While adinkraic representations are special among all representations of the 1-dimensional 𝒩\mathcal{N}-extended supersymmetry algebra, the idea is that the dimensional reduction of representations (supermultiplets) of higher dimensional supersymmetry algebras down to 1d are of this form, at least for dimensional reduction from d=4d = 4 and N=4N = \mathbf{4} (GGMPPRW 2009, section 3, Gates, Hubsch & Stiffler 2014).

The classification of adinkras, and hence of adinkraic representations, turns out to be controlled by linear codes (Doran, Faux, Gates et al. 2011) and to be related to certain special super Riemann surfaces via dessins d'enfants (Doran, Iga, Landweber & Mendez-Diez 13, Doran, Iga, Kostiuk & Mendes-Diez 2016).

Definition

For general background, see at geometry of physics – supersymmetry.

For N∈ℕN \in \mathbb{N}, write ℝ 1|N\mathbb{R}^{1 \vert N} for the super Lie algebra over the real numbers that is spanned by a single generator PP in even (i.e., bosonic) degree and NN generators Q IQ_I, I∈{1,2,⋯,N}I \in \{1, 2, \cdots, N\} in odd (i.e., fermionic) degree, whose only non-trivial components of the super-Lie bracket are

[Q I,Q J]=2δ IJP={2P | I=J 0 | otherwise [Q_I, Q_J] \,=\, 2 \delta_{I J} P \,=\, \left\{ \array{ 2 P & \vert & I = J \\ 0 & \vert & \text{otherwise} } \right.

This is the 1-dimensional NN-extended super translation super Lie algebra. We may think of this as the super-translational symmetry of 1-dimensional NN-extended super Minkowski spacetime.

Consider then super Lie algebra representations of ℝ 1|N\mathbb{R}^{1 \vert N} on super vector spaces of smooth superfields on ℝ 1|N\mathbb{R}^{1 \vert N} (regarded as a supermanifold) and such that the bosonic generator PP acts as the derivative operator on smooth functions on ℝ 1\mathbb{R}^1 in each component. If in addition the representation is such that in the canonical linear basis the odd generators Q IQ_I send even/odd basis elements ϕ i\phi_i to single odd/even basis elements ψ j\psi_j (as opposed to linear combinations of them), hence if the Q IQ_I act apart from degree-shift and possibly differentiation by permutations on the components of the superfields, then this representation of ℝ 1|N\mathbb{R}^{1\vert N} is called adinkraic. (Zhang 2013, p. 16).

The corresponding adinkra is the bipartite graph which expresses these permutations:

table grabbed from Doran, Iga, Landweber & Mendez-Diez 2013, p. 7

graphics grabbed from Iga & Zhang 2015, p. 3

Classification

The topology of an adinkra graph together with its edge coloring in {1,2,⋯,N}\{1,2, \cdots, N\} is called its chromotopology.

The set of adinkra chromotopologies is equivalent to the set of colored NN-cubs modulo doubly even length-NN linear codes (Doran, Faux, Gates, Hubsch, Iga, Landweber & Miller 2011).

(A linear code of length NN is a linear subspace of (𝔽 2) N(\mathbb{F}_2)^N for 𝔽 2\mathbb{F}_2 the prime field with two elements and it is doubly even if every element has weight a multiple of 4.)

see Zhang 2013, chapter 2

graphics grabbed from Iga & Zhang 2015, p. 4

From supermultiplets in higher dimensions

The dimensional reduction of the smallest supermultiplets of d=4,N=4d = 4, N = \mathbf{4} supersymmetry down to 1d yield adinkraic representations (GGMPPRW 2009, section 3, Gates, Hubsch & Stiffler 2014).

Corresponding adinkras for the chiral scalar supermultiplet (CM), the vector multiplet (VM) and the tensor multiplet (TM) look as follows:

References

For an introduction to adinkras, see the talk

Lutian Zhao, What is an Adinkra? (2014) [slides pdf]

The concept of adinkras was introduced into supersymmetry representation theory in:

Michael Faux, Jim Gates, Adinkras: A Graphical Technology for Supersymmetric Representation Theory, Phys. Rev. D 71 (2005) 065002 [hep-th/0408004, doi:10.1103/PhysRevD.71.065002]

and further developed in the article

Charles Doran, Michael Faux, Jim Gates, Tristan Hübsch, Kevin Iga, Greg Landweber, On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields, Int. J. Mod. Phys. A 22 , (2007) 869-930 [arXiv:math-ph/0512016, doi:10.1142/S0217751X07035112]

and many more (“DFGHIL collaboration”). For instance the relation to Clifford supermodules is discussed in

Charles Doran, Michael Faux, Jim Gates, Tristan Hübsch, Kevin Iga, Greg Landweber, Off-shell supersymmetry and filtered Clifford supermodules, Algebr Represent Theor 21 (2018) 375–397 &lbraclarXiv:math-ph/0603012, doi:10.1007/s10468-017-9718-8]

kinetic terms are discussed in

Charles Doran, Michael Faux, Jim Gates, Tristan Hübsch, Kevin Iga, Greg Landweber, Adinkras and the Dynamics of Superspace Prepotentials [arXiv:hep-th/0605269, InSpire:717895]

The classification of adinkras in terms of graphs and linear codes is due to

Charles Doran, Michael Faux, Jim Gates, Tristan Hübsch, Kevin Iga, Greg Landweber, R. L. Miller, Codes and Supersymmetry in One Dimension, Adv. Theor. Math. Phys. 15 (2011) 1909-1970 [arXiv:1108.4124, doi;10.4310/ATMP.2011.v15.n6.a7]

and discussed in mathematical detail in

Yan X Zhang, Adinkras for Mathematicians (arXiv:1111.6055)
Yan X Zhang: The combinatorics of Adinkras, PhD thesis, MIT (2013) [pdf]

The dimensional reduction of the standard supermultiplets of D=4,𝒩=1D = 4, \mathcal{N} = 1 supersymmetry to adinkraic representations of D=1,𝒩=4D = 1, \mathcal{N}=4 is due to

Jim Gates, J. Gonzales, B. MacGregor, J. Parker, R. Polo-Sherk, V. G. J. Rodgers, L. Wassink, 4D4D, 𝒩=1\mathcal{N} = 1 Supersymmetry Genomics (I), JHEP 0912:008 (2009) [arXiv:0902.3830, doi:10.1088/1126-6708/2009/12/008]
Jim Gates, Tristan Hübsch, Kory Stiffler, Adinkras and SUSY Holography, Int. J. Mod. Phys. A 29 7 (2014) 1450041 [arXiv:1208.5999, doi;10.1142/S0217751X14500419]