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 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 N=4,8 and global supermultiplets (k,N,N−k), the exceptional superalgebras D(2, 1;α) are recovered for N=4, with a relation between α and the scaling dimension given by α= (2 −k)λ. For N=8 and k≠ 4 all four N=8 finite superconformal algebras are recovered, at the critical values λk=(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 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 S3 group action on α, coupled with the relation between α and the scaling dimension λ, induces non-trivial constraints on the admissible models of N=4 superconformal mechanics. The existence of new superconformal models is pointed out. For example, coupled (1, 4, 3) and (3, 4, 1) supermultiplets generate an N=4 superconformal mechanics if λ is related to the golden ratio. The relation between classical versus quantum D-module representations is presented.