Introduction and Overview E. Bick, F.D. Steffen 1 Topology and Geometry in Physics 2 An Outline of the Book 3 Complementary Literature Topological Concepts in Gauge Theories F. Lenz 1 Introduction 2 Nielsen-Olesen Vortex 2.1 Abelian Higgs Model 2.2 Topological Excitations 3 Homotopy 3.1 The Fundamental Group 3.2 Higher Homotopy Groups 3.3 Quotient Spaces 3.4 Degree of Maps 3.5 Topological Groups 3.6 Transformation Groups 3.7 Defects in Ordered Media 4 Yang-Mills Theory 5 \'t Hooft-Polyakov Monopole 5.1 Non-Abelian Higgs Model 5.2 The Higgs Phase 5.3 Topological Excitations 6 Quantization of Yang-Mills Theory 7 Instantons 7.1 Vacuum Degeneracy 7.2 Tunneling 7.3 Fermions in Topologically Non-trivial Gauge Fields 7.4 Instanton Gas 7.5 Topological Charge and Link Invariants 8 Center Symmetry and Confinement 8.1 Gauge Fields at Finite Temperature and Finite Extension 8.2 Residual Gauge Symmetries in QED 8.3 Center Symmetry in SU(2) Yang-Mills Theory 8.4 Center Vortices 8.5 The Spectrum of the SU(2) Yang-Mills Theory 9 QCD in Axial Gauge 9.1 Gauge Fixing 9.2 Perturbation Theory in the Center-Symmetric Phase 9.3 Polyakov Loops in the Plasma Phase 9.4 Monopoles 9.5 Monopoles and Instantons 9.6 Elements of Monopole Dynamics 9.7 Monopoles in Diagonalization Gauges 10 Conclusions Aspects of BRST Quantization J. W. van Holten 1 Symmetries and Constraints 1.1 Dynamical Systems with Constraints 1.2 Symmetries and Noether\'s Theorems 1.3 Canonical Formalism 1.4 Quantum Dynamics 1.5 The Relativistic Particle 1.6 The Electro-magnetic Field 1.7 Yang-Mills Theory 1.8 The Relativistic String 2 Canonical BRST Construction 2.1 Grassmann Variables 2.2 Classical BRST Transformations 2.3 Examples 2.4 Quantum BRST Cohomology 2.5 BRST-Hodge Decomposition of States 2.6 BRST Operator Cohomology 2.7 Lie-Algebra Cohomology 3 Action Formalism 3.1 BRST Invariance from Hamilton\'s Principle 3.2 Examples 3.3 Lagrangean BRST Formalism 3.4 The Master Equation 3.5 Path-Integral Quantization 4 Applications of BRST Methods 4.1 BRST Field Theory 4.2 Anomalies and BRST Cohomology Appendix. Conventions Chiral Anomalies and Topology J. Zinn-Justin 1 Symmetries, Regularization, Anomalies 2 Momentum Cut-Off Regularization 2.1 Matter Fields: Propagator Modification 2.2 Regulator Fields 2.3 Abelian Gauge Theory 2.4 Non-Abelian Gauge Theories 3 Other Regularization Schemes 3.1 Dimensional Regularization 3.2 Lattice Regularization 3.3 Boson Field Theories 3.4 Fermions and the Doubling Problem 4 The Abelian Anomaly 4.1 Abelian Axial Current and Abelian Vector Gauge Fields 4.2 Explicit Calculation 4.3 Two Dimensions 4.4 Non-Abelian Vector Gauge Fields and Abelian Axial Current 4.5 Anomaly and Eigenvalues of the Dirac Operator 5 Instantons, Anomalies, and 0-Vacua 5.1 The Periodic Cosine Potential 5.2 Instantons and Anomaly: CP(N-1) Models 5.3 Instantons and Anomaly: Non-Abelian Gauge Theories 5.4 Fermions in an Instanton Background 6 Non-Abelian Anomaly 6.1 General Axial Current 6.2 Obstruction to Gauge Invariance 6.3 Wess-Zumino Consistency Conditions 7 Lattice Fermions: Ginsparg-Wilson Relation 7.1 Chiral Symmetry and Index 7.2 Explicit Construction:Overlap Fermions 8 Supersymmetric Quantum Mechanics and Domain Wall Fermions 8.1 Supersymmetric Quantum Mechanics 8.2 Field Theory in Two Dimensions 8.3 Domain Wall Fermions Appendix A. Trace Formula for Periodic Potentials Appendix B. Resolvent of the Hamiltonian in Supersymmetric QM Supersymmetric Solitons and Topology M. Shifman 1 Introduction 2 D = 1+1; N=1 2.1 Critical (BPS) Kinks 2.2 The Kink Mass (Classical) 2.3 Interpretation of the BPS Equations. Morse Theory 2.4 Quantization. Zero Modes: Bosonic and Fermionic 2.5 Cancelation of Nonzero Modes 2.6 Anomaly I 2.7 Anomaly II (Shortening Supermultiplet Down to One State) 3 Domain Walls in (3+1)-Dimensional Theories 3.1 Superspace and Superfields 3.2 Wess-Zumino Models 3.3 Critical Domain Walls 3.4 Finding the Solution to the BPS Equation 3.5 Does the BPS Equation Follow from the Second Order Equation of Motion? 3.6 Living on a Wall 4 Extended Supersymmetry in Two Dimensions: The Supersymmetric CP(1) Model 4.1 Twisted Mass 4.2 BPS Solitons at the Classical Level 4.3 Quantization of the Bosonic Moduli 4.4 The Soliton Mass and Holomorphy 4.5 Switching On Fermions 4.6 Combining Bosonic and Fermionic Moduli 5 Conclusions Appendix A. CF(l) Model- 0(3) Model (N=1 Superfields N) Appendix B. Getting Started (Supersymmetry for Beginners) B.1 Promises of Supersymmetry B.2 Cosmological Term B.3 Hierarchy Problem Forces from Connes\' Geometry T. Schücker i Introduction 2 Gravity from Riemannian Geometry 2.1 First Stroke: Kinematics 2.2 Second Stroke: Dynamics 3 Slot Machines and the Standard Model 3.1 Input 3.2 Rules 3.3 The Winner 3.4 Wick Rotation 4 Connes’ Noncommutative Geometry 4.1 Motivation: Quantum Mechanics 4.2 The Calibrating Example: Riemannian Spin Geometry 4.3 Spin Groups 5 The Spectral Action 5.1 Repeating Einstein\'s Derivation in the Commutative Case 5.2 Almost Commutative Geometry 5.3 The Minimax Example 5.4 A Central Extension 6 Connes’ Do-It-Yourself Kit 6.1 Input 6.2 Output 6.3 The Standard Model 6.4 Beyond the Standard Model 7 Outlook and Conclusion Appendix A.1 Groups A.2 Group Representations A.3 Semi-Direct Product and Poincaré Group A.4 Algebras Index
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