Session 4
Mathematical Structures
Chair: Hiroshi Ooguri, Caltech, USA
Rapporteur: Robbed Dijlcgrauf, University of Amsterdam, the Netherlands
Scientific secretaries: Philippe Spindel (Universitk de Mons-Hainaut) and Laurent
Houad (Universiti: Libre de Bruxelles)4.1 Rapporteur talk: Mathematical Structures, by Robbert Dijk-
graaf
4.1.1 Abstract
The search for a quantum theory of gravity has stimulated many developments in
mathematics. String theory in particular has had a profound impact, generating
many new structures and concepts that extend classical geometry and give indica-
tions of what a full theory of quantum gravity should entail. I will try to put some
of these ingredients in a broader mathematical context.
4.1.2 Quantum Theory and Mathematics
Over the years the search for a theory of quantum gravity has both depended on
and enriched many fields of mathematics. String theory [l] in particular has had an
enormous impact in mathematical thinking. Subjects like algebraic and differential
geometry, topology, representation theory, infinite dimensional analysis and many
others have been stimulated by new concepts such as mirror symmetry [a], [3],
quantum cohomology [4] and conformal field theory [5]. In fact, one can argue
that this influence in mathematics will be a lasting and rewarding impact of string
theory, whatever its final role in fundamental physics. String theory seem to be the
most complex and richest mathematical object that has so far appeared in physics
and the inspiring dialogue between mathematics and physics that it has triggered
is blooming and spreading in wider and wider circles.
This synergy between physics and mathematics that is driving so many de-
velopments in modern theoretical physics, in particularly in the field of quantum91