Critical scaling dimension of D-module representations of N=4,7,8 Superconformal Algebras and constraints on Superconformal Mechanics

At critical values of the scaling dimension $\lambda$, supermultiplets of the global ${\cal N}$-Extended one-dimensional Supersymmetry algebra induce $D$-module representations of finite superconformal algebras (the latters being identified in terms of the global supermultiplet and its critical scaling dimension). For ${\cal N}=4,8$ and global supermultiplets $(k, {\cal N}, {\cal N}-k)$, the exceptional superalgebras $D(2,1;\alpha)$ are recovered for ${\cal N}=4$, with a relation between $\alpha$ and the scaling dimension given by $\alpha= (2-k)\lambda$. For ${\cal N}=8$ and $k\neq 4$ all four ${\cal N}=8$ finite superconformal algebras are recovered, at the critical values $\lambda_k = \frac{1}{k-4}$, with the following identifications: D(4,1) for $k=0,8$, F(4) for $k=1,7$, A(3,1) for $k=2,6$ and D(2,2) for $k=3,5$. The ${\cal N}=7$ global supermultiplet $(1,7,7,1)$ induces, at $\lambda= -1/4$, a $D$-module representation of the exceptional superalgebra G(3). $D$-module representations are applicable to the construction of superconformal mechanics in a Lagrangian setting. The isomorphism of the $D(2,1;\alpha)$ algebras under an $S_3$ group action on $\alpha$, coupled with the relation between $\alpha$ and the scaling dimension $\lambda$, induces non-trivial constraints on the admissible models of ${\cal N}=4$ superconformal mechanics. The existence of new superconformal models is pointed out. E.g., coupled $(1,4,3)$ and $(3,4,1)$ supermultiplets generate an ${\cal N}=4$ superconformal mechanics if $\lambda$ is related to the golden ratio. The relation between classical versus quantum $D$-module representations is presented.