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Advances in High Energy Physics
Volume 2016 (2016), Article ID 3980613, 12 pages
http://dx.doi.org/10.1155/2016/3980613
Research Article

Structural Theory and Classification of 2D Adinkras

1Natural Science Division, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, USA
2Department 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 SCOAP3.

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 [13]. 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 [49]. 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 35 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.

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 if and a solid edge if . The grading is represented by the vertical height as indicated on the axis on the left.
2.2. Structural Aspects of 1D Adinkras

Let be a 1D Adinkra with colors, with vertex set . For all , define such that, for all , is the unique vertex joined to by an edge of color . In [6], it was shown that the map is a graph isomorphism (in fact, an involution) from the underlying graph of to itself which preserves colors. ’s commute with each other. These facts can be used to combine the maps into an action of on the graph underlying the Adinkra in the following way.

Definition 2. The action of on the graph underlying the Adinkra is given on vertices by 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 the ’s commute implies that the order of the colors does not matter.
The Adinkra is connected if and only if action is transitive on the vertex set of . 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 , the code of the Adinkra , to be this stabilizer. This is a binary linear code of length (i.e., a linear subspace of ). As these are the only types of codes we use, from now on we simply say code to mean “binary linear code.”
We call the elements of a code codewords. The weight of a codeword is the number of 1’s in the word. A code is called even if all its codewords have even weight. A code is called doubly even if all its codewords have weight divisible by 4. An example of a doubly even code is the span , which has elements. An example of a code that is even but not doubly even is the 1-dimensional code .
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: let , the Hamming cube, be the graph with vertices labeled by strings of length using the alphabet , with an edge between two vertices and if and only if they differ in exactly one place. There is a natural coloring on : just color each edge by the coordinate where the two vertices differ. Now, codes in act on by bitwise addition modulo 2, and these are isomorphisms that preserve colors. A natural operation to consider on a colored graph and a group acting on via graph isomorphisms that preserve colors is the quotient , where the vertices are defined to be orbits in , and we have an -colored edge if and only if there is at least one -colored edge with in the orbit and in the orbit .

Theorem 3. is the colored graph of some 1D Adinkra if and only if 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 does not necessarily retain nice properties of ; 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 is a doubly even code) between graph properties of the quotient and properties of the code .

Figure 2 provides an example of a quotient of by a code that obeys this theorem. Encoded within the proof of Theorem 3 is the fact that if 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 graph . (b) The quotient . (c) As the code is doubly even, there exists an Adinkra with the quotient as its underlying graph by Theorem 3.
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 Adinkra , and a vertex of , we define vertex switching at to be the operation on that returns a new Adinkra with the same vertices, edges, coloring, and grading but a new dashing so thatWe leave to the reader to check that is still an admissible dashing; since the vertices, edges, coloring, and grading(s) are the same, remains an Adinkra (Figure 3). We also use a vertex switching of to refer to a composition of vertex switchings at various vertices of .

Figure 3: A vertex switching at turns the Adinkra on the left into the Adinkra on the right. The two Adinkras have the same dashing except precisely the edges that are incident to . Note that in both cases each face of the cube has an odd number of dashed edges.

In [14], vertex switching was first applied to dashings in Adinkras from a point of view inspired by Seidel’s two 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 on signed 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 developed D Adinkras to study D 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 by mass dimension, which is , and spin, which is . Mass dimension is the units of mass associated with the field, where and spin is the eigenvalue of .)

A D 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). Let and be nonnegative integers. A 2D Adinkra with colors is a 1D Adinkra with colors, and two grading functions and so that we have the following: (i).(ii)Let be an edge. If then is called a left-moving edge; if then it is called a right-moving edge. Similarly, the first colors are called left-moving colors and the last colors are called right-moving colors.(iii)if is a left-moving edge, then and . If is a right-moving edge, then and .

See Figure 4 for an example of a D Adinkra. The main goal of this paper is to follow the program set out in [12] and completely characterize D Adinkras. As a first step, we define the natural notion of products in the following section.

Figure 4: A D Adinkra with colors. The grading coordinates are given next to the nodes as .

4. Products

One important way to produce a D Adinkra with colors is to take the product of two 1D Adinkras (one with colors, and the other with colors), using the following construction.

Construction 6. Let and be nonnegative integers. Let be a 1D Adinkra with colors and let be a 1D Adinkra colors. We define the product of these Adinkras as the following 2D Adinkra with colors: where and there are two kinds of edges in :(i)For every edge in connecting vertices and , and for every vertex , we have an edge in between vertices and in of color and dashing .(ii)For every edge in connecting vertices and and for every vertex , we have an edge in between vertices and in of color and dashing   . See Figure 5 for an example.

Figure 5: Constructing the D Adinkra (b) from two smaller D 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 .

This definition is intended to be a graph-theoretic version of the tensor product construction in -graded representations (see Appendix for more details). The edges that come from give rise to left-moving edges, and the edges that come from give rise to right-moving edges. It follows easily that for every vertex in and for every color in there is a unique edge in incident to . The fact that two-colored simple cycles have length 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 Adinkras and , is a 2D Adinkra.

Definition 8 (extending codes). Let and be nonnegative integers and let . Define to be the function that appends zeros, so that, for instance, if and , then . Likewise, define to be the function that prepends zeros.
Our most common use of this notation is as follows: if is a binary block code of length , we write for the image under . Likewise, if is a binary block code of length , we write for the image under .

Proposition 9. Let and be as above. Then

Proof. Let . Let . We can write , where is zero in the last bits and is zero in the first bits. Now This means that if and only if and . So if and only if and .

5. Codes for 2D Adinkras

Let be a connected D Adinkra with colors. Then there is a doubly even code associated with . But as a D Adinkra, we make a distinction between the first colors and the last colors, which for a code translates to the first bits and the last bits. A natural question is “knowing that a D Adinkra can be enriched into a D 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 vector , the weight of , denoted by , is the number of ’s in . Likewise, is the the number of ’s in the first bits and is the number of ’s in the last bits of . Let a code , along with the parameters , be called an even-split doubly even (ESDE) code if is doubly even and all codewords in have both and even.

This definition of ESDE codes is due to [12], which also proves the following.

Theorem 11. If is a connected 2D Adinkra with colors, then is an ESDE.

We now prove the converse of this theorem. That is, given an ESDE code, there exists connected D Adinkra with that code. This procedure is analogous to the Valise Adinkras in 1D [4, 13], in that the possible values of each component of the bigrading are as small as possible, that is, two values.

Construction 12. Let be an ESDE code. We will describe a construction that provides a D Adinkra with code , called the Valise 2D Adinkra. First, since is doubly even, there exists a connected 1D Adinkra with code by Theorem 3. Fix a vertex of . Now for every vertex there is a vector so that . Then define    and   . Note that these functions are well-defined since is ESDE. Then is a bigrading for , making it a D Adinkra. An example of the kind of D Adinkra that arises from this construction is Figure 4.

We therefore have the following.

Theorem 13. For a code , there exists a 2D Adinkra with if and only if is a ESDE code.

The structure of the ESDE relates to interesting features of the colored graph of . Let be the 1D Adinkra with colors that consists of only the left-moving edges of . Let be the 1D Adinkra with colors that consists of only the right-moving edges of (where we shift the colors so that they range from 1 to instead of to ). Pick a vertex in . Let be the connected component of containing and let be the connected component of containing .

We now see that the codes for and (with an appropriate number of s added to the left or right as necessary) provide important linear subspaces of .

Proposition 14. The following hold:

In other words, the codewords that are zero in the last bits are precisely the codewords from with zeros appended to the right; and the codewords that are zero in the first bits are precisely the codewords from with zeros prepended to the left.

Proof. If , so that for some , Then trivially . Furthermore, in . In , does exactly the same thing, so in . Therefore .
Conversely if , then by the definition of the group action there is a path in from to following the colors corresponding to the 1s in . Since , we have that this path only consists of left-moving colors and so lies in .
The proof for is similar.

Corollary 15. The following hold:

Comparing with Proposition 9, this corollary says that the code for is a linear subspace of the code for .

Simply for brevity (and thus, readability) in later descriptions, we define using the product construction above and the code . Also, we write for . Then and .

Lemma 16. There exists a binary linear block code so that

Proof. From Corollary 15 and basic linear algebra, there exists a vector subspace of that is a vector space complement of in .

Note that is not necessarily uniquely defined. It is, however, uniquely defined up to adding vectors in . So a more invariant approach would be to use instead of , but has the advantage of being a code, therefore more concrete for computational purposes.

The interpretation of can be obtained by examining the set . Since has only left-moving edges and has only right-moving edges, has no edges at all: only vertices. Furthermore, for every , we have and so all of the vertices in have the same bigrading.

We now show that there is a bijection between and . As before, for every , we write , where is zero in the last bits and is zero in the first bits. Using this notation, we have the following theorem.

Theorem 17. The map given by is a bijection.

Proof. If , then , so . So .
To prove is one-to-one, suppose . Then . Therefore . Likewise . So . Since , we have that is one-to-one.
To prove is onto, let . Since , there exists with last bits zero, so that in . Likewise there exists with first bits zero, so that in . Then . Since , we can write , where and . By the fact that , we have that and so . Then

Example 18. Let and and let the generating matrix for be Then has generating vector and , the trivial code. We therefore see that is a D Adinkra with four colors with code with generating vector and is a 1D Adinkra with two colors with trivial code. The code can be chosen to be generated by . Another choice for would have been the code generated by See Figure 6 for this example. To use Theorem 17, we start at and follow an edge of color and then an edge of color , which uses the left-moving edges in . This brings us to a new vertex, which is in . This vertex and itself are the two elements of , corresponding to the two elements of .

Figure 6: (a) A D Adinkra with dashes omitted. (b) Picking any vertex and looking at connected components give us a pair of 1D Adinkras and , the product of which contains as a quotient the original Adinkra; in this case we quotient via a 1-dimensional code of size , obtaining the desired vertices.

Example 19. Let and consider the following generating matrix for : If we let be the rows of this matrix, we see that 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 has generating matrix/vector and has generating matrix/vector
Likewise we can find which is the unique nonzero codeword with all zeros on the left side, and so has generating matrix/vector and has generating matrix Then and are both 1D Adinkras with colors with code generated by , and can be taken to be, for instance, Standard arguments in linear algebra allow us to choose the generating basis for to consist of a generating basis for , then a generating basis for , and then a generating basis for . In this case, we would write where the horizontal lines separate the three subspaces. Each line has weight a multiple of , where the first two lines have the 1s all on one side or the other, while the last two lines (the ones responsible for ) have the 1s split on both sides in a way that both sides have even weight.
Then has four elements: , , , and .

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 into . In particular, instead of insisting that the left-moving colors be written as the first bits, we partition into such that and . We fix and a doubly even code and then characterize which partitions into and make an ESDE.

Theorem 20. Given a doubly even code of length , there is a bijection between codewords in and (ordered) partitions that make into an ESDE. There are such partitions.

Proof. Consider a partition that makes into an ESDE, and consider the codeword that is defined to have 1 at all the positions in and at all the positions in . By definition of ESDE codes, all codewords in have an even number of 1’s in the support of , which is equivalent to saying that is orthogonal to all the codewords in . Thus, . Conversely, for any , is orthogonal to all codewords in 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 and 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 connected D Adinkra with colors. As in Section 5 we pick a vertex in and define and . We use Construction 6 to define and let and . Let be a code such that (the existence of which is guaranteed by Lemma 16).

Theorem 21. Let be a connected 2D Adinkra. Then there is a vertex switching and an action of the code on that preserves colors, dashing, and bigrading, such that as an isomorphism of 2D Adinkras (i.e., as an isomorphism of graphs that preserves colors, dashing, and bigrading).

We prove Theorem 21 in two steps, by first constructing a (color and bigrading preserving) graph isomorphism and then finding a suitable vertex switching .

Theorem 22. The code acts on via color preserving isomorphisms to produce a quotient . There is a color preserving graph epimorphism that sends to . This descends to to produce a color preserving graph isomorphism .

Proof. Let be the colored Hamming cube: that is, a graph with vertex set and two vertices are connected with an edge of color if they differ only in bit . Recall from Theorem 3 that every connected Adinkra is, as a colored graph, the quotient of by the code for the Adinkra. So we have and . These are isomorphisms as colored graphs. They can be chosen so that is sent to in and in , respectively.
Now is a doubly even code, and , so acts on and on in a way that nontrivial elements of move vertices a distance greater than . By the content of the proof of the extension of Theorem in [9], this means we can quotient the colored graph and thus , by . We then have the following commutative diagram of colored graphs:

A standard argument gives and adding this to the above commutative diagram, we then have

where and . Then is an isomorphism of colored graphs, and is an epimorphism of colored graphs. Standard diagram chasing shows that .

It will be useful to have the following result.

Lemma 23. If , then

Proof. For , the vector that is in component and otherwise, this lemma is the statement that is color-preserving. By composing many maps of this type, we get the statement for all vectors .

Lemma 24. For all , , and for all , . In particular, restricted to is an isomorphism onto its image and likewise for restricted to .

Proof. Let be such that . By Lemma 23, . The proof for is similar.

Lemma 25. Let , with . Then

Proof. We have from Lemma 24. Acting on both sides with , and using Lemma 23, the result follows.

Lemma 26. The graph epimorphism preserves the bigrading.

Proof. Let with . By Lemma 25, we get Since follows right-moving colors, this does not affect , and so the above is equal to . In , this is . Therefore preserves . The fact that it preserves is proved similarly.

Lemma 27. If , and , then has the same bigrading as .

Proof. This follows from Lemma 26 and the fact that , so if , then .

Theorem 28. The code acts on via color and bigrading preserving isomorphisms to produce a quotient . The map (resp., ) is a color and bigrading preserving epimorphism (resp., isomorphism).

Proof. This theorem builds on Theorem 22. Lemma 27 means that the action of on preserves the bigrading. Lemma 26 provides the rest of this theorem.

Unfortunately, it is too much to expect to preserve the dashing or even that the dashing on is invariant under the action of (so that could have an obviously well-defined dashing). However, if we allow the operation of vertex switching, then we can basically accomplish these goals, giving of Theorem 21.

Consider the dashing on . This restricts to and , and Construction 6 produces a dashing on . The graph homomorphism pulls back the dashing to on . While and can be different, they agree on the following parts of the Adinkra.

Lemma 29. The dashings and agree on and on .

Proof. The construction of gives each edge in the same dashing as in under the association of every edge with . Lemma 24 shows that the same is true for . Therefore and agree on , likewise for .

Lemma 30. Two dashings and on an Adinkra 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 and agree on cycles is equivalent to in cohomology.) if and only if there is a vertex switching on that turns into .

Proof. Since vertex switching preserves parity on any cycle, the “if” direction is trivial and it suffices to prove the other direction.
Assume that and have the same parity on all cycles. It suffices to prove the statement for connected, since we can repeat our argument on each connected component of .
Next, we will prove that, for any tree that is a subgraph of , there exists a vertex switching on so that and agree on . This can be proved by induction of the number of vertices in . The base case of one vertex is trivial. If has more than one vertex, then there is a leaf in incident to only one edge in . Let be the tree with the vertex and edge omitted. Then has one fewer vertex than so by the inductive hypothesis, there is a vertex switching on so that and agree on . If , then let be the vertex switching followed by a vertex switching at ; otherwise let . Then and agree on , so agrees with on all of .
Now in the case where is a spanning tree (so that it is maximal), we claim that and agree on all of . Consider any edge not in . This edge completes at least one cycle with edges in (otherwise was not a spanning tree). Since the two cycles have the same parity in and by assumption, and the and agree on all edges in the cycle except for , they must agree on as well. Thus, on all of .

Now we return to the two dashings and on . Based on what we just proved, the following lemma will assure the existence of a vertex switching that sends to . It uses the fact that these two dashings agree on and (Lemma 29). In terms of the cubical cohomology, this result is a kind of Künneth theorem.

Lemma 31. The parities of and agree on all cycles of .

Proof. Let our cycle be with . We first consider the case where .
For this proof, we define a color sequence of a path to be the sequence of colors of edges along the path. Note that, given a starting vertex and a color sequence, there is a unique path that starts at with that color sequence (Recall in Section 2.2 we treated this sequence of colors as a action on the underlying graph, 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 of the definition of an Adinkra.
We begin with the color sequence for the cycle . 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 an adjacent swap results in a color sequence Modifying a color sequence in this way leads to a new path from . The path is unchanged up to , but by the definition of Adinkras, property , the path returns to so it is only that has changed (see Figure 7). Thus, the new path is still a cycle starting at . The effect on the parity of any dashing is, by property , to add 1 modulo 2. In particular, and 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 from , stays in , and then follows right-moving edges, ending in . Since the right-moving edges end in , it must be that the right-moving edges are in . By Lemma 29, and 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 loops where . Since is connected, there is a path from to . Take the path , followed by , then followed by (meaning traversed in the opposite sense). This is a loop starting and ending in , but the parity of a dashing is the same as that of , since every new edge in is counterbalanced by a new edge in . Therefore the parities of and agree on all loops.

Figure 7: An adjacent swap of colors. If we swap colors in position and , vertex will be replaced by vertex , and the path gets modified from the upper path to the lower path in the diagram. The parity of the path changes by exactly modulo 2 because the square above has odd parity.

We are now ready to put everything together and prove our main theorem.

Proof of Theorem 21. Let be a connected D Adinkra, and define , , , , , , and as above.
Use Construction 6 to construct the D Adinkra with dashing . By Theorems 22 and 28, there is a graph homomorphism that preserves colors and bigrading. If we take the dashing from and pull it back using to a dashing on , then Lemmas 31 and 30 together give the existence of a vertex switching sending to .
By Theorem 28, acts on to produce , a well-defined D Adinkra without dashing and an isomorphism that preserves colors and the bigrading. Since is invariant under , we obtain , a well-defined D Adinkra with dashing. Since and only differ in dashing, is still an isomorphism that preserves colors and bigrading. Since is obtained by pulling back from , this isomorphism preserves dashing as well.

7. The Structure of 2D Adinkras

Theorem 21 is very powerful; we immediately know a lot about what a D Adinkra must look like. Let the support of a D Adinkra (and/or its bigrading function ) be defined as the range of , its bigrading function. Then we have the following.

Corollary 32. Let be a connected 2D Adinkra. The support of is a rectangle. That is, there exist integers , , , and such that the support is

Proof. Since is a connected D Adinkra and edges change by 1, there are integers and so that the range of the grading is . Likewise has a range of grading for some integers and .
By Construction 6, the support of is Since preserves the bigrading, this is the support of as well.

Proposition 33. Let be a connected 2D Adinkra. All connected components of (and, resp., ) are isomorphic as graded posets.

Proof. Consider a connected component of . Suppose is a vertex in this connected component. Then there is a so that . Then the map with is a color-preserving graph isomorphism. Thus, it sends connected components onto connected components and, in particular, to the connected component containing .

From these results and the results of the previous section, we now fully know what a D Adinkra looks like. All of the graphical data of a D 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 encodes); then these “slices” are put together into a rectangle.

7.1. Constructing 2D Adinkras

An alternate way to view Theorem 21 is as a way to construct all 2D Adinkras.

Construction 34. First choose any doubly even code and any codeword in its orthogonal complement. Theorem 20 shows that this is exactly the amount of data we need to create an ESDC code . Write where and . Use the quotient construction involved in Theorem 3 to create the D Adinkras and , where is projection onto the first bits and is projection onto the last bits.
Second, we need a grading on that is invariant under and a grading on that is invariant under . 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 . Construct the quotient using Construction 6. This produces a graph with colors and a bigrading. The invariance of the gradings and makes this bigrading well defined under the quotient.
Finally, we put an admissible dashing on . There is again a finite number of possible dashings, so this is doable via an exhaustive process. We can obtain dashings on the D Adinkra and use to pull them back to . Recall the discussion after Theorem 3 for relevant results.

Theorem 35. Every 2D Adinkra can be obtained by this construction.

Proof. Given any D Adinkra , there is an ESDC code . Pick a vertex and define and as in Section 5.
Restrict the gradings and onto and . Note that if , then , and so and . Therefore restricted to is invariant under . Likewise restricted to is invariant under . The dashings, as described in [7], can be obtained by choosing the specific quotient . Theorem 21 gives a description of in terms of this construction.

Example 36. Consider the code given by the generating matrix/vector . Then and are trivial, with , and is generated by
As graphs, Adinkras and are both isomorphic to . There are normally ways (up to relabeling of vertices) to put a rank function on :

However, the height function must be invariant under . In other words, moving once with an edge of both colors should not change the grading. Thus, both and 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 product , which has vertices:

To construct , recall that is generated by , 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 vertices. If we put an admissible dashing on it, we obtain a complete D 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 D 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 -dimensional or higher-dimensional supermultiplets dimensionally reduced to dimension. Likewise, the present work allows us to study the dimensional reduction to 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 that 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 Theorem 20:(i)For example, it would be good to know how many ESDE’s are “compatible” with a D Adinkra (i.e., there is some D Adinkra with the same underlying graph and rankings as , with the required ESDE as its code) or with the family of D Adinkras with the same underlying graph.(ii)Enumerating all D Adinkras is also a natural goal, though fairly ambitious (We still do not know how to count D Adinkras with as underlying graph beyond small . For with nontrivial , 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 .), but counting all D Adinkras under some natural constraints may be fruitful.(iii)How often does the main theorem require no additional vertex switching (i.e., 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.

Appendix

Relation to Hübsch’s Original Language

The statement of Theorem 21 is a bit different from the statement of the conjecture in [12]. There, the language was partly in terms of representations of the 2D SUSY algebra instead of graphs.

Construction A.1 ((off-shell) [12, Construction  2.1]). Let and denote off-shell representations of two copies of the D SUSY algebra with and colors, respectively, and let be a symmetry of , as a representation. The -quotient of the tensor product is then an off-shell representation of D SUSY algebra with colors.

The conjecture in [12] then says that every Adinkraic representation of D SUSY with colors is obtained by this construction.

The relation between this conjecture and Theorem 21 will be apparent once we establish the following relationships. None of these facts are new to this paper, but this information is collected here for the convenience of the reader:(i)The relationship between off-shell representations of SUSY and Adinkras is as follows: this is the central idea behind the original paper on Adinkras [13], and so we do not go into detail here. The idea is that each vertex of the Adinkra corresponds to a field in the SUSY representation, and acts on fields by the edge of color , with possible derivatives depending on the grading or bigrading (as the case may be), and with an extra minus sign if the corresponding edge is dashed.(ii)The relationship between and is as follows: the definition of as a representation of D SUSY that if and if . This is the standard way in which tensor products are defined in -graded algebras [19, 20]. Then Construction 6 mimics this definition on the level of Adinkras.(iii)A vertex switching at corresponds to replacing with . Then all equations involving or will get an extra minus sign. This reverses all dashings on edges connected to .(iv)A quotient defined in Theorem 21 is a symmetry of the representation.

Conflict of Interests

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

Acknowledgments

The authors wish to thank Charles Doran, Sylvester Gates, Tristan Hübsch, and Richard Eager for helpful conversation.

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