The link invariants of the Chern-Simons field theory. New developments in topological quantum field theory
type: Item
instance of: scholarly article
title: The link invariants of the Chern-Simons field theory. New developments in topological quantum field theory
zbMATH Open document ID: 0812.57002
author: E. Guadagnini
published in: De Gruyter Expositions in Mathematics
publication date: 1994-05-18T00:00:00Z
review text: The book under review is an extended version of a lecture course given by the author for physicists, and inherits all merits and demerits of the source. On the one hand, a free lecture style and an appeal to a physical intuition enable the reader to get a quick approach to deep aspects of the quantum Chern-Simons field theory. On the other hand, sufficiently good knowledge of quantum physics is required while the lack of rigorous definitions and proofs makes the book difficult even for qualified mathematicians. The book can be decomposed into three parts. In the first part, properties of the quantum Chern-Simons field theory in \(R^ 3\) are studied. The observables of the theory, associated with knots and links in \(R^ 3\), are described in topological terms. In the second part, the exact solution of the model is obtained. The expectation values of the observables represent the values of an ambient isotopy invariant of framed links in \(R^ 3\). The invariant generalizes the famous Jones polynomial. It satisfies some simple properties that are similar to the usual skein relations and serve as calculus rules for computations. The properties may be considered as defining relations for the invariant. As usual, the proof of internal consistency for the relations is non- trivial, the author restricts himself with the argument that the quantum Chern-Simons field theory does exist. In the last part, the solution of the theory in any closed orientable 3- manifold is constructed by means of the operator surgery method. The 3- manifold invariant naturally associated with the Chern-Simons theory is defined and its values are computed for several examples. The invariant is close to Reshetikhin-Turaev invariant. In conclusion note, that the book is worth reading especially for topologists, since it explains hidden reasons why different constructions of new knot and 3-manifold invariants give nearly identical results.
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 81T10
Mathematics Subject Classification ID: 81Q20
Mathematics Subject Classification ID: 81-02
zbMATH DE Number: 570283
zbMATH Keywords: topological quantum field theory
zbMATH Keywords: state sum invariants
zbMATH Keywords: quantum Chern- Simons field theory in \(R^ 3\)
zbMATH Keywords: knots and links in \(R^ 3\)
zbMATH Keywords: Jones polynomial
zbMATH Keywords: skein relations
zbMATH Keywords: 3-manifold invariant
zbMATH Keywords: Reshetikhin-Turaev invariant
reviewed by: Sergei Vladimirovich Matveev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 570283