From worldline to quantum superconformal mechanics with/without oscillatorial terms: $D(2,1;\alpha)$ and $sl(2|1)$ models

In this paper we quantize superconformal $\sigma$-models defined by worldline supermultiplets. Two types of superconformal mechanics, with and without a DFF term, are considered. Without a DFF term (Calogero potential only) the supersymmetry is unbroken. The models with a DFF term correspond to deformed (if the Calogero potential is present) or undeformed oscillators. For these (un)deformed oscillators the classical invariant superconformal algebra acts as a spectrum-generating algebra of the quantum theory. Besides the $osp(1|2)$ examples, we explicitly quantize the superconformally-invariant worldine $\sigma$-models defined by the ${\cal N}=4$ $(1,4,3)$ supermultiplet (with $D(2,1;\alpha)$ invariance, for $\alpha\neq 0,-1$) and by the ${\cal N}=2$ $(2,2,0)$ supermultiplet (with two-dimensional target and $sl(2|1)$ invariance). The parameter $\alpha$ is the scaling dimension of the $(1,4,3)$ supermultiplet and, in the DFF case, has a direct interpretation as a vacuum energy. In the DFF case, for the $sl(2|1)$ models, the scaling dimension $\lambda$ is quantized (either $\lambda=\frac{1}{2}+{\mathbb Z}$ or $\lambda={\mathbb Z}$). The ordinary two-dimensional oscillator is recovered, after imposing a superselection restriction, from the $\lambda=-\frac{1}{2}$ model. In particular a single bosonic vacuum is selected. The spectrum of the unrestricted two-dimensional theory is decomposed into an infinite set of lowest weight representations of $sl(2|1)$. Extra fermionic raising operators, not belonging to the original $sl(2|1)$ superalgebra, allow (for $\lambda=\frac{1}{2}+{\mathbb Z}$) to construct the whole spectrum from the two degenerate (one bosonic and one fermionic) vacua.