Three-Tier CFTs from Frobenius Algebras

Abstract

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.