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In supergravity and supersymmetric representation theory, Adinkra symbols are a graphical representation of supersymmetric algebras. Mathematically they can be described as colored finite connected simple graphs, that are bipartite and n-regular. Their name is derived from Adinkra symbols of the same name, and they were introduced by Michael Faux and Sylvester James Gates in 2004.

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

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. S2CID 18179363.
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. Zhang, Yan X. (2011). "Adinkras for Mathematicians". arXiv:1111.6055 .

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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