April 24, 2014

Coxeter Lecture Series 2000


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