Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 6

Advening to adynkrafields: Young tableaux to component fields of the 10D, $\mathcal{N}=1$ scalar superfield

Pages: 1449 – 1547

DOI: https://dx.doi.org/10.4310/ATMP.2021.v25.n6.a3

Authors

S. James Gates, Jr. (Brown Theoretical Physics Center and Department of Physics, Brown University, Providence, Rhode Island, U.S.A.)

Yangrui Hu (Brown Theoretical Physics Center and Department of Physics, Brown University, Providence, Rhode Island, U.S.A.)

S.-N. Hazel Mak (Brown Theoretical Physics Center and Department of Physics, Brown University, Providence, Rhode Island, U.S.A.)

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 → Young Tableaux → 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.

Published 24 June 2022