# 4D, $\mathcal{N}=1$ Matter Gravitino Genomics

## TL;DR

This paper classifies 4D, $
=1$ supersymmetric multiplets, including matter gravitino formulations, using adinkras, isomer- and holoraumy-equivalence classes, advancing the supersymmetry genomics project with computational tools.

## Contribution

It analyzes two matter gravitino multiplets in terms of their adinkra-based classifications, completing the characterization of spin-two representations in 4D, $
=1$ supersymmetry.

## Key findings

- Classified matter gravitino multiplets using adinkra isomer- and holoraumy-equivalence.
- Extended the supersymmetry genomics framework to include spin-two multiplets.
- Provided open-source tools and data for further research.

## Abstract

Adinkras are graphs that encode a supersymmetric representation's transformation laws that have been reduced to one dimension, that of time. A goal of the supersymmetry ``genomics'' project is to classify all 4D, $\mathcal{N}=1$ off-shell supermultiplets in terms of their adinkras. In~previous works, the genomics project uncovered two fundamental isomer adinkras, the cis- and trans-adinkras, into which all multiplets investigated to date can be decomposed. The number of cis- and trans-adinkras describing a given multiplet define the isomer-equivalence class to which the multiplet belongs. A further refining classification is that of a supersymmetric multiplet's holoraumy: the commutator of the supercharges acting on the representation. The one-dimensionally reduced, matrix representation of a multiplet's holoraumy defines the multiplet's holoraumy-equivalence class. Together, a multiplet's isomer-equivalence and holoraumy-equivalence classes are two of the main characteristics used to distinguish the adinkras associated with different supersymmetry multiplets in higher dimensions. This paper focuses on two matter gravitino formulations, each with 20 bosonic and 20 fermionic off-shell degrees of freedom, analyzes them in terms of their isomer- and holoraumy-equivalence classes, and compares with non-minimal supergravity which is also a 20x20 multiplet. This analysis fills a missing piece in the supersymmetry genomics project, as now the isomer-equivalence and holoraumy-equivalence for representations up to spin two in component fields have been analyzed for 4D, $\mathcal{N}=1$ supersymmetry. To handle the calculations of this research effort, we have used the Mathematica software package called Adinkra.m. This package is open-source and available for download at a GitHub Repository. Data files associated with this paper are also published open-source at a Data Repository also on GitHub.