M&I: Electronic mail #3 of 10

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This is a message that was circulated throughout the mathematical community announcing the introduction of the Seiberg-Witten equations. You may know about it since it was written up in Science, NY Times and LA Times (front page) as well as others. Notice that the mathematical notation is written in tex format (see message #1 for a discussion of tex) - it is a convenient way for mathematicians to express complicated notation in e-mail even when they aren't sending a whole tex document.

Taubes did talk a few hours ago, and his talk was absolutely sensational. 
If Witten's idea is really correct, the Yang-Mills equations are no longer
needed to study the topology of 4-manifolds.  There are now new, and much
easier ways to prove all of Donaldson's theorems.  He gave us, for
example, a 3 minute proof of Donaldson's main theorem, in the case where
the intersection form is even (that is, you want to conclude the second
homology is zero).  Then another 3 minute proof that K squared is a
diffeomorphism invariant of algebraic surfaces.  He claimed also that the
results of Kronheimer-Mrowka could be recovered, and the Thom conjecture
proved in general.  It all depends on introducing a new equation, as

On a spin compact oriented 4-manifold X, fix a hermitian line bundle L
with chern class divisible by 2.  (More generally, one has recourse to
spin-c structures.) Let S+, S- be the spinor bundles, t the Clifford map
from S+ to S- valued 2-forms, and p projection on the self-dual part.  Then

(1) a connection A on L and
(2) a section f of S+ tensor L


(a) D(A) f = 0 where D(A) is the Dirac operator;

(b) p F(A) = , "Witten's magical equation". 

The moduli space M of gauge-equivalence classes of such pairs has expected 

   I = -3/4 signature -1/2 euler char. + 1/4 c_1(L).c_1(L)

by the usual elliptic theory.  Also, for generic metrics, M is smooth at 
nonzero f, and has the expected dimension.  But! IT IS ALWAYS COMPACT.  
This followed from a Weitzenbock argument I didn't entirely get.  Singer 
said, it's essentially an extension of Lichnerowicz.  Integrals of norm 
squares of F(A) & del f, and 4th power of f,  can be bounded in this way, 
and one uses uniform boundedness.

One then gets an invariant if I = 0 just by counting points, with sign. 
The total is well-defined up to sign, since an orientation at one point
determines it at others.  For I > 0, look at the larger moduli space N
given by dividing by the smaller gauge group which is fixed at a marked 
point.  Then N has a U(1) action with quotient M; the fixed points are 
exactly where f = 0.  But (essentially by Uhlenbeck) this can be avoided 
for generic metrics provided c_1(L).c_1(L) isn't 0.  So N is a U(1) bundle 
over M, and one gets an integer again by multiplying up its chern class 
and integrating it.  This is proved to be an invariant in the usual way.

So, proof of the Donaldson theorem: we have a spin manifold with b+ = 0, 
and want to show b- = 0.  By Rochlin's theorem, b- = 16 m for some m.
Let L be the trivial bundle, A the trivial connection, f the zero 
section.  Then I = 4m - 1, and (A,f) is the unique singular point.  Cut 
out a ball around it in N; then the boundary down on M is S^{4m-1}/S^1 = 
CP^{2m-1}, and the chern class of N restricts to a generator.  The 
integral of its top power over CP^{2m-1} is therefore 1, but since the 
chern class extends over all of M minus the singular point, this 
contradicts Stokes's theorem.


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