Adinkras: A graphical technology for supersymmetric representation theory

Abstract

We present a symbolic method for organizing the representation theory of one-dimensional superalgebras. This relies on special objects, which we have called adinkra symbols, which supply tangible geometric forms to the still-emerging mathematical basis underlying supersymmetry.

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.…”

“…(1) The "spin sum rule" is independent of the "dashing rule" [9,10] whereby the number of dashed edges in any quadrangle within any Adinkra must be odd, and which stems from the anticommutivity of the D's; see below, at the end of this section. In fact, the above "spin sum rule" is unaffected by changes in the solid/dashing assignments.…”

4D, $ \mathcal{N} $ = 1 supersymmetry genomics (II)

“…Related to this, the graph theoretic tool of Adinkras were introduced in 2004 [10]. Adinkras are graphic representations of the 1D 'shadows' of supersymmetric representations.…”

Hierarchy of mechanics models

“…For instance, in the case of N=4 and N=8 supersymmetry there exist off-shell multiplets with field contents (4, 4, 0) and (8,8,0), respectively (so-called "root" [32,33] or "extreme" multiplets). These cannot be recovered by dimensional reduction because the suitable d>1 off-shell multiplets always include auxiliary fields.…”