ABSTRACT (original article): Adinkras are graphical representations of the gauge invariant field components in supersymmetric theories and their orbits under the action of supersymmetry (SUSY) generators in the context of supermultiplets. A discussion is given that provides a thorough review of the concepts of holoraumy, permutahedra, and gadgets. One consequence of these concepts, the additional concept of hopper operators, is discussed. These play a particularly important role that ignites the processes needed for the study of adinkras related to minimal 4D,
=1
supermultiplets (chiral, vector, tensor and complex linear supermultiplets) through the prism of very simple cellular automata following Wolfram’s `New Kind of Science’ paradigm. CITATION (original article): S. James Gates Jr., Youngik (Tom) Lee (2024), A Précis: Minimal Four Color Holoraumy and Wolfram’s “New Kind of Science” Paradigm, arXiv:2408.09342. https://doi.org/10.48550/arXiv.2408.09342

1. Introduction

This post is the supplement Mathematica code handbook for the article : https://arxiv.org/pdf/2408.09342 (For convenience we will call this article as A2408). The details conceptual explanation for each term in this post can be found in A2408 and this post is focused on the what kind of supporting Mathematica code is used for each calculations in A2408.

2. Bosonic Holoraumy/Jumpers matrix calculations

(A2408: Chapter 4,5 : Minimal Supermultiplets and CLS)

2-1 Minimal Supermultiplets

For example the X, Y, Z, W-matrix for Chiral supermultiplets can be calculated as below. (arXiv:0902.3830v4 [hep-th])
Other Minimal supermultiplets also can be calculated as same process.
In[]:=
L1={{1,0,0,0},{0,0,0,-1},{0,1,0,0},{0,0,-1,0}};​​L2={{0,1,0,0},{0,0,1,0},{-1,0,0,0},{0,0,0,-1}};​​L3={{0,0,1,0},{0,-1,0,0},{0,0,0,-1},{1,0,0,0}};​​L4={{0,0,0,1},{1,0,0,0},{0,0,1,0},{0,1,0,0}​​};
In[]:=
X1=L2.Inverse[L1];X2=L3.Inverse[L2];X3=L4.Inverse[L3];X4=L1.Inverse[L4];​​Y1=X2.Inverse[X1];Y2=X3.Inverse[X2];Y3=X4.Inverse[X3];Y4=X1.Inverse[X4];​​Z1=Y2.Inverse[Y1];Z2=Y3.Inverse[Y2];Z3=Y4.Inverse[Y3];Z4=Y1.Inverse[Y4];​​W1=Z2.Inverse[Z1];W2=Z3.Inverse[Z2];W3=Z4.Inverse[Z3];W4=Z1.Inverse[Z4];
W1//MatrixForm​​W2//MatrixForm​​W3//MatrixForm​​W4//MatrixForm
Out[]//MatrixForm=
-1
0
0
0
0
-1
0
0
0
0
-1
0
0
0
0
-1
Out[]//MatrixForm=
-1
0
0
0
0
-1
0
0
0
0
-1
0
0
0
0
-1
Out[]//MatrixForm=
-1
0
0
0
0
-1
0
0
0
0
-1
0
0
0
0
-1
Out[]//MatrixForm=
-1
0
0
0
0
-1
0
0
0
0
-1
0
0
0
0
-1

2-2 CLS representation

The same logic is used, and one can reference the L- and R-matrix information of CLS.

CLS L-matrix

L1=
;​​L2=
;​​L3=
;​​L4=
;
In[]:=
X1=L2.Inverse[L1];X2=L3.Inverse[L2];X3=L4.Inverse[L3];X4=L1.Inverse[L4];​​Y1=X2.Inverse[X1];Y2=X3.Inverse[X2];Y3=X4.Inverse[X3];Y4=X1.Inverse[X4];​​Z1=Y2.Inverse[Y1];Z2=Y3.Inverse[Y2];Z3=Y4.Inverse[Y3];Z4=Y1.Inverse[Y4];​​W1=Z2.Inverse[Z1];W2=Z3.Inverse[Z2];W3=Z4.Inverse[Z3];W4=Z1.Inverse[Z4];
W1//MatrixForm​​W2//MatrixForm​​W3//MatrixForm​​W4//MatrixForm
Out[]//MatrixForm=
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
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-1
0
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-1
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-1
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-1
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-1
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-1
0
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-1
0
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0
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0
0
0
0
-1
0
0
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0
0
0
0
0
-1
Out[]//MatrixForm=
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
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-1
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-1
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-1
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-1
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-1
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-1
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0
0
0
-1
0
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0
0
0
0
-1
Out[]//MatrixForm=
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
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-1
0
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-1
0
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-1
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-1
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-1
0
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0
0
0
0
0
0
0
-1
0
0
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-1
0
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-1
0
0
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0
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0
0
0
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0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
Out[]//MatrixForm=
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
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0
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-1
0
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-1
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-1
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-1

CLS R-matrix

R1=
;​​R2=
;​​R3=
;​​R4=
;​​​​X1'=R2.Inverse[R1];X2'=R3.Inverse[R2];X3'=R4.Inverse[R3];X4'=R1.Inverse[R4];​​Y1'=X2'.Inverse[X1'];Y2'=X3'.Inverse[X2'];Y3'=X4'.Inverse[X3'];Y4'=X1'.Inverse[X4'];​​Z1'=Y2'.Inverse[Y1'];Z2'=Y3'.Inverse[Y2'];Z3'=Y4'.Inverse[Y3'];Z4'=Y1'.Inverse[Y4'];​​W1'=Z2'.Inverse[Z1'];W2'=Z3'.Inverse[Z2'];W3'=Z4'.Inverse[Z3'];W4'=Z1'.Inverse[Z4'];
W1'//MatrixForm​​W2'//MatrixForm​​W3'//MatrixForm​​W4'//MatrixForm
Out[]//MatrixForm=
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
0
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-1
0
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0
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-1
Out[]//MatrixForm=
-1
0
0
0
0
0
0
0
0
0
0
0
0
-1
0
0
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0
-1

3. Adinkra Representation on Wolfram Model (A2408: Chapter 6)

The basic introduction code sample for Adinkra generation by using Bosonic Holoraumy matrix (X-matrix).
Also one can reference the code at : https://community.wolfram.com/groups/-/m/t/2576265

3-1 L-matrix representation

Let us use the chiral supermultiplet as a toy model.

Brute Forcing Adinkra

We can generate Adinkra as brute forcibly assign the each connection between the arbitrary two nodes.

Rewriting Rules Adinkra

However we can also generate Adinkra by using the rewriting rules as below.
Thus the main argument of this chapter is how to find the rewriting rules for particular supermultiplet Adinkra.
And we use the G-matrix as the key object for Adinkra generation.

3-1. Chiral

3-2. Vector

3-3. Tensor

4. G-matrix Calculation (A2408: Chapter 7, Appendix A)

This chapter one can see how to design the G-matrix code on Mathematica.
when list is empty, it returns empty list

3-1. Inverse square method

MatrixPower function is one of the useful tool that one need to find the square-root or the multiple-square matrix. If we need to calculate A^(1/p), MatrixPower is efficient way to calculate the solution. However it returns one answer among the possible solutions.
Also when MatrixPower function do square-root calculation for the matrix, it using the method of diagonalizing matrix, which is
(1) A=P D P^(-1) (Here D is a diagonalize matrix of its eigenvalues)
(2) A^(1/p)= P D^(1/p) P^(-1)
So we can also get the result with complex numbers

3-2. Brute-force method

In this project, when we calculate the A^(1/p) we want to find the solution matrix that consist with 0, -1, 1, and also the all possible solution. So one of the available way to do is using the Brute-force method
And we can see that the Brute-force method return the integer solution when we compare with inverse-square method

Reference

[1] S. James Gates, Jr., Y. Lee, "A Precis: Minimal Four Color Holoraumy and Wolfram’s “New Kind of Science” Paradigm", arXiv:2408.09342
[2] S. James Gates, Jr. , James Gonzales, Boanne MacGregor, James Parker, Ruben Polo-Sherk, Vincent G. J. Rodgers, Luke Wassink, "4D, N = 1 Supersymmetry Genomics (I)", arXiv:0902.3830v4 [hep-th]
[3] M. Gillis and D. Chester, 2022, “Wolfram Summer School Community Post, Physics” https://community.wolfram.com/groups/-/m/t/2576265

CITE THIS NOTEBOOK

A Précis: Minimal Four Color Holoraumy and Wolfram's "New Kind of Science" Paradigm​
by Sylvester James Gates and Youngik (Tom) Lee​
Wolfram Community, STAFF PICKS, August 26, 2024
​https://community.wolfram.com/groups/-/m/t/3257119