Gauge theories on noncommutative spaces

AlbertSchwarz

Abstract.IreviewmyresultsaboutnoncommutativegaugetheoriesandabouttherelationofthesetheoriestoM(atrix)theoryfollowingmylectureonICMP2000.

InmylectureonICMP2000Igaveashortreviewofmyresultsonnoncom-mutativegaugetheoriesandtalkedinmoredetailaboutmyrecentpaper[9].HereI’llskipalldetailsreferringtopapers[1]-[13].I’lllistonlymainresultsofthesepapers.

Inthepaper[1]itwasshownthatgaugetheoriesonnoncommutativetoriappearnaturallyinconsiderationofcompactificationsofM(atrix)theory.Thesamelogiccanbeusedtoobtaingaugetheoriesonnoncommutativetoroidalorbifolds[14],[15],[11],[12].

Moreprecisely,ifGisasubgroupofthegroupofsymmetriesofanymodelwecanrestrictourselvestofieldsthatareG-invariantuptogaugeequivalence.ThismeansthatthechangeofafieldAundertheactionofanelementγ∈GcanbecompensatedforbygaugetransformationUγ.Formatrixmodels(i.e.inthecasewhenAisacollectionofmatricesandgaugetransformationsareunitarytransformations)thismeansthat(1)

γ(A)=UγAUγ−1

.

Usuallyfinitesizematricesdon’tsatisfythisequation;oneshouldreplace(Hermit-ian)matricesby(Hermitian)operatorsininfinite-dimensionalHilbertspaceEand

considerUγasunitaryoperatorsinthisspace.ThereexistsnoreasontoexpectthatUγλ=UγUλ,buttakingintoaccountthat(2)

(Uγλ−1·UγUλ)A(Uγλ−1·UγUλ)=A

itisnaturallytoassumethat(3)

Uγλ=eiπθ(γ,λ)UγUλ

Onecansay,thattheoperatorsUγspecifyaprojectiverepresentationofthe

groupG.InthecasewhenG=ZdtheassociativealgebraTd

algebraoffunctionsond-dimensionalθgeneratedbyoper-atorsUγcanbeinterpretedasthenoncom-mutativetorus.InotherwordsthespaceEcanbeconsideredasaTθd-module.

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Wealwaysconsiderfinitelygeneratedprojectivemodules(directsummandsinfree

dn

modules(Tθ)).Innoncommutativegeometrythismeansthatweconsider”vectorbundles”overnoncommutativetori.

ThetorusTθisspecifiedbymeansofbilinearformθ(γ,λ)onZd;withoutlossofgeneralityonecanassumethatthisformisantisymmetric.Itwillbemorecon-venientforustosaythatanoncommutativetorusisdeterminedbyantisymmetricmatrixθjkcorrespondingtotheformθ(γ,λ)insomebasisofZd.IntermsofthismatrixnoncommutativetoruscanbeinterpretedasanalgebrawithunitarygeneratorsU1,...,UdsatisfyingrelationsUjUk=e2πiθjkUkUj.IfA=(A1,...,Ad)andthegroupZdactsonAbymeansoftranslations(i.e.γ(A)=A+γ),thenthesolutionoftheequation(1)canbeconsideredasaconnectiononnoncommu-tativetorusTθinthesenseofA.Connes[16],[17].(Thenotionofconnectionisdiscussedindetailattheendofthepaper.)IfourstartingpointisBFSSorIKKTmatrixmodel[18],[19],thentheaboveconstructionleadstoSUSYYang-Millsthe-oryonnoncommutativetorus[1].ReplacingZdwithsemidirectproductofZdandfinitegroupweobtaingaugetheoriesonnoncommutativetoroidalorbifolds.TheappearanceofnoncommutativegeometrycanbeexplainednotonlyfromtheviewpointofM(atrix)theory,butalsofromtheviewpointofstringtheoryaswasshowninaseriesofpapers[20]-[24],culminatingbySeiberg-Wittenpaper[25]thatcontainsverydetailedanalysisofrelationbetweenstringtheoryandgaugetheoryonnoncommutativespaces.

GaugetheoriesonnoncommutativetoriwerestudiedbyA.ConnesandM.Rieffel,especiallyintwo-dimensionalcase[26]-[28].Iobtainednewresultsaboutthesetheoriesfocusingmyattentiononproblemsrelatedtophysics.Alreadyin[1]itwasconjecturedthatMoritaequivalenceofalgebrasisrelatedtodualityin

ˆifthephysics.OnesaysthatanalgebraAisMoritaequivalenttothealgebraA

ˆ-modules.Inotherwords,categoryofA-modulesisequivalenttothecategoryofA

ˆ-modulesandAˆ-modulesintoA-weshouldbeabletotransferA-modulesintoA

modules;thiscorrespondenceshouldbenatural(foreveryA-linearmapϕ:E→E′

ˆ-linearmapϕˆ→Eˆ′ofcorrespondingAˆ-ofA-modulesshouldbedefinedanAˆ:E

modules;onerequiresthatthecorrespondenceϕ→ϕˆtransformscompositionofmapsintocompositionofmaps).However,toprovethatgaugetheoriesoverAare

ˆweshouldbeabletotransferalsoconnectionsonA-relatedtogaugetheoriesonA

ˆ-modules.IintroducedanewnotionofgaugeMoritamodulestoconnectionsonA

equivalence(inoriginalpaper[2]Iusedtheterm”completeMoritaequivalence”)andprovedthatgaugeMoritaequivalenceofalgebrasimpliesphysicalequivalence

d

ofcorrespondinggaugetheories.Itisprovedin[2]thatnoncommutativetoriTθanddTθˆaregaugeMoritaequivalentifandonlyifthereexistsamatrix

󰀁󰀂AB

(4)

CDbelongingtoSO(d,d,Z)andobeying(5)

ˆ=(Aθ+B)(Cθ+D)−1θ

HereA,B,C,Dared×dmatricesandSO(d,d,Z)standsforthegroupof2d×2d

matriceswithintegerentriesthatareorthogonalwithrespecttoquadratieformx1xd+1+...+xdx2dhavingsignature(d,d).(Thefactthattherelation(5)impliesMoritaequivalencewasprovedinearlierpaper[4]writtentogetherwithM.Rieffel.)

GAUGETHEORIESONNONCOMMUTATIVESPACES3

EquivalenceofgaugetheoriesonnoncommutativetoriTθandTθˆwasstudiedindetailin[2],[6],[7],[8].ItiscloselyrelatedtoT-dualityinstringtheory;thisrelationwasthoroughlyanalyzedin[5].Thisanalysisled,inparticular,tothediscoveryofpossibilitytotradenoncommutativityparameterforbackgroundfieldintheexpressionsforBPSenergies.(Almostsimultaneouslythisfactwasfoundin[25]atthelevelofactionfunctionals;itwascalledbackgroundindependence.)

Thepapers[6],[7],[8],[13]aredevotedtothestudyofBPSfieldsandBPSstatesinSUSYgaugetheoriesonnoncommutativetori.AnalysisofBPSspectrabymeansofsupersymmetryalgebrawasperformedin[7].AnotherwaytostudyBPSstatesisbasedongeometricquantizationofmodulispacesofclassicalconfigurationshavingsomesupersymmetry(BPSfields).Onecanidentify1

states.

InstantonsonnoncommutativeR4wereanalyzedin[3]bymeansofgener-alizationofADHMconstruction.Themoststrikingfeatureofnoncommutativeinstantonsistheabsenceofsmallinstantonsingularityinmodulispaceofnoncom-mutativeinstantons.Instantonsonnoncommutativetoriwerestudiedin[10];inparticular,weconstructedanoncommutativeanalogofNahmtransform.Instan-tonscanbecharacterizedas1

4BPS

fieldsand1

2BPS

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assumethateverylinearcombinationofmatrixelementsofθhavingintegercoeffi-cientsisirrational.)Thisremarkcanbeusedalsoinmanyothercases;itconfirmstheideathatnoncommutativetoriwithirrationalθaresimplerthatcommutativetori.

Gaugetheoriesonnoncommutativetoroidalorbifoldswerestudiedin[11],[12].Fairlycompleteanalysisofmodules,ofconstantcurvatureconnectionsandcorre-d

spondingmodulispaces,ofMoritaequivalenceisgivenforTθ/Z2;however,themethodsdevelopedin[11],[12]workalsoforothertoroidalorbifolds.

AllresultswementionedarebasedonthenotionofconnectiononA-module.Thereexistdifferentdefinitionsofthisnotion,butallofthemarebasedonthesameidea:aconnectionshouldsatisfyLeibnizrule.Ifann-dimensionalLiealgebraLactsonassociativealgebraAbymeansofinfinitesimalautomorphisms(derivations)wecandefineaconnectionon(left)A-moduleEasacollectionofnlinearoperators∇i:E→E,i=1,...,nobeyingtheLeibnizrule:

∇i(ae)=a·∇ie+δia·e,

wherea∈A,e∈E,andδ1,...,δnarederivationscorrespondingtoelementsofabasisofLiealgebraL.(Notice,thatoperators∇idon’tcommutewithmulti-plicationbya∈A,i.e.theyareC-linear,butnotA-linear.However,if∇i,∇′i

′

(i=1,...,n)aretwoconnectionsthedifference∇i−∇iisA-linear;inotherwords∇′i−∇iisanendomorphismofE.)

Whenweconsidernoncommutativetoriweshoulddefineconnectionsusing

d

d-dimensionalcommutativeLiealgebraactingonTθbymeansoftranslations.

IfwewouldliketodefineconnectionsintermsofcovariantdifferentialinsteadofcovariantderivativeweshouldassumethatthealgebraAisaZ2-gradedassociativealgebraequippedwithaparityreversingderivationQ:A→A.ThestandardassumptionisthatQ2=0(thenAiscalledagradeddifferentialalgebra).Howeveritisshownin[9]thatonecanrelaxthisassumptionrequiringonlythatQ2a=[ω,a].(HereωisafixedelementofAobeyingQω=0.)IfAisanassociativealgebraequippedwithanoperatorQofthiskind(aQ-algebraistheterminologyof[9])wecandefineaconnectionon(left)A-moduleEasanlinearoperator∇:E→EobeyingtheLeibnizrule:

D(ae)=(−1)degaaDe+Qa·e.

Thestandardtheoryofconnections(includingthenotionofCherncharacter)canbegeneralizedtothecaseofmodulesoveraQ-algebra.IfPisamoduleoverQ-algebraAand∇PisaconnectiononPwecandefineastructureofQ-algebra

ˆ=EndAPbytheformulaQϕ˜=[∇P,ϕ].(HereEndAPstandsforanalgebraonA

ofendomorphismsofA-moduleP,i.e.foranalgebraofA-linearmapsofPinto

ˆisMoritaequivalenttoA,itself.)UndercertainconditionsonPthealgebraA

ˆ-modulesandviceversa.(IfAhasaunitwei.e.wecantransferA-modulesintoA

shouldrequirethatAconsideredasleftA-moduleisadirectsummandofPNforsomeNandPisprojective)Usingtheconnection∇Pwecantransferconnections

ˆ-modules.Thisoperationper-onA-modulestoconnectionsoncorrespondingonA

ˆ-modulesmitsustoextendtheequivalencebetweencategoriesofA-modulesandA

toanequivalenceofcorrespondinggaugetheories.Thisgivesaverygeneraldual-itytheorem;SO(d,d,Z)dualityofgaugetheoriesonnoncommutativetoricanbederivedfromthisgeneraltheorem[9].

GAUGETHEORIESONNONCOMMUTATIVESPACES5

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DepartmentofMathematics,UniversityofCaliforniaatDavis,Davis,CA95616E-mailaddress:schwarz@math.ucdavis.edu