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 λ k = 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 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 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 N = 4 superconformal mechanics if λ is related to the golden ratio.The relation between classical versus quantum D-module representations is presented.

Pure" homogeneous linear supermultiplets (minimal and non-minimal) of the N = 4-Extended one-dimensional Supersymmetry Algebra are classified. "Pure" means that they admit at least one graphical presentation (the corresponding graph/graphs are known as "Adinkras").We further prove the existence of "entangled" linear supermultiplets which do not admit a graphical presentation, by constructing an explicit example of an entangled N = 4 supermultiplet with field content (3,8,5). It interpolates between two inequivalent pure N = 4 supermultiplets with the same field content. The one-dimensional N = 4 sigma-model with a three-dimensional target based on the entangled supermultiplet is presented.The distinction between the notion of equivalence for pure supermultiplets and the notion of equivalence for their associated graphs (Adinkras) is discussed.Discrete properties such as "chirality" and "coloring" can discriminate different supermultiplets. The tools used in our classification include, among others, the notion of field content, connectivity symbol, commuting group, node choice group and so on.In this paper we present, for N = 4, the classification of the homogeneous (minimal and nonminimal) linear pure supermultiplets of the global N -extended one-dimensional Supersymmetry Algebra (the dynamical Lie superalgebra of the Supersymmetric Quantum Mechanics [1])A pure supermultiplet admits at least one graphical presentation (the corresponding graph is known as "Adinkra", [2]). We further prove the existence of the conjectured non-adinkrizable supermultiplets [3] (we prefer to call them here"entangled supermultiplets") which do not admit a graphical presentation, by explicitly introducing a supermultiplet, proved to be entangled, which interpolates between two pure supermultiplets (the interpolation is measured by an angle).We also construct a one-dimensional N = 4 supersymmetric σ-model with three target coordinates, defined in terms of the given entangled supermultiplet.This work is based on several advances, made in the course of the last decade, concerning the classification of homogeneous linear supermultiplets of the one-dimensional (1) N -extended superalgebra [4]- [19]. The classification of the N = 4 pure supermultiplets is made possible by combining different tools introduced in several works [11]-[18]. These tools (which include, among others, the notions of engineering dimension, length of a supermultiplet, mirror symmetry duality, connectivity symbol, node choice group, commuting group, possible chirality and/or coloring of the supermultiplets) are revised in an Appendix and further discussed in the text, when needed.A key issue concerns the distinction between the equivalence class of pure supermultiplets and the equivalence class of their associated graphs (we recall that fields are represented by vertices, while supersymmetry transformations are represented by colored edges, solid or dashed according to their sign, see the Appendix). Equivalent graphs are related by two types of moves: i) local moves, ba...

We construct the non-minimal linear representations of the N = 4 Extended Supersymmetry in one-dimension. They act on 8 bosonic and 8 fermionic fields. Inequivalent representations are specified by the mass-dimension of the fields and the connectivity of the associated graphs. The oxidation to minimal N = 5 linear representations is given. Two types of N = 4 σ-models based on non-minimal representations are obtained: the resulting off-shell actions are either manifestly invariant or depend on a constrained prepotential. The connectivity properties of the graphs play a decisive role in discriminating inequivalent actions. These results find application in partial breaking of supersymmetric theories.

We suggest a method to compute the correlation functions in conformal quantum mechanics (CFT 1 ) for the fields that transform under a non-local representation of sl(2) basing on the invariance properties. Explicit calculations of 2-and 3-point correlation functions are presented.

We couple dual pairs of N = 8 superconformal mechanics with conical targets of dimension d and 8−d. The superconformal coupling generates an oscillator-type potential on each of the two target factors, with a frequency depending on the respective dual coordinates. In the case of the inhomogeneous (3,8,5) model, which entails a monopole background, it is necessary to add an extra supermultiplet of constants for half of the supersymmetry. The N = 4 analog, joining an inhomogeneous (1,4,3) with a (3,4,1) multiplet, is also analyzed in detail.