Table Of Content

GroupTheory December10,2002 Group Theory Lie’s, Tracks, and Exceptional Groups Predrag Cvitanovic´ GONE WITH THE WIND PRESS ATLANTA AND COPENHAGEN GroupTheory December10,2002 dedicatedtothememoryof BorisWeisfeilerandWilliamE.Caswell GroupTheory December10,2002 Contents Chapter1. Introduction 1 Chapter2. Apreview 5 2.1 Basicconcepts 5 2.2 Firstexample: SU(n) 9 2.3 Secondexample: E6family 12 Chapter3. Invariantsandreducibility 15 3.1 Preliminaries 15 3.2 Invariants 19 3.3 Invariancegroups 23 3.4 Projectionoperators 24 3.5 Furtherinvariants 25 Chapter4. Diagrammaticnotation 27 4.1 Birdtracks 27 4.2 Clebsch-Gordancoefficients 29 4.3 Zero-andone-dimensionalsubspaces 31 4.4 Infinitesimaltransformations 32 4.5 Liealgebra 35 4.6 OtherformsofLiealgebracommutators 37 4.7 Irrelevancyofclebsches 38 4.8 Abriefhistoryofbirdtracks 39 Chapter5. Recouplings 41 5.1 Couplingsandrecouplings 41 5.2 Wigner3n-jcoefficients 44 5.3 Wigner-Eckarttheorem 45 Chapter6. Permutations 49 6.1 Symmetrization 49 6.2 Antisymmetrization 51 6.3 Levi-Civitatensor 53 6.4 Determinants 55 6.5 Characteristicequations 57 6.6 Fully(anti)symmetrictensors 57 Chapter7. Casimiroperators 59 GroupTheory December10,2002 ii CONTENTS 7.1 CasimirsandLiealgebra 60 7.2 Independentcasimirs 61 7.3 Casimiroperators 63 7.4 Dynkinindices 64 7.5 Quadratic,cubiccasimirs 69 7.6 Quarticcasimirs 70 7.7 Sundryrelationsbetweenquarticcasimirs 72 7.8 Identicallyvanishingtensors 75 7.9 Dynkinlabels 75 Chapter8. Groupintegrals 79 8.1 Groupintegralsforarbitraryreps 80 8.2 Characters 82 8.3 Examplesofgroupintegrals 83 Chapter9. Unitarygroups 85 9.1 Two-indextensors 85 9.2 Three-indextensors 86 9.3 Youngtableaux 87 9.4 Youngprojectionoperators 90 9.5 Reductionoftensorproducts 93 9.6 3-jsymbols 94 9.7 Characters 97 9.8 Mixedtwo-indextensors 97 9.9 Mixeddefining⊗adjointtensors 99 9.10 Two-indexadjointtensors 101 9.11 CasimirsforthefullysymmetricrepsofSU(n) 105 9.12 SU(n),U(n)equivalenceinadjointrep 107 Chapter10. Orthogonalgroups 115 10.1 Two-indextensors 116 10.2 Mixedadjoint⊗definingreptensors 117 10.3 Two-indexadjointtensors 118 10.4 Three-indextensors 121 10.5 Gravitytensors 124 10.6 SO(n)Dynkinlabels 127 Chapter11. Spinors 129 11.1 Spinography 130 11.2 Fierzingaround 133 11.3 Fierzcoefficients 137 11.4 6-jcoefficients 138 11.5 Exemplaryevaluations,continued 140 11.6 Invarianceofγ-matrices 140 11.7 Handedness 142 11.8 Kahanealgorithm 143 Chapter12. Symplecticgroups 145 12.1 Two-indextensors 146 GroupTheory December10,2002 CONTENTS iii Chapter13. Negativedimensions 149 13.1 SU(n)=SU(−n) 150 13.2 SO(n)=Sp(−n) 151 Chapter14. Spinors’Sp(n)sisters 153 14.1 Spinsters 153 14.2 Racahcoefficients 158 14.3 Heisenbergalgebras 159 Chapter15. SU(n)familyofinvariancegroups 161 15.1 RepresentationsofSU(2) 161 15.2 SU(3)asinvariancegroupofacubicinvariant 163 15.3 Levi-CivitatensorsandSU(n) 166 15.4 SU(4)-SO(6)isomorphism 167 Chapter16. G2 familyofinvariancegroups 169 16.1 Jacobirelation 171 16.2 Alternativityandreductionoff-contractions 171 16.3 Primitivityimpliesalternativity 174 16.4 Sextonians 176 16.5 CasimirsforG2 178 16.6 Hurwitz’stheorem 179 16.7 RepresentationsofG2 180 Chapter17. E8 familyofinvariancegroups 181 17.1 Two-indextensors 182 17.2 DecompositionofSym3A 186 17.3 Decompositionof ⊗ 188 17.4 Diophantineconditions 190 17.5 Recentprogress 190 Chapter18. E6 familyofinvariancegroups 193 18.1 Reductionoftwo-indextensors 193 18.2 Mixedtwo-indextensors 194 18.3 DiophantineconditionsandtheE6family 196 18.4 Three-indextensors 197 18.5 Defining⊗adjointtensors 199 18.6 Two-indexadjointtensors 202 18.7 DynkinlabelsandYoungtableauxforE6 206 18.8 CasimirsforE6 207 18.9 SubgroupsofE6 210 18.10Springerrelation 211 Chapter19. F4 familyofinvariancegroups 213 19.1 Two-indextensors 213 19.2 Defining⊗adjointtensors 216 19.3 Two-indexadjointtensors 218 19.4 JordanalgebraandF4(26) 219 GroupTheory December10,2002 iv CONTENTS Chapter20. E7 familyanditsnegativedimensionalcousins 221 20.1 Theantisymmetricquarticinvariant 222 20.2 FurtherDiophantineconditions 224 20.3 Liealgebraidentification 225 20.4 Symmetricquarticinvariant 228 20.5 Theextendedsupergravitiesandthemagictriangle 229 Chapter21. Exceptionalmagic 235 21.1 Magictriangle 235 21.2 Landsberg-Manivelconstruction 238 21.3 Epilogue 238 Chapter22. Magicnegativedimensions 243 22.1 E7andSO(4) 243 22.2 E6andSU(3) 243 AppendixA. Recursivedecomposition 245 AppendixB. PropertiesofYoungprojections 247 B.1 UniquenessofYoungprojectionoperators 247 B.2 Normalization 248 B.3 Orthogonality 249 B.4 Thedimensionformula 249 B.5 Literature 251 AppendixC. G2 calculations 253 C.1 EvaluationrulesforG2 253 C.2 G2,furthercalculations 255 Bibliography 259 GroupTheory December10,2002 CONTENTS v ACKNOWLEDGMENTS IwouldliketothankTonyKennedyforcoauthoringtheworkdiscussedinchapters onspinors,spinstersandnegativedimensions;HenrietteElvangforcoauthoringthe chapteronrepresentationsofU(n); DavidPritchardformuchhelpwiththeearly versionsofthismanuscript;RogerPenroseforinventingbirdtracks(andthusmak- ingthemrespectable)whileIwasstrugglingthroughgradeschool; PaulLauwers forthebirdtracksrock-around-the-clock; Feza Gürseyand PierreRamond forthe first lessons on exceptional groups; Susumu Okubo for inspiring correspondence; BobPearsonforassortedbirdtrack,Youngtableauxandlatticecalculations;Bernard JuliaforcommentsthatIhopetounderstandsomeday(aswellaswhydoeshenot citemyworkonthemagictriangle?);P.HoweandL.Brinkforteachingmehowto count(supergravitymultiplets),W.Siegelformanyhelpfulcriticismsofmysuper- gravitymultipletcounting,E.CremmerforhospitalityatÉcoleNormaleSuperieure, M.KontsevichforbringingtomyattentionthemorerecentworkofDeligne,Cohen anddeMan;J.Landsbergforacknowledgingmysufferings;R.Abdelatif,G.M.Ci- cuta, A. Duncan, E. Eichten, B. Durhuus, R. Edgar, M. Günaydin, K. Oblivia, G. Seligman, A. Springer, L. Michel, R.L. Mkrtchyan, P.G.O. Freund, T. Gold- man, R.J. Gonsalves, H. Harari, D. Milicˇic´, H. Pfeiffer, C. Sachrayda P. Sikivie, S.A.Solla,G.Tiktopoulos,andB.W. Westburyfordiscussions(orcorrespondence). Theappellation“birdtracks”isduetoBerniceDurandwhoenteredmyoffice,saw theblackboardandaskedpuzzled: “Whatisthis? Footprintsleftbybirdsscurrying alongasandybeach?” I am grateful to Dorte Glass for typing most of the manuscript, to the Aksel Tovborg Jensens Legat, Kongelige Danske Videnskabernes Selskab, for financial supportwhichmadethetransformationfromhand-drawntodraftedbirdtrackspos- sible,and,mostofall,tothegoodAndersJohansenwhodidmostofdrawingofthe birdtracksandwithoutwhomthisbookwouldstillbebutacollectionofsquiggles inmyfilingcabinet. CarolMonsrudandCecileGourgueshelpedwithtypingthe earlyversionofthismanuscript. The manuscript was written in stages, in Chewton-Mendip, Paris, Bures-sur- Yvette,Rome,Copenhagen,Frebbenholm,Røros,Juelsminde,Göteborg-Copenhagen train,CathayPacific(HongKong-Paris)andinnumerableotherairportsandplanes, SjællandsOdde,Göteborg,Miramare,KurkelaandVirginiaHighlands. Iamgrate- fultoT.Dorrian-Smith, R.delaTorre, BDC,N.-R.Nilsson, E.Høsøinen, family Cvitanovic´, U.SelmerandfamilyHerlinfortheirkindhospitalityalongthislong way. IwouldlovetothankW.E.Caswellforaperspicaciousobservation,andB.Weis- feilerfordelightfuldiscussionsatI.A.S,butitcannotbedone. In1985,whilehiking inAndes,BoriswasallegedlykidnappedbytheChileanstatesecurity,andtortured and executed by Nazis at Colonia Dignidad, Chile. Bill boarded flight AA77 on September11,2001andwasflownintoPentagonbyanolesscharminggroupof Islamicfanatics. Thisbookisdedicatedtotheirmemory. GroupTheory December10,2002 GroupTheory December10,2002 Chapter One Introduction This monograph offers a derivation of all classical and exceptional semi-simple Lie algebras through a classification of “primitive invariants”. Using somewhat unconventionalnotationinspiredbytheFeynmandiagramsofquantumfieldtheory, the invariant tensors are represented by diagrams; severe limits on what simple groupscouldpossiblyexistarededucedbyrequiringthatirreduciblerepresentations be of integer dimension. The method provides the full Killing-Cartan list of all possiblesimpleLiealgebras,butfailstoprovetheexistenceofF ,E ,E andE . 4 6 7 8 Onesimplequantumfieldtheoryquestionstartedthisproject;whatisthegroup theoreticfactorforthefollowingQuantumChromodynamicsgluonself-energydi- agram = ? (1.1) I first computed the answer for SU(n). There was a hard way of doing it, using Gell-Mann f and d coefficients. There was also an easy way, where one ijk ijk coulddoodleoneselftotheanswerinafewlines. Thisisthe“birdtracks”method whichwillbedescribedhere. ItworksnicelyforSO(n)andSp(n)aswell. Out of curiosity, I wanted the answer for the remaining five exceptional groups. This engenderedfurtherthought,andthatwhichIlearnedcanbebetterunderstoodasthe answertoadifferentquestion. Supposesomeonecameintoyourofficeandasked, “OnplanetZ,mesonsconsistofquarksandantiquarks,butbaryonscontainthree quarksinasymmetriccolorcombination. Whatisthecolorgroup?” Theanswer isneithertrivial,norwithoutsomebeauty(planetZ quarkscancomein27colors, andthecolorgroupcanbeE ). 6 Onceyouknowhowtoanswersuchgroup-theoreticalquestions,youcananswer manyothers. Thismonographtellsyouhow. Likethebrain,itisdividedintotwo halves;theploddinghalfandtheinterestinghalf. Theploddinghalfdescribeshowgrouptheoreticcalculationsarecarriedoutfor unitary,orthogonalandsymplecticgroups,chapters3–15. Exceptforthe“negative dimensions”ofchapter13andthe“spinsters”ofchapter14,noneofthatisnew,but themethodsarehelpfulincarryingoutdailychores, suchasevaluatingQuantum Chromodynamics group theoretic weights, evaluating lattice gauge theory group integrals,computing1/N corrections,evaluatingspinortraces,evaluatingcasimirs, implementingevaluationalgorithmsoncomputers,andsoon. The interesting half, chapters16–21, describes the “exceptional magic” (a new constructionofexceptionalLiealgebras),the“negativedimensions”(relationsbe- tweenbosonicandfermionicdimensions). Openproblemsandpersonalconfessions arerelegatedtotheepilogue, sect.21.3. Themethodsusedareapplicabletofield GroupTheory December10,2002 2 CHAPTER1 theoreticmodelbuilding. Regardlessoftheirpotentialapplications,theresultsare sufficiently intriguing to justity this entire undertaking. In what follows we shall forgetaboutquarksandquantumfieldtheory,andofferinsteadasomewhatunortho- doxintroductiontothetheoryofLiealgebras. IfthestyleisnotBourbaki,itisnot sobyaccident. There are two complementary approaches to group theory. In the canonical approach one chooses the basis, or the Clebsch-Gordan coefficients, as simply as possible. ThisisthemethodwhichKilling[97]andCartan[20]usedtoobtainthe completeclassificationofsemi-simpleLiealgebras,andwhichhasbeenbroughtto perfectionbyDynkin[57]. Thereexistmanyexcellentreviewsofapplicationsof Dynkindiagrammethodstophysics,suchasthereviewbySlansky[145]. Inthetensorialapproachpursuedhere,thebasesarearbitraryandeverystatement isinvariantunderchangeofbasis. Tensorcalculusdealsdirectlywiththeinvariant blocksofthetheoryandgivestheexplicitformsoftheinvariants,Clebsch-Gordan series,evaluationalgorithmsforgrouptheoreticweights,etc. The canonical approach is often impractical for computational purposes, as a choiceofbasisrequiresaspecificcoordinatizationoftherepresentationspace. Usu- ally,nothingthatwewanttocomputedependsonsuchacoordinatization;physical predictions are pure scalar numbers (“color singlets”), with all tensorial indices summedover. However,thecanonicalapproachcanbeveryusefulindetermining chains of subgroup embeddings, we refer the reader to the Slansky review [145] forsuchapplications. Hereweshallconcentrateontensorialmethods,borrowing from Cartan and Dynkin only the nomenclature for identifying irreducible repre- sentations. Extensive listings of these are given by McKay and Patera [109] and Slansky[145]. Toappreciatethesenseinwhichcanonicalmethodsareimpractical,letusconsider using them to evaluate the group-theoretic factor associated with diagram (1.1) for the exceptional group E . This would involve summations over 8 structure 8 constants. TheCartan-Dynkinconstructionenablesustoconstructthemexplicitly; anE structureconstanthasabout2483/6elements,andthedirectevaluationofthe 8 group-theoreticfactorfordiagram(1.1)istediousevenonacomputer. Anevaluation in terms of a canonical basis would be equally tedious for SU(16); however, the tensorialapproachillustratedbytheexampleofsect.2.2yieldstheanswerforall SU(n)inafewsteps. Simplicityofsuchcalculationsisonemotivationforformulatingatensorialap- proachtoexceptionalgroups. Theotheristhedesiretounderstandtheirgeometrical significance. TheKilling-CartanclassificationisbasedonamappingofLiealgebras onto a Diophantine problem on the Cartan root lattice. This yields an exhaustive classificationofsimpleLiealgebras,butgivesnoinsightintotheassociatedgeome- tries. In the 19th century, the geometries or the invariant theory were the central question and Cartan, inhis1894 thesis, made an attempttoidentifytheprimitive invariants. MostoftheentriesinhisclassificationweretheclassicalgroupsSU(n), SO(n)andSp(n). Ofthefiveexceptional algebras, Cartan[21]identifiedG as 2 thegroupofoctonionisomorphisms,andnotedalreadyinhisthesisthatE hasa 7 skew-symmetricquadraticandasymmetricquarticinvariant. Dickinson[51]char- acterized E as a 27-dimensional group with a cubic invariant. The fact that the 6