### Adinkras

#### Posted by Urs Schreiber

Late one night, a while ago, Blake Stacey noticed that there are “category diagrams” which, in the twilight, look like Adinkras

– at least to some people:

Doran, C. F.; Faux, M. G.; Gates, S. J.; Hubsch, T.; Iga, K. M.; Landweber, G. D.
*On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields*

math-ph/0512016

The simplest such Adinkra appearing in the study of $N$-extended $d=1$ supersymmetry looks like $\array{ \bullet \\ \downarrow \\ \circ } \,.$ This corresponds to $N=1$. A slightly more interesting one is obtained for $N=2$: $\array{ && \bullet \\ &\swarrow && \searrow \\ \circ &&&& \circ \\ & \searrow && \swarrow \\ && \bullet } \,.$

I was struck, since to me these look like the categorical super-point and the categorical 2-super point.

Let me try to explain…

**Odd Flow Lines**

For any category $C$, it seems to make good sense to address the category of morphism
from the fat point
$\mathbf{pt} = \{\bullet \stackrel{\sim}{\to} \circ \}$
into $C$, which fix the body $(\mathrm{pt} = \{\bullet\}) \hookrightarrow \mathbf{pt}$ of the fat point as the categorical tangent bundle
$p : T C \to \mathrm{Obj}(C)
\,.$
The sections of $T C$ form a group, due to the canonical embedding
$\Gamma(T C) \hookrightarrow T_{\mathrm{Id}_C}(\mathrm{End}(C))
\,.$
For $G$ any group, a *$G$-flow* on $C$ is a group homomorphism
$G \to \Gamma(T C)$
in this sense.

In particular, if $C = P_1(X)$ is the path groupoid of a smooth space $X$, then smooth $\mathbb{R}$-flows $v : \mathbb{R} \to \Gamma(T P_1(X))$ are ordinary vector fields on $X$.

That’s nice, because using this arrow-theoretic differential theory, we now immediately know a vast generalization of the concept of a vector in all kinds of exotic ways.

For instance by replacing $\mathbb{R}$ by the finite group $\mathbb{Z}_2$. A $\mathbb{Z}_2$-flow on $C$ is a group homomorphism $\mathbb{Z}_2 \to \Gamma(T C) \,.$ And it might be useful to think of this as a categorical version of an odd vector field. I’ll come to that.

The simplest category with a nontrivial $\mathbb{Z}_2$-flow on it is the fat point $\mathbf{pt} = \{\bullet \stackrel{\sim}{\to} \circ \}$ itself. Looking at it this way, I am inclinded to call this the categorical superpoint.

Following the authors of the *Adinkra*-papers (to which I shall come), I will restrict attention to superpoints – but possibly with higher supersymmetry.

Since there is precisely one nontrivial $\mathbb{Z}_2$-flow on $\mathbf{pt}$, let’s call it the

$N=1$ superpoint : $\mathbf{pt}_{N=1}$

What would be the $N=2$ superpoint, then?

It’s the categorical “wedge” product
$\mathbf{pt}_{N=2} := \mathbf{pt}_{N=1} \otimes \mathbf{pt}_{N=1}
\,.$
I claim. Here “$\otimes$” is the *true* tensor product on $n$-categories, the one which sends an $n$-category $C_n$ and an $m$-category $D_m$ to the $(n+m)$-category
$C_n \otimes D_m
\,.$
That’s described in full generality in the beautiful work

Sjoerd Crans
* Pasting schemes for the monoidal biclosed structure on $\omega-\mathrm{Cat}$*

(ps)

but it is useful to also look at the discussion in

Sjoerd Crans
*A tensor product for Gray-categories*

(ps).

(I am grateful to Todd Trimble for pointing me to this stuff.)

Notice that, on p. 3, Crans mentions our $N=1$-superpoint under the name $2_1 \,.$ What I here call the $(N=2)$-superpoint $\mathrm{pt}_{N=2} = \left\lbrace \array{ (\bullet,\bullet) &\to& (\bullet,\circ) \\ \downarrow &\Downarrow^\sim& \downarrow \\ (\circ,\bullet) &\to & (\circ,\circ) } \right\rbrace$ is closely related to what Sjoerd Crans would call $2_2 \,.$ And so on.

And indeed, there are precisely *two* different nondegenerate $\mathbb{Z}_2$-flows
$\mathbb{Z}_2 \to T_{\mathrm{Id}_{\mathbf{pt}_{N=2}}(\mathrm{End}(\mathbf{pt}_{N=2}))}$
on the 2-super point.

**Adinkras**

In

Michael Faux, S. J. Gates Jr
*Adinkras: A Graphical Technology for Supersymmetric Representation Theory*

hep-th/0408004

the authors argue that it is useful to draw diagrams essentially of the above form for thinking about representations of the 1-dimensional but $N$-extended local super-Poincaré algebra.

This means we need to be thinking in terms of superfields on the line, on which we want to act with the super-Lie algebra whose single even generator is the ordinary derivative $\partial_t : \phi \mapsto \phi'$ and whose single odd generator $q$ has the graded bracket $[Q,Q] = \partial_t \,.$

The standard representation of this algebra is spanned by an even field $\phi$ and an odd field $\psi$ with the action of $Q$ given by $\begin{aligned} Q \phi &\propto \psi \\ Q \psi & \propto \partial_t \phi \end{aligned} \,.$

Faux and Gates propose to associate with this representation the diagram $\bullet^{\phi} \to \circ^{\psi}$ (I have switched $\bullet \leftrightarrow \circ$ with respect to their notation) in order to indicate that there are

- two fields in the game (hence two vertices)

- one of which is even (a filled verted)

- and one of which is odd (an open vertex)

- and where the action of $Q$ on one of these involves $\partial_t$, while on the other it does not (and this is indicated by the arrow pointing from the latter to the former).

This diagram they call an *Adinkra*.

Thinking carefully about what superfields really are can prove to be quite tricky. Since it is quite late at night here (but recall, as Blake Stacey notes: maybe a necessary prerequisite for *Adinkras* to look like categorical diagrams) I won’t try to give a meaningful mathematical exegesis of the above formulas.

Suffice it, for the moment, to notice that not only does the *Adinkra*
$\bullet^{\phi} \to \circ^{\psi}$
look a lot like the $N=1$ superpoint (which *a priori* might very well be just a coincidence of simple structures), but it also encodes a similar mechanism: an arrow which indicates a transformation that exchanges something with its “odd” partner.

Adrinkas are supposed to become useful as one studies $N$-extended supersymmetry algebras of higher $N$.

In the $N=2$-version of the above setup, we’d have *two* even fields
$\phi_1\,, \phi_2$
and *two* odd fields
$\psi_1\,, \psi_2$
and *two* different odd vector fields
$Q_1 \,, Q_2$
whose action on $\phi_i$ and $\psi_i$ is pretty much as before, but now may also involve swapping the subscript of the fields.

Accordingly, we draw *two* solid vertices
$\array{
&&
\bullet^{\phi_1}
\\
\\
\\
\\
&&
\bullet^{\phi_2}
}$
and *two* open ones
$\array{
\circ^{\psi_1} &&&& \circ^{\psi_2}
}$
and connect them by arrows going in *two* perpendicular directions
$\array{
&&
\bullet^{\phi_1}
\\
&{}^1\swarrow && \searrow^2
\\
\circ^{\psi_1} &&&& \circ^{\psi_2}
\\
& {}_2\searrow && \swarrow_1
\\
&&
\bullet^{\phi_2}
}
\,.$

There are various ways this swapping of subscripts may be combined with the general susy pattern $q : \phi \mapsto \propto \psi$ and $q : \psi \mapsto \propto \phi'$. The *Adinkra* as just drawn, for instance, encodes the transformation behaviour
$\begin{aligned}
Q_1 : & \phi_1 \mapsto \propto \psi_1
\\
Q_2 : & \phi_1 \mapsto \propto \psi_2
\\
\\
Q_1 : & \psi_1 \mapsto \propto \partial_t \phi_1
\\
Q_2 : & \psi_1 \mapsto \propto \phi_2
\end{aligned}$

The theory of representations of $N$-extended $d=1$ supersymmetry, hence also that of these *Adinkra*s, becomes more interesting as one moves to higher $N$.

And I am wondering: maybe, if looked at from the right angle, *Adinkras* look like categories not just in twilight.

(The Adinkra graphics here are from the site Adinkra Symbols.)