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