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1、GEOMETRYOF2DTOPOLOGICALFIELDTHEORIESBorisDUBROVINSISSA,TriestePreprintSISSA-89/94/FMAbstract.Theselecturenotesaredevotedtothetheoryofequationsofassociativitydescribinggeometryofmodulispacesof2Dtopological eldtheories.arXiv:hep-th/9407018v105Jul19941Co
2、ntents.Introduction.Lecture1.WDVVequationsandFrobeniusmanifolds.AppendixA.PolynomialsolutionsofWDVV.AppendixB.SymmetriiesofWDVV.TwistedFrobeniusmanifolds.AppendixC.WDVVandChazyequation.Aneconnectionsoncurveswithprojectivestructure.Lecture2.Topologica
3、lconformal eldtheoriesandtheirmoduli.Lecture3.SpacesofisomonodromydeformationsasFrobeniusmanifolds.AppendixD.Geometryof
atpencilsofmetrics.AppendixE.WDVVandPainleve-VI.AppendixF.Branchingofsolutionsoftheequationsofisomonodromicdeformationsandbraidgro
4、up.AppendixG.MonodromygroupofaFrobeniusmanifold.AppendixH.GeneralizedhypergeometricequationassociatedtoaFrobeniusmanifoldanditsmon-odromy.AppendixI.DeterminationofasuperpotentialofaFrobeniusmanifold.Lecture4.FrobeniusstructureonthespaceoforbitsofaCoxe
5、tergroup.AppendixJ.ExtendedcomplexcrystallographicgroupsandtwistedFrobeniusmanifolds.Lecture5.Di erentialgeometryofHurwitzspaces.Lecture6.Frobeniusmanifoldsandintegrablehierarchies.Couplingtotopologicalgravity.References.2Introduction.Intheselectureno
6、tesIconsideroneremarkablesystemofdi erentialequationsthatappearedinthepapersofphysisistsontwo-dimensionaltopological eldtheory(TFT)inthebeginningof'90[148,39].Roughlyspeaking,theproblemisto ndaquasihomogeneousfunctionF=F(t)ofthevariablest=(t1;:::;tn)s
7、uchthatthethirdderivativesofit@3F(t)c(t):=@t@t@tforanytarestructureconstantsofanassociativealgebraAtwithat-independentunity(thealgebrawillbeautomaticallycommutative)(seeLecture1forthepreciseformulationoftheproblem).ForthefunctionF(t)oneobtainsaverycom
8、plicatedoverdeterminedsystemofPDEs.IcallitWDVVequations.InthephysicalsettingthesolutionsofWDVVdescribemodulispaceoftopologicalconformal eldtheories.Oneoftheprojectsoftheselecturesistotrytoreconstructthebuildingof2DTFTonethebaseofWDVVequation.F
