On Dimensional Extension of Supersymmetry: From Worldlines to Worldsheets

TL;DR

This paper investigates conditions under which off-shell worldline supermultiplets can be extended to worldsheet supersymmetry, providing explicit methods to re-engineer Adinkras for such extensions based on twin theorems.

Contribution

It introduces twin theorems that characterize when worldline supermultiplets extend to worldsheet supersymmetry and demonstrates how to re-engineer Adinkras accordingly.

Findings

Twin theorems characterize extension conditions.

Explicit re-engineering of Adinkras shown.

Extension criteria are necessary for higher-dimensional supersymmetry.

Abstract

There exist myriads of off-shell worldline supermultiplets for (N{\leq}32)-extended supersymmetry in which every supercharge maps a component field to precisely one other component field or its derivative. A subset of these extends to off-shell worldsheet (p,q)-supersymmetry and is characterized by the twin theorems 2.1 and 2.2 in this note. The evasion of the obstruction defined in these theorems is conjectured to be sufficient for a worldline supermultiplet to extend to worldsheet supersymmetry; it is also a necessary filter for dimensional extension to higher-dimensional spacetime. We show explicitly how to "re-engineer" an Adinkra---if permitted by the twin theorems 2.1 and 2.2---so as to depict an off-shell supermultiplet of worldsheet (p,q)-supersymmetry.