Date: Wed, 2 Nov 94 19:26:53 EST From: benzvi (David Dror Ben-Zvi) Return-Receipt-To: benzvi (David Dror Ben-Zvi) To: T.Tokieda@newton.ac.cam.uk, benzvi, aknaton@math.mit.edu, burchard@math.utah.edu, memerton Subject: The end of Donaldson Theory IT's official, and coming out of the mouth of Taubes himself (though paraphrased): Donaldson theory is over. In his talk today entitled Witten's Magical Equation, Taubes outlined how an equation proposed by Witten as an afterthought at the end of his recent MIT talk on Donaldson theory and N=2 supersymmetric QFT, where he conjectured it should give the Donaldson invariatns, essentially trivializes all of the work done in Donaldson theory, from the start to the recent Kronheimer-Mrowka stuff to a whole slew of problems solved in the last couple of weeks, including the Thom conjecture on CP2. As all of you understand this stuff better than me, excuse me for the goofs and I'll be sparing on detail -it'll all be completely revised and different within a week anyway. In any case, Witten gave an equation in terms of very classical information (basically just spinors and line bundles), which gives rise to a new set of invariants, which he conjectures agree with the Donaldson ones. So far this has not been proven generally but is known to agree for all known four-manifolds. In any case, regardless of the relation to Donaldson invariatnts, these Witten invariants give trivial proofs (several of which he outlined in the talk) of the major theorems of Donaldson theory, with, as Taubes said, about 1/1000 of the length. The setup is you have X a compact 4d oriented surface with L a complex line bundle, c1(L) mod 2 = w2(TX) - the chern class of L agrees with the StiefelWhitney clas of TX mod 2. If X is spin (w2=0) consider also the spinor bundle, otherwqise we take a spin-c bundle - principal fiber bundle for (Spin 4 x S1)/Z2 instead of sp(4)=su2 x su2. We then tensor this bundle with the square root of L, and take this as the auxilliary bundle for our construction. The data are a connection A on L, and a section Psi of the self-dual part of the auxilliary bundle. There are two equations : one is that Psi be a harmonic spinor (the Dirac equation for Psi, DPsi=0). The second is that the self-dual projection of the curvature of A be given in coordinates a,b as (p*F)(a,b)=-1/2 with e^a and e^b being Clifford multiplication acting on our section Psi.. OK I don't understand this equation so excuse me for the vagueness.. Our moduli space will be pairs A, Psi solving these equations mod the action of Aut L on A. (note that our bundles and all our fixed, and everything is just in terms of spinors and line bundles..) The linearization of our equation, whose solutions give the (formal) tangent space to M, give an elliptic Fredholm operator, a very nice and benign creature. FOR A GENERIC METRIC (all of this is done with choosing some arbitrary metric, necessary to define the Dirac operator etc.) our moduli space will be smooth, with dimension given by the index of this operator (i.e. dim coker=0).... Wait there's more - M is always compact!!! this and the nice linear Fredholm stuff make this infinitely better than Donaldson theory - compact so no bubbling off, no hard analysis, PLUS things are really easy to compute.. Furthermore there are only finitely many L (up to iso) with Ind>=0 solutions.. all these proofs were done essentially in lecture using a simple Weitzenbok-Bochner type formula - DPsi=0 =>integral of norm^2 DPsi =0, rewrite as some inner product, use Clifford multiplication and other equation to get integral (Norm GradPsi^2+scalar curvature time norm^2 Psi + some other (positive) curvature terms =0.. in particular no solutions for scalar curvature >=0, or if norm psi is too big - this gives uniform estimates for psi, its derivatives etc => compactnes. Examples include when I=0 where one simply counts (with orientation) the number of points of M to get an invariant.... Because of this elliptic Fredholm stuff, everything behaves as if it's finite dimensional.. and everything becomes easy! e.g. Taubes conjectured that within a year there will be a combinatorial approach to Donaldson theory. He then went on to give quick and easy proofs of a bunch of Donaldson's theorems.. There was also a LOT of historical/philosophical talk (mainly between him and Singer and Bott -- all of the geometrs at HArvard and MIT were there) about why this wasn't discovered in the 1960's and how one should give proper credit to Donaldson theory because no-one would have thought of this before as possibly leading to something interesting + the experience with really hard nonlinear gauge theory (here the geroup is U(1)..) made this all trivial and easy to do.. in the past couple of weeks a frenzied collection of results has been pouring through by Taubes, Kronheimer, Mrowka, Fintushel, Stern etc. though no one wants to take too much credit for these theorems- all a bit too trivial! Within a very very short time the whole 4-manifold scene is expected to look completely different. They were also complaining that some young person should have stumbled on it and become really famous - they can't give Witten yet another Fields medal! This definitely seems like one of Witten's greater contributions to math.. Taubes was totally in awe of him. Anyway, I'll let you all have it sink in.. but this is a rather momentous occasion! Anyway later, and don't forget to keep me informed! David