Clifford algebras, spin structures and dirac operators

Abstract

First, we present the algebraic basics: the theory of Clifford algebras, spinor groups and their representations. Next, we study spin- and Spinc -structures on Riemannian manifolds. The central concept of this chapter is that of a Dirac bundle endowed with a natural first order differential operator called the Dirac operator. We study the theory of Dirac operators in a systematic way, including the Hodge Decomposition Theorem, Weitzenboeck formulae and the classical elliptic complexes. The chapter culminates in a proof of the Atiyah-Singer Index Theorem via the heat kernel method and includes applications to the classical complexes.