I would like to discuss a class of three-dimensional topological field theories called Reshetikhin-Turaev theories. Examples are Chern-Simons theories with compact gauge group. There are no point-like observables in these theories, and one typically considers line-observables, called Wilson lines. However, one can also discuss observables associated to surfaces. We will see how to describe such observables in Reshetikhin-Turaev TQFTs and look at some applications.

Two-dimensional conformal field theory can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. Generalising what one finds for 2d topological field theories, a solution to these constraints can be obtained from a symmetric special Frobenius algebra in the appropriate braided monoidal category.

Given two conformal field theories, one of which is defined on the upper half plane and the other on the lower half plane, one can ask for conformally invariant ways to join them along the real line. The resulting interface is called a defect line. These defects contain interesting information about the CFT, such as its symmetries, order-disorder dualities and T-dualities. They also provide relations between string theories on different target spaces.

TBA

In this seminar we will look at two places where two-dimensional conformal field theory is related to the study of matrix models. The first is that a matrix model may be thought of as a statistical model on a fluctuating lattice. In the continuum limit, that is in the limit of large matrix size N, this yields a c less than 1 CFT coupled to gravity. The second relation is that the matrix model can, even before the continuum limit, be described in terms of free bosons. The free boson CFT turns out to be useful in understanding the large N behaviour of the matrix model.