Description
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.
The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.
The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.
While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies
Chapter
3. Reduction of Theorem 2.5 to lemmas
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
Chapter III. Foundations of topological quantum field theory
1. Axiomatic definition of TQFT’s
2. Fundamental properties
3. Isomorphisms of TQFT’s
5. Hermitian and unitary TQFT’s
6. Elimination of anomalies
Chapter IV. Three-dimensional topological quantum field theory
1. Three-dimensional TQFT: preliminary version
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
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
Part II. The Shadow World
1. Algebraic approach to 6j -symbols
3. Symmetrized multiplicity modules
5. Geometric approach to 6j -symbols
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
Chapter VIII. Generalities on shadows
2. Miscellaneous definitions and constructions
5. Bilinear forms of shadows
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
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
7. Invariants of framed graphs from the shadow viewpoint
8. Proof of Theorem VII.4.2
9. Computations for graph manifolds
Part III. Towards Modular Categories
Chapter XI. An algebraic construction of modular categories
1. Hopf algebras and categories of representations
2. Quasitriangular Hopf algebras
4. Digression on quasimodular categories
6. Quantum groups at roots of unity
7. Quantum groups with generic parameter
Chapter XII. A geometric construction of modular categories
1. Skein modules and the Jones polynomial
3. The Temperley-Lieb algebra
4. The Jones-Wenzl idempotents
6. Refined skein category
7. Modular and semisimple skein categories
9. Hermitian and unitary skein 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
Quantum Invariants of Knots and 3-Manifolds
( De Gruyter Studies in Mathematics )
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