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

Resumo

In this paper we quantize superconformal $\sigma$-models defined by worldline supermultiplets. \par Two types of superconformal mechanics, with and without a DFF term, are considered. \par Without a DFF term (Calogero potential only) the supersymmetry is unbroken. \par 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.\par
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 from $\lambda=-\frac{1}{2}$. The spectrum of the theory is decomposed into an infinite set of lowest weight representations of $sl(2|1)$. Surprisingly, extra fermionic raising operators, not belonging to $sl(2|1)$, allow to construct the whole spectrum from a single
(for $\lambda=\frac{1}{2}+{\mathbb Z}$) bosonic vacuum.