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 CP^{n - 1} with the Fubini–Study symplectic structure. These systems are labeled by an integer p∈ { 1 , … , 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.