Quantum invariants of knots and 3-manifolds / Vladimir G. Touraev.

Format:

Book

Author/Creator:

Turaev, V. G. (Vladimir G.), 1954- author.

Series:

De Gruyter studies in mathematics ; 18.

de Gruyter Studies in Mathematics, 0179-0986 ; 18

Language:

English

Subjects (All):

Quantum field theory.

Knot theory.

Three-manifolds (Topology).

Invariants.

Mathematical physics.

Physical Description:

1 online resource (608 p.)

Edition:

Third edition.

Place of Publication:

Berlin, [Germany] ; Boston, [Massachusetts] : Walter de Gruyter GmbH, 2016.

Language Note:

English

Summary:

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-manifoldsFoundations of topological quantum field theoryThree-dimensional topological quantum field theoryTwo-dimensional modular functors6j-symbolsSimplicial state sums on 3-manifoldsShadows of manifolds and state sums on shadowsConstructions of modular categories

Contents:

Frontmatter

Preface

Contents

Introduction

Part I. Towards Topological Field Theory

Chapter I. Invariants of graphs in Euclidean 3-space

Chapter II. Invariants of closed 3-manifolds

Chapter III. Foundations of topological quantum field theory

Chapter IV. Three-dimensional topological quantum field theory

Chapter V. Two-dimensional modular functors

Part II. The Shadow World

Chapter VI. 6j-symbols

Chapter VII. Simplicial state sums on 3-manifolds

Chapter VIII. Generalities on shadows

Chapter IX. Shadows of manifolds

Chapter X. State sums on shadows

Part III. Towards Modular Categories

Chapter XI. An algebraic construction of modular categories

Chapter XII. A geometric construction of modular categories

Appendix I. Dimension and trace re-examined

Appendix II. Vertex models on link diagrams

Appendix III. Gluing re-examined

Appendix IV. The signature of closed 4-manifolds from a state sum

References

Subject index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

3-11-043456-3

3-11-043522-5

OCLC:

958054778