Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)

Abstract

Q-systems describe "extensions" of an infinite von Neumann factor $N$, i.e., finite-index unital inclusions of $N$ into another von Neumann algebra $M$. They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of $N$. We review the relation between Q-systems, their modules and bimodules as structures in a category on one side, and homomorphisms between von Neumann algebras on the other side. We then elaborate basic operations with Q-systems (various decompositions in the general case, and the centre, the full centre, and the braided product in braided categories), and illuminate their meaning in the von Neumann algebra setting. The main applications are in local quantum field theory, where Q-systems in the subcategory of DHR endomorphisms of a local algebra encode extensions $A(O)\subset B(O)$ of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [arXiv:1405.7863, 1410.8848].

v1: 54 pages. v2: 90 pages. Title changed by request of editor, numerous material added, especially an overview of applications in Quantum Field Theory; some corrections. v3: minor corrections to match the published version; Springer Briefs in Mathematical Physics, vol. 3, 2015

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Cited by in corpus (40)

• On gauging finite subgroups
• Tensor categories for vertex operator superalgebra extensions
• Topological Defect Lines in Two Dimensional Fermionic CFTs
• Conformal Field Theory, Tensor Categories and Operator Algebras
• The Morita theory of quantum graph isomorphisms
• Drinfeld center and representation theory for monoidal categories
• Generalized Orbifold Construction for Conformal Nets
• Minimal index and dimension for 2-$C^{\ast }$-categories with finite-dimensional centers
• Spontaneous symmetry breaking from anyon condensation
• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Variational approach to relative entropies (with application to QFT)
• Phase boundaries in algebraic conformal QFT
• Q-systems and extensions of completely unitary vertex operator algebras
• Two-dimensional topological order and operator algebras
• Algebraic conformal quantum field theory in perspective
• Non-Equilibrium Thermodynamics and Conformal Field Theory
• Infinite index extensions of local nets and defects
• Fusion and positivity in chiral conformal field theory
• K-theoretic classification of inductive limit actions of fusion categories on AF-algebras
• Galois Correspondence and Fourier Analysis on Local Discrete Subfactors
• A few remarks on the tube algebra of a monoidal category
• Quantum Operations on Conformal Nets
• Wightman fields for two-dimensional conformal field theories with pointed representation category
• Braided categories of endomorphisms as invariants for local quantum field theories
• Anyonic Chains -- $\alpha$-Induction -- CFT -- Defects -- Subfactors
• Quantum cellular automata and categorical dualities of spin chains
• von Neumann Subfactors and Non-invertible Symmetries
• From vertex operator superalgebras to graded-local conformal nets and back
• Algebraic structures in two-dimensional conformal field theory
• Entanglement-symmetries of covariant channels
• Minimal index and dimension for inclusions of von Neumann algebras with finite-dimensional centers
• Bisognano-Wichmann property for rigid categorical extensions and non-local extensions of conformal nets
• On Induction for Twisted Representations of Conformal Nets
• A characterization of a finite-dimensional commuting square producing a subfactor of finite depth
• Quantum graphs, subfactors and tensor categories I
• A relative tensor product of subfactors over a modular tensor category
• Separable algebras in multitensor C${}^{\ast }$-categories are unitarizable
• A planar algebraic description of conditional expectations
• The relative Drinfeld commutant of a fusion category and $\alpha$-induction
• On a Connes Fusion Approach to Finite Index Extensions of Conformal Nets