Chapter 5 The Atiyah-Singer index theorem

Abstract

The basic facts concerning Clifford algebras and spin structures are reviewed in this chapter. The spectral theory of self-adjoint elliptic partial differential operators and the Hodge decomposition theorem are discussed. The classical elliptic complexes such as de Rham, signature, spin, spin C , Yang–Mills, and Dolbeault are defined in the chapter; these elliptic complexes are all of Dirac type. In addition, various characteristic classes for vector bundles are defined such as Chern forms, Pontrjagin forms, Chern character, Euler form, Hirzebruch L polynomial, and Todd polynomial. The characteristic classes for principal bundles are also described. The Atiyah–Singer index theorem is presented; the Chern–Gauss–Bonnet formula, the Hirzebruch signature formula, and the Riemann–Roch formula are special cases of the index theorem. The equivariant index theorem and the index theorem for manifolds with boundary are described. The Dolbeault complex is the holomorphic analogue of the de Rham complex.