Topological Quantum Field Theories from Subfactors  book cover
1st Edition

Topological Quantum Field Theories from Subfactors

ISBN 9781138442108
Published September 30, 2020 by Chapman & Hall
136 Pages

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Book Description

Pure mathematicians have only recently begun a rigorous study of topological quantum field theories (TQFTs). Ocneanu, in particular, showed that subfactors yield TQFTs that complement the Turaev-Viro construction. Until now, however, it has been difficult to find an account of this work that is both detailed and accessible.

Topological Quantum Field Theories from Subfactors provides a self-contained, explicit description of Ocneanu's construction It introduces and discusses its various ingredients with the distinct advantage of employing only genuine triangulations. The authors begin with axioms for a TQFT, go through the Turaev-Viro prescription for constructing such a TQFT, and finally work through Ocneanu's method of starting with a finite depth hyperfinite subfactor" and obtaining the data needed to invoke the Turaev-Viro machine.

The authors provide a very concise treatment of finite factors of type and their bimodules and include details and calculations for all constructions. They also present, perhaps for the first time in book form, notions such as quantization functors and fusion algebras. Accessible to graduate students and others just beginning to explore this intriguing topic, Topological Quantum Field Theories from Subfactors will also be of interest to researchers in both mathematics and theoretical physics.

Table of Contents

Preface -- 1 Preliminaries on TQFTs -- 1.1 Introduction -- 1.2 Quantisation factors -- 1.3 Fusion algebras -- 2 The invariant -- 2.1 The construction -- 2.2 Invariance -- 3 von Neumann algebras -- 3.1 Preliminaries on II1 factors -- 3.2 Bases and M-valued inner products -- 3.3 Contragredients and tensor products -- 3.4 Frobenius reciprocity -- 3.5 TQFTs from subfactors -- 4 The group example -- 4.1 The TQFT -- 4.2 The quantum double -- A Appendix -- A .l Some topological preliminaries -- A.2 Simplicial complexes -- Bibliography -- Index.

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Kodiyalam, Vijay