Mathematical Structures 161
find a small cosmological constant. So, I would like to know what the status of
statistics is in the heterotic context.
S. Kachru I can say one thing. One of the huge advances in the mid-nineties was
this duality revolution. After the duality revolution, Freedman, Morgan, Witten
and many others developed very powerful techniques to take heterotic strings
on certain Calabi-Yau three-folds, elliptic Calabi-Yam, and Fourier transform
them over to type I1 strings, F-theory or type IIB string theory. Actually the
groups that have been making the most progress in constructing realistic GUT
models in the heterotic string, the Penn group for instance, worked precisely
in this elliptic Calabi-Yaus. Now you can then ask: “What happens if you
dualise these over to the type I1 context where people have started counting
vacua with fluxes?” It seems quite clear that the same kind of structure will
emerge in the heterotic theory. The reason that it is much easier to study in
the type I1 context is that what these fluxes in the type I1 theory map back
to in the heterotic theory correspond to deformations of the Calabi-Yau into a
non-Kahler geometry. As should have been clear from Yau’s talk, although that
is a very interesting subject, we know almost nothing about it. That makes it
clear that you can import the best features of GUTS into the type I1 context
and the best features of the type I1 context back into the heterotic theory. But
different sides are definitely better suited to describing different phenomena.
A. Strominger I think that is very misleading. There is no reason to believe that
there are not also non-Kahler types of geometries on the type I1 side. We just
are sticking with those because we are looking under the lamp post.
S. Kachru I completely agree with what you have said.
A. Polyakov Two brief comments. First of all it seems to me that one should not
be overfixated on Calabi-Yau compactifications, or on compactifications at all.
We do not really know how string theory applies to the real world. It is quite
possible that we have some non-critical string theory working directly in four
dimensions. I think that the important thing to do is to realise what mecha-
nisms, what possibilities, we have in string theory, and it might be premature to
try to get too much, to directly derive the standard model, etc. That is the first
comment. The second comment is about a physical problem. It is just to inform
you about things that are not widely known. There is an interesting application
of the methods which we developed in string theory and conformal field theory
in the theory of turbulence. There are recent numerical results which indicate
that, in two dimensional turbulence, in some cases there are very conclusive
signs of conformal theories. So that might be repaying debts which Nekrasov
mentioned.
B. Julia I would like just to make a general remark in the same spirit of being
useful. I think the most interesting thing I have learned, and everybody has
learned, here is that the classical world does not exist. There is no unique
classical limit. I think we should really develop more powerful tools to decide
