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Application to Symplectic Geometry

  • J. J. Duistermaat
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

In this chapter we will describe how, starting from a manifold M with a symplectic form σ, which satisfies an integrality condition such that it is the Chern form of a connection of a suitable complex line bundle L over M, one can get all the structures needed for the definition of a spin-c Dirac operator D, acting on sections of EL. Furthermore a Hamiltonian action of a torus T in (M, σ) can be lifted to L in such a way that it preserves all the previously introduced structures. If M is compact, then one obtains a fixed point formula for the virtual character of the representation in the finite-dimensional null space of D. The ingredients in this formula which come from the topology of the line bundle L, and the T-action on L at the fixed points of the T-action in M, are expressed in terms of the symplectic form σ and the momentum mup μ of the Hamiltonian action, respectively.

Keywords

Line Bundle Symplectic Form Symplectic Manifold Connection Form Momentum Mapping 
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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