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 herein by evading an obstruction specified visually and computationally by the "bow-tie" and "spin sum rule" twin theorems. The evasion of this obstruction is proven to be both necessary and 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-so as to depict an off-shell supermultiplet of worldsheet (p, q)-supersymmetry. This entails starting from an Adinkra depicting a specific type of supermultiplet of worldline (p+q)-supersymmetry, judiciously re-defining a subset of component fields and e-print archive: http://lanl.arXiv.org/abs/1104.0722v2

Weaving worldsheet supermultiplets from the worldlines within

“…PACS: 11.30.Pb, 12.60.Jv And there in the warp and the woof is the proof of it.-Elwyn Brooks White ("Charlotte's Web") 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.The paper is organized as follows: The remainder of this introduction presents the requisite definitions, and Section 2 then presents the three constructions of off-shell and on the half-shell representations of worldsheet (p, q)-supersymmetry. Section 3 discuses the role of esDE error-correcting codes in the proposed framework for classifying off-shell representations of worldsheet supersymmetry; in particular, Section 3.3 catalogs the maximal such codes-and thus the minimal such supermultiplets-for p+q 8.…”

“…-Elwyn Brooks White ("Charlotte's Web") 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.…”

“…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.…”

Adinkras from ordered quartets of BC4 Coxeter group elements and regarding 1,358,954,496 matrix elements of the Gadget

“…Another approach [34], somewhat related, was taken with respect to the simpler problem of providing a dimensional extension of adinkras into the construction of 2D Minkowski space supersymmetric representations. In particular, this work uncovered the "no two-color ambidextrous bow-tie" rule which governs the lifting of the adinkra into a 2D Minkowski space supermultiplet.…”

“…An example of an image given the name of an "adinkra graph" [7] and mathematically defined in subsequent works [24][25][26][27][28][29] is shown in figure 1.…”