Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)
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arXiv
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| Format: | Preprint |
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2014
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| author | Bischoff, Marcel Longo, Roberto Kawahigashi, Yasuyuki Rehren, Karl-Henning |
| author_facet | Bischoff, Marcel Longo, Roberto Kawahigashi, Yasuyuki Rehren, Karl-Henning |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1407_4793 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory) Bischoff, Marcel Longo, Roberto Kawahigashi, Yasuyuki Rehren, Karl-Henning Operator Algebras Mathematical Physics Category Theory 47L90, 81T05, 81T40 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]. |
| title | Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory) |
| topic | Operator Algebras Mathematical Physics Category Theory 47L90, 81T05, 81T40 |
| url | https://arxiv.org/abs/1407.4793 |