geometry of 2d topological field theories

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1、GEOMETRYOF2DTOPOLOGICALFIELDTHEORIESBorisDUBROVINSISSA,TriestePreprintSISSA-89/94/FMAbstract.Theselecturenotesaredevotedtothetheoryofequationsofassociativitydescribinggeometryofmodulispacesof2Dtopologicaleldtheories.arXiv:hep-th/9407018v105Jul19941Conten

2、ts.Introduction.Lecture1.WDVVequationsandFrobeniusmanifolds.AppendixA.PolynomialsolutionsofWDVV.AppendixB.SymmetriiesofWDVV.TwistedFrobeniusmanifolds.AppendixC.WDVVandChazyequation.Aneconnectionsoncurveswithprojectivestructure.Lecture2.Topologicalconform

3、aleldtheoriesandtheirmoduli.Lecture3.SpacesofisomonodromydeformationsasFrobeniusmanifolds.AppendixD.Geometryof atpencilsofmetrics.AppendixE.WDVVandPainleve-VI.AppendixF.Branchingofsolutionsoftheequationsofisomonodromicdeformationsandbraidgroup.AppendixG

4、.MonodromygroupofaFrobeniusmanifold.AppendixH.GeneralizedhypergeometricequationassociatedtoaFrobeniusmanifoldanditsmon-odromy.AppendixI.DeterminationofasuperpotentialofaFrobeniusmanifold.Lecture4.FrobeniusstructureonthespaceoforbitsofaCoxetergroup.Appendi

5、xJ.ExtendedcomplexcrystallographicgroupsandtwistedFrobeniusmanifolds.Lecture5.DierentialgeometryofHurwitzspaces.Lecture6.Frobeniusmanifoldsandintegrablehierarchies.Couplingtotopologicalgravity.References.2Introduction.IntheselecturenotesIconsideroneremar

6、kablesystemofdierentialequationsthatappearedinthepapersofphysisistsontwo-dimensionaltopologicaleldtheory(TFT)inthebeginningof'90[148,39].Roughlyspeaking,theproblemistondaquasihomogeneousfunctionF=F(t)ofthevariablest=(t1;:::;tn)suchthatthethirdderivativ

7、esofit@3F(t)c(t):=@t@t@tforanytarestructureconstantsofanassociativealgebraAtwithat-independentunity(thealgebrawillbeautomaticallycommutative)(seeLecture1forthepreciseformulationoftheproblem).ForthefunctionF(t)oneobtainsaverycomplicatedoverdeterminedsystem

8、ofPDEs.IcallitWDVVequations.InthephysicalsettingthesolutionsofWDVVdescribemodulispaceoftopologicalconformaleldtheories.Oneoftheprojectsoftheselecturesistotrytoreconstructthebuildingof2DTFTonethebaseofWDVVequation.F