Friday, October 05, 2012

Adinkras and SUSY

















Jim Gates is back with another paper on Adinkras, which are graphs describing supersymmetric equations.  Just as a weight-space diagram is a graphical representation that precisely encodes the mathematical relations between the members of SU(3) families, so an adinkra is a graphical representation that precisely encodes the mathematical relations between the members of supersymmetry families.

The extra fun comes from seeing SUSY through the lens of error correcting codes. In the figure below, pick the bottom dot as a starting point and assign it an address of (0000). To move to any of the dots at the second level requires traversing one of the coloured links. There are four distinct ways in which this can be done. To move to any dot at the third level from the bottom dot requires the use of two different coloured links, and so on for the rest of the adinkra. In this way, every dot is assigned an address, from (0000) to (1111). These sequences of ones and zeros are binary computer words. 

  

To accomplish the folding that maintains the SUSY property in the associated equations, we must begin by squeezing the bottom dot together with the upper dot. When their addresses are added bit-wise to one another, this yields the sequence (1111). If we continue this folding process, always choosing pairs of dots so that their associated "words" sum bit-wise to (1111), we can transform the adinkra on the left-hand side to the one on the right in the figure below.



 Thus, maintaining the equations' SUSY property requires that the particular sequence of bits given by (1111) be used in the folding process. The process used to meet this criterion happens to correspond to the simplest member of the family containing the check-sum extended Hamming code. 


1 comment:

ThePeSla said...

Ahh... this begins to look a lot like the directions I have long envisioned, well close to the beginning of it all anyway.

L. Edgar Otto ThePeSla pesla.blogspot.com