Application to Symplectic Geometry

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.