A Note On Exemplary Off-Shell Constructions Of 4D, ${\mathbf {\cal N}}$ = 2 Supersymmetry Representations

TL;DR

This paper investigates the combinatorial rules and algebraic structures, specifically permutation symmetries and matrix invariants, that determine how 4D, $ ext{N}=1$ supermultiplets can be combined to form off-shell 4D, $ ext{N}=2$ supermultiplets, confirming known supermultiplet structures.

Contribution

It introduces the use of Height Yielding Matrix Numbers (HYMN) and permutation analysis within adinkras to identify valid off-shell supermultiplet combinations for 4D, $ ext{N}=2$ supersymmetry.

Findings

HYMN accurately distinguishes valid supermultiplet combinations.

Only chiral + vector and chiral + tensor combinations form valid off-shell 4D, $ ext{N}=2$ supermultiplets.

Results align with the established structure of 4D, $ ext{N}=2$ supermultiplets.

Abstract

We continue the search for rules that govern when off-shell 4D, $\cal N$ = 1 supermultiplets can be combined to form off-shell 4D, $\cal N$ = 2 supermultiplets. We study the ${\mathbb S}_8$ permutations and Height Yielding Matrix Numbers (HYMN) embedded within the adinkras that correspond to these putative 4D, $\cal N$ = 2 supermultiplets off-shell supermultiplets. Even though the HYMN definition was designed to distinguish between the raising and lowering of nodes in one dimensional valises supermultiplets, they are shown to accurately select out which combinations of off-shell 4D, $\cal N$ = 1 supermultiplets correspond to off-shell 4D, $\cal N$ = 2 supermultiplets. Only the combinations of the chiral + vector and chiral + tensor are found to have valises in the same class. This is consistent with the well known structure of 4D, $\cal N$ = 2 supermultiplets.