From Quantum Groups to Unitary Modular Tensor Categories

Abstract

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

16 pages, final version to appear in Contemporary Mathematics. Version 2: minor typos fixed, references updated, Theorem 4.5 clarified. Version 3: major addition deriving generating functions for ranks, paper reorganized

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• Simulation of topological field theories by quantum computers
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Vertex operator algebras, the Verlinde conjecture and modular tensor categories
• The N-eigenvalue Problem and Two Applications

Cited by in corpus (20)

• Fermionic Modular Categories and the 16-fold Way
• On classification of modular categories by rank
• Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs
• Higher central charges and Witt groups
• Unitarizablity of premodular categories
• On Arithmetic Modular Categories
• Quantization via Linear homotopy types
• Rank 4 Premodular Categories
• Braid representations from quantum groups of exceptional Lie type
• Unitary Braid Representations with Finite Image
• Algebraic number fields generated by Frobenius-Perron dimensions in fusion rings
• Finite Linear Quotients of $\B_3$ of Low Dimension
• Classification of Rank 6 Modular Categories with Galois Group $⟨(012)(345)⟩$
• G-Extensions of Quantum Group Categories and Functorial SPT
• On the classification of non-self-dual modular categories
• Unitary coboundary weak quasi-Hopf algebras in the Kazhdan-Lusztig-Finkelberg theorem
• Topological-Like Features in Diagrammatical Quantum Circuits
• Modular categories, integrality and Egyptian fractions
• Modular tensor categories, subcategories, and Galois orbits
• Generating functions for ranks of pre-modular categories