Adinkra symbols (physics)

Graphical representation of supersymmetric algebras

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

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

{ Q I , Q J } = 2 i δ I J ∂ τ {\displaystyle \\{Q_{I},Q_{J}\\}=2i\delta _{IJ}\partial _{\tau }} .

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

Q ϕ = i ψ {\displaystyle Q\phi =i\psi } ,
Q ψ = ∂ τ ϕ {\displaystyle Q\psi =\partial _{\tau }\phi } .

Since we have just one supersymmetry generator in this case, the superalgebra relation reduces to  Q 2 = i ∂ τ {\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

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* Feynman diagram

## References

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1. ^ a b 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 [math.CO].

## External links

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* 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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