Advances in High Energy Physics

Volume 2016 (2016), Article ID 3980613, 12 pages

http://dx.doi.org/10.1155/2016/3980613

## Structural Theory and Classification of 2D Adinkras

^{1}Natural Science Division, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, USA^{2}Department of Mathematics, University of California, 970 Evans Hall, Berkeley, CA 94720-3840, USA

Received 6 August 2015; Revised 3 January 2016; Accepted 11 January 2016

Academic Editor: Torsten Asselmeyer-Maluga

Copyright © 2016 Kevin Iga and Yan X. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Adinkras are combinatorial objects developed to study (1-dimensional) supersymmetry representations. Recently,* 2D Adinkras* have been developed to study -dimensional supersymmetry. In this paper, we classify all D Adinkras, confirming a conjecture of T. Hübsch. Along the way, we obtain other structural results, including a simple characterization of Hübsch’s* even-split doubly even codes*.

#### 1. Introduction

Despite supersymmetry being of theoretical interest since the 1970s, there has not been a careful mathematical classification of off-shell supersymmetric field theories. Many supermultiplets have been discovered in an ad hoc fashion. Many of these theories are only known on-shell, and it was not clear which of these had off-shell counterparts [1–3]. The approach in [3] was to consider 1D theories (i.e., representations of the super-Poincaré algebra in one dimension, i.e., supersymmetric quantum mechanics). This is reasonable for several reasons: first, it makes sense to solve a problem starting with simpler cases, and 1D has a trivial Lorentz group structure, not to mention lack of gauge fields. Second, supermultiplets in higher dimensions (perhaps dimensions of interest like 4 or 10) can be dimensionally reduced to 1D, and so the 1D reduction can serve as a starting point for classifying higher dimensional theories. Third, this 1D classification is a compelling mathematical question, in its own right.

In 2004, Faux and Gates developed Adinkras (what we call* 1D Adinkras* in this paper) to study off-shell supermultiplets in one dimension. There have been a number of developments that have led to the classification of 1D Adinkras [4–9]. From this, the classification of off-shell 1D supersymmetric theories was outlined in [10].

Based on the success of this program, there have been a few recent approaches to using Adinkra-like ideas to study the super-Poincaré algebra in two dimensions. Note that many of the motivations for studying 1D SUSY apply here as well. In the progression from simple to difficult, this is a logical next step. It also has a very easy Lorentz group symmetry and a lack of gauge fields, while incorporating a few elements that are of interest in higher dimensions. Two-dimensional SUSY also is of interest to superstring theory.

One approach to off-shell 2D SUSY is to study the process of dimensional reduction from 2 to 1 dimension and to use the results from the 1D classification, to determine graphical objects that capture the relevant representation theoretic data in this new setting. The graphical “calculus” idea is very useful because once the fundamental physics ideas are instilled into the definitions, we only need to perform combinatorial manipulations and very little algebra. This has led to the development of 2D Adinkras [11, 12].

In this paper, we completely characterize 2D Adinkras, guided by the approach and conjectures set forth in [12]. The main result settles Hübsch’s Conjecture (the formulation in [12] is slightly different: see Appendix for details) in Theorem 21. Essentially, this says that these 2D Adinkras come from two 1D Adinkras: one describing the left-moving supersymmetries, and the other describing the right-moving supersymmetries. Every D Adinkra is a product of these 1D Adinkras, followed by vertex switches and a quotienting operation. This allows us to use our knowledge of 1D Adinkras to completely understand D Adinkras.

We begin in Section 2 by recalling the definition of (1D) Adinkras and some of their features, reviewing the* code* associated with an Adinkra [6] and the concept of* vertex switching* [8, 9]. As this paper is a mostly self-contained work of combinatorial classification, we do not discuss (or require from the reader) the physics and representation theory background relating to 1D Adinkras; the interested reader may see Appendix and the aforementioned references for more information along these lines. Instead, Section 2’s goal is to provide the minimum background to understand and manipulate Adinkras as purely combinatorial objects.

Then, Sections 3–5 discuss 2D Adinkras: the definition, some basic constructions, and characterizing their codes. In Section 6, we prove the main theorem, Hübsch’s Conjecture mentioned above.

Finally, Section 7, guided by the main theorem, summarizes the basic structure of 2D Adinkras, including a (computable but impractical due to combinatorial explosion) scheme to generate all 2D Adinkras. We end with some remarks in Section 8.

#### 2. Preliminaries

##### 2.1. 1D Adinkras

*Adinkras* in [4, 9, 13] will be referred to as* 1D Adinkras* in this paper, since they relate to supersymmetry in 1 dimension. In this section, we review a definition of 1D Adinkras and give some tools from previous work on their structural theory. The material in this section is mainly found in [6, 9], with minor paraphrasing.

*Definition 1 (1D Adinkras). *Let be a nonnegative integer. A* 1D Adinkra* with colors is where(1) is a finite undirected graph (called the* underlying graph* of the Adinkra) with vertex set (in [4, 13], there is also a bipartition of the vertices, where some vertices are represented by open circles and called bosons, and other vertices are represented by filled circles and called fermions. This is not necessary to include in our definition, because the bipartition can be obtained directly by taking the grading modulo 2, which is a bipartition by property below) and edge set ,(2) is a map called the* coloring*. We require that, for every and , there exists exactly one so that and . We also require that every two-colored simple cycle be of length (A* simple* cycle is one which does not repeat vertices other than the starting vertex; A* two-colored* cycle is one where the set of colors of the edges has cardinality ),(3) is a map called the* dashing*. The* parity* of on a cycle given by vertices is defined as the sum We require the parity of on every two-colored simple cycle to be odd. Such a dashing is called* admissible*,(4) is a map called the* grading*. We require that if , then . Equivalently, provides a height function that makes into the Hasse diagram of a ranked poset.

Figure 1 gives an example of a 1D Adinkra.