Magnetic Monopoles

by Shnir, Yakov M.; Shnir, Ya. M.

• ISBN13: 9783540252771
• ISBN10: 3540252770
• Format: Hardcover
• Copyright: 2005-10-16
• Publisher: Springer Verlag

Summary

Surveys the monopole problem on a few different levels, from classical electrodynamics up to N=2 SUSY Yang-Mills theory. and presents a compact, 'bird's eye view' on the entire set of problems related with very notion of monopole including actual stand of the problem, related historical remarks and comprehensive bibliography.Presents original results obtained by the author in collaboration with other researches are presented as well as it summarizes the present status of the theory of monopoles and provides an introduction to the field.

Table of Contents

Part I Dirac Monopole
1 Magnetic Monopole in Classical Theory	3	(24)
1.1 Non-Relativistic Scattering on a Magnetic Charge	3	(7)
1.2 Non-Relativistic Scattering on a Dyon	10	(2)
1.3 Vector Potential of a Monopole Field	12	(3)
1.4 Transformations of the String	15	(5)
1.5 Dynamical Symmetries of the Charge-Monopole System	20	(2)
1.6 Dual Invariance of Classical Electrodynamics	22	(5)
2 The Electron–Monopole System: Quantum-Mechanical Interaction	27	(40)
2.1 Charge Quantization Condition	27	(4)
2.2 Spin-Statistics Theorem in a Monopole Theory	31	(2)
2.3 Charge-Monopole System: Quantum-Mechanical Description	33	(7)
2.3.1 The Generalized Spherical Harmonics	34	(3)
2.3.2 Solving the Radial SchrOdinger Equation	37	(3)
2.4 Non-Relativistic Scattering on a Monopole: Quantum Mechanical Description	40	(2)
2.5 Charge-Monopole System: Spin in the Pauli Approximation	42	(11)
2.5.1 Dynamical Supersymmery of the Electron-Monopole System	44	(2)
2.5.2 Generalized Spinor Harmonics: j > or equal to μ + 1/2	46	(2)
2.5.3 Generalized Spinor Harmonics: j = μ - 1/2	48	(1)
2.5.4 Solving the Radial Pauli Equation	49	(4)
2.6 Charge-Monopole System: Solving the Dirac Equation	53	(14)
2.6.1 Zero Modes and Witten Effect	55	(6)
2.6.2 Charge Quantization Condition and the Group SL(2, Z)	61	(6)
3 Topological Roots of the Abelian Monopole	67	(42)
3.1 Abelian Wu—Yang Monopole	67	(3)
3.2 Differential Geometry and Topology	70	(23)
3.2.1 Notions of Topology	70	(11)
3.2.2 Notions of Differential Geometry	81	(8)
3.2.3 Maxwell Electrodynamics and Differential Forms	89	(4)
3.3 Wu—Yang Monopole and the Fiber-Bundle Topology	93	(16)
3.3.1 Fiber Bundles	93	(4)
3.3.2 Principal Bundle and Connection	97	(5)
3.3.3 Wu—Yang Monopole Bundle	102	(1)
3.3.4 Hopf Bundle	103	(6)
4 Abelian Monopole: Relativistic Quantum Theory	109	(32)
4.1 Two Types of Charges	110	(2)
4.2 Two-Potential Formulation of Electrodynamics	112	(6)
4.2.1 Energy-Momentum Tensor and Angular Momentum	115	(3)
4.3 Canonical Quantization	118	(7)
4.3.1 Relativistic Invariance of Two-Charge Electrodynamics	121	(4)
4.4 Renormalization of QED with a Magnetic Charge	125	(3)
4.5 Vacuum Polarization by a Dyon Field	128	(4)
4.6 Effective Lagrangian of QED with a Magnetic Charge	132	(9)
Part II Monopole in Non-Abelian Gauge Theories
5 't Hooft–Polyakov Monopole	141	(32)
5.1 SU(2) Georgi—Glashow Model and the Vacuum Structure	141	(16)
5.1.1 Non-Abelian Wu—Yang Monopole	141	(2)
5.1.2 Georgi—Glashow Model	143	(3)
5.1.3 Topological Classification of the Solutions	146	(2)
5.1.4 Definition of Magnetic Charge	148	(3)
5.1.5 't Hooft—Polyakov Ansatz	151	(3)
5.1.6 Singular Gauge Transformations and the Connection between't Hooft—Polyakov and Dirac monopoles	154	(1)
5.1.7 Dyons	155	(2)
5.2 The Bogomol'nyi Limit	157	(6)
5.2.1 Gauge Zero Mode and the Electric Dyon Charge	161	(2)
5.3 Topological Classification of Non-Abelian Monopoles	163	(7)
5.3.1 SO(3) vs SU(2)	163	(2)
5.3.2 Magnetic Charge and the Topology of the Gauge Group	165	(1)
5.3.3 Equivalence of Topological and Magnetic Charge	166	(2)
5.3.4 Topology of the Dyon Sector	168	(2)
5.4 The 9 Term and the Witten Effect Again	170	(3)
6 Multimonopole Configurations	173	(68)
6.1 Multimonopoles Configurations and Singular Gauge Transformations	174	(4)
6.1.1 Singular SU(2) Monopole with Charge g = ngo	174	(2)
6.1.2 Magnetic Dipole	176	(2)
6.2 Rebbi—Rossi Multimonopoles, Chains of Monopoles and Closed Vortices	178	(14)
6.3 Interaction of Magnetic Monopoles	192	(9)
6.3.1 Monopole in External Magnetic Field	192	(2)
6.3.2 The Interaction Energy of Monopoles	194	(3)
6.3.3 Classical Interaction of Two Widely Separated Dyons	197	(4)
6.4 The η-Monopole Configuration in the BPS Limit	201	(26)
6.4.1 BPS Multimonopoles: A Bird's Eye View	201	(2)
6.4.2 Projective Spaces and Twistor Methods	203	(3)
6.4.3 The η-Monopole Twistor Construction	206	(8)
6.4.4 Hitchin Approach and the Spectral Curve	214	(3)
6.4.5 Nahm Equations	217	(3)
6.4.6 Solution of the Nahm Equations	220	(3)
6.4.7 The Nahm Data and Spectral Curve	223	(4)
6.5 Moduli Space and Low-Energy Multimonopoles Dynamics	227	(14)
6.5.1 Zero Modes Lagrangian and the Moduli Space Metric	227	(5)
6.5.2 Metric on the Space M2	232	(4)
6.5.3 Low-Energy Scattering of Two Monopoles	236	(5)
7 SU(2) Monopole in Quantum Theory	241	(34)
7.1 Field Fluctuations on Monopole Background	241	(13)
7.1.1 Generalized Angular Momentum and the Spectrum of Fluctuations	245	(5)
7.1.2 Quantum Correction to the Mass of a Monopole	250	(4)
7.2 Non-Abelian Monopole: Quasiclassical Quantization	254	(13)
7.2.1 Collective Coordinates and Constraints	254	(4)
7.2.2 Quantum Mechanics on the Moduli Space	258	(5)
7.2.3 Evaluation of the Generating Functional	263	(4)
7.3 gg Pair Creation in an External Magnetic Field	267	(8)
7.3.1 Dynamics of Non-Abelian Monopole in Weak External Field	267	(2)
7.3.2 Metastable Vacuum Decay and Monopole Pair Creation in an External Field	269	(6)
8 Monopoles Beyond SU(2) Group	275	(44)
8.1 SU (N) Monopoles	276	(25)
8.1.1 Generalization of the Charge Quantization Condition	276	(1)
8.1.2 Towards Higher Rank Gauge Groups	277	(2)
8.1.3 Montonen—Olive Conjecture	279	(3)
8.1.4 Cartan—Weyl Basis and the Simple Roots	282	(2)
8.1.5 SU(3) Cartan Algebra	284	(3)
8.1.6 SU(3) Monopoles	287	(14)
8.2 Massive and Massless Monopoles	301	(5)
8.2.1 Pathologies of Non-Abelian Gauge Transformations	303	(3)
8.3 SU(3) Monopole Moduli Space	306	(13)
8.3.1 SU(3) Monopoles: Nahm Equations	314	(5)
9 Monopoles and the Problem of Confinement	319	(40)
9.1 Quark Confinement in QCD	319	(9)
9.1.1 Dual Superconductor	324	(4)
9.2 Monopoles in the Lattice QCD	328	(11)
9.2.1 Compact QED and Lattice Monopoles	331	(3)
9.2.2 Lattice Duality	334	(5)
9.3 Abelian Projection	339	(10)
9.3.1 "Monopoles" from Abelian Projection	339	(7)
9.3.2 Maximal Abelian Gauge	346	(3)
9.4 Polyakov Solution of Confinement in the d = 3 Georgi—Glashow Model	349	(10)
9.4.1 Dilute Gas of Monopoles in the d = 3 Georgi—Glashow Model	349	(6)
9.4.2 Wilson Loop Operator in d = 3 Georgi—Glashow Model	355	(4)
10 Rubakov—Callan Effect	359	(48)
10.1 Dirac Hamiltonian on the Non-Abelian Monopole Background	359	(19)
10.1.1 Fermionic Zero Modes	363	(4)
10.1.2 Zero Modes and the Index Theorem	367	(6)
10.1.3 S-Wave Fermion Scattering on a Monopole	373	(5)
10.2 Anomalous Non-Conservation of the Fermion Number	378	(12)
10.2.1 Axial Anomaly and the Vacuum Structure	378	(1)
10.2.2 Effective Action of Massless Fermions	379	(6)
10.2.3 Properties of the Anomalous Fermion Condensate	385	(3)
10.2.4 Properties of Other Condensates	388	(2)
10.3 Monopole-Fermion Scattering in the Bosonisation Technique	390	(17)
10.3.1 Vertex Operator and Bosonization of the Free Model	391	(6)
10.3.2 Monopole Catalysis of the Proton Decay	397	(3)
10.3.3 Monopole Catalysis of the Proton Decay: Semiclassical Model	400	(7)
Part III Supersymmetric Monopoles
11 Supersymmetric Yang-Mills Theories	407	(30)
11.1 What is Supersymmetry?	407	(8)
11.1.1 Poincare Group and Algebra of Generators	408	(4)
11.1.2 Algebra of Generators of Supersymmetry	412	(3)
11.2 Representations of SUSY Algebra	415	(5)
11.2.1 N = 1 Massive Multiplets	415	(2)
11.2.2 N = 1 Massless Multiplets	417	(1)
11.2.3 N = 2 Extended SUSY	418	(2)
11.3 Local Representations of SUSY	420	(9)
11.3.1 N = 1 Superspace	420	(4)
11.3.2 N = 1 Superfields	424	(4)
11.3.3 Non-Abelian Multiplets	428	(1)
11.4 N = 1 SUSY Lagrangians	429	(8)
12 Magnetic Monopoles in the N = 2 Supersymmetric Yang-Mills Theory	437	(28)
12.1 N = 2 Supersymmetric Lagrangian	437	(6)
12.1.1 Praise of Beauty of N = 2 SUSY Yang-Mills	441	(2)
12.2 N = 2 Supersymmetric SU(2) Magnetic Monopoles	443	(3)
12.2.1 Construction of N = 2 Supersymmetric SU(2) Monopoles	443	(3)
12.3 Central Charges in the N = 2 SUSY Yang-Mills	446	(3)
12.4 Fermionic Zero Modes in Supersymmetric Theory	449	(2)
12.5 Low Energy Dynamics of Supersymmetric Monopoles	451	(2)
12.6 N = 2 Supersymmetric Monopoles beyond SU(2)	453	(12)
12.6.1 SU(3) N = 2 Supersymmetric Monopoles	458	(7)
13 Seiberg-Witten Solution of N = 2 SUSY Yang-Mills Theory	465	(36)
13.1 Moduli Space	466	(12)
13.1.1 Moduli Space and its Parameterization	466	(6)
13.1.2 Quantum Moduli Space of N = 2 SUSY Yang-Mills Theory	472	(6)
13.2 Global Parametrization of the Quantum Moduli Space	478	(7)
13.2.1 Transformation of Duality for N = 2 Low-Energy Effective Theory	478	(5)
13.2.2 BPS Bound Reexamined	483	(2)
13.3 Seiberg-Witten Explicit Solution	485	(13)
13.3.1 Monodromies on the Moduli Space	485	(7)
13.3.2 Solution of the Monodromy Problem	492	(4)
13.3.3 Confinement and the Monopole Condensation	496	(2)
13.4 Concluding Remarks	498	(3)
A Representations of SU(2)	501	(4)
B Quaternions	505	(4)
C SU(2) Transformations of the Monopole Potential	509	(4)
References	513	(16)
Index	529