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1、THESELF-DUALITYEQUATIONSONARIEMANNSURFACEN.J.HITCHIN[Received15September1986]IntroductionInthispaperweshallstudyaspecialclassofsolutionsoftheself-dualYang-Millsequations.Theoriginalself-dualityequationswhicharoseinmathematicalphysicsweredefinedonEuclidean4-space.Thephysi
2、callyrelevantsolutionsweretheoneswithfiniteaction—theso-called'instantons'.ThesameequationsmaybedimensionallyreducedtoEuclidean3-spacebyimposinginvarianceundertranslationinonedirection.Theseequationsalsohavephysicalrelevance—thesolutionswhichhavefiniteactioninthreedimens
3、ionsarethe'magneticmonopoles'.Ifwetakethereductionprocessonestepfurtherandconsidersolutionswhichareinvariantundertwotranslations,weobtainasetofequationsintheplane.Here,however,thereisnoclearphysicalmeaningand,indeed,attemptstofindfiniteactionsolutionshavefailed.Neverthel
4、ess,thesearetheequationsweshallconsider.2DespitethelackofinterestingsolutionsinU,theequationshavetheimportantproperty—conformalinvariance—whichallowsthemtobedefinedon2manifoldsmodelledonUbyconformalmaps,namelyRiemannsurfaces.Weshallconsiderheresolutionsoftheself-dualitye
5、quationsdefinedonacompactRiemannsurface.Thereareinfactsolutions,asweshallshow,andthemodulispaceofallsolutionsturnsouttobeamanifoldwithanextremelyrichgeometricstructurewhichwillbethefocusofourstudy.ItbringstogetherinaharmoniouswaythesubjectsofRiemanniangeometry,topology,a
6、lgebraicgeometry,andsymplecticgeometry.Illuminatingallthesefacetsofthesameobjectaccountsforthelengthofthispaper.Theself-dualityequationsareequationsfromgaugetheory;geometricallytheyaredefinedintermsofconnectionsonprincipalbundles.Whilethegroupoftheprincipalbundlemaybecho
7、senarbitrarilyfortheequationstomakesense,werestrictattentionheretothesimplestcaseofSU(2)orSO(3).Therearetworeasonsforthis.Thefirst,andmostobvious,isthatitsimplifiescalculationsandavoidstheuseofinductiveprocesseswhichareinherentintheconsiderationofageneralLiegroupofhigher
8、rank.ThesecondreasonisthatsolutionsforSU(2)haveanintimaterelationshipwiththeinternalstructureoftheRiema
