# A three-dimensional superconformal quantum mechanics with $sl(2|1)$ dynamical symmetry

## TL;DR

This paper constructs a three-dimensional superconformal quantum mechanics model with $sl(2|1)$ symmetry, analyzing its spectrum, eigenstates, and dimensional reductions, revealing complex energy patterns and connections to lower-dimensional oscillators.

## Contribution

It introduces a novel 3D superconformal quantum mechanics with $sl(2|1)$ symmetry, including a detailed spectral analysis and dimensional reduction to 2D and 1D models.

## Key findings

- Spectrum exhibits recursive zigzag vacuum energy pattern.
- Degeneracy grows linearly up to $E \,\sim\, \beta$ and quadratically thereafter.
- Eigenstates are expressed via Laguerre polynomials and spin spherical harmonics.

## Abstract

We construct a three-dimensional superconformal quantum mechanics (and its associated de Alfaro-Fubini-Furlan deformed oscillator) possessing an $sl(2|1)$ dynamical symmetry. At a coupling parameter $\beta\neq 0$ the Hamiltonian contains a $\frac{1}{r^2}$ potential and a spin-orbit (hence, a first-order differential operator) interacting term. At $\beta=0$ four copies of undeformed three-dimensional oscillators are recovered. The Hamiltonian gets diagonalized in each sector of total $j$ and orbital $l$ angular momentum (the spin of the system is $\frac{1}{2}$). The Hilbert space of the deformed oscillator is given by a direct sum of $sl(2|1)$ lowest weight representations. The selection of the admissible Hilbert spaces at given values of the coupling constant $\beta$ is discussed. The spectrum of the model is computed. The vacuum energy (as a function of $\beta$) consists of a recursive zigzag pattern. The degeneracy of the energy eigenvalues grows linearly up to $E\sim \beta$ (in proper units) and quadratically for $E>\beta$. The orthonormal energy eigenstates are expressed in terms of the associated Laguerre polynomials and the spin spherical harmonics. The dimensional reduction of the model to $d=2$ produces two copies (for $\beta$ and $-\beta$, respectively) of the two-dimensional $sl(2|1)$ deformed oscillator. The dimensional reduction to $d=1$ produces the one-dimensional $D(2,1;\alpha)$ deformed oscillator, with $\alpha$ determined by $\beta$.