Abstract
We study complex integrable systems on quiver varieties associated with the cyclic Noquiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero–Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type A n−1 and B n , as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.
Highlights
The integrable n-particle systems of Toda [38], Calogero-Moser [7, 30], and Ruijsenaars-Schneider [35] have a remarkable tendency to maintain many of their interesting properties when being extended in various ways1
We focused on establishing superintegrability of complex generalisations of the rational CM system associated with cyclic quivers
These various systems are allowed to admit different types of spin variables or a harmonic oscillator potential term, which are completely determined by the underlying quivers
Summary
The integrable n-particle systems of Toda [38], Calogero-Moser [7, 30], and Ruijsenaars-Schneider [35] have a remarkable tendency to maintain many of their interesting properties when being extended in various ways. This paper reinforces the above-mentioned phenomenon by proving the superintegrability of (spin) Calogero-Moser type systems attached to cyclic quivers. A superintegrable Hamiltonian system with N degrees of freedom, that is a 2N -dimensional symplectic manifold (M, ω) with a smooth function H ∈ C∞(M ) of special importance, has 2N − 1 globally defined, independent constants of motion. Such systems are usually referred to as maximally superintegrable in the literature [43].
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