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

TL;DR

This paper quantizes superconformal worldline sigma-models with and without DFF oscillatorial terms, revealing their spectrum structure, superalgebra actions, and parameter quantizations, especially for models with $D(2,1; heta)$ and $sl(2|1)$ symmetry.

Contribution

It provides explicit quantization of superconformal sigma-models with novel analysis of their spectra and symmetry representations, including parameter quantization and vacuum structure.

Findings

Supersymmetry remains unbroken without DFF terms.

Models with DFF terms correspond to deformed or undeformed oscillators.

Spectrum decomposes into lowest weight representations of superalgebras.

Abstract

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.