An essential ingredient in the construction of the Green-Schwarz superstring [Green, Schwarz, Witten] is the spinor identity \begin{equation} \epsilon^{klm} \gamma^\mu \Psi_k \bar\Psi_l \gamma_\mu \Psi_m = 0 \label{Super} \end{equation} for anticommuting spinors $\Psi_k$, $\Psi_l$, $\Psi_m$, where $\epsilon^{klm}$ indicates total antisymmetrization. This identity can be viewed as a special case of a Fierz rearrangement. An analogous identity holds for commuting spinors $\Psi$, namely \begin{equation} \gamma^\mu \Psi \bar\Psi \gamma_\mu \Psi = 0 \label{Commute} \end{equation}

It is well-known that the $3$-$\Psi$ Rule holds for Majorana spinors in three dimensions, Majorana or Weyl spinors in four dimensions, Weyl spinors in six dimensions, and Majorana-Weyl spinors in ten dimensions. Thus, the Green-Schwarz superstring exists only in those cases [Green, Schwarz, Witten] . As was shown by Fairlie and Manogue \cite{FairlieI}, the $3$-$\Psi$ Rule in all these cases is equivalent to an identity on the $\gamma$-matrices, which holds automatically for the natural representation of the $\gamma$-matrices in terms of the four division algebras $\RR,\CC,\HH,\OO$, corresponding precisely to the above four types of spinors. Manogue and Sudbery then showed how to rewrite these spinor expressions in terms of $2\times2$ matrices over the appropriate division algebra, thus eliminating the $\gamma$-matrices completely.

We show here that the $3$-$\Psi$ Rule is in fact equivalent to an associator identity over the division algebras.

Let $U$, $V$, $W$ be arbitrary octonionic vectors, that is, $1\times n$ octonionic matrices. The vector associator \begin{equation} [U,V,W] = (U V^\dagger) W - U (V^\dagger W) \end{equation} satisfies \begin{equation} [W,V,V] \equiv 0 \label{Master} \end{equation} which is established by direct computation using alternativity. Setting $W=V$ yields \begin{equation} [V,V,V] \equiv 0 \label{Cube} \end{equation}

Essentially the same argument establishes a similar formula when $W$ is replaced by an octonionic scalar $\lambda$, namely \begin{equation} [\lambda,V,V] = 0 \end{equation} where we have implicitly defined yet another associator, namely \begin{equation} [\lambda,V,W] = (\lambda V^\dagger) W - \lambda (V^\dagger W) \end{equation} An interesting consequence of this result is the Hermitian conjugate relation \begin{equation} V^\dagger (V \bar\lambda) = (V^\dagger V) \bar\lambda \end{equation} which has applications to the right eigenvalue problems considered in § 13.

We can polarize (\ref{Master}) to obtain \begin{equation} [U,V,W]+[U,W,V] \equiv 0 \label{LemmaIII} \end{equation} a special case of which is \begin{equation} [V,V,W]+[V,W,V] \equiv 0 \end{equation} obtained by setting $U=V$. A further special case of (\ref{LemmaIII}) is \begin{align} &[U,V,W]+[U,W,V]+[V,W,U] \nonumber\\ &\qquad +[V,U,W]+[W,U,V]+[W,V,U] \equiv 0 \label{SixPsi} \end{align} obtained by adding cyclic permutations of (\ref{LemmaIII}), or alternatively, without requiring (\ref{Master}), by repeated polarization of (\ref{Cube}).

In the commuting case, the (unpolarized) $3$-$\Psi$ Rule can be written in terms of a 2-component octonionic “vector” (really a spinor) $V$ as [arXiv:hep-th/9407045] \begin{equation} \tilde{(VV^\dagger)} \, V = 0 \label{3Psi} \end{equation} where \begin{equation} \tilde{A} = A - \tr{A} \end{equation} corresponding to time reversal. It is straightforward to check that (\ref{3Psi}) holds by alternativity, or by using the identity \begin{equation} \tr(V V^\dagger) = \tr(V^\dagger V) = V^\dagger V \in \RR \end{equation} to write \begin{equation} \tilde{(VV^\dagger)} \, V = (VV^\dagger - V^\dagger V) \, V = (VV^\dagger) V - V (V^\dagger V) = [V,V,V] \end{equation} and we see that the $3$-$\Psi$ Rule for commuting spinors is just (\ref{Cube}).

Equivalently, we can rewrite the (polarized) $3$-$\Psi$ Rule in terms of 2-component “vectors” $U$, $V$, $W$ as \begin{equation} (\tilde{UV^\dagger}+\tilde{VU^\dagger}) \, W + (\tilde{VW^\dagger}+\tilde{WV^\dagger}) \, U + (\tilde{WU^\dagger}+\tilde{UW^\dagger}) \, V = 0 \label{ThreePsi} \end{equation} Using the identity \begin{equation} \tr{(U V^\dagger + V U^\dagger)} = \tr{(V^\dagger U + U^\dagger V)} = V^\dagger U + U^\dagger V \end{equation} we have \begin{equation} \tilde{UV^\dagger}+\tilde{VU^\dagger} = (U V^\dagger + V U^\dagger) - (U^\dagger V + V^\dagger U) \end{equation} where the last term is Hermitian and hence real. We thus have \begin{equation} (\tilde{UV^\dagger}+\tilde{VU^\dagger}) W = (U V^\dagger + V U^\dagger) W - W (U^\dagger V + V^\dagger U) \end{equation} which implies that (\ref{ThreePsi}) is precisely the same as (\ref{SixPsi}).

An analogous argument can be given for anticommuting spinors; this is essentially the approach used in Combining these results, the $3$-$\Psi$ Rule can be written without $\gamma$-matrices in terms of 2-component octonionic spinors $\psi_\alpha$ as \begin{align} & [\psi_1,\psi_2,\psi_3] \pm [\psi_1,\psi_3,\psi_2] + [\psi_2,\psi_3,\psi_1] \nonumber\\ & \qquad \pm [\psi_2,\psi_1,\psi_3] + [\psi_3,\psi_1,\psi_2] \pm [\psi_3,\psi_2,\psi_1] \equiv 0 \end{align} for both commuting ($+$) or anticommuting ($-$) spinors. Both of these identities follow from the identity (\ref{Cube}) applied to 2-component octonionic vectors, which is a special case of the more general identity (\ref{Master}) which holds for octonionic vectors of arbitrary rank.