  These are lecture notes of a course given at the Summer School on Topology
and Field Theories held at the Centre for Mathematics of the University of
Notre Dame, Indiana, from May 29 to June 2, 2012. The idea of extending quantum
field theories to manifolds of lower dimension was first proposed by Dan Freed
in the nineties. In the case of conformal field theory (CFT), we are talking of
an extension of the Atiyah-Segal axioms, where one replaces the bordism
category of Riemann surfaces by a suitable bordism bicategory, whose ob jects
are points, whose morphisms are 1-manifolds, and whose 2-morphisms are pieces
of Riemann surface. There is a beautiful classification of full (rational) CFTs
due to Fuchs, Runkel and Schweigert, which roughly says the following. Fix a
chiral algebra A (= vertex algebra). Then the set of full cfts whose left and
right chiral algebras agree with A is classified by Frobenius algebras internal
to Rep(A). A famous example to which one can successfully apply this is the
case where the chiral algebra A is affine su(2) at level k, for some k in N. In
that case, the Frobenius algebras in Rep (A) are classified by A_n, D_n, E_6,
E_7, E_8, and so are the corresponding CFTs. Recently, Kapustin and Saulina
gave a conceptual interpretation of the FRS classification in terms of
3-dimensional Chern-Simons theory with defects. Those defects are also given by
Frobenius algebra ob ject in Rep(A). Inspired by the proposal of Kapustin and
Saulina, we will (partially) construct the three-tier CFT associated to a
Frobenius algebra object.
