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 λ=0, the D-module representations of the A(1,1) superalgebra are obtained. For $λ\neq 0$, the D-module representations of the D(2,1;α) superalgebras are obtained, with $α$ determined in terms of the scaling dimension $λ$ according to: $α=-2λ$ for k=4, i.e. the (4,4) supermultiplet, $α=-λ$ for k=3, i.e. (3,4,1), and $α=λ$ for k=1, i.e. (1,4,3). For $λ\neq 0$ 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 $λ=1/4$. 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).

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