Yakov M. Shnir. Magnetic Monopoles. 4y Springer

Transcription

1 Yakov M. Shnir Magnetic Monopoles 4y Springer

2 Contents Part I Dirac Monopole Magnetic Monopole in Classical Theory Non-Relativistic Scattering on a Magnetic Charge Non-Relativistic Scattering on a Dyon Vector Potential of a Monopole Field Transformations of the String Dynamical Symmetries of the Charge-Monopole System Dual Invariance of Classical Electrodynamics 22 The Electron Monopole System: Quantum-Mechanical Interaction Charge Quantization Condition Spin-Statistics Theorem in a Monopole Theory Charge-Monopole System: Quantum-Mechanical Description The Generalized Spherical Harmonics Solving the Radial Schrodinger Equation Non-Relativistic Scattering on a Monopole: Quantum Mechanical Description Charge-Monopole System: Spin in the Pauli Approximation Dynamical Supersymmery of the Electron-Monopole System Generalized Spinor Harmonics: j> /i + 1/ Generalized Spinor Harmonics: j ; = fi 1/ Solving the Radial Pauli Equation Charge-Monopole System: Solving the Dirac Equation Zero Modes and Witten Effect Charge Quantization Condition and the Group SL(2, Z) 61 Topological Roots of the Abelian Monopole Abelian Wu-Yang Monopole Differential Geometry and Topology Notions of Topology Notions of Differential Geometry 81

3 XIV Contents Maxwell Electrodynamics and Differential Forms Wu-Yang Monopole and the Fiber-Bundle Topology Fiber Bundles Principal Bundle and Connection Wu-Yang Monopole Bundle Hopf Bundle Abelian Monopole: Relativistic Quantum Theory Two Types of Charges Two-Potential Formulation of Electrodynamics Energy-Momentum Tensor and Angular Momentum Canonical Quantization Relativistic Invariance of Two-Charge Electrodynamics Renormalization of QED with a Magnetic Charge Vacuum Polarization by a Dyon Field Effective Lagrangian of QED with a Magnetic Charge 132 Part II Monopole in Non-Abelian Gauge Theories 5 't Hooft-Polyakov Monopole SU(2) Georgi-Glashow Model and the Vacuum Structure Non-Abelian Wu-Yang Monopole Georgi-Glashow Model Topological Classification of the Solutions Definition of Magnetic Charge 't Hooft-Polyakov Ansatz Singular Gauge Transformations and the Connection between't Hooft-Polyakov and Dirac monopoles Dyons The Bogomol'nyi Limit Gauge Zero Mode and the Electric Dyon Charge Topological Classification of Non-Abelian Monopoles SO(3) vs SU{2) Magnetic Charge and the Topology of the Gauge Group Equivalence of Topological and Magnetic Charge Topology of the Dyon Sector The 9 Term and the Witten Effect Again 170

4 Contents XV Multimonopole Configurations Multimonopoles Configurations and Singular Gauge Transformations Singular SU(2) Monopole with Charge g = ngo Magnetic Dipole Rebbi Rossi Multimonopoles, Chains of Monopoles and Closed Vortices Interaction of Magnetic Monopoles Monopole in External Magnetic Field The Interaction Energy of Monopoles Classical Interaction of Two Widely Separated Dyons The n-monopole Configuration in the BPS Limit BPS Multimonopoles: A Bird's Eye View Projective Spaces and Twistor Methods The n-monopole Twistor Construction Hitchin Approach and the Spectral Curve Nahm Equations Solution of the Nahm Equations The Nahm Data and Spectral Curve Moduli Space and Low-Energy Multimonopoles Dynamics Zero Modes Lagrangian and the Moduli Space Metric Metric on the Space M Low-Energy Scattering of Two Monopoles 236 SU(2) Monopole in Quantum Theory Field Fluctuations on Monopole Background Generalized Angular Momentum and the Spectrum of Fluctuations Quantum Correction to the Mass of a Monopole Non-Abelian Monopole: Quasiclassical Quantization Collective Coordinates and Constraints Quantum Mechanics on the Moduli Space Evaluation of the Generating Functional gg Pair Creation in an External Magnetic Field Dynamics of Non-Abelian Monopole in Weak External Field Metastable Vacuum Decay and Monopole Pair Creation in an External Field 269 Monopoles Beyond SU(2) Group SU(N) Monopoles Generalization of the Charge Quantization Condition Towards Higher Rank Gauge Groups Montonen-Olive Conjecture 279

5 XVI Contents Cartan-Weyl Basis and the Simple Roots (3) Cartan Algebra SU{3) Monopoles Massive and Massless Monopoles Pathologies of Non-Abelian Gauge Transformations /7(3) Monopole Moduli Space t/(3) Monopoles: Nahm Equations Monopoles and the Problem of Confinement Quark Confinement in QCD Dual Superconductor Monopoles in the Lattice QCD Compact QED and Lattice Monopoles Lattice Duality Abelian Projection "Monopoles" from Abelian Projection Maximal Abelian Gauge Polyakov Solution of Confinement in the d = 3 Georgi-Glashow Model Dilute Gas of Monopoles in the d = 3 Georgi-Glashow Model Wilson Loop Operator in d = 3 Georgi-Glashow Model Rubakov-Callan Effect Dirac Hamiltonian on the Non-Abelian Monopole Background Fermionic Zero Modes Zero Modes and the Index Theorem S-Wave Fermion Scattering on a Monopole Anomalous Non-Conservation of the Fermion Number Axial Anomaly and the Vacuum Structure Effective Action of Massless Fermions Properties of the Anomalous Fermion Condensate Properties of Other Condensates Monopole-Fermion Scattering in the Bosonisation Technique Vertex Operator and Bosonization of the Free Model Monopole Catalysis of the Proton Decay Monopole Catalysis of the Proton Decay: Semiclassical Model 400

6 Contents XVII Part III Supersymmetric Monopoles 11 Supersymmetric Yang-Mills Theories What is Supersymmetry? Poincare Group and Algebra of Generators Algebra of Generators of Supersymmetry Representations of SUSY Algebra N = 1 Massive Multiplets N = 1 Massless Multiplets N = 2 Extended SUSY Local Representations of SUSY N = 1 Superspace N = 1 Superfields Non-Abelian Multiplets AT = 1 SUSY Lagrangians Magnetic Monopoles in the N = 2 Supersymmetric Yang Mills Theory N = 2 Supersymmetric Lagrangian Praise of Beauty oi N = 2 SUSY Yang-Mills N = 2 Supersymmetric SU{2) Magnetic Monopoles Construction of N = 2 Supersymmetric SU(2) Monopoles Central Charges in theat = 2 SUSY Yang-Mills Fermionic Zero Modes in Supersymmetric Theory Low Energy Dynamics of Supersymmetric Monopoles N = 2 Supersymmetric Monopoles beyond SU{2) SU{3) N = 2 Supersymmetric Monopoles Seiberg-Witten Solution of N = 2 SUSY Yang-Mills Theory Moduli Space Moduli Space and its Parameterization Quantum Moduli Space of N = 2 SUSY Yang-Mills Theory Global Parametrization of the Quantum Moduli Space Transformation of Duality for N = 2 Low-Energy Effective Theory BPS Bound Reexamined Seiberg-Witten Explicit Solution Monodromies on the Moduli Space Solution of the Monodromy Problem Confinement and the Monopole Condensation Concluding Remarks 498

7 XVIII Contents A Representations of SU(2) 501 B Quaternions 505 C SU(2) Transformations of the Monopole Potential 509 References 513 Index 529