D-module Representations of N=2,4,8 Superconformal Algebras and Their Superconformal Mechanics

The linear (homogeneous and inhomogeneous) (k, N, N-k) supermultiplets of the N-extended one-dimensional Supersymmetry Algebra induce D-module representations for the N=2,4,8 superconformal algebras. For N=2, the D-module representations of the A(1,0) superalgebra are obtained. For N=4 and scaling dimension \lambda=0, the D-module representations of the A(1,1) superalgebra are obtained. For $\lambda\neq 0$, the D-module representations of the D(2,1;\alpha) superalgebras are obtained, with $\alpha$ determined in terms of the scaling dimension $\lambda$ according to: $\alpha=-2\lambda$ for k=4, i.e. the (4,4) supermultiplet, $\alpha=-\lambda$ for k=3, i.e. (3,4,1), and $\alpha=\lambda$ for k=1, i.e. (1,4,3). For $\lambda\neq 0$ the (2,4,2) supermultiplet induces a D-module representation for the centrally extended sl(2|2) superalgebra. For N=8, the (8,8) root supermultiplet induces a D-module representation of the D(4,1) superalgebra at the fixed value $\lambda=1/4$. A Lagrangian framework to construct one-dimensional, off-shell, superconformal invariant actions from single-particle and multi-particles D-module representations is discussed. It is applied to explicitly construct invariant actions for the homogeneous and inhomogeneous N=4 (1,4,3) D-module representations (in the last case for several interacting supermultiplets of different chirality).