c 2011 International Press

Adv. Theor. Math. Phys. 15 (2011) 1909–1970

Codes and supersymmetry in one
dimensionCharles F. Doran1 , Michael G. Faux2 , Sylvester James
Gates, Jr.3 , Tristan Hübsch4 , Kevin M. Iga5 ,Gregory D. Landweber6 and Robert L. Miller7,8

1 Department of Mathematical and Statistical Sciences, University ofAlberta, Edmonton, Alberta T6G 2G1, Canada
doran@math.ualberta.ca
2 Department of Physics, State University of New York, Oneonta,
NY 13825, USA
fauxmg@oneonta.edu
3 Center for String and Particle Theory, Department of Physics, Universityof Maryland, College Park, MD 20472, USA
gatess@wam.umd.edu
4 Department of Physics and Astronomy, Howard University, Washington,
DC 20059, USA
thubsch@howard.edu
5 Natural Science Division, Pepperdine University, Malibu, CA 90263, USA
Kevin.Iga@pepperdine.edu
6 Mathematics Program, Bard College, Annandale-on-Hudson, NY
12504-5000, USA
gregland@bard.edu
7 Department of Mathematics, University of Washington, Seattle, WA
98105, USA
rlmill@math.washington.edu
8 Quid, Inc., 733 Front Street, C1A, San Francisco, CA 94111, USA
rmiller@quid.com
e-print archive: http://lanl.arXiv.org/abs/1108.4124v2

1910

CHARLES F. DORAN ET AL.
Abstract
Adinkras are diagrams that describe many useful supermultiplets in
D = 1 dimensions. We show that the topology of the Adinkra is uniquelydetermined by a doubly even code. Conversely, every doubly even code
produces a possible topology of an Adinkra. A computation of doubly
even codes results in an enumeration of these Adinkra topologies up to
N = 28, and for minimal supermultiplets, up to N = 32.

1

Introduction, review and synopsis

Although many supersymmetric theories have been known since the 1970s,
there is still no overarching classiﬁcation of supermultiplets, even in one
dimension (time). In fact, supersymmetry in one dimension has been thesubject of several investigations, for instance the development of theGR(d, N ) algebras [1–3], the development of Adinkras [4], and to othereﬀorts [5–14].
An Adinkra is a directed graph with various colorings and other markingson vertices and edges, which in a pictoral way encode all details of the supersymmety transformations on the component ﬁelds within a supermultiplet
in one dimension [4]. The main purpose of the present work is to determine
the kinds of graphs (i.e., the topology) that can be used for Adinkras.
The class of supermultiplets described by Adinkras is wide enough to contain many noteworthy superﬁelds. [4, 15] This paper will clarify the conditions for a supermultiplet in one dimension to be described using an Adinkra.
Beyond this, by showing that this class is large, we thereby show that theclass of supermultiplets is also large, and thus establish that the classiﬁcation problem of supersymmetry in one dimension is much more intricate
than it might appear at ﬁrst. We also thereby add many new supermultiplets to the literature that were previously unknown, and it is possible thatsome of these newly discovered supermultiplets may be useful or interestingin their own rights.
Finally, we hope that classifying Adinkras may help one better understandthe conditions under which a superﬁeld has an oﬀ-shell description. Subject
to a particular set of assumptions about dynamics, Siegel and Roček [16]had previously shown that not all supermultiplets have oﬀ-shell descriptions.
On the other hand, superﬁelds described by Adinkras are oﬀ-shell supermultiplets. Therefore, understanding the range of what Adinkras can describe
may shed some light on the question of which supermultiplets admit anoﬀ-shell description.

CODES AND SUPERSYMMETRY IN ONE DIMENSION
1.1

1911

Adinkras

In one dimension, with N supersymmetry generators Q1 , . . . , QN , thesupersymmetry algebra is{QI , QJ } = 2iδIJ ∂τ ,(1.1)

where ∂τ is the derivative in the time direction.
An Adinkra is a ﬁnite directed graph, with every vertex colored either
white or black, and with every edge colored one of N colors (each color
corresponds to one of the supersymmetries QI ), and each edge drawn witheither a solid or a dashed line. The vertices correspond to the component
ﬁelds (black for fermions, white for bosons) and the edges correspond to
the action of each of the QI , in a way that is reminiscent of the Cayley
diagram of a ﬁnitely generated group, or even more analogously, the Schreier
diagram of the set of cosets of a subgroup. Details of Adinkras and how theycorrespond to supermultiplets can be found in [4, 17]. The classiﬁcation of
Adinkras naturally falls into four steps:(1) Determine which topologies are possible (the topology of an Adinkra
is the underlying graph of vertices and edges without colorings, as, forinstance, in [17]).
(2) Determine the ways in which vertices and edges may be colored.
The topology of the Adinkra, together with the colorings of vertices
and edges, will be called the chromotopology of the Adinkra. It ischromotopologies that are classiﬁed in this paper.
(3) Determine the ways in which edges may be chosen as dashed or solid.
This is closely related to the well-known theory of Cliﬀord algebras,
and will be studied in a future eﬀort.
(4) Determine the ways in which arrows may be directed along each edge.
This issue is addressed in [17], and shown to be equivalent to thequestion of “hanging” the graph on a few sinks. Alternately, we canstart with an Adinkra where all arrows go from bosons to fermions,then perform a sequence of vertex raises to arrive at other choices ofarrow directions.
As it happens, it is convenient to do 1 and 2 together; that is, to classify chromotopologies. Herein, we show that the classiﬁcation of Adinkra
chromotopologies is equivalent to another interesting question from coding
theory: the classiﬁcation of doubly even codes. Much work has already been
done in this area [18–20], and the work described in this paper goes evenfurther in developing this classiﬁcation; see Appendix B.

1912

CHARLES F. DORAN ET AL.

We emphasize that we focus here on the representation theory, not the
dynamics. This is natural, as we need to ﬁrst know the full palette of supersymmetric representations before discussing the properties of the dynamics
in theories built upon such representations. For instance, presupposing astandard, uncoupled Lagrangian for the supermultiplets that we intend toclassify would necessarily limit the possibilities; there do exist supermultiplets which can only have interactive Lagrangians [21, 22]. Herein, we defer
the task of ﬁnding Lagrangians involving the supermultiplets considered inthis paper. In [23, 24], we have in fact started on such studies, and, using
Adinkras, have constructed supersymmetric Lagrangians for some of thesupermultiplets that are also discussed herein.
In units where  = 1 = c, all physical quantities may have at most unitsof mass, the exponent of which is called the engineering dimension and is an
essential element of physics analysis in general. The engineering dimensionof a ﬁeld φ(τ ) will be written [φ]; for more details, see Refs. [17, 25].
1.2

Main result

Our main result about the chromotopology types of Adinkras and the corresponding supermultiplets, up to direct sums, may be summarized as follows:
We deﬁne the function:
⎧
⎪
0
⎪
⎪
⎪
⎪
⎪
⎨1
κ(N ) := 2
⎪
⎪
⎪
3
⎪
⎪
⎪
⎩4 + κ(N −8)

for N < 4,
for N = 4, 5,
for N = 6,
for N = 7,
for N ≥ 8, recursively.

(1.2)(1) Every Adinkra can be separated into its connected components. (Thesupermultiplet corresponding to such an Adinkra breaks up into a
direct sum of other supermultiplets, each of which corresponds to oneof the connected components of the Adinkra).
(2) There is a one-to-one correspondence between possible chromotopologies of connected Adinkras and doubly even codes of length N .
(a) Each connected chromotopology has, associated to it, a doubly evencode of length N and dimension k ≤ κ(N ) that records which paths
connect a vertex to itself.
(b) The chromotopology is then the quotient of the colored N -dimensional cube by this code. (The colored N -cube is the set of vertices
and edges of the N -dimensional cube [0, 1]N , with colors on the

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1913

edges determined by which axis it is parallel to, and colors of vertices according to the number, modulo 2, of coordinates that are
1).
(c) This quotient can be viewed as an iterated k-fold Z2 -quotient.
(d) These chromotopologies really do come from supermultiplets in
D = 1 dimension, and if the arrows are chosen properly(one-hooked) we can arrange it so that it is easy to see that diﬀerent
codes give rise to diﬀerent supermultiplets.
(e) Permuting the columns of a code corresponds to permuting thecolors of the chromotopology, which in turn describes R-symmetriesof the supermultiplet.
(f) There are an enormous multitude of distinct doubly-even codes
for N ≤ 32, even when counting permutation equivalent codes as
the same code. Thus, there is an enormous multitude of Adinkra
chromotopologies.
This paper is organized as follows: Section 2 is a brief introduction tocodes, and Section 3 provides a review of Adinkras and their relationshipwith supermultiplets. The ﬁrst major result, in Section 4, is that each
Adinkra chromotopology gives rise to a doubly even code. It will be convenient to provide a few classes of examples of doubly even codes for our
discussions, and to give a sense for how many doubly even codes there are,so this is done in Section 5. We then turn to the second major result: thatevery doubly even code actually arises as the code for an Adinkra chromotopology for a supermultiplet. This is done in Sections 6 and 7. Sections 8
and 9 discuss some consequences and directions for further research.

2

Codes

We begin with a brief introduction to the theory of codes. For a morethorough introduction to the subject, see [18–20].
We think of {0, 1} as a group with the operation , which is addition
modulo 2, i.e., the group Z2 . For the purposes of this paper, a code of length
N means a subgroup of {0, 1}N .1 Although the standard notation for anelement of a cartesian product is (x1 , x2 , . . . , xN ), in practice we frequently
abandon the parentheses and the commas, so that the element (0, 1, 1, 0, 1)
may be written more succinctly as the codeword 01101. The componentsof such an N -tuple are called bits, and the N -tuple is called a word. This
word is called a codeword if it is in the code.
1

In the coding literature, there are sometimes other more general deﬁnitions of codes.
What we have described is a linear binary block code of length N .

1914

CHARLES F. DORAN ET AL.

Now, Z2 is not only a group; it is also a ﬁeld, so {0, 1}N can be viewed
as a vector space over {0, 1}. All the concepts of linear algebra then apply,
but with R replaced with Z2 . Elements of {0, 1}N may be thought of as
vectors, with vector addition the operation  of bitwise addition modulo 2.
Codes are then linear subspaces of {0, 1}N . Every code has a basis, called
a generating set, g1 , . . . , gk , so that every codeword can be written uniquely
as a sumk

xi gi ,

(2.1)

i=1

where the coeﬃcients x1 , . . . , xk are each either 0 or 1. The number k is thesame for every generating set for a given code, and is called the dimensionof the code. It is common to say we have an [N, k] linear code when N is thelength of the codewords and k is the dimension. It is traditional to denotea generating set as an k×N matrix, where each row is an element of thegenerating set.
If v ∈ {0, 1}N , we deﬁne the weight of v, written wt(v), to be the numberof 1s in v. For instance, the weight of 01101 is wt(01101) = 3.
A code is called even if every codeword in the code has even weight. It iscalled doubly even if every codeword in the code has weight divisible by 4.
Examples of doubly even codes are given in Section 5.1 below.
If v and w are in {0, 1}N , then v & w is deﬁned to be the “bitwise and”
of v and w: the ith bit of v & w is 1 if and only if the ith bit of v and theith bit of w are both 1. A basic fact in {0, 1}N iswt(v  w) = wt(v) + wt(w) − 2 wt(v & w).

(2.2)

There is a standard inner product. If we write v and w in {0, 1}N as(v1 , . . . , vN ) and (w1 , . . . , wN ), thenv, w ≡

N


vi wi

(mod 2).

(2.3)

i=1

We call v and w orthogonal if v, w = 0. This occurs whenever there are aneven number of bit positions where both v and w are 1. Note that v, v ≡wt(v) (mod 2), and thus, when wt(v) is even, v is orthogonal to itself. Also
note that v, w ≡ wt(v & w) (mod 2). One important consequence for us isthat if wt(v) and wt(w) are multiples of 4, then (2.2) implies that wt(v  w)
is a multiple of 4 if and only if v and w are orthogonal.

CODES AND SUPERSYMMETRY IN ONE DIMENSION

3

1915

Supersymmetric representations and adinkras

The N -extended supersymmetry algebra without central charges in one
dimension is generated by the time-derivative, ∂τ , and the N supersymmetry
generators, Q1 , . . . , QN , satisfying the following supersymmetry relations:{QI , QJ } = 2 i δIJ ∂τ ,[∂τ , QI ] = 0,

I, J = 1, . . . , N.

(3.1)

In this section, we determine some essential facts about the transformation
rules of these operators on ﬁelds for which it is possible to maintain thephysically motivated concept of engineering dimension. We note that since
the time-derivative has engineering dimension [∂τ ] = 1, the supersymmetry
relations (3.1) imply that the engineering dimension of the supersymmetry
generators is [QI ] = 12 .
3.1

Supermultiplets as representations of supersymmetry

A real supermultiplet M is a real, ﬁnite-dimensional, linear representationof the algebra (3.1), in the following sense: It is spanned by a basis of realbosonic and fermionic component ﬁelds, φ1 (τ ), . . . , φm (τ ) and ψ1 (τ ), . . . ,ψm (τ ), respectively; each component ﬁeld is a function of time, τ . The
supersymmetry transformations, generated by the Hermitian operators
Q1 , . . . , QN , act linearly on M while satisfying equations (3.1). The supermultiplet is oﬀ-shell if no diﬀerential equation is imposed on it2 . The numberof bosons as fermions is then the same, guaranteed by supersymmetry.

3.2

Building supermultiplets from Adinkras

The authors of [4, 17, 23] introduced and then studied Adinkras, diagrams
that encode the transformation rules of the component ﬁelds under theaction of the supersymmetry generators Q1 , . . . , QN .
Supermultiplets that can be described by Adinkras have a collection ofbosonic and fermionic component ﬁelds and a collection of supersymmetry2

Logically, it is possible for some — but not all — component ﬁelds to become subject
to a diﬀerential equation. This does not violate the literal deﬁnition of the oﬀ-shell supermultiplet. However, it does obstruct standard methods of quantization, which is our
eventual purpose for keeping supermultiplets oﬀ-shell. For an example in 4-dimensional
supersymmetry; see [15].

1916

CHARLES F. DORAN ET AL.

generators Q1 , . . . , QN , so that: (1) Given a bosonic ﬁeld φ and a supersymmetry generator QI , the transformation rule for QI of φ is of the form
either QI φ = ± ψ,
or QI φ = ± ∂τ ψ,

(3.2)
(3.3)

for some fermionic ﬁeld ψ. (2) Given instead a fermionic ﬁeld η and asupersymmetry generator QI , the transformation rule of QI on η is of theform
QI η = ± i B,

(3.4)

or QI η = ± i ∂τ B,(3.5)

eitherfor some bosonic ﬁeld B. In particular, these supersymmetry generators act
linearly using ﬁrst-order diﬀerential operators. Furthermore, the supersymmetry algebra requires that
QI φ = ± ψ⇐⇒

QI ψ = ± i ∂τ φ,(3.6)

QI ψ = ± i φ,

(3.7)

and
QI φ = ± ∂τ ψ⇐⇒

and where the ± signs are correlated to preserve equations (3.1).
More generally, suppose we label the bosons φ1 , . . . , φm and the fermionsψ1 , . . . , ψm . Choose an integer I with 1 ≤ I ≤ N , and an integer A with
1 ≤ A ≤ m. For each such pair of integers, we consider the transformation
rules for QI on the boson φA , and we might expect that these will be of theform
QI φA (τ ) = c ∂τλ ψB (τ ),(3.8)

where c = ±1, λ = 0 or 1, and B is an integer with 1 ≤ B ≤ m, so that ψBis some fermion; each of c, λ, B will, in general, depend on I and A. Notethat
1
λ = [φA ] − [ψB ] + ,
2

(3.9)

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1917

Table 1: The correspondences between the Adinkra components and supersymmetry transformation formulae: vertices ⇐⇒ component ﬁelds; vertex color ⇐⇒ fermion/boson; edge color/index ⇐⇒ QI ; edge dashed ⇐⇒c = −1; and orientation ⇐⇒ placement of ∂τ . They apply to all φA , ψB
within a supermultiplet and all QI -transformations amongst them.
AdinkraQ-action
  

ψ
iφ̇A
QI B =φA
ψB

AdinkraQI

QI

Q-action
 

ψB
−iφ̇A
=
φA
−ψB



 

φA
ψ̇B
=
ψB
iφA


QI

 

φA
−ψ̇B
=
ψB
−iφA

The edges are here labeled by the variable index I; for ﬁxed I, they are drawn in the I th color.

for φA and ψB to have a deﬁnite engineering dimension — provided thetransformation rules had only dimensionless constants as we assume throughout. For each such transformation rule, we will get a correspondingtransformation rule for the QI on the fermion ψB (τ ) that looks like this:QI ψB (τ ) =i 1−λ∂φA (τ ).
c τ

(3.10)

Equations (3.8)–(3.10) constitute all of the transformation rules on thebosons and fermions, respectively.
Deﬁnition 3.1. A supermultiplet, M , is adinkraic if all of its supersymmetric transformation rules are of the form (3.8) and (3.10).
For each adinkraic supermultiplet, its Adinkra, AM , is a directed graph,consisting of a set of vertices, V , a set of edges, E, a coloring C of the edges,
a set of their orientations, O, and a labeling D of each edge corresponding
to whether or not it is dashed.
Each component ﬁeld of M is represented by a vertex in AM : white forbosonic ﬁelds and black for fermionic ones, thus equipartitioning the vertex
set V → W . Every transformation rule of the form (3.8) is depicted by anedge connecting the vertex corresponding to φA to the vertex correspondingto ψB , and color the edge with the I th color. We use a dashed edge if c = −1,
and oriented it from φA to ψB if λ = 0 and the other way around if λ = 1.
Table 1 illustrates the four possibilities for an edge.
We can also use the Adinkra to reconstruct the adinkraic supermultiplet,since the Adinkra contains all the information necessary to write down the1918

CHARLES F. DORAN ET AL.

transformation rules. Thus, an Adinkra is simply a graphical depiction of
the transformation rules (3.2)– (3.5).

4Adinkra chromotopologies

It is the purpose of this paper to classify the possible topologies for Adinkras.
To this end, we will need the precise deﬁnition:
Deﬁnition 4.1. The topology of an Adinkra, T(AM ), is the graph (V, E)
consisting of only the (unlabeled) vertices and edges of the Adinkra; cf.
Deﬁnition 3.1. Also, T(M ) := T(AM ).
In particular, from Deﬁnition 3.1 of an Adinkra, we forget the bipartition(black or white) of the vertices and all additional information associated tothe edges, namely, the color of the edges (which records the QI corresponding to that edge), dashedness (c = ±1), and direction (the exponent of ∂τin (3.2)–(3.5)).
Deﬁnition 4.2. The chromotopology of an Adinkra is the topology of the
Adinkra, together with the vertex bipartition (coloring each vertex black orwhite), and the edge coloring (assigning a color to each edge).
In particular, from Deﬁnition 3.1 of an Adinkra, we forget the dashednessof the edges and the direction of the arrow along the edge.
Suppose we have an adinkraic multiplet with component ﬁelds (F1 (τ ), . . . ,F2m (τ )),3 and let A be its Adinkra. We wish to consider the chromotopology. That is, we wish to ignore the arrows on the edges and ignore whether
an edge is dashed. We can capture this information by taking the vertex
set of the Adinkra V = {v1 , . . . , v2m }, the coloring of these vertices, and theedge set E with its coloring. Since this graph is inherited from an Adinkra,
for every vertex and every I ∈ {1, . . . , N } there is an edge corresponding toapplying QI to the ﬁeld corresponding to that vertex. Applying this QI may
involve derivatives, a sign, and/or a factor of i, but it will be convenient for
now to suppress this. To this end, we deﬁne functions q1 , . . . , qN from thevertex set V = {v1 , . . . , v2m } to itself, such that whenever A, B, and I are

3

Previously, we labeled the bosons φ1 , . . . , φm and the fermions ψ1 , . . . , ψm . Here, itwill be convenient for notation to treat them on the same footing.

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1919

such that there is an equation of type (3.8) or (3.10),
QI FA (τ ) = c ∂τλ FB ,=⇒qI (vA ) = vB .

(4.1)

Note that in deﬁning qI from QI , we are forgetting the coeﬃcients c = ±1,
as well as the ∂τ ’s which encode the diﬀerences in engineering dimensions.
The supersymmetry algebra (3.1) then implies:
qI2 = 1l,i.e., qI qI (v) = v, for all v ∈ V ,

(4.2)

and
qI qJ = qJ qI , i.e., qI qJ (v) = qJ qI (v) , for all I, J, and all v ∈ V .
(4.3)
The vertex set V = {v1 , . . . , v2m }, the partition of V into bosons andfermions, and the maps qI : V → V then encode the chromotopology.
4.1

The colored N -cube chromotopology

The fundamental example of an Adinkra topology is that of the N -cube,
I N = [0, 1]N . It has 2N vertices and N ·2N −1 edges. We may embed it
in RN by locating the vertices at the points p = (p1 , . . . , pN ) ∈ RN , where
pI = 0, 1 in all 2N possible combinations. An edge connects two vertices
that diﬀer in precisely one coordinate. For every vertex, p, the weight of p,
written wt(p), equals the number of J ∈ {1, . . . , N } for which pJ = 1.
There is a natural chromotopology associated to the N -cube: we color
vertices of even weight white, and vertices of odd weight we color black.
We associate the numbers from 1 to N with N diﬀerent colors, and thencolor each edge of the N -cube that connects vertices which diﬀer only in the
Ith coordinate with the Ith color. The result is called the colored N -cube.
Using the notation from the previous section, we determine q1 , . . . , qN as
follows:
qI (p1 , . . . , pI−1 , pI , pI+1 , . . . , pN ) = (p1 , . . . , pI−1 , 1 − pI , pI+1 , . . . , pN ).
(4.4)
There are indeed Adinkras of supermultiplets that have this chromotopology, as we will see in Section 7.

1920
4.2

CHARLES F. DORAN ET AL.
The quotient of a colored cube by a code

We now introduce another chromotopology: given a colored N -cube and acode C of length N , we will create a chromotopology called [0, 1]N /C.
Before we do so, recall that C is a subset of {0, 1}N , and so codewords(elements of C) are vectors, which we can write as g. Likewise, the verticesof the colored N -cube are elements of {0, 1}N as well.
For each vertex v ∈ {0, 1}N , we identify it with all vertices of the formv  g for g ∈ C. The resulting vertex set we call V  . Likewise, consider anedge e colored I, connecting v to w (note that this means the v and w are
identical except in bit I, where they diﬀer). We identify e with the otheredges colored I that connect v  g with w  g (note that v  g and w  galso diﬀer precisely at bit I, and so there is a unique edge connecting them,and it is colored I). One way to view this is that the group C acts on thevertex set and the edge set by , and we are taking the orbit space of thisgroup action.
Now note that we have thereby created a graph with vertices and edges,
and the colors of the edges is determined by this construction. In order toconsistently label the colors of the vertices, we must only identify vertices
when they are of the same color (boson or fermion). Since the color of avertex in the colored N -cube is determined by its weight modulo 2, andsince equation (2.2) says thatwt(v  g) = wt(v) + wt(g) − 2 wt(v & g),(4.5)

we see that wt(v  g) agrees with wt(v) modulo 2 for all g if and only if
wt(g) is always even. Therefore, if C is an even code, this quotient of acolored N -cube gives a consistent coloring of the vertices.
This quotient may not necessarily be the chromotopology of the Adinkra
of a supermultiplet: we will see in the next section that for this, the code
must be doubly even. We will then see in Section 7 that this is suﬃcient.

4.3

The code of an Adinkra chromotopology

Suppose an Adinkra is not connected. Then the corresponding supermultiplet splits into a direct sum, each component of which corresponds to aconnected component of the Adinkra. So if we can determine the chromotopologies of connected Adinkras, the general chromotopology of an Adinkra

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1921

will be a ﬁnite disjoint union of such. Thus we focus our attention toconnected Adinkras.
Construction 4.1. Given a connected Adinkra with N edge colors, weconstruct a code of length N as follows.
Let V be the set of vertices. Deﬁne qI : V → V as above. Pick any vertexv∗ ∈ V and ﬁx it. Now deﬁnexN(v∗ ) = v∗ .
C = (x1 , . . . , xN ) | q1x1 · · · qN

(4.6)

Proposition 4.1. The set C deﬁned in this construction is a code.
Proof. We need to prove that the elements of C form a subgroup of {0, 1}N .
To prove closure, let x = (x1 , . . . , xN ) and y = (y1 , . . . , yN ) be elements of
C. By equation (4.3), we havexN yNyN
x N y1(v∗ ) = q1x1 · · · qN(v∗ ) = v∗
q1x1 y1 · · · qN
q1 · · · qN

(4.7)

and so x  y ∈ C.
0 (v ) = v .
To prove identity, note that q00 · · · qN∗∗

To prove inverse, note that in {0, 1}N , every element is its own inverse.



Proposition 4.2. The code C is independent of the choice of vertex v∗ .
Proof. Let v0 be any other vertex in the Adinkra. Since the Adinkra isconnected, there is a sequence of edges connecting v∗ to v0 . If the colors ofthese edges are I1 , . . . , Im , then by the deﬁnition of qI , we have
qI1 · · · qIm (v∗ ) = v0 .

(4.8)

Using equation (4.3), we can rearrange the qIi so that they are in increasingorder, and we can furthermore cancel pairs of qIi so that each qI appears at
most once. In this way, we obtain a vector t = (t1 , . . . , tN ) ∈ {0, 1}N so thattN(v∗ ) = v0 .
q1t1 · · · qN

(4.9)

Now suppose g is a codeword obtained by applying Construction 4.1 usingv∗ . ThengN(v∗ ) = v∗ .
q1g1 · · · qN

(4.10)

1922

CHARLES F. DORAN ET AL.

tN
to both sides, using equation (4.3) to commute the qI past
Apply q1t1 · · · qN
each other, and we get
gN t1
tN
tN
q1g1 · · · qN
(q1 · · · qN(v∗ )) = q1t1 · · · qN(v∗ ).

(4.11)

Then we plug in equation (4.9) and getgN(v0 ) = v0 .
q1g1 · · · qN

(4.12)

Therefore, any codeword for v∗ is a codeword for v0 . Applying the argument
with v0 and v∗ switched proves the converse.
Remark 4.1. Thus we can determine the code of an Adinkra as follows:ﬁrst pick any vertex v∗ . Then for each g = (g1 , . . . , gN ) ∈ {0, 1}N , write
down the set S of all colors I for which gI = 1, in some order. Then takethe path that starts at v∗ and goes along edges according to the colors inthe set S. If we return to v∗ , then g is in the code.
For example, consider the Adinkra in (7.9) in Section 7 below. If we takev∗ = φ(0000) , we can follow the colors black (color 1), then red (color 2), thengreen (color 3), then blue (color 4). This takes us on a path that goes toψ(1000) then φ(1100) , then ψ(1110) , then back to φ(0000) . From this we see that1111 is in the code. On the other hand, suppose we took red (color 2), thenblue (color 4). This takes us from φ(0000) to ψ(0100) to φ(1010) , which is not
φ(0000) . So we see that 0101 is not in the code. You can check that the codeis {0000, 1111} no matter which starting vertex v∗ we use.
Theorem 4.3. Every connected Adinkra chromotopology is isomorphic to aquotient of a colored N -dimensional cube by the code of the chromotopology.
Proof. Suppose we have a connected Adinkra with N edge colors, withvertex set V and edge set E. Pick any one bosonic vertex v∗ ∈ V and
ﬁx it. As in Construction 4.1, we deﬁnegN(v∗ ) = v∗ }.
C = {(g1 , . . . , gN ) | q1g1 · · · qN

(4.13)We then take the quotient of the colored N -cube by C. Let the vertex setof this quotient be called W , and let the edge set be called F . Now we
produce an isomorphism from the chromotopology of the quotient of thecolored N -cube to the chromotopology of the original Adinkra. By this wemean a bijection from W to V that preserves the colors of the vertices,
and a bijection from F to E that preserves the colors of the edges and therelationship of which edges are incident with which vertices.

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1923

To deﬁne the bijection f : W → V , let w ∈ W be any vertex in W . It
comes from a quotient of an N -cube, meaning it is the result of identifying
vectors in {0, 1}N , so there is at least one vector (w1 , . . . , wN ) ∈ {0, 1}N thatwas used in the identiﬁcation to produce w. We deﬁnewN(v∗ ).
f (w) = q1w1 · · · qN

(4.14)

We note that any other choice of N -tuple that produces w would be (w1 , . . . ,wN )  g where g ∈ C, andwN gNgN
wN g1
wN(v∗ ) = q1w1 · · · qN(v∗ ) = q1w1 · · · qN(v∗ ) = f (w),q1w1 g1 · · · qN
q1 · · · qN
(4.15)

so that f does not depend on the choice of (w1 , . . . , wN ).
We now prove f is injective. Suppose f (w) = f (x). ThenwN
xN(v∗ ) = q1x1 · · · qN(v∗ ).
q1w1 · · · qN

(4.16)

By using equation (4.3), we obtainwN xN(v∗ ) = v∗ ,q1w1 x1 · · · qN

(4.17)

which implies (w1 , . . . , wN )  (x1 , . . . , xN ) = (g1 , . . . , gN ) is in the code C.
Then (x1 , . . . , xN ) = (w1 , . . . , wN )  (g1 , . . . , gN ), so that the vertices(w1 , . . . , wN ) and (x1 , . . . , xN ) are identiﬁed in the construction of thequotient. Therefore w = x.
To see that f is surjective, let v ∈ V be any vertex. Recall that we areassuming that the Adinkra is connected. That is, every vertex v ∈ V isconnected, via a path of edges, to the ﬁxed v∗ ∈ V . In the Adinkra, theseedges have colors, forming a sequence, I1 , . . . , Ik , when tracing from thevertex v to v∗ . If we then apply to v∗ a corresponding sequence of qI ’s, we
obtain:qI1 · · · qIk (v∗ ) = v.

(4.18)

Using the commutativity of the q’s to put them in numerical order andequation (4.2) to eliminate those that appear more than once, we can write
this asxN(v∗ ) = v.
q1x1 · · · qN

(4.19)

However, the left hand side is just f (x1 , . . . , xN ). Therefore, f is surjective.

1924

CHARLES F. DORAN ET AL.

We now examine the edges. Let e be an edge in F with color I. It connects
two vertices v and w, so that qI (v) = w. We wish to show that there is anedge in E colored I connecting f (v) to f (w); that is, that qI (f (v)) = f (w).
Choose a vector (v1 , . . . , vN ) ∈ {0, 1}N that gets used in the identiﬁcation
to make v. For (w1 , . . . , wN ) we take(w1 , . . . , wN ) = (v1 , . . . , vI−1 , 1 − vI , vI+1 , . . . , vN ),(4.20)

which is possible because this is the edge of color I coming from (v1 , . . . , vN )in the original N -cube before quotienting. Now we apply qI to f (v):vN(v∗ )qI (f (v)) = qI q1v1 · · · qN
vI−1 1−vI vI+1
vN
qI
qI+1 · · · qN(v∗ )= q1v1 · · · qI−1
wN
w1
= q1 · · · qN (v∗ )= f (w).

(4.21)(4.22)(4.23)(4.24)

Therefore, there is an edge colored I connecting f (v) to f (w).
The fact that π sends bosons to bosons and fermions to fermions can now
be seen by the fact that it sends the boson (0, . . . , 0) ∈ {0, 1}N to the bosonv∗ , and the fact that every vertex in V is connected to v∗ by a sequence ofedges, each of which alternates between bosons and fermions.

We next consider what kinds of codes C can appear in Adinkras. We
have already noted in the previous section that in order to get a consistent vertex-coloring, C must be an even code. We now show that theedge-dashing of the original Adinkra (which we have been ignoring instudying chromotopologies) will force C to be doubly even; that is, for eachg ∈ C, wt(g) must be a multiple of 4.
Theorem 4.4. The code for an Adinkra must be doubly even.
Proof. To prove this, the qI ’s no longer suﬃce, and we will need to use the
QI ’s; in particular, we need to “remember” the scaling constants c encoding
the edge-dashedness in an Adinkra, and the equipartition of the vertices into
bosonic and fermionic ones.
The statement that x ∈ C means thatxN(v∗ ) = v∗ ,q1x1 · · · qN

(4.25)

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1925

which means, if F∗ is the component ﬁeld that is represented by v∗ , thenQx1 1 · · · QxNN F∗ (τ ) = c ∂τwt(x)/2 F∗ (τ )(4.26)

for some complex number c. The exponent of ∂τ follows simply fromcomparing engineering dimensions of the left-hand side and the right-handside.
Since this sequence of QI operators, corresponding to a closed path in the
Adinkra, must send bosons to bosons and fermions to fermions, it must bethat wt(x) is even, so that 12 wt(x) is indeed an integer and equation (4.26)
is well deﬁned.
Applying Qx1 1 · · · QxNN twice to F∗ (τ ), we ﬁnd:
Qx1 1 · · · QxNN · Qx1 1 · · · QxNN F∗ (τ ) = c2 ∂τwt(x) F∗ (τ ).

(4.27)On the left side, using the supersymmetry algebra (3.1), we cananti-commute the QI past each other. Rearrange these to regroup the result
into
wt(
x)
2xN
2 wt(x)
1F∗ (τ ),(−1)( 2 ) Q2x
1 · · · QN F∗ (τ ) = c ∂τ

(4.28)

which, using the supersymmetry algebra (3.1) again, becomeswt(x)(−1)( 2 ) iwt(x) ∂τwt(x) F∗ (τ ) = c2 ∂τwt(x) F∗ (τ ).

(4.29)

Comparing the exponents of ∂τ in equation (4.29) conﬁrms the exponent of∂τ in equation (4.26). Comparing the numerical coeﬃcients produces:
wt(
x)
2
c2 = (−1)( 2 ) iwt(x) = iwt(x)(wt(x)−1)+wt(x) = i(wt(x) ) .

(4.30)

Since wt(x) is even, we know that wt(x)2 is a multiple of 4. Thus, the left
hand side of this is 1.
Using (3.8) and (3.10) repeatedly, we see that c is ±1 if wt(x) = 0 (mod 4),
and ±i if wt(x) = 2 (mod 4). But c2 = 1 implies that c = ±1, and this canhappen only if wt(x) = 0 (mod 4).
Remark 4.2. The converse is also true. That is, if G is a doubly even code,
then there is a family of adinkraic supermultiplets, the chromotopology of
the Adinkra of which is a quotient by G of a colored N -cube. This will bedone in Section 7.

1926

5
5.1

CHARLES F. DORAN ET AL.

Finding doubly even codes
Examples of doubly even codes

Since doubly even codes classify chromotopologies, it is useful to consider afew examples of such codes. For each N there is a trivial doubly even code
{00 · · · 0} with one element, which we call tN ; its generating set is the empty
set. In addition, when N = 4, there is a code {0000, 1111}, called d4 . The
generating set is {1111}. More generally, for every even N ≥ 4, there is adoubly even code called dN , of length N and with N2 − 1 generators, withgenerating set⎡⎤
111100000 ··· 00000⎢0 0 1 1 1 1 0 0 0 · · · 0 0 0 0 0⎥⎢⎥
⎢0 0 0 0 1 1 1 1 0 · · · 0 0 0 0 0⎥
⎢
⎥.
⎢
⎥
..
⎣
⎦
.

(5.1)

000000000 ··· 01111
Note that this is a description of the generating set, so that the actual code
has more codewords, including the null-vector and all those constructed byadding (bitwise, modulo 2) any number of these generators together. For
example,⎡⎤11110000⎣0 0 1 1 1 1 0 0⎦00001111⎫v0 = 0 0 0 0 0 0 0 0⎪⎪⎪v1 = 1 1 1 1 0 0 0 0⎪⎪⎪⎪v2 = 0 0 1 1 1 1 0 0⎪⎪⎪⎬v3 = 0 0 0 0 1 1 1 1.
v1  v2 = 1 1 0 0 1 1 0 0⎪
⎪
⎪
⎪
⎪
⎪
⎪
v2  v3 = 0 0 1 1 0 0 1 1⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪

v
=
1
1
1
1
1
1
1
1
v
⎪
⎪
1
3
⎪
⎭
⎩v  v  v = 1 1 0 0 0 0 1 1⎪
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨

generates

1

2

(5.2)

3

Note that the same code, on the right-hand side of the display (5.2), is justas well generated by {v1 , v2 , (v1  v3 )} and several other choices. For generalN
N , the dN code contains 2 2 −1 codewords.
When N is congruent to 7 or 8 modulo 8, there is an important doubly
even code called eN , the generating set of which is that of dN (or t1 ⊕ dN −1
when N ≡ 7 (mod 8)) augmented by an additional generator of the form

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1927

101010 · · · . For instance,⎡⎤
1111000
e7 : ⎣0 0 1 1 1 1 0⎦ ,1010101⎡⎤
11110000⎢0 0 1 1 1 1 0 0⎥⎥
e8 : ⎢⎣0 0 0 0 1 1 1 1⎦ ,10101010(5.3)

and we then write:⎡⎤
111100000000000⎢0 0 1 1 1 1 0 0 0 0 0 0 0 0 0⎥⎢⎥
⎢0 0 0 0 1 1 1 1 0 0 0 0 0 0 0⎥⎢⎥⎥
e15 : ⎢
⎢0 0 0 0 0 0 1 1 1 1 0 0 0 0 0⎥ ,⎢0 0 0 0 0 0 0 0 1 1 1 1 0 0 0⎥⎢⎥
⎣0 0 0 0 0 0 0 0 0 0 1 1 1 1 0⎦101010101010101⎡⎤
1111000000000000⎢0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0⎥⎢⎥
⎢0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0⎥⎢⎥
⎢0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0⎥⎢
⎥ , (5.4)
e16 : ⎢
⎥
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
⎢
⎥
⎢0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0⎥⎢⎥
⎣0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1⎦
1010101010101010

and so on.
These are famous codes: e7 is known as the Hamming [7, 3] code, and e8
is the parity-extended Hamming code. Conway et al. [20], describe what it
calls “Construction A”, which determines a lattice as a subset of ZN of all
the points whose coordinates modulo 2 are in the code, and under this, weform the famous lattices e7 and e8 . The points that are of closest distance tothe origin form the root lattice for the Lie algebras E7 and E8 , respectively.
The previously deﬁned dN codes likewise relate to the root lattice for the
Lie algebras DN .
Besides the trivial doubly even code tN , {000 · · · 0}, for any N ≡ 0 (mod 4),
there is an [N, 1] doubly even code hN consisting of {000 · · · 0, 111 · · · 1}, thegenerating set of which is {111 · · · 1}.4 Note that h4 = d4 , but hN ⊂ dN for
N = 8, 12, 16, . . . .
There are many other doubly even codes, and the number grows quickly
as N becomes large; see Appendix B and [20, 26].

4This is the only code mentioned here not speciﬁcally named in [19].

1928
5.2

CHARLES F. DORAN ET AL.
Permutation equivalence and R-symmetry

It is also possible to permute the columns in a code. For instance, for e7 wemight swap the last two columns and obtain a generating set⎡⎤
1111000⎣0 0 1 1 1 0 1⎦ .
(5.5)
1010110
This is another doubly even code, and it is diﬀerent from e7 as given inequation (5.3). To verify this, one could write the 8 codewords in both casesand compare. More generally, any permutation of columns of a code willproduce another code, which is sometimes the same code, sometimes not.
An example where a column-permutation results in precisely the samecode again can be seen by taking the e7 generating set (5.3), and swapping
the ﬁrst and third column, then swapping the second and fourth column.
The result would be:⎡⎤⎡⎤
1111000
1111000⎣0 0 1 1 1 1 0⎦ −→ ⎣1 1 0 0 1 1 0⎦ .
(5.6)
1010101
1010101
The result does not look like the e7 generating set, but it generates the samecode. Indeed, replace the second generator with the sum of the ﬁrst and thesecond generator, and we recover exactly the original generator set for e7 .
If a permutation of the columns sends one code to another, the codes are
said to be permutation-equivalent. It is convenient for the classiﬁcation andnaming of codes to give one name for the permutation equivalence class, andrecognize the multiplicity of codes that the name represents.
Since the columns of a code correspond to the various QI , a permutation of
the columns of the code corresponds to a permutation of the QI , i.e.,to an Rsymmetry. For real N -extended supersymmetry, the group of R-symmetries
is O(N ); the permutation equivalences describe the subgroup of this matrix
group consisting of permutation matrices. Though this might suggest thatthe physically relevant question is permutation equivalence of codes, this is
not necessarily so: It may well be possible to construct a theory with two
types of supermultiplets, corresponding to two diﬀerent but permutationequivalent codes, coupled in a way that precludes rewriting the same theory
in terms of only one type of supermultiplet. Although diﬀerent in technicaldetail, the inextricable coupling of chiral and twisted-chiral supermultiplets
discovered in [27] is a conceptual paradigm of this possibility.

CODES AND SUPERSYMMETRY IN ONE DIMENSION1929

The columns also correspond to the colors of the Adinkra, so permutation equivalence classes give rise to Adinkra topologies (without the edge
colors). This raises the question: do permutation equivalence classes of doubly even codes classify connected Adinkra topologies? Certainly we have justdescribed a map from the set of permutation equivalence class of doubly evencodes to the set of connected Adinkra topologies. And certainly this map issurjective. But is it injective? That is, is it possible that two non-equivalent
doubly even codes will give rise to the same Adinkra topology? The answer
to this question is not clear, but luckily, in trying to classify Adinkras, we
can leapfrog the issue of classifying Adinkra topologies and instead use theclassiﬁcation of Adinkra chromotopologies, where the issue is clear.
We have created computer algorithms to ﬁnd and count permutationequivalence classes of doubly even codes. Table 2 provides a list of such
Table 2: A listing of permutation equivalence classes for N up to 11, with
the number of codes in the permutation equivalence class given in in theparentheses. Here, ⊕ denotes a vector space direct sum, so that if U ⊂ (Z2 )Nand V ⊂ (Z2 )M , then U ⊕ V ⊂ (Z2 )N ⊕ (Z2 )M ∼= (Z2 )N +M . The notationtM ∗ C denotes the direct sum together with at least one additional “glue”
codeword extending into the tM summand, similar to how a eN code isconstructed from the corresponding dN code.
N k =0
k =1
k =2
k =3
k =4
4 t4 (1) d4(1)
5 t5 (1) t1 ⊕ d4 (5)(15)
6 t6 (1) t2 ⊕ d4 (15) d6(30)
7 t7 (1) t3 ⊕ d4 (35) t1 ⊕ d6 (105) e7(30)
8 t8 (1) t4 ⊕ d4 (70) t2 ⊕ d6 (420) t1 ⊕ e7 (240) e8(1) d4 ⊕ d4 (35) d8(105)
h8
9 t9 (1) t5 ⊕ d4 (126) t3 ⊕ d6 (1260) t2 ⊕ e7 (1080) t1 ⊕ e8 (270)(315) t1 ⊕ d8 (945)
t1 ⊕ h8 (9) t1 ⊕d4 ⊕ d4
10 t10 (1) t6 ⊕ d4 (210) t4 ⊕ d6 (3150) t3 ⊕ e7 (3600) t2 ⊕ e8 (1350)(945)
t2 ⊕ h8 (45) t6 ∗ d6 (630) d4 ⊕ d6 (3150) d10(1575) t2 ⊕ d8 (4725)
t2 ⊕d4 ⊕ d4
11 t11 (1) t7 ⊕ d4 (330) t5 ⊕ d6 (6930) t4 ⊕ e7 (9900) t3 ⊕ e8 (4950)(6930) t1 ⊕
(34650) t1 ⊕ d10 (10395)
t3 ⊕ h8 (165) t1 ⊕t6 ∗ d 4
d4 ⊕ d6(5775) t3 ⊕ d8 (17325) d4 ⊕ e7 (9900)
t3 ⊕d4 ⊕ d4
t5 ∗ d6 (13860)

1930

CHARLES F. DORAN ET AL.

Table 3: Number of distinct permutation classes of doubly even [N, k]
codes. The “ * ” entry indicates codes that are still being enumerated;
see http://www.rlmiller.org/de codes/ for up-to-date results, including
links to listings of the actual codes.
k
N\ 1

2

3

4

5

6

7

8

9

10

11

12

13

14

4

1

5

1

6

1

1

7

1

1

1

8

2

2

2

1

9

2

2

2

1

10

2

3

3

2

11

2

3

4

3

12

3

5

7

7

2

13

3

5

8

8

4

14

3

7

12

14

9

15

3

7

15

20

15

8

16

4 10

23

38

36

23

9

2

17

4 10

25

45

50

34

14

3

18

4 13

34

72

94

79

35

9

19

4 13

40

94

146

141

75

19

20

5 17

57

158

295

353

231

84

21

5 17

63

194

439

629

494

198

38

22

5 21

83

298

812

1481

1465

740

187

25

23

5 21

95

387

1287

2970

3811

2362

714

119

24

6 27 129

607

2444

7287

12395

10048

3710

739

94

9

25

6 27 141

755

3808

15177

35916

38049

16039

2973

309

22

26

6 32 180 1114

6923

37455

128270

194626103527

20206

1829

103

27

6 32 202 1435 11320

86845

464579 1103023

817167174809

13578

525

28

7 39 263 2136 20812

224825 1917212 7631323 8948070 2550127

203178

7402

151

29

7 39 287 2693 34233

555804 8084014* 4837471 133563

1940

30

7 46 359 3866 61871 1477074

*

*

*

*

*

* 7074431

7 46 400 4972

*

*

*

*

*

*

*

*

* 2549732

8 55 506

*

*

*

*

*

*

*

*

*

*

4

5

6

7

8

9

10

11

12

13

N

/k 1

2

3

15 16

4
2

*

10

*

11

731
210

* 7689
14

85

15 16

for N ≤ 11, but beyond this the list is too long to include here. Instead,
we provide table 3, which provides the number of permutation equivalenceclasses of doubly even codes for N ≤ 32, except for a few cases where we do
not yet know the answer. Details on how these tables were created are in
Appendix B.

6Supersymmetry and Cliﬀord algebras

It has been understood for a long time that the supersymmetry algebra (3.1)
has a formal similarity with the Cliﬀord algebra generated by the Dirac ΓI
matrices5 :{ΓI , ΓJ } = 2 δIJ 1l.
5

(6.1)

In (3.1), I, J = 1, . . . , N count fermionic dimensions. Accordingly, the quadratic form,δIJ , occurring in the right-hand side of equation (6.1) is positive deﬁnite. This is unlike

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1931

One manifestation of this was the study of the spinning particle by Gates
and Rana [1, 2], resulting in the Scalar Supermultiplet (which we herein
rename Isoscalar Supermultiplet), deﬁned in terms of the GR(d, N ) algebra,
a form of the Cliﬀord algebra. We will review their work ﬁrst, and more
explicitly describe how it relates to Cliﬀord algebras; this will motivate themore general construction of relating supermultiplets to a representation of
the Cliﬀord algebra.
The main idea relating supermultiplets and Cliﬀord representations is
to note that the supersymmetry algebra in one dimension and the Cliﬀord
algebra diﬀer only in that in the former, there is a factor of i and a derivative∂τ . So, to turn a supermultiplet into a representation of the Cliﬀord algebra,
we can simply forget the factors of i and the derivatives ∂τ . This construction
is in a sense the same construction quotienting by the ideal (H − 1) in
Ref. [25], where H = i∂τ . This also removes the Z-grading aﬀorded by thenotion of engineering dimension. Conversely, to take a representation of theCliﬀord Algebra, we must re-insert factors of i and ∂τ . The factor of i must
go on the right-hand side of the transformation rules of the fermions (seeAppendix A.1) and if we declare that the ∂τ goes in that same place, we
end up with a supermultiplet called the Isoscalar supermultiplet.

6.1

The isoscalar supermultiplet and the cliﬀord algebra

The Isoscalar supermultiplet consists of component ﬁelds (φ1 , . . . , φm |ψ1 , . . . ,ψm ), written as column vectors:
⎡

⎤
φ1
⎢ ⎥
Φ = ⎣ ... ⎦

⎤
ψ1
⎢ ⎥
Ψ = ⎣ ... ⎦ .
⎡

and

φm

(6.2)

ψm

and with supersymmetry transformations given by the following Ansatz:
QI Φ = LI Ψ,(6.3)

QI Ψ = i RI ∂τ Φ,(6.4)

the typical ﬁeld theory use of the Cliﬀord/Dirac algebra, where I, J would count bosonicdimensions of spacetime, implying the Lorentzian signature for the quadratic form on theright-hand side of equation (6.1). Another point is that traditionally there would be aminus sign on the right-hand side; but this diﬀerence is equivalent to changing the metricfrom positive deﬁnite to negative deﬁnite. In the language of [28], this leads to the Cliﬀord
algebra Cl(0, N ) instead of Cl(N ) = Cl(N, 0).

1932

CHARLES F. DORAN ET AL.

where LI and RI are m × m real matrices to be determined. The supersymmetry algebra (3.1) then implies
LI RJ + LJ RI = 2 δIJ 1l,(6.5)

RI LJ + RJ LI = 2 δIJ 1l,(6.6)

where 1l is the m × m identity matrix. The I = J cases of these equations
imply that RI = L−1
I .
Remark 6.1. The additional requirement LI = ±RTI was also made in [1],but this is needed only when writing Lagrangians and will play no rôle inthis paper.
We create, for each I, a 2m × 2m real matrix ΓI of the form


0 LI
.
ΓI :=
RI 0

(6.7)

Equations (6.5) and (6.6) then imply that these Γ1 , . . . , ΓN indeed satisfy
equation (6.1), the algebra of the Dirac gamma matrices in dimension N ,whence the notation.
We do not insist that m be minimal here (a minimality criterion was used
in Refs. [1, 2]), nor do we adopt any particular convention for the type ofthe ΓI matrices, except that the entries must be real.
The property of being block oﬀ-diagonal corresponds to the existence ofa fermion number operator (−1)F , which anti-commutes with the ΓI . If wewrite our ﬁelds listing the bosons ﬁrst, followed by the fermions, then (−1)Fwill be a diagonal matrix of the form


1l 0
.
(6.8)
(−1)F =
0 −1l
If we deﬁne Γ0 to be (−1)F , then Γ0 anticommutes with the remaining ΓI :
Γ0 2 = +1l,

{Γ0 , ΓI } = 0.

(6.9)

The formal algebra generated by Γ0 , Γ1 , . . . , ΓN , is then deﬁned by the anticommutation relations:{ΓI , ΓJ } = 2 δIJ 1l,

for I, J = 0, 1, . . . , N(6.10)and therefore {Γ0 , . . . , ΓN } satisfy (6.1). The algebra generated by theΓ1 , . . . , ΓN is the Cliﬀord algebra Cl(0, N ), and the inclusion of Γ0 extends

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1933

Cl(0, N ) into the Cliﬀord algebra Cl(0, N +1). Any set of real matrices
representing these Γ0 , Γ1 , . . . , ΓN such that the above algebra closes is calleda real Cliﬀord representation [28].
Now suppose we are given a real Cliﬀord representation. We can split the
representation into the +1 and −1 eigenvalues for Γ0 , and if we choose abasis that respects this splitting, then Γ0 will be of the form given above,in (6.8). The Ansatz (6.3)–(6.4) then gives the transformation rules for the
Isoscalar multiplet corresponding to the Cliﬀord representation.

7

Constructing Adinkras from codes

The main construction in this paper is to take a description of an
N -dimensional cube [0, 1]N and a doubly even [N, k]-code C, and constructa real Cliﬀord representation. The Isoscalar supermultiplet corresponding to this real Cliﬀord representation will then be a supermultiplet for
N -extended supersymmetry in one dimension, and will have an Adinkra
with a chromotopology given by [0, 1]N /C.
7.1

Cliﬀord supermultiplets and cubical Adinkras

We start with the N -dimensional cube itself. To obtain an Adinkra withchromotopology the N -cube, we take the Cliﬀord Algebra itself, Cl(0, N ), asa Cliﬀord representation.
The Cliﬀord algebra Cl(0, N ) is a realN
2 -dimensional vector space, spanned by products of the form ΓI1 · · · ΓIk ,where I1 < · · · < Ik [28]. As a vector space, it splits as a direct sum of two
vector spaces: the even and odd parts. The even (resp. odd) part is thesubspace spanned by products of even (resp. odd) numbers of ΓI matrices.
The Cliﬀord algebra Cl(0, N +1) acts on Cl(0, N ) in the following way:for every 1 ≤ I ≤ N , ΓI acts by multiplication on the left. The operator
Γ0 multiplies the even Γ-monomials by 1 and the odd Γ-monomials by −1.
It is then easy to see that the Cliﬀord algebra (6.10) holds.
We take as a basis for Cl(0, N ) the products ΓI1 . . . ΓIk as above. Specifically, for every vertex of the cube x = (x1 , . . . , xN ) ∈ {0, 1}N , we deﬁne acomponent ﬁeld6ex = Γ1 x1 · · · ΓN xN .

(7.1)

6
Elements of a Cliﬀord algebra can be used to realize the component ﬁelds of asupermultiplet, by allowing the element of the Cliﬀord algebra to be a function of thetime-like coordinate τ . This construction was explicitly carried out in equation (67) of [3].

1934

CHARLES F. DORAN ET AL.

We note that ΓI ex = ΓI Γ1 x1 . . . ΓN xN can be transformed into the form (7.1),
with perhaps an overall minus sign: we use the anticommutation of the ΓI ’s
and, if xI = 1, the fact that ΓI 2 = 1. This results inΓI · ex = ±Γ1 x1 · · · ΓI−1 xI−1 ΓI 1−xI ΓI+1 xI+1 · · · ΓN xN .

(7.2)

The sign is +1 if the number of J with xJ = 1 and J < I is even, and is −1
otherwise. If we deﬁne (−1)|x<I| to be that sign, and deﬁne x  I to be thevector (x1 , . . . , xI−1 , 1 − xI , xI+1 , . . . , xN ), then this equation becomesΓI · ex = (−1)|x<I| ex  I .

(7.3)

We will also need the function wt(x), which equals the number of 1’s in x.
Thus, Cl(0, N ) is a representation of the Cliﬀord algebra Cl(0, N +1).
It corresponds to an Isoscalar supermultiplet, by replacing ex → φx (τ ) when
the weight of x is even, and ex → ψx (τ ) when the weight of x is odd.
Following the construction of the Isoscalar supermultiplet, we deﬁne LI and
RI to beLI ψx = (−1)|x<I| φx  I ,RI φx = (−1)|x<I| ψx  I ,(7.4)

QI φx = (−1)|x<I| i∂τ ψx  I .

(7.5)

and thus deﬁne
QI ψx = (−1)|x<I| φx  I ,The Adinkra for this is an N -dimensional cube. To see this, we notethat the basis elements are labeled by x ∈ {0, 1}N , the vertices of the N dimensional cube. The edges colored I connect x to x  I, which changes
the I th coordinate. The supermultiplet speciﬁed by equations (7.5) was
called the “base bosonic Cliﬀord Algebra superﬁeld” in [4]. To emphasize
its chromotopology, we will refer to this as the “Colored N -cube Cliﬀord
supermultiplet”.
7.2

The N = 4, D4 projection

In considering Adinkras that are not cubes, but rather quotients of cubes, it
is useful to ﬁrst consider a few examples. First, we consider the N = 4
example obtained by quotienting a four-dimensional cube by identifying
antipodal points. This Adinkra was ﬁrst described in Ref. [4], where it was
identiﬁed as the dimensional reduction of the D = 4 chiral superﬁeld. Here,
we present this example in a way that will motivate the general construction
to quotient cubes.

CODES AND SUPERSYMMETRY IN ONE DIMENSION
7.2.1

1935

One D 4 projection

Consider the Cliﬀord representation Cl(0, 4). Deﬁne the elementg = Γ1 Γ 2 Γ 3 Γ 4 .

(7.6)

Deﬁne the two linear transformations π+ , π− : Cl(0, N ) → Cl(0, N ) to be
π± (v) = v·

1±g
,
2

v, π± (v) ∈ Cl(0, N ).

(7.7)

Note that in the deﬁnition (7.7), we multiply v by a factor on its right.
This is important. It implies that for every ΓI and v ∈ Cl(0, N ), we havethat ΓI π± (v) = π± ΓI (v) .
The fact that g is even means that π± preserves the bosonic and fermionic
2 = π , π2 = π , π π =statistics. The fact that g 2 = 1 implies that π+
+
−
+ −
−
π− π+ = 0, and π+ + π− = 1l. This means that π+ and π− are a complete
set of projection operators, so that Cl(0, N ) = Im(π+ ) ⊕ Im(π− ) as Cliﬀord
representations, and π+ and π− project onto their corresponding components. Neither of these components are zero, since (1 + g)/2 and (1 − g)/2
are themselves non-zero elements of Cl(0, N ), which are in Im(π+ ) andIm(π− ), respectively.
This kind of projection is nothing new: the matrix g is, up to a scalar factor, the matrix known as γ5 in four-dimensional ﬁeld theory, andthe projection π± corresponds to the familiar projection to chiral spinors:
the left- and right-handed halves of the Dirac spinor.
It is also true that ker(π+ ) = Im(π− ) and vice-versa, so that we can also
describe these representations in terms of constraints: as “v such that v·(1 ∓g) = 0”. Then we can realize the Cliﬀord representation as a subspace ofCl(0, N ), rather than as a quotient. This accords with the idea that Cl(0, N ),being a representation for Cl(0, N +1), decomposes as a direct sum into
irreducibles.
We take the standard basis for Cl(0, 4) mentioned above, {ex : x ∈{0, 1}N },and apply π+ (resp. π− ) on it. The result spans Im(π+ ) (resp. Im(π− )),but there are duplications (up to sign). For instance, in Im(π+ ), a vertex
π+ (ex ) and π+ (ex ) g will be identiﬁed. Since, up to an overall sign,
π+ (e(x1 ,x2 ,x3 ,x4 ) )·g ∝ π+ (e(1−x1 ,1−x2 ,1−x3 ,1−x4 ) ),vertices are accordingly identiﬁed pairwise.

(7.8)

1936

CHARLES F. DORAN ET AL.

Constructing the Isoscalar supermultiplet from this, we get the following
Adinkra:(7.9)

The result (7.9) is the four-dimensional cube with opposite corners identiﬁed, as in [4]. This Adinkra is a quotient of the four-dimensional cube by
the code d4 generated by 1111. This corresponds to the fact that by either
doing nothing, or by reversing all four bits of a vertex, we return to thesame vertex (with perhaps a minus sign). In reference to the code name,the topology of the four-dimensional cube with opposite corners identiﬁed
will be called D4 .

7.2.2

The two inequivalent D4 quotients

Similarly, we can ﬁnd the Adinkra for the image of π− , using 1 − Γ1 Γ2 Γ3 Γ4instead. When we do so, we see that the image of π− and of π+ look similar— indeed, they have the same topology, the 4-cube with opposite corners
identiﬁed. But they have diﬀerent patterns of dashed edges. These patterns
cannot be made to coincide even when we redeﬁne some of the vertices
by replacing them with their negatives. There are, in fact, two distinct
irreducible representations of Cl(0, 5), and these are the two. Nevertheless,
we can easily describe the relationship between them: by replacing Γ4 →−Γ4 , for instance, which corresponds to replacing Q4 with −Q4 . This, inturn, results in reversing the sign associated to each edge with I = 4. The
Adinkras for these two are as follows:(7.10)We have suppressed the directions of the arrows: they are assumed to always
point upward. The replacement of Q4 by − Q4 is seen in the two Adinkras

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1937

above by noting that all solid blue lines in the left hand Adinkra are replacedby dashed blue lines in the in the right hand Adinkra (and vice versa).
Since the dimensional reduction of the N = 1 chiral superﬁeld in dimension D = 4 down to dimension D = 1 results in the image of π+ , Readers
might be tempted to think that the image of π− arises from the dimensional reduction of the N = 1, D = 4 antichiral superﬁeld. This, however,is not the case. Rather, it distinguishes diﬀerent ways of reconstituting theQ1 , . . . , Q4 into Qα and Q†α̇ , with α, α̇ = 1, 2, as usual. To go from one
to the other Adinkra (7.10), we must complex conjugate not both of thecomponents of Qα , but only half of them — which is impossible without
violating Lorentz symmetry in four dimensions. If we complex conjugate
all of Qα , we do swap chiral with antichiral superﬁelds, but we reverse bothQ3 and Q4 . This change can be reversed by a redeﬁnition of the real component ﬁelds, which swaps their complex combinations into the complex
conjugates. Graphically, the action of complex conjugation would requirethat two colors must be used to implement the dashed/solid exchanges.
So for example, both solid blue lines and solid green lines in the left hand
Adinkra are replaced by dashed blue lines and dashed green lines in the in
the right hand Adinkra (and vice-versa).
The distinction between the two non-isomorphic supermultiplets
depicted by the Adinkras (7.10) is thus more subtle. In fact, in more than
2-dimensional spacetimes, the QI ’s are not Lorentz-invariant, and the signof only one of them cannot be changed without violating Lorentz symmetry. In two-dimensional spacetime the supermultiplets depicted in (7.10)
are called chiral and twisted-chiral [27], and we adopt this nomenclature
also for the worldline N = 4 supersymmetry. In (2, 2)-supersymmetric theories in 2-dimensional spacetime, the transformation between the two supermultiplets (7.10) has been identiﬁed [29, 30] as the root of mirror symmetry [31–33].
Now, any Lagrangian term involving only one of these types of supermultiplets can just as well be written in terms of only the other type; in thissense they may be regarded as equivalent. However, these two supermultiplets may well mix in a Lagrangian, and in a way that prevents rewriting
the Lagrangian in terms of only one or the other type of supermultiplet,as has been done in [27]. This feature makes the two representations ofsupersymmetry, corresponding to two distinct irreducible representations ofCl(0, 5) and depicted by the Adinkras (7.10), usefully distinct.
Note that such two non-isomorphic irreducible representations of Cl(0,N + 1) exist precisely when N = 0 (mod 4), according to table 5, discussed
in Section 7.6.

1938
7.3

CHARLES F. DORAN ET AL.
Projecting twice: the D6 isoscalar supermultiplet

For N = 6, deﬁne the two elements
g1 = Γ1 Γ2 Γ3 Γ4 ,(7.11)

g2 = Γ3 Γ4 Γ5 Γ6 .

(7.12)As before, g12 = g22 = 1. Also note that g1 and g2 commute, and in fact,
g1 g2 = g2 g1 = −Γ1 Γ2 Γ5 Γ6 .

(7.13)

Analogously to the D4 example, we deﬁne the four projection operators
1 ± g1,2
1 ± g2π2± (v) = v·
2

π1± (v) = v·

(7.14)(7.15)and note that since g1 and g2 commute, so do π1+ and π2+ . We will now
project twice: once using π1+ (or π1− ) and then again, using π2+ (or π2− ) —a total of four choices. Using π1+ and π2+ (for instance) produces the Clifford representation Im(π1+ ◦ π2+ ). In this, and in what follows, we will use
π1+ and π2+ for notational deﬁniteness, but it is to be understood that these
may be replaced by π1− or π2− , respectively and independently, mutatis
mutandis.
The composition is1
1 + g2 1 + g1·
= v·(1 + g1 + g2 + g2 g1 )2
2
4

(7.16)

1= v·(1 + Γ1 Γ2 Γ3 Γ4 + Γ3 Γ4 Γ5 Γ6 − Γ1 Γ2 Γ5 Γ6 ).
4(7.17)(π1+ ◦ π2+ )(v) = v·

It is straightforward to prove that Im(π1+ ◦ π2+ ) = Im(π1+ ) ∩ Im(π2+ ) andthat this is a Cliﬀord representation. Writing this as ker(π1− ) ∩ ker(π2− ), we
can see that for all v in this image, v = v·g1 = v·g2 = v·g1 g2 . In particular,when g1 and g2 commute with v, we have that v = g1 v = g2 v = g1 g2 v.
Deﬁne
e0 := (π1+ ◦ π2+ )(1) = 14 (1 + Γ1 Γ2 Γ3 Γ4 + Γ3 Γ4 Γ5 Γ6 − Γ1 Γ2 Γ5 Γ6 ). (7.18)
Note that e0 commutes with g1 and g2 , and so e0 = g1 e0 = g2 e0 = g1 g2 e0 .
We successively apply the various Γ1 , Γ2 , Γ3 , Γ5 to e0 on the left and we geta collection of 16 ﬁelds that span Im(π1+ ◦ π2+ ), corresponding to vectors

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1939

of the form e(x1 ,x2 ,x3 ,0,x5 ,0) . Applying Γ4 will not generate any new vectors
because Γ4 = g1 Γ1 Γ2 Γ3 , and so for every ex ,Γ4 ex = g1 Γ1 Γ2 Γ3 · ex(7.19)= g1 Γ1 Γ2 Γ3 Γx1 1 · · · ΓxNN · e0
= ±Γ1 Γ2 Γ3 Γx1 1 · · · ΓxNN g1 · e0
= ±Γ1 Γ2 Γ3 Γx1 1 · · · ΓxNN · e0

(7.20)= ±Γ1 Γ2 Γ3 · ex .

(7.23)(7.21)(7.22)

Similarly, Γ6 will not generate new vectors, using Γ6 = g2 Γ3 Γ4 Γ5 .
The Isoscalar supermultiplet corresponding to this will have 8 bosons and8 fermions. The ex vectors correspond to the various φx and ψx .
If we do this, we get an Adinkra whose topology we call D6 , which is a
six-dimensional cube projected twice: once according to g1 = Γ1 Γ2 Γ3 Γ4 andthen according to g2 = Γ3 Γ4 Γ5 Γ6 :(7.24)
Here the colors are as before, with orange for Q5 and purple for Q6 .
The name D6 derives from the code d6 , generated by the k = 2 codewords
c1 = 111100 and c2 = 001111; to denote this, we assemble the generator
codewords as a matrix, called a generator matrix in coding theory:

111100001111,(7.25)

but here we simply regard it as a collection of the codewords listed in its rows.
The ﬁrst codeword corresponds to Γ1 Γ2 Γ3 Γ4 , the second with Γ3 Γ4 Γ5 Γ6 , andthe sum of these codewords modulo 2, 110011, corresponds to −Γ1 Γ2 Γ5 Γ6 .
More generally, addition in the code turns into multiplication of the corresponding products of ΓI matrices, perhaps with a minus sign, because theΓI can be anticommuted past each other, and when a particular ΓI appears
in both g1 and g2 , once anticommutation is done so that the two ΓI ’s are
adjacent, these two simplify to ΓI 2 = 1.

1940
7.4

CHARLES F. DORAN ET AL.
Constructing an isoscalar supermultiplet from a code

The procedure illustrated in the previous two examples can be generalized to the following construction, ﬁrst applied to Cliﬀord representations
by Dimakis. [34].
Construction 7.1. Suppose we are given N and a doubly-even code C ⊂(Z/2)N of length N , given by a generating set {c1 , . . . , ck } ⊂ C. Writing
each ci as (xi1 , . . . , xiN ) ∈ {0, 1}N , we associate to it gi := Γx1 i1 · · · ΓxNiN . For
instance, c1 = 101100100 would produce g1 = Γ1 Γ3 Γ4 Γ7 .
The wt(ci ) being even translates into gi being even. When wt(ci ) is even,wt(ci ) being a multiple of 4 is equivalent to gi 2 = 1. At the end of Section 2
we presented the standard fact that any two elements of the codes are orthogonal (that is, share an even number of 1s). As a result, all of the g1 , . . . , gk
commute with each other. They generate a group G under multiplicationthat is isomorphic as a group to the code C. The isomorphism is done
analogously to converting ci to gi , except that a minus sign is sometimes
required.7
For each gi we have πi± : Cl(0, N ) → Cl(0, N ) deﬁned as
πi± (v) = v·

1 ± gi.
2(7.26)As before, the operators πi± are homomorphisms. The evenness of gi implies
that πi± preserves the fermionic and bosonic statistics. The fact that gi 2 = 1
implies that πi+ and πi− are projection operators.
The fact that the gi all commute implies that the πi+ and πi− all commute.
The following table summarizes how the properties of the generators of
the code relate to the properties of the πi± .
gi
πi± (Projector)
ci (Codeword)
Even weight
Even
Preserves statistics
2
projection
weight is 0 (mod 4) gi = 1
commutepairwise orthogonal commute

7(7.27)

This construction is due to Wood, who used it to classify two-elementary abeliansubgroups of the spin groups [35].

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1941

We deﬁne8
πC := π1+ ◦ · · · ◦ πk+ ,(7.28)and then as in the previous example,
Im(πC ) = Im(π1+ ) ∩ · · · ∩ Im(πk+ ) = ker(π1− ) ∩ · · · ∩ ker(πk− )(7.29)

is the Cliﬀord representation we want. If we deﬁne
e0 = πC (1) =1 + gk1 
1 + g1
···
= k
g,
2
2
2

(7.30)g∈G

and then successively apply the various ΓI on the left to e0 , we obtain a setof elements of Cl(0, N ). Using the Dirac relations (6.1) shows that many ofthese are the same, up to an overall sign. Furthermore, for all gi , we havegi e0 = e0 , and more generally this is true of all elements of G. It therefore
follows that if we begin with an element v represented by a dv -dimensional
matrix, the application of πC results in a quantity representable by a 2−k dv dimensional matrix.
In fact, this will result in 2N −k+1 diﬀerent elements of Im(πC ), occurringin ± pairs. If we arbitrarily choose one from each ± pair, the result is acollection of 2N −k vectors that form a basis for Im(πC ).

7.5

Other examples

Example 7.1. The case N = 8 allows the code e8 , which has the following
generator set:
 11110000 e8 :800111100
00001111
10101010.

(7.31)

Strictly speaking, the notation (7.28)–(7.29) should specify for each gi which sign isbeing used. For illustrative purposes, herein we only use πi+ ’s. This choice will not aﬀect
chromotopology, but will aﬀect the dashedness of the edges. Precisely which of these 2k
distinct choices in π1± ◦ · · · ◦ πk± provide (non-)isomorphic representations akin to (7.10)
is a question we defer to a subsequent eﬀort.

1942

CHARLES F. DORAN ET AL.

The corresponding gi are:g1 = Γ1 Γ2 Γ3 Γ4 ,(7.32)

g2 = Γ3 Γ4 Γ5 Γ6 ,(7.33)

g3 = Γ5 Γ6 Γ7 Γ8 ,(7.34)

g4 = Γ1 Γ3 Γ5 Γ7 .

(7.35)

These produce the following group elements:
g1 g2 = −Γ3 Γ4 Γ5 Γ6 ,g1 g3 =

g1 g4 =

Γ2 Γ 4 Γ 5 Γ 7 ,g2 g3 = −Γ3 Γ4 Γ7 Γ8 ,(7.36b)g2 g4 =

Γ1 Γ 4 Γ 6 Γ 7 ,g3 g4 =

Γ1 Γ 3 Γ 6 Γ 8 ,(7.36c)g1 g2 g3 =

Γ3 Γ 4 Γ 7 Γ 8 ,g1 g2 g4 = −Γ1 Γ4 Γ6 Γ7 ,(7.36d)g1 g3 g4 =

Γ2 Γ 4 Γ 6 Γ 8 ,g2 g3 g4 = −Γ1 Γ3 Γ4 Γ8 ,(7.36e)g1 g2 g3 g4 =

Γ1 Γ 4 Γ 5 Γ 8 .

Γ1 Γ 2 Γ 3 Γ 4 Γ 5 Γ 6 Γ 7 Γ 8 ,(7.36a)(7.36f)Recall that each g must be a product of a doubly even number of ΓI ’s,
explaining why are we only now seeing g’s with diﬀering numbers of ΓI ’s.
The Adinkra is below. This has the feature that every boson is connected
to every fermion. So this is a K(8, 8) graph, and is denoted E8 . This wasintroduced in [36, 37] as the N = 8 spinning particle in relation to what was
described as a “supergravity surprise”. The new color, brown, corresponds
to Γ8 .

(7.37)As in the N = 4 case, there are two irreducible representations, and one
is obtained as above, while the other is obtained by reversing the sign on
one of the gi . The result is the same Adinkra topology, but with various
signs on the edges reversed. For instance, we can reverse the sign on g1 byreversing the signs on edges corresponding to Γ4 . This preserves the sign on
g2 , g3 , and g4 . Reversing various ΓI yields apparently diﬀerent Adinkras,
but all of these must fall into just two isomorphism classes.
8!
= 30 codes that are permutation equivalent to e8 .
There are 1344

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1943

Figure 1: N = 8 doubly even codes and their subset relationships: The
arrows connecting two codes are injections (possibly after permutation): thecode on the lower level is a subcode of the higher one. The trivial code t8
is at the bottom, and e8 is the unique maximal doubly even code for N = 8
and is drawn at the top. Every doubly even code in N = 8 is a subcode ofe8 . The code h8 is the one generated by 11111111.
As before, we can take a subgroup of e8 , but in this case, the situation
is a bit more interesting: there are multiple inequivalent choices for which
generator to remove. Figure 1 shows all the N = 8 doubly even codes, up
to permutation equivalence.
For instance, for k = 3, we could choose e7 ⊕ t1 , which results in an
Adinkra topology E7 × I 1 , or we could choose d8 , which results in a different Adinkra topology D8 :(7.38)(7.39)The only diﬀerence between these two Adinkras is the way one of the supersymmetries acts, the one represented here by orange edges. In the E7 × I 1 Adinkra (7.38), it acts within each of the two halves, leaving the Adinkra
1-color-decomposable: only the brown edges span the whole Adinkra, andit decomposes into two identical N = 7 Adinkras if the brown edges are
erased. Each of the halves has the E7 topology, except that edges of one

1944

CHARLES F. DORAN ET AL.

color (black) have their dashing reversed. This becomes clearer upon rearranging the nodes a little:(7.40)
In turn, in the D8 -Adinkra (7.39), the orange edges span the whole
Adinkra together with the brown ones. For the sake of comparison with
the E7 × I 1 -Adinkra (7.40), we also rearrange the nodes of the D8 -Adinkra:(7.41)
To decompose this second Adinkra, one would have to erase the edges ofat least two colors; we say it is two-color-decomposable. This n-colordecomposability property correlates with the fact that the e7 ⊕ t1 code has
one column of zeros, whereas d8 does not:
0000 1111
0000 1111
1
E7 × I ⇐⇒ e7 ⊕ t1 : 0011 1100 , vs. D8 ⇐⇒ d8 : 0011 1100 .
0101 0101

1111 0000(7.42)
Finally, both Adinkras (7.38) and (7.39) admit precisely one supersymmetrypreserving Z2 symmetry: for the former, it is encoded as 11110000 andgenerated by Γ1 Γ2 Γ3 Γ4 , while the latter is symmetric with respect to theaction of Γ2 Γ4 Γ6 Γ8 , encoded as 01010101. The projection of each Adinkra
by its respective symmetry then produces the E8 -Adinkra (7.37).
Similar relations exist between the D6 × I 2 - vs. D4 × D4 -Adinkras, andthe D4 × I 4 - versus H8 -Adinkra corresponding to the k = 2 and k = 1 rows
in the diagram in ﬁgure 1.
To summarize, table 4 gives the possible topologies for each N up to
N = 10. The cartesian product × refers to taking the cartesian product ofthe vertex set, and drawing edges between (v1 , w1 ) and (v1 , w2 ) whenever
there is an edge between w1 and w2 , and between (v1 , w1 ) and (v2 , w1 )whenever there is an edge between v1 and v2 . Exponentiation means iterated

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1945

Table 4: Adinkra topologies up to N = 10: The number of nodes in an
Adinkra with the indicated topology is shown to the right of each topology,
in the form (nB |nF ), where nB is the number of bosons and nF is the numberof fermions in the supermultiplet.
N
1
2
3
4
5
6
7
8

k=0
(1|1)
I2
(2|2)
I3
(4|4)
4
I
(8|8)
I 5 (16|16)
I 6 (32|32)
I 7 (64|64)
I 8 (128|128)

9

I9

10 I

†

k=1

k=2

k=3

k=4

I

10

D4
D4 × I 1
D4 × I 2
D4 × I 3
D4 × I 4
H8(256|256) D4 × I 5
H8 × I 1(512|512) D4 × I 6
H8 × I 1

(4|4)
(8|8)
(16|16)D6(8|8)(32|32)
D6 × I 1 (16|16)
E7
2(64|64)
D6 × I(32|32)
E7 × I 1(64|64)D4 × D4 (32|32)
D8(128|128)
D6 × I 3 (64|64)
E7 × I 2(128|128) D4 × D4 × I 1 (64|64)
D8 × I 1(256|256)
D6 × I 4 (128|128) E7 × I 3(256|256) D4 × D4 × I 2 (128|128) D8 × I 2(D4 ∗1 I 6 )† (128|128) D4 × D6

(8|8)
(16|16)
E8(16|16)(32|32) E8 × I 1(32|32)(64|64) E8 × I 2(64|64)
D10(64|64)(8|8)(16|16)(32|32)(32|32)

Herein, the code name “D4 ∗1 I 6 ” denotes that the D4 code is padded by six zeroes and aug-

mented by one additional, “glue” generator spanning the positions of the six added zeroes and asuﬃcient number (here, two) of 1’s within D4 so as to generate a doubly even code; it is generated by{1111 0000 00, 0011 1111 11}.

cartesian products. We note the Adinkra topologies D4 , D6 , D8 , and D10from the codes d4 , d6 , d8 , and d10 , as well as E7 and E8 from e7 and e8 ,respectively. In addition, there turns up a code with no standard name,generated by {1111000000, 0011111111}, which we tentatively call D4 ∗1 I 6 .
7.6

Comparison with Cliﬀord representations

Since Cliﬀord representations are already classiﬁed, it is worthwhile comparing what we have just found with the known classiﬁcation of Cliﬀord
representations. The abstract algebras Cl(0, N +1) are known for all N .
The following can be found on any standard text on Cliﬀord algebras, such
as Lawson and Michelsohn’s Spin Geometry [28].
The Cliﬀord algebras Cl(0, N ) and Cl(0, N +1) are given in table 5. The
description of these algebras has a modulo 8 sort of periodicity, so it isconvenient to write N = 8m + s, where m and s are integers and 0 ≤ s ≤ 7.
The notation R(n), C(n), and H(n) denotes the algebra of n × n matrices
with real, complex, and quaternionic coeﬃcients, respectively.
Note from table 5 that every Cliﬀord algebra is either a matrix algebra,
or a direct sum of two matrix algebras. It is a classical result that everyﬁnite-dimensional real representation of such an algebra decomposes into
irreducibles. For R(n), C(n), and H(n), there is up to isomorphism only

1946

CHARLES F. DORAN ET AL.

Table 5: Cliﬀord algebras Cl(0, N ) decomposed into irreducible representations of Cl(0, N +1): Here N = 8m + s where m and s are integers and0 ≤ s ≤ 7.
dimR
dimR
# (Irrep.) ins Cl(0, N )Cl(0, N +1)
Irrep. Cl(0, N ) Cl(0, N )max k
0 R(16m )R(16m ) ⊕ R(16m )
16m
162m
16m
4m
R(2 · 16m )2 · 16m 2 · 162m
16m
4m
1 R(16m ) ⊕ R(16m )
m
m
m
2m
m
C(2 · 16 )4 · 164 · 16164m
2 R(2 · 16 )H(2 · 16m )8 · 16m 8 · 162m
16m
4m
3 C(2 · 16m )H(2 · 16m ) ⊕ H(2 · 16m ) 8 · 16m 16 · 162m 2 · 16m
4m + 1
4 H(2 · 16m )16 · 16m 32 · 162m 2 · 16m
4m + 1
5 H(2 · 16m ) ⊕ H(2 · 16m ) H(4 · 16m )
m
m
m
2m
m
C(8 · 16 )16 · 1664 · 164 · 16
4m + 2
6 H(4 · 16 )R(16 · 16m )16 · 16m 128 · 162m 8 · 16m
4m + 3
7 C(8 · 16m )one irreducible real representation: Rn , Cn , or Hn , respectively. For R(n) ⊕R(n), there are two non-isomorphic irreducible representations: one that
ignores the second summand and is the standard representation on the ﬁrst,
and the other that ignores the ﬁrst summand. Likewise for C(n) ⊕ C(n) andH(n) ⊕ H(n).
The fourth column in table 5 counts the real dimension of the irreducible
representation. Based on the real dimension of Cl(0, N ), this determines thenumber of copies of the irreducible representation found in Cl(0, N ), whichis shown in column six.
The last column of the table is labeled “max k”, and refers to the maximal
dimension k of a doubly even code. This can be computed using Gaborit’s
mass formula, described in Appendix B.1 [38], which gives the number ofdoubly even codes for a given N and k. This number vanishes when k islarger than the numbers given in the table. To ﬁnd this value of k, recall
that in (1.2) we deﬁned the following function of N :⎧⎪0
for N < 4,
⎪
⎪
⎪
⎪
⎪
for N = 4, 5,⎨1
κ(N ) := 2
(7.43)
for N = 6,⎪⎪⎪
3
for N = 7,⎪⎪⎪
⎩4 + κ(N −8) for N ≥ 8, recursively.
Note that if we write N = 8m + s where 0 ≤ s ≤ 7, then
⎧
4m
⎪
⎪
⎪
⎨4m + 1
κ(N ) =⎪
4m + 2⎪⎪⎩4m + 3for s = 0, 1, 2, or 3,
for s = 4 or s = 5,
for s = 6,
for s = 7.

(7.44)

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1947

and Gaborit’s mass formula implies that for any doubly even code of length
N and dimension k, 0 ≤ k ≤ κ(N ).
Suppose we take an N -dimensional cubical Adinkra. It has 2N nodes.
If we quotient by a doubly even code of dimension k, so that the code has
2k elements, we have 2N −k nodes in the quotient Adinkra. The minimal
Adinkra then occurs when k is as large as possible; that is, for k = κ(N ).
Thus, a minimal Adinkra for a given N has 2N −κ(N ) nodes: half bosons,and the other half of them fermions. In turn, this gives rise to a representation of the Cliﬀord algebra, which in general will be some number ofcopies of the irreducible representations. The fact that the real dimensionof the irreducible representation, found in column 4, coincides with the real
dimension of the maximally quotiented Adinkra, 2N −κ(N ) , indicates that theirreducible representations of the Cliﬀord algebra can actually be obtainedby quotienting by maximal codes. In particular, the corresponding isoscalarmultiplet has a description in terms of Adinkras.
Incidentally, it may concern the reader that this table shows that there isa unique irreducible representation for the Cliﬀord algebra when N is not a
multiple of 4, and that there are precisely two irreducible representations for
the Cliﬀord algebra when N is a multiple of 4. This seems strange in contrast
to the multitude of codes of maximal k in table 4. This indicates that for
Isoscalar Adinkras at least, there are hidden isomorphisms between thesesupermultiplets. The one-hooked versions, however, have no isomorphisms,
and so are immune to this problem. This issue will be investigated in greater
detail in a future work.
7.7

One-hook hanging Adinkras

Since we have concluded that any connected Adinkra chromotopology is aquotient of the N -cube by a doubly even [N, k]-code C, the vertices of the
Adinkra (that is, the component ﬁelds of the supermultiplet) correspond tocosets of {0, 1}N , thought of as (Z2 )N , by the subgroup C. This immediately
implies that the Adinkra has 2N −k nodes, where k is the dimension of C.
Of these, one half represents bosonic component ﬁelds, and the other halffermionic component ﬁelds in the corresponding supermultiplet. We thus
have that the number of bosonic component ﬁelds, dB , and the number of
fermionic component ﬁelds, dF , after taking this quotient satisfydB = dF = 2N −k−1 .

(7.45)

In [17], we described a notion of hanging a graph by a one or more sources.
For instance, if we pick the vertex v∗ and hang the graph by it, we let all

1948

CHARLES F. DORAN ET AL.

the arrows on edges point from the vertices that are further away fromv∗ (as measured through the edge set) to vertices that are closer to v∗ .
Equivalently, for each vertex v we deﬁne the engineering degree of v to be1(7.46)[v] = [v∗ ] − dist(v, v∗ ),2where dist(v, v∗ ) is the length of the shortest path from v to v∗ in the edge
set. Then the arrows are drawn in the direction of increasing engineering
degree, upward.
For every doubly even code C, use the Construction above to ﬁnd an
Adinkra. Then choose one bosonic ﬁeld and create a one-hooked Adinkra
hooked on that ﬁeld. Call this Adinkra AC . In this way, we have assigned toevery doubly even code C an Adinkra AC . Now given two diﬀerent doubly
even codes C1 and C2 , we have two distinct Adinkras, A1 and A2 . They are
not isomorphic as chromotopologies because it is a diﬀerent set of sequencesof colors that take us from a vertex to itself in both cases. But because
these are one-hooked, we can say something stronger: the supermultiplets
they describe will not be isomorphic.
Theorem 7.1. Let C1 and C2 be diﬀerent doubly even codes, and suppose
A1 and A2 are one-hooked connected Adinkras with codes C1 and C2 , respectively. Then the supermultiplets from A1 and from A2 are not isomorphic.
Proof. For suppose there were such an isomorphism. Suppose φ1 is thehooked bosonic ﬁeld for A1 and φ2 is the hooked bosonic ﬁeld for A2 . Any
isomorphism preserves the number of degrees of freedom in each mass dimension, and since there are no ﬁelds with mass dimension less than [φ1 ] or[φ2 ], we must have [φ1 ] = [φ2 ]. Any isomorphism must send some linear
combination of component ﬁelds of A1 and their derivatives to φ2 . But φ2
is of lowest mass dimension, and derivatives increase mass dimension. So
the only candidate for what must be sent to φ2 is φ1 perhaps multiplied bya unit-less constant c.
The fact that an isomorphism commutes with QI means that for each I,
cQI φ1 gets sent to QI φ2 . We continue in this fashion, until the isomorphismof supermultiplets produces an isomorphism of the corresponding chromotopologies.
By Proposition 4.2, the code is an invariant of the chromotopoogy, and
therefore, this isomorphism of chromotopologies implies C1 = C2 .
One ﬁnal remark about one-hooked Adinkras: for such Adinkras, we can
examine how many component ﬁelds are in each engineering degree. By

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1949

equation (7.46), this is equivalent to ﬁnding how many vertices are of agiven distance from the highest vertex, v∗ . The corresponding notion incoding theory is the “coset weight enumerator” [19]. This is a polynomial of
the form  a x where a is the number of cosets whose distance to the 0coset is . Thus, the coset weight enumerator for a doubly even code can beused to ﬁnd the number of component ﬁelds in each engineering dimension
for a one-hooked Adinkra corresponding to that doubly even code.

8

Application to four- and higher-dimensional theories

We expect to ﬁnd, among D = 1 oﬀ-shell theories, the dimensional reductions of the oﬀ-shell theories of higher-dimensional supersymmetric theories.
For instance, oﬀ-shell theories in four dimensions with N -extended supersymmetry will dimensionally reduce to one-dimensional theories with
N = 4N supersymmetries, since the smallest irreducible spinor in fourdimensional spacetime has N = 4 components. In higher dimensions thesmallest irreducible spinors have dimension that is an integral multiple of
this.
Multiplets with minimal numbers of degrees of freedom are of interest,
not only because they presumably provide the simplest theories, but also
because they give us an idea of a lower bound to the size of a theory.
Minimal multiplets correspond to the irreducible representations of the Clifford Algebra, which are described in table 5. As was pointed out in Section 7.6, they also correspond to maximal codes for each N , and these arelisted in table 3 for N ≤ 32. In any case, the number of component ﬁelds
is 2N −κ(N ) so that the number of bosons is 2N −κ(N )−1 and likewise for thenumber of fermions.
The results are shown in table 6.
Table 6 can be used to make an argument about the size of the smallest
irreducible oﬀ-shell representation for a given value of N . If the smallest indecomposable representation coincides with the smallest irreducible
representation, then for a given value of N , the table above determines
the smallest oﬀ-shell representation. For each value of N , it is possible toconsider the representation that appears at the lowest level of that column.
All of these topologies are shown together with the number of bosonic andfermionic nodes in table 6.
For the cases of N = 1 and 2, the number of bosonic and fermionic degrees
of freedom are in agreement with the known minimal oﬀ-shell

1950

CHARLES F. DORAN ET AL.
Table 6: Minimal oﬀ-shell 4D, N ≤ 8 supermultiplets.

N
1
2
3
4
5
6
7
8
∗

N #(Bosons) #(Fermions)#(Adinkra topologies)
4
4
4
1 D4
8
8
8
1 E8
12
64
64
2 D12 , E8 × D4
16
128
128
2 E16 , E8 × E8
20
1,024
1,024
10 . . .∗
24
2,048
2,048
9 . . .∗
28
16,384
16,384151 . . .∗32
32,76832,76885 . . .∗For N ≥ 5 there are too many Adinkra topologies to be shown here; see

http://www.rlmiller.org/de codes/ for an up-to-date table with links to actual

codes.

supersymmetrical representations. The smallest 4D, N = 1 oﬀ-shell representations do indeed consist of 4 bosons and 4 fermions. In a similarmanner, the smallest 4D, N = 2 oﬀ-shell representations do indeed consistof 8 bosons and 8 fermions. The case of 4D, N = 3 oﬀ-shell representations
is not so widely known. Nevertheless, Siegel has presented an argument
about the oﬀ-shell structure of conformal 4D, N = 3 supergravity that indicates it describes 64 bosons and 64 fermions [39]. There exist, also, one
known oﬀ-shell example of a 4D, N = 4 supermultiplet in Salam–Strathdee
superspace. It is the conformal 4D, N = 4 supergravity supermultiplet ﬁeld
strength [40] and it consists of precisely 128 bosons and 128 fermions. Allof this agrees with the ﬁrst four ‘data’ points on table 6.
Precisely when N is a multiple of 8, the maximum value of k is N/2, andthese codes are self-dual (where the orthogonal space of the code equals thecode; doubly even implies that these codes are self-orthogonal). Thus, these
relate to even unimodular lattices. Indeed, the case N = 2, or N = 16, provides the two lattices, well known to string theorists: E8 × E8 , and SO(32),which we call E16 , since D16 ﬁts in the sequence of codes D2n such that
D16 ⊂ E16 :⎡⎤1111 0000 0000 0000
⎢0011 1100 0000 0000⎥⎢⎥⎢0000 1111 0000 0000⎥⎢⎥⎢⎥
D16 ⇐⇒ d16 : ⎢0000 0011 1100 0000⎥⎢⎥⎢0000 0000 1111 0000⎥⎢⎥⎣0000 0000 0011 1100⎦
0000 0000 0000 1111

versusCODES AND SUPERSYMMETRY IN ONE DIMENSION⎡⎤
1111 0000 0000 0000
⎢0011 1100 0000 0000⎥⎢⎥⎢0000 1111 0000 0000⎥⎢⎥⎢0000 0011 1100 0000⎥⎢⎥
E16 ⇐⇒ e16 : ⎢⎥.
⎢0000 0000 1111 0000⎥⎢⎥⎢0000 0000 0011 1100⎥⎢⎥⎣0000 0000 0000 1111⎦
0101 0101 0101 0101

1951

(8.1)

It is of interest to consider the ﬁnal case above, with N = 8, or N =32 supersymmetries. For this case we have 2N −16 = 216 = 65,536 total
nodes. The Adinkra associated with this topology has 32,768 bosonic nodes
and 32,768 fermionic nodes. By very diﬀerent arguments [37], the ﬁnal
result of min(dB ) = min(dF ) = 32, 768 in this table appeared very early in
our considerations of iso-spinning particles and “Garden Algebras” as the
spectrum generating algebras of spacetime supersymmetry.
There are 85 diﬀerent maximal codes in N = 32 (up to permutation equivalence). The attempt to classify these began with Conway and Pless [41]in 1980, though a correction was found to be necessary by Conway et al.,in 1990 [20]. Bilous and van Rees [26] in 2007 replicated these results byperforming a more systematic search and provided the list of all 85 codes on
the web site [42]. It is amusing to note that this was achieved not long ago,that N = 32 is the upper limit of what is known currently about self-dual
doubly even codes, and that 32 is the maximal N needed in applications tosuperstrings and their M - and F -theory extensions. Five of these 85 codes
have minimal weight 8, and it would be interesting to see if these ﬁve playa special role in N = 8 supersymmetries in four dimensions.

9

Toward a classiﬁcation of Adinkras and supermultiplets

An Adinkra is determined by its chromotopology, a hanging of the vertices,
and a choice of which edges are dashed. We have in this paper described
the classiﬁcation of chromotopologies of Adinkras: they are disjoint unions
of connected chromotopologies, each of which is described by a doubly evencode. Conversely, every doubly even code can be used to form an Adinkra.
As per Deﬁnition 3.1, to go from this to specifying an Adinkra would requirea choice of orientations for all the edges and a choice of dashing of each edge.
Concerning orienting the edges, we have discussed in this paper two
such choices that are always available: Isoscalar Adinkras and one-hooked

1952

CHARLES F. DORAN ET AL.

Adinkras. More generally, in [17], we have shown that the choices in orienting edges correspond to the ways of “hanging” the vertices of the graph at
various heights, subject to certain conditions. The Isoscalar Adinkra resultsfrom hanging the Adinkra on all of the bosons, which are placed all at thesame level. The one-hooked Adinkra results from hanging the Adinkra froma single boson.
So if we describe the various ways of dashing the edges (an eﬀort which
will be described in another paper), would this then classify one-dimensional
N -extended oﬀ-shell supermultiplets? Not quite. First, there is the question of whether every such supermultiplet comes from an Adinkra. Ourinvestigations in this area indicate that there do indeed exist non-adinkraic
supermultiplets, but even for these supermultiplets, Adinkras turn out to be
useful in their description. [15] Second, we might ask whether two Adinkras
may describe the same supermultiplet. But if we insist that the Adinkras
are one-hooked, then the corresponding representations are isomorphic if
and only if they have the same code, as we saw in Theorem 7.1. So by
one-hooking the Adinkras, we can be sure that these are all diﬀerent supermultiplets. Even considering these two issues, what is clear is that there isa surprising wealth of supermultiplets in one dimension as N is increased
towards N = 32.
Finally, we wish to draw attention to the “degeneracy” in constructing
even the minimal supermultiplets for various N , uncovered by the listing of
permutation equivalence classes of doubly even codes in table 3. Entries inthis table that are greater than 1 indicate situations where there is morethan one permutation class of doubly even codes that are possible for thatvalue of N and k. This is true even if we restrict our attention to minimal
supermultiplets, which have maximal k, when N ≥ 10. Note that as N
increases, this degeneracy increases, certainly for k non-maximal, but also
for k maximal. This clearly illustrates that this “degeneracy” amongst eventhe minimal supermultiplets grows extremely fast with N .

Acknowledgments
The research of S.J.G. is supported by the endowment of the John S. Toll
Professorship, the University of Maryland Center for String & Particle
Theory, and National Science Foundation Grant PHY-0354401. T.H. isindebted to the generous support of the Department of Energy throughthe grant DE-FG02-94ER-40854, as well as the Department of Physics,
University of Central Florida, Orlando FL, and the Physics Departmentof the Faculty of Natural Sciences of the University of Novi Sad, Serbia,

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1953

for recurring hospitality and resources. The work described in Appendix B
uses computer facilities supported by the National Science Foundation grant
no. DMS-0555776. Some Adinkras were drawn with the aid of the Adinkrac 2008 by G. Landweber. A special thanks goes to Martin Mueller,mat Erik Duamine, Morteza Zadimoghaddam, Richard Stanley, all at M.I.T.,
and especially Yan Zhang, Ricard Stanley’s student, for fruitful discussions.

Appendix A

A.1

Deferred
Adinkrasdetailsonsupersymmetry

andReal coeﬃcients

Real supermultiplets, M = (F1 (τ ), . . . , Fm (τ )) consist of real component†
ﬁelds: FA (τ ) = FA (τ ), and supersymmetry is assumed to preserve thiscondition. In this section, we will show that this condition implies that thecoeﬃcients c in equation (3.8) and equation (3.10) are real.
We use the convention whereby (XY )† = Y † X † , regardless whether X
and Y are bosonic (commuting) or fermionic (anticommuting) objects, as is
standard in the physics literature.
If φA (τ ) is real, then its supersymmetry transformδQ () := −iI QI(A.1)

must also be real. Applying this to equation (3.8) results in[φA ]+ 12 −[ψB ]δQ () φA (τ ) = −iI c ∂τ

ψB (τ ).

(A.2)

Thus
†
[φA ]+ 12 −[ψB ] †[φA ]+ 12 −[ψB ]= i ∂τ
ψB (τ ) c∗ I = −iI c∗ ∂τψB (τ ).
δQ () φA (τ )(A.3)
Comparing the right-hand sides of equations (A.2) and (A.3), we ﬁnd thatc∗ = c.
Thus the coeﬃcients c are real.

(A.4)

1954

CHARLES F. DORAN ET AL.

A.2

The proof that scaling factors c = ±1 are the only ones necessary

We now will show that the coeﬃcients in an adinkraic supermultiplet canbe chosen to be c = ±1 via a rescaling of ﬁelds by real numbers.
Proposition A.1. Suppose we have an adinkraic supermultiplet, withF1 (τ ), . . . , F2m (τ ) for component ﬁelds. There is a real rescaling of thesecomponent ﬁelds so that each non-zero coeﬃcient c in equations (3.8) and(3.10) is equal to 1 or −1.
Proof. We suppose we have component ﬁelds F1 (τ ), . . . , F2m (τ ),9 andsupersymmetry generators QI so that the supersymmetry transformation
rules are all of the form
QI FA = c ∂τλ FB ,(A.5)

where λ is either 0 or 1, and FB is another component ﬁeld, and c is acomplex number (real if FA is bosonic, pure imaginary if FA is fermionic).
We can draw an Adinkra-like graph for this supermultiplet: we again create vertices v1 , . . . , v2m to correspond to the component ﬁelds F1 , . . . , F2m— white for bosonic ﬁelds and black for fermionic ﬁelds. For each supersymmetry transformation like the one above, we draw an edge from thevertex vA to vB . The arrow goes from vA to vB if λ = 0, and the reverse
if λ = 1. As before, this is consistent with the corresponding equation for
QI FB . Instead of choosing between a dashed or solid line, we label the edgeby c if FA is bosonic and i/c if it is fermionic. It is easy to check that thisis consistent with the corresponding equation for QI FB .
We can work on each connected component of this Adinkra-like graphseparately. So for the following we assume our Adinkra-like graph is connected.
We wish to rescale the various component ﬁelds so that c has absolutevalue 1. A naive approach would be to simply rescale FB by c, so that cdisappears in equation (A.5). But remember that there are 2mN of theseequations, and only 2m component ﬁelds. More visually, imagine theAdinkra-like graph with separate c-labels on each edge. We can imagine
starting at one vertex, then going along an edge to the next vertex, rescaling the corresponding ﬁeld by whatever it takes to make c = 1 on that edge.
The trouble is that when we come back to a vertex we already saw, we are9

See footnote in the proof of Theorem 4.3 about the use of this notation.

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1955

Figure 2: On the left is an N = 3 cubical Adinkra. The dashedness of thelines are suppressed for clarity. Some edges (shown with thin lines) can bedeleted, until what is left is a tree, called the spanning tree (shown at right).

no longer free to rescale that ﬁeld without messing up other c’s. But as we
will see, it turns out that the supersymmetry algebra guarantees that when
we return to a previously-seen vertex, the edge will have |c| = 1.
We ﬁrst begin by removing all the loops, using the following standard
procedure from graph theory: if we choose an edge, and erase it from thegraph, the graph may either separate or stay connected. We ﬁrst ﬁnd anedge so that erasing it keeps the graph connected. By successively ﬁnding
such edges and erasing them, we obtain a minimal graph so that this graphis still connected. This resulting graph is a tree, so that for every pair ofvertices vi and vj , there exists a unique, non-backtracking path from vi to
vj in this tree (were it not unique, we could remove another edge). This
tree is called a spanning tree for the original graph. The spanning tree fora graph is not unique, but we select one. See ﬁgure 2.
Now pick any vertex in our graph, denote it v∗ and the corresponding
component ﬁeld F∗ (τ ). For simplicity of exposition we can assume v∗ isbosonic. If vi is any other vertex, there is a sequence of edges in the tree
connecting v∗ with vi , corresponding to a sequence of QI ’s, say QI1 , . . . , QIn .
We then apply equation (A.5) iteratively to F∗ (τ ), obtaining
QIn · · · QI1 F∗ (τ ) = Ci ∂τλ Fi (τ )(A.6)

for some non-negative integer λ (recording the number of arrows that pointin the opposite direction of the path), and some non-zero coeﬃcient Ci .
This coeﬃcient Ci is real if n ≡ 0, 1 (mod 4), and pure imaginary if n ≡ 2, 3

1956

CHARLES F. DORAN ET AL.

Figure 3: Using the same example as in ﬁgure 2, suppose we are considering
the vertices vi and vj as seen in the ﬁgure. The blue path shows a path P
in the spanning tree going from vj to v∗ , followed by a path in the spanning
tree going from v∗ to vi . One edge is retraced, but this is irrelevant forour purposes. The sequence of colors (black = 1, red = 2, green = 3) along P
indicate that we should consider Q3 Q1 Q2 Q2 Q3 vj which will be a constantmultiple of ∂τ3 vi . Because of our rescaling in equation (A.7), we know thisconstant has absolute value 1.
(mod 4). It is then possible to rescale Fi (τ ) as
F̃i (τ ) := |Ci | Fi (τ ),(A.7)

and using F̃i (τ ) instead of Fi (τ ) in the description of the supermultiplet.
This rescaling is by a real number. Do so for every vi = v∗ . We have now
done all the rescaling of the ﬁelds that we will do. The result is a new
Adinkra-like graph, where each Ci has absolute value 1.
If instead we want to go from vi back to v∗ in the tree, we can do thesupersymmetry transformations in reverse:
QI1 · · · QIn Fi =

1 n−λ
∂
F∗ .
Ci τ

(A.8)

Now suppose we choose two vertices vi and vj of the graph. We constructa path P from vj to vi by ﬁrst taking the non-retracing path in the spanning
tree from vj to v∗ , followed by the non-retracing path in the spanning tree
from v∗ to vi . See ﬁgure 3. Note that P might be partially retracing, if thelast few edges of the ﬁrst path is retraced backward by a beginning segmentof the second path. This retracing is immaterial for us.

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1957

Figure 4: Suppose we consider the orange edge from vi to vj , which is in
the original Adinkra but not in the spanning tree. We draw that path P
from vj to vi in the spanning tree as in ﬁgure 3, and then tack on the orange
edge. The resulting path P  , shown in blue, is a loop from vj to itself.
If the path P involves the sequence QJ1 · · · QJk of supersymmetry generators, we then use equation (A.6) and equation (A.8) to obtain
QJk · · · QJ1 Fj =Ci μ
∂ Fi ,Cj τ(A.9)

where μ is some non-negative integer, measuring the number of arrows
pointing against the path P . Note that Ci /Cj has absolute value 1.
In particular, if we consider any edge of our Adinkra that is in the
spanning tree, its label c now has absolute value 1.
We now examine an edge of the Adinkra that is not in the spanning tree.
Suppose it connects a bosonic vertex vi to a fermionic vertex vj , i.e., there
is a QI so that
QI vi = c ∂τλ vj(A.10)

for some real c and λ = 0 or 1. Now take the path P in the spanning tree
described above, namely, the one that goes from vj to v∗ then to vi . This
corresponds to a sequence of supersymmetry generators QI1 , . . . , QIn . We
now close the path P into a closed loop, P  by adding to P the new edge
going back from vi to vj (see ﬁgure 4).

1958

CHARLES F. DORAN ET AL.

Using equation (A.10) together with equation (A.9) we get, for somer ∈ N,QI QIn · · · QI1 Fj (τ ) = c(−i)rCi λ+μ∂Fj .
Cj τ(A.11)As this is a closed loop, we can apply this equation to itself, and get
2

r 2



2

Ci
Cj

( QI QIn · · · QI1 ) Fj (τ ) = (−1) c∂τ2(λ+μ) Fj (τ ).

(A.12)

Using equation (3.1) we can anti-commute the QI ’s past each other,
contracting the repeated QI ’s, until we obtain±(i∂τ )n+1

r 2

Fj (τ ) = (−1) c



Ci
Cj

2
∂τ2(λ+μ) Fj (τ ).

(A.13)The ± sign is determined by n and how many repetitions there are in the
QI sequence. Note that n + 1 is even, since each edge connects vertices of
opposite statistics, and a sequence of n + 1 of these edges go from vj toitself. Thus, the left hand side of equation (A.13) is actually real. We canthen match coeﬃcients in equation (A.13) to get
c2 = ±(−1)r+(n+1)/2

Cj2
Ci2

= ±1.

(A.14)

From this we see that c must have complex absolute value 1. Since c is real,c = ±1. Furthermore, we also note from equation (A.13) that 2(λ + μ) =n + 1, indicating that the loop P  involved precisely (n + 1)/2 arrows goingagainst the path, and thus, (n + 1)/2 arrows going along the path. Thisimplies that the absence of central charge excludes Eschericloops [4].
This also proves that Adinkraic supermultiplets with no central charge
are engineerable [17]: it is possible to assign engineering dimensions to allcomponent ﬁelds, consistently with the formulas for the supersymmetry
generator action of the form equations (3.2)– (3.5) of the supersymmetry
algebra (3.1), and without having to introduce parameters of non-zero engineering dimension in either of these.

CODES AND SUPERSYMMETRY IN ONE DIMENSION

Appendix B

1959

Computing doubly even codes

For some classiﬁcation problems, there are theorems that answer the question once and for all. For example, the theorem on the classiﬁcation ofﬁnitely generated abelian groups describes the isomorphism class of such agroup via a simple sequence of integers. In most cases in combinatorics, one
cannot hope for such a theorem. Instead one must settle for an exhaustive
enumeration up to a certain size. It turns out that there are more than 1.1
trillion doubly even [32, k] codes, which without compression will take somenumber of terabytes to store. Using methods including those described inthis appendix, we have already computed over 60,000 of these codes.
The ﬁrst section of this appendix gives the formula describing the number of codes we are interested in and a lower bound for the number of
permutation-isomorphism classes of codes. The next section explains thecomputations necessary for computing the automorphism group and canonical representative of a code, which are critical for the exhaustive enumeration. In the third section, some optimizations for the particular situation at
hand are discussed, which are used in an implementation written speciﬁcallyfor this purpose. The fourth section describes the overarching algorithm
designed to produce the intended enumeration.
The methods used to tackle this problem, namely partition reﬁnement
and canonical augmentation, were originally developed by McKay [43, 44],
who describes an eﬀective method for computing the automorphism groupof a graph and for generating a unique representative for each isomorphismclass. These methods were adapted by Leon in [45] to tackle similar problems, including several group theory questions (which are graph isomorphism
complete) as well as the problem at hand, namely computing the automorphism group of a linear code. McKay and Leon have both written optimized
software packages implementing these methods. The ﬁrst software package,
which remains the standard program for determining graph isomorphism,
is called nauty, and is available online through McKay’s website [46]. The
second is now licensed under the GPL, thanks to the eﬀorts of David Joyner
and Vera Pless, and is available as a part of GUAVA [47], a GAP package forcoding theory. GPL implementations for the cases of graphs of any size andcodes of length up to 32 (or 64 on many machines) can be found in Sage [48](sage.graphs.graph_isom and sage.coding.binary_code). For more details
about the latter, written speciﬁcally for the classiﬁcation of Adinkra topologies, the Reader can consult Section B.3. Sections 2 and 4 are essentially asummary of the techniques described in [43, 44], respectively, which provide
the full story. The Reader should also be aware of the improvements to thisalgorithm in the sparse case in [49].

1960
B.1

CHARLES F. DORAN ET AL.
The mass formula

The following formula, due to Gaborit [38], gives the number σ(N, k) of
distinct doubly even [N, k] codes, depending on the residue of N modulo 8:⎧
k−1
 2N −2i−2 + 2 N2 −i−1 − 1
⎪
⎪
⎪
⎪
,
⎪
i+1 − 1
⎪
2
⎪
⎪ i=0
⎪
⎪
N
N
⎪
k−1
⎪
⎪  2 2 −i−1 + 1 2 2 −i−1 − 1
⎪
⎪
,
⎪
⎪
2i+1 − 1
⎪
⎪
i=0
⎪
⎪
⎪
⎨k−1
 2N −2i−2 − 2 N2 −i−1 − 1σ(N, k) =,⎪2i+1 − 1
⎪
i=0
⎪
⎪
⎪
⎪
⎪k−2
 2N −2i−2 + 2 N2 −i−1 − 2⎪⎪⎪· + (N, k),⎪
i+1 − 1
⎪
2
⎪
⎪
i=0
⎪
⎪
⎪
N
k−2
⎪

⎪
2N −2i−2 + 2 2 −i−1 − 2⎪⎪· − (N, k),⎪⎩2i+1 − 1

if N ≡ 1, 7 (mod 8),
if N ≡ 2, 6 (mod 8),
if N ≡ 3, 5 (mod 8),
if N ≡ 0 (mod 8),
if N ≡ 4 (mod 8),

i=0

(B.1)
± (N, k) :=

1

+
2k−1

N
−k
2

2N −2k ± 2
2k − 1

−2

,

(B.2)

We are interested in those codes with N ≤ 32. According to the mass formula above, we ﬁnd the most such codes when N = 32 and k = 10. In thiscase,
σ(32, 10) = 162, 953, 548, 221, 364, 911, 292, 708, 847, 668, 107, 902,
745, 573, 601, 875≈ 1.6 × 1047 .

(B.3)

However, we are interested not in the codes themselves, but rather theirequivalence classes under permutations of the columns. To establish a lower
bound on the number of equivalence classes, suppose that every such code
has a trivial automorphism group and thus its orbit is as large as possible,
namely 32!. Hence there are at leastσ(32, 10)= 619, 287, 158, 132
32!

(B.4)

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1961

distinct classes of [32, 10] codes. If we carry out a similar computation foreach case of interest, we ﬁnd that there are at leastκ(32) 

k=0

σ(32, k)= 1, 117, 005, 776, 858 ≈ 1.1 × 1012
32!

(B.5)

distinct classes (note that for smaller N , we simply add zero columns to
obtain an equivalent code of degree 32). Since the automorphism group of
a generic code is small, this lower bound is a reasonable estimate for theactual number of codes.
Regardless of the speed at which an individual processor can generate
codes, the procedure for generating the codes can be run in many parallel
processes. We can not only divide by the rate at which we can generate
codes per processor, but we can also divide by the number of processors
working on the job. This is what makes the computation feasible; speed
increases almost linearly as a function of the number of processors.
B.2

Computing the automorphism group

Computing the automorphism group of a code is closely related to the graphisomorphism problem. Given a code C ⊂ {0, 1}N , deﬁne a bipartite graph
G(C), with the vertices partitioned into “left” and “right”, as follows: The
words of the code are the left vertices, and the set {1, 2, . . . , N } forms theright set of vertices. Given a word w on the left, and a number j on theright, there is an edge between the two if and only if w has a 1 in thejth place. Consider the automorphism group Aut(G(C)) of the graph, butallow only those permutations that map left vertices to left vertices andright vertices to right vertices. This gives a subgroup Aut(G(C))b which
is isomorphic in a canonical way to the permutation automorphism groupof the code, simply by considering each permutation’s action on the right
set of vertices, which is identiﬁed with the set of columns of the code. Our
approach is to think of the codes in terms of their corresponding bipartite
graphs, with additional structure coming from the linear, self-orthogonal
code. The algorithm described in this section computes the automorphismgroup of the code, as well as a canonical representative of the code, which
is an arbitrary but ﬁxed representative of the isomorphism class.
Suppose G is a graph, and Π is a partition of the vertices V (G). The
partition Π is called an equitable partition if for every pair of cells C1 , C2
in Π, the number of edges {u, v} such that v ∈ C2 is constant as u ranges
over vertices in C1 . By considering the orbits of vertices of the graph under

1962

CHARLES F. DORAN ET AL.

any subgroup of the automorphism group, one obtains a partition whichis always equitable. However, the converse is not always true; there are
equitable partitions which do not arise in this way. One partition Π1 iscoarser than another partition Π2 (or Π2 is ﬁner than Π1 ) if every cell of Π2
is a subset of a cell of Π1 . The discrete partition is the one where every cellis of size one, and the unit partition is the one where there is only one cell.
Given a graph G and a partition Π of V (G), denote by EG (Π) the coarsest
equitable partition of G which is ﬁner than Π. In particular, if Π is equitablewith respect to G, then EG (Π) = Π. The algorithm described in this section takes as input a graph G and a partition Π0 of V (G), and it returns
the subgroup Aut(G)Π0 of the automorphism group consisting of permutations that respect Π0 , i.e., that do not carry any vertex of one cell of
Π0 into another. From here on, consider all partitions to be ordered, sothat for example ({1, 2, 3}, {4, 5}) = ({4, 5}, {1, 2, 3}). In particular, a discrete ordered partition is simply an ordering on the vertices of G. For v ∈V (G), if Π = (C1 , . . . , Cr ) and v ∈ Ci , deﬁne R(G, Π, v) = EG (Π ), whereΠ = (C1 , . . . , Ci−1 , {v}, Ci−1 \ {v}, Ci+1 , . . . , Cr ). This deﬁnes R(G, Π, v)
up to reordering of the cells of the partition. For full details, see Ref. [43],Algorithm 2.5, which deﬁnes the ordering.
Deﬁne a rooted tree T = T (G, Π0 ) consisting of equitable partitions of G
ﬁner than Π0 as follows: The root of T is the partition EG (Π0 ), and for any
node Π of T , its children are the partitions R(G, Π, v) for which v is not yet
in a singleton cell of Π. The group Aut(G)Π0 acts on the tree T by taking
a sequence of nested partitions Π0 , . . . , Πk to the resulting nested sequenceof partitions by letting Aut(G)Π0 act on the elements of the cells; since thissequence is deﬁned by a sequence v1 , . . . , vk , the sequence γ(v1 ), . . . , γ(vk )deﬁnes the image under γ ∈ Aut(G)Π0 . Further, the subgroup of permutations respecting a partition Π acts on any subtree of T rooted at Π. Because
this action is faithful, the structure of T can be used to calculate a set ofgenerators for Aut(G)Π0 .
The algorithm itself is a backtrack algorithm that successively reﬁnes thepartitions to explore the tree T . Any leaf of the tree is a discrete orderedpartition ({v1 }, {v2 }, . . . , {vn }), which deﬁnes an ordering of the verticesof G. Given two leaves of the tree T , one has two orderings v1 , . . . , vn and
v1 , . . . , vn , which we think of as a permutation deﬁned by vi → vi . This is themeans by which the algorithm ﬁnds automorphisms, and it uses the presenceof automorphisms to deduce when diﬀerent parts of the tree are equivalent.
At this point the algorithm backtracks towards the root until there is a newpart of the tree to explore which is not yet known to be equivalent to a partof the tree already traversed. In practice, the part of the tree traversed ismuch smaller than the entire tree, and once one backtracks oﬀ of the root,

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1963

one has a set of generators for Aut(G)Π0 . Invariants and orderings can also
be used to ﬁnd a leaf of the tree T which is maximal amongst the nodes withlargest invariants, which is uniquely deﬁned independently of reordering the
inputs. This leaf is a “canonical label” for the pair (G, Π0 ). In particular,
if γ ∈ Aut(G)Π0 , then the canonical label for (Gγ , Πγ0 ) is the same as thatof (G, Π0 ). For details, the Reader is once again directed to the theory in
Ref. [43] and the code in Ref. [48].
The outcome is an algorithm to eﬃciently compute the group Aut(C) =Aut(G(C))b (here “=” means canonically isomorphic) as well as a uniquerepresentative of each isomorphism class which we will denote c(C), both ofwhich will be used in the algorithm to generate all the permutation classesof doubly even codes. Also note that by the use of the bipartite graphconstruction, not only is c(C) a code, but it also contains an ordering of its
words. In practice, c(C) will be output as an actual generator matrix.

B.3

New software for special circumstances

It is a curious coincidence that the limit on N for this application is 32.
This implies that the size of the words in a code are at most the size of themachine words on a 32-bit system. The analogy goes further: the operationof adding two vectors in a code becomes the single clock tick of taking an
XOR on the machine words representing them.
When analyzing codes in terms of their corresponding graphs, we cantake advantage of the linear orthogonal structure of the codes to optimize
standard graph algorithms. For example, given an equitable partition Π ofthe bipartite graph G(C), the set of words appearing in their own singleton
cells of Π is closed under binary addition. This is immediate from thedeﬁnition of equitable partition and of binary addition, and this fact greatly
reduces the expected height of the tree to be searched in computing c(C).
Further, every permutation acts linearly, so to check if some permutation isindeed an automorphism, we need only check it on a basis. It is also possible
to use the linear structure to derive variants on the reﬁnement procedure,
depending on the size and type of codes being generated.
In the generation algorithm, since we are interested only in self-orthogonal
codes, we can use more linear algebra to eﬃciently enumerate the possible children of a code. We start by writing the code in standard form,
C = [Ik |C  ], and then extend the basis given by the rows to a basis for C ⊥ .
This basis can be taken so that the ﬁrst k positions of each additional vector
are zero, and we can also shuﬄe the pivot columns to the front, so that we1964

CHARLES F. DORAN ET AL.

obtain this basis as the rows of a matrix of the form


Ik
C 
,
IN −2k ∗0where C  is a rearrangement of the columns! of C . The rowspan of thebottom part of the matrix, 0 IN −2k ∗ , then forms a set of uniquecoset representatives of the original code. Furthermore, (2.2) implies thefollowing:

Corollary B.1. Suppose C is a code spanned by {v1 , . . . , vk }, and thatwt(vi ) ≡ 0 (mod 4) for each i = 1, . . . , k. If either of the following hold for
all i = j, then C is doubly even:(a) vi , vj  = 0,(b) wt(vi + vj ) ≡ 0 (mod 4).
Proof. These results are special to binary codes, since every vector in C is
the sum of distinct vectors in {v1 , . . . , vk }. Equation (2.2) implies that bothconditions are equivalent, so we use b). Suppose x, y, z ∈ C are such thatwt(x) ≡ wt(y) ≡ wt(z) ≡ wt(x + y) ≡ wt(x + z) ≡ wt(y + z) ≡ 0 (mod 4).
Then again by (2.2),
wt(x + y + z) ≡ wt(x + y) + wt(z) − 2x + y, z
≡ − 2x, z − 2y, z
≡ − 2 wt(x) − 2 wt(z) + 2 wt(x + z)
− 2 wt(y) − 2 wt(z) + 2 wt(y + z)
≡ 0 (mod 4).
By induction on the number of basis elements in a linear combination, C isdoubly even.
Going back to the situation at hand, this means that we need
! only examine
the doubly even vectors in the rowspan of 0 IN −2k ∗ , since the code
we already have is doubly even and self-orthogonal, and any vector to beconsidered is already orthogonal to the code. These vectors are in one-to-one
correspondence with the set of doubly even codes of one dimension higher
containing (our rearrangement of) C . We
! can do even better by startingwith the even subcode of 0 IN −2k ∗ , since the sum of two even words
is even.

CODES AND SUPERSYMMETRY IN ONE DIMENSION
B.4

1965

Exhaustively generating the codes

Here we use another algorithm developed by McKay [44], called canonicalaugmentation. The ingredients for this are a question like the one we areconsidering, a canonical labeling function as described, a way of computing
the automorphism group of a code, and a hereditary structure on the objects
to be generated.
We obtain a hereditary structure by thinking of a certain code’s children
as the set of codes constructible by adding a single word not already in thecode. Thus the dimension of a code is one less than the dimension of allits children, and every code can be built up from the zero dimensional codeby a limited number of augmentations. In our application, this puts all thedesired codes on a tree of height 16. At this point one can consider the very
naive algorithm of generating all the children for each code, keeping a list andthrowing out isomorphs. However, the obvious problem with this approach isthe excessive isomorphism computation, which is expensive. This problem
is not really ameliorated by storing the canonical representative of eachcode, as the canonical representative calculation is almost as expensive as
isomorphism testing.
The idea behind canonical augmentation is that instead of requiring theobjects generated be in canonical form, we require simply that the augmentation itself be canonical. The deﬁnition of an augmentation is simply anordered pair (C, C  ), where C  is a child of C. An isomorphism of augmentations is a permutation γ such that (C γ , C  γ ) = (D, D ). For example,
γ(C, C  ) ∼= (C, C  ) implies that there is a γ ∈ Aut(C) such that C  = C  .
Suppose we have a function p which takes codes C of dimension k > 0
to codes p(C) of dimension k − 1, such that C is a child of p(C). Suppose
further that p satisﬁes the following property: if C ∼= D, then (p(C), C) ∼=(p(D), D). If we have such a function, we can deﬁne that an augmentation(C, D) is canonical if (C, D) ∼= (p(D), D). Now suppose that we have two
augmentations (C, C  ) and (D, D ) such that C  ∼= D . If they were bothcanonical augmentations then we would have
(C, C  ) ∼= p(C  ), C  ∼= p(D ), D ∼= (D, D ).
In other words, if both augmentations are canonical, then C  ∼= D implies
that C ∼= D. Thus any repeated isomorphs would have to be due to theparents being repeated. If we assume that there are no isomorphs on theparent level, then this implies that C = D, and (C, C  ) ∼= (C, D ). In otherwords, under the framework of canonical augmentation, repeated isomorphs

1966

CHARLES F. DORAN ET AL.

arise only due to automorphisms of the parents. Thus we have Algorithm 1
below.
Algorithm 1 Generate all doubly even codes of degree 32, isomorph-free.
C_0 := the dimension zero code, of degree 32
traverse(C_0)
procedure traverse(C):report C
children := the children of C
representatives := {}for D in children:
if D is minimal in its orbit under Aut(C):add D to representatives
for D in representatives:
if (C, D) is a canonical augmentation:
traverse(D)The only remaining question is to produce such a function p. This iswhere we use the canonical label deﬁned in the last section. If C is a codeof dimension k, let γ be the permutation taking C to c(C). Recall thatc(C) came with a particular generator matrix determined by the orderingof the words of the code. Removing the last row of that generator matrix
gives a code C  of the same dimension as C and applying γ −1 to that,
we arrive at p(C). Suppose that C ∼= D, which in particular implies thatc(C) = c(D). Let γC , γD be the permutations taking C, D to c(C), c(D),−1respectively. These are isomorphisms, so in fact, γD◦ γC is an isomorphismfrom C to D taking p(C) to p(D) by the deﬁnition of p. This proves theproperty that (p(C), C) ∼= (p(D), D). Further, since we have already done
the computation c(C), we can obtain not only the canonical label but also
generators for the automorphism group Aut(C) for free. Thus when we arechecking whether (C  , C) ∼= (p(C), C), we can simply look for an element ofAut(C) that takes C to p(C).
The ﬁnal task of enumerating the codes for storage to disk will approximate the experience of a harvest. Many separate worker processes will beworking in parallel, each examining the pairs (C, D) as in the second to lastline of Algorithm 1, for a ﬁxed C and many D. Each worker will receive
one code at a time, and perform all the steps in Algorithm 1, only insteadof recursively calling the function on the last line, they will record the augmented codes that succeed in a table, which will each in turn become thefodder for another worker. As codes are given to workers to augment on,
they will be ﬂagged as searched, and once all the codes have been ﬂagged

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1967

and all the workers are done, all the desired codes will be in the table. The
intended end result of all this is a searchable online database, which categorizes the codes by N and k, automorphism group size, weight distribution,and perhaps other parameters.

References[1] S.J. Gates Jr. and L. Rana, A theory of spinning particles for large
N -extended supersymmetry, Phys. Lett. B 352(1–2) (1995), 50–58,[arXiv:hep-th/9504025].
[2] S.J. Gates Jr. and L. Rana, A theory of spinning particles for large
N -extended supersymmetry. II, Phys. Lett. B 369(3–4) (1996), 262–
268, [arXiv:hep-th/9510151].
[3] S.J. Gates Jr. W.D. Linch and J. Phillips, III, When superspace is notenough, [arXiv:hep-th/0211034][4] M. Faux and S.J. Gates Jr., Adinkras: a graphical technology for supersymmetric representation theory, Phys. Rev. D 71(3) (2005), 065002,[arXiv:hep-th/0408004].
[5] M. Faux, D. Kagan and D. Spector, Central charges andextra
dimensions
insupersymmetricquantummechanics,[arXiv:hep-th/0406152].
[6] M. Faux and D. Spector, Duality and central charges in supersymmetric quantum mechanics, Phys. Rev. D (3) 70(8) (2004), 085014,
5, [arXiv:hep-th/0311095].
[7] M. Faux and D. Spector, A BPS interpretation of shape invariance,
J. Phys. A 37 (2004), 10397–10407, [arXiv:quant-ph/0401163].
[8] Z. Kuznetsova, M. Rojas and F. Toppan, Classiﬁcation of irreps
and invariants of the N -extended supersymmetric quantum mechanics,
J. High Energy Phys. 03 (2006), 098, [arXiv:hep-th/0511274].
[9] S. Bellucci, S. Krivonos, A. Marrani and E. Orazi, ‘Root’ action for
N = 4 supersymmetric mechanics theories, Phys. Rev. D 73 (2006),
025011, [arXiv:hep-th/0511249].
[10] S. Bellucci, J. Gates, S. James and E. Orazi, A journey
through garden algebras, Lect. Notes Phys. 698 (2006), 1–47,[arXiv:hep-th/0602259].
[11] S. Bellucci, S. Krivonos, O. Lechtenfeld and A. Shcherbakov, Superﬁeld
formulation of nonlinear N = 4 supermultiplets, [arXiv:0710.3832[hep-th]].

1968

CHARLES F. DORAN ET AL.

[12] Z. Kuznetsova and F. Toppan, Reﬁning the classiﬁcation of the irrepsof the 1d n-extended supersymmetry, Mod. Phys. Lett. A 23 (2008),
37–51, [arXiv:0701225].
[13] M. Gonzales, M. Rojas and F. Toppan, One-dimensional sigma-models
with n = 5, 6, 7, 8 oﬀ-shell supersymmetries, Int. J. Mod. Phys. A 24
(2009), 4317–4326, [arXiv:0812.3042].
[14] Z. Kuznetsova and F. Toppan, Extended supersymmetric quantum
mechanics of Fierz and Schur type, [arXiv:1012.1005].
[15] C.F. Doran, M.G. Faux, S.J. Gates Jr., T. Hübsch, K.M. Iga and G.D.
Landweber, On the matter of n = 2 matter, Phys. Lett. B 659 (2008),
441–446, [arXiv:0710.5245].
[16] W. Siegel and M. Roček, On oﬀ-shell supermultiplets, Phys. Lett. B
105(4) (1981), 275–277.
[17] C.F. Doran, M.G. Faux, S.J. Gates Jr., T. Hübsch, K.M. Iga and G.D.
Landweber, On graph-theoretic identiﬁcations of Adinkras, supersymmetry representations and superﬁelds, Int. J. Mod. Phys. A 22 (2007),
869–930, [arXiv:math-ph/0512016].
[18] F.J. MacWilliams and N.J.A. Sloane, The theory of error-correcting
codes, North-Holland, Amsterdam, 1977.
[19] W.C. Huﬀman and V. Pless, Fundamentals of error-correcting codes,
Cambridge University Press, 2003.
[20] J.H. Conway, V. Pless and N.J.A. Sloane, The binary self-dual codes oflength up to 32: a revised enumeration, J. Comb. Theory, Series A 60
(1992), 183–195.
[21] T. Hübsch, Haploid (2, 2)-superﬁelds in 2-dimensional space-time, Nucl.
Phys. B 555(3) (1999), 567–628.
[22] R.Q. Almukahhal and T. Hübsch, Gauging Yang-Mills symmetries in(1+1)-dimensional space-time, Int. J. Mod. Phys. A 16(29) (2001),
4713–4768.
[23] C.F. Doran, M.G. Faux, S.J. Gates Jr., T. Hübsch, K.M. Iga and
G.D. Landweber, Adinkras and the dynamics of superspace
prepotentials, Adv. Stud. Theor. Phys. (2007, in press),[arXiv:hep-th/0605269].
[24] C.F. Doran, M.G. Faux, S.J. Gates Jr., T. Hübsch, K.M. Iga and G.D.
Landweber, Super-Zeeman embedding models on N-supersymmetricworld-lines, J. Phys. A 42 (2008), 065402, [arXiv:0803.3434].
[25] C.F. Doran, M.G. Faux, S.J. Gates Jr., T. Hübsch, K.M. Iga and G.D.
Landweber, Oﬀ-shell supersymmetry and ﬁltered Cliﬀord supermodules,[arXiv:math-ph/0603012].

CODES AND SUPERSYMMETRY IN ONE DIMENSION

1969[26] R.T. Bilous and G.H.J. van Rees, An enumeration of binary self-dual
codes of length 32, Des. Codes Cryptogr. 26 (2002), 61–86.
[27] S.J. Gates Jr., C.M. Hull and M. Roček, Twisted multiplets and newsupersymmetric nonlinear sigma modelsnd new supersymmetric nonlinear sigma models, Nucl. Phys. B 248 (1984), 157.
[28] H.B. Lawson and M.-L. Michelsohn Jr., Spin geometry, Princeton Mathematical Series, 38, Princeton University Press, Princeton, NJ, 1989.
[29] S.J. Gates Jr., Vector multiplets and the phases of n=2 theories in 2d:
Through the looking glass, Phys. Lett. B 352 (1995), 43–49.
[30] D.R. Morrison and M.R. Plesser, Towards mirror symmetry as duality
for two dimensional abelian gauge theories, Nucl. Phys. Proc. Suppl.
46 (1996), 177–186, [arXiv:hep-th/9508107].
[31] S.-T. Yau (Ed.), Mirror manifolds, International Press, 1990.
[32] B. Greene and S.-T. Yau (Eds.), Mirror manifolds II, International
Press, 1996.
[33] D.H. Phong, L. Vinet and S.-T. Yau (Eds.), Mirror manifolds III,American Mathematical Society, Providence, RI, 1999.
[34] A. Dimakis, A new representation of Cliﬀord algebras, J. Phys. A 22
(1989), 3171–3193.
[35] J.A. Wood, Self-orthogonal codes and the topology of spinor groups,
Coding Theory and Design Theory 20 (1990), 219–239.
[36] S.J. Gates and L. Rana Jr., Ultramultiplets: a new representation of
rigid 2-d, N =8 supersymmetry, Phys. Lett. B 342 (1995), 132–137,[arXiv:hep-th/9410150].
[37] S.J. Gates Jr., W. Linch, J. Phillips and L. Rana, The fundamental supersymmetry challenge remains, Gravit. Cosmol. 8(1–2) (2002),
96–100, [arXiv:hep-th/0109109].
[38] P. Gaborit, Mass formulas for self-dual codes over Z4 and Fq + uFq
rings, IEEE Trans. Inf. Theory 42(4) (1996), 1222–1228.
[39] W. Siegel, On-shell O(N) supergravity in superspace, Nucl. Phys. B 177(1981), 325.
[40] E. Bergshoeﬀ, M. de Roo, and B. de Wit, Extended conformal supergravity, Nucl. Phys. B 182 (1981), 173.
[41] J.H. Conway and V. Pless, On the enumeration of self-dual codes,
J. Comb. Theory, Series A 28 (1980), 26.
[42] R.T. Bilous and G.H.J. van Rees, Self-dual code lists, http://www.cs.
umanitoba.ca/~umbilou1/SelfDualCodes/toc.html.
[43] B. McKay, Practical graph isomorphism, Congr. Numer. 30 30 (1981),
45–87.

1970

CHARLES F. DORAN ET AL.

[44] B. McKay, Isomorph-free exhaustive generation, J. Algorithms 26(1998), 306324.
[45] J.S. Leon, Permutation group algorithms based on partition, I: theory
and algorithm, J. Symbolic Comput. 12 (1991), 533583.
[46] B. McKay, The nauty page, http://cs.anu.edu.au/~bdm/nauty/(2008).
[47] R. Baart, T. Boothby, J. Cramwinckel, W. Joyner, R. Miller,
E. Minkes, E. Roijackers, L. Ruscio, and C. Tjhai, Gap package guava,
http://opensourcemath.org/guava/ (2008).
[48] W. Stein, Sage: open source mathematical software (version 2.10.3),
the Sage group, http://www.sagemath.org (2008).
[49] P. Darga, M. Liﬃton, K. Sakallah, and I. Markov, Exploiting structure
in symmetry detection for CNF, Proceedings of 41st Design Automation
Conference, 2004, http://vlsicad.eecs.umich.edu/BK/SAUCY/