Picard groups in rational conformal field theory

Abstract

Algebra and representation theory in modular tensor categories can be combined with tools from topological field theory to obtain a deeper understanding of rational conformal field theories in two dimensions: It al- lows us to establish the existence of sets of consistent correlation functions, to demonstrate some of their properties in a model-independent manner, and to derive explicit expressions for OPE coecients and coecients of partition functions in terms of invariants of links in three-manifolds. We show that a Morita class of (symmetric special) Frobenius algebras A in a modular tensor category C encodes all data needed to describe the correla- tors. A Morita-invariant formulation is provided by module categories over C. Together with a bimodule-valued fiber functor, the system (tensor category + module category) can be described by a weak Hopf algebra. The Picard group of the category C can be used to construct examples of symmetric special Frobenius algebras. The Picard group of the category of A-bimodules describes the internal symmetries of the theory and allows one to identify generalized Kramers-Wannier dualities.