Superconformal mechanics in SU(2|1) superspace

Using the worldline SU(2|1) superfield approach, we construct N=4 superconformally invariant actions for the d=1 multiplets (1, 4, 3) and (2, 4, 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 N=4 superconformal algebra D(2,1;$\alpha$) and its central-extended $\alpha$=0 and $\alpha$=-1 psu(1,1|2)$\oplus$su(2) descendants. We capitalize on the observation that D(2,1;$\alpha$) at $\alpha\neq$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 $\mu$, the parameter of contraction to the flat N=4, d=1 superspace. This closure property and its $\alpha$=0 analog suggest a simple criterion for the SU(2|1) invariant actions to be superconformal: they should be even functions of $\mu$. We find that the superconformal actions of the multiplet (2, 4, 2) exist only at $\alpha$=-1, 0 and are reduced to a sum of the free sigma-model type action and the conformal superpotential yielding, respectively, the oscillator potential $\sim \mu^2$ 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 non-zero central charge in the superalgebra su(1,1|2).