An Introduction to Dirac Operators on Manifolds

Dirac operators play an important role in several domains of
mathematics and physics, for example: index theory, elliptic
pseudodifferential operators, electromagnetism, particle physics, and
the representation theory of Lie groups.
In this essentially self-contained work, the basic ideas underlying
the concept of Dirac operators are explored. Starting with Clifford
algebras and the fundamentals of differential geometry, the text
focuses on two main properties, namely, conformal invariance, which
determines the local behavior of the operator, and the unique
continuation property dominating its global behavior. Spin groups and
spinor bundles are covered, as well as the relations with their
classical counterparts, orthogonal groups and Clifford bundles.
The chapters on Clifford algebras and the fundamentals of
differential geometry can be used as an introduction to the above
topics, and are suitable for senior undergraduate and graduate
students. The other chapters are also accessible at this level so that
this text requires very little previous knowledge of the domains
covered. The reader will benefit, however, from some knowledge of
complex analysis, which gives the simplest 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.