Advening to Adynkrafields: Young Tableaux to Component Fields of the 10D, N = 1 Scalar Superfield

TL;DR

This paper introduces a novel graphical and algebraic framework called adynkrafields, which uses Young Tableaux and Dynkin Labels to efficiently describe component fields in 10D, N=1 scalar supermultiplets, bypassing traditional superfield methods.

Contribution

It develops adynkrafields, a new method combining Young Tableaux and Dynkin Labels, providing a more efficient computational approach to supermultiplet component fields in higher dimensions.

Findings

Allows rapid conversion from adynkras to component fields

Eliminates the need for $ heta$-expansions and $\sigma$-matrices

Offers an algorithmically superior expansion method

Abstract

Starting from higher dimensional adinkras constructed with nodes referenced by Dynkin Labels, we define "adynkras." These suggest a computationally direct way to describe the component fields contained within supermultiplets in all superspaces. We explicitly discuss the cases of ten dimensional superspaces. We show this is possible by replacing conventional $\theta$-expansions by expansions over Young Tableaux and component fields by Dynkin Labels. Without the need to introduce $\sigma$-matrices, this permits rapid passages from Adynkras $\to$ Young Tableaux $\to$ Component Field Index Structures for both bosonic and fermionic fields while increasing computational efficiency compared to the starting point that uses superfields. In order to reach our goal, this work introduces a new graphical method, "tying rules," that provides an alternative to Littlewood's 1950 mathematical results which proved branching rules result from using a specific Schur function series. The ultimate point of this line of reasoning is the introduction of mathematical expansions based on Young Tableaux and that are algorithmically superior to superfields. The expansions are given the name of "adynkrafields" as they combine the concepts of adinkras and Dynkin Labels.