On Dimensional Extension of Supersymmetry: From Worldlines to Worldsheets

Abstract

There exist myriads of off-shell worldline supermultiplets for (N ≤ 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.

Weaving worldsheet supermultiplets from the worldlines within

“…Constructions 2.1 and 2.3 for off-shell representations, and Construction 2.2 for unidextrous (on the half-shell) representations of worldsheet (p, q)-supersymmetry, and their listing for p+q 8; 2. the definition (and a p+q 8 listing) of even-split (binary) doubly even linear block (esDE) codes (see Section 3.1) that encode possible Z 2 quotients of tensor product supermultiplets, many of which not themselves tensor products (see Section 3.4); 3. the definition of a twisted Z 2 symmetry in Adinkras, which implies a complex structure; 4. a demonstration that some worldsheet supermultiplets depicted by topologically inequivalent Adinkras are nevertheless equivalent, and by (super)field redefinition only; 5. a demonstration that the same Adinkra may depict distinct supermultiplets of the same (p, q)-supersymmetry, though at least some of them can be shown to be equivalent, and by (super)field redefinition only; 6. an independent confirmation of the conclusion of Ref. [16], that ambidextrous off-shell supermultiplets of ambidextrous supersymmetry must have at least three levels [23,9], i.e., their component (super)fields must have at least three distinct, adjacent engineering dimensions.…”

“…α− 's (which square to i∂ = ). As shown in (2), component fields themselves acquire spin, and the necessary and sufficient condition for an Adinkra to depict a worldsheet supermultiplet [16] insures that all component fields can be assigned a spin consistently with the D α+ -and D .…”

“…Ref. [16] provides a simple criterion for extending worldline off-shell supermultiplets to worldsheet supersymmetry, which then suffices for many string theory applications. This also significantly enhances the efficiency of the numerical criteria of Ref.…”

“…Complementary to the filtering approach of Ref. [16], the constructive approach presented herein produces for all p, q 0: 1. off-shell (ambidextrous) supermultiplets of worldsheet (p, q)-supersymmetry, and 2. on the half-shell (unidextrous) supermultiplets of ambidextrous (p, q)-supersymmetry, by tensoring a left-and a right-handed copy of worldline supermultiplets, and projecting to their quotients by certain discrete symmetries. It is gratifying to note that the lists obtained in such complementary ways in fact coincide, at least for the low enough values of p+q, where comparisons could be made by inspection.…”

“…[10,11,45], being an encryption theory consequence of the filtering condition of Ref. [16]. Very much like in the case of worldline supermultiplets [10,11], passing to the quotient of such a Z 2 symmetry provides a new, half-sized supermultiplet, and one may do so repeatedly using mutually commuting such Z 2 symmetries.…”