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Texts and Monographs in Physics SeriesEditors: R.Balian,Gif-sur-Yvette,France W.Beiglböck,Heidelberg,Germany H.Grosse,Wien,Austria W.Thirring,Wien,Austria Yakov M. Shnir Magnetic Monopoles ABC Dr.YakovM.Shnir InstituteofPhysics CarlvonOssietzkyUniversityOldenburg 26111Oldenburg Germany E-mail:[email protected] LibraryofCongressControlNumber:2005930438 ISBN-10 3-540-25277-0SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-25277-1SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com (cid:1)c Springer-VerlagBerlinHeidelberg2005 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandTechBooksusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11398127 55/TechBooks 543210 To Marina with love 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,beautyisnotentirely“intheeyeofthebeholder”.It seemstoberelatedtothesymmetryoftheobject.Fromaphysicalviewpoint, 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,ourimperfect(atleastatlow-energyscale)worldisfullof 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 violationandhowthesymmetrymayberecovered;thenourpictureofNature 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. Overthepast25yearsthetheoryofmagneticmonopoleshassurprisingly become closely connected with many actual directions of theoretical physics. ThisincludestheproblemofconfinementinQuantumChromodynamics,the problemofprotondecay,astrophysicsandevolutionoftheearlyUniverse,and the supersymmetrical extension of the Standard Model, to name just a few. Itseemsplausiblethattheanswertothequestion:“Whydomagneticmono- poles not exist?” is a key to understanding the very foundations of Nature. Furthermore, the mathematical problem of construction and investigation of VIII Preface the exact multimonopole configurations is at the frontier of the most fasci- nating directions of modern field theory and differential geometry. The tech- niques developed in this area of theoretical physics find many other applications and have become very important mathematical tools. Thetheoryofmonopolesseemstobetailor-madefordemonstratingbeau- tifulinterplaybetweenmathematicsandphysics.Therefore,Ibelievethatan introductiontothebasicideasandtechniquesthatarerelatedtothedescrip- tion and construction of monopoles may be useful to physicists and mathe- maticiansinterestedinthemoderndevelopmentsinthisdirection.Moreover, thereisasecondaspectofthemonopoles.Theseobjectsariseinmanydiffer- ent 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. Ihavethereforetriedtogivearestrictedintroductiontotheclassicaland 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 andanelectriccharge.Thequantummechanicalconsiderationintermsofthe 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. Chapter4isdevotedtothegeneralizationofQED,whichincludesthemono- poles. Part II forms the core of the book. There I discuss the theory of non- Abelian monopoles, construction of the multimonopole solutions, and some applications. In Chapter 5 the famous ’t Hooft–Polyakov solution, the sim- plestspecimenofthemonopolefamily,isdiscussed.Thisisthefirststepinside the monopole core. I review the basic properties of the classical non-Abelian monopoles,whichariseinspontaneouslybrokenSU(2)gaugetheory,andthe relationthatexistsbetweenthemagneticchargeoftheconfigurationandthe Preface IX topological charge. The Bogomol’nyi–Prasad–Sommerfield (BPS) monopole appears here for the first time as a particular analytic solution with vanish- ing potential. Here I also give a brief account of the gauge zero mode and commentonitsrelationtotheelectriccharge.Chapter6containsasurveyof 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,inChapter8theconsiderationisextendedtoamoregeneralclassof SU(3)theoriescontainingdifferentlimitsofsymmetrybreaking.Itturnsout thatthemultimonopoleconfigurationsarenaturalinamodelwiththegauge group of higher rank. Here I discuss fundamental and composite monopoles and consider the limiting situation of the massless states. Chapter9containsabriefsurveyoftherolethatthemonopolesmayplay inthephenomenonofconfinement.Idiscussherethecompactlatticeelectro- dynamics,formalismofAbelianprojectioningluodynamicsandthePolyakov 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 previousdiscussion.Indeed,thespectrumofstatesofN =2supersymmetric (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 modulispace 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–Millstheory.Chapter11isanintroductoryaccountofsupersymmetry. 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 toavoid such adiscussion.The readerwill definitely find this topic well presented elsewhere. Letusmentionsomeomissions.Anobviousgapisthecurrentexperimen- tal 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]. How- ever, 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 X Preface by-product topics like, for example, the conception of the Berry phase. Nei- ther do I consider some specific mathematical problems of the Abelian theory of monopoles (e.g., singularities and regularization). In considering con- structionoftheBPSmultimonopoles,Ihavemadenoattempttodiscussone of the approaches that is related to the application of the inverse scattering method to the linearized Bogomol’nyi equation. Instead, the discussion con- centratesonthemoderndevelopmentduetotheNahmtechniqueandtwistor 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–Higgstheory.Ipaymoreattentiontothegeneralprop- erties of the non-Abelian monopoles, namely, to their topological nature. Couplingwithgravityyieldsanumberofclassicalsolutionsthatarenotpre- sented 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 tur- moil. 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, Ju¨rgen 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 Non- nenmacher, 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. Tolka- chev,whoweremy teachersand advisors,fortheir valuable support,encouragement,andguidance.Theyawakenedmyinterestinthemonopoleproblem. Many of the ideas discussed here are due to Nick Manton, who played a veryimportantroleinmyunderstandingofthemonopoles,boththroughhis papersandinprivatediscussions.Hecommandsmydeepestpersonalrespect andgratitute.TheyearIspentinCambridgeinhisgroupstronglyinfluenced 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 Preface XI 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 Blagojević, Maxim Chernodub, Adriano Di Giacomo, Fridrich W. Hehl, and Valentine Zakharov for reading a pre- liminary 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 Théorique, CEA-Saclay, the Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut), Mu¨nchen, 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 occa- sions. Oldenburg, Yakov Shnir June 2005 Contents Part I Dirac Monopole 1 Magnetic Monopole in Classical Theory .................. 3 1.1 Non-Relativistic Scattering on a Magnetic Charge .......... 3 1.2 Non-Relativistic Scattering on a Dyon..................... 10 1.3 Vector Potential of a Monopole Field ..................... 12 1.4 Transformations of the String ............................ 15 1.5 Dynamical Symmetries of the Charge-Monopole System ..... 20 1.6 Dual Invariance of Classical Electrodynamics............... 22 2 The Electron–Monopole System: Quantum-Mechanical Interaction ......................... 27 2.1 Charge Quantization Condition .......................... 27 2.2 Spin-Statistics Theorem in a Monopole Theory............. 31 2.3 Charge-Monopole System: Quantum-Mechanical Description . 33 2.3.1 The Generalized Spherical Harmonics ............... 34 2.3.2 Solving the Radial Schro¨dinger Equation ............ 37 2.4 Non-Relativistic Scattering on a Monopole: Quantum Mechanical Description......................... 40 2.5 Charge-Monopole System: Spin in the Pauli Approximation.. 42 2.5.1 Dynamical Supersymmery of the Electron-Monopole System................... 44 2.5.2 Generalized Spinor Harmonics: j ≥µ+1/2 .......... 46 2.5.3 Generalized Spinor Harmonics: j =µ−1/2 .......... 48 2.5.4 Solving the Radial Pauli Equation.................. 49 2.6 Charge-Monopole System: Solving the Dirac Equation ...... 53 2.6.1 Zero Modes and Witten Effect ..................... 55 2.6.2 Charge Quantization Condition and the Group SL(2,Z)........................... 61 3 Topological Roots of the Abelian Monopole............... 67 3.1 Abelian Wu–Yang Monopole ............................. 67 3.2 Differential Geometry and Topology ...................... 70 3.2.1 Notions of Topology .............................. 70 3.2.2 Notions of Differential Geometry ................... 81