lectures on topological quantum field theory

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1、CERN-TH/97-250US-FT-30/97hep-th/9709192September1997LecturesonTopologicalQuantumFieldTheory1J.M.F.Labastidaa;bandCarlosLozanobaTheoryDivision,CERN,CH-1211Geneva23,Switzerland.bDepartamentodeFsicadePartculas,UniversidadedeSantiagodeCompostela,E-15706SantiagodeCompostela,Spain.Abstr

2、actIntheselectureswepresentageneralintroductiontotopologicalquantumeldtheories.ThesetheoriesarediscussedintheframeworkoftheMathai-Quillenfor-malismandinthecontextoftwistedN=2supersymmetrictheories.WediscussarXiv:hep-th/9709192v126Sep1997indetailtherecentdevelopmentsinDonaldson-Witten

3、theoryobtainedfromtheap-plicationofresultsbasedondualityforN=2supersymmetricYang-Millstheories.ThisinvolvesadescriptionofthecomputationofDonaldsoninvariantsintermsofSeiberg-Witteninvariants.GeneralizationsofDonaldson-Wittentheoryarereviewed,andthestructureofthevacuumexpectationvalueso

4、ftheirobservablesisanalyzedinthecontextofdualityforthesimplestcase.1LecturesgivenattheworkshopTrendsinTheoreticalPhysics",CERN-SantiagodeCompostela-LaPlataMeeting,heldinLaPlata,Argentina,April28-May61997.Contents1Introduction12TopologicalQuantumFieldTheory53TheMathai-Quillenformalism

5、123.1Finite-dimensionalcase..........................133.2Innite-dimensionalcase.........................203.2.1Supersymmetricquantummechanics..............213.2.2Topologicalsigmamodels....................254Donaldson-WittenTheory284.1TwistofN=2supersymmetry.....................304.2

6、Perturbativeapproach..........................334.3Non-perturbativeapproach........................344.3.1Donaldsoninvariants:abstractapproach............354.3.2Donaldsoninvariants:concreteapproach............385GeneralizedDonaldson-WittenTheory506Finalremarks561IntroductionTopologica

7、lquantumeldtheory(TQFT)emergedintheeightiesasanewrelationbetweenmathematicsandphysics.Thisrelationconnectedsomeofthemostadvancedideasinthetwoelds.Theninetieshavebeencharacterizedbyitsdevelopment,originatingunexpectedresultsintopologyandtestingsomeofthemostfundamentalideasinquantume

8、ldthe