Resonance Gyrons and Quantum Geometry

We describe irreducible representations, coherent states and star-products for algebras of integrals of motions (symmetries) of two-dimensional resonance oscillators. We demonstrate how the quantum geometry (quantum K"ahler form, metric, quantum Ricci form, quantum reproducing measure) arises in this problem. We specifically study the distinction between the isotropic resonance $1:1$ and the general $l:m$ resonance for arbitrary coprime $l,m$. Quantum gyron is a dynamical system in the resonance algebra. We derive its Hamiltonian in irreducible representations and calculate the semiclassical asymptotics of the gyron spectrum via the quantum geometrical objects.