## Coxeter Lecture Series 2000

### ARTICLE

Coxeter Lectures: Graeme Segal

In the first week of December 2000 Prof. Graeme Segal (Oxford University)
gave three lectures as part of the Coxeter Lecture Series, entitled
"The Idea of Space in String Theory". The general theme of
the lectures was quantum field theory; roughly, a two-dimensional field
theory associates a complex vector space H to a compact one-dimensional
manifold (in other words, a circle) and associates a linear map from
H p to H q to a compact oriented two manifold whose boundary consists
of a number of copies of the circle, p of which are positively oriented
and q negatively. If is equipped with a conformal structure and the
output data are unchanged by diffeomorphisms of preserving the conformal
structure, then the theory is called a conformal field theory.

Examples may be constructed using the loop space of a Riemannian manifold
X (the spacetime) and the space of smooth maps from into X. In some
illuminating examples the space H is a representation of the loop group
associated to a compact Lie group.

The lectures (which surveyed ideas based in part on joint work with
M. Atiyah and G. Moore) outlined the idea of a B-field (or equivalently
gerbe with connection) which features prominently in recent string theory
and is the analogue of the parallel transport specified by a connection.
A B-field gives rise to a closed 3-form, which is the analogue of the
2-form representing the curvature of a connection. A distinguished class
of B-fields are those satisfying the natural set of field equations
(analogues of the Einstein-Maxwell equations). Two examples of Riemannian
manifolds X equipped with B-fields where the field equations are satisfied
are flat Riemannian tori and compact Lie groups equipped with bi-invariant
metrics. The lectures also described the concept of a D-brane, which
is a submanifold of X equipped with additional structure (for example
vector bundles with connections). Finally the lectures outlined recent
joint work with G. Moore in which D-branes are related to a form of
K-theory (twisted K-theory).

Lisa Jeffrey, University of Toronto