An Integration Formula for the Moment Maps of Circle Actions

TL;DR

This paper derives explicit formulas for integrating the exponential of the square of the moment map in circle actions, using stationary phase and Duistermaat-Heckman methods, and computes related cohomological pairings.

Contribution

It introduces a new integration formula for the moment map of circle actions and compares two different computational approaches.

Findings

Derived explicit formulas for the integral using stationary phase and Duistermaat-Heckman methods.

Computed cohomological pairings on symplectic quotients explicitly.

Analyzed the asymptotic behavior of the formulas to understand volume and polynomial properties.

Abstract

The integration of the exponential of the square of the moment map of the circle action is studied by a direct stationary phase computation and by applying the Duistermaat-Heckman formula. Both methods yield two distinct formulas expressing the integral in terms of contributions from the critical set of the square of the moment map. The cohomological pairings on the symplectic quotient, including its volume (which was known to be a piecewise polynomial), are computed explicitly using the asymptotic behavior of the two formulas.