Generating All 36,864 Four-Color Adinkras via Signed Permutations and Organizing into <i>ℓ</i>- and <i>ℓ</i>˜-Equivalence Classes

• Main Authors: S. James Gates, Kevin Iga, Lucas Kang, Vadim Korotkikh, Kory Stiffler
• Format: Article
• Language: English
• Published: MDPI AG 2019-01-01
• Series: Symmetry
• Subjects: adinkras; equivalence classes; holography; holoraumy; representation theory; supersymmetry; sigma models
• Online Access: https://www.mdpi.com/2073-8994/11/1/120

Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras with four colors, four bosons, and four fermions have been computed. In this paper, we further analyze these results in terms of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>C</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> </inline-formula>, the signed permutation group of three elements, and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>C</mi> <mn>4</mn> </msub> </mrow> </semantics> </math> </inline-formula>, the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>C</mi> <mn>4</mn> </msub> </mrow> </semantics> </math> </inline-formula> boson &#215;<inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>C</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> </inline-formula> color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the <i>gadget</i>, which is used to distinguish adinkras. We show how 96 equivalence classes of adinkras that are based on the gadget emerge in terms of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>C</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>C</mi> <mn>4</mn> </msub> </mrow> </semantics> </math> </inline-formula>. We also comment on the importance of the gadget as it relates to separating out dynamics in terms of K&#228;hler-like potentials. Thus, on the basis of the complete analysis of the supersymmetrical representations achieved in the preparatory first four sections, the final comprehensive achievement of this work is the construction of the universal <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>C</mi> <mn>4</mn> </msub> </mrow> </semantics> </math> </inline-formula> non-linear <inline-formula> <math display="inline"> <semantics> <mi>&#963;</mi> </semantics> </math> </inline-formula>-model.