Ribbon categories and (unoriented) CFT: Frobenius algebras, automorphisms, reversions

TL;DR

This paper explores the algebraic structures underlying conformal field theories, focusing on Frobenius algebras in modular tensor categories, their automorphisms, and the additional structures needed for unoriented world sheets.

Contribution

It introduces a group homomorphism from automorphisms of Frobenius algebras to the Picard group, and refines Morita equivalence to include algebras with reversion for unoriented cases.

Findings

Established a group homomorphism from Aut(A) to the Picard group.

Refined Morita equivalence to account for algebras with reversion.

Clarified the role of automorphisms and reversions in unoriented CFTs.

Abstract

A Morita class of symmetric special Frobenius algebras A in the modular tensor category of a chiral CFT determines a full CFT on oriented world sheets. For unoriented world sheets, A must in addition possess a reversion, i.e. an isomorphism from A^opp to A squaring to the twist. Any two reversions of an algebra A differ by an element of the group Aut(A) of algebra automorphisms of A. We establish a group homomorphism from Aut(A) to the Picard group of the bimodule category C_AA, with kernel consisting of the inner automorphisms, and we refine Morita equivalence to an equivalence relation between algebras with reversion.