preparation for gauge theory

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1、PreparationforGaugeTheoryGeorgeSvetlichny1February19,1999arXiv:math-ph/9902027v312Mar19991DepartamentodeMatematica,PontifciaUniversidadeCatolica,RuaMarqu^esdeS~aoVicente225,22453-900Gavea,RiodeJaneiro,RJ,Brazile-mail:svetlich@mat.puc-rio.brContents1Introduction12Preliminaries,Notation,Conv

2、entions23Groups33.1GroupAction...........................33.2LieGroupsandLieGroupActions...............63.3GroupandLieAlgebraRepresentations.............83.4AneActions...........................104Bundles104.1FiberBundles...........................104.2G-bundles.............................184

3、.3StructuredFibers.........................204.4PrincipalBundles.........................214.5BundleOperations........................254.6Connections............................264.7GaugeTransformations......................324.8ParallelTransport.........................344.9Curvature.........

4、....................404.10CovariantDerivatives.......................435Manifolds495.1Pseudo-RiemannianManifolds..................495.2TheLevi-CivitaConnection...................556LagrangianTheories576.1Lagrangians............................576.2MinimalCoupling.........................637Elec

5、tromagnetism657.1Maxwell'sEquations.......................657.2Dirac'sMagneticMonopole...................678Spin698.1CliordAlgebras.........................698.2SpinGroups............................76i8.3SpinBundles...........................798.4TheDiracOperator........................848.5T

6、heSeiberg-WittenEquations..................89ABasicConventions90BParameterizedMaps91CVector-valuedDierentialForms92ii1IntroductionThesearetheclassnotesforacoursepreparatorytoclassicalgaugethe-orygivenattheMathematicsDepartmentofthePontifciaUniversidadeCatolicaofRiodeJaneiroduringthe(souther

7、n-hemisphere)springtermof1997.Theypurporttoprovidethenecessarymathematicalbackgroundatabeginninggraduatelevelforsomeoneinterestedingaugetheoryandwhohasanelementaryunderstandingofdierentiablemanifolds,Liegroups,Lieal-gebras,andmul