Annual Report 2005 - Fields Institute - University of Toronto

The Geometry of String Theory: September 2004–June 2005
Retrospective on the Thematic Program
Organizing Committee: Kentaro Hori and Lisa Jeffrey
(Toronto), Mikhail Kapranov (Yale and Toronto), Boris
Khesin (Toronto), Rob Myers (Perimeter), and Amanda
Peet (Toronto)
Scientific Committee: Alexander Beilinson (Chicago), Jim
Bryan (UBC), D. Freed (Austin), Kentaro Hori (Toronto),
Jacques Hurtubise (McGill), Lisa Jeffrey (Toronto), Mikhail
Kapranov (Yale and Toronto), Sheldon Katz (Urbana-
Champaign), Boris Khesin (Toronto), Robert Myers
(Perimeter Institute), Amanda Peet (Toronto), Edward
Witten (I.A.S.) and Noriko Yui (Queen’s)
This year’s thematic program on the Geometry of String
Theory was hosted jointly by the Fields Institute and Perimeter
Institute for Theoretical Physics (in nearby Waterloo).
The activities were essentially divided equally between
Fields and Perimeter.
The central idea defining string theory is that, when viewed
with sufficiently high resolution, all elementary particles
will appear to be extended one-dimensional objects, i.e.,
strings. From this relatively simple starting point emerges
an extraordinarily rich mathematical structure. For
example, the internal consistency of the theories requires
that the strings propagate in a ten-dimensional spacetime.
So, six of the dimensions must be curled up on a compact
geometry in order to reproduce the four-dimensional
physics which we observe. In fact it is the intricacies of this
internal geometry which are responsible for the complex
physical interactions which emerge in the four-dimensional
world. This is a simple example which illustrates the central
role which geometry plays in string theory.
Further, string theory incorporates ‘supersymmetry’, a
symmetry which changes the spins of elementary particles
Organizers: Lisa Jeffrey, Mikhail Kapranov, Boris Khesin, Robert Myers, and Amanda Peet
T h e m a t i c P r o g r a m s
pairing each fermion with a boson – this is the origin of
the ‘super’ in superstrings. On the physical side, supersymmetry
plays an important role in taming high energy
divergences which appear in theories of point particles. On
the mathematical side, this symmetry is fundamental in
constructing new topological invariants, e.g., in Seiberg-
Witten theory. More recently, it has been realized that
string theory is more that just a theory of strings. That
is, there are other kinds of extended objects, known as
Dirichlet-branes or D-branes, which play an important
role. In fact in the past ten years, D-branes have been the
source of a profusion of new ideas and remarkable progress
in our understanding of string theory. In particular, Dbranes
played a crucial role in constructing new relations
or ‘dualities’ between what were previously seen as five
different consistent superstring theories. Instead it is now
believed that they represent different phases of a single unified
framework, commonly known as M-theory. Of course,
mathematics played an important role in this progress as
well. For example, the proper classification of D-branes
comes from K-theory.
String theory originated from a ‘failed’ attempt in the
1960’s to describe the nuclear interactions. Soon after, it
evolved towards a very much more ambitious goal of being
‘the theory of everything’– that is a providing a unified
framework to describe all of the elementary particles and
fundamental forces in nature. Combining Einstein’s theory
of gravity with the standard quantum theory used to
describe physics at subatomic scales has lead to perplexing
inconsistencies which have mystified physicists for over
fifty years. Hence finding a quantum theory of gravity is
often seen as the holy grail of theoretical physics. String
theory is seen by many as the leading contender in this
quest since as well as containing the appropriate structures
Fields Institute 2005 ANNUAL REPORT 9