Morita classes of algebras in modular tensor categories

TL;DR

This paper studies algebras in modular tensor categories, establishing a Morita equivalence criterion via full centres, with implications for boundary conditions in rational conformal field theory.

Contribution

It introduces the full centre construction for algebras in modular tensor categories and proves a Morita equivalence criterion based on isomorphism of these centres.

Findings

Morita equivalence characterized by isomorphic full centres

Full centre construction for algebras in modular tensor categories

Implication for boundary conditions in conformal field theory

Abstract

We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.