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SUPERSYMMETRIC SOLITONS

In the last decade methods and techniques based on supersymmetry have provided deep insights in quantum chromodynamics and other non-supersymmetric gauge theories at strong coupling. This book summarizes major advances in critical solitons in supersymmetric theories, and their implications for understanding basic dynamical regularities of non-supersymmetric theories. After an extended introduction on the theory of critical solitons, including a historical introduction, the authors focus on three topics: non-Abelian strings and confined monopoles; reducing the level of supersymmetry; and domain walls as D brane prototypes. They also provide a thorough review of issues at the cutting edge, such as non-Abelian flux tubes. The book presents an extensive summary of the current literature so that researchers in this field can understand the background and related issues.

Mikhail Shifman is the Ida Cohen Fine Professor of Physics at the University of Minnesota, and is one of the world leading experts on quantum chromodynamics and non-perturbative supersymmetry. In 1999 he received the Sakurai Prize for Theoretical Particle Physics, and in 2006 he was awarded the Julius Edgar Lilienfeld Prize for outstanding contributions to physics. He is the author of several books, over 300 scientiﬁc publications, and a number of popular articles and articles on the history of high-energy physics.

AlexeiYung is a Senior Researcher in the Theoretical Department at the Petersburg Nuclear Physics Institute, Russia, and a Visiting Professor at the William I. Fine Theoretical Physics Institute. His research interests lie in non-perturbative dynamics of non-Abelian supersymmetric gauge theories and its interplay with string theory, and the problem of color conﬁnement in non-Abelian gauge theories. Many of his recent advances in these areas are included in this book.

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Supersymmetric Solitons

M.SHIFMAN William I. Fine Theoretical Physics Institute University of Minnesota

A.YUNG William I. Fine Theoretical Physics Institute University of Minnesota Petersburg Nuclear Physics Institute Institute of Theoretical and Experimental Physics

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First published 2009

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Library of Congress Cataloging-in-Publication Data

Shifman, M. (Mikhail), 1949– Supersymmetric solitons / M. Shifman and A. Yung. p. cm. Includes index. ISBN 978-0-521-51638-9 (hardback) 1. Solitons. 2. Supersymmetry. I. Yung, A. (Alexei) II. Title. QC174.26.W28S44 2009 510.12 4–dc22 2008038901

ISBN 978-0-521-51638-9 hardback

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Caricature of Alyosha Yung and Misha Shifman by Andrey Feldshteyn, 2007.

Contents

Acknowledgments Page xiii List of abbreviations xiv

1 Introduction 1

I SHORT EXCURSION

2 Central charges in superalgebras 7 2.1 History 7 2.2 Minimal supersymmetry 8 2.2.1 D = 29 2.2.2 D = 39 2.2.3 D = 410 2.3 Extended SUSY 11 2.3.1 N = 2inD = 211 2.3.2 N = 2inD = 312 2.3.3 On extended supersymmetry (eight supercharges) in D = 413

3 The main building blocks 15 3.1 Domain walls 15 3.1.1 Preliminaries 15 3.1.2 Domain wall in the minimal Wess–Zumino model 18 3.1.3 D-branes in gauge ﬁeld theory 24 3.1.4 Domain wall junctions 29 3.1.5 Webs of walls 31 3.2 Vortices in D = 3 and ﬂux tubes in D = 432 3.2.1 SQED in 3D 33

x Contents 3.2.2 Four-dimensional SQED and the ANO string 40 3.2.3 Flux tube junctions 41 3.3 Monopoles 43 3.3.1 The Georgi–Glashow model: vacuum and elementary excitations 43 3.3.2 Monopoles – topological argument 45 3.3.3 Mass and magnetic charge 45 3.3.4 Solution of the Bogomol’nyi equation for monopoles 47 3.3.5 Collective coordinates (moduli) 49 3.3.6 Singular gauge, or how to comb a hedgehog 54 3.3.7 Monopoles in SU(N) 55 3.3.8 The θ term induces a fractional electric charge for the monopole (the Witten effect) 59 3.4 Monopoles and fermions 60 3.4.1 N = 2 super-Yang–Mills (without matter) 61 3.4.2 Supercurrents and the monopole central charge 62 3.4.3 Zero modes for adjoint fermions 65 3.4.4 Zero modes for fermions in the fundamental representation 66 3.4.5 The monopole supermultiplet: dimension of the BPS representations 67 3.5 More on kinks (in N = 2CP(1) model) 67 3.5.1 BPS solitons at the classical level 69 3.5.2 Quantization of the bosonic moduli 71 3.5.3 The kink mass and holomorphy 72 3.5.4 Fermions in quasiclassical consideration 74 3.5.5 Combining bosonic and fermionic moduli 76

II LONG JOURNEY

Introduction to Part II 81

4 Non-Abelian strings 85 4.1 Basic model: N = 2 SQCD 85 4.1.1 SU(N)×U(1) N = 2 QCD 88 4.1.2 The vacuum structure and excitation spectrum 89 4.2 ZN Abelian strings 92 4.3 Elementary non-Abelian strings 98 4.4 The world-sheet effective theory 99 4.4.1 Derivation of the CP(N − 1) model 100 4.4.2 Fermion zero modes 104

Contents xi 4.4.3 Physics of the CP(N − 1) model with N = 2 108 4.4.4 Unequal quark masses 110 4.5 Confined monopoles as kinks of the CP(N − 1) model 115 4.5.1 The first-order master equations 118 4.5.2 The string junction solution in the quasiclassical regime 120 4.5.3 The strong coupling limit 123 4.6 Two-dimensional kink and four-dimensional Seiberg–Witten solution 126 4.7 More quark flavors 130 4.8 Non-Abelian k-strings 135 4.9 A physical picture of the monopole confinement 137

5 Less supersymmetry 142 5.1 Breaking N = 2 supersymmetry down to N = 1 144 5.1.1 Deformed theory and string solutions 144 5.1.2 Heterotic CP(N − 1) model 149 5.1.3 Large-N solution 153 5.1.4 Limits of applicability 157 5.2 The M model 159 5.3 Conﬁned non-Abelian monopoles 163 5.4 Index theorem 166

6 Non-BPS non-Abelian strings 171 6.1 Non-Abelian strings in non-supersymmetric theories 171 6.1.1 World-sheet theory 172 6.1.2 Physics in the large-N limit 174 6.2 Non-Abelian strings in N = 1∗ theory 182

7 Strings on the Higgs branches 188 7.1 Extreme type-I strings 189 7.2 Example: N = 1 SQED with the FI term 192

8 Domain walls as D-brane prototypes 196 8.1 N = 2 supersymmetric QED 197 8.2 Domain walls in N = 2 SQED 199 8.3 Effective ﬁeld theory on the wall 202 8.4 Domain walls in the U(N) gauge theories 206

9 Wall-string junctions 209 9.1 Strings ending on the wall 209

xii Contents 9.2 Boojum energy 212 9.3 Finite-size rigid strings stretched between the walls. Quantizing string endpoints 214 9.4 Quantum boojums. Physics of the world volume theory 219

10 Conclusions 223

Appendix A Conventions and notation 228

Appendix B Many faces of two-dimensional supersymmetric CP(N − 1) model 234

Appendix C Strings in N = 2 SQED 243

References 248 Index 256 Author index 258

Acknowledgments

We are grateful to Adam Ritz, David Tong, and Arkady Vainshtein for useful discussions. The work of M.S. was supported in part by DOE grant DE-FG02-94ER408. The work of A.Y. was supported by FTPI, University of Minnesota, by RFBR Grant No. 06-02-16364a and by Russian State Grant for Scientiﬁc School RSGSS- 11242003.2.

xiii

Abbreviations

AdS Anti de Sitter ANO Abrikosov–Nielsen–Olesen BPS Bogomol’nyi–Prasad–Sommerﬁeld CC Central Charge CMS Curve(s) of the Marginal Stability CFT Conformal Field Theory CFIV Cecotti–Fendley–Intriligator–Vafa FI Fayet–Iliopoulos IR Infrared NSVZ Navikov–Shifman–Vainshtein–Zakharov QCD Quantum Chromodynamics SUSY Supersymmetry, Supersymmetric SQCD Supersymmetric Quantum Chromodynamics SQED Supersymmetric Quantum Electrodynamics UV Ultraviolet VEV Vacuum Expectation Value

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