Correspondences of ribbon categories

TL;DR

This work develops a braided generalisation of correspondences by exploiting commutative symmetric Frobenius algebras in ribbon categories, and it constructs a robust framework of local modules, centers, and induction that mirrors and extends classical induction in symmetric settings. The authors introduce endofunctors $E_A^{l/r}$ and local induction $\,\ell$-Ind$_A^{l/r}$, establish central Frobenius-structure on centers, and prove a pivotal ribbon-equivalence between local modules over centers and ambichiral bimodules. A central achievement is a category-theoretic coset correspondence, showing that modular tensor categories are trivialisable and that one can express a target category as a local induction from a product with a dual category. The results illuminate coset constructions in conformal field theory, provide tools for analyzing chiral-extensions and modular invariants, and unify subfactor and vertex-algebraic approaches within a common categorical language.

Abstract

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.

Theorems & Definitions (133)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Lemma 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Lemma 2.9
• Definition 2.11