Adinkra symbols (physics)

In supergravity and supersymmetric representation theory, Adinkra symbols (Named for the Adinkra Symbols of the Akan people of Ghana) 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 they were introduced by Michael Faux and Sylvester James Gates in 2004.^[1]

A small Adinkra graph.

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.

See also

• Feynman diagram