Adinkra symbols

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 they were introduced by Michael Faux and Sylvester James Gates in 2004.^[1]

Overview

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

[math]\displaystyle{ \{Q_I, Q_J\} = 2 i \delta _{I J} \partial_\tau }[/math].

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

[math]\displaystyle{ Q \phi= i \psi }[/math],

[math]\displaystyle{ Q \psi= \partial_\tau \phi }[/math].

Since we have just one supersymmetry generator in this case, the superalgebra relation reduces to [math]\displaystyle{ Q^2 = i \partial _\tau }[/math], 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

References

External links

• http://golem.ph.utexas.edu/category/2007/08/adinkras.html
• https://www.flickr.com/photos/science_and_thecity/2796684536/
• https://www.flickr.com/photos/science_and_thecity/2795836787/
• http://www.thegreatcourses.com/courses/superstring-theory-the-dna-of-reality.html

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