Magnetic monopoles

Texts and Monographs in Physics
Series Editors:
R. Balian, Gif-sur-Yvette, France
W. Beiglböck, Heidelberg, Germany
H. Grosse, Wien, Austria
W. Thirring, Wien, Austria


Yakov M. Shnir

Magnetic Monopoles

ABC
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Dr. Yakov M. Shnir
Institute of Physics
Carl von Ossietzky University Oldenburg
26111 Oldenburg
Germany
E-mail:

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To Marina with love

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Preface

“One would be surprised if
Nature had made no use of it.”

P.A.M. Dirac
According to some dictionaries, one meaning of the notion of “beauty” is
“symmetry”. Probably, beauty is not entirely “in the eye of the beholder”. It
seems to be related to the symmetry of the object. From a physical viewpoint,
this definition is very attractive: it allows us to describe a central concept of
theoretical physics over the last two centuries as being a quest for higher
symmetry of Nature. The more symmetric the theory, the more beautiful it
looks.
Unfortunately, our imperfect (at least at low-energy scale) world is full of
nasty broken symmetries. This has impelled physicists to try to understand
how this happens. In some cases, it is possible to reveal the mechanism of
violation and how the symmetry may be recovered; then our picture of Nature
becomes a bit more beautiful.
One of the problems of the broken symmetry that we see is that, while
there are electric charges in our world, their counterparts, magnetic monopoles, have not been found. Thus, in the absence of the monopoles, the symmetry between electric and magnetic quantities is lost. Can this symmetry
be regained?
In the history of theoretical physics, the hypothesis about the possible
existence of a magnetic monopole has no analogy. There is no other purely
theoretical construction that has managed not only to survive, without any
experimental evidence, in the course of more than a century, but has also
remained the focus of intensive research by generations of physicists.
Over the past 25 years the theory of magnetic monopoles has surprisingly
become closely connected with many actual directions of theoretical physics.
This includes the problem of confinement in Quantum Chromodynamics, the
problem of proton decay, astrophysics and evolution of the early Universe, and
the supersymmetrical extension of the Standard Model, to name just a few.
It seems plausible that the answer to the question: “Why do magnetic monopoles not exist?” is a key to understanding the very foundations of Nature.
Furthermore, the mathematical problem of construction and investigation of

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the exact multimonopole configurations is at the frontier of the most fascinating directions of modern field theory and differential geometry. The techniques developed in this area of theoretical physics find many other applications and have become very important mathematical tools.
The theory of monopoles seems to be tailor-made for demonstrating beautiful interplay between mathematics and physics. Therefore, I believe that an
introduction to the basic ideas and techniques that are related to the description and construction of monopoles may be useful to physicists and mathematicians interested in the modern developments in this direction. Moreover,
there is a second aspect of the monopoles. These objects arise in many different contexts running through all levels of modern theoretical physics, from
classical mechanics and electrodynamics to multidimensional branes. This
provides an alternative point of view on the subject, which may be of interest to readers.
My original motivation was to provide a comprehensive review on the
monopole that would capture the current status of the problem, something
which could be entitled “Everything you always wanted to know about the
monopole but did not have time to ask”. However, it soon became clear that
such a project was too ambitious. An estimate of the related literature approaches 6000 papers. The original paper by Dirac [200] has been quoted
more than 1000 times and the citation index of the papers by ’t Hooft and
Polyakov [270, 428] is approaching 1400.
I have therefore tried to give a restricted introduction to the classical and
quantum field theory of monopoles, a more or less compact review, which
could give a “bird’s eye view” on the entire set of problems connected with
the field theoretical aspects of the monopole.
The book is divided into three parts. This approach reproduces in some
sense that used by S. Coleman in his famous lectures [43]; that is, I start
the discussion with a simple classical consideration of a monopole as seen at
large distances and then go on to its internal structure.
In Part I, the monopole is considered “from afar”, at the large distances
where pure electrodynamical description works well. In the first chapter, I
review some features of the classical interaction between a static monopole

and an electric charge. The quantum mechanical consideration in terms of the
Dirac potential is described in Chapter 2. Next, in Chapter 3 the notions of
topology, which are closely related to the theory of monopole, are described.
Chapter 4 is devoted to the generalization of QED, which includes the monopoles. Part II forms the core of the book. There I discuss the theory of nonAbelian monopoles, construction of the multimonopole solutions, and some
applications. In Chapter 5 the famous ’t Hooft–Polyakov solution, the simplest specimen of the monopole family, is discussed. This is the first step inside
the monopole core. I review the basic properties of the classical non-Abelian
monopoles, which arise in spontaneously broken SU (2) gauge theory, and the
relation that exists between the magnetic charge of the configuration and the

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topological charge. The Bogomol’nyi–Prasad–Sommerfield (BPS) monopole
appears here for the first time as a particular analytic solution with vanishing potential. Here I also give a brief account of the gauge zero mode and
comment on its relation to the electric charge. Chapter 6 contains a survey of
the classical multimonopoles, both in the BPS limit and beyond. A powerful
formalism for investigation of the low-energy dynamics of the BPS monopoles is the moduli space approach, which arises from consideration of the
monopole collective coordinates. In Chapter 7 some of the results related to
the quantum field theory of the SU (2) monopoles are reviewed.
Next, in Chapter 8 the consideration is extended to a more general class of
SU (3) theories containing different limits of symmetry breaking. It turns out
that the multimonopole configurations are natural in a model with the gauge
group of higher rank. Here I discuss fundamental and composite monopoles
and consider the limiting situation of the massless states.
Chapter 9 contains a brief survey of the role that the monopoles may play
in the phenomenon of confinement. I discuss here the compact lattice electrodynamics, formalism of Abelian projection in gluodynamics and the Polyakov

solution of confinement in the 2+1-dimensional Georgi–Glashow model. In
Chapter 10 the original Yang–Mills–Higgs system is extended by inclusion
of fermions. Here I consider the details of the monopole–fermion interaction,
especially the role of the fermionic zero modes of the Dirac equation. In this
context, I briefly describe the current status of the Rubakov–Callan effect.
The last part of the book reveals the intersection of many lines of the
previous discussion. Indeed, the spectrum of states of N = 2 supersymmetric
(SUSY) Yang–Mills theory includes the monopoles. There the arguments of
duality become well-founded and the BPS mass bound arises in a new context. Moreover, the geometrical moduli space approach, which was originally
developed to describe the dynamics of BPS monopoles, turns out to be a
key element of the Seiberg–Witten solution of the low-energy N = 2 SUSY
Yang–Mills theory. Chapter 11 is an introductory account of supersymmetry.
Construction of the N = 2 SU (2) supersymmetric monopoles is described
in Chapter 12 and the Seiberg–Witten solution is presented in Chapter 13.
Evidently, this is a separate topic, which has been intensively discussed in
recent years. However, the very structure of the book does not make it possible to avoid such a discussion. The reader will definitely find this topic well
presented elsewhere.
Let us mention some omissions. An obvious gap is the current experimental situation. I do not venture to discuss the numerous experiments directed
to the search for a monopole. This must be the subject of a separate survey.
I would like to point the reader to the very good reviews [47, 48, 50]. However, the most important thing we know from experiment is that there are
probably no monopoles around.
I do not consider the astrophysical aspects of monopoles, the problem of relic monopoles, or other related directions. I do not discuss some

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by-product topics like, for example, the conception of the Berry phase. Neither do I consider some specific mathematical problems of the Abelian theory of monopoles (e.g., singularities and regularization). In considering construction of the BPS multimonopoles, I have made no attempt to discuss one
of the approaches that is related to the application of the inverse scattering
method to the linearized Bogomol’nyi equation. Instead, the discussion concentrates on the modern development due to the Nahm technique and twistor
approach. I would like to draw attention to the recent excellent monograph
by N. Manton and P. Sutcliffe, “Topological Solitons” [54], which provides
the reader with a solid framework of modern classical theory of solitons, not
only monopoles, in a very general context.
Because of the restricted size of the book, I do not consider the very
interesting properties of gravitating monopoles, which are solutions of the
Einstein–Yang–Mills–Higgs theory. I pay more attention to the general properties of the non-Abelian monopoles, namely, to their topological nature.
Coupling with gravity yields a number of classical solutions that are not presented in flat space, so that the related discussion becomes rather involved.
Another omission is the Kaluza–Klein monopole and, more generally, the
analysis of multidimensional theories. For more rigor and broader discussion
I refer the reader to the original publications.
Though extensive, the list of references at the end of the book cannot
be considered an exhaustive bibliography on monopoles. I apologize to those
authors whose contributions are not mentioned here.
The work on this project coincided with a period of serious personal turmoil. I am grateful to all my friends and colleagues who supported me. I am
deeply indebted to Ana Achucarro, Emil Akhmedov, Alexander Andrianov,
Dmitri Antonov, Jă
urgen Baacke, Pierre van Baal, Askhat Gazizov, Dmitri
Diakonov, Conor Houghton, Iosif Khriplovich, Viktor Kim, Valerij Kiselev,
Ken Konishi, Boris Krippa, Steffen Krusch, Dieter Maison, Stephane Nonnenmacher, Alexander Pankov, Murray Peshkin, Victor Petrov, Lutz Polley,
Mikhail Polikarpov, Maxim Polyakov, Kirill Samokhin, Ruedi Seiler, Andrei
Smilga, Joe Sucher, Paul Sutcliffe, Tigran Tchrakian, Arthur Tregubovich,
Andreas Wipf, and Wojtek Zakrzewski for many useful discussions, critical
interest and remarks. I am very thankful to L.M. Tomilchik and E.A. Tolkachev, who were my teachers and advisors, for their valuable support, encouragement, and guidance. They awakened my interest in the monopole problem.
Many of the ideas discussed here are due to Nick Manton, who played a
very important role in my understanding of the monopoles, both through his
papers and in private discussions. He commands my deepest personal respect

and gratitute. The year I spent in Cambridge in his group strongly influenced
my life.
This book originates from work in collaboration with Per Osland which,
unfortunately, was not completed. Without his support and encouragement
I would never have started to work on this extended project. A draft version

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of the first five chapters was prepared in collaboration with him during my
stays at the Institute of Physics, University of Bergen.
I am deeply indebted to Burkhard Kleihaus and Jutta Kunz for collaboration and help in numerous ways. The support I received in Oldenburg has
been invaluable.
My special thanks go to Milutin Blagojevi´c, Maxim Chernodub, Adriano
Di Giacomo, Fridrich W. Hehl, and Valentine Zakharov for reading a preliminary version of several chapters and providing many helpful comments,
suggestions, and remarks.
I would like to acknowledge the hospitality I received at the Service
de Physique Th´eorique, CEA-Saclay, the Max-Planck-Institut fă
ur Physik
(Werner-Heisenberg-Institut), Mă
unchen, and the Abdus Salam International
Center for Theoretical Physics, Trieste, where some parts of this work were
carried out. A substantial part of the work on the manuscript was done in
1999–2002 at the Institute of Theoretical Physics, University of Cologne.
Some chapters of the book are elaborations of lectures given on several occasions.
Oldenburg,

June 2005

Yakov Shnir

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Contents

Part I Dirac Monopole
1

Magnetic Monopole in Classical Theory . . . . . . . . . . . . . . . . . .
1.1 Non-Relativistic Scattering on a Magnetic Charge . . . . . . . . . .
1.2 Non-Relativistic Scattering on a Dyon . . . . . . . . . . . . . . . . . . . . .
1.3 Vector Potential of a Monopole Field . . . . . . . . . . . . . . . . . . . . .
1.4 Transformations of the String . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Dynamical Symmetries of the Charge-Monopole System . . . . .
1.6 Dual Invariance of Classical Electrodynamics . . . . . . . . . . . . . . .

2

The Electron–Monopole System:
Quantum-Mechanical Interaction . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Charge Quantization Condition . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Spin-Statistics Theorem in a Monopole Theory . . . . . . . . . . . . .
2.3 Charge-Monopole System: Quantum-Mechanical Description .
2.3.1 The Generalized Spherical Harmonics . . . . . . . . . . . . . . .
2.3.2 Solving the Radial Schră
odinger Equation . . . . . . . . . . . .

2.4 Non-Relativistic Scattering on a Monopole:
Quantum Mechanical Description . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Charge-Monopole System: Spin in the Pauli Approximation . .
2.5.1 Dynamical Supersymmery
of the Electron-Monopole System . . . . . . . . . . . . . . . . . . .
2.5.2 Generalized Spinor Harmonics: j ≥ µ + 1/2 . . . . . . . . . .
2.5.3 Generalized Spinor Harmonics: j = µ − 1/2 . . . . . . . . . .
2.5.4 Solving the Radial Pauli Equation . . . . . . . . . . . . . . . . . .
2.6 Charge-Monopole System: Solving the Dirac Equation . . . . . .
2.6.1 Zero Modes and Witten Effect . . . . . . . . . . . . . . . . . . . . .
2.6.2 Charge Quantization Condition
and the Group SL(2, Z) . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Topological Roots of the Abelian Monopole . . . . . . . . . . . . . . .
3.1 Abelian Wu–Yang Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Differential Geometry and Topology . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Notions of Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Notions of Differential Geometry . . . . . . . . . . . . . . . . . . .

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3.2.3 Maxwell Electrodynamics and Differential Forms . . . . .

3.3 Wu–Yang Monopole and the Fiber-Bundle Topology . . . . . . . .
3.3.1 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Principal Bundle and Connection . . . . . . . . . . . . . . . . . . .
3.3.3 Wu–Yang Monopole Bundle . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Hopf Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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103

Abelian Monopole:
Relativistic Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Two Types of Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Two-Potential Formulation of Electrodynamics . . . . . . . . . . . . .
4.2.1 Energy-Momentum Tensor and Angular Momentum . .
4.3 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Relativistic Invariance of Two-Charge Electrodynamics
4.4 Renormalization of QED with a Magnetic Charge . . . . . . . . . .
4.5 Vacuum Polarization by a Dyon Field . . . . . . . . . . . . . . . . . . . . .
4.6 Effective Lagrangian of QED with a Magnetic Charge . . . . . . .

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Part II Monopole in Non-Abelian Gauge Theories
5

’t Hooft–Polyakov Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 SU (2) Georgi–Glashow Model and the Vacuum Structure . . . .
5.1.1 Non-Abelian Wu–Yang Monopole . . . . . . . . . . . . . . . . . . .
5.1.2 Georgi–Glashow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Topological Classification of the Solutions . . . . . . . . . . .
5.1.4 Definition of Magnetic Charge . . . . . . . . . . . . . . . . . . . . .
5.1.5 ’t Hooft–Polyakov Ansatz . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.6 Singular Gauge Transformations
and the Connection between’t Hooft–Polyakov
and Dirac monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.7 Dyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The Bogomol’nyi Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Gauge Zero Mode and the Electric Dyon Charge . . . . .
5.3 Topological Classification of Non-Abelian Monopoles . . . . . . . .
5.3.1 SO(3) vs SU (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Magnetic Charge and the Topology
of the Gauge Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Equivalence of Topological and Magnetic Charge . . . . .
5.3.4 Topology of the Dyon Sector . . . . . . . . . . . . . . . . . . . . . . .
5.4 The θ Term and the Witten Effect Again . . . . . . . . . . . . . . . . . .

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7

8

Multimonopole Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Multimonopoles Configurations

and Singular Gauge Transformations . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Singular SU (2) Monopole with Charge g = ng0 . . . . . .
6.1.2 Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Rebbi–Rossi Multimonopoles, Chains
of Monopoles and Closed Vortices . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Interaction of Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Monopole in External Magnetic Field . . . . . . . . . . . . . . .
6.3.2 The Interaction Energy of Monopoles . . . . . . . . . . . . . . .
6.3.3 Classical Interaction
of Two Widely Separated Dyons . . . . . . . . . . . . . . . . . . .
6.4 The n-Monopole Configuration in the BPS Limit . . . . . . . . . . .
6.4.1 BPS Multimonopoles: A Bird’s Eye View . . . . . . . . . . . .
6.4.2 Projective Spaces and Twistor Methods . . . . . . . . . . . . .
6.4.3 The n-Monopole Twistor Construction . . . . . . . . . . . . . .
6.4.4 Hitchin Approach and the Spectral Curve . . . . . . . . . . .
6.4.5 Nahm Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.6 Solution of the Nahm Equations . . . . . . . . . . . . . . . . . . . .
6.4.7 The Nahm Data and Spectral Curve . . . . . . . . . . . . . . . .
6.5 Moduli Space and Low-Energy Multimonopoles Dynamics . . .
6.5.1 Zero Modes Lagrangian and the Moduli Space Metric .
6.5.2 Metric on the Space M2 . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Low-Energy Scattering of Two Monopoles . . . . . . . . . . .

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SU (2) Monopole in Quantum Theory . . . . . . . . . . . . . . . . . . . . .
7.1 Field Fluctuations on Monopole Background . . . . . . . . . . . . . . .
7.1.1 Generalized Angular Momentum and the Spectrum
of Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Quantum Correction to the Mass of a Monopole . . . . . .
7.2 Non-Abelian Monopole: Quasiclassical Quantization . . . . . . . .
7.2.1 Collective Coordinates and Constraints . . . . . . . . . . . . . .
7.2.2 Quantum Mechanics on the Moduli Space . . . . . . . . . . .
7.2.3 Evaluation of the Generating Functional . . . . . . . . . . . .
7.3 g¯
g Pair Creation in an External Magnetic Field . . . . . . . . . . . .
7.3.1 Dynamics of Non-Abelian Monopole
in Weak External Field . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3.2 Metastable Vacuum Decay and Monopole Pair
Creation in an External Field . . . . . . . . . . . . . . . . . . . . . .

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Monopoles Beyond SU (2) Group . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 SU (N ) Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Generalization of the Charge Quantization Condition .
8.1.2 Towards Higher Rank Gauge Groups . . . . . . . . . . . . . . . .
8.1.3 Montonen–Olive Conjecture . . . . . . . . . . . . . . . . . . . . . . .

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8.1.4 Cartan–Weyl Basis and the Simple Roots . . . . . . . . . . . .
8.1.5 SU (3) Cartan Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.6 SU (3) Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Massive and Massless Monopoles . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Pathologies of Non-Abelian Gauge Transformations . . .
8.3 SU (3) Monopole Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 SU (3) Monopoles: Nahm Equations . . . . . . . . . . . . . . . .

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Monopoles and the Problem of Confinement . . . . . . . . . . . . . .
9.1 Quark Confinement in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Dual Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Monopoles in the Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Compact QED and Lattice Monopoles . . . . . . . . . . . . . .
9.2.2 Lattice Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Abelian Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 “Monopoles” from Abelian Projection . . . . . . . . . . . . . . .
9.3.2 Maximal Abelian Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4 Polyakov Solution of Confinement
in the d = 3 Georgi–Glashow Model . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 Dilute Gas of Monopoles in the d = 3
Georgi–Glashow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Wilson Loop Operator in d = 3
Georgi–Glashow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Rubakov–Callan Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Dirac Hamiltonian
on the Non-Abelian Monopole Background . . . . . . . . . . . . . . . .
10.1.1 Fermionic Zero Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.2 Zero Modes and the Index Theorem . . . . . . . . . . . . . . . .
10.1.3 S-Wave Fermion Scattering on a Monopole . . . . . . . . . .
10.2 Anomalous Non-Conservation of the Fermion Number . . . . . . .
10.2.1 Axial Anomaly and the Vacuum Structure . . . . . . . . . . .
10.2.2 Effective Action of Massless Fermions . . . . . . . . . . . . . . .
10.2.3 Properties of the Anomalous Fermion Condensate . . . .
10.2.4 Properties of Other Condensates . . . . . . . . . . . . . . . . . . .
10.3 Monopole-Fermion Scattering
in the Bosonisation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.3.1 Vertex Operator and Bosonization
of the Free Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Monopole Catalysis of the Proton Decay . . . . . . . . . . . .
10.3.3 Monopole Catalysis of the Proton Decay:
Semiclassical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359

9

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349
349
355

359
363
367
373
378
378
379
385
388
390
391
397
400

Contents

XVII

Part III Supersymmetric Monopoles
11 Supersymmetric Yang-Mills Theories . . . . . . . . . . . . . . . . . . . . .
11.1 What is Supersymmetry? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.1 Poincar´e Group and Algebra of Generators . . . . . . . . . .
11.1.2 Algebra of Generators of Supersymmetry . . . . . . . . . . . .
11.2 Representations of SUSY Algebra . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 N = 1 Massive Multiplets . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 N = 1 Massless Multiplets . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.3 N = 2 Extended SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Local Representations of SUSY . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 N = 1 Superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.2 N = 1 Superfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 Non-Abelian Multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 N = 1 SUSY Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Magnetic Monopoles in the N = 2
Supersymmetric Yang–Mills Theory . . . . . . . . . . . . . . . . . . . . . .
12.1 N = 2 Supersymmetric Lagrangian . . . . . . . . . . . . . . . . . . . . . . .
12.1.1 Praise of Beauty of N = 2 SUSY Yang–Mills . . . . . . . . .
12.2 N = 2 Supersymmetric SU (2) Magnetic Monopoles . . . . . . . . .
12.2.1 Construction of N = 2 Supersymmetric
SU(2) Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Central Charges in the N = 2 SUSY Yang–Mills . . . . . . . . . . . .
12.4 Fermionic Zero Modes in Supersymmetric Theory . . . . . . . . . .
12.5 Low Energy Dynamics of Supersymmetric Monopoles . . . . . . .
12.6 N = 2 Supersymmetric Monopoles beyond SU (2) . . . . . . . . . . .

12.6.1 SU (3) N = 2 Supersymmetric Monopoles . . . . . . . . . . .
13 Seiberg–Witten Solution
of N = 2 SUSY Yang–Mills Theory . . . . . . . . . . . . . . . . . . . . . . .
13.1 Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1.1 Moduli Space and its Parameterization . . . . . . . . . . . . . .
13.1.2 Quantum Moduli Space
of N = 2 SUSY Yang–Mills Theory . . . . . . . . . . . . . . . . .
13.2 Global Parametrization of the Quantum Moduli Space . . . . . .
13.2.1 Transformation of Duality
for N = 2 Low-Energy Effective Theory . . . . . . . . . . . . .
13.2.2 BPS Bound Reexamined . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Seiberg–Witten Explicit Solution . . . . . . . . . . . . . . . . . . . . . . . . .
13.3.1 Monodromies on the Moduli Space . . . . . . . . . . . . . . . . .
13.3.2 Solution of the Monodromy Problem . . . . . . . . . . . . . . . .
13.3.3 Confinement and the Monopole Condensation . . . . . . . .
13.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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407
407
408
412
415
415
417
418
420
420
424

428
429

437
437
441
443
443
446
449
451
453
458

465
466
466
472
478
478
483
485
485
492
496
498


XVIII Contents

A

Representations of SU (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

B

Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

C

SU (2) Transformations of the Monopole Potential . . . . . . . . 509

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

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Part I

Dirac Monopole

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1 Magnetic Monopole in Classical Theory

1.1 Non-Relativistic Scattering on a Magnetic Charge
One could set up a naive definition of a monopole as being just a point-like
particle with a magnetic charge instead of an electric one. Then almost all

non-trivial features caused by its presence would manifest themselves in the
process of interaction between a monopole and “normal” electrically charged
particles. One can see these features already on the level of classical mechanics by comparing the electric-charge-monopole scattering and the standard Coulomb problem. Historically that problem was first considered by
H. Poincar´e in the context of interaction of an electron beam and the pole
of a very long and very thin magnet already more than a century ago, in
1896 [425]. This work could be considered a first brick in the foundation of
the modern history of the monopole. Nevertheless, one should say that for a
long time before H. Poincar´e’s work, the question about the possible existence
of a single magnetic pole was raised many times.1
In this section, we will consider the classical non-relativistic motion of
a charge in an external field. That is why it would be correct to define a
magnetic charge g as a source of a static Coulomb-like magnetic field
B=g

r
.
r3

(1.1)

Then the equation of motion of an electrically charged particle e in such a
field is
d2 r
eg dr
×r ,
(1.2)
m 2 = e [v × B] = 3
dt
r dt
where a static monopole is situated at the origin and the vector r defines the

position of the electric charge (see Fig. 1.1). For the sake of simplicity we
will use units such that the speed of light c is equal to 1 and in this section
consider only positive values of both electric and magnetic charges.
1

A very detailed description of the “stone age history” of the monopole problem
is given in [35], where the genesis of it has been traced up to the notes by Petrus
Pelegrinius, written at the Crusades in 1269! We will not go into this fascinating
story.

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4

1 Magnetic Monopole in Classical Theory

z

v
e

θ

g

r

-eg r
θ

L

L

Fig. 1.1. Motion of an electric charge in the monopole field

One could obtain the corresponding integrals of motion just by making
use of (1.2). Scalar multiplication of (1.2) by a vector of velocity v gives:
1 d
mv 2 = 0 ,
2 dt

(1.3)

so that the kinetic energy of an electric charge in a monopole field is a constant:
mv 2
E=
= const. ,
(1.4)
2
as is the absolute value v of the velocity vector.
On the other hand, the scalar product of the equation of motion (1.2) and
the radius vector r gives:
r·

1 d2 2
d2 r
≡
r − v2 = 0 .

2
dt
2 dt2

Taking into account the conservation of energy (1.3), one can write
r=

v 2 t 2 + b2 ,

(1.5)

and therefore r · (dr/dt) = r · v = v 2 t. Thus, there is no closed orbit in the
charge-monopole system: the electric charge is falling down from infinitely
far away onto the monopole, approaching a minimal distance b and reflected
back to infinity (so-called “magnetic mirror” effect).
A very special feature of such a motion is that the conserved angular
momentum is different from the ordinary case. Indeed, one can see that the
absolute value of the vector of ordinary angular momentum

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1.1 Non-Relativistic Scattering on a Magnetic Charge

L = r × mv

5

(1.6)

is conserved, because the cross product of r and (1.2) is
d
dL
eg
[r × mv] ≡
=
L×r .
dt
dt
mr3

(1.7)

Scalar multiplication of this equation with the vector L gives
d
|L| = 0 ,
dt

(1.8)

and, because the absolute value of the velocity vector is a constant, one can
write
L ≡ |L| = mvb .
(1.9)
The very important difference from the ordinary Coulomb problem is
that now the direction of the vector of angular momentum is not a constant,
because from (1.7) it follows that
d
dL
r

=
= 0,
L − eg
dt
r
dt

(1.10)

where the generalized angular momentum is an integral of motion:
L = [r × mv] − eg

r
= L − egˆr .
r

(1.11)

Let ˆr be a unit vector in the direction of r. Taking into account (1.9) one can
write (see Fig. 1.1)
L2 ≡ L2 = L2 + e2 g 2 = (mvb)2 + (eg)2 .

(1.12)

As was demonstrated by J.J. Thompson already in 1904 [13, 500], the
appearance of an additional term in the definition of the angular momentum
(1.11) originates from a non-trivial field contribution. Indeed, since a static
monopole is placed at the origin, its magnetic field is given by (1.1). Then
the classical angular momentum of the electric field of a point-like electric
charge, whose position is defined by its radius vector r, and the magnetic

field of a monopole is a volume integral involving the Poynting vector
Leg =

1
4π

d3 r [r × (E × B)] = −

g
4π

d3 r (∇ · E) ˆr = −egˆr ,

(1.13)

where we perform the integration by parts, take into account that the fields
vanish asymptotically and invoke the Maxwell equation
(∇ · E) = 4πe δ (3) (r − r ) .

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6

1 Magnetic Monopole in Classical Theory

At first sight, this conclusion looks rather paradoxical. Indeed, according to
(1.13) even a static charge-monopole system has a non-zero angular momentum.
Notice that this formula could easily be generalized to the case of a pair
of dyons, dual charged particles having both electric and magnetic charges,

(e1 , g1 ) and (e2 , g2 ), respectively [549]. Let one of the dyons be placed at the
origin and the position of the other one be given by the vector r. Then the
fields are
r
r
B = g1 3 + B(g2 ) ,
E = e1 3 + E(e2 ),
r
r
and by analogy with (1.13) one has
Ldd =

1
4π

d3 r [r ×(E × B)]

=

1
4π

d3 r

=

e1
4π

d3 r [∇ · B(g2 )] ˆr −

r × e1

r
× B(g2 )
r3
g1
4π

+ r × E(e2 ) × g1

r
r3

d3 r [∇ · E(e2 )] ˆr

= (e1 g2 − g1 e2 )ˆr .

(1.14)

Later we will come back to the definition of the generalized angular momentum by making use of standard variational procedure. Here we would like
only to note that the conservation of the magnitude of the velocity together
with the constant modulus of the angular momentum vector means that the
impact parameter of the scattering problem coincides with the minimal separation b between the monopole and the electric charge. Also note that the
energy of a charge in a monopole field (1.4) can be written as
E=

L2 − (eg)2
mr˙ 2
+

= const. ,
2
2mr2

(1.15)

where we make use of the definition (1.12).
Thus, unlike the standard problem of charge scattering in a Coulomb field,
now the trajectory does not lie in the plane of scattering that is orthogonal
to the vector L. To define the character of the motion note that
|L · ˆr| = eg = const. ,

(1.16)

i.e., the angle between the vectors L and r is a constant and the electric charge
is moving on the surface of a cone whose axis is directed along −L with the
cone angle θ, which can be defined using simple geometry (see Fig. 1.2) as
cot θ =

eg
|L|

=

eg
,
mvb

or

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(1.17)


1.1 Non-Relativistic Scattering on a Magnetic Charge

7

V+

8

V-

8

z

Θ
θ

∆φ
2

y

∆φ
2

x
Fig. 1.2. Geometry of scattering of an electron by a monopole

sin θ =

L
=
L

mvb
(mvb)2

+

(eg)2

,

cos θ =

eg
=
L

eg
(mvb)2 + (eg)2

.

(1.18)

Thus, the motion becomes planar only in the limit g → 0, or θ = π, which
corresponds to the degeneration of the cone.
In the same way the ordinary vector of angular momentum L is precessing
on the surface of a cone with a different cone angle but the same axis, because
L · L = L2 = (mvb)2 = const.
As was noted already by H. Poincar´e [425], the existence of the integrals of
motion (1.11) and (1.7) links the system of interacting electric and magnetic
charges with a simple mechanical analog, a spherical top. One can understand
it as a rotating disk with a thin rod of variable length as an axis of rotation.
The charge and the monopole are sitting at the opposite ends of the rod.
Finally, the cross product of L (1.11) and the radius vector r, together
with (1.5), yields
v=

1
v2 t
1
dr
ˆr =
=
[L
×
r]
+
[L × r] +
dt
mr2
r
mr2

v
1 + (b/vt)2

= [ω × r] + vr ˆr ,

ˆr
(1.19)

where the angular and radial components of the velocity vector are
ω=

L
,
mr2

vr =

v
1 + (b/vt)2

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.

(1.20)


8

1 Magnetic Monopole in Classical Theory

Hence, asymptotically
ω

t=±∞

= 0,

vr

t=±∞

= v.

At the turning point of the path, where the distance between the charge and
the monopole is minimal
ω

t=0

(mvb)2 + (eg)2
,
mb2

=

vr

t=0

= 0.

Thus, because the angular velocity is defined as ω = dϕ/dt, the azimuthal
angle ϕ as a function of time can be obtained by simple integration2
ϕ(t) =

vt
1
arctan ,
sin θ
b

(1.21)

where we made use of (1.5) and fix the boundary condition to ϕ = 0 at t = 0.
Furthermore, θ is given by (1.18).
Since asymptotically
ˆ
v

t=±∞

=

± sin θ cos

∆ϕ
,
2

sin θ sin

∆ϕ
,
2

± cos θ ,

where ∆ϕ = ϕ(∞) − ϕ(−∞) = π/ sin θ (see Fig. 1.2), we can now calculate
the angle of scattering on a monopole
ˆ
cos Θ = v

t=−∞

ˆ
·v

or
cos

t=+∞

Θ
2

= 2 sin2 θ sin2

π

− 1,
2 sin θ

π
2 sin θ

= sin θ sin

,

(1.22)

(1.23)

where θ is a function of the impact parameter b, (1.18).
Unlike the standard problem of scattering in a Coulomb field, the angle of
scattering Θ is not a monotonous function of the impact parameter b [462].
The dependence Θ(b) is depicted in Fig. 1.3, where the impact parameter b is
rescaled in units of the parameter eg/mv. That is why, in order to calculate
the effective cross-section, one has to take into account the contributions
from all values of the impact parameter (or, equivalently, from all values of
the cone angles θi ), leading to scattering into the surface element dσ:
bdb
dσ
=
=
dΩ
d(cos Θ)

θi

eg
mv

2

sin 2θdθ
1
.
2 cos4 θ sin ΘdΘ

Here we made use of (1.17).
2

Remember, v is constant, but r˙ is not.

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(1.24)


1.1 Non-Relativistic Scattering on a Magnetic Charge

9

Fig. 1.3. Dependence of the scattering angle Θ on the impact parameter b

One can see that the effective cross-section of an electric charge (1.24) on
the monopole is singular if sin Θ = 0 or dΘ/dθ = 0. In the scattering theory
these two situations are referred to as the glory and rainbow respectively

[462]. The first case corresponds to the back scattering, where3 Θ = π, while
the cone is not degenerated, i.e., θ = π. The formula (1.22) allows us to define
corresponding “critical” values of the cone angles [131, 462]:4
sin θn =

1
,
2n

n = 1, 2, 3 . . .

(1.25)

or θ1 = 0.5236, θ2 = 0.2527, θ3 = 0.1674 . . .
The rainbow scattering corresponds to cone angles θr being the solutions
of the transcendental equation
tan

π
2 sin θr

=

π
.
2 sin θr

(1.26)

These angles are θI = 0.3571, θII = 0.2048, θIII = 0.1446 . . . Note that

in both situations of glory and rainbow scattering the singularities of the
cross-section are integrable and the total cross-section for scattering on a
monopole is well defined. Note that such singularities are absent for smallangle scattering, defined by the condition Θ ≈ π − 2θ = 2eg/mvb
1. In
such a case the differential cross-section is
3
4

The case Θ → 0, or θ → π/2, would correspond to eg → 0.
Other authors use the complementary angle π/2 − θ.

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10

1 Magnetic Monopole in Classical Theory

1
dσ
= 4
dΩ
Θ

2eg
mv

2

,

(1.27)

which is evidently analogous to the Rutherford formula.

1.2 Non-Relativistic Scattering on a Dyon
Let us generalize the results of the previous section to the case of classical
non-relativistic scattering of an electrically charged particle on a static dyon
having both electric (Q) and magnetic (g) charges5 . For simplicity we restrict
our consideration to the case of an attractive electrostatic potential, i.e., suppose that V = eQ/r, where eQ < 0. A qualitative analysis suggests that
unlike the charge-monopole scattering, described above, there are closed trajectories in such a system. Indeed, let us consider the corresponding equation
of motion (cf. (1.2))
m

r
eg
dr
d2 r
= eQ 3 − 3 r ×
.
2
dt
r
r
dt

(1.28)

Obviously, the generalized angular momentum L given by (1.11) is still an
integral of motion. Also, the projection of the total angular momentum onto

the radial direction Lr = |L · ˆr| = eg, as well as the magnitude of the orbital
angular momentum L = mbv0 , where v0 is the initial velocity of the electric
charge given at an infinitely large distance from the scattering center, are
conserved. Thus the motion is restricted to the same surface of a cone with
a cone angle cot θ = eg/mbv0 , as it was in the case of charge-monopole
scattering. The difference is that now the magnitude of the velocity is no
longer an integral of motion, because unlike (1.4) the total energy conserved
is now
L2
eQ
mr˙ 2
mv2
eQ
+
=
+
= const.
(1.29)
E=
+
2
r
2
2mr2
r
Here, one of the basic features of the interaction between a monopole
and an electrically charged particle manifests itself: if the radial part of the
Hamiltonian is determined by a Coulomb interaction, then the interaction
of a charge and a monopole is described by its angular part. Indeed, we have
seen that the magnitude of the radius vector of a charge moving in a magnetic

Coulomb field depends on time just as in the case of free motion (see (1.5)).
Hence, in the system of reference, which rotates with the angular velocity ω(t)
5

The problem of charge motion in a monopole (dyon) field was probably considered first by S.A. Boguslavsky [128], who also derived an expression for a
vector potential of a monopole field a decade before the celebrated paper by
P.A.M. Dirac [200]. The author is grateful to E.A. Tolkachev and L.M. Tomilchik
for kindly informing him about that undeservedly forgotten paper [497]. Other
references include [114, 388].

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1.2 Non-Relativistic Scattering on a Dyon

11

(cf. (1.20)), the equation of motion is trivial: in the rotating plane orthogonal
to the vector L the electric charge is moving with a constant velocity v along
a straight line. Just in the same way, in the case of motion in a dyon field, the
time dependence of the magnitude of the radius vector of the electric charge,
is the same as for the ordinary interaction of two electric charges e and Q.
Indeed, from (1.29) follows that

r˙ =

2
m

E+

L2
|eQ|
−
r
2mr2

.

(1.30)

For a bound motion E < 0 and according to the standard procedure (see for
example [18]) we can write
m
2|E|

t=

rdr

,

(1.31)

−r2 + (|eQ|r)/|E| − L2 /(2m|E|)

where the constant of integration can be chosen to fix the parameters t0 = 0
and r0 = d. The latter denotes the minimal distance between the charge e
and the dyon, which unlike the problem of charge-monopole scattering is no
longer equal to the impact parameter.

The elementary integration of (1.31) allows us to find the parametric
dependence of coordinates on time. Putting
a=

|eQ|
,
2|E|

b=

L
2m|E|

,

1−

ε=

b2
=
a2

1−

2|E|L2
,
me2 Q2

(1.32)

the integral (1.31) can be rewritten as
t=

m
2|E|

rdr
a2 − b2 − (r − a)2

=

m
2|E|

rdr
a2 ε2 − (r − a)2

,

which gives the parametric equation
t=

ma2
(ξ − ε sin ξ) , r = a (1 − ε cos ξ) .
2|E|

(1.33)

An azimuthal angle ϕ as a function of time could be defined in the same

way. Because the angular velocity of a charge in a dyon field is given by
(1.20), the elementary integration of this relation gives
|L|
ϕ(t) = √
2m

dr
r2

−|E| + (|eQ|)/r − L2 /(2mr2 )

or

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,

(1.34)