Lecture Notes on Quantum Teichmuller Theory

arXiv:0710.2051v1[math.AG]10Oct2007LECTURENOTESONQUANTUMTEICHM¨ULLERTHEORY.LEONIDCHEKHOV.NotescollectedbyM.MazzoccoAbstract.ThesenotesarebasedonalecturecoursebyL.ChekhovheldattheUniversityofManchesterinMay2006andFebruary-March2007.Theyaredivulgativeincharacter,andinsteadofcontainingrigorousmathematicalproofs,theyillustratestatementsgivinganintuitiveinsight.Weintentionallyremovemostbibliographicreferencesfromthebodyofthetextdevotingaspecialsectiontothehistoryofthesubjectattheend.Contents1.CombinatorialdescriptionofTeichm¨ullerspaces12.CoordinatesintheTeichm¨ullerspace83.Modulargroupaction134.Poissonbrackets174.1.Relationsbetweengeodesicfunctions185.Quantization206.Bibliographicreferences23AppendixA.Thehyperbolicmetric24AppendixB.Poissonbracketsbetweengeodesicfunctions25References271.CombinatorialdescriptionofTeichm¨ullerspacesInthislectureweconcentrateontheTeichm¨ullerspaceTsgofRiemannsurfacesFg,sofgenusgwithsholes.Example1.1.Thesimplestcaseisthetoruswithonehole,F1,1(seefigure1).Figure1.Atoruswithonehole12LEONIDCHEKHOV.Theholescanbeconsideredindifferentparameterizations.ForthislecturecourseweconcentrateonthePoincar´euniformization.Thereisanotheruniformiza-tionwhichisduetoStrebel,butinthePoincar´eonethereisagooddescriptionofthePoissonstructureandofthequantization.WhatisthePoincar´eunformization?OneachFg,swemaydefineseveralmetrics.Weconsidertwometricstobeequivalentiftheyaremappedonetoanotherbyadiffeomorphism.Withineachequivalenceclass,wechosetherepresentativetobeametricwithlocalconstantcurvature−1.Observethatforg1therealwaysexistssuchrepresentativeelement(i.e.wemaypicks=0).Forg=1weneedatleastonehole,otherwiseitisimpossibleastheuniversalcoverofthetorusisflat.Forg=0,weneedatleasts=3.ThisiscalledPoincar´euniformization:ourRiemannsurfaceFr,sismappedtoasurfacewithlocalconstantcurvature−1,namelyFg,s=H/Δg,s,whereHdenotestheupperhalfplaneandΔg,sisaFuchsiangroup,i.e.afinitelygenerateddiscretesubgroupoftheisometrygroupPSL(2,R)ofH:Δg,s=hγ1,...,γ2g+si,whereγ1,...,γ2g+sarehyperbolicelements,i.e.theyhavetwodistinctfixedpointsontheabsolute,i.e.R∪{∞}.RecallthatonHwehavethemetricds2=dx2+dy2y2andthegeodesicsareeithersemi–circleswithcentreonthex–axisorhalf–linesparalleltothey–axis(seefigure2).Tofamiliarizethereaderwiththehyperbolicmetric,wehavediscussedsomeexamplesinAppendixA.Figure2.SomegeodesicsintheupperhalfplaneObservethatgivenageodesicγandapointPinHnotbelongingtoit,thereareinfinitelymanygeodesicsthroughPwhichdonotcrossγ(thereforeEuclid’spostulatedoesnotholdtrueinhyperbolicgeometry).Thex–axisisinfinitelydistantfromthepointsinH,thisiswhywesaythatitbelongstotheabsolute.Notethatactually,no“points”lieontheabsolute;insteadweconsiderclassesofgeodesicsterminatingatsuchapointinasensethatallofthembecomeasymp-toticallycloseatlargeproperdistances:thecollectionofsuch“points”istheopenreallinetogetherwiththeinfinitypoint(definedbyupperendsofverticalgeodesiclines:alltheselinesinFig.2areasymptoticallyclose).TheequivalentandoftenusedpictureisthePoincar´edisc:theabsoluteistheboundaryofthediskandthegeodesicsarearcs.QUANTUMTEICHM¨ULLERTHEORY3LetA∈PSL(2,R)withA=abcd.ThisactsonHbytheM¨obiustransfor-mationγA:z7→az+bcz+d.Thisactionistransitive,andtheM¨obiustransformationhastwofixedpoints(pos-siblyatinfinity),givenbyz±=12ca−d±pT2−4,whereT=trace(A).Ifonefixedpointisat∞(i.e.ifc=0)thentheotherisatb/(d−a)∈R.NoticethatTisnotwell-definedonPSL(2,R)butT2is.Anal-ogously,theratioa−dciswelldefinedinPSL(2,R),sothefixedpointsareindeeduniquelydeterminedbytheelementinPSL(2,R).Twonon–identityelementsinPSL(2,R)commuteifandonlyiftheyhavethesamefixedpoints.TheelementA∈PSL(2,R)issaidtobehyperbolicifT24,ellipticifT24andparabolicifT2=4.Parabolicelementshaveauniquefixedpointontheabsolute,andareconjugateto1101.Wewillalmostexclusivelybeinterestedinhyperbolicelements.Thefixedpointsofsuchelementsarereal(lieontheabsolute).SincetheM¨obiustransformationγAisahyperbolicisometry,itfollowsthatitpreservestheuniquegeodesicbetweenitstwofixedpoints.Wecallsuchgeodesicinvariantaxis.TheeigenvaluesofthematrixAaregivenbyλ±=12T±pT2−4.ThelinearizationoftheM¨obiustransformationγA,atthefixedpointz±isthecomplexlinearmapwitheigenvalueλ2±.Givenanyelementγ∈PSL(2,R),γ=abcd,ad−bc=1,wecanuniquelydetermineitbyitseigenvaluesanditseigenvectors.Wearenowgoingtocharac-terizeγbytwootherobjects:aclosedgeodesicanditslength.Infact,sincethedeterminantofγisone,theeigenvaluescanbeexpressedasexp±lγ2.Inthebasisoftheeigenvectors,γ(z)=exp(lγ)z,soγisadilationandbecauseitmustbeanisometry,itmapsthewholegeodesicthoughztoasecondgeodesic(seefigure3).Figure3.4LEONIDCHEKHOV.Ourdiagonalizedelementγhastwodistinctfixedpointswhichare0and∞.Generallyifγisnotdiagonal,thepositionofthefixedpointsisuniquelydeterminedbytheeigenvectors.Theyalwayslieontheabsolute.Example1.2.Considerγ=22121,ithaseigenvaluesλ±=123±√5
andeigenvectorsv±=(1∓√5,1).Sothefixedpointslieatz=1∓√5.Theothergeodesicsaremappedasinfigure4.;Figure4.IfweidentifypointsonHbytheactionofγ,itmeansthatwehavetoidentifytheinitialgeodesicanditsimageunderγ,sothatweobtainaninfinitehyperboloid.Le