THEORY
JORGEPLAZAS
Abstract.Inthisshortcommunicationwesurveysomeknownresultsrelat-ingnoncommutativegeometrytotheclassfieldtheoryofnumberfields.Theseresultsappearwithinthecontextofquantumstatisticalmechanicswheresomearithmeticpropertiesofagivennumberfieldcanberealizedintermsofthestructureofequilibriumstatesofaquantumstatisticalmechanicalsystem.
1.Introduction
Inthelasttenyearssomeveryinterestingresultsrelatingnoncommutativege-ometrytoclassfieldtheoryhaveemerged.ThefirstinstanceofthisconnectionwasexploredbyBostandConnesin[2]wheretheyrelatedtheclassfieldtheoryofthefieldofrationalnumbersQtothestructureofequilibriumstatesofaparticularquantumstatisticalmechanicalsystem.Variousresultsgeneralizingsomeaspectsofthisconstruction,basedonquantumstatisticalmechanicalsystemsrelatedtoothernumberfields,haveappearedsincethen([1,4,12,13,14]).InparticulartheproblemoffindingquantumstatisticalmechanicalsystemsthatencodetheexplicitclassfieldtheoryofquadraticimaginaryextensionsofQhasbeensolvedrecently([7,8]).Theexistenceofquantumstatisticalmechanicalsystemswithricharith-meticalpropertiesopensanewapproachtothestudyofexplicitclassfieldtheoryusingthetoolsofquantumstatisticalmechanics.Thepurposeofthisarticleistogiveabriefintroductiontothistopic.
Iwanttothanktheorganizersofthe2005summerschool“Geometricandtopo-logicalmethodsforquantumfieldtheory”andMaxPlanckInstitutefortheirsup-port.
2.Basicsinclassfieldtheory
Themainobjectsofstudyinalgebraicnumbertheoryarenumberfields,bydefinitionanumberfieldKisafinitedegreenumbersQ.Oncewefixanalgebraic¯extensionofthefieldofrational
ofKwewouldtheabsoluteGaloisgroupGal(K
¯closureK
liketounderstand|K),whichturnsouttobeverydifficultevenwhenK=Q.Onestepinunderstandingthegroup¯|K)consistsinstudyingitsabelianizationGal(K
¯Gal(K
|K)ab,thisabeliangroupistheGaloisgroupofKab,themaximalabelianextensionofK,sowehave:
Gal(Kab|K)=Gal(K
¯|K)ab2JORGEPLAZAS
ThefieldKabsionsofKinK
¯maybeobtainedasthelimitoverallfinitenormalabelianexten-.AbelianclassfieldtheorystudiestheseabelianextensionstogetherwiththecorrespondingGaloisgroups.Oneofthemainresultsofthetheorychar-acterizesthegroupGal(Kabringconstructedfromthefield|K)asKaalone.quotientThisofthetopologicalgroupofringunitsisoftheatopologicalid`eleclassgroupCK.Letusbrieflyrecallitsconstruction.
LetKbeanumberfieldandletOKbeitsringofintegers,i.e.OKconsistsoftheelementsinKwhicharerootsofmonicpolynomialswithcoefficientsinZ.ThereisaonetoonecorrespondencebetweentheprimeidealsofOKandthefiniteplacesofK,whichareequivalenceclassesofnonarchimedeanvaluations||onK.Forinstanceeveryprimenumberp∈OQ=Zgivesrisetoafiniteplacecorrespondingtothep-adicabsolutevalue||pinQ.GivenafiniteplaceνofKwedenotebyKνthecompletionofKwithrespecttothemetricinducedbyνanddenotebyOνtheringofintegersofKν.ForeachνtheringOνisanopencompactsubringofKνandwemayformtherestrictedtopologicalproduct
Af,K=
(Kν:Oν)
ν
whereνrunsoverthefiniteplacesproductfinitenumberofcoordinateslieofOK.Thisrestrictedproductisbydefinition
thesubsetofthecartesianνKνconsistingofelementsforwhichallbutainthesubringsOν.Itisgiventheweakesttopologyforwhichthesets
ν∈FKν×ν∈/FOν,forFafinitesetofplaces,areopen.ThetopologicalringAf,Kiscalledtheringoffinitead`elesofK.IfweaddtoAf,KtheproductofthecompletionsofKwithrespecttotheinfiniteplaceswegetAK,theringofad`elesofK.Infiniteplacesaregivenbyequivalenceclassesofarchimedeanvaluationsandcorrespondtothe[K:Q]differentembeddingsofKinC.
ConsidernowGl1(AK),thegroupofunitsofAK.K∗canbeembeddeddi-agonallyasadiscretesubgroupofGl1(AK).ThequotientCK=Gl1(AK)/K∗iscalledtheid`eleclassgroupofK.ItturnsoutthattheabelianextensionsofKcorrespondtothenormalsubgroupsofCK.Inparticularthefollowingimportanttheoremholds(cf.[9],SectionII.3.7):
Theorem2.1.LetDKbetheconnectedcomponentoftheidentityinCK.Thereisacanonicalisomorphism:
θ:CK/DK→Gal(Kab|K)
ForthecaseK=Qalotmoreisknown.BytheKroneckerWebertheorem(cf.[18]Section20)everyabelianextensionofQiscontainedinacyclotomicextension,thatis,insomeextensionoftheformQ(ζn)whereζnisaprimitiven-throotofunityforsomen.ThecyclotomicextensionQ(ζn)isabeliananditsGaloisgroupGal(Q(ζn)|Q)isisomorphicto(Z/nZ)∗.ThemaximalabelianextensionthefieldQcyclobtainedbyadjoiningtoQallrootsofunityandCQ/DQ=Z
ˆofQis
∗whereZ
ˆ=lim←n
Z/nZ.ForageneralnumberfieldK,onewouldalsoliketoknowwhichelementsofK
¯generateKaboverKandhowtheGaloisgroupGal(Kabtors.ThisisthecontentofHilbert’s12thproblemknown|Kalso)actsastheontheseexplicitgenera-classfieldtheoryproblem.UptonowtheonlynumberfieldsapartfromQforwhichacompletesolutiontotheexplicitclassfieldtheoryproblemisknownarequadraticimaginaryfields,thatisfieldsoftheformK=Q(
√QUANTUMSTATISTICALMECHANICSANDCLASSFIELDTHEORY3
integer.Inthiscasetheanswer√isgivenbythetheoryofcomplexmultiplicationwhichcharacterizesthefieldQ(−d)֒→CtheringofintegersofQ(√
ofananalyticfunctionFonEatthepointsoffinite−d)abis
generatedbythevaluesorderofE.ThefunctionFisgivenbyPuwherePistheWeierstrassfunctionofEanduistheorderofthegroupofautomorphismsofE.
3.Basicsinquantumstatisticalmechanics
Quantumstatisticalmechanicsstudiesstatisticalensemblesofquantummechan-icalsystems.Fromamathematicalpointofview,aquantumstatisticalmechanicalsystemconsistsofasetofobservablesAhavingthestructureofaC∗-algebraandatimeevolutionofthesystemgivenbyaone-parametergroupofautomorphismsσtofthealgebraofobservables,σtsystemisspecifiedbyaprobabilitymeasure∈Aut(Aon),thet∈phaseR.Classicallyspace,integrationthestateofofanaobservableagainstthismeasuregivesitsmeanvaluecorrespondingtotheparticu-larstate.Inaquantummechanicalsetting,probabilitymeasuresarereplacedbystatesonthealgebraofobservables.BydefinitionastateonaunitalC∗-algebraAisalinearmapϕ:A→Csatisfying:
•ϕ(a∗a)≥0foralla∈A.The•appropriateϕ(1)=1.
definitionofequilibriumstatesinthiscontextwasgivenbyHaag,HugenholtzandWinninkin[11].Givenaquantumstatisticalmechanicalsystem(A,σt)anequilibriumstatewillbeastateϕonthealgebraAsatisfyingcertaincompatibilityconditionwithrespecttothetimeevolutionofthesystem.Thiscondition,knownastheKMScondition(afterKubo,MartinandSchwinger)dependsonathermodynamicparameterβ=1
4JORGEPLAZAS
atimeevolutiononAby
σt(a)=eitHae−itH,
a∈A
Iftheoperatore−βHistraceclassthen
ϕ(a)=
1
ϕ(ρ(1))ϕ
◦ρ.Innerendomorphismscomingfromisometries
invariantunderthetimeevolutionacttriviallyonequilibriumstates.SymmetriesinducedbyendomorphismswereintroducedinthecontextofsuperselectionsectorsdevelopedbyDoplicher,Haagandroberts.
4.TheBost-Connessystem
In[2]BostandConnesconstructedaremarkablequantumstatisticalmechanicalsystem(A,σt)inwhichthestructureofequilibriumstateswiththeclassfieldtheoryofQ.ThegroupCQ/DQ=Z
ˆisrelatedinadeepway
∗actsassymmetriesofthissystemandthealgebraicnumbersgeneratingthemaximalabelianextensionofQcanberecoveredasvaluesoftheequilibriumstatesatzerotemperatureontheobservablescorrespondingtoanarithmeticsubalgebraofA.WhatismoreremarkableaboutthissystemisthefactthatthevaluescommuteswiththeactionofCQ/DQ=Z
ˆactionoftheGaloisgrouponthis
∗ontheequilibriumstates.Inthissectionwedescribethissystemanditsmainfeatures.
LetAbetheC∗-algebrageneratedbytwosetsofelements{e(r)|r∈Q/Z}and{µn(1)|n∈µ∗N+}withrelations:
nµn=1∀n∈N+(2)µnµk=µnk(3)(0)=1,e(1
r)∗∀=n,ek(−∈rN+
e),e(r)e(s)=e(r+s)
∀r,s∈Q/Z
(4)
µne(r)µ∗n=
QUANTUMSTATISTICALMECHANICSANDCLASSFIELDTHEORY5
DefineatimeevolutionontheC∗-algebraAbytaking
σt(µn)=nitµn,n∈N+,t∈Rσt(e(r))=e(r),
r∈Q/Z,t∈R
Wewillrefertothequantumstatisticalmechanicalsystem(A,σt)astheBost-Connessystem.Wesummarizethemainresultsof[2]aboutthestructureofthissystem.AsabovewedenotebyE∞thesetofextremalKMS∞states.Theorem4.1(Bost,Connes).
•ThegroupCQ/DQ=Z
ˆ∗actsasagroupofsymmetriesofthesystem(A,σt).•LetAQbetheQ-subalgebraofAgeneratedoverQbythesetsofelements
{e(r)|r∈Q/Z}and{µn,µ∗∗
n|n∈N}.Thenforeveryϕ∈E∞andeverya∈AQthevalueϕ(a)isalgebraicoverQ.Moreoverforanyϕ∈E∞onehasϕ(AQ)⊂QabandQabisgeneratedbynumbersoftheformϕ(a)withϕ∈E∞anda∈AQ.
•Forallϕ∈E∞,γ∈Gal(Qab|Q)anda∈AQonehas
γϕ(a)=ϕ(θ−1(γ)a)
whereθ:Z
ˆ∗→Gal(Qab|Q)istheclassfieldtheoryisomorphism.Thesystem(A,σt)wasoriginallyintroducedbyBostandConnesinthecontext
ofHeckealgebras.TheinclusionofringsZ⊂QinducesaninclusionPZ+⊂PQ+wherePQ+={b0a|b∈Q,a∈Q∗+}andPZ+
={matrix1c01
Thisinclusionofmatrixgroupsisalmostnormalinthesensethatthegroups
1orbits|c∈Z}of.PZ+actingontheleftonthesetofrightcosetsPQ+/PZ+arefiniteandviceversa.
OnecanthenformaconvolutionalgebraH(PZ+,PQ+
)givenbyfinitelysupported
C-valuedfunctionsonPZ+\\PQ+/PZ+
,thisalgebraiscalledtheHeckealgebraofthe
pair(PZ
+
,PQ+).H(PQ+;PZ+)canbecompletedtoaC∗-algebrawithanaturaltimeevolution.Itisshownin[2]thatinthiswayoneobtainsthesystem(A,σt)describedabove.
Let(ǫn)n∈N+bethecanonicalbasisoftheHilbertspacel2(N+).Foranyelementγ∈Gal(Qab|Q)onecandefinearepresentationπγofthealgebraAinl2(N+)by
πγ(µn)ǫk=ǫnk,
n,k∈N+
πγ(e(r))ǫk=γ(e2πikr)ǫk,
k∈N+,r∈Q/Z
ForanyoftheserepresentationsthetimeevolutionofthesystemisimplementedbytheoperatorHǫk=(logk)ǫk,k∈N+andsothepartitionfunctionofthesystemistheRiemannzetafunction
Z(β)=Tr(e−βH)=
1
k
6JORGEPLAZAS
5.Somegeneralizationstoothernumberfields
TheworkofBostandConneshasinspiredseveralauthorstoconstructquantumstatisticalmechanicalsystemsassociatedtoothernumberfields.Theseconstruc-tionsgeneralizeindifferentdirectionssomeoftheresultsin[2].
In[13]M.LacaandI.RaeburnrealizedthealgebraofobservablesoftheBost-Connessystemasasemigroupcrossedproductalgebra.GivenadiscretegroupΓitsgroupalgebraCΓcanbecompletedtoaC∗-algebraC∗(Γ)byconsideringallunitaryirreduciblerepresentationsofCΓasanalgebraofoperatorsonsomeHilbertspace.IfasemigroupSactsbyendomorphismsonthealgebraC∗(Γ)onecantwisttheproductandtheconvolutiononC∗(Γ)bytheactionofSgettingacrossedproductC∗-algebraC∗(Γ)⋊S.TheC∗-algebraoftheBost-ConnessystemisthengivenbyasemigroupcrossedproductC∗(Q/Z)⋊N+wheretheactionofN+onC∗(Q/Z)isarightinversetotheactioncomingfromthenaturalmultiplication(n,r)→nrwheren∈N+andr∈Q/Z.
Arledge,LacaandRaeburnconsideredin[1]crossedproductsoftheformC∗(K/O)⋊O×forKanarbitrarynumberfieldandOitsringofintegers.TheycharacterizethefaithfulrepresentationsofthesealgebrasandrealizedthemasHeckealgebras.AsinthecaseofQonehasthatforanynumberfieldKtheringinclusion
++
O⊂KinducesanalmostnormalinclusionofmatrixgroupsPO⊂PK.HarariandLeichtnamstudiedin[12]aquantumstatisticalmechanicalsystemcorrespondingto
++
theHeckealgebraH(PK;PO).Theequilibriumstatesofthissystemsharemany
∗
ˆ∗=ofthepropertiesoftheBost-Connessystem.ThegroupOνOν,whereν
runsoverthefiniteplacesofK,actsasasymmetrygroupofthesystemandthepartitionfunctionofthesystemistheDedekindzetafunctionofthenumberfieldK
1
ζK(s)=
1−N(p)−s
awithafinitenumberoffactorsremovedfromtheproduct.InthisexpressionarunsovertheintegralidealsofO,prunsovertheprimeidealsofOandN(a)=[O:a]istheabsolutenormoftheideala.Thissystemexhibitsspontaneoussymmetrybreakingatthepoleofthisfunction;for0<β≤1thesystemadmitsonlyoneequilibriumstateandthesymmetrygroupofthesystemactstriviallyonthisstate.
ˆ∗andthesymmetrygroupactsfreelyForβ>1thespaceEβisparameterizedbyO
andtransitivelyonthisspace.Theresultsof[12]holdforarbitraryglobalfields.Theseincludenumberfieldsandfiniteseparableextensionsofthefieldofrationalfunctionsoverafinitefield.
Aquantumstatisticalmechanicalsystemsimilartotheoneof[12]wasintroducedin[4]byCohen.HersystemhastheadvantageofrecoveringthefullDedekindzetafunctionofthenumberfieldKaspartitionfunctionofthesystem.
LacaandvanFrankenhuijsenstudiedin[14]theHeckealgebracorrespondingtothealmostnormalinclusionofmatrixgroups
1O1K
⊂
0O∗0K∗foranarbitrarynumberfieldKwithringofintegersO.Theequilibriumstates
ofthecorrespondingsystemareanalyzedinthecaseoffieldswithclassnumberone.Forquadraticimaginaryfieldsofclassnumberonethesymmetrygroupofthe
QUANTUMSTATISTICALMECHANICSANDCLASSFIELDTHEORY7
systemisisomorphictotheGaloisgroupofthemaximalabelianextensionofthefieldandtheactionofthisgrouponextremalequilibriumstatesistransitive.
6.Fabulousstatesfornumberfields
TheworkofBostandConnestogetherwithitsvariousgeneralizationsopensthepossibilityofapproachingtheproblemofexplicitclassfieldtheorywithintheframeworkofnoncommutativegeometry.FollowingtheselinesonewouldliketohaveforanarbitrarynumberfieldKaquantumstatisticalmechanicalsystemwhichfullyincorporatesitsclassfieldtheory,thenotionoffabulousstatesfornumberfields,introducedin[5]byConnesandMarcolli,encodesthisaim.
GivenanumberfieldKtogetherwithanembeddingK֒→Cthe“problemoffabulousstates”asksfortheconstructionofaquantumstatisticalmechanicalsystem(A,σt)suchthat:
•ThegroupCK/DKactsasagroupofsymmetriesofthesystem(A,σt).•ThereexistsaK-subalgebraϕ(a)isalgebraicAKofAsuchthatforeveryϕ∈E∞andeverya∈AKthevalueoverK.MoreoverKabisgeneratedoverKbynumbersofthisform.
•Forallϕ∈E∞,γ∈Gal(Kab|K=)ϕand(θ−1a(γ∈)aAKonehas
γϕ(a))
whereθ:CK/DKThegroundstatesofsuch→Galasystem(Kab|Kare)isrefereedtheclassasfield“fabulous”theoryinisomorphism.viewofitsricharithmeticalproperties.
7.Q-lattices,quadraticextensionsandcomplexmultiplication
ForthecaseofquadraticimaginaryfieldsK=Q(
√
8JORGEPLAZAS
naturalchoiceofanarithmeticstructuregivenbyaQ-subalgebraofC∗(Ln/R∗+).InthiswayonerecoverstheBost-Connessystem[5].
ThegroupC∗actsbyrescalingonthesetoftwodimensionalQ-lattices.AsintheonedimensionalcasetheC∗-algebraC∗(Ln/C∗)correspondingtothespaceofcommensurabilityclassesoftwodimensionalQ-latticesuptoscalinghasanaturaltimeevolutionandanaturalchoiceofanarithmeticstructure.ThestructureofequilibriumstatesofthecorrespondingquantumstatisticalmechanicalsystemisrelatedtotheGaloistheoryofthefieldofmodularfunctions.Wereferto[6]and[16]forasurveyoftheseresults.√ConsidernowaquadraticimaginaryfieldK=Q(
d),whered∈N+
isasquarefreepositiveinteger.In[15]Maninproposedtheuseofnoncommutativegeometryasageometricframeworkforthestudyofabelianclassfieldtheoryofrealquadraticfields.Thisisthesocalled“realmultiplicationprogram”.Themainideaisthatnoncommutativetorimayplayaroleinthestudyofrealquadraticfieldsanalogoustotheroleplayedbyellipticcurvesinthestudyofimaginaryquadraticfields.SomepossiblerelationsbetweenManin’sprogramandQ-latticesarediscussedin[16].
References
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[4]P.Cohen,AC∗-dynamicalsystemwithDedekindzetapartitionfunctionandspontaneous
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Bost-Connesaucasdescorpsglobauxquelconques.SelectaMath.(N.S.)3(1997)205-243.[13]M.Laca,I.Raeburn,Asemigroupcrossedproductarisinginnumbertheory.J.London
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[14]M.Laca,M.vanFrankenhuijsen,PhasetransitionsonHeckeC*-algebrasandclass-field
theoryoverQ.preprintmath.OA/0410305
[15]Y.ManinRealmultiplicationandnoncommutativegeometry.In:“ThelegacyofNielsHenrik
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SocietyStudentTexts,50.CambridgeUniversityPress,Cambridge(2001).MaxPlanckInstituteforMathematics,Vivatsgasse7,Bonn53111,GermanyE-mailaddress:plazas@mpim-bonn.mpg.de
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