Symplectic reduction at zero angular momentum

TL;DR

This paper investigates the symplectic reduction of a phase space with zero angular momentum, providing algebraic descriptions, symplectomorphism classifications, and analyzing singularities and Gorenstein properties of the resulting quotients.

Contribution

It offers a detailed algebraic and geometric analysis of symplectic quotients at zero angular momentum, including relations, symplectomorphisms, and singularity properties.

Findings

Describes the ideal of relations for the ring of regular functions.

Establishes graded symplectomorphisms among O_n and SO_n quotients.

Shows the zero fiber has rational singularities when n ≤ k.

Abstract

We study the symplectic reduction of the phase space describing $k$ particles in $\mathbb{R}^n$ with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of $\operatorname{O}_n$ on $k$ copies of $\mathbb{C}^n$ at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate $\mathbb{Z}^+$-graded regular symplectomorphisms among the $\operatorname{O}_n$- and $\operatorname{SO}_n$-symplectic quotients and determine which of these quotients are graded regularly symplectomorphic to linear symplectic orbifolds. We demonstrate that when $n \leq k$, the zero fibre of the moment map has rational singularities and hence is normal and Cohen-Macaulay. We also demonstrate that for small values of $k$, the ring of regular functions on the symplectic quotient is graded Gorenstein.