Quantum Invariants of Knots and 3-Manifolds / Vladimir G. Turaev.

Format:

Book

Author/Creator:

Turaev, Vladimir G., Author.

Series:

De Gruyter studies in mathematics ; Volume 18.

De Gruyter Studies in Mathematics ; 18

Language:

English

Subjects (All):

Quantum field theory.

Physical Description:

1 online resource (600 pages).

Edition:

Reprint 2020

Other Title:

Quantum invariants of knots and three-manifolds.

Place of Publication:

Berlin ; Boston : De Gruyter, [2020]

Language Note:

In English.

Summary:

This monograph provides a systematic treatment of topological quantum field theories (TQFT's) in three dimensions, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and self-contained exposition of new fundamental algebraic and topological concepts that emerged in this theory. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFT's and 2-dimensional modular functors from so-called modular categories. This gives new knot and 3-manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules. This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.

Contents:

Frontmatter

Contents

Introduction

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

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

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

Problems

References

Subject index

Notes:

Includes bibliographical references and index.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 06. Apr 2020)

ISBN:

9783110883275

3110883279

OCLC:

1149402321