geometry of 2d topological field theories

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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.DierentialgeometryofHurwitzspaces.Lecture6.Frobeniusmanifoldsandintegrablehierarchies.Couplingtotopologicalgravity.References.2Introduction.Intheselectureno

6、tesIconsideroneremarkablesystemofdierentialequationsthatappearedinthepapersofphysisistsontwo-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