Shimura-Taniyama-Weil conjecture - you can look at the textbooks on elliptic curves listed in "week13". But I won't say anything about this, since I don't understand it. Instead, I want to talk about how elliptic curves show up in string theory. For more on how these two applications fit together, try:

1) Yuri I. Manin, Reflections on arithmetical physics, in Conformal Invariance and String Theory, eds. Petre Dita and Vladimir Georgescu, Academic Press, 1989.

Let me just quote the beginning:

The development of theoretical physics in the last quarter of the twentieth century is guided by a very romantic system of values. Aspiring to describe fundamental processes at the Planck scale, physicists are bound to lose any direct connection with the observable world. In this social context the sophisticated mathematics emerging in string theory ceases to be only a technical tool needed to calculate some measurable effects and becomes a matter of principle.

Today at least some of us are again nurturing an ancient Platonic feeling that mathematical ideas are somehow predestined to describe the physical world, however remote from reality their origins seem to be.

From this viewpoint one should perversely expect number theory to become the most applicable branch of mathematics."

I think this remark wisely summarizes both the charm and the dangers of physics that relies more heavily on criteria of mathematical elegance than of experimental verification.

Anyway, I don't want to get too deep into the theory of elliptic curves; just enough so we see why the number 24 is so important in string theory. You may remember that bosonic string theory works best in 26 dimensions (while the physically more important superstring theory, which includes spin-1/2 particles, works best in 10). Why is this true? Well, there are various answers, but one is that if you think of the string as wiggling in the 24 directions perpendicular to its own 2-dimensional surface - two real dimensions, that is! - various magical properties of the number 24 conspire to make things work out.

What are these magical properties of the number 24? Well,

12 + 22 + 32 + ... + 242

is itself a perfect square, and 24 is the only integer with this property besides silly ones like 0 and 1. As described in "week95", this has some very profound relationships to string theory. Unfortunately, I don't know any way to deduce from this that bosonic string theory works best in 26 dimensions.

One reason bosonic string theory works best in 26 dimensions is that

1 + 2 + 3 + .... = -1/12

and 2 x 12 = 24. Of course, this explanation is unsatisfactory in many ways. First of all, you might wonder what the above equation means! Doesn't the sum diverge???

Actually this is the least unsatisfactory feature of the explanation. Although the sum diverges, you can still make sense of it. The Riemann zeta function is defined by

ζ(s) = 1-s + 2-s + 3-s + ....

whenever the real part of s is greater than 1, which makes the sum converge. But you can analytically continue it to the whole complex plane, except for a pole at 1. If you do this, you find that

ζ(-1) = -1/12.

Thus we may jokingly say that 1 + 2 + 3 + .... = -1/12. But the real point is how the zeta function shows up in string theory, and quantum field theory in general. (It's also big in number theory.)

Unfortunately, the details quickly get rather technical; one has to do some calculations and so on. That's the really unsatisfactory part. I want something that clearly relates strings and the number 24, something so simple even a child could understand it, and which, when you work out all the implications, implies that bosonic string theory only makes sense in 26 dimensions. I don't expect a child to be able to figure out all the implications... but I want the essence to be childishly simple.

Here it is. Suppose the string worldsheet is an elliptic curve. Then we can make it by taking a "lattice" of parallelograms in the complex plane:

and identifying each point in each parallelogram with the corresponding points on all the others. This rolls the plane up into a torus. Now, two lattices are more symmetrical than the rest. One of them is the square lattice:
             *     *     *     *

             *     *     *     *

             *     *     *     *
which has 4-fold rotational symmetry. The other is the lattice with lots of equilateral triangles in it:
             *       *      *      *

                 *       *      *                

             *       *      *      *

which has 6-fold rotational symmetry. The magic property of the number 24, which makes string theory work so well in 26 dimensions, is that
                       4 x 6 = 24 !!!
Okay, great. But if you're anything like me, at this point you're wondering how the heck this actually helps. Why should string theory care about these specially symmetrical lattices? And why should we multiply 4 and 6? So far everything I've said has been flashy but insubstantial. Next week I'll fill in some of the details. Of course, I'll need to turn up the sophistication level a notch or two.

In the meantime, you can read a bit more about this stuff in the following article on Richard Borcherds, who won the Fields medal for his work relating bosonic string theory, the Leech lattice in 24 dimensions, and the Monster group:

2) W. Wayt Gibbs, Monstrous moonshine is true, Scientific American, November 1998, 40-41. Also available at http://www.sciam.com/1998/1198issue/1198profile.html.

Gibbs asked me to come up with a simple explanation of the j invariant for elliptic curves; you can judge how well I succeeded. For a more detailed attempt to do the same thing, see "week66", which also has more references on the Monster group. By the way, John McKay didn't actually make his famous discovery relating the j invariant and Monster while reading a 19th-century book on elliptic modular functions; he says "It was du Val's Elliptic Functions book in which j is expanded incorrectly as a q-series - very much a 20th century book." Apart from that, the article seems accurate, as far as I can tell.

If you really want to understand how elliptic curves are related to strings, you need to learn some conformal field theory. For that, try:

3) Phillippe Di Francesco, Pierre Mathieu, and David Senechal, Conformal Field Theory, Springer, 1997.

This is a truly wonderful tour of the subject. It's 890 pages long, but it's designed to be readable by both mathematicians and physicists, so you can look at the bits you want. It starts out with a 60-page introduction to quantum field theory and a 30-page introduction to statistical mechanics. The reason is that when we perform the substitution called the "Wick transform":

it/ħ → k/T,

quantum field theory turns into statistical mechanics, and a nice Lorentzian manifold may turn into a Riemannian manifold - in other words, "spacetime" turns into "space". And this gives conformal field theory a double personality.

First, conformal field theory studies quantum field theories in 2 dimensions that are invariant under all conformal transformations - transformations that preserve angles but not necessarily lengths. These are important in string theory because we can think of them as transformations of the string worldsheet that preserve its complex structure.

Secondly, if we do a Wick transform, these quantum field theories become 2-dimensional statistical mechanics problems that are invariant under all conformal transformations. This may seem an esoteric concern, but thin films of material can often be treated as 2-dimensional for all practical purposes, and conformal invariance is typical at "critical points" - boundaries between two or more phases for which there is no latent heat, such as the boundary between the magnetized and unmagnetized phases of a ferromagnet. In 2 dimensions, one can use conformal field theory to thoroughly understand these critical points.

After this warmup, the book covers the fundamentals of conformal field theory proper, including:

By now we're at page 486. I'm getting tired just summarizing this thing!

Anyway, the book then turns to conformal field theories having Lie group symmetries: in particular, the so-called Wess-Zumino-Witten or "WZW" models. Pure mathematicians are free to join here, even amateurs, because we are now treated to a wonderful 78-page introduction to simple Lie algebras, starting from scratch and working rapidly through all sorts of fun stuff, skipping all the yucky proofs. Then we get a 54-page introduction to affine Lie algebras, which are infinite- dimensional generalizations of the simple Lie algebras, and play a crucial role in string theory. Finally, we get a detailed 143-page course on WZW models - which are basically conformal field theories where your field takes values in a Lie group - and coset models - where your field takes values in a Lie group modulo a subgroup. It sounds like all minimal models can be described as coset models, though I'm not quite sure.

Whew! Believe it or not, the authors plan a second volume! Anyway, this is a wonderful book to have around. I was just about to buy a copy in Chicago last spring - on sale for a mere $50 - when I discovered I'd lost my credit card. Sigh. The big ones always get away....

There are various formalisms for doing conformal field theory that aren't covered in the above text. For example, the theory of "vertex operator algebras", or "vertex algebras" is really popular among mathematicians studying conformal field theory and the Monster group.

The standard definition of a vertex operator algebra is long and complicated: it summarizes a lot of what you'd want a conformal field theory to be like, but it's hard to learn to love it unless you already know some other approaches to conformal field theory. There's another definition using operads that's much nicer, which will eventually catch on - some people complain that operads are too abstract, but that's just hogwash. But anyway, there is a definite need for more elementary texts on the subject. Here's one:

4) Victor Kac, Vertex Algebras for Beginners, American Mathematical Society, University Lecture Series vol. 10, 1997.

And then of course there is string theory proper. How do you learn that? There's always the bible by Green, Schwarz and Witten (see "week118"), but a lot of stuff has happened since that was written. Luckily, Joseph Polchinski has come out with a "new testament"; I haven't seen it yet but physicists say it's very good:

5) Joseph Polchinski, String Theory, 2 volumes, Cambridge U. Press, 1998.

There are also other textbooks, of course. Here's one that's free if you print it out yourself:

6) E. Kiritsis, Introduction to Superstring Theory, 244 pages, to be published by Leuven University Press, preprint available as hep-th/9709062.

For a more mathematical approach, you might want to try this when it comes out:

7) Quantum Fields and Strings: A Course for Mathematicians, eds. P. Deligne, P. Etinghof, D. Freed, L. Jeffrey, D. Kazhdan, D. Morrison and E. Witten, American Mathematical Society, to appear.

Finally, when you get sick of all this new-fangled stuff and want to read about the good old days when physicists predicted new particles that actually wound up being observed, you can turn to this book about Dirac and his work:

8) Abraham Pais, Maurice Jacob, David I. Olive, and Michael F. Atiyah, Review of Paul Dirac: The Man and His Work, Cambridge U. Press, 1998.

Also try this:

9) Michael Berry, Paul Dirac: the purest soul in physics, Physics World, February 1998, pp. 36-40.

First, and above all for Dirac, the logic that led to the theory was, although deeply sophisticated, in a sense beautifully simple. Much later, when someone asked him (as many must have done before) "How did you find the Dirac equation?" he is said to have replied "I found it beautiful." - Michael Berry

© 1998 John Baez