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I got some news today from Allen Knutson. Briefly, it appears that Witten has come up with a new way of doing Donaldson theory that is far easier than any previously known. According to Taubes, many of the main theorems in Donaldson theory should now have proofs that are 1/1000th as long!
I suppose to find this exciting one must already have some idea of what Donaldson theory is. Briefly, Donaldson theory is a theory born in the 1980s that revolutionized the study of smooth 4-dimensional manifolds by using an idea from physics, namely, the self-dual Yang-Mills equations. The Yang-Mills equations describe most of the forces we know and love (not gravity), but only in 4 dimensions can one get solutions of them of a special form, known as self-dual solutions. (In physics these self-dual solutions are known as instantons, and they were used by 't Hooft to solve a problem plaguing particle physics, called the U(1) puzzle.)
Mathematically, 4-dimensional manifolds are very different from manifolds of any other dimension! For example, one can ask whether R^n admits any smooth structure other than the usual one. (Technically, a smooth structure for a manifold is a maximal set of coordinate charts covering the manifold which have smooth transition functions. Loosely, it's a definition of what counts as a smooth function.) The answer is no - EXCEPT if n = 4, where there are uncountably many smooth structures! These "exotic R^4's" were discovered in the 1980's, and their existence was shown using the work of Donaldson using the self-dual solutions of the Yang-Mills equation, together with work of the topologist Freedman. More recently, a refined set of invariants of smooth 4-manifolds, the Donaldson invariants, have been developed using closely related ideas.
Some references are:
1) "The Geometry of Four-Manifolds," by Simon K. Donaldson and P. B. Kronheimer, Oxford University Press, Oxford, 1990.
Polynomial invariants for smooth four-manifolds, by S. K. Donaldson, Topology 29 (1990), 257-315.
"Instantons and Four-Manifolds," by Daniel S. Freed and Karen K. Uhlenbeck, Springer-Verlag, New York (1984).
"Differential Topology and Quantum Field Theory," by Charles Nash, Academic Press, London, 1991.
This is an extremely incomplete list, but it should be enough to get started. Or, while you wait for the new, simplified treatments to come out, you could make some microwave popcorn and watch the following video:
2) Geometry of four dimensional manifolds, by Simon K. Donaldson, videocassette (ca. 60 min.), color, 1/2 in, American Mathematical Society, Providence RI, 1988.
Now, what follows is my interpretation of David Dror Ben-Zvi's comments on a lecture by Clifford Taubes entitled "Witten's Magical Equation", these comments being kindly passed on to me by Knutson. I have tried to flesh out and make sense of what I received, and this required some work, and I may have screwed up some things. Please take it all with a grain of salt. I only hope it gives some of the flavor of what's going on!
So, we start with a compact oriented 4-manifold X with L a complex line bundle over X having first Chern class equal to w2, the second Stiefel-Whitney class of TX, modulo 2. If X is spin (meaning that the w2 = 0), take the bundle of spinors over X. Otherwise, pick a Spin-c bundle and take the bundle of complex spinors over X. Note that Spin-c structure is enough to define complex spinors on X, and it will always exist if w2 is the mod 2 reduction of an integral characteristic class. For more on this sort of stuff, try:
3) "Spin Geometry," by H. Blaine Lawson, Jr. and Marie-Louise Michelson, Princeton U. Press, Princeton, 1989.
In either case, take our bundle of spinors, tensor it with the square root of L, and call the resulting bundle B. (Perhaps someone can explain to me why L has a square root here; it's obvious if X is spin, but I don't understand the other case so well.) The data for our construction are now a connection A on L, and a section Ψ of the self-dual part of B. (Note: I'm not sure what the "self-dual part of B" is supposed to mean. I guess it is something required to make the right-hand side of the formula below be self-dual in the indices a,b.) Consider now two equations. The first is the Dirac equation for Ψ. The second is that the self-dual part F^+ of the curvature of A be given in coordinates as
F^+_{ab} = -1/2 <Ψ, e^a e^b Ψ>
where the basis 1-forms e^a, e^b act on Ψ by Clifford multiplication.
Next form the moduli space M of solutions (A, Ψ) modulo the action of the automorphisms of L. The wonderful fact is that this moduli space is always compact, and for generic metrics it's a smooth manifold. Still more wonderfully (here I read the lines between what was written), it is a kind of substitute for the moduli space normally used in Donaldson theory, namely the moduli space of instantons. It is much nicer in that it lacks the singularities characteristic of the other space.
What this means is that everything becomes easy! Apparently Taubes, Kronheimer, Mrowka, Fintushel, Stern and the other bigshots of Donaldson theory are frenziedly turning out new results even as I type these lines. On the one hand, the drastic simplifications are a bit embarassing, since the technical complications of Donaldson theory were the stuff of many erudite and difficult papers. On the other hand, Donaldson invariants were always notoriously difficult to compute. Taubes predicted that a purely combinatorial formula for them may be around within a year. (Here it is interesting to note the work of Crane, Frenkel, and Yetter in that direction; see "week2" and "week38".) This is sure to lead to a deeper understanding of 4-dimensional topology, and quite possibly, 4-dimensional physics as well.
© 1994 John Baez
baez@math.removethis.ucr.andthis.edu
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