Codes and supersymmetry in one dimension

Abstract

Adinkras are diagrams that describe many useful supermultiplets in D = 1 dimensions. We show that the topology of the Adinkra is uniquely determined by a doubly even code. Conversely, every doubly even code produces a possible topology of an Adinkra. A computation of doubly even codes results in an enumeration of these Adinkra topologies up to N = 28, and for minimal supermultiplets, up to N = 32.

On dimensional extension of supersymmetry: from worldlines to worldsheets

“…In this spirit, the authors of [9][10][11][12][13][14][15] developed a detailed classification of a huge class (∼ 10 12 for no more than 32 supersymmetries) of worldline supermultiplets wherein each supercharge maps each component field to precisely one other component field or its derivative, and which are faithfully represented by graphs called Adinkras; see also [16][17][18][19][20][21][22]. The subsequently intended dimensional extension has been addressed recently [7,8], and the purpose of the present note is to complement this effort and identify an easily verifiable obstruction to dimensional extension.…”

“…In particular, the combinatorial explosion of worldline supermultiplets [12][13][14][15] owes also to the fact that replacing a worldline field with its derivative, φ → . φ = (∂ τ φ), changes only the engineering dimension of the field and produces only minor, though important changes in the supersymmetry relations [42].…”

“…Also, the horizontal rearrangements of the nodes within the one-hooked Adinkra, generates another (sub)group of evident equivalences, and has Adinkras with more complicated height-arrangements depict supermultiplets with more complicated choices of relative engineering dimension assignments for the component fields. The numbers of such inequivalent Adinkras then grows combinatorially with N ; the "node choice group" of [13,15] was defined to encode the symmetries in arbitrary Adinkras.…”

“…Herein, this feature is employed as a filter to distinguish those worldline off-shell supermultiplets, say from the huge 4 collection of [12][13][14][15] that do extend to worldsheet off-shell supermultiplets, some of which would possibly further extend to higher-dimensional off-shell supermultiplets.…”

From Diophantus to supergravity and massless higher spin multiplets

“…The adinkras are being used in order to generate a set of adjacency matrices associated with them which provide the various representations of suppersymmetry. These adjacency matrices satisfy a set of algebraic relations called "Garden Algebra" [26][27][28][29]. In [24] it was shown that one can use the Garden algebra in order to find off-shell completions of supersymmetric theories, by generating a system of quadratic equations.…”