ABSTRACT
This chapter gives the heart of the proof of the Index Theorem. We will study the so-called symbolic calculus for operators on bundles of Clifford modules. The idea is to provide a systematic way of investigating the ‘top order part' of an operator or a family of operators. For instance, our proof of the Weyl asymptotic formula 8.16 was based on our knowledge that the 'top order part’ of the heat kernel on a manifold is simply the heat kernel on Euclidean space. Getzler’s innovation was the introduction of a sophisticated notion of ‘order’, with respect to wrhich the index form — discussed at the end of the last chapter — naturally appears as a ‘top order part’.
