116 The Quantum Structure of Space and TimeComputing the BPS states through geometric quantization we find that
abrane(p) = H*(Mp).
The cohomology of these moduli spaces have been computed [30] with the result
that
@ qNH*(HilbN(T4)) = Fq.
N>OThis gives the final result
Bbrane (p) = F(/.2/2) Bstring (P)
where p and p are related by an S0(5,5, Z) transformation.
This is just a simple example to show that indeed the same mathematical struc-
tures (representation theory of affine lie algebras, Virasoro algebras, etc.) can ap-
pear both in the perturbative theory of strings and non-perturbative brane systems.
Again this is a powerful1 hint that a more unified mathematical structure underlies
quantum gravity.
4.1.6 The Role of Mathematics
In this rapporteur talk we have surveyed some deep connections between physics
and mathematics that have stimulated much intellectual activity. Let me finish to
raise some questions about these interactions.
0 First of all, it must be said that despite all these nice results, there does seem
to operate a principle of complementarity (in the spirit of Niels Bohr) that
makes it difficult to combine physical intuition with mathematical rigor. Quite
often, deep conjectures have been proven rigorously, not be making the physical
intuition more precise, but by taking completely alternative routes.0 It is not at all clear what kind of mathematical structure Nature prefers. Here
there seem to be two schools of thoughts. One the one hand one can argue that
it is the most universal structures that have proven to be most successful. Here
one can think about the formalism of calculus, Riemannian manifolds, Hilbertspaces, etc. On the other hand, the philosophy behind a Grand Unified Theory
or string theory, is that our world is very much described by a single unique(?)
mathematical structure. This point of view seems to prefer exceptional mathe-
matical objects, such as the Lie algebra E8, Calabi-Yau manifolds, etc. Perhaps
in the end a synthesis of these two points of view (that roughly correspond to
the laws of Nature versus the solutions of these laws) will emerge.
0 Continuing this thought it is interesting to speculate what other mathematical
fields should be brought into theoretical physics. One could think of number
theory and arithmetic geometry, or logic, or even subjects that have not been