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Book Cover
Author Turaev, V. G. (Vladimir G.), 1954-

Title Quantum invariants of knots and 3-manifolds / V.G. Turaev.

Published Berlin ; New York : W. de Gruyter, 1994.


Location Call No. Status
Physical description x, 588 pages : illustrations ; 25 cm.
Series De Gruyter studies in mathematics ; 18.
De Gruyter studies in mathematics ; 18.
Bibliography Includes bibliographical references (pages [573]-584) and index.
Contents Pt. I. Towards Topological Field Theory. Ch. I. Invariants of graphs in Euclidean 3-space. Ch. II. Invariants of closed 3-manifolds. Ch. III. Foundations of topological quantum field theory. Ch. IV. Three-dimensional topological quantum field theory. Ch. V. Two-dimensional modular functors -- Pt. II. The Shadow World. Ch. VI. 6j-symbols. Ch. VII. Simplicial state sums on 3-manifolds. Ch. VIII. Generalities on shadows. Ch. IX. Shadows of manifolds. Ch. X. State sums on shadows -- Pt. III. Towards Modular Categories. Ch. XI. An algebraic construction of modular categories. Ch. 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.
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.
Subject Quantum field theory.
Knot theory.
Three-manifolds (Topology)
Mathematical physics.
Other title Quantum invariants of knots and three-manifolds.
ISBN 3110137046 (acid-free paper)