Mathematical Structures 121the effective Lagrangian is to all orders in R2, we still know what the answer is
for g-string equals zero, that is on the sphere. For such systems, we sometimes
even know the answer on the torus and for higher genus surfaces. In this way
we can somehow circumvent the a‘-correction problem.
R. Dijkgraaf I do not really know what to say in response. Concerning your
first remark, we have indeed often asked ourselves “What is string theory?”.
But now, we have to ask “What is not string theory?” That is really a big
question because, in some sense, finding the upper bound, finding structures
which definitely are not connected in any way, is probably now more challenging
than the other way round.P. Ramond You talked, glibly, about exceptional structures. The next frontier
in a sense is that, besides non-commutativity, non-associativity occurs with
specific systems. Have you encountered any need for this? Does it come up
geometrically?R. Dijkgraaf That is again speculative. Roughly, look at the three levels. At
the first level of particles, we have gauge fields. At the second level where we
have strings, we have these kind of B-fields. That is a 2-form field that is notrelated to a gauge field but to something that is called gerbes. And then, in
M-theory, there is the 3-form field. So the B-field is clearly related to non-
commutativity. But the 3-form field, if with anything, it might be related to
non-associativity. But again, there have been many isolated ideas, but I donot think there is something that ties everything together. In fact, it is also a
question to the mathematicians. For instance, in string theory, the B-field is
intimately connected to K-theory. It is related to some periodicity mod 2. The
2 is the same 2 of the B-field. In fact, in M-theory, there is almost something
like a periodicity mod 3. So again, the question is what replaces this structure.I do not think we have any hint. There are some suggestions that E8 plays an
important r61e. So there is a direct connection between gauge theories of Es
and 3-forms. Greg Moore has very much pushed this. But again, these are only
quest ions.A. Polyakov This fusion of physics and mathematics which occurred in the last
few decades and which you discussed so beautifully is quite amazing of course.
But it is actually bothersome to me because it seems that we start following the
steps of mathematicians. I believe we are supposed to invent our own mathe-
matics even though modern mathematics is very seductive. Mathematicians areclever enough to do their own job, I suppose. Anyway that is general philosophy.
More concretely, I think there is an unexplored domain which is important forunderstanding gauge-string duality. It is differential geometry in loop space.
In gauge theory we get some interesting differential operators in loop space,
which are practically unexplored. Very few, primitive, things are known about
these operators and the loop equations for gauge theories. There is obviously
some deep mathematical structure. There could be some Lax pairs or inte-