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Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background

Non-relativistic conformally invariant systems in a rotating cosmic string (conical) spacetime are analyzed at the classical and quantum levels by means of the gravitoelectromagnetic interpretation of the background. Solutions of the equations of motion are found by employing a local canonical transformation, that leads to their natural interpretation in terms of Riemann surfaces. The cone parameter $α$ and the angular velocity $Ω$ of the background determine the existence of hidden symmetries. Globally defined higher order integrals associated with perihelion of geodesic orbits appear at rational values of $α$. For the harmonic oscillator system with frequency $ω$, the integrals responsible for the trajectory closure arise only for rational values of $α$ and $|γ|=|Ω/ω|$, with $|γ|=1$ corresponding to the Landau problem. We face a quantum anomaly problem since the hidden symmetry operators can only be constructed when $α$ is integer. Such operators are non-local in the case of the free particle. For the harmonic oscillator, the symmetry generators are obtained with the help of the conformal bridge transformation. We also study a multi-particle version of the harmonic oscillator system with $|γ|=1$ using the mean-field theory and find that the emerging vortex structure respects a singular point of the background.