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Tensor Categories and Endomorphisms of von Neumann Algebras

with Applications to Quantum Field Theory

  • Book
  • © 2015

Overview

Authors:
  1. Marcel Bischoff

    1. Universität Göttingen, Institut für Theoretische Physik, Göttingen, Germany

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  2. Yasuyuki Kawahigashi

    1. Department of Mathematical Sciences and Kavli IPMU (WPI), The University of Tokyo, Tokyo, Japan

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  3. Roberto Longo

    1. Dipartimento di Matematica, Università di Roma "Tor Vergata", Rome, Italy

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  4. Karl-Henning Rehren

    1. Universität Göttingen, Institut für Theoretische Physik, Göttingen, Germany

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Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 3)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren
    Pages 1-4

  3. Homomorphisms of von Neumann Algebras

    • Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren
    Pages 5-13

  4. Frobenius Algebras, Q-Systems and Modules

    • Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren
    Pages 15-40

  5. Q-System Calculus

    • Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren
    Pages 41-76

  6. Applications in QFT

    • Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren
    Pages 77-91

  7. Conclusions

    • Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren
    Pages 93-94
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Keywords

About this book

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).

Reviews

“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)

Authors and Affiliations

  • Universität Göttingen, Institut für Theoretische Physik, Göttingen, Germany

    Marcel Bischoff, Karl-Henning Rehren

  • Department of Mathematical Sciences and Kavli IPMU (WPI), The University of Tokyo, Tokyo, Japan

    Yasuyuki Kawahigashi

  • Dipartimento di Matematica, Università di Roma "Tor Vergata", Rome, Italy

    Roberto Longo

Bibliographic Information

  • Book Title: Tensor Categories and Endomorphisms of von Neumann Algebras

  • Book Subtitle: with Applications to Quantum Field Theory

  • Authors: Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren

  • Series Title: SpringerBriefs in Mathematical Physics

  • DOI: https://doi.org/10.1007/978-3-319-14301-9

  • Publisher: Springer Cham

  • eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)

  • Copyright Information: The Author(s) 2015

  • Softcover ISBN: 978-3-319-14300-2Published: 23 January 2015

  • eBook ISBN: 978-3-319-14301-9Published: 13 January 2015

  • Series ISSN: 2197-1757

  • Series E-ISSN: 2197-1765

  • Edition Number: 1

  • Number of Pages: X, 94

  • Number of Illustrations: 138 b/w illustrations

  • Topics: Quantum Field Theories, String Theory, Mathematical Physics, Algebra

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