Contents -- Preface -- 1. Quantization of Discretized Gauge Theories -- 1.1 Continuum gauge theory -- 1.1.1 Configurations and observables -- 1.1.2 Yang-Mills theory and BF theory -- 1.2 Discretized gauge theory -- 1.2.1 Discretized manifold and connection -- 1.2.2 Yang-Mills theory -- 1.2.3 BF theory -- 1.2.4 BF-Yang-Mills -- 1.3 Path integral quantization -- 1.4 Circuit diagrams - baby version -- 1.4.1 Basics -- 1.4.2 Key identities -- 1.5 Diagrammatic representation -- 1.5.1 Partition function -- 1.5.2 Wilson loop observables -- 1.5.3 Gauge fixing -- 1.6 An exact solution: 2-dimensional lattice gauge theory -- 1.7 Strong-weak duality -- 1.8 The dual of abelian lattice gauge theory -- 2. Topology: Decomposition of Manifolds -- 2.1 Simplicial decompositions -- 2.1.1 Simplices -- 2.1.2 Euclidean complexes -- 2.1.3 Simplicial complexes -- 2.1.4 Triangulations -- 2.1.5 Pachner moves -- 2.2 Cellular decompositions -- 2.2.1 Cells and decompositions -- 2.2.2 Cellular moves -- 3. Categories and Diagrams -- 3.1 Monoidal categories with structure -- 3.2 Diagrammatic isotopy invariance -- 3.3 Semisimplicity and circuit diagrams -- 3.4 Key identities -- 3.5 Purification -- 4. Representation Theory: Groups & Hopf Algebras -- 4.1 Groups -- 4.1.1 Representation categories -- 4.1.2 Representative functions -- 4.1.3 Diagrams for representative functions -- 4.1.4 Integration and semisimplicity -- 4.1.5 Recovering the baby diagrams -- 4.2 Hopf algebras and quantum groups -- 4.2.1 Hopf algebras with extra structure -- 4.2.2 Representation categories -- 4.2.3 "Representative functions" -- 4.2.4 Integration and semisimplicity -- 4.2.5 Supergroups -- 4.3 Examples -- 4.3.1 Duality -- 4.3.2 Abelian groups -- 4.3.3 SU(2) -- 4.3.4 Supergroups -- 4.3.5 q-deformations -- 5. Cellular Gauge Theory -- 5.1 Symmetric cellular gauge theory.

5.2 Non-symmetric cellular gauge theory -- 5.2.1 Dimension 2 - pivotal categories -- 5.2.2 Dimension larger than 2 -- 5.2.3 Dimension 3 - spherical categories -- 5.2.4 Dimension 4 - ribbon categories -- 5.3 Wilson networks -- 5.3.1 Symmetric case -- 5.3.2 Non-symmetric case -- 5.4 Gauge fixing -- 5.4.1 Symmetric case -- 5.4.2 Non-symmetric case -- 6. Topological Quantum Field Theory -- 6.1 Boundaries and gluing -- 6.1.1 Boundary circuit diagram -- 6.1.2 Gluing manifolds and diagrams -- 6.1.3 Sum over labelings -- 6.1.4 Wilson networks -- 6.2 Topological quantum field theory -- 6.2.1 The delta move -- 6.2.2 The anomaly move -- 6.2.3 Invariants of manifolds -- 6.2.4 Invariants of ribbon knots -- 6.2.5 Cobordisms -- 7. Related Constructions -- 7.1 Spin networks -- 7.2 Spin foams and nj-symbols -- 7.3 Turaev-Viro invariant and Crane-Yetter invariant -- 7.4 The trace formalism -- 7.5 Modular categories and chain mail -- 8. Applications to Lattice Models and Quantum Gravity -- 8.1 Two theories with SU(2) -- 8.2 The dual of non-abelian lattice gauge theory -- 8.3 The chiral model -- 8.4 Spin networks in loop quantum gravity -- 8.5 Spin foam models of quantum gravity -- 8.6 The Barrett-Crane model -- 8.7 Three-dimensional quantum supergravity -- 8.8 Generating field theory -- 8.8.1 From Feynman diagrams to triangulations -- 8.8.2 Generating BF theory -- 8.8.3 Beyond BF theory -- 8.8.4 A version of lattice gauge theory -- 8.8.5 The Barrett-Crane model -- 8.9 Renormalization -- 8.9.1 Standard renormalization -- 8.9.2 General discretizations -- 8.9.3 Cellular decompositions and moves -- 8.9.4 Examples -- References -- Index.