Hopf and Frobenius algebras in conformal field theory

Abstract: This thesis contains results relevant for two different classes of conformal field theory. We partly treat rational conformal field theory, but also derive results that aim at a better understanding of logarithmic conformal field theory. For rational conformal field theory, we generalize the proof that the construction of correlators, via three-dimensional topological field theory, satisfies the consistency conditions to oriented world sheets with defect lines. We also derive a classifying algebra for defects. This is a semisimple commutative associative algebra over the complex numbers whose one-dimensional representations are in bijection with the topological defect lines of the theory. Then we relax the semisimplicity condition of rational conformal field theory and consider a larger class of categories, containing non-semisimple ones, that is relevant for logarithmic conformal field theory. We obtain, for any finite-dimensional factorizable ribbon Hopf algebra H, a family of symmetric commutative Frobenius algebras in the category of bimodules over H. For any such Frobenius algebra, which can be constructed as a coend, we associate to any Riemann surface a morphism in the bimodule category. We prove that this morphism is invariant under a projective action of the mapping class group of the Riemann surface. This suggests to regard these morphisms as candidates for correlators of bulk fields of a full conformal field theory whose chiral data are described by the category of left-modules over H.