Superconformal mechanics inSU(2|1)superspace

Abstract

Using the worldline $SU(2|1)$ superfield approach, we construct $\mathcal{N}=4$ superconformally invariant actions for the $d=1$ multiplets ($\mathbf{1},\mathbf{4},\mathbf{3}$) and ($\mathbf{2},\mathbf{4},\mathbf{2}$). The $SU(2|1)$ superfield framework automatically implies the trigonometric realization of the superconformal symmetry and the harmonic oscillator term in the corresponding component actions. We deal with the general $\mathcal{N}=4$ superconformal algebra $D(2,1;\ensuremath{\alpha})$ and its central-extended $\ensuremath{\alpha}=0$ and $\ensuremath{\alpha}=\ensuremath{-}1$ $psu(1,1|2)\ensuremath{\bigoplus}su(2)$ descendants. We capitalize on the observation that $D(2,1;\ensuremath{\alpha})$ at $\ensuremath{\alpha}\ensuremath{\ne}0$ can be treated as a closure of its two $su(2|1)$ subalgebras, one of which defines the superisometry of the $SU(2|1)$ superspace, while the other is related to the first one through the reflection of $\ensuremath{\mu}$, the parameter of contraction to the flat $\mathcal{N}=4,d=1$ superspace. This closure property and its $\ensuremath{\alpha}=0$ analog suggest a simple criterion for the $SU(2|1)$ invariant actions to be superconformal: they should be even functions of $\ensuremath{\mu}$. We find that the superconformal actions of the multiplet ($\mathbf{2},\mathbf{4},\mathbf{2}$) exist only at $\ensuremath{\alpha}=\ensuremath{-}1,0$ and are reduced to a sum of the free sigma-model-type action and the conformal superpotential yielding, respectively, the oscillator potential $\ensuremath{\sim}{\ensuremath{\mu}}^{2}$ and the standard conformal inverse-square potential in the bosonic sector. The sigma-model action in this case can be constructed only on account of nonzero central charge in the superalgebra $su(1,1|2)$.