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 CPn−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.