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Clifford Algebras, Spin Structures and Dirac Operators

  • Gerd RudolphEmail author
  • Matthias Schmidt
Chapter
  • 3k Downloads
Part of the Theoretical and Mathematical Physics book series (TMP)

Abstract

First, we present the algebraic basics: the theory of Clifford algebras, spinor groups and their representations. Next, we study spin- and \(Spin^{c}\)-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.

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany

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