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📙 Quantum invariants of knots and 3-manifolds by Vladimir G. Turaev — free pdf


Some quite amazing results have appeared in the last two decades that connect two seemingly different fields of knowledge, namely topology and quantum field theory. Topological considerations have played a role in quantum field theory for quite some time, due to the role of instantons, but quantum field theory did not shed any light on topological questions until Edward Witten, Vaughan Jones, and other physicists and mathematicians showed that it can be a powerful tool of investigation in 3-dimensional and 4-dimensional topology.

This book, written for the mathematician, does not follow the physical line of reasoning that has been employed to obtain invariants of knots and 3-manifolds. Instead, it endeavors to remain as rigorous as possible, and thus the approaches using conformal field theory or Chern-Simons field theory are not developed by the author (this is not to say that one cannot see the influence of these areas in the book). The author incorporates not only the developments from the current research literature up to the date the book was written, but also interjects some original results of his own. The book could also be viewed as a textbook, as there are exercises put in at various places in the book.

Topological quantum field theory is defined rigorously in this book, but is put in the context of what are called modular categories and modular functors by the author.Modular categories are finite dimensional modules over a Hopf algebra. Hence, one should think of the designation 'quantum field theory' in the book as being one that indicates only its historical roots. A fully operational quantum field theory always needs infinite dimensions to gain its predictive power. These modular categories are constructed from modular functors, the latter arising from closed oriented surfaces with a distinguished Lagrangian subspace and a finite set of marked points, or "colors" (the reader versed in conformal field theory will see the origins of these ideas). Choosing a particular modular category and set of colors will give the familiar Jones polynomial.

After defining an isotopy invariant for colored frame oriented links in Euclidean space, topological invariants for closed oriented 3-manifolds are defined by doing surgery on the standard 3-sphere along a framed link. The dependence on the link is removed by employing Kirby calculus, which gives the sequence of moves needed to relate one link to another. The resulting quantum invariant is thus dependent on the link diagrams, but an intrinsic definition computed from the manifold is via a state sum on a triangulation of the manifold. Most interesting is that this state sum is computed using the 6j-symbols, familiar to physicists in the quantum theory of angular momentum. The actual invariant requires the computation of a product over the manifold and one equal to it except taking the opposite orientation. In addition, the computation is done inside an arbitrary compact oriented piecewise-linear 4-manifold bounded by the manifold. This computation utilizes the concept of a "shadow" of a 4-manifold, which are topological objects related to 6j-symbols.The illumination (no pun intended) by the author of the theory of shadows is done in great detail and occupies most of the space in the book.

The existence of modular categories is related to the theory of representations of quantum groups at roots of unity, these quantum groups being Hopf algebras over the complex numbers which are constructed via 1-parameter deformations of the universal enveloping algebra of simple Lie algebras. The author though sticks with the general language of categories, and algebraic and geometric constructions of them are discussed in detail by the author.

All of the results in this book are interesting, but the author admits that their connection with low-dimensional topology and the classical invariants of 3-manifolds is not readily apparent, especially their connection with homotopy via the fundamental group. Such a connection would possibly shed light on the one of the most nagging questions in 3-dimensional topology: the Poincare conjecture.

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