Symplectic reduction at zero angular momentum

Joshua  Cape; Hans-Christian  Herbig; Christopher  Seaton

doi:10.3934/jgm.2016.8.13

Article Contents

2016, Volume 8, Issue 1: 13-34. Doi: 10.3934/jgm.2016.8.13

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Symplectic reduction at zero angular momentum

1.
Department of Applied Mathematics and Statistics, Johns Hopkins University, 3400 N. Charles St, Baltimore, MD 21218
2.
Departamento de Matemática Aplicada, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C, CEP: 21941-909 - Rio de Janeiro
3.
Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112

Received: March 31, 2015

Revised: July 31, 2015

Published: February 2016

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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 $O_n$ on $k$ copies of $T^\ast\mathbb{R}^n$ 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 $O_n$- and $SO_n$-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.

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Mathematics Subject Classification: Primary: 53D20, 13A50; Secondary: 57S15, 37J15, 20G20.

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