Quantum Invariants of Knots and 3-Manifolds

Description

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 Preface Contents Introduction Part I. Towards Topological Field Theory Chapter I. Invariants of graphs in Euclidean 3-space 1. Ribbon categories 2. Operator invariants of ribbon graphs 3. Reduction of Theorem 2.5 to lemmas 4. Proof of lemmas Notes Chapter II. Invariants of closed 3-manifolds 1. Modular tensor categories 2. Invariants of 3-manifolds 3. Proof of Theorem 2.3.2. Action of SL(2;Z) 4. Computations in semisimple categories 5. Hermitian and unitary categories Notes Chapter III. Foundations of topological quantum field theory 1. Axiomatic definition of TQFT’s 2. Fundamental properties 3. Isomorphisms of TQFT’s 4. Quantum invariants 5. Hermitian and unitary TQFT’s 6. Elimination of anomalies Notes Chapter IV. Three-dimensional topological quantum field theory 1. Three-dimensional TQFT: preliminary version 2. Proof of Theorem 1.9 3. Lagrangian relations and Maslov indices 4. Computation of anomalies 5. Action of the modular groupoid 6. Renormalized 3-dimensional TQFT 7. Computations in the renormalized TQFT 8. Absolute anomaly-free TQFT 9. Anomaly-free TQFT 10. Hermitian TQFT 11. Unitary TQFT 12. Verlinde algebra Notes Chapter V. Two-dimensional modular functors 1. Axioms for a 2-dimensional modular functor 2. Underlying ribbon category 3. Weak and mirror modular functors 4. Construction of modular functors 5. Construction of modular functors continued Notes Part II. The Shadow World Chapter VI. 6j -symbols 1. Algebraic approach to 6j -symbols 2. Unimodal categories 3. Symmetrized multiplicity modules 4. Framed graphs 5. Geometric approach to 6j -symbols Notes Chapter VII. Simplicial state sums on 3-manifolds 1. State sum models on triangulated 3-manifolds 2. Proof of Theorems 1.4 and 1.7 3. Simplicial 3-dimensional TQFT 4. Comparison of two approaches Notes Chapter VIII. Generalities on shadows 1. Definition of shadows 2. Miscellaneous definitions and constructions 3. Shadow links 4. Surgeries on shadows 5. Bilinear forms of shadows 6. Integer shadows 7. Shadow graphs Notes Chapter IX. Shadows of manifolds 1. Shadows of 4-manifolds 2. Shadows of 3-manifolds 3. Shadows of links in 3-manifolds 4. Shadows of 4-manifolds via handle decompositions 5. Comparison of bilinear forms 6. Thickening of shadows. 7. Proof of Theorems 1.5 and 1.7–1.11 8. Shadows of framed graphs Notes Chapter X. State sums on shadows 1. State sum models on shadowed polyhedra 2. State sum invariants of shadows 3. Invariants of 3-manifolds from the shadow viewpoint 4. Reduction of Theorem 3.3 to a lemma 5. Passage to the shadow world 6. Proof of Theorem 5.6 7. Invariants of framed graphs from the shadow viewpoint 8. Proof of Theorem VII.4.2 9. Computations for graph manifolds Notes Part III. Towards Modular Categories Chapter XI. An algebraic construction of modular categories 1. Hopf algebras and categories of representations 2. Quasitriangular Hopf algebras 3. Ribbon Hopf algebras 4. Digression on quasimodular categories 5. Modular Hopf algebras 6. Quantum groups at roots of unity 7. Quantum groups with generic parameter Notes Chapter XII. A geometric construction of modular categories 1. Skein modules and the Jones polynomial 2. Skein category 3. The Temperley-Lieb algebra 4. The Jones-Wenzl idempotents 5. The matrix S 6. Refined skein category 7. Modular and semisimple skein categories 8. Multiplicity modules 9. Hermitian and unitary skein categories Notes 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 References Subject index