Conformal bridge in a cosmic string background

TL;DR

This paper explores hidden symmetries of non-relativistic systems in a cosmic string background using the conformal bridge transformation, revealing classical integrals and quantum anomalies related to the geometrical parameter.

Contribution

It demonstrates the relation between free particles and harmonic oscillators on a cosmic string background via the conformal bridge, identifying conditions for well-defined symmetries and uncovering quantum anomalies.

Findings

Classical integrals exist only at specific rational values of the geometrical parameter.

Quantum hidden symmetry operators are well-defined only when the parameter is an integer.

A quantum anomaly is revealed for non-integer values of the geometrical parameter.

Abstract

Hidden symmetries of non-relativistic $\mathfrak{so} (2,1)\cong \mathfrak{sl}(2, {\mathbb R})$ invariant systems in a cosmic string background are studied using the conformal bridge transformation. Geometric properties of this background are analogous to those of a conical surface with a deficiency/excess angle encoded in the "geometrical parameter" $\alpha$, determined by the linear positive/negative mass density of the string. The free particle and the harmonic oscillator on this background are shown to be related by the conformal bridge transformation. To identify the integrals of the free system, we employ a local canonical transformation that relates the model with its planar version. The conformal bridge transformation is then used to map the obtained integrals to those of the harmonic oscillator on the cone. Well-defined classical integrals in both models exist only at $\alpha=q/k$ with $q,k=1,2,\ldots,$ which for $q>1$ are higher-order generators of finite nonlinear algebras. The systems are quantized for arbitrary values of $\alpha$; however, the well-defined hidden symmetry operators associated with spectral degeneracies only exist when $\alpha$ is an integer, that reveals a quantum anomaly.