# Structural Theory and Classification of 2D Adinkras

Authors: Kevin Iga; Yan X. Zhang
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

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


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## Body

## 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 call1D 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 2D 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 2D Adinkras.We begin in Section2 by recalling the definition of (1D) Adinkras and some of their features, reviewing thecode associated with an Adinkra [6] and the concept ofvertex 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, Sections3–5 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, Section7, 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 as1D 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).
Letn be a nonnegative integer. A1D Adinkra with n colors is (V,E,c,μ,h) where(1)
(V,E) is a finite undirected graph (called theunderlying 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 h modulo 2, which is a bipartition by property (4) below)V and edge set E,(2)
c:E→[n]:={1,…,n} is a map called thecoloring. We require that, for every v∈V and i∈[n], there exists exactly one w∈V so that (v,w)∈E and c(v,w)=i. We also require that every two-colored simple cycle be of length 4 (Asimple cycle is one which does not repeat vertices other than the starting vertex; Atwo-colored cycle is one where the set of colors of the edges has cardinality 2),(3)
μ:E→Z2={0,1} is a map called thedashing. Theparity of μ on a cycle given by vertices (v0,…,vk) is defined as the sum (1)∑i=0k-1μvi,vi+1mod2,We require the parity ofμ on every two-colored simple cycle to be odd. Such a dashing μ is calledadmissible,(4)
h:V→Z is a map called thegrading. We require that if (v,w)∈E, then hv-hw=1. Equivalently, h provides a height function that makes (V,E) into the Hasse diagram of a ranked poset.Figure1 gives an example of a 1D Adinkra.Figure 1
Example of a 1D Adinkra with 3 colors. The coloring is represented by colors on the edges. The dashing is represented by having a dashed edge ifμ(e)=1 and a solid edge if μ(e)=0. The grading is represented by the vertical height as indicated on the axis on the left.


### 2.2. Structural Aspects of 1D Adinkras

LetA be a 1D Adinkra with n colors, with vertex set V. For all i∈[n], define qi:V→V such that, for all v∈V, qi(v) is the unique vertex joined to v by an edge of color i. In [6], it was shown that the map qi is a graph isomorphism (in fact, an involution) from the underlying graph of A to itself which preserves colors. qi’s commute with each other. These facts can be used to combine the q1,…,qn maps into an action of Z2n on the graph (V,E) underlying the Adinkra in the following way.Definition 2.
The action ofZ2n on the graph (V,E) underlying the Adinkra is given on vertices by (2)x1,…,xnv=q1x1∘⋯∘qnxnv.
Intuitively, the action of a sequence of bits, for instance, 11001, on a vertex is obtained by following edges with colors that correspond to 1’s in the sequence (in this case, colors 1, 2, and 5). The fact that theqi’s commute implies that the order of the colors does not matter.
The AdinkraA is connected if and only if Z2n action is transitive on the vertex set of A. In this case the stabilizers of all vertices are equal (in general the stabilizers of two points in the same orbit are conjugate; here we know more since the group is abelian). Define C(A), thecode of the AdinkraA, to be this stabilizer. This is abinary linear code of length n (i.e., a linear subspace of Z2n). As these are the only types of codes we use, from now on we simply saycode to mean “binary linear code.”
We call the elements of a codecodewords. Theweight of a codeword w is the number of 1’s in the word. A code is calledeven if all its codewords have even weight. A code is calleddoubly even if all its codewords have weight divisible by 4. An example of a doubly even code is the span 〈111100,001111〉, which has 22=4 elements. An example of a code that is even but not doubly even is the 1-dimensional code 〈11〉.
Codes are surprisingly relevant to the structural theory of Adinkras; in fact, one should basically think of the underlying graph of an Adinkra as a doubly even code, as we now see.


### 2.3. Quotients

We now know that the stabilizer of our action on the graph is a code. We can also go in the opposite direction: letIn, theHamming cube, be the graph with 2n vertices labeled by strings of length n using the alphabet {0,1}, with an edge between two vertices v and w if and only if they differ in exactly one place. There is a natural coloring on In: just color each edge by the coordinate where the two vertices differ. Now, codes in Z2n act on In by bitwise addition modulo 2, and these are isomorphisms that preserve colors. A natural operation to consider on a colored graph Γ=(V,E,c) and a group C acting on V via graph isomorphisms that preserve colors is thequotientΓ/C, where the vertices are defined to be orbits in V/C, and we have an c-colored edge (v,w) if and only if there is at least one c-colored edge (v′,w′)∈E with v′ in the orbit v and w′ in the orbit w.Theorem 3.
I
n
/
C is the colored graph of some 1D Adinkra if and only if C is a doubly even code.See [6] for the original proof of this result. See [9] for a more general treatment of quotienting by a code and a slightly extended correspondence (note that the quotient Γ/C does not necessarily retain nice properties of Γ; it does not even have to be a simple graph. It may also have edges with different colors between two vertices. Part of the work here is to show that these pathologies do not happen when C is a doubly even code) between graph properties of the quotient In/C and properties of the code C.Figure2 provides an example of a quotient of I4 by a code that obeys this theorem. Encoded within the proof of Theorem 3 is the fact that if C is a doubly even code; then there exists an admissible dashing. A constructive proof of existence can be found in [7]. See [9] for an enumeration of all admissible dashings for any doubly even code.Figure 2
(a) The colored graphI4. (b) The quotient I4/{0000,1111}. (c) As the code {0000,1111} is doubly even, there exists an Adinkra with the quotient as its underlying graph by Theorem 3.
(a)
(b)
(c)


### 2.4. Vertex Switching

Vertex switching was first introduced in the context of Adinkras in [13] and is more thoroughly set in its context in [8, 9].Definition 4 (vertex switching).
Given an AdinkraA, and a vertex v of A, we definevertex switching at v to be the operation on A that returns a new Adinkra A- with the same vertices, edges, coloring, and grading but a new dashing μ- so that(3)μ-e=1-μe,ifeisincidenttov,μe,otherwise.We leave to the reader to check that μ- is still an admissible dashing; since the vertices, edges, coloring, and grading(s) are the same, A- remains an Adinkra (Figure 3). We also use avertex switching ofA to refer to a composition of vertex switchings at various vertices of A.Figure 3
A vertex switching atv turns the Adinkra A on the left into the Adinkra A- on the right. The two Adinkras have the same dashing except precisely the edges that are incident to v. Note that in both cases each face of the cube has an odd number of dashed edges.
(a)
(b)In [14], vertex switching was first applied to dashings in Adinkras from a point of view inspired by Seidel’stwo graphs [15]. (In Seidel’s setting,vertex switching switched the existence of edges, not the sign of edges; this can be seen as equivalent to our definition applied to the complete graph. The type of vertex switching we do in this paper is sometimes called vertex switching onsigned graphs in literature for disambiguation.) An enumeration of vertex switching classes leading to counting the number of dashings of any 1D Adinkra can be found in [9].


## 2.1.
1D Adinkras

Adinkras in [4, 9, 13] will be referred to as1D 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).
Letn be a nonnegative integer. A1D Adinkra with n colors is (V,E,c,μ,h) where(1)
(V,E) is a finite undirected graph (called theunderlying 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 h modulo 2, which is a bipartition by property (4) below)V and edge set E,(2)
c:E→[n]:={1,…,n} is a map called thecoloring. We require that, for every v∈V and i∈[n], there exists exactly one w∈V so that (v,w)∈E and c(v,w)=i. We also require that every two-colored simple cycle be of length 4 (Asimple cycle is one which does not repeat vertices other than the starting vertex; Atwo-colored cycle is one where the set of colors of the edges has cardinality 2),(3)
μ:E→Z2={0,1} is a map called thedashing. Theparity of μ on a cycle given by vertices (v0,…,vk) is defined as the sum (1)∑i=0k-1μvi,vi+1mod2,We require the parity ofμ on every two-colored simple cycle to be odd. Such a dashing μ is calledadmissible,(4)
h:V→Z is a map called thegrading. We require that if (v,w)∈E, then hv-hw=1. Equivalently, h provides a height function that makes (V,E) into the Hasse diagram of a ranked poset.Figure1 gives an example of a 1D Adinkra.Figure 1
Example of a 1D Adinkra with 3 colors. The coloring is represented by colors on the edges. The dashing is represented by having a dashed edge ifμ(e)=1 and a solid edge if μ(e)=0. The grading is represented by the vertical height as indicated on the axis on the left.


## 2.2. Structural Aspects of 1D Adinkras

LetA be a 1D Adinkra with n colors, with vertex set V. For all i∈[n], define qi:V→V such that, for all v∈V, qi(v) is the unique vertex joined to v by an edge of color i. In [6], it was shown that the map qi is a graph isomorphism (in fact, an involution) from the underlying graph of A to itself which preserves colors. qi’s commute with each other. These facts can be used to combine the q1,…,qn maps into an action of Z2n on the graph (V,E) underlying the Adinkra in the following way.Definition 2.
The action ofZ2n on the graph (V,E) underlying the Adinkra is given on vertices by (2)x1,…,xnv=q1x1∘⋯∘qnxnv.
Intuitively, the action of a sequence of bits, for instance, 11001, on a vertex is obtained by following edges with colors that correspond to 1’s in the sequence (in this case, colors 1, 2, and 5). The fact that theqi’s commute implies that the order of the colors does not matter.
The AdinkraA is connected if and only if Z2n action is transitive on the vertex set of A. In this case the stabilizers of all vertices are equal (in general the stabilizers of two points in the same orbit are conjugate; here we know more since the group is abelian). Define C(A), thecode of the AdinkraA, to be this stabilizer. This is abinary linear code of length n (i.e., a linear subspace of Z2n). As these are the only types of codes we use, from now on we simply saycode to mean “binary linear code.”
We call the elements of a codecodewords. Theweight of a codeword w is the number of 1’s in the word. A code is calledeven if all its codewords have even weight. A code is calleddoubly even if all its codewords have weight divisible by 4. An example of a doubly even code is the span 〈111100,001111〉, which has 22=4 elements. An example of a code that is even but not doubly even is the 1-dimensional code 〈11〉.
Codes are surprisingly relevant to the structural theory of Adinkras; in fact, one should basically think of the underlying graph of an Adinkra as a doubly even code, as we now see.


## 2.3. Quotients

We now know that the stabilizer of our action on the graph is a code. We can also go in the opposite direction: letIn, theHamming cube, be the graph with 2n vertices labeled by strings of length n using the alphabet {0,1}, with an edge between two vertices v and w if and only if they differ in exactly one place. There is a natural coloring on In: just color each edge by the coordinate where the two vertices differ. Now, codes in Z2n act on In by bitwise addition modulo 2, and these are isomorphisms that preserve colors. A natural operation to consider on a colored graph Γ=(V,E,c) and a group C acting on V via graph isomorphisms that preserve colors is thequotientΓ/C, where the vertices are defined to be orbits in V/C, and we have an c-colored edge (v,w) if and only if there is at least one c-colored edge (v′,w′)∈E with v′ in the orbit v and w′ in the orbit w.Theorem 3.
I
n
/
C is the colored graph of some 1D Adinkra if and only if C is a doubly even code.See [6] for the original proof of this result. See [9] for a more general treatment of quotienting by a code and a slightly extended correspondence (note that the quotient Γ/C does not necessarily retain nice properties of Γ; it does not even have to be a simple graph. It may also have edges with different colors between two vertices. Part of the work here is to show that these pathologies do not happen when C is a doubly even code) between graph properties of the quotient In/C and properties of the code C.Figure2 provides an example of a quotient of I4 by a code that obeys this theorem. Encoded within the proof of Theorem 3 is the fact that if C is a doubly even code; then there exists an admissible dashing. A constructive proof of existence can be found in [7]. See [9] for an enumeration of all admissible dashings for any doubly even code.Figure 2
(a) The colored graphI4. (b) The quotient I4/{0000,1111}. (c) As the code {0000,1111} is doubly even, there exists an Adinkra with the quotient as its underlying graph by Theorem 3.
(a)
(b)
(c)


## 2.4. Vertex Switching

Vertex switching was first introduced in the context of Adinkras in [13] and is more thoroughly set in its context in [8, 9].Definition 4 (vertex switching).
Given an AdinkraA, and a vertex v of A, we definevertex switching at v to be the operation on A that returns a new Adinkra A- with the same vertices, edges, coloring, and grading but a new dashing μ- so that(3)μ-e=1-μe,ifeisincidenttov,μe,otherwise.We leave to the reader to check that μ- is still an admissible dashing; since the vertices, edges, coloring, and grading(s) are the same, A- remains an Adinkra (Figure 3). We also use avertex switching ofA to refer to a composition of vertex switchings at various vertices of A.Figure 3
A vertex switching atv turns the Adinkra A on the left into the Adinkra A- on the right. The two Adinkras have the same dashing except precisely the edges that are incident to v. Note that in both cases each face of the cube has an odd number of dashed edges.
(a)
(b)In [14], vertex switching was first applied to dashings in Adinkras from a point of view inspired by Seidel’stwo graphs [15]. (In Seidel’s setting,vertex switching switched the existence of edges, not the sign of edges; this can be seen as equivalent to our definition applied to the complete graph. The type of vertex switching we do in this paper is sometimes called vertex switching onsigned graphs in literature for disambiguation.) An enumeration of vertex switching classes leading to counting the number of dashings of any 1D Adinkra can be found in [9].


## 3.
2D Adinkras

Just as 1D Adinkras were used to study 1D supersymmetry, Gates and Hübsch developed2D Adinkras to study 2D supersymmetry [11, 12]. We use a definition here that is equivalent to the one found there. (The main notational difference is a kind of change of coordinates: there, nodes are labeled bymass dimension, which is hL+hR, andspin, which is hR-hL. Mass dimension is the units of mass associated with the field, where c=ħ=1 and spin is the eigenvalue of x∂t+t∂x.)A2D Adinkra is similar to a 1D Adinkra except that some colors are called “left-moving” and the other colors called “right-moving.” Edges are called “left-moving” if they are colored by left-moving colors and are called right-moving otherwise. Furthermore, there are two gradings, one that is affected by the left-moving edges and the other for the right-moving edges. More formally, we have the following.Definition 5 (2D Adinkras).
Letp and q be nonnegative integers. A2D Adinkra with(p,q)colors is a 1D Adinkra (V,E,c,μ,h) with p+q colors, and two grading functions hL:V→Z and hR:V→Z so that we have the following: (i)
h(v)=hL(v)+hR(v).(ii)
Lete be an edge. If c(e)≤p then e is called aleft-moving edge; if c(e)>p then it is called aright-moving edge. Similarly, the first p colors are calledleft-moving colors and the last q colors are calledright-moving colors.(iii)
if(v,w) is a left-moving edge, then hLv-hLw=1 and hR(v)=hR(w). If (v,w) is a right-moving edge, then hRv-hRw=1 and hL(v)=hL(w).See Figure4 for an example of a 2D Adinkra. The main goal of this paper is to follow the program set out in [12] and completely characterize 2D Adinkras. As a first step, we define the natural notion ofproducts in the following section.Figure 4
A2D Adinkra with (2,2) colors. The grading coordinates are given next to the nodes as (hL,hR).


## 4. Products

One important way to produce a2D Adinkra with (p,q) colors is to take the product of two 1D Adinkras (one with p colors, and the other with q colors), using the following construction.Construction 6.
Letp and q be nonnegative integers. Let A1=(V1,E1,c1,μ1,h1) be a 1D Adinkra with p colors and let A2=(V2,E2,c2,μ2,h2) be a 1D Adinkra q colors. We define theproduct of these Adinkras A1×A2 as the following 2D Adinkra with (p,q) colors: (4)A1×A2=V,E,c,μ,h1,h2,where V=V1×V2 and there are two kinds of edges in E:(i)
For every edgee in E1 connecting vertices v and w∈V1, and for every vertex x∈V2, we have an edge in E between vertices (v,x) and (w,x) in V=V1×V2 of color c1(e) and dashing μ1(e).(ii)
For every edgee in E2 connecting vertices v and w∈V2 and for every vertex x∈V1, we have an edge in E between vertices (x,v) and (x,w) in V=V1×V2 of color p+c2(e) and dashing μ2e+h1x  mod2.
See Figure5 for an example.Figure 5
Constructing the2D Adinkra (b) from two smaller 1D Adinkras (a) as a product. Note that the dashings are all “consistent” with the smaller Adinkras, except for the right-moving edges on the upper-left “boundary” of the rectangle; these correspond to right-moving edges where the grading corresponding to the first Adinkra has height 1.
(a)
(b)This definition is intended to be a graph-theoretic version of the tensor product construction inZ2-graded representations (see Appendix for more details). The edges that come from E1 give rise to left-moving edges, and the edges that come from E2 give rise to right-moving edges. It follows easily that for every vertex v in A1×A2 and for every color in [n] there is a unique edge in A1×A2 incident to v. The fact that two-colored simple cycles have length 4 follows from following cases, depending on whether the colors are both left-moving, both right-moving, or one of each. The parity condition for an Adinkra also follows from considering these cases. The properties related to the bigrading are straightforward. We then have the following.Proposition 7.
Given AdinkrasA1 and A2, A1×A2 is a 2D Adinkra.Definition 8 (extending codes).
Letp and q be nonnegative integers and let n=p+q. Define ZL:Z2p→Z2n to be the function that appends q zeros, so that, for instance, if p=4 and q=3, then ZL(1011)=1011000. Likewise, define ZR:Z2q→Z2n to be the function that prepends p zeros.
Our most common use of this notation is as follows: ifC is a binary block code of length p, we write ZL(C) for the image under ZL. Likewise, if C is a binary block code of length q, we write ZR(C) for the image under ZR.Proposition 9.
LetA1 and A2 be as above. Then(5)CA1×A2=ZLCA1⊕ZRCA2.Proof.
Let(v1,v2)∈A1×A2. Let x∈Z2N. We can write x=xL+xR, where xL is zero in the last q bits and xR is zero in the first p bits. Now (6)xv1,v2=xL+xRv1,v2=xLv1,xRv2.This means that x(v1,v2)=(v1,v2) if and only if xLv1=v1 and xRv2=v2. So x∈C(A1×A2) if and only if xL∈ZL(C(A1)) and xR∈ZR(C(A2)).


## 5. Codes for 2D Adinkras

LetA be a connected 2D Adinkra with (p,q) colors. Then there is a doubly even code C(A) associated with A. But as a 2D Adinkra, we make a distinction between the first p colors and the last q colors, which for a code translates to the first p bits and the last q bits. A natural question is “knowing that a 1D Adinkra can be enriched into a 2D Adinkra, what else can we say about its code?” In this section, we address this question.Definition 10 (weights for left-moving and right-moving colors; ESDE codes).
Recall that, for any vectorx∈Z2n, theweight of x, denoted by wt(x), is the number of 1’s in x. Likewise, wtL(x) is the the number of 1’s in the first p bits and wtR(x) is the number of 1’s in the last q bits of x. Let a code C, along with the parameters (p,q), be called aneven-split doubly even (ESDE) code if C is doubly even and all codewords x in C have both wtL(x) and wtR(x) even.This definition of ESDE codes is due to [12], which also proves the following.Theorem 11.
IfA is a connected 2D Adinkra with (p,q) colors, then C(A) is an ESDE.We now prove the converse of this theorem. That is, given an ESDE code, there exists connected2D Adinkra with that code. This procedure is analogous to the Valise Adinkras in 1D [4, 13], in that the possible values of each component (hL,hR) of the bigrading are as small as possible, that is, two values.Construction 12.
LetC be an ESDE code. We will describe a construction that provides a 2D Adinkra with code C, called theValise 2D Adinkra. First, since C is doubly even, there exists a connected 1D Adinkra A with code C(A)=C by Theorem 3. Fix a vertex 0¯ of A. Now for every vertex v there is a vector x∈Z2n so that x0¯=v. Then define hL(v)=wtL(x)  mod2 and hR(v)=wtR(x)  mod2. Note that these functions are well-defined since C is ESDE. Then (hL,hR) is a bigrading for A, making it a 2D Adinkra. An example of the kind of 2D Adinkra that arises from this construction is Figure 4.We therefore have the following.Theorem 13.
For a codeC⊂Z2n, there exists a 2D Adinkra A with C(A)=C if and only if C is a ESDE code.The structure of the ESDE relates to interesting features of the colored graph ofA. Let AL be the 1D Adinkra with p colors that consists of only the left-moving edges of A. Let AR be the 1D Adinkra with q colors that consists of only the right-moving edges of A (where we shift the colors so that they range from 1 to q instead of p+1 to p+q). Pick a vertex 0¯ in A. Let AL0 be the connected component of AL containing 0¯ and let AR0 be the connected component of AR containing 0¯.We now see that the codes forAL0 and AR0 (with an appropriate number of 0s added to the left or right as necessary) provide important linear subspaces of C(A).Proposition 14.
The following hold:(7)ZLCAL0=CA∩ZLZ2p,ZRCAR0=CA∩ZRZ2q.In other words, the codewords that are zero in the lastq bits are precisely the codewords from C(AL0) with q zeros appended to the right; and the codewords that are zero in the first p bits are precisely the codewords from C(AR0) with p zeros prepended to the left.Proof.
Ifx∈ZL(C(AL0)), so that x=ZL(y) for some y∈C(AL0), Then trivially x∈ZL(Z2p). Furthermore, y0¯=0¯ in AL0. In A, ZL(y)=x does exactly the same thing, so x0¯=0¯ in A. Therefore x∈C(A).
Conversely ifx∈C(A)∩ZL(Z2p), then by the definition of the group action there is a path in A from 0¯ to 0¯ following the colors corresponding to the 1s in x. Since x∈ZL(Z2p), we have that this path only consists of left-moving colors and so lies in AL0.
The proof forZR(C(AR0)) is similar.Corollary 15.
The following hold:(8)ZLCAL0⊕ZRCAR0⊂CA.Comparing with Proposition9, this corollary says that the code for AL0×AR0 is a linear subspace of the code for A.Simply for brevity (and thus, readability) in later descriptions, we define using the product construction aboveA′=AL0×AR0 and the code C′=ZL(C(AL0))⊕ZR(C(AR0)). Also, we write C for C(A). Then C(A′)=C′ and C′⊂C.Lemma 16.
There exists a binary linear block codeK so that (9)C=C′⊕K.Proof.
From Corollary15 and basic linear algebra, there exists a vector subspace K of Z2n that is a vector space complement of C′ in C.Note thatK is not necessarily uniquely defined. It is, however, uniquely defined up to adding vectors in C′. So a more invariant approach would be to use C/C′ instead of K, but K has the advantage of being a code, therefore more concrete for computational purposes.The interpretation ofK can be obtained by examining the set V0=AL0∩AR0. Since AL0 has only left-moving edges and AR0 has only right-moving edges, V0 has no edges at all: only vertices. Furthermore, for every v∈V0, we have hL(v)=hL(0¯) and hR(v)=hR(0¯) so all of the vertices in V0 have the same bigrading.We now show that there is a bijection betweenK and V0. As before, for every x∈Z2n, we write x=xL+xR, where xL is zero in the last q bits and xR is zero in the first p bits. Using this notation, we have the following theorem.Theorem 17.
The mapΨ:K→V0 given by Ψ(x)=xL0¯ is a bijection.Proof.
Ifx∈K⊂C(A), then (xL+xR)0¯=0¯, so xL0¯=xR0¯. So Ψ(x)∈AL0∩AR0=V0.
To proveΨ is one-to-one, suppose Ψ(x)=Ψ(y). Then xL0¯=yL0¯. Therefore xL+yL∈C(AL0). Likewise xR+yR∈C(AR0). So x+y∈C′. Since C′∩K={0}, we have that Ψ is one-to-one.
To proveΨ is onto, let v∈V0. Since v∈AL0, there exists xL with last q bits zero, so that xL0¯=v in AL0. Likewise there exists xR with first p bits zero, so that xR0¯=v in AR0. Then x=xL+xR∈C(A). Since C(A)=C′⊕K, we can write x=c+k, where c∈C′ and k∈K. By the fact that C′=ZL(C(AL0))⊕ZR(C(AR0)), we have that cL∈ZL(C(AL0)) and so cL0¯=0¯. Then Ψ(k)=kL0¯=(xL-cL)0¯=xL(cL(0¯))=xL(0¯)=v.Example 18.
Letp=4 and q=2 and let the generating matrix for C be (10)111100001111.Then ZL(C(AL0)) has generating vector m=[1111∣00] and ZR(C(AR0))={0}, the trivial code. We therefore see that AL0 is a 1D Adinkra with four colors with code with generating vector 1111 and AR0 is a 1D Adinkra with two colors with trivial code. The code K can be chosen to be generated by k=0011∣11. Another choice for K would have been the code generated by (11)k+m=110011.
See Figure6 for this example. To use Theorem 17, we start at 0¯ and follow an edge of color 3 and then an edge of color 4, which uses the left-moving edges in 001111. This brings us to a new vertex, which is in V0. This vertex and 0¯ itself are the two elements of V0, corresponding to the two elements of K.Figure 6
(a) A2D Adinkra with dashes omitted. (b) Picking any vertex and looking at connected components give us a pair of 1D Adinkras AL0 and AR0, the product of which contains as a quotient the original Adinkra; in this case we quotient via a 1-dimensional code K of size 21=2, obtaining the desired 8×4/2=16 vertices.
(a)
(b)Example 19.
Letp=q=4 and consider the following generating matrix for C: (12)11001100111111110011110010101010.If we let x1,…,x4 be the rows of this matrix, we see that x1+x3 has zeros on the right side of the vertical line. Other than the zero word, no other combination has all zeros on the right side, so ZL(C(AL0)) has generating matrix/vector 1111∣0000 and C(AL0) has generating matrix/vector 1111.
Likewise we can findx1+x2+x3 which is the unique nonzero codeword with all zeros on the left side, and so ZR(C(AR0)) has generating matrix/vector (13)00001111and C(AR0) has generating matrix 1111. Then AL0 and AR0 are both 1D Adinkras with 4 colors with code generated by 1111, and K can be taken to be, for instance, (14)0011110010101010.Standard arguments in linear algebra allow us to choose the generating basis for C to consist of a generating basis for ZL(C(AL0)), then a generating basis for ZR(C(AR0)), and then a generating basis for K. In this case, we would write (15)
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
0
0
1
1
1
1
0
0
1
0
1
0
1
0
1
0
,where the horizontal lines separate the three subspaces. Each line has weight a multiple of 4, where the first two lines have the 1s all on one side or the other, while the last two lines (the ones responsible for K) have the 1s split on both sides in a way that both sides have even weight.
ThenV0 has four elements: 0¯, (00110000)0¯, (10100000)0¯, and (10010000)0¯.While not necessary for proving the main theorem of this paper, [12] also asked how to classify ESDE codes. It turns out that the answer is fairly concise. To do this, it is useful to extend the notion of the splitting of n into p+q. In particular, instead of insisting that the left-moving colors be written as the first p bits, we partition [n] into [n]=L∪R such that L=p and R=q. We fix n and a doubly even code C and then characterize which partitions into L and R make C an ESDE.Theorem 20.
Given a doubly even codeC of length n, there is a bijection between codewords in C⊥ and (ordered) partitions L∪R that make C into an ESDE. There are 2n-k such partitions.Proof.
Consider a partitionL∪R that makes C into an ESDE, and consider the codeword w that is defined to have 1 at all the positions in L and 0 at all the positions in R. By definition of ESDE codes, all codewords in C have an even number of 1’s in the support of w, which is equivalent to saying that w is orthogonal to all the codewords in C. Thus, w∈C⊥. Conversely, for any w∈C⊥, w is orthogonal to all codewords in C and thus gives an ESDE. Thus, there is a bijection between the two sets. Note these are ordered partitions; the codeword which has 1 at all the positions in R and 0 otherwise would give the same partition, but in reversed order.


## 6. Proof of Main Theorem

In this section we prove the main theorem of the paper. This refers to a connected2D Adinkra A with (p,q) colors. As in Section 5 we pick a vertex 0¯ in A and define AL0 and AR0. We use Construction 6 to define A′=AL0×AR0 and let C=C(A) and C′=C(A′)=ZL(C(AL0))⊕ZR(C(AR0)). Let K be a code such that C=C′⊕K (the existence of which is guaranteed by Lemma 16).Theorem 21.
LetA be a connected 2D Adinkra. Then there is a vertex switching F and an action of the code K on F(A′) that preserves colors, dashing, and bigrading, such that (16)A≅FA′Kas an isomorphism of 2D Adinkras (i.e., as an isomorphism of graphs that preserves colors, dashing, and bigrading).We prove Theorem21 in two steps, by first constructing a (color and bigrading preserving) graph isomorphism Φ~:A′/K→A and then finding a suitable vertex switching F.Theorem 22.
The codeK acts on A′ via color preserving isomorphisms to produce a quotient A′/K. There is a color preserving graph epimorphism Φ:A′→A that sends (0¯,0¯) to 0¯. This descends to A′/K to produce a color preserving graph isomorphism Φ~:A′/K→A.Proof.
LetIn be the colored Hamming cube: that is, a graph with vertex set {0,1}n and two vertices are connected with an edge of color i if they differ only in bit i. Recall from Theorem 3 that every connected Adinkra is, as a colored graph, the quotient of In by the code for the Adinkra. So we have In/C≅A and In/C′≅A′. These are isomorphisms as colored graphs. They can be chosen so that 0=(0,…,0) is sent to 0¯ in A and (0¯,0¯) in A′, respectively.
NowK is a doubly even code, and K∩C′=0, so K acts on In/C′ and on A′ in a way that nontrivial elements of K move vertices a distance greater than 2. By the content of the proof of the extension of Theorem 2.3 in [9], this means we can quotient the colored graph In/C′ and thus A′, by K. We then have the following commutative diagram of colored graphs:
A standard argument gives(17)In/C′K≅InC′⊕K=InC,and adding this to the above commutative diagram, we then have
whereΦ~=i3∘i4∘i2-1 and Φ=Φ~∘π2. Then Φ~ is an isomorphism of colored graphs, and Φ is an epimorphism of colored graphs. Standard diagram chasing shows that Φ(0¯,0¯)=0¯.It will be useful to have the following result.Lemma 23.
Ifx∈Z2n, then Φ(x(v1,v2))=xΦ(v1,v2).Proof.
Forx=ei, the vector that is 1 in component i and 0 otherwise, this lemma is the statement that Φ is color-preserving. By composing many maps of this type, we get the statement for all vectors x∈Z2n.Lemma 24.
For allv∈AL0, Φ(v,0¯)=v, and for all w∈AR0, Φ(0¯,w)=w. In particular, Φ restricted to AL0×{0¯} is an isomorphism onto its image and likewise for Φ restricted to {0¯}×AR0.Proof.
Letx∈Z2p be such that x0¯=v. By Lemma 23, Φ(v,0)=Φ(ZL(x)(0¯,0¯))=ZL(x)Φ(0¯,0¯)=ZL(x)0¯=v. The proof for w is similar.Lemma 25.
Let(v,w)∈A′, with w=x0¯. Then Φ(v,w)=ZR(x)v.Proof.
We haveΦ(v,0¯)=v from Lemma 24. Acting on both sides with ZR(x), and using Lemma 23, the result follows.Lemma 26.
The graph epimorphismΦ preserves the bigrading.Proof.
Let(v,w)∈A′ with w=x0¯. By Lemma 25, we get (18)hLΦv,w=hLZRxv.Since ZR(x) follows right-moving colors, this does not affect hL, and so the above is equal to hL(v). In A′=AL0×AR0, this is hL(v,w). Therefore Φ preserves hL. The fact that it preserves hR is proved similarly.Lemma 27.
Ifx∈K, and (v1,v2)∈AL0×AR0, then x(v1,v2) has the same bigrading as (v1,v2).Proof.
This follows from Lemma26 and the fact that K⊂C(A), so if x∈K, then xΦ(v1,v2)=Φ(v1,v2).Theorem 28.
The codeK acts on A′ via color and bigrading preserving isomorphisms to produce a quotient A′/K. The map Φ (resp., Φ~) is a color and bigrading preserving epimorphism (resp., isomorphism).Proof.
This theorem builds on Theorem22. Lemma 27 means that the action of K on A′ preserves the bigrading. Lemma 26 provides the rest of this theorem.Unfortunately, it is too much to expectΦ to preserve the dashing or even that the dashing on A′=AL0×AR0 is invariant under the action of K (so that A′/K could have an obviously well-defined dashing). However, if we allow the operation ofvertex switching, then we can basically accomplish these goals, giving F of Theorem 21.Consider the dashingμ on A. This restricts to AL0 and AR0, and Construction 6 produces a dashing μ1 on A′=AL0×AR0. The graph homomorphism Φ:A′→A pulls back the dashing μ to μ2 on A′. While μ1 and μ2 can be different, they agree on the following parts of the Adinkra.Lemma 29.
The dashingsμ1 and μ2 agree on AL0×{0¯} and on {0¯}×AR0.Proof.
The construction ofμ1 gives each edge in AL0×{0¯} the same dashing as in AL0 under the association of every edge (v,w) with ((v,0¯),(w,0¯)). Lemma 24 shows that the same is true for μ2. Therefore μ1 and μ2 agree on AL0×{0¯}, likewise for {0}×AR0.Lemma 30.
Two dashingsμ and μ- on an Adinkra A have the same parity on all cycles (This type of result has a natural reformulation with homological algebra, done in independent ways by the first author’s work using cubical cohomology [8] and the second author’s work using CW-complexes [9]. In either formulation, having parities of μ and μ- agree on cycles is equivalent to μ-μ-2=0 in cohomology.) if and only if there is a vertex switching on A that turns μ into μ-.Proof.
Since vertex switching preserves parity on any cycle, the “if” direction is trivial and it suffices to prove the other direction.
Assume thatμ and μ- have the same parity on all cycles. It suffices to prove the statement for A connected, since we can repeat our argument on each connected component of A.
Next, we will prove that, for any treeT that is a subgraph of A, there exists a vertex switching F on A so that F(μ) and μ- agree on T. This can be proved by induction of the number of vertices in T. The base case of one vertex is trivial. If T has more than one vertex, then there is a leaf v in T incident to only one edge e in T. Let T0 be the tree with the vertex v and edge e omitted. Then T0 has one fewer vertex than T so by the inductive hypothesis, there is a vertex switching F0 on A so that F0(μ) and μ- agree on T0. If F0(μ)(e)≠μ-(e), then let F be the vertex switching F0 followed by a vertex switching at v; otherwise let F=F0. Then F(μ) and F0(μ) agree on T0, so F(μ) agrees with μ- on all of T.
Now in the case whereT is a spanning tree (so that it is maximal), we claim that F(μ) and μ- agree on all of A. Consider any edge e not in T. This edge completes at least one cycle with edges in T (otherwise T was not a spanning tree). Since the two cycles have the same parity in F(μ) and μ- by assumption, and the F(μ) and μ- agree on all edges in the cycle except for e, they must agree on e as well. Thus, F(μ)=μ- on all of A.Now we return to the two dashingsμ1 and μ2 on A′. Based on what we just proved, the following lemma will assure the existence of a vertex switching that sends μ1 to μ2. It uses the fact that these two dashings agree on AL0×{0¯} and {0¯}×AR0 (Lemma 29). In terms of the cubical cohomology, this result is a kind of Künneth theorem.Lemma 31.
The parities ofμ1 and μ2 agree on all cycles of A′.Proof.
Let our cycle be(v0,v1,…,vk) with v0=vk. We first consider the case where v0=0¯.
For this proof, we define acolor sequence of a path to be the sequence of colors (c(v0,v1),c(v1,v2),…,c(vk-1,vk)) of edges along the path. Note that, given a starting vertex v0 and a color sequence, there is a unique path that starts at v0 with that color sequence (Recall in Section 2.2 we treated this sequence of colors as a Z2n action on the underlyinggraph, where the order did not matter. In this proof we are not just traversing the graph but also keeping track of the sign of the dashings, so we have to keep in mind the order.) This follows by applying induction to property (2) of the definition of an Adinkra.
We begin with the color sequence for the cycle(v0,…,vk). We will now describe a series of modifications to this cycle, described by modifying the color sequence. The idea is to perform a “bubble sort,” by iteratively swapping adjacent colors until the left-moving colors are all at the beginning and the right-moving colors are all at the end.
First, given a color sequence(19)c1,…,cj-1,cj,cj+1,cj+2,…,ck,an adjacent swap results in a color sequence (20)c1,…,cj-1,cj+1,cj,cj+2,…,ck.Modifying a color sequence in this way leads to a new path from 0¯. The path is unchanged up to vj-1, but by the definition of Adinkras, property (2), the path returns to vj+1 so it is only vj that has changed (see Figure 7). Thus, the new path is still a cycle starting at 0¯. The effect on the parity of any dashing is, by property (3), to add 1 modulo 2. In particular, μ1 and μ2 are both affected in the same way.
It is straightforward to find a series of adjacent swaps so that the left-moving colors are moved to the beginning of the color sequence. Then the resulting path starts from0¯, stays in AL0×{0}, and then follows right-moving edges, ending in 0¯. Since the right-moving edges end in 0¯, it must be that the right-moving edges are in {0}×AR0. By Lemma 29, μ1 and μ2 are equal here, and so their parities on this modified path are the same. Therefore, their parities on the original loop were the same.
Now we consider loopsp where v0≠0¯. Since AL0×AR0 is connected, there is a path p0 from 0¯ to v0. Take the path p0, followed by p, then followed by p0-1 (meaning p0 traversed in the opposite sense). This is a loop starting and ending in 0¯, but the parity of a dashing is the same as that of p, since every new edge in p0 is counterbalanced by a new edge in p0-1. Therefore the parities of μ1 and μ2 agree on all loops.Figure 7
An adjacent swap of colors. If we swap colors in positionj and j+1, vertex vj will be replaced by vertex vj′, and the path gets modified from the upper path to the lower path in the diagram. The parity of the path changes by exactly 1 modulo 2 because the square above has odd parity.We are now ready to put everything together and prove our main theorem.Proof of Theorem21.
LetA be a connected 2D Adinkra, and define AL0, AR0, A′, C′, K, Φ, and Φ~ as above.
Use Construction6 to construct the 2D Adinkra A′=AL0×AR0 with dashing μ1. By Theorems 22 and 28, there is a graph homomorphism Φ:A′→A that preserves colors and bigrading. If we take the dashing μ from A and pull it back using Φ to a dashing μ2 on A′, then Lemmas 31 and 30 together give the existence of a vertex switching F sending μ1 to μ2.
By Theorem28, K acts on A′ to produce A′/K, a well-defined 2D Adinkra without dashing and an isomorphism Φ~:A′/K→A that preserves colors and the bigrading. Since μ2 is invariant under K, we obtain F(A′)/K, a well-defined 2D Adinkra with dashing. Since F(A′) and A′ only differ in dashing, Φ~ is still an isomorphism that preserves colors and bigrading. Since μ2 is obtained by pulling back μ from A, this isomorphism preserves dashing as well.


## 7. The Structure of 2D Adinkras

Theorem21 is very powerful; we immediately know a lot about what a 2D Adinkra must look like. Let thesupport of a 2D Adinkra (and/or its bigrading function (hL,hR)) be defined as the range of (hL,hR), its bigrading function. Then we have the following.Corollary 32.
LetA be a connected 2D Adinkra. The support of A is a rectangle. That is, there exist integers x0, x1, y0, and y1 such that the support is (21)i,j∈Z2∣x0≤i≤x1,y0≤j≤y1.Proof.
SinceAL0 is a connected 1D Adinkra and edges change hL by 1, there are integers x0 and x1 so that the range of the grading is {i∣x0≤i≤x1}. Likewise AR0 has a range of grading {j∣y0≤j≤y1} for some integers y0 and y1.
By Construction6, the support of A′=AL0×AR0 is (22)i,j∣x0≤i≤x1,y0≤j≤y1.Since Φ preserves the bigrading, this is the support of A as well.Proposition 33.
LetA be a connected 2D Adinkra. All connected components of AL (and, resp., AR) are isomorphic as graded posets.Proof.
Consider a connected component ofAL. Suppose v is a vertex in this connected component. Then there is a x∈Z2n so that x0¯=v. Then the map f:AL→AL with f(w)=x(w) is a color-preserving graph isomorphism. Thus, it sends connected components onto connected components and, in particular, AL0 to the connected component containing v.From these results and the results of the previous section, we now fully know what a2D Adinkra looks like. All of the graphical data of a 2D Adinkra is basically dictated by the connected components (one from left-moving colors and one from right-moving colors) of any single vertex and how they are glued together (this is what K encodes); then these “slices” are put together into a rectangle.


### 7.1. Constructing 2D Adinkras

An alternate way to view Theorem21 is as a way to construct all 2D Adinkras.Construction 34.
First choose any doubly even code and any codeword in its orthogonal complement. Theorem20 shows that this is exactly the amount of data we need to create an ESDC code C. Write (23)C=CL⊕CR⊕K,where CL=C∩ZL(Z2p) and CR=C∩ZR(Z2q). Use the quotient construction involved in Theorem 3 to create the 1D Adinkras A1=Ip/πL(CL) and A2=Iq/πR(CR), where πL:Z2n→Z2p is projection onto the first p bits and πR:Z2n→Z2q is projection onto the last q bits.
Second, we need a gradingh1 on A1 that is invariant under πL(K) and a grading h2 on A2 that is invariant under πR(K). There are a finite number of rank functions to consider for each graph, so it is definitely possible in principle to generate all gradings (For specific algorithms, one can use either the “hanging gardens” construction in [4] or consider the vertices of the “rank family poset” from [9].) though this is expected to be a large set for higher n. Construct the quotient A=(A1×A2)/K using Construction 6. This produces a graph with colors and a bigrading. The invariance of the gradings h1 and h2 makes this bigrading well defined under the quotient.
Finally, we put an admissible dashing onA. There is again a finite number of possible dashings, so this is doable via an exhaustive process. We can obtain dashings on the 1D Adinkra In/C and use Φ to pull them back to A. Recall the discussion after Theorem 3 for relevant results.Theorem 35.
Every 2D Adinkra can be obtained by this construction.Proof.
Given any2D Adinkra A, there is an ESDC code C. Pick a vertex 0¯ and define AL0 and AR0 as in Section 5.
Restrict the gradingshL and hR onto AL0 and AR0. Note that if g∈C, then πL(g)v=πR(g)v, and so hL(πL(g)v)=hL(v) and hR(πL(g)v)=hR(v). Therefore hL restricted to AL0 is invariant under πL(K). Likewise hR restricted to AR0 is invariant under πR(K). The dashings, as described in [7], can be obtained by choosing the specific quotient In/C. Theorem 21 gives a description of A in terms of this construction.Example 36.
Consider the code given by the generating matrix/vector11∣11. Then CL and CR are trivial, with p=q=2, and K is generated by 1111.
As graphs, AdinkrasA1 and A2 are both isomorphic to I2. There are normally 2 ways (up to relabeling of vertices) to put a rank function on I2:
However, the height function must be invariant underπL(C)=πR(C)=〈11〉. In other words, moving once with an edge of both colors should not change the grading. Thus, both A1 and A2 can only be graded via the rank function depicted on the right. So they must look like this, after assigning colors:
We now take the productA1×A2, which has 4×4=16 vertices:
To constructA1×A2/K, recall that K is generated by 1111, which means that each vertex is identified with the vertex that is obtained by following all four colors once. We should now get a graph with 16/2=8 vertices. If we put an admissible dashing on it, we obtain a complete 2D Adinkra. One such choice recovers our Adinkra from Figure 4:


## 7.1. Constructing 2D Adinkras

An alternate way to view Theorem21 is as a way to construct all 2D Adinkras.Construction 34.
First choose any doubly even code and any codeword in its orthogonal complement. Theorem20 shows that this is exactly the amount of data we need to create an ESDC code C. Write (23)C=CL⊕CR⊕K,where CL=C∩ZL(Z2p) and CR=C∩ZR(Z2q). Use the quotient construction involved in Theorem 3 to create the 1D Adinkras A1=Ip/πL(CL) and A2=Iq/πR(CR), where πL:Z2n→Z2p is projection onto the first p bits and πR:Z2n→Z2q is projection onto the last q bits.
Second, we need a gradingh1 on A1 that is invariant under πL(K) and a grading h2 on A2 that is invariant under πR(K). There are a finite number of rank functions to consider for each graph, so it is definitely possible in principle to generate all gradings (For specific algorithms, one can use either the “hanging gardens” construction in [4] or consider the vertices of the “rank family poset” from [9].) though this is expected to be a large set for higher n. Construct the quotient A=(A1×A2)/K using Construction 6. This produces a graph with colors and a bigrading. The invariance of the gradings h1 and h2 makes this bigrading well defined under the quotient.
Finally, we put an admissible dashing onA. There is again a finite number of possible dashings, so this is doable via an exhaustive process. We can obtain dashings on the 1D Adinkra In/C and use Φ to pull them back to A. Recall the discussion after Theorem 3 for relevant results.Theorem 35.
Every 2D Adinkra can be obtained by this construction.Proof.
Given any2D Adinkra A, there is an ESDC code C. Pick a vertex 0¯ and define AL0 and AR0 as in Section 5.
Restrict the gradingshL and hR onto AL0 and AR0. Note that if g∈C, then πL(g)v=πR(g)v, and so hL(πL(g)v)=hL(v) and hR(πL(g)v)=hR(v). Therefore hL restricted to AL0 is invariant under πL(K). Likewise hR restricted to AR0 is invariant under πR(K). The dashings, as described in [7], can be obtained by choosing the specific quotient In/C. Theorem 21 gives a description of A in terms of this construction.Example 36.
Consider the code given by the generating matrix/vector11∣11. Then CL and CR are trivial, with p=q=2, and K is generated by 1111.
As graphs, AdinkrasA1 and A2 are both isomorphic to I2. There are normally 2 ways (up to relabeling of vertices) to put a rank function on I2:
However, the height function must be invariant underπL(C)=πR(C)=〈11〉. In other words, moving once with an edge of both colors should not change the grading. Thus, both A1 and A2 can only be graded via the rank function depicted on the right. So they must look like this, after assigning colors:
We now take the productA1×A2, which has 4×4=16 vertices:
To constructA1×A2/K, recall that K is generated by 1111, which means that each vertex is identified with the vertex that is obtained by following all four colors once. We should now get a graph with 16/2=8 vertices. If we put an admissible dashing on it, we obtain a complete 2D Adinkra. One such choice recovers our Adinkra from Figure 4:


## 8. Conclusion and Future Work

In this work, we have continued in the vein of [12] to study 2D Adinkras and provided stuctural results to study them combinatorially. Describing already-known worldsheet supermultiplets in these terms could lead to new insights about these supermultiplets and lead to the discovery of new worldsheet supermultiplets.One of the motivations for [13] was the idea of studying 4-dimensional or higher-dimensional supermultiplets dimensionally reduced to 1 dimension. Likewise, the present work allows us to study the dimensional reduction to 2 dimensions, which might carry important information about the original supermultiplet.As in [16], we could also consider nonadinkraic worldsheet supermultiplets. It would be interesting to see if, as in the case of one dimension, there is a continuum of worldsheet supermultiplets. Another direction to extend these results is local supersymmetry, as in the case of supergravity or superconformal theories, which could be of importance to superstrings.Of course, one obvious step is to go to three or four dimensions. It is expected thatSO(1,d-1) representations will play a role, and this work may begin to intersect [17, 18]. (Higher dimensions is also a natural place to consider gauge fields, which have not yet played a role in this discussion. In one dimension gauge fields can be gauged away to zero. In two dimensions this is not the case, but the corresponding field strengths automatically satisfy Bianchi identities, and so the Adinkra formalism works well in this case. But in higher dimensions, gauge fields will be more difficult to avoid.) Beyond this, it is hoped that this work will generally help develop our knowledge of supersymmetry in two dimensions.Many mathematicians may appreciate Adinkras simply as nice combinatorial objects with lots of structure, with surprising links to coding theory, switching graphs, and graph coloring. This view presents additional questions, less relevant to the physics but still mathematically interesting, in the spirit of Theorem20:(i)
For example, it would be good to know how many ESDE’s are “compatible” with a1D Adinkra A (i.e., there is some 2D Adinkra A′ with the same underlying graph and rankings as A, with the required ESDE as its code) or with the family of 1D Adinkras with the same underlying graph.(ii)
Enumerating all2D Adinkras is also a natural goal, though fairly ambitious (We still do not know how to count 1D Adinkras with In as underlying graph beyond small n. For In/C with nontrivial C, we have almost no data! The work in [9] basically settles dashings completely and gives some structural results on rankings, but counting rankings completely remains a very difficult problem, related to the chromatic polynomial for some families of C.), but counting all 2D Adinkras under some natural constraints may be fruitful.(iii)
How often does the main theorem require no additional vertex switching (i.e.,F is the identity)? In general, when given a dashing, we can ask related questions about the minimum number of vertex switches needed to produce a dashing with a well-defined quotient.



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