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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.

Several recent studies [

The classification of Adinkras is a natural goal of this work, and various aspects of this have been dealt with extensively in previous studies [

The GAAC (Garden Algebra/Adinkra/Codes) Program began [

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

The Adinkra graphs dealt with in this work are simple, undirected, bipartite, edge-

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

A

A graph is

A graph

Finally, a

Adinkra graphs were introduced in [

An Adinkra graph

Each vertex in

a colouring

each vertex being given a

The edges are also coloured in two ways:

a partition into

an edge parity assignment

Finally, the connections in the graph are essentially binary in the following manner. Every path of length two having edge colours

We refer to the valence of an Adinkra as its

The above conditions imply several additional properties. The edge-

The number of vertices of any Adinkra is a power of two.

Since the Adinkra is

When drawing Adinkra graphs, we represent the bipartition by black and white vertices. As in previous work by Doran et al. [

The Adinkras of Figure

(i) A 3-cube Adinkra and (ii) a

It is also worth mentioning at this point that the questions of whether dashed edges correspond to even or odd parity, and which of the black or white node sets correspond to bosons or fermions, have been left ambiguous, as they do not impact on the following analysis and often represent a symmetry in the system. Also note that the absolute height values of height assignments are also currently ambiguous, as only relative values of height will be relevant to the following work. Finally, we note that since this work is concerned only with questions of equivalence and isomorphism of Adinkras, the assumption of connectedness will be made throughout to simplify the analysis, without loss of generality.

There are other particularly useful ways of defining and modelling Adinkras. Before describing these it will be appropriate to introduce some more terminology, specifically relating to linear codes and Clifford algebras. We must also consider how to define notions of equivalence and isomorphism between Adinkras.

Given an Adinkra

Two operations relative to a definition of equivalence have been defined on Adinkras [

The operation of

This switching operation is analogous to Seidel switching of graphs (see [

Any Adinkras in the same switching class will be considered isomorphic. Adinkras related via vertex raising/lowering operations will be considered equivalent, but not necessarily isomorphic.

Hence the definitions of equivalence and isomorphism considered here differ slightly. Permuting the vertex labels, switching and raising/lowering operations all preserve equivalence, whereas the only operations preserving isomorphism are those of switching and permutations of vertex labels.

All three Adinkras drawn in Figure

Three Adinkras belonging to the same equivalence class. The first two are isomorphic, whereas the third belongs to a separate isomorphism class.

An

Note that as the ultimate goal of introducing Adinkras is a greater understanding of representations of off-shell supermultiplets, useful definitions of equivalence and isomorphism regarding Adinkras should be related to the underlying supermultiplets that they encode. However notions of isomorphism applied to supermultiplets are inherently context-specific. A pertinent discussion relating to automorphisms of supermultiplets can be found in Appendix B of [

Transformations acting on a supermultiplet are grouped into two classes,

A transformation induces an automorphism when the result is indistinguishable from the original when its image and preimage are the same. However “indistinguishability” is inherently contextual, and so there are several notions of equivalence between supermultiplets. Whether an outer transformation induces an automorphism can depend on the particular model being studied; hence permutations of the

Consequently, the definitions of equivalence and isomorphism provided above for Adinkras are not the only possible definitions, and as such a general classification of Adinkras must address a broader class of possible equivalences. One such alternative which is considered in Section

To introduce the tools necessary to classify the automorphism group properties of Adinkras, it will first be necessary to give a brief description of linear codes and Clifford algebras.

A binary linear

A

The

A

By applying a permutation of the coordinates of codewords and considering an alternative generating set, the code

Note that the freedom to interchange coordinates means that there are numerous standard form generating sets that we could produce, which in turn would have different codes associated with them. Given the nature of the standard form however, any code with an associated standard form generating set will have a

Note also that generating sets will be assumed to be minimal in that the codewords comprising the set are linearly independent. Thus, a generating set with

For any

The notion of codewords can be effectively used to furnish a further organisation of the edge and vertex sets of an Adinkra. If we order the edge colours of an

Applying this codeword representation to the vertices of a 3-cube Adinkra yields the graph of Figure

A 3-cube Adinkra shown together with a codeword representation for each vertex.

An important theorem of [

Every

This quotienting process works as follows. We start with a doubly even

If we take the 4-cube Adinkra, with codewords given by the elements of

One will consider the

It will be convenient to view the 4-cycle of Adinkras in terms of the anticommutativity property of Clifford generators. If we consider each position in the codeword representation of a vertex of an

For example, a vertex with codeword representation (01100) corresponds to the product of Clifford generators

In connection with this Clifford generator notation, we define a standard form for the switching state of an

A

The 3-cube Adinkra of Example

As we have seen, many Adinkras can be formed by quotienting the

Start with the standard form

Find a standard form generating set for

Associate the

The

The factor of

Define

Given a doubly even

As we are only considering equivalence classes, the height assignments will be ignored.

Consider an

Isolating any set of

Since this graph-wide factor of

As odd-weight vertices are considered to be fermions, the

For case (ii), the antipodal points are

Case (iii) proceeds similarly to case (ii).

Hence the graphs produced via this construction method are indeed

A

The smallest nontrivial

In this case the two Adinkras formed belong to different equivalence classes, as we demonstrate in the following section.

Two

This construction provides a much simpler practical method of constructing Adinkras, and the standard form will be convenient in the proof of later results. Note that all

We now have sufficient tools to derive the main result of this part: classifying the automorphism group of an Adinkra with respect to the local parameters of the graph, namely, the associated doubly even code. We start with several observations regarding the automorphism properties of Adinkras.

Ignoring height assignments, the

Note that equivalently, we may consider Adinkras with only two heights to correspond to the assignment of bosons and fermions, and we call these valise Adinkras. Those with more than two heights are called nonvalise Adinkras.

It suffices to show that any

To show this, we begin with a standard form

Then the switching state of edge

Note that if

The

Consider any

Given some

In particular, fixing the parity of the

This reduces the problem of finding automorphisms of a general

Consider an

Consider without loss of generality the case where

We wish to know when

For edge dimension

The value of

Now

Two vertices having the same relative inner products with respect to each element of a set

Conversely to Theorem

Theorem

In [

An

The automorphism group of a valise

In the work of Gates et al. [

Throughout the following section we will consider an

The

Consider the 3-cube Adinkra of Example

If we order the vertex set into bosons and fermions, this Adinkra can be represented by the matrix:

We define

Further details regarding these objects can be found in [

In order to generalise (

The cycles of an

The cycles of an

each edge dimension is traversed

at least one edge dimension is traversed

In fact, the codeword

Consider the product of

The elements of the main diagonal of

trivially, if each edge dimension contained in

if

Consider the case

In case (ii) of Lemma

Consider the product of Clifford generators corresponding to path

Assume (i) holds. Then

One immediate corollary of the preceding lemma is that the trace of

Given an

The trace of

The

Note that if

The results up to this point deal with equivalence classes of valise Adinkras, where we consider only 2-level Adinkras. For the nonvalise case, we require a method of partitioning general

Two equivalent, nonisomorphic Adinkras with the same values of

Suppose we are given an

Consider

Order the neighbours of

Repeat this for vertices at distance 2, beginning with the neighbours of

Repeat this similarly for vertices at each distance, until all vertices have been assigned an ordering in

In other words,

To formalise the partitioning of heights discussed above, consider an

Consider two Adinkras

If

The preceding theorem provides an efficient method of classifying Adinkras according to their isomorphism class. Note that the classification is relative to a given ordering of the edge colours and requires a knowledge of the associated doubly even code. In cases where the associated code is unknown, Lemma

Given an

We provide an example of the above certificates below. Consider the two Adinkras of Figure

Two equivalent, nonisomorphic

To analyse the above Adinkras, we first order the edge colours from green to purple, such that, in the codeword representation, green corresponds to the last coordinate and purple to the first. Note that henceforth we will order the right-most coordinate to the left-most coordinate, consistent with a bit-string interpretation of codewords in the computational application. Alternatively, the green edges are associated with the first Clifford generator and the purple edges with the fifth generator. As these are

In [

Firstly however, we note that a different definition of isomorphism is considered in [

Allowing permutations of the edge colours to preserve isomorphism, an

In other words, any permutations that correspond to symmetries of the code will extend naturally to automorphisms of the Adinkra. Conversely, if a permutation of the codeword is not in the automorphism group of the code, then trivially it cannot be an automorphism of the Adinkra. Note that Lemma

Hence, Klein flip degeneracy no longer necessarily exhibits for Adinkras containing the all-1 codeword. Instead, note that any automorphism of the associated doubly even code corresponding to an

Somewhat fortuitously, a recent result regarding doubly even codes sheds some light on this situation. Doran et al. [

An

However, the question of whether any other Adinkras with Klein flip degeneracy (again allowing for edge-colour permutations) exist still remains. This is equivalent to asking whether any doubly even codes exist, which are both not self-dual and contain no nontrivial even-permutation automorphisms, and is the subject of continuing work. Furthermore, if the automorphism group of doubly even codes turns out to be sufficiently rich, a greatly simplified certificate encoding the isomorphism class of an Adinkra could be developed, as will be discussed briefly in Section

In [

Two nonisomorphic height 3,

Applying the methods described in the previous section to this pair proceeds as in the analysis of the Adinkras of Figure

The second pair of Adinkras in [

Two nonisomorphic height 3,

In this case, after ordering the edge colours from dark blue to green, as for the top left node of the first Adinkra, both Adinkras have the associated code generated by

The automorphism group results of the preceding sections were also verified numerically, independently to the analytical results. All

all

all 1-level

All doubly even

Then given a vertex

Permute the ordering of the vertices relative to their connections to

Switch vertices

Repeat for neighbours of vertex

Repeat for each vertex, in the ordering above, such that edges appearing lexicographically earlier in the adjacency matrix are of even parity where possible.

This defines a unique switching state of the Adinkra, up to isomorphism. Furthermore it is a canonical form, in that any two vertices belonging to the same orbit result in identical matrix forms. To illustrate the above process, we consider a

Standard form valise Adinkra.

Consider the process of converting this Adinkra to canonical form, relative to vertex (0011) (the fifth white vertex from the left in Figure

Adinkra after step (i).

Steps (ii) and (iii) correspond to a set of switching operations, permuting the switching state in the following stages, as shown in Figure

Adinkra after steps (ii) and (iii).

Step (iv) then involves the remaining set of transformations, shown in Figure

Adinkra after step (iv).

At this point the Adinkra is now in canonical form. The particular switching state of this resulting Adinkra is unique to choices of source vertices in the same orbit, in this case the set of vertices

This work provides a graph theoretic characterisation of Adinkras, in particular classifying their automorphism groups according to an efficiently computable set of local parameters. In the current work, a number of comparisons are made to previous work. The connection between Adinkras and codes [

All nonvalise Adinkras through a series of node raising and lowering can be brought to the form of a valise Adinkra. In this sense

Additionally, the classification method used here is robust in the sense that it can be readily adapted to address a variety of definitions of isomorphism/equivalence, relevant to the particular supersymmetric system of interest. For instance, in Section

It is the work of future investigations to explore whether these tools (

Note that when edge-colour permutations are allowed, the calculation of the certificate encoding the isomorphism class of an Adinkra, as detailed in Section

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Akihiro Munemasa for detailed comments and suggestions on this paper, as well as Cheryl Praeger, Cai-Heng Li, Gordon Royle, and John Bamberg for several useful discussions. This work was partially supported by the National Science Foundation Grant PHY-13515155. The authors would also like to thank the Institute of Advanced Studies at The University of Western Australia for providing travel support, which enabled initial discussions among the authors.