Hopf and Frobenius algebras in conformal field theory

TL;DR

This thesis advances the mathematical understanding of conformal field theories by generalizing correlator constructions to non-semisimple categories and establishing algebraic structures related to defects and bulk fields.

Contribution

It generalizes correlator construction proofs to oriented world sheets with defect lines and introduces a classifying algebra for defects, extending to non-semisimple categories relevant for logarithmic CFT.

Findings

Correlator construction satisfies consistency conditions with defect lines.

Classifying algebra for defects is a semisimple commutative algebra.

Morphisms associated with Riemann surfaces are invariant under mapping class group actions.

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.