Annual Report 2005 - Fields Institute - University of Toronto

compactification of ten-dimensional superstring theories to
physically interesting four-dimensional settings. Hence her
perspective was one of working with the low-energy supergravity
equations, i.e., the stringy equivalent of the Einstein
equations, to describe these compactifications. Of course,
Calabi-Yau manifolds have long been used by physicists to
“curl up” the extra dimensions of string theory. Recently,
however, interest has turned to giving these internal spaces
“fluxes”, such as a nontrivial B-field. Mariana and her collaborators
have found that Hitchin’s generalized Calabi-Yau
manifolds emerge from the supersymmetry equations of
supergravity. In particular, the supersymmetry transformations
for type II theories on six-manifolds can be written
as differential conditions on a pair of pure spinors, the
exponentiated Kähler form and the holomorphic form.
Interestingly, mirror symmetry appears as a symmetry of
these equations under exchange of the two pure spinors
and a choice of even or odd-rank RR-fields, which are additional
flux fields which can appear in string theory. Further
these RR-fluxes appear as an obstruction to the integrability
of one of the pure spinors appearing in the generalized
geometries. One might take this result and Mariana’s
work that string theory actually uses further richer generalizations
of generalized geometries which remain to be
explored and understood.
It was perhaps appropriate that this mini-course on such
a young area of research was delivered by such a youthful
set of lecturers as Marco, Yi and Mariana. They all showed
great enthusiasm for the mini-course and their research
by working hard to prepare and deliver a very clear set of
lectures. In particular, while Yi Li is still “only” a graduate
student, everyone at the lectures would agree that he should
be commended on his superb command of the material and
the remarkable maturity which he showed in delivering his
lectures. Of course, the lecturers’ enthusiasm was matched
by that of the audience. As well as drawing a strong attendance
for all of the lectures, the mini-course also generated
many intense discussions of various aspects of generalized
geometries over lunch or in Perimeter’s many discussion
areas throughout the weeks of the course. As well as the
Perimeter researchers, the audience included a strong
contingent of both mathematicians and physicists coming
from Toronto and other area universities. The lectures even
attracted one student, Sven Rinke, all the way from the
Physics Department at Duke University.
As seen in the mini-course, generalized geometries are currently
at center stage in the interplay of mathematics and
physics. It is a topic, which one continued to hear about in
T h e m a t i c P r o g r a m s
the seminars and workshops throughout the remainder of
the String Theory program.
Minicourse on Toda Lattices: Basics and Perspectives
March–May 2005
Instructors: Boris Khesin (Toronto), Michael Gekhtman
(Notre Dame) and Andrei Marshakov (Moscow)
Toda lattice theory, the subject of the minicourse given
at the Fields Institute concurrently with the course on
“Symplectic Topology and Integrable Systems”, is one of the
main junctions in the theory of integrable equations. The
minicourse consisted of three lectures given by three lecturers
on different aspects of the Toda theory and covered
motivations and basic properties of the system, the integrability
of Toda flows on orbits of semi-simple Lie algebras
and the rather unexpected appearance of Toda lattices in
the Dijkgraaf-Vafa theory of matrix integrals. Although it
was selfcontained, this minicourse naturally complemented
the graduate course on integrable systems. It gave yet
another very popular and well studied model, described
along with its applications in field theory.
Workshop on Forms of Homotopy Theory: Elliptic
Cohomology and Loop Spaces
September 27–October 2, 2004
Held at the Fields Institute
Organizers: Matthew Ando (UIUC), Michael Hopkins
(MIT), Haynes Miller (MIT) and Jack Morava (Johns Hopkins)
This conference brought together mathematicians working
on problems in algebraic topology which are related to
string theory.
Michael J. Hokins, Jack Morava, Matthew Ando and Haynes R. Miller
Fields Institute 2005 ANNUAL REPORT 20