For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism of H, we establish the existence of the following structure: an H-bimodule F ω and a bimodule morphism Z ω from Lyubashenko's Hopf algebra object K for the bimodule category to F ω . This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to F ω . We further show that the bimodule F ω can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal category of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple. The bimodules K and F ω can both be characterized as coends of suitable bifunctors. The morphism Z ω is obtained by applying a monodromy operation to the coproduct of F ω ; a similar construction for the product of F ω exists as well. Our results are motivated by the quest to understand the bulk state space and the bulk partition function in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple.

We demonstrate that topological defects in a rational conformal field theory can be described by a classifying algebra for defects -a finite-dimensional semisimple unital commutative associative algebra whose irreducible representations give the defect transmission coefficients. We show in particular that the structure constants of the classifying algebra are traces of operators on spaces of conformal blocks and that the defect transmission coefficients determine the defect partition functions.

Lyubashenko's construction associates representations of mapping class groups Map g:n of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H. For any such Hopf algebra we find an invariant of Map g:n for all values of g and n. More generally, we obtain such invariants for any pair (H, ω), where ω is a ribbon automorphism of H. Our results are motivated by the quest to understand higher genus correlation functions of bulk fields in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple, so-called logarithmic conformal field theories.

It is known that for any full rational conformal field theory, the correlation functions that are obtained by the TFT construction satisfy all locality, modular invariance and factorization conditions, and that there is a small set of fundamental correlators to which all others are related via factorization -provided that the world sheets considered do not contain any nontrivial defect lines. In this paper we generalize both results to oriented world sheets with an arbitrary network of topological defect lines.

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S 2 × S 1 . Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.