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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 $λ$, 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 $D(2,1;α)$ are recovered for ${\cal N}=4$, with a relation between $α$ and the scaling dimension given by $α= (2-k)λ$. For ${\cal N}=8$ and $k\neq 4$ all four ${\cal N}=8$ finite superconformal algebras are recovered, at the critical values $λ_k = \frac{1}{k-4}$, 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 $λ= -1/4$, 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 $D(2,1;α)$ algebras under an $S_3$ group action on $α$, coupled with the relation between $α$ and the scaling dimension $λ$, 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 $λ$ is related to the golden ratio. The relation between classical versus quantum $D$-module representations is presented.