Adinkra symbols (physics)

A small adinkra graph, that is graph isomorphic to a cube. (Dashed lines have been omitted)

In supergravity theory 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 Ashanti symbols of the same name (Adinkra).^[7]

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

$\{Q_I, Q_J\} = 2 i \delta _{I J} \partial_\tau$.

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

$Q \phi= i \psi$,

$Q \psi= \partial_\tau \phi$.

References

1. Faux, M.; Gates, S. J. (2005). "Adinkras: A graphical technology for supersymmetric representation theory". Physical Review D. 71 (6). 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. http://arxiv.org/pdf/1111.6055.pdf
7. http://www.onbeing.org/blog/symbols-power-adinkras-and-nature-reality/2438

External links

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