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We use world-line methods for pseudo-supersymmetry to construct$sl(2|1)$-invariant actions for the $(2,2,0)$ chiral and ($1,2,1)$ realsupermultiplets of the twisted $D$-module representations of the $sl(2|1)$superalgebra. The derived one-dimensional topological conformal $\sigma$-modelsare invariant under nilpotent operators. The actions are constructed for bothparabolic and hyperbolic/trigonometric realizations (with extra potential termsin the latter case). The scaling dimension $\lambda$ of the supermultipletsdefines a coupling constant, $2\lambda+1$, the free theories being recovered at$\lambda=-\frac{1}{2}$. We also present, generalizing previous works, the$D$-module representations of one-dimensional superconformal algebras inducedby ${\cal N}=(p,q)$ pseudo-supersymmetry acting on $(k,n,n-k)$ supermultiplets.Besides $sl(2|1)$, we obtain the superalgebras $A(1,1)$, $D(2,1;\alpha)$,$D(3,1)$, $D(4,1)$, $A(2,1)$ from $(p,q)= (1,1), (2,2), (3,3), (4,4), (5,1)$,at given $k,n$ and critical values of $\lambda$.

A world-line framework for 1D Topological Conformal sigma-models