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,
We want to evaluate the oscillatory integral
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
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© 2000 Springer-Verlag Berlin Heidelberg
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(2000). Equivariant Cohomology and the Localization Principle. In: Equivariant Cohomology and Localization of Path Integrals. Lecture Notes in Physics Monographs, vol 63. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46550-2_2
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DOI: https://doi.org/10.1007/3-540-46550-2_2
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