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arXiv:1407.4793

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

Marcel Bischoff, Longo, Roberto, Kawahigashi, Yasuyuki, Karl-Henning Rehren

Published Jul 17, 2014• 31 citations
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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].

DOI
10.48550/arxiv.1407.4793

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Q-systems describe "extensions" of an infinite von Neumann factor NNN, i.e., finite-index unital inclusions of NNN into another von Neumann algebra MMM. They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of NNN. 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)⊂B(O)A(O)\subset B(O)A(O)⊂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].
The subfactor theory provides a tool to analyze and construct extensions of Quantum Field Theories, once the latter are formulated as local nets of von Neumann algebras. We generalize some of the results of [62] to the case of extensions with infinite Jones index. This case naturally arises in physics, the canonical examples are given by global gauge theories with respect to a compact (non-finite) group of internal symmetries. Building on the works of Izumi–Longo–Popa [44] and Fidaleo–Isola [30], we consider generalized Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite von Neumann algebras, which generalize ordinary Q-systems introduced by Longo [58] to the infinite index case. We characterize inclusions which admit generalized Q-systems of intertwiners and define a braided product among the latter, hence we construct examples of QFTs with defects (phase boundaries) of infinite index, extending the family of boundaries in the grasp of [7].
A subtheory of a quantum field theory specifies von Neumann subalgebras A(phi) (the 'observables' in the space-time region phi) of the von Neumann algebras B(phi) (the 'fields' localized in phi). Every local algebra being a (type III1) factor, the inclusion A(phi) subset of B(phi) is a subfactor. The assignment of these local subfactors to the space-time regions is called a 'net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the 'relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.
The article arXiv:2410.21454 constructed a braided W∗\mathrm{W^*}W∗-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category B\mathcal{B}B in arXiv:2506.19969 is equivalent to Hilb(B)op\mathsf{Hilb}(\mathcal{B})^{\rm op}Hilb(B)op, the opposite of the unitary ind-completion of B\mathcal{B}B, as a braided W∗\mathrm{W^*}W∗-tensor category.
A subtheory of a quantum field theory specifies von Neumann subalgebras [Formula: see text] (the ‘observables’ in the space-time region [Formula: see text]) of the von Neumann algebras [Formula: see text] (the 'field' localized in [Formula: see text]). Every local algebra being a (type III1) factor, the inclusion [Formula: see text] is a subfactor. The assignment of these local subfactors to the space-time regions is called a ‘net of subfactors’. The theory of subfactors is applied to such nets. In order to characterize the ‘relative position’ of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions ℬ of a given theory [Formula: see text] in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.
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