Topology Types of Adinkras and the Corresponding Representations of N-Extended Supersymmetry

TL;DR

This paper advances the classification of supermultiplets in N-extended supersymmetry by linking Adinkra diagrams to binary error-correcting codes, enabling enumeration of equivalence classes up to 26 supercharges.

Contribution

It establishes a novel connection between Adinkra topologies and binary error-correcting codes, providing a systematic classification method for supermultiplets.

Findings

Classified Adinkra diagrams using coding theory.

Enumerated equivalence classes up to 26 supercharges.

Identified maximal codes for minimal supermultiplets up to 32 supercharges.

Abstract

We present further progress toward a complete classification scheme for describing supermultiplets of N-extended worldline supersymmetry, which relies on graph-theoretic topological invariants. In particular, we demonstrate a relationship between Adinkra diagrams and quotients of N-dimensional cubes, where the quotient groups are subgroups of $(Z_2)^N$. We explain how these quotient groups correspond precisely to doubly even binary linear error-correcting codes, so that the classification of such codes provides a means for describing equivalence classes of Adinkras and therefore supermultiplets. Using results from coding theory we exhibit the enumeration of these equivalence classes for all cases up to 26 supercharges, as well as the maximal codes, corresponding to minimal supermultiplets, for up to 32 supercharges.