TFT construction of RCFT correlators III: Simple currents

TL;DR

This work provides a rigorous, cohomology-driven framework for constructing RCFT correlators using simple currents as invertible objects in modular tensor categories. By developing braided Picard theory, classifying haploid symmetric special Frobenius algebras, and translating these algebras into theta-category data and Kreuzer–Schellekens bihomomorphisms, the authors classify modular invariants, boundary conditions, and defects arising from simple currents. The results establish finiteness and Morita equivalence reductions, connect to known Kreuzer–Schellekens invariants, and furnish explicit boundary and defect formalisms via module and bimodule theories. Altogether, the paper provides a cohesive, cohomological foundation for simple-current constructions of RCFT correlators and their boundary/defect structures.

Abstract

We use simple currents to construct symmetric special Frobenius algebras in modular tensor categories. We classify such simple current type algebras with the help of abelian group cohomology. We show that they lead to the modular invariant torus partition functions that have been studied by Kreuzer and Schellekens. We also classify boundary conditions in the associated conformal field theories and show that the boundary states are given by the formula proposed in hep-th/0007174. Finally, we investigate conformal defects in these theories.

Theorems & Definitions (39)

• Definition 2.1
• Remark 2.3
• Proposition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Lemma 2.8
• Definition 2.9
• Remark 2.10
• Proposition 2.11