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C* tensor categories are a point of contact where Operator Algebras and Quantum Field Theory meet. They are the underlying unifying concept for homomorphisms of (properly infinite) von Neumann algebras and representations of quantum observables.
The present introductory text reviews the basic notions and their cross-relations in different contexts. The focus is on Q-systems that serve as complete invariants, both for subfactors and for extensions of quantum field theory models.
It proceeds with various operations on Q-systems (several decompositions, the mirror Q-system, braided product, centre and full centre of Q-systems) some of which are defined only in the presence of a braiding.
The last chapter gives a brief exposition of the relevance of the mathematical structures presented in the main body for applications in Quantum Field Theory (in particular two-dimensional Conformal Field Theory, also with boundaries or defects).
List of contents
Introduction.- Homomorphisms of von Neumann algebras.- Endomorphisms of infinite factors.- Homomorphisms and subfactors.- Non-factorial extensions.- Frobenius algebras, Q-systems and modules.- C* Frobenius algebras.- Q-systems and extensions.- The canonical Q-system.- Modules of Q-systems.- Induced Q-systems and Morita equivalence.- Bimodules.- Tensor product of bimodules.- Q-system calculus.- Reduced Q-systems.- Central decomposition of Q-systems.- Irreducible decomposition of Q-systems.- Intermediate Q-systems.- Q-systems in braided tensor categories.- a-induction.- Mirror Q-systems.- Centre of Q-systems.- Braided product of Q-systems.- The full centre.- Modular tensor categories.- The braided product of two full centres.- Applications in QFT.- Basics of algebraic quantum field theory.- Hard boundaries.- Transparent boundaries.- Further directions.- Conclusions.
Summary
C* tensor categories are a point of contact where Operator Algebras and Quantum Field Theory meet. They are the underlying unifying concept for homomorphisms of (properly infinite) von Neumann algebras and representations of quantum observables.
The present introductory text reviews the basic notions and their cross-relations in different contexts. The focus is on Q-systems that serve as complete invariants, both for subfactors and for extensions of quantum field theory models.
It proceeds with various operations on Q-systems (several decompositions, the mirror Q-system, braided product, centre and full centre of Q-systems) some of which are defined only in the presence of a braiding.
The last chapter gives a brief exposition of the relevance of the mathematical structures presented in the main body for applications in Quantum Field Theory (in particular two-dimensional Conformal Field Theory, also with boundaries or defects).
Additional text
“The volume gives a coherent overview of some recent mathematical developments in the study of endomorphisms of von Neumann algebras and their applications in algebraic quantum field theory. … every chapter has its own list of references, which points the reader to more detailed literature. … Anyone who wishes to understand the recent advances in our understanding of endomorphisms of von Neumann algebras … should find this book a valuable resource.” (Ko Sanders, Mathematical Reviews, January, 2016)
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"The volume gives a coherent overview of some recent mathematical developments in the study of endomorphisms of von Neumann algebras and their applications in algebraic quantum field theory. ... every chapter has its own list of references, which points the reader to more detailed literature. ... Anyone who wishes to understand the recent advances in our understanding of endomorphisms of von Neumann algebras ... should find this book a valuable resource." (Ko Sanders, Mathematical Reviews, January, 2016)
