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 E ⊗ L. 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.
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© 2011 Springer Science+Business Media, LLC
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Duistermaat, J.J. (2011). Application to Symplectic Geometry. In: The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator. Modern Birkhäuser Classics. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8247-7_15
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DOI: https://doi.org/10.1007/978-0-8176-8247-7_15
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-8246-0
Online ISBN: 978-0-8176-8247-7
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