Trigonometric and elliptic Ruijsenaars-Schneider systems on the complex projective space

TL;DR

This paper constructs compact forms of trigonometric and elliptic Ruijsenaars-Schneider systems on complex projective space, extending previous work by allowing a broader range of coupling parameters and using Hamiltonian reduction.

Contribution

It provides a direct construction of these systems with the phase space as complex projective space, generalizing earlier compactification approaches and including a wider parameter range.

Findings

Constructed compact real forms on $ ext{CP}^{n-1}$

Extended Ruijsenaars's compactification to larger parameter ranges

Built on Hamiltonian reduction methods

Abstract

We present a direct construction of compact real forms of the trigonometric and elliptic $n$-particle Ruijsenaars-Schneider systems whose completed center-of-mass phase space is the complex projective space $\mathbb{CP}^{n-1}$ with the Fubini-Study symplectic structure. These systems are labelled by an integer $p\in\{1,\dots,n-1\}$ relative prime to $n$ and a coupling parameter $y$ varying in a certain punctured interval around $p\pi/n$. Our work extends Ruijsenaars's pioneering study of compactifications that imposed the restriction $0<y<\pi/n$, and also builds on an earlier derivation of more general compact trigonometric systems by Hamiltonian reduction.