Four types of (super)conformal mechanics: D-module reps and invariant actions

Resumo

(Super)conformal mechanics in one dimension is induced by parabolic or hyperbolic/trigonometric
transformations, either homogeneous (for a scaling dimension $\lambda$) or inhomogeneous (at $\lambda=0$, with $\rho$ an inhomogeneity parameter).  Four types of inequivalent
(super)conformal actions are thus obtained. With the exclusion of the homogeneous parabolic case, dimensional constants are present. \par
Both the inhomogeneity and the insertion of $\lambda$ generalize the construction of Papadopoulos [CQG 30 (2013) 075018; arXiv:1210.1719].\par
Inhomogeneous $D$-module reps are presented for the $d=1$ superconformal algebras $osp(1|2)$,
$sl(2|1)$, $B(1,1)$ and $A(1,1)$. For centerless superVirasoro algebras $D$-module reps are presented
(in the homogeneous case for ${\cal N}=1,2,3,4$; in the inhomogeneous case for ${\cal N}=1,2,3$).\par
The four types of $d=1$ superconformal actions are derived for ${\cal N}=1,2,4$ systems.  When ${\cal N}=4$, the homogeneously-induced  actions are $D(2,1;\alpha)$-invariant ($\alpha$ is critically linked to $\lambda$); the inhomogeneously-induced actions are $A(1,1)$-invariant.\par
In $d=2$, for a single bosonic field, the homogeneous transformations induce a conformally invariant power-law action, while the inhomogeneous transformations induce the conformally invariant Liouville action.