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Abstract

The main goal of this chapter is Theorem 6.1, which says that the square of the spin-c Dirac operator D is equal to the Laplace operator plus a zero order term, given by curvature expressions. The contribution from the curvature of L will be responsible for the Chern characters ch (L j ) in Proposition 13.2. On the other hand, the term with one half of the curvature of K* leads, by combining the corresponding factors in (11.17) and (12.12) with the real determinants, to the complex determinants in Proposition 13.1 and Proposition 13.2, respectively. Theorem 6.1 is followed by a comparison of the spin-c Dirac operator with the spinor Dirac operator, which exists if Mis provided with a spin structure. We conclude this chapter with the description, in Proposition 6.1, of what happens with the formula for D2 in the Kähler case when D is equal to the Dolbeault-Dirac operator.

Keywords

Vector Bundle Line Bundle Dirac Operator Double Covering Principal Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • J. J. Duistermaat
    • 1
  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands

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