The Generalized Atiyah Singer Index Theorem And Noncommutative Residue

The present thesis is devoted to studying the generalization and proof of local Atiyah-Singer index theorem and the noncommutative residues. As a frontier research subject,Atiyah-Singer index theorem closely connected the two big branches of mathematics:analysis and topology, which are seemingly independent of each other. These theorems also connected to the differential geometry, partial differential equation, operator alge-bra, number theory et al. many research fields, which have important theoretical and application values.In 1961, Atiyah and Singer worked together to solve the Israel Gel'fand conjecture:the index of linear elliptic operators can be described by the topological invariants of manifolds. On the basis of these works they established Atiyah-Singer index theorem.Atiyah-Singer index theorem can be described as: the analytical index of any linear elliptic operator is equal to its topological index on compact manifolds.Since the Atiyah and Singer announced their index theorem in 1963 there has been occurred a variety of generalizations and proof methods of Atiyah-Singer index theorem,of which the heat kernel method got the attention of many scholars. Patodi for the first time using the heat kernel method proved the local version of the Gauss-Bonnet-Chern theorem. Several years later, Getzler and Bismut gave the pure analytic proof of the local index theorem by heat kernel method. This motivated many generalizations of local index theorems and emergence of several different heat kernel proofs, of which Ponge using Getzler rescaling and Greiner's approach of the heat kernel asymptotics gave the proof of local index theorem, which is the mainly used method in this thesis.Ponge and Wang using Getzler rescaling and Greiner 's approach of the heat kernel asymptotics proved the local equivariant index theorem. Lafferty, Yu and Zhang presented a simple and direct geometric proof of the Lefschetz fixed point formula for an orientation preserving isometry on an even dimensional spin manifold by Clifford asymptotics of heat kernel. Wang using Getzler rescaling and Greiner's approach of the heat kernel asymp-totics proved the family of local index theorem and family of local equivariant index the-orem. Before the results from Freed, Lefschetz theory all deal with orientation-preserving isometries for which there is no nontrivial Lefschetz formula in odd dimensions. Freed considered the case of an orientation reversing involution acting on an odd dimensional spin manifold and gave the associated Lefschetz formulas by the K-theoretical way. Wang constructed an even spectral triple by the Dirac operator and the orientation-reversing involution and computed the Chern-Connes character for this spectral triple. On the basis of Freed's work, Liu and Wang proved an equivariant odd index theorem for Dirac operators with involution parity for odd dimensional spin manifolds. Inspired by above proof methods and conclusions, Chapter 2 using Getzler rescaling and Greiner 's approach of the heat kernel asymptotics proved the family odd dimensional local index theorem and the family odd dimensional local equivariant index theorem of Dirac operator.Zhang introduced the Sub-Signature operators and proved a local index formula for these operators. Ma, Zhang and Dai et al. using Sub-Signature operators gave new proof of the Riemann-Roch-Grothendieck type formula. Chapter 3 using Getzler rescaling and Greiner's approach of the heat kernel asymptotics proved the local equivariant index theorem.In addition, As a very active research fields, noncommutative geometry have impor-tant influence on geometry?topology?number theory, as well as physics. Noncommuta-tive residues comes from the important research results from Adler and Wodzicki, and play an important role in noncommutative geometry. Connes made a challenging observation that the noncommutative residue of the square of the inverse of the Dirac operator was proportional to the Einstein-Hilbert action, which was called Kastler-Kalau-Walze theo-rem later. Kastler, Kalau and Walze gave independent proof of this theorem respectively.Recently, Ponge defined lower dimensional volumes of Riemannian manifolds by the Wodzicki residue. Fedosov et al. defined a noncommutative residue on a manifold with boundary by heat kernel expansion. Wang defined the lower dimensional volumes on the manifolds with boundary about Dirac operators and proved the associated Kastler-Kalau-Walze type theorem. Wang et al. computed Wres[(??D-2)2] for 7-dimensional manifolds with boundary,and proved a Kastler-Kalau-Walze type theorem. Ackermann and Tolksdorf proved a generalized version of the known Lichnerowicz formula for the square of the most general Dirac operator with torsion on an even dimensional spin manifolds associated to a metric connection with torsion. Wang et al. got a Kastler-Kalau-Walze type theorem associated with Dirac operators with torsion on compact Spin manifolds with boundary, and derived the gravitational action on the boundary. Chapter 4 establish a Kastler-Kalau-Walze type theorem associated with Dirac operators with torsion for 7-dimensional manifolds with boundary, and derived the gravitational action of the manifolds.The thesis is comprised of four chapters. Chapter 1 mainly recalls Greiner's ap-proach of the heat kernel asymptotics, orthogonal frame and normal coordinate, and some important concepts and conclusions about noncommutative residues on manifolds with boundary.Chapter 2 using Getzler rescaling and Greiner's approach of the heat kernel asymp-totics proved the family odd dimensional local index theorem and the family odd dimen-sional local equivariant index theorem of Dirac operators.Chapter 3 using Getzler rescaling and Greiner's approach of the heat kernel asymp-totics proved the local equivariant index theorem.Chapter 4 establish a Kastler-Kalau-Walze type theorem associated with Dirac op-erators with torsion for 7-dimensional Spin manifolds with boundary, and derived the gravitational action of the manifolds.