Mathematical Structures 151
4.6 Discussion
B. Greene Just in the spirit of the Mathematics-Physics interface which is the
theme of the session, I might note that Dine and Seiberg some time ago have
showed that the existence of certain R-symmetries allows you to prove that
there can be exact flat directions. These directions can give rise to the kinds of
deformations that Yau was talking about, and in particular, the one example
on the three generation Calabi-Yau, the deformation T @ 0 @ 0. You can in
fact realise an example of that sort using the R-symmetries and prove that such
a solution would exist. So you can have a physics proof, if you will, of that
particular example that Yau was discussing.
A. Strominger Perhaps the most interesting new things are the ones that are not
obtained by deformations like this last example and really cannot be understood
by any such arguments. I have a question for Yau. Twenty years ago you made
your famous estimate that there were ten thousand Calabi-Yau spaces. How
many of these things do you think there are?
S.T. Yau More, I think, that is all I can say at this moment.
N. Seiberg I have two questions and I am glad that most of the relevant experts
are in the audience. The first question is: What is the status of the non-
perturbative existence of the topological string? The second question is: We
have learned about many new Calabi-Yau spaces. How many of them look like
the real world?
H. Ooguri I would like to personally respond to the first question. I can see at
least two independent non-pertubative completions of the topological string in
certain situations. In the case when you have an open string dual you can
often use a matrix model to give a non-pertubative completion in the sense
that you have a convergent matrix integral whose perturbative expansion gives
rise to topological string amplitudes in the close string dual. On the other
hand you can also propose to define topological string amplitudes in terms of
black hole entropy, where the counting of number of states of black holes is
well defined and the perturbative expansion of this counting, in particular the
generating function, gives rise again to topological string partition function via
the OSV conjecture. You can see in particular examples that these two give
rise to different non-pertubative completions. One possible view is that the
topological string is a tool to address various interesting geometric programs
in physics. Depending on situations, there can be different non-perturbative
completions. But there might be people with other views on that.
N. Seiberg I am not aware of an example where you have two systems which are
the same to all orders in perturbation theory and their D-branes are the same,
in the sense that you can probe the system with large classical field excursions,
and yet they have more than one non-pertubative completion.
H. Ooguri Yes, so this might be a counterexample to that.
