A Unified Enumeration of 1-dimension Garden Algebras and Valise Adinkras

TL;DR

This paper provides a comprehensive enumeration formula for 1D supersymmetry garden algebras and valise Adinkras, extending previous special case studies to arbitrary dimensions and numbers of generators.

Contribution

It introduces a unified enumeration method for garden algebras and valise Adinkras applicable to all N and d, broadening the scope beyond prior specific cases.

Findings

Derived a general enumeration formula for garden algebras and Adinkras.

Extended previous enumerations from special cases to arbitrary N and d.

Streamlined the enumeration process using group-theoretic and computational methods.

Abstract

In the study of supersymmetry in one dimension, various works enumerate sets of generators of garden algebras $GR(d,N)$ (and equivalently, valise Adinkras) for special cases $N = d = 4$ and $N = d = 8$, using group-theoretic methods and computer computation. We complement this work by enumerating the objects for arbitrary $N$ and $d$ via a formula in a streamlined manner.