Advances in Mathematical Physics

Volume 2015 (2015), Article ID 584542, 17 pages

http://dx.doi.org/10.1155/2015/584542

## Automorphism Properties and Classification of Adinkras

^{1}School of Physics, University of Western Australia, Perth, WA 6009, Australia^{2}Center for String and Particle Theory, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA

Received 16 March 2015; Accepted 7 July 2015

Academic Editor: Pavel Kurasov

Copyright © 2015 B. L. Douglas et al. 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.

#### Abstract

Adinkras are graphical tools for studying off-shell representations of supersymmetry. In this paper we efficiently classify the automorphism groups of Adinkras relative to a set of local parameters. Using this, we classify Adinkras according to their equivalence and isomorphism classes. We extend previous results dealing with characterization of Adinkra degeneracy via matrix products and present algorithms for calculating the automorphism groups of Adinkras and partitioning Adinkras into their isomorphism classes.

#### 1. Introduction

Several recent studies [1–10] have introduced and developed a novel approach to the off-shell problem of supersymmetry. In particular, a graph theoretic tool has emerged to tackle this problem by encoding representations of supersymmetry into a family of graphs termed Adinkras. This graphical encoding has the advantage of allowing convenient manipulation of these objects, with the goal of achieving a deeper understanding of the underlying representations.

The classification of Adinkras is a natural goal of this work, and various aspects of this have been dealt with extensively in previous studies [2, 4, 8, 11], in which many of the properties of Adinkra graphs have been ascertained. The works of [2, 3] relate topological properties of Adinkras to doubly even codes and Clifford algebras. The work in [11] is particularly striking as apparently Betti numbers and functions similar to those of catastrophe theory seem on the horizon. However, we will not direct our current effort toward these observations.

The GAAC (Garden Algebra/Adinkra/Codes) Program began [12–15] with a series of observations of what appeared to be universal matrix algebra structures that seem to occur in* all* off-shell supersymmetrical theories. Two unexpected transformations have occurred since this start. First, there emerged Adinkras providing a graphical technology to represent these matrix algebras and second the connection of Adinkras to codes. As the ultimate goal of this program is to provide a definitive classification of* all* off-shell supersymmetrical theories, the appearance of these new unexpected discoveries continues to be encouraging that the goal can be reached despite the pessimism surrounding this over thirty-year-old unsolved problem.

The goal of this paper is to present a method of efficiently distinguishing Adinkras, in other words, a way of efficiently solving the graph isomorphism problem, restricted to this particular class of graphs. This is achieved via a classification of their automorphism group, together with an efficient algorithm for computing this automorphism group for any given Adinkra. In particular, we specify the automorphism group of an Adinkra in terms of its associated doubly even code. These codes are characterized in terms of local properties of Adinkras, such that the codes, and hence the associated equivalence and isomorphism classes of these graphs, can be efficiently computed.

The structure of the paper is as follows: Section 2 provides a formal graph theoretic definition of Adinkras, introducing some graph theoretic terms related to their study and discussing some basic properties that are derived from the definition. Section 3 defines the notions of equivalence and isomorphism on the class of Adinkras. We relate Adinkra graphs to doubly even codes, citing a result from [2] that all Adinkras have an associated code. The work of [3] relating properties of Adinkras to Clifford algebras is also discussed, and we define a standard form for Adinkras, used in the proof of later results. Section 4 establishes the main result of this work, classifying the automorphism group of valise Adinkras in terms of the related doubly even code. In Section 5 these results are used to generalize some results of [5] and provide a polynomial that partitions Adinkras into their equivalence classes. In Section 6 this is extended to isomorphism classes of nonvalise Adinkras, and we provide an associated algorithm that accomplishes this partitioning. Section 7 details some of the associated numerical methods and results and provides some additional examples.

#### 2. Adinkra Graphs

##### 2.1. Graph-Theoretic Notation

The Adinkra graphs dealt with in this work are simple, undirected, bipartite, edge-*N*-partite, and edge- and vertex-colored graphs. Note that in previous work [1–3] Adinkras are considered to be directed graphs. However, as this information is naturally encoded into the height assignment component of the vertex coloring, we remove the edge directions here to simplify the analysis.

We define a few of the graph theoretic terms below. For a more complete treatment, see [16].

A* simple, undirected graph * consists of a vertex set together with an edge set of unordered pairs of .

A graph is* bipartite* if its vertex set can be partitioned into two disjoint sets such that no edges lie wholly within either set. Equivalently, this is a graph containing no odd-length cycles.

A graph is* edge-N-partite* if its edge set can be partitioned into disjoint sets, such that every vertex is incident with exactly one edge from each of these sets. In other words, a graph is edge--partite if it has a 1-factorisation.

Finally, a* coloring* of the edge or vertex set of a graph is a partitioning of these sets into different* colour classes*. Formally, this restricts the automorphism group of the graph to the subset that setwise stabilises these colour classes, the subset that does not map vertices (resp., edges) in one colour class to vertices (resp., edges) in another.

##### 2.2. Definition and Properties

Adinkra graphs were introduced in [1] to study off-shell representations of supersymmetry. Thorough definitions are provided in [2, 3, 8], although, as mentioned above, they vary slightly from the definition presented here in that for the purposes of this study we will consider them to be undirected graphs. The most current and complete definition of Adinkras can be found in Definition 3.2 in the work of [2]. With that in mind, we use the following definition of Adinkras for the remainder of this work.

*Definition 1. *An Adinkra graph is a bipartite, vertex- and edge-coloured graph with the following properties.

Each vertex in is coloured in two ways: (i)a colouring corresponding naturally to the bipartition (labelled as* bosons* and* fermions*);(ii)each vertex being given a* height assignment*, hgt: such that adjacent vertices are at adjacent heights.

The edges are also coloured in two ways: (i)a partition into colour classes corresponding to an edge--partition of the graph;(ii)an edge parity assignment ; we term these two edge types* dashed* and* solid*.

Finally, the connections in the graph are essentially binary in the following manner. Every path of length two having edge colours defines a unique 4-cycle with edge colours . That is to say, of all 4-cycles including these two edges, only one has the edge colours . Such 4-cycles all have an odd number of dashed edges.

We refer to the valence of an Adinkra as its* dimension*. Hence an Adinkra with edge colours (not counting edge parity) is an -dimensional Adinkra. There will generally be an assumed ordering of the edge colours from 1 to , with the term ’th* edges* or ’th* edge dimension* referring to all edges of the ’th colour. The edge parity is often referred to as* dashedness* (see, e.g., [5]). In this work we also refer to it as the* switching state* of an edge or set of edges, motivated by a forthcoming analogy to switching and two graphs. Relative to the edge-colour ordering, the switching state of an edge of colour will be denoted alternately by , , or (referring to the ’th edge of the vertex or or simply the edge ).

The above conditions imply several additional properties. The edge--partite restriction requires equal numbers of bosons and fermions; hence the bipartition must consist of two sets of equal size. Moreover, we have the following important property.

Lemma 2. *The number of vertices of any Adinkra is a power of two.*

*Proof. *Since the Adinkra is -regular and edge--partite and satisfies the 4-cycle property, there exists some , such that, for any -length subset of the edge colours (i.e., removing all of the edges whose colour does not lie within this subset), the Adinkra is simply an -dimensional hypercube or -cube. Hence we have that the number of vertices is equal to . Note that if , then ignoring edge and vertex colouring the resulting graph is simply the -cube. Hence we term the corresponding Adinkra the -*cube Adinkra* (in Section 4 we show that this is unique to ). Where , we denote the corresponding Adinkra to be an * Adinkra*, where .

When drawing Adinkra graphs, we represent the bipartition by black and white vertices. As in previous work by Doran et al. [3], the height assignments will be represented by arranging the vertices in rows, incrementally, according to height.

*Example 3. *The Adinkras of Figure 1 satisfy all requirements listed above. Note that nodes in the same bipartition (bosons or fermions) must be at the same height of modulo 2 and that every 4-cycle containing only two edge colours has an odd number of dashed/solid edges. In case (i), this is simply every 4-cycle; however case (ii) also contains 4-cycles with four edge colours.