Picard groups in rational conformal field theory

Abstract

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 allows 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 coefficients and coefficients 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 correlators. 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.

Reflection and transmission for conformal defects

“…One can also consider a topological defect X close to a conformal boundary B. Since X commutes with the stress tensor, moving the defect against the boundary gives rise to a new conformal boundary condition X ⋆ B [10,15,38]. This defines an action of topological defects on boundary conditions.…”

The graphical calculus for ribbon categories: Algebras, modules, Nakayama automorphisms

“…A large number of nontrivial examples of Frobenius algebras is provided by so-called Schellekens algebras [9,6]; as objects they are direct sums of invertible objects, and they are classified in terms of the cohomology of the Picard group (the group of isomorphism classes of invertible objects) of C and of its subgroups. For a Frobenius algebra, any left module (Ṁ , ρ) gives rise to a left comodule (and vice versa), namely (Ṁ , ̺) with ̺ := (idȦ ⊗ ρ) • ((∆•η) ⊗ idṀ ).…”