There is a beautiful classification of full (rational) CFT due to
Fuchs, Runkel and Schweigert. The classification says roughly the
following. Fix a chiral algebra A (= vertex algebra). Then the set of
full CFT 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 when the chiral
algebra A is affine su(2): 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-dimentional Chern-Simons theory with
defects. Those defects are also given by Frobenius algebras in Rep(A).
Inspired by the proposal of Kapustin and Saulina, we will (partially)
construct the three-tier CFT associated to a given Frobenius algebra.