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. For the reason that we want to include the case that (M, σ) itself is a reduced phase space, which usually is an orbifold rather than a smooth manifold, we actually will apply the orbifold version of the fixed point formula of Chapter 14. The detailed results are presented in Section 15.5, which the reader may consult first in order to get a quick impression.