# Adinkra symbols (physics)

A small Adinkra graph.

In supergravity and supersymmetric representation theory, Adinkra symbols are a graphical representation of supersymmetric algebras.[1][2][3][4][5] Mathematically they can be described as colored finite connected simple graphs, that are bipartite and n-regular.[6] Their name is derived from Adinkra symbols of the same name, and are the brainchild of Sylvester James Gates.[7][8]

## Overview

One approach to the representation theory of super Lie algebras is to restrict attention to representations in one space-time dimension and having ${\displaystyle N}$ supersymmetry generators, i.e., to ${\displaystyle (1|N)}$ superalgebras. In that case, the defining algebraic relationship among the supersymmetry generators reduces to

${\displaystyle \{Q_{I},Q_{J}\}=2i\delta _{IJ}\partial _{\tau }}$.

Here ${\displaystyle \partial _{\tau }}$ denotes partial differentiation along the single space-time coordinate. One simple realization of the ${\displaystyle (1|1)}$ algebra consists of a single bosonic field ${\displaystyle \phi }$, a fermionic field ${\displaystyle \psi }$, and a generator ${\displaystyle Q}$ which acts as

${\displaystyle Q\phi =i\psi }$,
${\displaystyle Q\psi =\partial _{\tau }\phi }$.

Since we have just one supersymmetry generator in this case, the superalgebra relation reduces to ${\displaystyle Q^{2}=i\partial _{\tau }}$, which is clearly satisfied. We can represent this algebra graphically using one solid vertex, one hollow vertex, and a single colored edge connecting them.

## References

1. ^ Faux, M.; Gates, S. J. (2005). "Adinkras: A graphical technology for supersymmetric representation theory". Physical Review D. 71 (6): 065002. arXiv:hep-th/0408004. Bibcode:2005PhRvD..71f5002F. doi:10.1103/PhysRevD.71.065002.
2. ^ S. James Gates Jr.: "Superstring Theory: The DNA of Reality Archived September 26, 2007, at the Wayback Machine" (The Teaching Company)
3. ^ S.J. Gates, Jr.: "Symbols of Power, Physics World, Vol. 23, No 6, June 2010, pp. 34 - 39" Archived July 26, 2011, at the Wayback Machine
4. ^ S.J. Gates, Jr.: "Quarks to Cosmos Archived March 19, 2011, at the Wayback Machine"
5. ^ S.J. Gates, Jr., and T. Hubsch, "On Dimensional Extension of Supersymmetry: From Worldlines to Worldsheets"
6. ^ Faux, Michael; Gates Jr, S. J. (2011). "Adinkras for Mathematicians". arXiv:1111.6055 [math.CO].
7. ^ "Symbols of Power: Adinkras and the Nature of Reality", S. J. Gates
8. ^