Morita classes of algebras in modular tensor categories

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.

28 pages, several figures, version to appear in Adv. Math

References in corpus (18)

• TFT construction of RCFT correlators I: Partition functions
• Duality and defects in rational conformal field theory
• Multi-interval Subfactors and Modularity of Representations in Conformal Field Theory
• TFT construction of RCFT correlators III: Simple currents
• Correspondences of ribbon categories
• TFT construction of RCFT correlators II: Unoriented world sheets
• D-branes and K-theory in 2D topological field theory
• Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
• Local fields in boundary conformal QFT
• Full field algebras
• Some Comments on Branes, G-flux, and K-theory
• Open-string vertex algebras, tensor categories and operads
• The fusion algebra of bimodule categories
• Cardy condition for open-closed field algebras
• Full field algebras, operads and tensor categories
• Open-closed field algebras
• Trace formulas for Annuli
• Rigidity and modularity of vertex tensor categories

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• Cardy algebras and sewing constraints, I
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• Invertible defects and isomorphisms of rational CFTs
• Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers
• A mathematical theory of gapless edges of 2d topological orders. Part II
• The Morita theory of quantum graph isomorphisms
• Fusion Rules of the ${\mathcal{W}}_{p,q}$ Triplet Models
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Cardy algebras and sewing constraints, II
• Pointed Drinfeld center functor
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• Tensor categories: A selective guided tour
• A note on permutation twist defects in topological bilayer phases
• Algebraic conformal quantum field theory in perspective
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• D-branes at multicritical points
• Algebraic Structures in Euclidean and Minkowskian Two-Dimensional Conformal Field Theory
• Bulk from boundary in finite CFT by means of pivotal module categories
• CFT Correlators for Cardy Bulk Fields via String-Net Models
• The three-dimensional origin of the classifying algebra
• Module categories for permutation modular invariants
• The rank of $G$-crossed braided extensions of modular tensor categories
• Full centre of an H -module algebra
• On the structure of the Witt group of braided fusion categories
• Hopf and Frobenius algebras in conformal field theory
• A relative tensor product of subfactors over a modular tensor category
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