160 The Quantum Structure of Space and Tame
ematical space to some extent has to be explored. Then, just to give a glib
answer to Seiberg's question: if you make the zeroth order sorts of estimatesthat I made in my work of two years ago and if you would incorporate both
the issues that Lust talked about, then the difficulty of getting the standard
model is one in a billion. It is one in lo2'' taking into account the cosmological
constant and other factors. You then decide that out of this number of Calabi-
Yaus and fluxes and the rest, something like one in lo2'' to lo3'' should work.
Although you may say: "How can you say such a thing?", the arguments have
the virtue that they are very simple. They are using very little input and so onecan take them as a zeroth order starting point. I certainly hope that is not the
right answer and that there are far more features that have not been exploited
yet in terms of the structure of the string vacua. To the extent that there
is structural peaking, then there might be far fewer that match the standard
model, or far more. Both of those possibilities would be interesting. It could bethat there is information from early cosmology. Just the difficulty of getting a
viable cosmology that will fit the data, or these more speculative considerations
about the wave function measure factors and so forth, all this could drasticallyaffect this calculation. All I am saying by throwing out a number like this
is the following: here is a framework in which to think about the problem of
combining these many disparate ingredients and talk about them together.D. Gross It seems to be clear that the one thing that was missing from this session,
in which we focused partly on mathematical structures that might reveal the
nature of space and time, was time, except in the last talk which could have
been delivered back in 1911. It is clear that elliptic equations are easier than
hyperbolic, and Euclidean metrics easier than Lorentzian, and ignoring time
easier than taking it into account. But for example the discussion of what
some people mistakenly call vacua, which are really metastable states, should
illustrate that they are discussing things which are of course time dependent.
Yet, nothing is known about the time evolution. This indicates that, surely,
the big open issue in string theory is time. What does it mean for time to
be emergent? Non-locality in time, how do we deal with that? How do we
make or have a causal structure? How do we discuss metastable states whosebeginning we know nothing about, and so on. It seems that a lot of tools we
are focusing on are avoiding the tough questions because the mathematics issimpler. As I said, we prefer to study elliptic equations rather than the hard
case of hyperbolic equations.N. Arkani-Hamed I just wanted to ask something about the status of large num-
bers of vacua in the heterotic context. It seems that a lot of the studies of
the statistics for type IIB vacua for example, which are definitely interesting,
are less likely to apply to the real world if we take the hint from gauge cou-
pling unification seriously. Everything seemed to be going swimmingly in the