Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev

Published by De Gruyter 2016

Quantum Invariants of Knots and 3-Manifolds

Vladimir G. Turaev

Volume 18 in the series De Gruyter Studies in Mathematics

https://doi.org/10.1515/9783110435221

Overview
Contents

About this book

Due to the strong appeal and wide use of this monograph, it is now available in its third revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups.
The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space.
This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics.

Contents:
Invariants of graphs in Euclidean 3-space and of closed 3-manifolds
Foundations of topological quantum field theory
Three-dimensional topological quantum field theory
Two-dimensional modular functors
6j-symbols
Simplicial state sums on 3-manifolds
Shadows of manifolds and state sums on shadows
Constructions of modular categories

Author / Editor information

Vladimir G. Turaev, Indiana University, Bloomington, USA.

Reviews

"Its treatment of modular categories, of modular functors, and of TQFT has stood the test of time and in many ways is still unsurpassed. [...] In the reviewer’s opinion, this book continues to be essential reading for anyone who wants to be a quantum topologist." Zentralblatt für Mathematik (review of the second edition)

"It is still the latest and greatest go-to source for information on quantum field theories in three dimensions [...]." Zentralblatt für Mathematik

Supplementary Materials

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Frontmatter Publicly Available Download PDF	I
Preface Requires Authentication Unlicensed Licensed Download PDF	VII
Contents Publicly Available Download PDF	IX
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Part I. Towards Topological Field Theory
Chapter I. Invariants of graphs in Euclidean 3-space Requires Authentication Unlicensed Licensed Download PDF	17
Chapter II. Invariants of closed 3-manifolds Requires Authentication Unlicensed Licensed Download PDF	72
Chapter III. Foundations of topological quantum field theory Requires Authentication Unlicensed Licensed Download PDF	118
Chapter IV. Three-dimensional topological quantum field theory Requires Authentication Unlicensed Licensed Download PDF	152
Chapter V. Two-dimensional modular functors Requires Authentication Unlicensed Licensed Download PDF	236
Part II. The Shadow World
Chapter VI. 6j-symbols Requires Authentication Unlicensed Licensed Download PDF	301
Chapter VII. Simplicial state sums on 3-manifolds Requires Authentication Unlicensed Licensed Download PDF	345
Chapter VIII. Generalities on shadows Requires Authentication Unlicensed Licensed Download PDF	367
Chapter IX. Shadows of manifolds Requires Authentication Unlicensed Licensed Download PDF	394
Chapter X. State sums on shadows Requires Authentication Unlicensed Licensed Download PDF	435
Part III. Towards Modular Categories
Chapter XI. An algebraic construction of modular categories Requires Authentication Unlicensed Licensed Download PDF	493
Chapter XII. A geometric construction of modular categories Requires Authentication Unlicensed Licensed Download PDF	518
Appendix I. Dimension and trace re-examined Requires Authentication Unlicensed Licensed Download PDF	561
Appendix II. Vertex models on link diagrams Requires Authentication Unlicensed Licensed Download PDF	563
Appendix III. Gluing re-examined Requires Authentication Unlicensed Licensed Download PDF	565
Appendix IV. The signature of closed 4-manifolds from a state sum Requires Authentication Unlicensed Licensed Download PDF	568
References Requires Authentication Unlicensed Licensed Download PDF	571
Subject index Requires Authentication Unlicensed Licensed Download PDF	593

(Deutsch)

Language: English
Publisher: De Gruyter
Edition: 3rd corr. ed.
Audience: Researchers, Graduate Students of Mathematics; Academic Libraries.
Pages
- Front matter: 12
- Main content: 596
Keywords: Mannigfaltigkeit; Knotentheorie; Topologie; Topologische Algebra; Topologische Gruppe

eBook

Published: July 11, 2016
ISBN: 9783110435221

Hardcover

Published: July 11, 2016
ISBN: 9783110442663