; TeX output 1999.06.21:1502bn_Bn_{ˍtff[$$ffȍ"V cmbx10Comm9unicationsTinMathematicalPhysicsmanuscriptNo. 8K`y cmr10(willUUbGeinsertedbytheeditor)Z34$ffffff[\(Nff cmbx12QuantumffMonos3dromyinIntegrableSystemsSanTVQ~;uNgoG.c^ٓRcmr71|s^ 0ercmmi7;\s^2:ፍ-:Aacmr61j|{Ycmr8InstitutXFJourierUMR5582,B.P.74,38402SainÎt-Martind'Hneres,FJrance. jE-mail:XSan.VJu-Ngo cmmi10P1|s(h);:::;Pnq~(h)bGeasetofcommutingself-adjointh-pseudo- Ddi erentialopGeratorsonann-dimensionalmanifold.IfthejointprincipalsymbGolpispropGer,itisknownfromtheworkofColindeV*erdiGere[6]andCharbGonnel[3]that=inaneighbGourhood=ofanyregularvqalueofp,thejointspGectrumlocallyhasthestructureofananeintegrallattice.Thisleadstotheconstructionof%anaturalinvqariant%ofthespGectrum,calledthequantummonodromy*.Wepresentthisconstructionhere,andshowthatthisinvqariantisgivenbytheclassicalYmonoGdromyoftheunderlyingLiouvilleintegrablesystem,asintroGducedbyDuistermaat[9].Themoststrikingapplicationofthisresultisthatalltwodegree>Roffreedomquantumintegrablesystemswitha%': cmti10fo}'cus-focusssingularity>RhavetheEsamenon-trivialquantummonoGdromy*.Forinstance,thisprovesEaconjectureofUUCushmanandDuistermaat[7]concerningthequantumsphericalpGendulum.(71. F?In9troQductionՍObstructionstotheexistenceofglobalaction-anglecoGordinatesforcompletelyintegrable*systemsarewellknownsinceDuistermaat'sarticle[9].ItwasthennaturaltoraisethequestionabGouttheimpactoftheseobstructionsonquantumintegrabletmsystems,atleastforthe(semi)-classicalpseudo-di erentialquanti-sation[oncotangentbundles.The rstattemptsinthisdirectionwere[7]and[11],>bGothofthemconcerningthemonodromyinvqariantfortheexampleofthesphericalHpGendulum.Thissystemisindeedoneofthesimplest(alongwiththeChampagnebGottle[1])thatexhibitsanon-trivialmonodromy*.The rstofthesearticlesUQ[7]propGosedaparticularlyinterestingwayofdetectingthemonoGdromybyXobservingashiftinthelatticestructureofthejointspGectrum.Itisthepur-pGoseUUofthisarticletostate,proveUUandexplainthisidea.*bn_20S.XVmono}'dromyinvqariant"foranysetthatsharesthesepropGerties(Sect.2).ThenhweproveinSect.3that,forspGectra,thequantummonoGdromyispreciselygiven>bytheclassicalmonoGdromyoftheunderlyingclassicalHamiltoniansystem.TheresultisappliedinSect.4totheparticularlyinterestingcaseofsystemsadmittingXafo}'cus-focusysingularity*.XThelastSect.5 nallyshowshowtoreado !themonoGdromyfromapictureofthespectrum.Asanexample,weusethespGectrumUUoftheChampagnebottlecomputedbyChild[4]. ΍2. F?ConstructionToftheQuan9tumMonoQdromyILet' !", cmsy10U%RbGeanopensubsetof msbm10R^nq~,letHbeasetofpositiverealnumbGersaccumu-latingUUat0,andforanyhinH%Slet(h)bGeadiscretesubsetofU9. IfBisanopGensubsetofU9,afamily(f(h))h O!cmsy72HdofsmoothfunctionsonBwithvqaluesinR^n 3iscalledasymb}'ol(oforderzero)ifitadmitsanasymptoticexpansionUUoftheformՍn\f(h)=f0S+8hf1+h2|sf2+for&smoGothfunctionsfid:BG!R^nq~.&Morepreciselywerequirethatforany`0, DforanyNٲ0,andforanycompactKٵBq,thereisaconstantC`;NQ;KlsuchthatUUforallh2H,a4'u cmex10 a4' a4' a4' a4' faf(h)8ѴN EşX k+B=0ohk됵fk   `bC`;NQ;KvhN,+1; ]Lwhere%k:k` denotestheC^`'norminK.ThesymbGolf(h)iselliptic۲ifitsprincipalpart;f0SisaloGcaldi eomorphismofBQintoR^nq~.Thevqalueoff(h)atapointc2BƲwillUUbGedenotedbyf(h;c). A)family9(rG(h))h2Hofelementsofa nitedimensionalvectorspaceissaidtobGenO(h^1x)ifforanyNd0thereisaconstantC>d0suchthatkrG(h)kdCh^N,uniformly forallhݛ2H. IfS(h)isanyfamilyofsetsdepGendingonh,thentheÊnotationf(h)~ĸ2S(h)X+OG(h^1x)Êmeansthatthefunctiondist7(f(h);S(h))isOG(h^1x). W*ewillsaythat(h)hasthestructureofan\asymptoticanelattice"whenever<(itcanbGedescribedwithalocally nitesetof\asymptoticaneinte-gralUUcharts",inthefollowingsense:De nitionT1.C@((h);U9) `isan\asymptoticanelattic}'e"ifforanyc2U, `ther}'eexistszasmallop}'enballBGUyaroundc,andanellipticsymbolf(h)A:BG!R^nofor}'derzerosuchthat,foranyfamily(h)2BX: bn_QuanÎtumXMonoTheHlackofinjectivityforH,ismeasuredbyx䍑MH^0K(U9;GL(n;Z))H:one PcanVcheckthattwocoGcycles[kP]and[k^0в]inx䍑H^1(U9;Z^nq~)yieldthesameelementofx䍑9H^1 q(U9;GA(n;Z))VifandonlyifthereisanM2x䍑 mkH^0 ܲ(U;GL(n;Z))Vsuchthat[kP^0в]=[MO8kP].yLetUUusnowgivevqariousinterpretationsofthequantummonoGdromyM. S ThegactionofGA(n;Z)onZ^n ,bGeinge ective,itisastandardfactthatthecohomologyrsetx䍑H^1(U9;GA(n;Z))rclassi estheisomorphismclassesof brebun-dlesoverUwithstructuregroupGA(n;Z)and breZ^n I(seeforinstance[12,pp.40{41]).yLetLbGesuchalatticebundleassociatedtoM.TheelementsA justUUde nethetransitionfunctionsbGetweenUUtwoadjacenttrivialisationsofL. S SincetthesetrivialisationfunctionsareloGcallyconstant,thereisanaturallyde nedޡparalleltranspGort 8:pofapointp2Lcalongޡapath lٲinthebaseU9., bn_QuanÎtumXMonoOballs(BiTL;Bj6),andB1|s;:::;B`5enumerateelementsofacoverofUvBecauseofDe nition1,anyopGenballaroundccontains,forhsmallenough,atWleastoneelementof(h).Therefore,thereexistsafamily(h).q2(h)b7\BsuchUUthat limph!0k(h)=c:LetUUk2Z^nӲandlet^09(h)bGeafamilyofelementsof(h)8\BƲsuchUUthatWf(h;(h))=f(h;09(h))8+hkw+OG(h1x):.2Then,UUashtendstozero,&hr0s(h)(h)ʉfe(MϺh1_ tendstowardsUUalimitv"2R^nӲsatisfying k=dUUf0|s(c)v(recallUUthatf0Ȳdenotestheprincipalpartoff(h)). Yd SinceUU(h)and^09(h)arein(h),thereisafamilykP^0в(h)2Z^nӲsuchUUthatJȟ^<$Sqog[ٲ(h;^09(h))8g(h;(h))Sqowfeci (֍/9h ^Ǘ=kP0в(h)8+OG(h1x):>bn_60S.XVBecauseKofthediscretenessofGA(n;Z),PropGosition1impliesthatthereAisanh0͚>Q'0suchthatthetransitionelementAisuniquelyde nedby(g[ٲ(h0|s)=h0)vs(f(h0)=h0)^12actingona nitesubsetofZ^nq~.Therefore,whenre-stricted}toanyopGensubsetofU|withcompactclosureinU9,thecocycle[M]isreallyaquantumiob8ject,inthesensethat\youdon'tneedtolethtendtozero"toUUde neit.Nbn_QuanÎtumXMonocpj6(h;x;uDz)=p1ɍjxݍ0|s(x;)8+hp1ɍjxݍ1|s(x;)+h2|sp1ɍjxݍ2(x;)8+8׵:yBecausegNtheprincipalsymbGolsp^1l0|s;:::;p^nl0̲commutewithrespGecttothesymplecticPoissonUUbracketonTc^sX,themapFsTcsX3(x;uDz)UU fd@ά- qp늲(p10|s(x;);:::;pn፱0q~(x;))2RnisPamomentummapfortheloGcalHamiltonianactionofR^nonTc^sXvde nedby ačtheHamiltonian owsofthep1ɍjxݍ0|s.W*ewillalwaysassumethatpispr}'oper,sothattheUUlevelsetscOc=p1 t(c)arecompact.Moreover,weaskthattheselevelsetsbGec}'onnected.Conclusionsfornon-connectedqKc,canbGeobtainedbyseparatelystudyingthedi erentconnectedcompGonents. , LetUr|bGetheopensubsetofregularvqaluesofthemomentummapp,andletUSbGeUUanopensubsetofUrwithcompactclosure. ItIfollowsfromtheArnold-LiouvilletheoremthatpU isasmoGoth brationwhose& bresareLagrangiantori.Thestructureofthis brationissemi-globally(i.e.inaneighbGourhoodofa bre)describGedwiththehelpofaction-angleco-ordinates._9However,the at brebundleH1|s(c;Z)=!c2U]r(with breZ^nq~)mayhavenon-trivialmonoGdromy*,preventingtheconstructionofglob}'al actionvqariablesGonp^1 t(U9)(seeDuistermaat[9]).W*ewilldenoteby[Mcl](classical PmonoGdromy)UUthecocycleinx䍑H^1Ʋ(U9;GL(n;Z))UUassociatedtothislatticebundle. OnWDtheotherhand,let(h)bGetheintersectionwithUU}ofthejointspGectrumofthehopGeratorsP1|s(h);:::;Pnq~(h).Itisknownfrom[3]thatthisspectrumisdiscreteandBforsmallhiscompGosedofsimpleeigenvqalues.Moreover,thefollowingresultholds: dPropQositionT2H([3]).T(h)isanasymptoticanelattic}'eonU9.9W*eldenoteby[Mq@Lu $]$2x䍑ͲH^1(U9;GA(n;Z))lthequantummonoGdromyofthespGec-trumUUonU9,givenbyDe nition2. RecallthatdenotestheinclusionofGL(n;R)intoGA(n;R)suchthatforanyUUM32GL(n;R),(M)leavestheorigin02R^nӲinvqariant. TheډrelationbGetweenډ[Mq@Lu $]andtheclassicalmonodromy[Mcl]isthengivenbyUUthefollowingtheorem: dTheoremT1.> Thequantummono}'dromyis\dual"totheclassic}'almonodromyinthefollowingsense:y}&[Mq@Lu $]=(jtV[Mcl]1 t):[ubn_80S.XVTheCfactthattheanelnatureofquantummonoGdromyisherenat-urally\reducedtoanactionoftheline}'argroupGL(n;Z)isduethetheglobalexistence)ofaprimitiveofthesymplecticformonTc^sX,namelytheLiouville1-formUU z. kbn_QuanÎtumXMonocus-FocuseSingularit9yItAisprobablynotworthdiscussingmonoGdromyinarbitrarydegreesoffreedom, DforUUitisatypicalphenomenonof4-dimensionalsymplecticmanifolds(see[13]). Morecprecisely*,letX,bGea2-dimensionalmanifold,andletP1|s(h),P2(h)bGetwocommutingself-adjointh-pseudo-di erentialopGeratorsonX.Asbefore,supposethatthemomentummapp=(p^1l0|s;p^2l0)de nedbytheprincipalsymbGolsisproperwithUUconnectedlevelsets. W*efshallmakethefollowinghypGothesis.Thereexistsacriticalpointm2Tc^sXofpofmaximalcorank(i.e.bGothp^1l0x&andp^2l0arecriticalatm)suchthat,insomeloGcalsymplecticcoordinates(x;y[;u;),theHessians(p^1l0|s)^0N9^0r(m)and(p^2l0)^0N9^0r(m)(thereafterfdenotedbyH(p^1l0|s)andH(p^2l0|s))generatea2-dimensionalsubalgebraofUthealgebraQ(4)ofquadraticformsin(x;y[;u;)UunderPoissonbracketthatadmitsUUthefollowingbasis(q1|s;q2):JPq1C=x+8y[;+JPq2C=x8y[u:Suchasingularitymiscalledafo}'cus-focusRsingularity*.ThepGointmisthen DisolatedamongstcriticalpGointsofp.Therefore,wecanchoGoseUQSUr@tobeasmallzpunctureddiscaroundoD=p(m).zFinally*,weshallalwaysassumethatmisUUtheonlycriticalpGointofthecriticallevelset0C=p^1 t(o). Itisknown(probablysince[15];seeforinstance[14]or[8]fordiscussionsandmorereferencesonthistopic)thatthe brationpU hasnon-trivialmonoGdromy*,andUUcanbGedescribedinthefollowingway: Nearm,weknowfrom[10]thattheintegrableHamiltoniansystem(p^1l0|s;p^2l0)can/bGebroughtintoanormalformgivenby(q1|s;q2)./InotherwordsthereexistsaUUloGcaldi eomorphismF*:(R^2|s;0)!(R^2;o)UUsuchthatM(p10|s;p20)=Fc(q1|s;q2):This,allowsonetode netransversalvector eldsX1andX2tangenttothe bres Dc.1thatPareequaltotheHamiltonianvector eldsXq1 ,andXq2nearm.NotethatUUX2ȲispGeriodicUUofpGeriodUU2[ٲ. Around8eachc2U9,8wecannowde nethefollowingsmoGothbasis( 1|s(c); 2(c))ofUUH1|s(c;Z)'1(c):zw{ 2|s(c)UUisasimpleintegralloGopofX2.{ T*akenapGointon 2|s(c);letitevolveunderthe owofX1|s.Aftera nitetime, itUUgoGesbackon 2|s(c).Closeitupon 2(c).Thisde nes 1(c).~lPropQositionT3H([15]).}L}'etc=2U9.Withr}'especttothebasis( 1|s(c); 2(c)),theactionAoftheclassic}'almonodromymap U^cl L}'et˅P1|s(h);P2(h)b}'eaquantumintegrablesystemwithafocus-focussingularity.Thenther}'eexistsasmallpuncturedneighbourhoodUofthecriticalvalueosuchthatforanyc2U9,iff(h)isananechartofthejointsp}'ectrum(h)ar}'oundchavingprincipalpart?q \ <$~y1{ wfe  (֍2cZ y 1 (c) z;<$j1۟wfe  (֍2՟cZY y 2 (c)*q \!Jص;jthenD~thevalueofthequantummono}'dromyD~map1ɍ Xq@Luv XfD2GA(2;Z)atasimplelo}'op ఍'21|s(U9;c)enclosingoisgivenbythematrix=1ɍ9q@Luv9f 0(`)=^č 1 D 0:1X^(t:>Her}'eisthesignofdet˵M,whereM32GL(2;R)istheuniquematrixsuchthat:\O]ײ(H(p10|s);H(p20))=MO8(H(q1);H(q2)):5. F?Ho9wTtoDetectQuantumMonoQdromy+5.1.T{Intr}'oduction.Theoremm31wouldn'tbGeofmuchinterestifonecouldnot\reado "UUthequantummonoGdromyfromapictureofthejointspGectrum. Thisisactuallyeasytodo,atleastinaheuristicway*.Therigorousmathe-maticalUUformulationmayhoweverloGokslightlyawkward. Thé rstideaisthefollowing.GivenastraightlatticeZ^nq~,andanytwopGointsAqandBpinZ^nq~,thereisanaturalparalleltranslationfromAtoBactingonZ^nq~, namelyUUthetranslationbytheintegralvector6„fd Wۍ !AB돲. bn_QuanÎtumXMono0suchthatforanyh~˵B2\.(hZ^nq~),thetranslationbythevectorsBfd  I[ !x䍑7~Px䍑(~ QtakesanypGointofx䍑[>~B2\.(hZ^nq~) ԍintox䍑~Bq^0ϸ\(hZ^nq~)(Fig.4).LetusdenotebyB0anopGenballinR^n !suchthatf(h;Bq)ix䍑~B .P{Pullingbackby\qw~f Ia(h),onethusde nesthe\paralleltranspGort" hbn_120S.XV0, DindepGendentUUofh,suchthatforanyA2(h)8\B{dbjjfd!/P!Q]zMfd fp;!P:Q '(A)&jj0.If}fe@ZU iscompact,asweshallalwaysassume,thiscanbGedonewitha nitenumberofsuchballsB1|s;:::;B`,orderedinawaythatUUforeach1i<`,UUBi,\8Bi+1;g6=;. In`thefollowing,takehtobGelessthanminNiMhiTL.LetPWb2ӵ(h)\B0 ӲandQ2(h)\B`.There)isauniquesymbGolg[ٲ(h)de nedontheuniversalcoverx䍑~U ofUthat@:isananechartfor(h)andthatcoincideswithf(h)abGoveB0|s.ThenQ ԍcanUUbGeseenasthelift 8:P*2x䍑~U r.Thepointisnowthat@T|ŵg[ٲ(h;Q)=x䍑X~Q ʲ+8OG(h1x):F*or?anyP*2x䍑~U r,andforany UP21|s(U9),thereisauniqueP( 8)2GA(n;Z)suchthatQg[ٲ(h; 8:Pc)=h=P( )(g[ٲ(h;Pc)=h)8+OG(h1x):ByEde nition,wehaveP( 8 ^0\q)= n9:P }( 8^0)P( 8).EButonecanshowthatforanyloGopUU ㍲suchthat 8:P*=Q,thenj0Q;( 80\q)=P( 8)P( 0\q)P( )1 t:Therefore,UUP Disactuallyahomomorphism.PropGosition5justsaysthatP r=䍑 <1K[ cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5O line10u cmex10