Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 6

On dimensional extension of supersymmetry: from worldlines to worldsheets

Pages: 1619 – 1667



Sylvester J. Gates, Jr. (Center for String and Particle Theory, Department of Physics, University of Maryland, College Park, Md., U.S.A.)

Tristan Hübsch (Department of Physics and Astronomy, Howard University, Washington, D.C., U.S.A.)


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 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 partitioning the worldline $(p, q)$-supersymmetry action into a proper worldsheet $(p, q)$-supersymmetry action.

Published 28 May 2013