Table Of ContentHarmonic Superspace
This is the first pedagogical introduction to the harmonic superspace method in
extended (N > 1) supersymmetry. Inspired by recent exciting developments
in superstring theory, it provides a systematic treatment of the quantum field
theories with N = 2 and N = 3 extended supersymmetry in harmonic
superspace. This new kind of superspace, proposed and developed by the
authors, is now generally recognized as the most natural geometric framework
for such theories.
The authors present the harmonic superspace approach as a means of pro-
viding a concise and covariant off-shell description of all of the N = 2 super-
symmetric theories, at both the classical and quantum levels. Furthermore, they
show how it offers a unique way to construct an off-shell formulation of a theory
with higher supersymmetry, namely, the N = 3 supersymmetric Yang–Mills
theory. On shell it is equivalent to the famous N = 4 super-Yang–Mills theory
which was the first example of an ultraviolet-finite supersymmetric model.
Harmonic superspace makes manifest many remarkable geometric properties
of the N = 2 theories, for example, the one-to-one correspondence between
N = 2 supersymmetric matter, and hyper-Ka¨hler and quaternionic manifolds.
Moreover, this methodology offers new and important insights into the geometry
of these complex Riemannian manifolds.
This book will be of great interest to researchers and graduate students
working in the areas of supersymmetric quantum field theory, string theory and
complex geometries.
ALEXANDER GALPERIN, EVGENY IVANOV, VICTOR OGIEVETSKY (deceased)
and EMERY SOKATCHEV invented harmonic superspace whilst working together at the
Joint Institute for Nuclear Research at Dubna, Russia, during the 1980s. They are world
experts on the subject and developed the formalism to the present state of the art. Since
that time, the authors have dispersed across the globe.
After working at Imperial College, London, and the Johns Hopkins University,
Baltimore, Galperin is now employed by a finance company in Washington DC, but
remains interested in the field. Ivanov is a professor at the Bogoliubov Laboratory
of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, where he heads
the research group ‘Problems of Supersymmetry’. He succeeded Professor Ogievetsky
who was the head of this research group for many years, until passing away in 1996.
Ogievetsky was a pioneer in the theory of supersymmetry and during his lifetime was
honoured with the I. E. Tamm Gold Medal of the Academy of Sciences of the USSR
and the von Humboldt Foundation Award. Since leaving Dubna, Sokatchev has worked
at CERN, the Bulgarian Academy of Sciences, and the University of Bonn, and is now
a professor at the Laboratoire d’Annecy-le-Vieux de Physique The´orique (CNRS) and
at the Universite´ de Savoie, France – where he teaches.
Between them, Galperin, Ivanov, Ogievetsky and Sokatchev have authored several
hundred papers in the fields of supersymmetry, gauge theories, string theory, super-
gravity and integrable systems.
CAMBRIDGE MONOGRAPHS ON
MATHEMATICAL PHYSICS
General editors: P. V. Landshoff, D. R. Nelson, D. W. Sciama, S. Weinberg
J. Ambjørn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach
A. M. Anile Relativistic Fluids and Magneto-Fluids
J. A. de Azca´rraga and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications
†
in Physics
V. Belinski and E. Verdaguer Gravitational Solitons
J. Bernstein Kinetic Theory in the Early Universe
G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems
N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Space†
S. Carlip Quantum Gravity in 2 + 1 Dimensions
J. C. Collins Renormalization†
M. Creutz Quarks, Gluons and Lattices†
P. D. D’Eath Supersymmetric Quantum Cosmology
F. de Felice and C. J. S. Clarke Relativity on Curved Manifolds†
P. G. O. Freund Introduction to Supersymmetry†
J. Fuchs Affine Lie Algebras and Quantum Groups†
J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for
Physicists
A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. Sokatchev Harmonic Superspace
R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity†
M. Go¨ckeler and T. Schu¨cker Differential Geometry, Gauge Theories and Gravity†
C. Go´mez, M. Ruiz Altaba and G. Sierra Quantum Groups in Two-dimensional Physics
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 1: Introduction†
M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 2: Loop Amplitudes,
Anomalies and Phenomenology†
S. W. Hawking and G. F. R. Ellis The Large-Scale Structure of Space-Time†
F. Iachello and A. Aruna The Interacting Boson Model
F. Iachello and P. van Isacker The Interacting Boson–Fermion Model
C. Itzykson and J.-M. Drouffe Statistical Field Theory, volume 1: From Brownian Motion to
Renormalization and Lattice Gauge Theory†
C. Itzykson and J.-M. Drouffe Statistical Field Theory, volume 2: Strong Coupling, Monte Carlo
Methods, Conformal Field Theory, and Random Systems†
J. I. Kapusta Finite-Temperature Field Theory†
V. E. Korepin, A. G. Izergin and N. M. Boguliubov The Quantum Inverse Scattering Method and
Correlation Functions†
M. Le Bellac Thermal Field Theory†
N. H. March Liquid Metals: Concepts and Theory
I. M. Montvay and G. Mu¨nster Quantum Fields on a Lattice†
A. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization†
R. Penrose and W. Rindler Spinors and Space-Time, volume 1: Two-Spinor Calculus and Relativistic
Fields†
R. Penrose and W. Rindler Spinors and Space-Time, volume 2: Spinor and Twistor Methods in
Space-Time Geometry†
S. Pokorski Gauge Field Theories, 2nd edition
J. Polchinski String Theory, volume 1: An Introduction to the Bosonic String
J. Polchinski String Theory, volume 2: Superstring Theory and Beyond
V. N. Popov Functional Integrals and Collective Excitations†
R. G. Roberts The Structure of the Proton†
J. M. Stewart Advanced General Relativity†
A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects†
R. S. Ward and R. O. Wells Jr Twistor Geometry and Field Theories†
†
Issued as a paperback
Harmonic Superspace
A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky, E. S. Sokatchev
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
© A. S. Galperin, E. A. Ivanov, V. I. O gievetsky, E. S. Sokatchev 2004
First published in printed format 2001
ISBN 0-511-04145-4 eBook (netLibrary)
ISBN 0-521-80164-8 hardback
Contents
Preface xi
1 Introductory overview 1
1.1 Brief motivations 1
1.2 Brief sumary 4
1.3 Spaces and superspaces 4
1.4 Chirality as a kind of Grasman analyticity 6
1.5 N = 1 chiral superfields 6
1.6 Auxiliary fields 7
1.7 Why standard superspace is not adequate for N = 2 supersymmetry 8
1.8 Search for conceivable superspaces (spaces) 9
1.9 N = 2 harmonic superspace 10
2
1.10 Dealing with the sphere S 10
1.10.1 Comparison with the standard harmonic analysis 11
1.1 Why harmonic superspace helps 13
1.12 N = 2 supersymmetric theories 15
1.12.1 N = 2 mater hypermultiplet 15
1.12.2 N = 2 Yang–Mils theory 16
1.12.3 N = 2 supergravity 18
1.13 N = 3 Yang–Mils theory 19
1.14 Harmonics and twistors. Self-duality equations 20
1.15 Chapters of the bok and their abstracts 23
2 Elements of supersymmetry 27
2.1 Poincare´ and conformal symmetries 27
2.1.1 Poincare´ group 27
2.1.2 Conformal group 28
2.1.3 Two-component spinor notation 29
2.2 Poincare´ and conformal superalgebras 29
vii
viii Contents
2.2.1 N = 1 Poincare´ superalgebra 29
2.2.2 Extended supersymmetry 30
2.2.3 Conformal supersymmetry 31
2.2.4 Central charges from higher dimensions 32
2.3 Representations of Poincare´ supersymmetry 3
2.3.1 Representations of the Poincare´ group 3
2.3.2 Poincare´ superalgebra representations. Massive case 34
2.3.3 Poincare´ superalgebra representations. Massless case 36
2.3.4 Representations with central charge 37
2.4 Realizations of supersymmetry on fields. Auxiliary fields 38
2.4.1 N = 1 mater multiplet 38
2.4.2 N = 1 gauge multiplet 41
2.4.3 Auxiliary fields and extended supersymmetry 41
3 Superspace 4
3.1 Coset space generalities 4
3.2 Coset spaces for the Poincare´ and super Poincare´ groups 46
3.3 N = 2 harmonic superspace 50
3.4 Harmonic variables 54
3.5 Harmonic covariant derivatives 58
3.6 N = 2 superspace with central charge coordinates 60
3.7 Reality properties 61
3.8 Harmonics as square rots of quaternions 63
4 Harmonic analysis 6
4.1 Harmonic expansion on the two-sphere 6
4.2 Harmonic integrals 67
2
4.3 Differential equations on S 69
4.4 Harmonic distributions 70
5 N = 2 mater with infinite sets of auxiliary fields 74
5.1 Introduction 74
5.1.1 N = 1 mater 74
5.1.2 N = 2 mater multiplets on shel 76
+
5.1.3 Relationship between q and ω hypermultiplets 77
5.1.4 Off-shel N = 2 mater before harmonic superspace 78
5.2 Fre of-shel hypermultiplet 79
5.2.1 The Fayet–Sohnius hypermultiplet constraints as analyticity
conditions 79
+
5.2.2 Fre of-shel q action 82
+
5.2.3 Relationship between q and ω hypermultiplets off shell 85
+
5.2.4 Masive q hypermultiplet 86
5.2.5 Invariances of the free hypermultiplet actions 87
Contents ix
5.3 Hypermultiplet self-couplings 90
+
5.3.1 General action for q hypermultiplets 90
+
5.3.2 An example of a q self-coupling: The Taub–NUT sigma
model 91
5.3.3 Symmetries of the general hypermultiplet action 95
5.3.4 Analogy with Hamiltonian mechanics 98
+
5.3.5 More examples of q self-couplings: The Eguchi–Hanson
sigma model and al that 9
6 N = 2 matter multiplets with a finite number of auxiliary fields.
N = 2 duality transformations 107
6.1 Introductory remarks 107
6.2 N = 2 tensor multiplet 109
6.3 The relaxed hypermultiplet 111
6.4 Further relaxed hypermultiplets 112
6.5 Non-linear multiplet 14
6.6 N = 2 duality transformations 117
6.6.1 Transforming the tensor multiplet 118
6.6.2 Transforming the relaxed hypermultiplet 121
6.6.3 Transforming the non-linear multiplet 122
6.6.4 General criterion for equivalence between hypermultiplet
and tensor multiplet actions 123
6.7 Conclusions 126
7 Supersymmetric Yang–Mils theories 128
7.1 Gauge fields from mater couplings 128
7.1.1 N = 0 gauge fields 128
7.1.2 N = 1 SYM gauge prepotential 129
7.1.3 N = 2 SYM gauge prepotential 131
7.2 Superspace diferential geometry 134
7.2.1 General framework 135
7.2.2 N = 1 SYM theory 136
7.2.3 N = 2 SYM theory 138
++
7.2.4 V versus Mezincescu’s prepotential 143
7.3 N = 2 SYM action 14
8 Harmonic supergraphs 148
8.1 Analytic delta functions 148
8.2 Green’s functions for hypermultiplets 150
8.3 N = 2 SYM: Gauge fixing, Green’s functions and ghosts 152
8.4 Feynman rules 156
8.5 Examples of supergraph calculations. Absence of harmonic
divergences 160
