Equivariant Index and the Moment Map for Completely Integrable Torus Actions

Abstract

Consider a completely integrable torus action on a compact Spin c manifold. The equivariant index of the Dirac operator is a virtual representation of the torus and is determined by the multiplicities of the weights which occur it in. We prove that these multiplicities are equal to values of the density function for the Duistermaat–Heckman measure, once this is defined appropriately. (The two-form that we take is half of the curvature of the line bundle which is associated to the Spin c structure. It is closed and invariant but not necessarily symplectic.) We deduce that these multiplicities are equal to the topological degree of the “descended moment map” Φ : M / T → t *, which in nice cases can be described as sums of certain winding numbers.