Quantum Invariants of Knots and 3-Manifolds

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

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

"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