Annual Report 2005 - Fields Institute - University of Toronto

to describe elementary particle physics; gravity is required
by the internal consistency of the theory. However, even
after more than three decades of intense investigation, our
current understanding of string theory remains inadequate
to understand whether or not it consistently describes
physical phenomena at all energies, i.e., from the Planck
scale, where the effects of quantum gravity and strings are
manifest, through to energy scales accessible in present-day
experiments.
Perhaps in an analogous way, string theory may also be
fostering a unification of mathematics. For example, the
‘mirror symmetry’ of strings on Calabi-Yau manifolds
displays a close connection between symplectic geometry
and algebraic geometry. However, it must be said that
the full mathematical implications of strings, supersymmetry,
dualities and D-branes remain to be understood.
There seem to be hints of new connections between such
diverse areas of mathematics as derived categories, elliptic
cohomology, geometric Langlands correspondence,
quantum cohomology, differential geometry varieties with
special holonomy and of special Lagrangian varieties. It is
also clear that the full physics potential of this remarkable
theory will only be realized once significant progress has
been made in understanding its mathematical structure.
Of course, superstring theory has long stood at centre stage
in the interplay between mathematics and physics and has
already proven to be a phenomenal source of new ideas
for both fields. Some of the most notable aspects of mathematics
where the interplay with string theory is relevant
include:
1. Algebraic topology (as with the first workshop at Fields
on Elliptic Cohomology and Loop Spaces: loop spaces are a
subject with obvious connections with string theory)
2. Algebraic geometry (particularly enumerative geometry,
which has received a great deal of input from mirror symmetry)
3. Riemannian geometry (Riemannian metrics, connections
and curvature belong in the toolkit of a string
theorist, and most string theories are framed in terms of
Calabi-Yau manifolds, those complex manifolds for which
the canonical bundle is trivial. The differential geometry of
manifolds with special holonomy and special Lagrangian
manifolds has attracted a great deal of attention)
4. Homological algebra (as treated in the graduate course by
Ragnar Buchweitz)
T h e m a t i c P r o g r a m s
5. Category theory (topics such as derived categories have
found many applications in string theory)
6. Representation theory (representations are ubiquitous in
string theories, and some nonstandard versions such as the
representation theory of affine Lie algebras and the geometric
Langlands correspondence have strong connections to
string theory)
7. Symplectic geometry and symplectic topology (as treated
in the graduate course by Boris Khesin)
8. Algebraic combinatorics (which formed the focus of the
Workshop on Schubert Varieties). This subject has to do
with computing the Gromov-Witten invariants (quantum
cohomology) of various target spaces (for example flag
manifolds), and is an offshoot of the A model of topological
string theory
9. Integrable systems (a subject with strong links to string
theory, as in the KdV and KP hierarchies, the lectures of
Boris Dubrovin on Frobenius manifolds, and in the minicourse
on Toda theory
10. Twisted K-theory, gerbes and other topics associated to
equivariant cohomology (appearing for example in Nigel
Hitchin’s Coxeter lecture series)
The aim of this year’s program was to foster new progress
in understanding the foundations of string/M-theory. As
well as the Fields Institute’s strong reputation in the mathematical
community, the program capitalized on the recent
creation of string theory groups in the Physics Department
at the University of Toronto and at Perimeter Institute in
Waterloo. In particular, running the program jointly with
Perimeter Institute produced a particularly active year with
over 800 participants, two parallel seminar series (one at
each site), seven workshops, four graduate courses, three
mini-courses and a graduate summer school. Further
the program concluded in July by hosting Strings05, the
premier international conference in string theory. The program
attracted many of the world leaders in their respective
fields. The most exciting topics at the forefront of research
in string theory were reported on through all of the various
activities, and innumerable collaborations and interactions
were stimulated by all of this activity.
The activities in the fall term (between September and
January) were planned to emphasize mathematical aspects,
while those in the spring term (February to July) were
planned to emphasize physical aspects. The highlights of
the program were the seven workshops (see the list later in
this section, and the individual workshop reports).
Fields Institute 2005 ANNUAL REPORT 10