Advances in Lie Superalgebras PDF
Preview Advances in Lie Superalgebras
Springer INdAM Series Volume 7 Editor-in-Chief V.Ancona SeriesEditors P.Cannarsa C.Canuto G.Coletti P.Marcellini G.Patrizio T.Ruggeri E.Strickland A.Verra Forfurthervolumes: http://www.springer.com/series/10283 Maria Gorelik • Paolo Papi Editors Advances in Lie Superalgebras Editors MariaGorelik PaoloPapi DepartmentofMathematics DipartimentodiMatematica TheWeizmannInstituteofScience Sapienza–UniversitàdiRoma Rehovot,Israel Roma,Italy ISSN:2281-518X ISSN:2281-5198(electronic) SpringerINdAMSeries ISBN978-3-319-02951-1 ISBN978-3-319-02952-8(eBook) DOI10.1007/978-3-319-02952-8 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013951834 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplicationofthispub- licationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’s location,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permis- sionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliable toprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublica- tion,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrors oromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespecttothe materialcontainedherein. CoverDesign:RaffaellaColombo,GiochidiGrafica,Milano,Italy TypesettingwithLATEX:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany(www.ptp-berlin.de) SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface ThefirstexamplesofLiesuperalgebrasappearinalgebraictopologyinthelate40’s (theWhiteheadproductonhomotopygroupsisaLiesuperalgebrabracket)andin the context of deformation theory of complex structures (Nijenhuis, Frölicher and Nijenhuis) in the late 50’s. Shortly after, Gerstenhaber, in a series of fundamental papers,shednewlightontheroleofLiesuperalgebrasinhistheoryofdeformation ofringsandalgebras,whileSpencerandhiscollaboratorsdevelopedapplicationsto pseudogroupstructuresonmanifolds. A renewed interest in Lie superalgebras came from Physics in the early 70’s: manyexamplesarisenaturallyas“supersymmetries”forquantumfieldtheories,e.g. intheWess-Zuminomodel1.ItwashoweverKac’slandmarkAdvancespaper2which establishedthestudyofLiesuperalgebrasasabranchofAlgebrainitsownright. Since then the subject has received dramatic developments, so that up to now morethan900papershaving“Liesuperalgebra”intheirtitlecanbecountedinthe MathSciNet. Thisvolumeoriginatesfrom“LieSuperalgebras”,heldinRoma,IstitutoNazio- nalediAltaMatematica“FrancescoSeveri”,December14-19,2012. Itconsistsoforiginalpapersand/orextendedexpositionsofthetalksdeliveredat theconference. Webelievethatthecontributions,kindlyofferedbytheinvitedspeakers,clearly illustrate one of the most remarkable features of the theory of Lie superalgebras whichis,theastonishingrangeofitsconnectionswithotherbranchesofMathemat- icsandMathematicalPhysics. Itisourpleasure tothank ProfessorVincenzo Ancona, PresidentofIndam,the ScientificCommitteeandtheentirestaffofIndam,forallowingustheopportunity togathersomanyspecialistsinsuchahighlystimulatingmeeting. 1AthoroughdiscussionoftheroleofLiesuperalgebrasuptothemid70’scanbefoundinCorwin- Ne’eman-Sternberg,Rev.Mod.Ph.,47,575–603(1975). 2Kac,V.G.:Liesuperalgebras.AdvancesinMath.26(1)8–96(1977). v vi Preface Byhappycoincidence,thepublicationofthisvolumecoincideswithVictorKac’s seventiethbirthday.ItwouldbehardtobelievethatthetheoryofLiesuperalgebras wouldhaveprogressedsofarwithouthiscontributioninthefield.Withtheconsent ofallcontributingauthors,wewouldliketodedicatethisvolumetohim. RehovotandRoma MariaGorelik September2013 PaoloPapi Contents Superbosonisation,Rieszsuperdistributions,andhighestweightmodules 1 AlexanderAlldridgeandZainShaikh Homologicalalgebraforosp(1/2n) ................................. 19 KevinCoulembier Finitenessandorbifoldvertexoperatoralgebras ..................... 35 AlessandroD’Andrea OnclassicalfiniteandaffineW-algebras ............................ 51 AlbertoDeSole Q-typeLiesuperalgebras ......................................... 67 MariaGorelikandDimitarGrantcharov WeightmodulesofDDD(((222,,,111,,,ααα))) ..................................... 91 CrystalHoyt OnSUSYcurves ................................................ 101 RitaFioresiandStephenDiwenKwok DiracoperatorsandtheverystrangeformulaforLiesuperalgebras..... 121 VictorG.Kac,PierluigiMösenederFrajriaandPaoloPapi ParaboliccategoryO forclassicalLiesuperalgebras .................. 149 VolodymyrMazorchuk OnKostant’stheoremforLiesuperalgebras......................... 167 ElenaPoletaeva ClassicalLiesuperalgebrasatinfinity .............................. 181 VeraSerganova ClassicalW-algebraswithinthetheoryofPoissonvertexalgebras ....... 203 DanieleValeri vii viii Contents Vertexoperatorsuperalgebrasandoddtracefunctions................ 223 JethrovanEkeren SerrepresentationsofLiesuperalgebras ............................ 235 RuibinZhang Superbosonisation, Riesz superdistributions, and highest weight modules AlexanderAlldridgeandZainShaikh Abstract Superbosonisation, introduced by Littelmann–Sommers–Zirnbauer, is a generalisationofbosonisation,withapplicationsinRandomMatrixTheoryandCon- densedMatterPhysics.Inthissurvey,welinkthesuperbosonisationidentitytoRep- resentationTheoryandHarmonicAnalysisandexplaintwonewproofs,oneviathe Laplacetransformandonebasedonamultiplicityfreenessstatement. 1 Introduction Supersymmetry (SUSY)has its origins in Quantum Field Theory. It is usually as- sociated with High Energy Physics, especially with SUGRA,where the fermionic fields correspond to physical quantities, the mathematical incarnation of a (as yet, hypothetical)fundamentalphenomenon.However,beyondthisfascinatinganddeep theory, and its independent mathematical interest, SUSY also has applications in quitedifferentareasofphysics,notably,inCondensedMatter. Here,thegeneratorsofsupersymmetrydonotcorrespondtophysicalquantities. Rather, they appear as effective symmetries of models for low-temperature limits of the fundamental Quantum Field Theory. This idea goes under the name of the SupersymmetryMethod,andwasdevelopedbyEfetovandWegner[9]. Itsparticularmeritisthepossibilitytoderive,bytheuseofHarmonicAnalysison certainsymmetricsuperspaces,preciseclosedformexpressionsforstatisticalquan- tities–suchasthemomentsoftheconductanceofametalwithimpurities[29,30]– inaregimewherethesystembecomescritical,forinstance,exhibitingatransition fromlocalisationtodiffusion,whichisnottractablebyothermethods. A.Alldridge( ) MathematicalInstitute,UniversityofCologne,Weyertal86–90,50931Köln,Germany e-mail:[email protected] Z.Shaikh MathematicalInstitute,UniversityofCologne,Weyertal86–90,50931Köln,Germany e-mail:[email protected] M.Gorelik,P.Papi(eds.):AdvancesinLieSuperalgebras.SpringerINdAMSeries7, DOI10.1007/978-3-319-02952-8_1,©SpringerInternationalPublishingSwitzerland2014 2 A.AlldridgeandZ.Shaikh Inconnectionwiththephysicsofthinwires,thesubjectwaswellstudiedinthe 1990s;ithasrecentlygainedsubstantialnewinterest,sincethe‘symmetryclasses’ investigatedinthiscontext[4,14,31]havebeenfoundtooccuras‘edgemodes’of certain2Dsystemsdubbed‘topologicalinsulators’(resp.superconductors)[12]. Mathematically, several aspects of the method beg justification. One both sub- tleandsalientpointisthetransformationofcertainintegralsoverflatsuperspacein highdimensionN→∞,whichoccurasexpressionsforstatisticalGreen’sfunctions, intointegralsoveracurvedsuperspaceoffixedrankanddimension–thelatterbeing moreamenabletoasymptoticanalysis(bysteepestdescentorstationaryphase).Tra- ditionally,thisstepisperformedbytheuseoftheso-calledHubbard–Stratonovich transformation,whichisbasedonacarefuldeformationoftheintegrationcontour. Thisposessevereanalyticalproblems,whichtothepresentdayhaveonlybeen overcomeincasesderivedfromrandommatrixensemblesthatfollowthenormaldis- tribution[15].ToextendtheSupersymmetryMethod’srangebeyondGaussiandis- order,forinstancetoestablishuniversalityforinvariantrandommatrixensembles, acomplementarytoolwasintroduced,basedonideasofFyodorov[13]:theSuper- bosonisation Identity of Littelmann–Sommers–Zirnbauer [21]. (A more complete accountofthehistoryofsuperbosonisationistobefoundintheintroductionof[3].) We now proceed to describe this identity. In general, it holds in the context of unitary,orthogonal,andunitary-symplecticsymmetry.Werestrictourselvestothe firstcase(ofunitarysymmetry),althoughourmethodscarryovertotheothercases. One considers the spaceW :=Cp|q×p|q of square super-matrices and a certain subsupermanifold Ωof purely even codimension, whose underlying (Riemannian symmetric) manifold Ω is the product of the positive Hermitian p×p matrices 0 withtheunitaryq×qmatrices.Let f beasuperfunctiondefinedandholomorphic onthetubedomainbasedonHerm+(p)×Herm(q).Thesuperbosonisationidentity states (cid:2) (cid:2) |Dv|f(Q(v))=C |Dy|Ber(y)nf(y), (1) Cp|q×n⊕Cn×p|q Ω forsomefinitepositiveconstantC,provided f hassufficientdecayatinfinityalong themanifoldΩ.Here,QisthequadraticmapQ(v)=vv∗,|Dv|istheflatBerezinian 0 density,and|Dy|isaBereziniandensityonΩ,invariantunderacertainnaturaltran- sitivesupergroupactionwewillspecifybelow. Remark that any GL(n,C)-invariant superfunction on Cp|q×n⊕Cn×p|q may be writtenintheform f(Q(v)).Thus,anotablefeatureoftheformulaisthatitputsthe ‘hidden supersymmetries’ (from GL(p|q,C)) into evidence through the invariant integral over the homogeneous superspace Ωwhere ‘manifest symmetries’ (from GL(n,C))enterviasomecharacter(namely,Ber(y)n). Aremarkablespecialcaseoccurswhen p=0.ThenEq.(1)reducesto (cid:2) (cid:2) |Dv|f(Q(v))=C |Dy|det(y)−nf(y), C0|q×n⊕Cn×0|q U(q) whichisknownastheBosonisationIdentityinphysics.Noticethattheleft-handside isapurelyfermionicBerezinintegral,whereastheright-handsideispurelybosonic.
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