## 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.

## Keywords

Line Bundle Symplectic Form Symplectic Manifold Connection Form Momentum Mapping## Preview

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