Introduction to Index Theory

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1、UniversityofParis11IntroductiontotheIndexTheoryCoursegivenbyJ.M.Bismut1ContentsChapter1.Introduction11.Analyticaspectsofindextheory:Fredholmoperators12.Geometricaspectsofindextheory2Chapter2.Algebraicpreliminaries51.Liealgebras52.Z2-gradedalgebras53.Supertrace7Chapter3.Superconnections91.Co

2、nnectionsonvectorbundles92.ConnectionsonHermitianvectorbundles123.Superconnections134.Gaugetheory15Chapter4.Chern-Weiltheory171.Invariantpolynomials172.ClassicalChern-Weiltheory193.PontryaginclassandEulerclass234.Quillen'stheory245.Superconnectionsandheatequation276.Koszulcomplexes29Chapter

3、5.Cliordalgebras331.Cliordalgebras332.Antisymmetricendomorphismsandc2(V)373.Quaternionsandc(R2)394.Spinors415.Constructionsofspinors436.SupertraceontheCliordalgebra467.DeformationsoftheCliordalgebra478.Pin(n)andSpin(n)48Chapter6.SpinstructuresandtheDiracoperator531.Principalandassociat

4、edbundles532.Spinvectorbundles553.Spinorbundles574.TheDiracoperator595.TheLichnerowiczformula616.TheAtiyah-Singerindextheorem63Chapter7.Fredholmoperators651.BasicpropertiesofFredholmoperators652.Topologicalpropertiesoftheindex693.Sobolevspaces704.TheDiracoperatorisFredholm7134CONTENTSChapte

5、r8.TheindexoftheDiracoperator731.TheMcKean-Singerformula732.ComputingtheasymptoticofTrS[Pt(x;x)]743.AstudyofMt744.Mehler'sformulafortheharmonicoscillator74Chapter9.Applicationsofthelocalindextheorem751.ThedeRhamcomplex752.TheChern-Gauss-Bonnetformula763.Poincareduality774.TheHirzebruchsign

6、atureformula785.TheRiemann-Roch-Hirzebruchtheorem79Bibliography81CHAPTER1Introduction1.Analyticaspectsofindextheory:FredholmoperatorsLetH+andHbeHilbertspaces.Definition1.1.AcontinuouslinearoperatorD+:H+!HisaFredholmoperatorifthekernelkerD+andcokernelcokerD+(cokerD+=H=imD+)areofnitedime

7、nsion.LetusnotethattheimageofD+iseasilyseentobeclosed.TheindexofD+,denotedbyInd(D+),isdenedbyInd(D+)=dimkerD+dimcokerD+:Example1.2.LetusassumeH+andHarenitedimensional.LetD+bealinearoperatorfromH+toH.ThenD+isacontinuousFredholmoperatorsinceH+andHare