122 The Quantum Structure of Space and Time
grability in the loop space. Understanding what is the right way to construct
these operators is not standard mathematics, but I believe, it is important for
physics.
H. Ooguri From that point of view, string field theory has been one of the direc-
tions which has been pursued in order to take these ideas seriously. Do you
have any comment about that?
A. Polyakov Of course it is all very closely related to string theory. It is essentially
the question how the Schrodinger equation for string theory looks like, how to
effectively write it down correctly. For example, how should one write the
Schrodinger equation when the boundary data are given at infinity, like in Ads
space. How should one write the Wheeler-DeWitt equation? Normally, we
have some boundary and the Wheeler-DeWitt equation is the equation for the
partition function as a functional of this boundary. But if this boundary is at
infinity then, instead of the usual Laplacian, there should be some first order
operator like the loop Laplacian. How should one make this concrete? I think
these are important questions, both for string theory and for gauge theory, of
course.
L. Faddeev After the “insult” to mathematics from my friend, I must say that I
think there are two intuitions. There is the physical intuition and the math-
ematical intuition. And now, in this field, they compete and they help each
other. There was a try to make a kind of consensus in Princeton five years ago,
not with real success. But I think we have still to work somehow together to
try to have some consensus of these two different intuitions.
H. Ooguri We will hear from three panelists now and then I will open the floor
for discussions. I would like to suggest that the discussion today should focus
on mathematical structures of string vacua, and in particular on what we have
learnt about the space of vacua and on what we need to do to better under-
stand it. We should leave the discussion on the physical implications, such
as the multi-universes and the anthropic principle to the session on cosmology
tomorrow.
