An Introduction to Dirac Operators on Manifolds

Dirac operators play an important part in several domains of mathematics and mathematical physics, for example: index theory, theory of elliptic pseudodifferential operators, theory of electromagnetism, particle physics, and representation theory of Lie groups. In this work, the basic theories underlying the concept of Dirac operators are explored. Starting with the requisite material on Clifford algebras and differential geometry, the text focuses on the two main properties of Dirac operators: conformal invariance, which determines the local behavior of the operator, and the unique continuation property dominating global behavior. Spin groups and spin or bundles are covered, as well as the relations with their classical counterparts, orthogonal groups and Clifford bundles. The chapters on Clifford algebra and the fundamentals of differential geometry can be used as an introduction to the above topics, and are suitable for senior undergraduates and graduates. The other chapters are also accessible at this level. Thus, this self-contained book requires very little previous knowledge of the domains covered, although the reader will benefit from knowledge of complex analysis, which gives the basic example of a Dirac operator. More advanced readers - mathematical physicists, physicists and mathematicians from diverse areas - will appreciate the fresh approach to the theory, as well as the new results on boundary value theory.