Subject: This Week's Finds 45 - Donaldson Theory Update
From: [email-masked] (John Baez)
Newsgroups: sci.physics.research,sci.physics,sci.math

This Week's Finds in Mathematical Physics (Week 45)
John Baez

In the previous edition of "This Week's Finds" I mentioned a burst of
recent work on Donaldson theory. I provocatively titled it "The End of
Donaldson Theory?", since the rumors I was hearing tended to be phrased
in such terms.  But I hope I made it clear at the conclusion of the
article that this recent work should lead to a lot of *new* results in
4-dimensional topology!  An example is Kronheimer and Mrowka's proof of
the Thom conjecture.  

Many thanks to my network of spies for obtaining a preprint of the
following paper:

1) The genus of embedded surfaces in the projective plane, by 
P. B. Kronheimer and T. S. Mrowka, 10 pages.

Let me simply quote the beginning of the paper:

"The genus of a smooth algebraic curve of degree d in CP^2 is given by
the formula g = (d-1)(d-2)/2.  A conjecture sometimes attributed to Thom
states that the genus of the algebraic curve is a lower bound for the
genus of any smooth 2-manifold representing the same homology class. The
conjecture has previously been proved for d <= 4 and for d =6, and less
sharp lower bounds for the genus are known for all degrees [references
omitted].  In this note we confirm the conjecture.
 
Theorem 1.  Let S be an oriented 2-manifold smoothly embedded in CP^2 so
as to represent the same homology class as an algebraic curve of degree
d.  Then the genus g of S satisfies g >= (d-1)(d-2)/2.

Very recently, Seiberg and Witten [references below] introduced new
invariants of 4-manifolds, closely related to Donaldson's polynomial
invariants [reference omitted], but in many respects much simpler to
work with.  The new techniques have led to more elementary proofs of
many theorems in the area.  Given the monopole equation and the
vanishing theorem which holds when the scalar curvature is positive
(something which was pointed out by Witten), the rest of the argument
presented here is not hard to come by.  A slightly different proof of
the Theorem, based on the same techniques, has been found by Morgan,
Szabo and Taubes."

The reference to Donaldson's polynomial invariants appears in "week44".
The references to the new Seiberg-Witten invariants are:

2) Monopoles and four-manifolds, by Edward Witten, in preparation.  
 
Electric-magnetic duality, monopole condensation, and confinement in N=2
supersymmetric Yang-Mills theory, by Edward Witten and Nathan Seiberg,
45 pages, available as hep-th/9407087.

Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric
QCD, by Edward Witten and Nathan Seiberg, 89 pages, available as
hep-th/9408099.  

Differential geometers attempting to read the second two papers will
find that they contain no instance of the term "Donaldson theory", and
they may be frustrated to find that these are very much *physics* papers.
They concern the ground states of supersymmetric Yang-Mills theory in 4
dimensions with gauge group SU(2).  The "ground states" of a field
theory are its least-energy states, which represent candidates for the
physical vacuum.  In certain theories there is not a unique ground
state, but instead a "moduli space" of ground states.  Seiberg and
Witten study these moduli spaces of ground states in both the classical
and quantum versions of SU(2) supersymmetric Yang-Mills theory in 4
dimensions.  They also consider the theory coupled to spinor fields,
which they call "quarks", using the analogy of the theory to quantum
chromodynamics, aka "QCD".

I haven't had time to go through their papers, since this isn't my main
focus of interest.  Perhaps the most useful thing I can do at this point
is to use Kronheimer and Mrowka's clear description of their moduli
space (which is presumably closely related to Seiberg and Witten's
moduli spaces) to simplify and fill in the holes of what I wrote in
"week44".  I will aim my exposition to mathematicians, but make some
elementary digressions on physics to spice things up.

We start with a compact oriented Riemannian 4-manifold X, and assume we
are given Spin-c structure on X.  Recall the meaning of this.  First,
the orthonormal frame bundle of X has structure group SO(4), and a spin
structure would be a double cover of this which is a principal bundle
with structure group given by the double cover of SO(4), namely SU(2) x
SU(2).  Thus we get two principal bundles with structure group SU(2),
the left-handed and right-handed spin bundles.  Using the fundamental
representation of SU(2), we obtain two vector bundles called the bundles
of left-handed and right-handed spinors.  This "handedness" or
"chirality" phenomenon for spinors is of great importance in physics,
since neutrinos are left-handed spinors --- meaning, in down-to-earth
terms, that they always spin clockwise relative to their direction of
motion.  The fact that the laws of nature lack chiral symmetry came as
quite a shock when it was first discovered, and part of Seiberg and
Witten's motivation in their second paper is to study mechanisms for
"spontaneous breaking" of chiral symmetry.  This means simply that while
the theory has chiral symmetry, its ground states need not.

A Spin-c structure is a bit more subtle, but it allows us to define
bundles of left-handed and right-handed spinors as U(2) bundles,
which Kronheimer and Mrowka denote by W+ and W-.  The determinant
bundle L of W+ is a line bundle on X.  The first big ingredient of the
theory is a hermitian connection A on L.  In physics lingo this is the
vector potential of a U(1) gauge field.  This gives a Dirac operator D_A
mapping sections of W+ to sections of W-.  The connection A has
curvature F, and the self-dual part F+ of F can be identified with a
section of sl(W+).  (This is just a global version of the isomorphism
between the self-dual part of Lambda^2 C^4 and sl(2,C).)

The second big ingredient of the theory is a section Psi of W+, i.e. a 
left-handed spinor field.  There is a way to pair two sections of 
W+ to get a section of sl(W+), which we write as sigma(.,.) and which is 
conjugate-linear in the first argument and linear in the second.
This is a global version of the similar pairing 

                     sigma(.,.): C^2 x C^2 -> sl(2,C)

where sigma(v,w) given by taking the traceless part of the 2x2 matrix 
v* tensor w.  Here v* is the element of the dual of C^2 coming from 
v via the inner product on C^2.

To get the magical moduli space, we consider solutions (A,psi) of

                       D_A psi = 0
                       F+ = i sigma(psi,psi).

Here we are thinking of F+ as a section of sl(W+).  These are pretty
reasonable equations for some sort of massless left-handed spinor field
coupled to a U(1) gauge field.  Let M be the space of solutions modulo
gauge transformations.  Kronheimer and Mrowka show the "moduli space" M
is compact.   

One can also perturb the equations above as follows.  If we have
any self-dual 2-form delta on X we can consider

                       D_A psi = 0
                       F+ + i delta = i sigma(psi,psi).

and get a moduli space M(delta).  This will still be compact if delta is
nice (here I gloss over issues of analysis).  

Now, if X has an almost complex structure, Kronheimer and Mrowka show
that one can pick a Spin-c structure for X such that, for "good" metrics
and generic small delta, M(delta) is a compact 0-dimensional manifold.
Using this fact and some geometrical yoga, it follows that the number n of
points in M(delta), counted mod 2, is independent of (such) delta.
(This is essentially a glorified version of the fact that, when you look
at the multiple images of an object in a warped mirror and slowly bend
the mirror, the images generically appear or disappear in pairs.)
Moreover, if the self-dual Betti number b+ of X is > 1, the space of
good metrics is path-connected, and n mod 2 is independent of the choice
of good metric.  Kronheimer and Mrowka call this a "simple mod 2 version
of the invariants of Seiberg and Witten".  It is one ingredient of their
proof of the Thom conjecture.  

--------------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics (as well as some of my research papers) can be
obtained by anonymous ftp from math.ucr.edu; they are in the directory
"baez."  The README file lists the contents of all the papers.  On the
World-Wide Web, you can attach to the address

http://info.desy.de/user/projects/Physics.html 

to access these files and more on physics.  Please do not ask me how to
use hep-th or gr-qc; instead, read the file preprint.info.

From baez Thu Dec  1 12:13:57 1994
To: [email-masked] articles/week45

From: [email-masked] (John Baez)
Newsgroups: 
Subject: This Week's Finds 44 - special edition: the end of Donaldson theory?
Approved: [email-masked]

This Week's Finds in Mathematical Physics (Week 44)
John Baez

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 Psi 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 Psi.
The second is that the self-dual part F^+ of the curvature of A be given
in coordinates as

F^+_{ab} = -1/2 <Psi, e^a e^b Psi>

where the basis 1-forms e^a, e^b act on Psi by Clifford multiplication.

Next form the moduli space M of solutions (A, Psi) 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.

--------------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics (as well as some of my research papers) can be
obtained by anonymous ftp from math.ucr.edu; they are in the
directory "baez."  The README file lists the contents of all the papers.
On the World-Wide Web, you can attach to the address
http://info.desy.de/user/projects/Physics.html to access these files and
more on physics.  Please do not ask me how to use hep-th or gr-qc;
instead, read the file preprint.info.