Equivariant Cohomology and the Localization Principle

Abstract

To help motivate some of the abstract and technical formalism which follows, we start by considering the evaluation of a rather simple integral. Consider the 2-sphere S ² of unit radius viewed in Euclidean 3-space ℝ³ as a sphere standing on end on the xy-plane and centered at z = a symmetrically about the z-axis. We introduce the usual spherical polar coordinates x = sin θ cos φ, y = sin θ sin φ and z = a - cos φ for the embedding of the sphere in 3-space as S ² = (x,y,z) ∈ ℝ³: x ² + y ² + (z - a)² = 1, where 0 ≤ θ < π and 0 ≤ φ ≤ 2π. The height of the sphere off of the xy-plane is given by the height function z in ℝ³ restricted to S ²,

$$ h_0 \left( {\theta ,\phi } \right) = a - \cos \theta $$

(2.1)

We want to evaluate the oscillatory integral

$$ Z_0 (T) = \int\limits_0^\pi {\int\limits_0^{2\pi } {d\theta {\text{ }}d\phi {\text{ }}\sin \theta {\text{ }}e^{iTh_0 (\theta ,\phi )} } } $$

(2.2)

which represents a ‘toy’ version of (1.1). The integration measure in (2.2) is the standard volume form on S ², i.e. that which is obtained by restriction of the measure dx dy dz of ℝ³ to the sphere, and T is some real-valued parameter. It is straightforward to carry out the integration in (2.2) to get

$$ \begin{gathered} Z_0 \left( T \right) = \begin{array}{*{20}c} {2\pi \int\limits_{ - 1}^{ + 1} {d\cos \theta } } & {e^{iT\left( {a - \cos \theta } \right)} } \\ \end{array} \hfill \\ = \frac{{2\pi i}} {T}\left( {e^{ - iT\left( {1 + a} \right)} - e^{iT\left( {1 - a} \right)} } \right) = \frac{{4\pi }} {T}e^{ - iTa} \sin T \hfill \\ \end{gathered} $$

(2.3)