- So, now we have the second the lecture
on the series of Talks are Read
by CMSA in Harvard jointly
with Tsinghua University,
the Med Center there.
And this is a serious talk on history
and literature in math science.
And we will cover many
different disciplines.
And today right now, Simon Donaldson
will cover in the subject
of Yang-Mills theory,
which is important in both geometry
and topology and Simon has been the master
of this subject in the last 40 years,
since the first time I
met him in Princeton,
he was a rising star at the time.
But now he is a master of this subject,
you load geometric topology, geometry,
and high dimensional use and construction
and all this has been leading
us for such a long time.
And I'm very happy that he is reading
to give this talk on the history
of ADHM constructs on Yang-Mills
instantons, okay, Simon.
- Hello everybody, thank
you to Professor Yao
for the invitation to give this talk
and for the introduction.
And thank you all for coming
to join this online lecture.
The topic I've chosen
focuses on work from more
than 40 years ago of a Atiyah,
Drinfeld, Hitchin and Manin,
but I'll begin by setting some of
the background for that work.
And then later in the talk,
I will discuss some
developments after that work.
So, let's begin.
I want to describe
the problem which they solved.
I'm sure a lot of this
will be extremely well
known to many of you.
So apologies to those people.
Let's just recall in electromagnetism,
we are working on space-time
on Minkowski space,
we can put the electric and
magnetic fields together
into a skew symmetric tensor Fij
and this can be represented
in terms of a potential.
So in according to the expression,
so, going back to the individual electric
and magnetic description, the potential A,
will be made up of the familiar
electrostatic potential
and magnetic potential vector fields.
But in the four-dimensional picture,
this, the skew symmetric
tensor is a two form.
And the Maxwell equations, the equations
of electromagnetism are generated
as the Euler graph Raj
equations of a functional,
in the case of the assignment.
- I saw some slides is not moving.
It's still the first page.
- [Man] It looks like you're
still on the first page.
- I'm seeing the second page.
You don't see it?
- [Woman] No.
- [Man] Do you want to try
Stopping your share and--
- I'll stop the share and then
do it again, let's try that.
- Okay, good.
- [Man] There we go.
- So we're seeing a formula for Fij,
expressed in terms of the
derivatives of the potential AI.
Can everyone see?
- Yeah.
- And then the functional which generates
the Maxwell equations
is just the extra detail
to know the indefinite
analog of that, we call it E.
So in 1954, Yang and Mills studying
more sophisticated particle physics,
considered the possibility
of letting the AI
take values in the Lie
algebra of Lie group.
So, for our purposes, we
will usually take this group
to be the unitary group, a compact group.
But sometimes we'll want to consider
the full complex group
of all complex matrices.
Anyway, the key thing is that they added
a nonlinear term to the formula,
involving the bracket in the Lie Algebra.
Between the components of the potential.
To find a functional as before,
the Yang-Mills functional
and this generates
Euler-Lagrange equations,
the Yang-Mills equations.
Some years later, it was realized
that this formula with a nonlinear term,
is exactly a very familiar
formula in differential geometry
for the curvature of a connection.
So, in this point of view,
the As represent a connection on a bundle
with structure group chi,
we'll actually talk about a vector bundle.
Will be the most convenient formulation.
And then this the F is just
the curvature of this connection.
Very familiar in differential geometry.
These developments both have the notion
of a connection and curvature.
And also the the link to particle physics
through the Yang-Mills equations,
was discussed in the electro
Professor Yang in this series.
Any case, I don't want to go in to review
the whole theory of connections,
but let's just recall
that if we think about
the covariant derivative of operation
defined by the As.
So, we just take the ordinary derivative
d by dxi and add on a zero thought a term,
given by multiplication by the matrix AI,
then the curvature
appears as the commutation
of these covariant derivatives
in different coordinate directions.
So, going to the next slide,
not a very professional
picture, I'm afraid.
The idea is that if we,
for example going back
to electromagnetism, we talk about
the wave function in quantum mechanics,
but really, that shouldn't
be thought of as a function,
but a section of a vector bundle
and we should replace
the ordinary derivatives
by these co-variant derivatives,
involving the contributions
from the electromagnetic potential.
This is the way we should
modify the usual quantum theory
in the presence of an
electromagnetic field.
So all of that discussion
was in Minkowski, space,
space time, that indefinite signature,
but we're going to work on
Euclidean four dimensional space R4.
That we now look at the
space of two phones,
as before the thing, then,
a special feature of four dimensions,
is that the split into the
sum of the self-dual forms,
lambda two plus and the
anti-self-dual-forms,
lambda two minus, this correspond
to the eigenspaces of the star operator.
Just reminder, for example,
if we take the star dx1, dx2,
that means we take the
complementary indices,
we get dx3 dx4 so, dx1, dx2 plus dx3, dx4,
would be an example of
a self-dual, two form.
Plus we can do the same at any,
we can do the same in higher dimensions,
we will get a splitting of the forms
in the middle dimension
in a similar fashion.
A special feature of dimension four,
is this mission occurs for two forms,
which have a particular significance
in terms of differential geometry
through the notion of curvature.
Any case, so in our situation, then,
the curvature we can write
as a sum of self-dual
and anti-self-dual components F+ plus F-.
And Yang-Mills functional is just the sum
of the squares of the
norms of those two things.
On the hand, if you take the difference
of the norms of those things,
then you find that this functional
is in a sense trivial with respect
to just compactly supported variations.
If you take a compactly
supported variation in your data,
then you can write the
variation of the integrand
as the derivative of something,
and so the integral is zero.
So, follows just from that,
without going into detail formula.
But if you have a connection
with F plus is zero,
then that is a solution of
the Yang-Mills equations.
It's a critical point of the functional E.
A solution of the Euler Lagrange equations
defined by the functional E.
So, these are the central topic,
these connections are called
anti-self-dual, Yang-Mills instantons.
Now, special solutions of
the Yang-Mills equations,
which occur in the case of dimension four.
Just to be clear, these
things are the solutions
of a nonlinear partial
differential equation,
for our connection
represented by the data A,
because the formula for F+,
involves that quadratic term,
with the bracket of,
involving the brackets in the Lie algebra.
So, the study of these things
is studying a nonlinear PDE.
So, beginning we talked about putting
the electric and magnetic fields together,
to get this object F, just
to see in a different way,
what the meaning of this
empty cell fuel condition is,
if we went back to choosing
a space and time description
of R4 dimensional space.
But now in the Euclidean version,
then we would represent our curvature
in terms of electric
and magnetic components.
And the condition is just
that the electric field
is minus the magnetic field.
And if you take the
ordinary Maxwell equations,
then they are symmetric
between electricity
and magnetism up to some signs.
When we go to the Euclidean version,
then the signs change around
and we get to a perfect symmetry
between electricity and magnetism.
And these anti-self-dual instantons,
are the ones which are
symmetric in that sense.
We can also E equals +B,
but that would just correspond
to a change in orientation conventions.
It would make no real
difference to the discussion.
So these equations, they
are conformally invariant.
They actually only us CB,
the metric structure on R4 up to a scale.
So everything extends to
the the compactification
of R4, getting by adding
a point at infinity.
Should get the full sphere
that's the standard
round Riemannian metric
on the full sphere is
conformally equivalent to R4.
According to a famous analysis results
of Uhlenbeck in 1979, studying
solutions on the full sphere
is the same as studying solutions on R4,
with a natural decay behavior at infinity.
That the same as imposing
the energy E is finite.
So we can pass interchangeably,
from working on the compact manifold,
the full sphere, or working on R4
with suitable growth
conditions at infinity.
And from this point of
view on the full sphere,
then the functional we call Kappa,
you wrote down before is
a very well known thing.
It's gives a characteristic class
of the bundle we're working
with over the full sphere.
So, in the case of the unitary group,
open up to a factor of eight pi-squared,
Kappa would just
correspond to what's called
the second churn class of our bundle.
A topological invariant of the situation.
So, that sets up the
background to describe
the problem solved by ADHM.
The problem is to describe all of these,
anti-self-dual Yang-Mills
instantons, over the full sphere,
for, well let's say for the
compact classical groups
that's to say for the unitary group,
what we'll talk about,
but same techniques apply
to the other classical groups,
the symplectic and orthogonal groups.
So this is the problem.
As it was solved by Atiyah,
Hitchin, Drinfeld and Manin,
at some time in 1977.
And you can read a bit
about the story of that,
in a Atiyah's commentary on volume five
of his "Collected Works" written in 1986.
With the help of Nigel Hitchin,
I finally saw how Horrocks
method gave a very satisfactory
and explicit solution to the problem.
I remember our final
discussion one morning.
When we just seen how to fit together
the last pieces of the puzzle.
We broke off for lunch feeling
very pleased with ourselves.
When I returned, I found
a letter from Manin,
outlining essentially the same solution
to the problem and saying, no doubt,
you've already realized this.
We replied at once and proposed
that we should submit a short
note from the four of us.
So that was the story of the solution
of these problem essentially
by independent groups, Atiyah Hitchin
and Drinfeld are Manin.
But one of the attractive
features of that solution
is that while it goes through
sophisticated mathematics
of different kinds, the actual answer
can be expressed rather simply.
So let's, without more ado,
outline what the answer is.
We'll work on R4 and
standard coordinates xi.
So, we're going to consider a family
of linear maps depending
in a linear fashion
on the coordinate x.
So these are gonna be linear maps,
lambda x from C2k plus R,
the complex vector space of
dimension 2K plus R, to C2k.
So having the form specified,
so everything is determined by five,
2k by 2k plus our matrices, Li and M.
I going from one to four or zero to three.
So we want to assume that these maps
are surjective for all x.
And then there are going to
be some other conditions,
which we'll come to later.
But some algebraic
conditions on these matrices.
Then, the construction goes as follows.
We take at each point x, we
take the kernel of lambda x.
Because lambda x is surjective,
this is an R-dimensional vector space.
And as x moves over our form,
it defines a bundle a rank
our vector bundle over R4.
The statement is that they
induced connection on that bundle
is one of the anti-self-dual instantons,
but we're after the second churn class,
the topological invariant is the number k
and all the instantons that
rank up the UI instantaneous,
the second churn class
k arise in this way.
So, nonlinear partial
differential equation
is solved in terms of very
finite dimensional data
a collection of matrices
satisfying some algebraic equations.
And towards the end of the talk,
we will come to write these equations
in a more explicit fashion.
But let's just say what we
mean by the induced connection.
So, this is just the same construction
as appears in elementary and
classical differential geometry
when you're discussing the tangent bundle
of a sub-manifold in Euclidean space.
That's to say we define a connection
by using orthogonal projection.
So to say that more formally,
a bundle we're calling it E-Tilde.
So on the one hand, a
subbandle of the trivial bundle
with reg 2k plus R.
I'm using an underline to
denote a trivial bundle
corresponding to the given vector space.
But it will also, we have
orthogonal projection,
which defines a map pie from
the bigger bundle to E-Tilde.
So we have an inclusion map Iota,
where our projection map pie.
For now, to define a connection
or covariant derivative
on the sections of E-Tilda,
we first apply Iota,
we can consider them as
sections of the bigger bundle.
Well there we, this is
just a trivial bundle.
So, we have a Standard Flat connection.
Just ordinary differentiation
that will take us outside the subbandle,
but then we project to get put to,
we get back to the subbandle E-Tilde.
So, this is just exactly the same way
that you define the
derivative of vector fields
on a submanifold in classical
differential geometry.
A very simple construction.
But why do I choose to talk
about this sum, this work?
What is special significance of it?
Well, there are many
things that one could say.
On the one hand, it was one
of the first applications
of sophisticated techniques for
modern differential geometry
and algebraic geometry to a problem
of great interest in physics.
On the other hand, it's also
a beautiful piece of mathematics,
which can be approached
for many directions
and connected with many other things as,
perhaps we'll see some
of later in the talk.
One of the references the
note that Atiyah referred to,
that appeared in 1978, Physics
Letters for author note,
just two and a half pages.
A greatly extended no exposition
was written by Atiyah.
Soon after, a write up of
the family lectures and Pisa.
This is a booklet of close to 100 pages.
And this came out in 1979.
I started as a graduate student in 1980.
And so this was an important
part of my education.
This booklet you can learn a lot about
the geometry not just of this problem,
but of all kinds of ideas from
geometry from this booklet.
So I greatly recommend it to anyone.
As I mentioned, there
are many points of view
that we can take on this ADHM result.
The ADHM construction.
But for the main part of this talk,
I'm going to follow pretty
much the point of view,
that they took, I'll follow and this goes
through something which is
of independent interest.
It's called the Penrose Twistor theory.
So, this Twistor theory is in general,
it relates objects of interest
in mathematical physics,
a differential geometry
to complex geometry
on another space.
So, we will need some language
and basic ideas from complex geometry.
So, in that direction, let's just recall
to begin with that if we look
at take the rim and stare,
the simplest, compact complex manifold,
just complex numbers with
a pointed infinite at it.
Then we can study line bundles.
Holomorphic line bundles
over this Riemann sphere
and the basic facts is that those
are just given by the degree integer D,
with powers of the offline bundle.
And the notation we want to use
is O of D for the line bundle
which is the diff power
of the half line bundle.
So to describe something
about this Twistor theory,
we want to consider a three
dimensional complex manifold
it is said, which contains what we call
a line, that's to say, an embedded copy
of the Riemann sphere
holomorphically embedded
copy of the Riemann sphere.
And we want to suppose
that the normal bundle
and of this line is
isomorphic to the old one,
plus a while as to say,
the sum of two copies
of the half line bundle.
Interclass rhombus LZ
is three dimensional,
L the line is one dimensional.
So the normal bundle of
course is two dimensional.
So there's a theory going back to Cordera
of deforming sub manifolds
of a complex manifold.
And this tells us that our L,
will move in a family of lines,
call it stripped M.
With tangent space that the given point L
is isomorphic to H0(N).
So, we set the bottom, recall the H0(N),
just denotes the the holomorphic section
of our holomorphic bundle N over L.
So in this case you recall that we take
the sections of the offline bundle 0(1)
that is two dimensional.
So, the sections have N are
dimension two plus two is four.
So, this family of lines
is a four dimensional complex manifold.
Simple, but surprising thing is that
there is an intrinsic
holomorphic conformal structure
on this family of lines N.
And this comes about in a very simple way.
If we write slightly more variantly N
is S minus tensor of one.
So, S minus is a two
dimensional vector space.
Then, the space of sections,
is the tensor product of S
minus with the sections of 0(1).
So, just write S plus
are the sections of 0(1)
of the tangent bundle
is the tensor product
S plus tensor S minus of a pair
of two dimensional vector spaces.
The observation then, is that
if we have a tensor product
with two dimensional vector spaces,
that automatically has
a natural quadratic form
to find up to a factor.
This is just because, if we take the,
we have a skew symmetric map from S plus,
times S plus to land the two of S plus.
Similarly, for S minus and if we take
the tensor product of those,
then we get a symmetric map from TM,
tensor TM to a tensor
product of a pair of lines.
So, trivializing that
tensor product we get,
we can think of that as C,
we get a quadratic form
to find up to a factor.
So this is the simple but remarkable thing
that if we have a three
dimensional complex manifold,
containing a family of
lines of a suitable sort,
then on that family,
there's a intrinsic conformal structure.
Now, this picture here is supposed
to illustrate some of that, this is Z.
We have lines there.
Each line Lx corresponds to a point x
at R4 dimensional complex manifold M.
A more geometric way of expressing,
what we just said, is that
if we take infinitesimally,
if we look at infinitesimal
variations of x,
the null variations with
respect to our quadratic form
just correspond to the
variations of our line
into a line which intersects Lx.
So we take two, two generic lines
in our three dimensional manifold.
Won't intersect will be
a co-dimension one set
where they do and if we
take the tangent cone there
of that at x, that gives the null cone
of our quadratic form.
For the part of this Twistor theory,
which is most relevant here,
is what's called the ward
correspondence, I think,
they put it that yes,
that's the heading for the section.
This goes back to Walter mid '70s,
who observed that one could encode
the instanton equations,
at least initially a
complex notion of those
in this Twistor geometry.
So if we have a holomorphic vector bundle
on our complex manifold Z,
which is it happens to be
trivial on all the lines
in the family M.
Then we get a bundle.
So, the E-Tilde over M
was taking the fiber of E-Tilde.
But a point correspond to a line L
is just the space of holomorphic sections
of E over restricted to L.
There is a very simple construction
of going from a bundle over Z,
to a bundle over the
family of lines N to M.
So there's a way to define
a connection on E-Tilde,
in this some complex geometry framework.
A good way of discussing that is in terms
of formal neighborhoods,
if you have a section
of E over L, then you find
that there's a unique way
of extending that to the first
formal neighborhood of L.
And that goes over to the
notion of a connection
as an infinitesimal
trivialization of the bundle.
And then this is a holomorphic,
anti-self-dual connection over M,
with respect to the conformal
structure that we defined.
And conversely, any such
anti-self-dual connection,
comes from a holomorphic bundle.
So here we're talking
everything is complex geometry.
'Cause that's in a way easier.
So we're using the
rather obvious extension,
of everything we did in the real world,
four dimensional world to the
case of complex manifolds.
Let's just note to make it clearer,
this condition of being trivial on online
is an open condition.
So if we have any holomorphic bundle,
over Z, provide it's trivial on some line,
is trivial on a whole open set of lines.
So to which we could apply the discussion.
And for our problem we want
to get to real four manifolds
and compact structure groups.
And that can be encoded
in this Twistor theory.
Introducing what's called
a real structure on Z.
In fact there are different
ways of doing that.
though the most obvious thing to do,
would be to take, let me see.
Let's describe it the second way,
there are different ways of doing that,
which would correspond
to different signatures
of the quadratic form
on a real four manifold.
But the one that's relevant
to Riemannian geometry,
is to consider an
anti-holomorphic map sigma,
from Z to Z with no fixed points.
Then, where there're no fixed points,
we can look at fixed lines.
Sigma invariant lines.
So make such line, sigma will
just be the antypital map
on the two sphere which
is anti holomorphic
and has no fixed points.
Then complex conformal structure restricts
to the four dimensional manifold,
for real dimensional manifold
of sigma invariant lines.
To give a positive definite
conformal structure.
Then the world correspondence identifies
the anti-self-dual instantons on E,
with, it's called the
real homomorphic bundles.
That's to say bundles, which
when we pull back by sigma,
predict sigma star V,
and we take the conjugate
dual that's isomorphic
to the bundle E.
So, all of this was quite general.
It applies to quite a
number of four manifolds,
which can be described in this way,
in fact there're what are
called self-dual-fourmanifolds.
And particular condition on the
Riemannian curvature tensor.
But for our problems, we're
interested in the full sphere,
and for that, we take a complex manifold,
is a very familiar one is just
the three dimensional
complex projective space CP3.
So to get the real structure
that's relevant to this,
we think of the underlying
four dimensional
complex vector space as being two copies
of the quaternions H2.
So CP3 is the quotient
of H2 minus the origin
by multiplication by C star.
And we get our anti-holomorphic map,
is just multiplication by J.
If our complex structure corresponds
to the quaternion high, we
take multiplication by J,
anit-commutes with I and gives
our anit-polymorphic map.
So, in that case, a real
form MR is the full sphere.
And the position can be
described rather simply
from the fact that we can
also take the quotient
of H2 minus zero by multiplication,
by non-zero quaternions
and that will give us
the one dimensional quaternion,
you wanna project your spheres.
Which is just the full sphere.
So, the real lines are just the fibers
of the natural map from CP3 down to S4.
And these are lines in
the ordinary service,
they are complex line, projective line,
in our complex projecting free space.
And the instantons we seek,
correspond to certain
holomorphic vector bundles,
over CP3 which satisfy
the reality condition,
with respect to this
anti-holomorphic map sigma.
We're going back to what I was
starting to say a while ago,
a more obvious thing to do,
would be to take sigma to
be an anti-holomorphic map
given just by complex conjugation.
So, that will be different
because that has fixed points,
corresponding to RP3 inside CP3
and one can't do the whole
theory in that setting.
But then we will be
studying a quadratic form
on our RP3 of, sorry a quadratic form
on a four manifold with signature two two.
So, we would be working it
with another discussion,
in indefinite signature.
So there's another picture,
which is suppose to
summarize the discussion.
This is the full sphere.
which is what we're interested in.
On a bundle E-Tilde over the full sphere.
We introduce the Twistor space CP3,
which is in fact a fiber bundle,
fiber in only the full sphere,
the fibers gift this family of lines,
the real lines that each one is preserved
by this anti-holomorphic map sigma.
And I ward correspondence
gives a correspondence
between certain holomorphic
bundles over CP3,
and the instanton connections
on a bundle E-Tilde,
over the full sphere
that we want to study.
So that's, in the way that
the technical background
to set the scene for the work
of Atiyah, Drinfeld, Hitchin, and Manin.
From this point of view,
we want to know how
to construct holomorphic
vector bundles over CP3.
So again, let's introduce
a little bit more notation.
If I have a bundle E
over an protected space,
we can take the tensor
product with a line bundle,
O(p) and we just use the
abbreviated notation Efp for that.
Of course we don't, we can
always take the tensor product
with O minus P to recover E from EFP.
So we don't really lose information
of going to these twisted bundles.
So the key idea goes back to Horrocks.
as mentioned in that quote
from Atiyah a while ago.
And it's what's called a a monad.
Terminology introduced by Horrocks.
These could be considered more generally,
but for the case of hand,
we just want to consider
a very particular situation,
where we take three complex
vector spaces U, V and W.
And let's write underlying
U, underlying V,
and underlying W for
the corresponding trivial vector bundles
over, in this case CP3.
So supposing we have bundle
maps, holomorphic bundle maps,
from the twist of U minus one,
if a map A to the trivial
bundle with fiber V,
and then we have another
map B to the twist W(1).
And let's suppose A is
everywhere injective,
B is everywhere surjective and
the composite be with A zero.
That such a thing that's called a monad.
Then we get a bundle by
taking the cohomology,
we take back the kernel of B
and divide by the image of A.
Now this is somehow a combination
of the two more familiar things,
either considering subbundles
for trivial bundle,
or quotient bundles for trivial bundle.
Here we're doing the
combination of the two.
Any case, is this such a thing
as a very sense explicit object.
If we take homogeneous coordinates
for our projective space Zi,
so I goes from one to four,
then talking about a map
from this twist of u to v,
that just says A has to be
written as the sum of say Ai, Zi
and B is the sum of Ej, Zj.
The condition that the
composite B with A zero,
is just when we take Bj Ai, plus Bi, Aj
is zero for all pairs ij.
So, these monads just
correspond to solutions
of a rather explicit
set of matrix equations
for the collection of
matrices the As and the Bs.
For the the well known
theorem where these brings
these monads to the four
as a result a Barth,
it's a trip to Barth by Atiyah.
In the references I gave
it a more detailed proof
appeared in a paper of
Barth and Hulek, 1978.
And it says that if we
have any holomorphic bundle
over CP3, which satisfies
certain vanishing conditions
on its cohomology, then
such a bundle arises
from a monad of the
kind we were describing
where the dimension of U and V and W
are given by the churn class
and the rank of the bundle E.
So these cohomology groups,
micro shift cohomology
groups, the H1 and so on,
there are central, maybe technical
motion in this ADHM work.
For those who are not so
familiar with those things,
let's just imagine that just
like from a topological space,
we have homology groups
and cohomology groups,
where the H0 is just,
detects number of components
within the higher cohomology groups
detect more subtle phenomenon.
So, recall that if we
have a vector bundle,
or say with sheaf of sections,
then we have, we can
take the global sections
which should be H0, and then we have
higher cohomology groups in general
which detect some more subtle information.
So, what happens if we put together
this monad construction for a bundle
with the ward correspondence?
Then it gives exactly the
induced connection construction,
for our connection on a
bundle over the full sphere.
In fact we can say this in a
purely holomorphic language.
We don't need to introduce
metrics to discuss this,
although in the end for
bundles with real structures,
we can do that as in
the way we said before.
It's to say, supposing we take a line L
over which our bundle E is trivial,
then we take a pair of points P1 and P2,
distinct points on the line L.
Then, if you look at the monad,
you can see that the triviality
over L can be expressed
as a simple transversality condition
for the linear maps involved.
We have a bundle map say B1,
which we can evaluate at the point P1
if this is defined as an up tool scalar.
But if we make a scalar
ambiguity it won't matter here
and similarly for B2 and
similarly for A1 and A2,
so we have four linear maps
between three vector spaces
and the triviality condition
is that these are if they're
in general position in a sense,
so, the restriction of
E to L can be described
either as a subspace
of our vector space V,
the intersection of the
kernels or as a quotient of V,
the quotient by the sum of the images.
So, that those two descriptions
are just what we need to
describe the induced connection
in the way that we did.
We have a inclusion map of E-Tilde
into a trivial bundle and a projection map
from the trivial bundle on to E-Tidle.
So, with all that in place,
all we have to do is to
show that the bundle E,
which we have over CP3,
according to the ward correspondence,
correspond to an instanton satisfies
all those vanishing
criteria in Barth's theorem.
Remember they had this,
all the stuff about
the H0s and the H1 of E and E star.
So, the essential case, the
eight zeros are rather obvious.
But essentially the essential case
is the vanishing of the H1 of E minus two.
And similarly, replace E
by the dual bundle, E star.
The argument for this is interesting,
and it goes back to
the sort of the origins
of Penrose's Twistor theory.
And I said earlier
earlier part of the story
of the development of Twistor theory.
Well, that's some digress to explain that.
So we this is just a really
local discussion initially,
we just take an open set in R4,
we can have a take the
corresponding open set in CP3,
just the union of all
the lines corresponding
to our open set in R4.
So let's fill that script U.
We have a shift cohomology class in H1
of the script U with coefficients in
the bundle O minus two,
then we can restrict
that to any line in our family.
So for a line, the O minus,
the H1 of O minus two over a
line is just one dimensional.
So we can, this situation
we can identify the complex numbers.
So the restriction of this
sheave cohomology cluster lines,
gives a function on U.
And Penrose showed,
this function satisfies
the Laplace equation.
The ordinary Laplace equation in R4.
And conversely, any local solution
of the Laplace equation
arises in this way.
This is a very attractive
idea and unrelated
to some classical analysis things,
the classical representations
for harmonic functions
in various contexts in
terms of contour integration
of holomorphic functions.
And if we take this Penrose construction,
if we represent our H1 class
in terms of check cohomology,
so, we have a holomorphic function
of three complex variables
defined on an overlap
of two domains for suitable kind,
then, you find that explicitly,
the corresponding harmonic function
will be given by a contour integral
of our holomorphic function.
Around a suitable family of contours.
A formula for classical nature.
In the same way, going back to a situation
of our bundle E on CP3 corresponds
to an instanton connection on Tilde-E
over the full sphere,
then this cohomology group
of interest, the H1 of E minus two
can be identified with
the space of sections
of E-Tilde which satisfy an equation.
The equation is a variant
of the Laplace equation,
the case we were discussing before.
And that variant comes about in two ways.
One is that we're considering
sections of a bundle.
So, we take the covariant Laplacian,
formed using the covariant derivative.
And the other is we're working
now on the full sphere.
So what appears is what's called
the conformally invariant Laplacian,
where we add on a multiple
of the scalar curvature.
R over six, where R is
the scalar curvature.
This is something which is going
to make a conformal
transformation goes over
to the ordinary, the
standard Laplacian on R4.
Working globally since the
scalar curvature is positive.
This the first term Navistar Nabla,
is a positive operator, the
only solution to this equation
is S equals zero, so we find that the H1
of E minus two is zero as desired.
The same applies to E star and
we can apply Barth's theorem.
So all we need to do looking at the time,
what we need to do to
understand this ADHM results,
if you grant the various
assertions that I quoted,
is to understand how to
prove Barth's theorem.
Why do all the bundles satisfying
certain vanishing theorems
come from this rather
simple monad construction.
For the quickest slickest
proof of that nowadays,
comes from using a
construction of Beilenson
which appeared somewhat
after the ADHM worked.
Probably very likely
motivated by the ADHM work
and I think that develops
ideas of Greenfield and Manin.
So let me try to give some idea
of this Beilenson construction.
In fact, the Beilen
construction applies initially
to bundles over any projective space,
that we're going to apply
it to this specific case
to a bundle of a CP3.
So let's, to begin with,
let's fix a point x
in a projective space CPN.
Then there's a preferred section of V
of the tangent bundle
twisted by minus one,
which has a simple zero
at x and no other zeros.
So this is essentially
a very familiar thing.
If we took our fine coordinates,
so we took Cn sitting inside CPN,
then this V would appear
as an ordinary vector field
and it's just the radial vector field.
The sum of Zi d by d Zi.
The field generation was
trying to radial dilations.
The effect of twisting by minus one,
when you go to projector space,
is to make it non vanishing
at the hyperplane at infinity.
Now let's recall a very
basic simple thing.
If we have a vector space call it K,
a nonzero vector e, then
we get a an exact sequence,
which we start off
mapping with on the right,
we start off mapping k star to C,
just in the pairing with E.
And then we put all the
exterior powers of k star
and we map from one to the other
by using contraction with E.
And the basic part of
this sequence is exact,
provide E is nonzero of course it is zero,
all the maps vanish.
So I can apply this idea in the setting
of bundles and sheaves to get
what's called a puzzle complex,
which in an our situation is
an exact sequence of sheaves,
where we take the Omega R, twisted by R,
the omega O is just notation
for the exterior power
of the cotangent bundle, the
dual of the tangent bundle.
And then literally a contraction
with our section defines maps from omega
are twisted by R to omega R minus one,
twisted by R minus one, and so on.
At the end, we go from
omega one twisted by one
down to the functions R.
But because of the V vanishes at zero,
that's not surjective a functionalize
in the image of that map,
if and only if it vanishes at the origin.
So we put on the skyscraper sheet at x,
the sheaf by a copy of
C, so concentrated at x,
at the end of the sequence.
Now if we have a
holomorphic vector bundle E,
over a projective space,
we can take the tensor product with E,
with everything we did above,
then we get an exact sequence of sheaves.
So we'll write these as
script F for shorthand,
but ending up with the fiber of E over x.
So in this general situation,
if we have an exact sequence of sheaves,
then one has what's called
a hyper colon cohomology
spectral sequence.
In fact it has a pair
of special sequences,
which abut to the same limit.
So but in the case on hand,
one of those sequences is trivial.
And so what we get is a special sequence,
which begins begins with the cohomology
of all our sheaves and abuts
to the fiber of E at x.
Just the, back to space.
So again for people
who are not so familiar
with sheaves and so on the
spectral sequences on well,
what this means is that
we have starting with E,
we make a collection of other bundles
given by taking tensor product
with these standard bundles,
we have our projective space built
out of the tangent bundle that we got.
So we have all those bundles,
we take all that cardiology groups,
so we get a big chessboard
of cohomology groups.
But then we have lots of maps
into smile section between those.
And then there are more subtle maps
between the cones of the first lot of maps
and the images and so forth.
So we're going to hold
principle big collection
of linear algebra data, but in the end,
that gives some way of
describing the fiber of E over x.
So that's the general story.
Let's look at our case.
Our spectral sequence
begins with this chessboard
of vector spaces that cohomology HQ,
of E tensor, this twisted exterior paths
that the cotangent bundles of n is three.
In fact, we're going to, this could be in
to replace E by the twist E minus two.
So then what you'll find is that the rows
of our chessboard, I've
got the the cohomology HQ
of E minus three, the E tensor omega two,
the second disturbed power
of the cotangent bundle,
and E tensor omega one minus one
and then we get E minus two preparing him.
So here, we can see that the E minus two,
remember that was just the crucial thing
for the H1 of E minus two.
That's just the crucial thing
appearing in Barth's theorem,
that's when Q is one,
we also have Q equals two,
but then that the H2 or the minus two
is self-dual to the H1
of E star minus two.
So that also vanishes.
So putting together that and
various standard documents,
you find that out of this
big potential chessboard
of cohomology spaces are
very complicated description.
Actually, things become very simple,
very three terms in our E1 term of E1 page
of our spectral sequence
and the spectral sequence
just amounts to the statement,
that the fiber of E or that E minus two,
at our point x is given by the kernel
of a map B for one space divided
by the image of another map A.
Otherwise, just the kind of thing
that we are looking for from a monad.
And indeed, if we know that
the point x vary on CP3,
then we have to keep track of
the scalar ambiguity in the section v.
And also the fact that we made
this twist from E to F minus two.
But when you keep track of all that,
what you'll find is that the bundle E
is described as the cohomology of a monad
of the kind we want,
where the vector space u
is just the H2 of the minus three,
V is the H2 of E tensor omega two
and W is the H2 Have a tensor omega one
twisted by minus one.
So this Beilinson construction,
makes this result about monads
appear as part of a larger picture.
For any bundle whatsoever,
you get some kind
of potentially complicated
description of the bundle
in terms of linear algebra built out
of its cohomology groups.
But this could be in
the general situation,
system by Beilenson,
one should really talk
about the derived category of sheaves
and his theory gives
information about that.
Another way of saying the last part,
where we let x vary is
that we should really work
not on CP3, but on the
product CPT times CP3
and then run I'm working with a resolution
of the ideal sheaf of appoint,
we should work with the
resolution of the diagonal,
and CP3 times CP3.
And that does everything
in sort of one blow
but with a more complicated notation.
So that, I've be just
gone for about an hour
that completes the main part
of the talk giving an
account of the ADHM work.
More or less in the way they did it,
with this refinement by Beilenson.
Now we go on to the last
section of the talk,
where I'm going to talk about some other
points of view and further directions.
One of these goes back to mark
of Mukai published in 1981.
So more or less at the same time.
But this was from a completely
different direction.
Mukai was I think, had
no knowledge or interest
in instantons at that
time as far as I know.
He was studying holomorphic bundles
or more generally sheaves
over a complex torus.
Well, his theory applies
largely to any dimension,
but in the case at hand we want to take
a two dimensional complex
torus, two complex dimension.
So in that situation, we
have a dual torus T hat,
which parameterizes the
flat line bundles over T.
So if any point psi and T hat
we have a flatline bundle, Lpsi over T.
So if we start with the
holomorphic bundle E over T,
then in favorable circumstances,
we get a bundle Ê over T hat.
So, the fiber of Ê at the point psi
is given by taking the H one over T of E
twisted by the line bundle Lpsi.
So here, more precise the
favorable circumstances
mean a situation where we know
that the other cardiology groups,
the H0 AND the H2, both vanish.
Then Mukai shows that this
had a striking property closely analogous
to the Fourier transform to any width.
If we take the dual of
the dual of the torus,
we get back to our original torus.
So we can start with Ê and
do the same construction
and then we recover the
bundle E with up to a psi.
You might have to take them,
the involution of the torus,
we do have to involutional torus,
mapping x to minus x,
just as in the usual
Fourier immersion formula,
you need to put in a change of sign.
So there either quite precise analogies
between the Mukai construction
and the Fourier transform.
So, that was a complex geometry discussion
in terms of cohomology.
Though we can also make a
Riemannian geometry discussion,
which turns out to be equivalent
for other simple reasons.
That if we take a flat metric on T,
compatible with a complex structure,
then we can describe the same bundle Ê,
using the direct operator.
So, we take the direct all
patient mapping coupled
negative spinners to positive spinners,
but all twisted with
this line bundle Lpsi.
And the kernel of that, right,
well known standard
results can be identified
with the H1 in the previous setting.
So this description is independent
of the complex structure.
And using a natural
projection construction,
similar to what we said before,
we start with an instant
connection on the bundle E,
we get an induced
instanton connection on Ê.
So, while Mukai's work was in
the setting of complex geometry,
there's a translation
of that into the setting
of instanton geometry, with reason,
I'm speaking a Fourier transform
that takes instantons on one Torus
to instantaneous on another Torus.
But and so if this
interchanges essentially,
the two topological invariants the rank
and the churn class so it's a,
not at all an obvious construction.
So developing that line of ideas,
let's go back to the
instanton equation on R4,
and think about that.
So, we can write that equation
in terms of the covariance,
the components of the
covariant derivative.
As a member of the curvature is given by,
the commutators of these
kind of these components.
So and we've expressed the the
anti self duality condition,
in terms of commutations in this way.
So we, a convenient way
is to take coordinates
on R4 going from naught to three.
We take the following expression,
we take the commutation
of grad note with grad I
and add computated grad Chang, grad k,
when IJK run over the
cyclic permutations of 123.
So now let's take complex combinations.
Let's write D1 is grad
naught plus I times grad one,
squared to minus one and plus one.
And similarly for D two.
Is then instanton equations can be written
in terms of the Ds, the
same that D1 and D2 commute,
and then we take the bracket
of D one and D one star,
D two D two star, the sum to zero.
So, about the fact that we
get this commutator equation,
D one bracket D two is zero,
essentially, is another point
of view on the connection
to holomorphic structures.
If we fix complex coordinates,
corresponding to the
way we don't think so,
we take one complex
coordinate to be x naught,
plus ix one, the other
two x two plus ix three,
then D one and D two correspond
to the Cauchy-Riemann operators
in the two complex coordinate directions.
And this commutation condition
is just the interoperability condition
that the connection defines
a holomorphic structure.
On the other hand, here we fix
this particular kaiso complex structure.
We could have taken any complex structure,
compatible with our metric on R4,
and so we get a whole family
of similar interoperability conditions.
And essentially that goes back
to the point of view of the
homomorphic bundle on CP3.
They're also connections
with important connections
with integrable systems where,
when we'll talk about
the Lax representation
of a if I write, a nonlinear equation,
express it Lax form as
the commutator of two,
related linear operations.
And there's a big literature relating
these instanton equations
to integral systems,
such as the KdV equations.
Are one of the pioneering papers
was Mason on Sparling in that direction.
Any case, looking at these equations,
the D1 bracket D2 a zero and then we have
the other equation, involving the stars.
And the matrix equations arising
from the ADHM construction,
can be put in a similar shape.
And we didn't, we admitted to say exactly
what the equations came down to.
But now we're going to make up for that.
So you can put the equations
in the following form.
Your data consists of a pair
of linear maps from Ck to Ck.
Call it alpha one and alpha two,
a map from P from Cr to
Ck, and Q from Ck to Cr.
So there's four different
complex matrices is our data.
And then the equations,
all this monad theory need to,
can be expressed in the following way,
the bracket of alpha one and alpha two
is P times composed with Q.
And the sum of these other terms,
is PP star minus Q star Q.
So if we were just able
to set P and Q to zero,
we ignore the right hand side,
we would have exactly the same structure
as the equations for the
differential operators
that we were considering before.
But just to complete the story,
remember we, the the ADHM description
of the connection involved
this family of maps lambda x,
written in the following way.
And I've just written
down here how explicitly
to write out these lambda
x in terms of the data
that we're looking at now.
The alphas and the pieces of twos.
So working in R4, the vector space U,
that we said in the complex
geometry description
that appears as an H2, or appear as an H1.
But in R4 it appears of
the space of solutions
of the couple Dirac equation.
I can write down,
in fact, just directly
write down a formula
for all the ADHM data.
I have the vector space, we have
to run find ways of writing
down these linear maps
and directly confirm the ADHM results,
just by differential
geometric calculations on R4.
This was done first by
Corrigan and Goddard,
in the early '80s.
So, another development
from the same time,
this is an important development,
work of Werner Nahm of the early '80s.
So, this takes be look at
potentially multiple monopoles,
which are the solutions
of the instanton equations
which are invariant under
translations in one direction.
So they can be expressed
in terms of solutions
of some equations over R3.
And will study those with
suitable asymptotic conditions,
like infinity.
And Nahm showed that these,
there was a transform in the same vein
as the Mukai transform also
as the ADHM construction
which took the these
are invariants solutions
of instanton equations,
two R3 invariant solutions.
Solutions of equations which are invariant
under three translations.
So, these latter, the partial
differential equations
just come down to order
differential equations.
And one is just studying
solutions of Nahm's equations,
which are these system of ODEs
for matrix valued functions
of one variable S.
So, the general picture
is that in many contexts,
one gets a transform,
which takes one version
of the instanton equation to another one.
So we might call this the ADHM
Fourier Mukai Nahm transform,
that rises in many contexts.
In the ADHM case, so
this, we have the P and Q.
So it's a bit more subtle,
roughly if you did a
approximate calculation on R4,
you will get the equations
without the P and Q.
But what do you think the P and Q arise
because they're a certain contribution
from boundary conditions,
the boundary terms at infinity
in your calculation.
So that's where the times
one point at P and Q arise.
But the general picture is
that one has, say a transform,
in this differences
settings from one version
of the instanton equation to another one.
And this is useful survey of that.
This is by Marcos Jardim,
the reference, quoted.
We're coming in towards
the end of the talk.
What we have we have, we have
these UR instantons on us S4,
for each churn class.
Each is a positive number K.
So, we have a family of
these modular spaces MKR,
which grow in dimension.
The dimension is something like four Kr
or something like that.
So let's give two out of many examples
of application of this ADHM theory,
involving these modulized
spaces in different ways.
One is in the direction of topology,
is work of Boyer, Hutubise, Milgram, Mann,
and then also of Kirwan, a related work.
So this give roughly speaking calculations
of the homology groups
of these modular spaces
in a stable range, where
the K is very large
compared to the dimension
of the homology group,
boosted the rank of the homology
group you're interested in.
Actually maybe it would
be more precise to say,
rather than calculation,
but they show that
these homology groups are isomorphic
to homology groups of another space.
Which is well known in algebraic topology,
the third loop space of
the corresponding group.
So this was the confirmation of,
what we call the Atiyah
to Jones conjecture,
which was an influential problem
in the '80s and early '90s.
In other direction more
directly related physics,
Microsoft 2003 used the ADHM description
of the modular spaces
to find what's called
the cyber written pre potential,
Which I believe a important thing
in supersymmetric quantum field theory
by expressiveness in terms of integration
over the modularized spaces and then using
localization techniques
to evaluate the integrals.
So, there are many other
applications both related
to geometry and topology
and other stems of physics,
of this explicit description
of relatively explicit description
of the instanton modular spaces.
Finally, because we could
mention some current
and possibly future
directions of research,
which are involved many of the same ideas
that we've touched on here in one,
again, there'd be many things
that one could say here.
I'm just choosing two.
In one direction there's
increasing interest recently
in looking at instanton equationns
and variations those equations,
for non compact groups such
as SL (2, C) for example.
This comes from work of
propulsion work Taubes
and others in the last few years.
As we've said, the the basic
old correspondence supplies
just as well the non
compact structure groups.
So, one can describe these instantons
in terms of algebraic
geometry abundance on CP3,
although, least in principle.
Or I could ask whether to
take a general formula fold
one would reproduce the
same kind of phenomena,
that I can find through
algebraic geometry.
In the special case of the fourth there.
Another direction is the
study of holomorphic bundles
on projective space for various dimensions
but dimensions two three and four,
is important and in
different ways in the theory
of some certain higher
dimensional versions
of instanton equation which
are actively pursued nowadays,
which occur in the context of manifolds
with exceptional holomorny
or special holomorny.
So again, there's a lot of scope
for work in that direction in
the next few years, I think.
So thank you for your attention.
That finishes my presentation.
- [Man] Great, thank you very much, Simon.
Let's have some questions.
- [Man] So there are four
questions in Q&A section.
- Okay, so if I stop the show,
how do I get to the questions?
I see okay, right.
Okay, so if I, yeah, so the first question
was Manning's contribution
joint with Drinfelds
or was Drinfeld independent?
I understand that Drinfelds and Manin
were working together at that time.
Yeah so, okay, another question I see also
from Jim Starfish.
Yes, I apologize there's a misspelling,
Hurtubise has an R,
Jack Hurtubise has an R.
There's another question, so I apologize.
I didn't see this as
we were going through.
To say that sigma has no fixed points.
There's no point in the
case at hand in CP3,
which was fixed by sigma, so this,
the model will be to think
of the antypital map,
on the two sphere.
There's no point which is preserved
by the anipytal map.
So, I see another question
from Sergej Cherokees.
"Why it called a monad?"
I'm not quite sure I did look
at the paper from Horrocks.
I didn't find any explanation
of why he chose that name.
I think a monad means a sort of
a fundamental entity in some way.
So perhaps that was the idea he had.
So I see a question from Conan Liung.
"How are bundles over CPN three and CP4,
"related to special holomorny?"
Okay so,
if a bundle another point of view
on this instanton story, would be to say
that bundle over, instantons on R4,
and in fact correspond to
holomorphic bundles on CP2,
which are trivial on the line at infinity.
This is something related
to the Twistor story
and also other things.
In the same way, if you
look at say what are called
emission Yang-Mills equations on C3,
which have certain decay
properties at infinity.
Then those can be identified
with certain holomorphic
bundles over CP3 and similarly for CP4.
So those emission Yang-Mills solutions,
give models for solutions of these,
both higher dimensional
instantonic equations,
one would have over manifolds
of dimension 678, roughly speaking.
Hope that's one answer to the question.
- [Man] So, are there more questions?
- I see a question about the connection
to internal system file next pal.
So the question is, "With this
connection driven systems,
"does a spectral curve arise,
"which gave data about the instanton?"
So I'm not, I'd have to
think about that in the case
we were discussing for
the main part of the talk.
But in the variants in the
case of Nahm's equations,
then the answer is definitely yes.
As studied in many
papers of Nigel Hitchin,
there is a these monopoles on R3,
can be in fact described in
terms of a spectral curve,
which has a, which one can see
either through Nahm's
equations essentially
the joint built out of the eigenvalues
of the family of matrix matrices Ti,
we wrote down, all can
be described directly
in terms of the geometry
on R3 of the monopole.
- [Man] So, any more questions?
- [Ryan] Looks like someone
has their hand raised.
- [Man] Yeah, yeah, yeah, yeah, yeah.
- [Ryan] Jim is there,
I'll allow him to talk.
- So, are there more questions?
So yes, so there's a question,
there's a question from Marco Brotieri
about, asked him to comment
about how learning about this,
the instantons on R4, led him
to use his modernized spaces
to study four manifolds.
So, I think that the answer to that
is that, is in my case, it's natural,
just within this Yang-Mills setting
to the study compactification
of these modular spaces,
that sort of correspondence
to the analysis questions
about the behavior of
sequences of instantons.
I can say about, so once one starts that,
then you find was applications
to four manifolds,
are sort of sitting in your hands, I said.
I've never had a plan to sit down
and so I got to study four
manifolds using modular spaces.
We can see that the existence
of these modular spaces
will give you some information
about the topology of the four manifolds
that you're working with.
- [Man] Any more questions?
- [Jim] Yes can you hear me?
Can you hear me?
Yes, so I had a question.
So way back in the late '70s,
there was some sabotage
work by Edwin Denhavasto,
by me and Yaz and Greene.
Where we looked at their
a version of itself.
Looking at partners offer are always CP3,
we would like to partners over
what we call a Beatrice space.
What we got was a correspondence
of these partners with
with Yang-Mill solutions,
rather than self-dual Yang-Mill solutions.
The problem with that structure
is that nobody ever knew
anything about those models?
I'm just curious if you've
ever looked at that work
from, as I said, we did it in late '70s
arounds '79 and we are,
I also did with Phil
Yazget and Paul Kreig.
Are you familiar with that work at all?
- I'm sorry that this
sound is very unclear
so I couldn't really get much of your--
- [Jim] So, our work of
looking at a correspondence
are for Yang-Mills Mill's solutions
of a a Makowski space or
a complex Makowski space
with boundaries of what
we call arbitrary space
which is sort of a
diagonal of CP3 over CP3.
Can you hear me?
- So I think, so we were discussing
the idea of extending this
description of the instantons
to salute description of solutions
of the full second order
Yang-Mills equation.
- [Jim] Yes, that's right Joe.
There was a paper by Ed Witten,
and also separate paper by me and Yazgen.
- That's right, there was some
activity in that direction,
but it's not an area I really followed it.
It won't be interesting
perhaps to go back to that,
because somewhat some years after that,
people proved by other means,
the existence of global solutions
to these second order equations.
- [Jim] Yes.
- It'll be interesting to see
whether those could be found
using this geometrical construction.
But it's not something I've ever
really looked into myself much.
- [Jim] Okay, thank you.
- Okay, but there any other questions?
Okay, we should thank Simon
for this beautiful talk.
Really nice and inspiring.
And we should clap our hands.
(claps hands)
Tomorrow there will be two talks.
One on black hole, one on quantum groups.
The black hole will start
at 11 by Lydia Berrier.
And quantum growth will
be by Paul Edingough.
So I hope to see you tomorrow.
Thank you very much, Simon.
- My pleasure, thank you for watching.
Thank you.
- Thank you, beautiful talk.
Okay, see you guys.
Bye.
