Quantum charges of harmonic oscillators

We discuss Riemannian geometry of one-dimensional quantum harmonic oscillator. Its wavefunction is a holomorphic section of the complex line bundle L_v over the phase space R². We show that the energy eigenfunctions ψₙ with n≥ 1, corresponding to the energy levels Eₙ, are complex coordinates on orbifolds R²/Zₙ embedded into L_v, where Zₙ is the cyclic group of order n. In fact, ψₙ (t,z) is a standing wave on R²/Zₙ, where z is a complex coordinate on the phase space R². Oscillators are characterized by two quantum charges (qₗ^, q_v)=(n,1), where qₗ^=n is the winding number for the group U(1) acting on R²/Zₙ and q_v^=1 is the winding number for the U(1)-rotations on fibres of the bundle L_v², and Eₙ=ω(qₗ^+1/2 q_v). We also discuss ``antioscillators" with opposite quantum charges and the same positive energy.