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\begin{document}

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\section*{adinkra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{superalgebra}{}\paragraph*{{Superalgebra}}\label{superalgebra}

[[!include supergeometry - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{classification}{Classification}\dotfill \pageref*{classification} \linebreak
\noindent\hyperlink{from_supermultiplets_in_higher_dimensions}{From supermultiplets in higher dimensions}\dotfill \pageref*{from_supermultiplets_in_higher_dimensions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[super-algebra|super]]-[[representation theory]], what is called \emph{adinkras} (\hyperlink{FauxGates04}{Faux-Gates 04}) is a graphical tool for denoting those [[representations]] ([[super multiplets]]) of the $N$-extended [[supersymmetry]] algebras in one dimension ([[supersymmetric quantum mechanics]] with $N$ supersymmetries) for which the [[supersymmetry]] generators act, up to [[derivatives]] and prefactors, by [[permutation]] of [[superfield]] components. These are called \emph{adinkraic} representations (\hyperlink{Zhang13}{Zhang 13, p. 16}).

While adinkraic representations are special among all representations of the 1-dimensional $N$-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 $d = 4$ and $N = \mathbf{4}$ (\hyperlink{GGMPPRW09}{GGMPPRW 09, section 3}, \hyperlink{GatesHubschStiffler14}{Gates-Hubsch-Stiffler 14}).

The classification of adinkras, and hence of adinkraic representations, turns out to be controled by [[linear codes]] (\hyperlink{DoranFauxGatesHubschIgaLandweberMiller11}{Doran \& Faux \& Gates \& HubschIgaLandweberMiller 11}) and to be related to certain special [[super Riemann surfaces]] via [[dessins d'enfants]] (\hyperlink{DoranIgaLandweberMendez-Diez13}{Doran \& Iga \& Landweber \& Mendez-Diez 13}, \hyperlink{DoranIgaKostiukMendes-Diez16}{Doran \& Iga \& Kostiuk \&Mendes-Diez 16}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For background, see at \emph{[[geometry of physics -- supersymmetry]]}.

For $N \in \mathbb{N}$, write $\mathbb{R}^{1 \vert N}$ for the [[super Lie algebra]] over the [[real numbers]] that is spanned by a single generator $P$ in even (i.e., bosonic) degree and $N$ generators $Q_I$, $I \in \{1, 2, \cdots, N\}$ in odd (i.e., fermionic) degree, whose only non-trivial components of the super [[Lie bracket]] is

\begin{displaymath}
[Q_I, Q_J]  = 2 \delta_{I J} P
  =
  \left\{
   \itexarray{
      2 P & \vert \; I = J
      \\
      0 & \vert \; \text{otherwise}
   }
  \right.
\end{displaymath}
This is the 1-dimensional $N$-extended [[super translation super Lie algebra]]. We may think of this as the super-translational symmetry of 1-dimensional $N$-extended [[super Minkowski spacetime]].

Consider then super [[Lie algebra representations]] of $\mathbb{R}^{1 \vert N}$ on [[super vector spaces]] of smooth [[superfields]] on $\mathbb{R}^{1 \vert N}$ (regarded as a [[supermanifold]]) and such that the bosonic generator $P$ acts as the [[derivative]] operator on [[smooth functions]] on $\mathbb{R}^1$ in each component. If in addition the representation is such that in the canonical [[linear basis]] the odd generators $Q_I$ send even/odd basis elements $\phi_i$ to single odd/even basis elements $\psi_j$ (as opposed to [[linear combinations]] of them), hence if the $Q_I$ act apart from degree-shift and possibly [[differentiation]] by [[permutations]] on the components of the [[superfields]], then this representation of $\mathbb{R}^{1\vert N}$ is called \emph{adinkraic}. (\hyperlink{Zhang13}{Zhang 13, p. 16}).

The corresponding \emph{adinkra} is the [[bipartite graph]] which expresses these permutations:



\begin{quote}%
table grabbed from \hyperlink{DoranIgaLandweberMendez-Diez13}{Doran \& Iga \& Landweber \& Mendez-Diez 13, p. 7}


\end{quote}


\begin{quote}%
graphics grabbed from \hyperlink{IgaZhang15}{Iga-Zhang 15, p. 3}


\end{quote}
\hypertarget{classification}{}\subsection*{{Classification}}\label{classification}

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

The set of adinkra chromotopologies is equivalent to the set of colored $N$-cubs modulo doubly even length-$N$ [[linear codes]] (\hyperlink{DoranFauxgatesHubschIgaLandweberMiller11}{Doran-Faux-Gates-Hubsch-Iga-Landweber-Miller 11})

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

see \hyperlink{Zhang13}{Zhang 13, chapter 2}



\begin{quote}%
graphics grabbed from \hyperlink{IgaZhang15}{Iga-Zhang 15, p. 4}


\end{quote}
\hypertarget{from_supermultiplets_in_higher_dimensions}{}\subsection*{{From supermultiplets in higher dimensions}}\label{from_supermultiplets_in_higher_dimensions}

The [[dimensional reduction]] of the smallest [[supermultiplets]] of $d = 4, N = \mathbf{4}$ supersymmetry down to 1d yield adinkraic representations (\hyperlink{GGMPPRW09}{GGMPPRW 09, section 3}, \hyperlink{GatesHubschStiffler14}{Gates-Hubsch-Stiffler 14}).

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



\hypertarget{references}{}\subsection*{{References}}\label{references}

For an introduction to adinkras, see the talk

\begin{itemize}%
\item [[Lutian Zhao]], \emph{What is an Adinkra?}, 2014 (\href{http://math.sjtu.edu.cn/conference/Bannai/2014/data/20141213B/slides.pdf}{slides})

\end{itemize}
The concept of adinkras was introduced into [[supersymmetry]] [[representation theory]] in

\begin{itemize}%
\item [[Michael Faux]], [[Jim Gates]], \emph{Adinkras: A Graphical Technology for Supersymmetric Representation Theory}, Phys.Rev. D71 (2005) 065002 (\href{https://arxiv.org/abs/hep-th/0408004}{hep-th/0408004})

\end{itemize}
and further developed in the article

\begin{itemize}%
\item [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], \emph{On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields}, Int. J. Mod. Phys. A22: 869-930, 2007 (\href{https://arxiv.org/abs/math-ph/0512016}{arXiv:math-ph/0512016})

\end{itemize}
and many more (``DFGHIL collaboration''). For instance the relation to Clifford supermodules is discussed in

\begin{itemize}%
\item [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], \emph{Off-shell supersymmetry and filtered Clifford supermodules} (\href{https://arxiv.org/abs/math-ph/0603012}{arXiv:math-ph/0603012})

\end{itemize}
kinetic terms are discussed in

\begin{itemize}%
\item [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], \emph{Adinkras and the Dynamics of Superspace Prepotentials} (\href{https://arxiv.org/abs/hep-th/0605269}{arXiv:hep-th/0605269})

\end{itemize}
The classification of adinkras in terms of [[graphs]] and [[linear codes]] is due to

\begin{itemize}%
\item [[Charles Doran]], [[Michael Faux]], [[Jim Gates]], [[Tristan Hübsch]], [[Kevin Iga]], [[Greg Landweber]], R. L. Miller, \emph{Codes and Supersymmetry in One Dimension}, Adv. in Th. Math. Phys. 15 (2011) 1909-1970 (\href{https://arxiv.org/abs/1108.4124}{arXiv:1108.4124})

\end{itemize}
and discussed in mathematical detail in

\begin{itemize}%
\item [[Yan X Zhang]], \emph{Adinkras for Mathematicians} (\href{https://arxiv.org/abs/1111.6055}{arXiv:1111.6055})


\item [[Yan X Zhang]], \emph{The combinatorics of Adinkras}, PhD thesis, MIT (2013) (\href{http://math.mit.edu/~yanzhang/math/thesis_adinkras.pdf}{pdf})



\end{itemize}
The [[dimensional reduction]] of the standard [[supermultiplets]] of $d = 4, N = 1$ supersymmetry to adinkraic representations of $d = 1, N=4$ is due to

\begin{itemize}%
\item [[Jim Gates]], J. Gonzales, B. MacGregor, J. Parker, R. Polo-Sherk, V.G.J. Rodgers, L. Wassink, \emph{$4D$, $N = 1$ Supersymmetry Genomics (I)}, JHEP 0912:008,2009 (\href{https://arxiv.org/abs/0902.3830}{arXiv:0902.3830})


\item [[Jim Gates]], [[Tristan Hübsch]], Kory Stiffler, \emph{Adinkras and SUSY Holography}, Int. J. Mod. Phys. A29 no. 7, (2014) 1450041 (\href{https://arxiv.org/abs/1208.5999}{arXiv:1208.5999})



\end{itemize}
See also

\begin{itemize}%
\item [[Kevin Iga]], [[Yan Zhang]], \emph{Structural Theory of 2-d Adinkras} (\href{https://arxiv.org/abs/1508.00491}{arXiv:1508.00491})

\end{itemize}
The relation of adinkras to special [[super Riemann surfaces]] via [[dessins d'enfants]] is due to

\begin{itemize}%
\item [[Charles Doran]], [[Kevin Iga]], [[Greg Landweber]], [[Stefan Méndez-Diez]], \emph{Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras} (\href{https://arxiv.org/abs/1311.3736}{arXiv:1311.3736})


\item [[Charles Doran]], [[Kevin Iga]], Jordan Kostiuk, [[Stefan Méndez-Diez]], \emph{Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras II} (\href{https://arxiv.org/abs/1610.09983}{arXiv:1610.09983})



\end{itemize}
Further developments includes

\begin{itemize}%
\item Mathew Calkins, D. E. A. Gates, [[Jim Gates]] Jr., Kory Stiffler, \emph{Adinkras, 0-branes, Holoraumy and the SUSY QFT/QM Correspondence} (\href{https://arxiv.org/abs/1501.00101}{arXiv:1501.00101})

\end{itemize}
Discussion in the context of [[spectral triples]] is in

\begin{itemize}%
\item [[Matilde Marcolli]], Nick Zolman, \emph{Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples} (\href{https://arxiv.org/abs/1606.04463}{arXiv:1606.04463})

\end{itemize}
See also

\begin{itemize}%
\item Wes Caldwell, Alejandro Diaz, Isaac Friend, [[Jim Gates]], Jr., Siddhartha Harmalkar, Tamar Lambert-Brown, Daniel Lay, Karina Martirosova, Victor Meszaros, Mayowa Omokanwaye, Shaina Rudman, Daniel Shin, Anthony Vershov, \emph{On the Four Dimensional Holoraumy of the $4D$, $\mathcal{N} = 1$ Complex Linear Supermultiplet} (\href{https://arxiv.org/abs/1702.05453}{arXiv:1702.05453})


\item Wikipedia, \emph{}



\end{itemize}
[[!redirects adinkras]]



\end{document}