Double Coset Construction of Moduli Space of Holom

ESITheErwinSchrodingerInternationalPasteurgasse6/7InstituteforMathematicalPhysicsA-1090Wien,AustriaDoubleCosetConstructionofModuliSpaceofHolomorphicBundlesandHitchinSystemsA.LevinM.OlshanetskyVienna,PreprintESI336(1996)May15,1996SupportedbyFederalMinistryofScienceandResearch,AustriaAvailableviaanonymousftporgopherfromFTP.ESI.AC.ATorvia:urMathematik,Bonny{OnleavefromInternationalInstituteforNonlinearStudiesatLan-dauInst,Vorob’iovskoesch.2,Moscow,117940,Russiaz{OnleavefromITEP,Bol.Cheremushkinskaya,25,Moscow,117259,Russia1E-mailaddress:[e-mail]2E-mailaddress:[e-mail]12,1996AbstractWepresentadescriptionofthemodulispaceofholomorphicvectorbundlesoverRiemanncurvesasadoublecosetspacewhichisdierfromthestandardloopgroupconstruction.Ourapproachisbasedonequivalentdenitionsofholomorphicbundles,basedonthetransitionmapsorontherstorderdierentialoperators.UsingthisapproachwepresenttwoindependentderivationsoftheHitchinintegrablesystems.WedeneasuperfreeupstairssystemsfromwhichHitchinsystemsareobtainedbythreestephamiltonianreductions.AspecialattentionisbeinggivenontheSchottkyparameterizationofcurves.1IntroductionThemodulispaceofholomorphicvectorbundlesoverRiemannsurfacesarepopularsub-jectinalgebraicgeometryandnumbertheory.Inmathematicalphysicstheywereinvesti-gatedduetorelationswiththeYang-Millstheory[1]andtheWess-Zumino-Wittentheory[2,3].TheconformalblocksintheWZWtheorysatisfytheWardidentitieswhichtakeaformofdierentialequationsonthemodulispace[4,5].InthisapproachthemodulispaceisdescribedasadoublecosetspaceofaloopgroupdenedonasmallcircleonaRiemannsurface[6].ThemaingoalofthepaperisanalternativedescriptionofthemodulispaceandtheHitchinintegrablesystems[7]basedonthisconstruction.WestartwithaspecialgroupvaluedeldonaRiemannsurfacewhichisdenedasamapfromaholomorphicbasisinavectorbundletoaC1basis.ThiseldisananalogousofthetetradeeldintheGeneralRelativityandwecallittheGeneralizedTetradeField(GTF).TheholomorphicstructurescanbeextractedfromGTF.Theyaredescribedviatheholomorphictransitionmaps,orbymeansoftheoperatorsd00.TheformerareinvariantundertheactionoftheglobalC1transformations,whilethelaterundertheactionofthelocalholomorphictransformations.ItallowstodenethemodulispaceasadoublecosetspaceofGTFwithrespecttotheactionsofthelocalholomorphictransformationsandtheglobalC1transformations.WeintroduceacotangentbundletoGTFandinvariantsymplecticstructureonit.Thecotangentbundletothemoduliofholomorphicbundlescanbeobtainedbythesymplecticfactorizationsovertheactionoftwotypesofcommutinggaugetransformations.ThiscotangentbundleisaphasespaceoftheHitchinintegrablesystems[7].ThetetradeeldsintheirturnsaresectionsoftheprinciplebundleovertheRiemannsurface,whichsatisfysomeconstraintsequations.Weinterpretthemasmomentconstraintsinabigsuperfree1systemwithaspecialgaugesymmetry.Thisspaceisacotangentbundletotheprinciplebundle.ThustheHitchinsystemsareobtainedbythethreestepsymplecticreductionsfromthisspace.WeinvestigatespeciallyourreductionsintermsofSchottkyparameterization,whichisaparticularcaseofthegeneralconstruction.ThisparameterizationwasusedtoderivetheKnizhnik-Zamolodchikov-Bernardequationsonthehighergenuscurves[3,8,9].OntheotherhandthequantumsecondorderHitchinHamiltonianscoincidewiththemonthecriticallevel.2ModuliofholomorphicvectorbundlesLet=gbeanondegenerateRiemanncurveofgenusgwithg1.Wewillconsiderinthissectionasetofstableholomorphicstructuresoncomplexvectorbundlesover[1].Todenethemweproceedintwowaysbasedonthe^CechandtheDolbeaultcohomologies.Eventually,wecometothemodulispaceLofstableholomorphicbundlesovergandrepresentthemasadoublecosetspace(Proposition2.3).1.ConsideravectorbundleVoverg.TobemoreconcreteweassumethatthestructuregroupofVisGL(N;C).LetUa;a=1;:::beacoveringofgbyopensubsets.WeconsidertwobasesinVtheholomorphicfeholgbasisandthesmoothC1feC1gone.Inlocalcoordinates(za2Ua)ehola=ehol(za);eC1a=eC1(za;za):Lethbethetransitionmapbetweenthemha=h(za;za).ThenlocallyinUawehavehaeC1a=ehola:(2.1)Wecanconsiderhaasthesections0C1(Ua;P)oftheadjointbundleP=AutV.Wecalltheeldhageneralizedtetradeeld(GTF).ItfollowsfromthedenitionsofthebasesthatthereexistsaglobalsectionforeC1eC1a(za;za)=eC1b(zb(za);zb(za));za2Uab=Ua\Ub6=;;(2.2)wherezb=zb(za)areholomorphicfunctionsdeningacomplexstructureong.Ontheotherhandthetransformationsofeholareholomorphicmapsehola(za)=gab(za)eholb(zb(za));gba(zb)=g1ab(za(zb))(2.3)gab20hol(Uab;AutV);(@gab=0;@=@za):ThesematrixfunctionsdenetheholomorphicstructureinthevectorbundleV.WecandescribethesameholomorphicstructureworkingwiththesmoothbasiseC1inV.LetAa=h1a@ha:(2.4)ThenthebasiseC1isannihilatedbytheoperatord00AjUa=@+Aa(@+Aa)eC1a=0:2TheGTFtransformationshin(2.1)bynomeansfree.LetRbethesubsetofsectionsinPwhichsatisesthefollowingconditionsR=fh20C1(Ua;P)jh1a@hajUab=h1b@hbjUab;8Uab6=;;a;b=1;:::g;(2.5)(