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

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 $\mathcal{N}=4$ $(1,4,3)$ supermultiplet (with $D(2,1;\alpha )$ invariance, for $\alpha \ne 0,-1$) and by the $\mathcal{N}=2$ $(2,2,0)$ 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.

31 pages; final version to appear in Phys. Rev. D

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• New aspects of the Z${}_{\text{2}}$ $\times$ Z${}_{\text{2}}$-graded 1D superspace: induced strings and 2D relativistic models
• Deformed supersymmetric quantum mechanics with spin variables
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• Higher Spin Symmetries and Deformed Schrödinger Algebra in Conformal Mechanics
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• Quantum SU$(2|1)$ supersymmetric ${\mathbb{C}}^N$ Smorodinsky--Winternitz system
• The octonionically-induced $\mathcal{N}=7$ exceptional $G(3)$ superconformal quantum mechanics