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

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 $pπ/n$. Our work extends Ruijsenaars's pioneering study of compactifications that imposed the restriction $0<y<π/n$, and also builds on an earlier derivation of more general compact trigonometric systems by Hamiltonian reduction.

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