On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields

TL;DR

This paper introduces a graph-theoretic framework called Adinkras to classify off-shell representations of extended supersymmetry in one dimension, expanding the understanding of supermultiplets via superfield constraints.

Contribution

It develops techniques to generate and analyze Adinkras, linking them to superfield constraints, and extends the classification of supersymmetric multiplets beyond previous methods.

Findings

Adinkras can be generated from N-dimensional cubes using superderivative constraints.

The framework broadens the range of supermultiplets describable by superspace formalism.

A cyclical main sequence of Adinkras for N=1 and N=2 is constructed.

Abstract

In this paper we discuss off-shell representations of N-extended supersymmetry in one dimension, ie, N-extended supersymmetric quantum mechanics, and following earlier work on the subject codify them in terms of certain graphs, called Adinkras. This framework provides a method of generating all Adinkras with the same topology, and so also all the corresponding irreducible supersymmetric multiplets. We develop some graph theoretic techniques to understand these diagrams in terms of a relatively small amount of information, namely, at what heights various vertices of the graph should be "hung". We then show how Adinkras that are the graphs of N-dimensional cubes can be obtained as the Adinkra for superfields satisfying constraints that involve superderivatives. This dramatically widens the range of supermultiplets that can be described using the superspace formalism and organizes them. Other topologies for Adinkras are possible, and we show that it is reasonable that these are also the result of constraining superfields using superderivatives. The family of Adinkras with an N-cubical topology, and so also the sequence of corresponding irreducible supersymmetric multiplets, are arranged in a cyclical sequence called the main sequence. We produce the N=1 and N=2 main sequences in detail, and indicate some aspects of the situation for higher N.