Pure spinors, intrinsic torsion and curvature in odd dimensions
Arman Taghavi-Chabert
{Kotl\'{a}\v{r}sk\'{a} 2, 611 37 Brno, Czech Republic }
We develop a spinor calculus for a $(2m+1)$-dimensional complex Riemannian manifold $(\mcM,g)$ equipped with a preferred holomorphic projective pure spinor field $\langle \xi \rangle$. Such a spinor defines a holomorphic distribution $\mcN$ of $m$-planes, totally null with respect to $g$. The stabiliser of $\langle \xi \rangle$ is a parabolic Lie subgroup $P$ of $\SO(2m+1,\C)$, and using its algebraic properties, we give $P$-invariant classifications of the curvature tensors of the Levi-Civita connection. In particular, we generalise the Petrov-Penrose classification of the Weyl tensor to odd dimensions. From a spinorial point of view, this generalises the notion of principal spinors. We also classify the intrinsic torsion of $\langle \xi \rangle$ in terms of $P$-irreducibles, which gives an algebraic measure of the failure of $\langle \xi \rangle$ to be parallel with respect to the Levi-Civita connection. In particular, we interpret the integrability properties of the associated null $m$-plane distribution $\mcN$ and its orthogonal complement $\mcN^\perp$ in terms of the algebraic types of the intrinsic torsion of $\langle \xi \rangle$. The conformal invariance of these classes are studied and we give curvature conditions for a number of cases. We then investigate the geometric properties of a number of spinorial differential equations. Notably, we relate the integrability properties of a null $m$-plane distribution to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We also give necessary and sufficient conditions for a pure conformal Killing spinor to be foliating. This leads in particular to a conjecture refining the author's odd-dimensional generalisation of the Goldberg-Sachs theorem. This work can also be applied to a smooth real pseudo-Riemannian manifold of split signature equipped with a preferred projective real pure spinor field.
Algebra.tex
\section{Algebraic background}\label{sec-algebra}
In this section, we review the background on Clifford algebras, spinor representations, and the algebraic properties of pure spinors. The approach will be to synthesize aspects of the theory of spinors given notably in \cites{Cartan1981,Penrose1986,Budinich1989,Harnad1995,Kopczy'nski1997} with the representation theory of parabolic Lie subalgebras of \cites{Baston1989,vCap2009}. We shall adopt the abstract index notation of \cites{Penrose1984,Penrose1986} throughout, in particular, that of Clifford algebras and spinors.

\subsection{Clifford algebras and spinor representations}
Let $\mfV$ be a $n$-dimensional complex vector space. Elements of $\mfV$ and of its dual $\mfV^*$ will carry upstairs and downstairs lower-case Roman indices respectively, e.g. $V^a$ and $\alpha_a$ respectively. This notation extends to tensor products of $\mfV$ and $\mfV^*$, i.e. we write $T \ind{_{ab}^c_d}$ for an element of $\otimes^2 \mfV^* \otimes \mfV \otimes \mfV^*$. We equip $\mfV$ with a non-degenerate symmetric bilinear form $g_{ab} = g_{(ab)} \in \odot^2 \mfV^*$. Here, as elsewhere, symmetrisation is denoted by round brackets, while skew-symmetrisation by squared brackets, e.g. $\alpha_{abc} = \alpha_{[abc]} \in \wedge^3 \mfV^*$. The inner product $g_{ab}$ together with its inverse $g^{ab}$ establishes an isomorphism between $\mfV$ and $\mfV^*$, so that one will lower or raise the indices of tensorial quantities as needed.

The Clifford algebra $\Cl ( \mfV , g )$ of $(\mfV , g )$ is defined as the quotient algebra $\bigotimes^\bullet \mfV / \mfI$ where $\mfI$ is the ideal generated by elements of the form $v \otimes v + g(v,v)$ where $v \in \mfV$. This means that $\Cl ( \mfV , g)$ and the exterior algebra $\wedge^\bullet \mfV$ are isomorphic as vector spaces, the wedge product being now replaced with the Clifford product $\cdot : \Cl ( \mfV , g) \times \Cl ( \mfV , g) \rightarrow \Cl ( \mfV , g)$ defined by
\begin{align*}
v \cdot w & := v \wedge w - g (v,w) \, ,
\end{align*}
for $v,w \in \mfV$ viewed as elements of $\Cl(\mfV,g)$.

Henceforth, we assume $n=2m+1$, in which case the Clifford algebra can be shown to be isomorphic to a direct sum of two copies of the algebra of $2^m \times 2^m$-matrices over $\C$. Each copy acts on a $2^m$-dimensional complex vector space, which is known as the \emph{spinor space of $(\mfV,g )$}. These spinor spaces are inequivalent irreducible representations of the double cover $\Spin(2m+1,\C)$ of $\SO(2m+1,\C)$. From now on, we shall work with one of these spinor representations, which we shall denote $\mfS$.

\begin{rem}\label{rem-spinor-form1}
 One can give an explicit construction of the spinor representation $\mfS$ in terms of forms as follows. Choose a totally null $m$-dimensional subspace $\mfN$ of $\mfV$, and fix a dual $\mfN^*$ of $\mfN$ so that $\mfV \cong \mfN \oplus \mfN^* \oplus \mfU$ where $\mfU$ is the (one-dimensional) complement of $\mfN \oplus \mfN^*$ in $\mfV$. Now, for any $(v,w,u) \in \mfV$, the action of the Clifford algebra on $\mfS \cong \wedge^\bullet \mfN$ is given by
\begin{align}\label{eq-spinor-action}
 (v,w,u) \cdot \xi = v \wedge \xi - w \lrcorner \xi \pm \epsilon u \xi \, ,
\end{align}
where
\begin{align*}
 \epsilon & =
\begin{cases}
 +1 \, , & \mbox{if $\xi \in \wedge^m \mfN \oplus \wedge^{m-2} \mfN \oplus \wedge^{m-4} \mfN \oplus \ldots \, ,$} \\
 -1 \, , & \mbox{if $\xi \in \wedge^{m-1} \mfN \oplus \wedge^{m-3} \mfN \oplus \wedge^{m-5} \mfN \oplus \ldots \, .$}
\end{cases}
\end{align*}
The choice of sign in \eqref{eq-spinor-action} reflects the choice of inequivalent spinor representations.
\end{rem}

Elements of $\mfS$ will carry upstairs upper-case Roman indices, e.g. $\xi^A$, and similarly for elements of the dual spinor space $\mfS^*$, with downstairs indices, e.g. $\eta_A$. From the definition of the Clifford algebra, one can show that $\Cl(\mfV,g )$ is generated by the \emph{$\gamma$-matrices $\gamma \ind{_a _A ^B}$} which satisfy the Clifford property
\begin{align}\label{eq-red_Clifford_property}
 \gamma \ind{_{\lp{a}} _A ^C} \gamma \ind{_{\rp{b}} _C ^B} & = - g_{ab} \delta \ind*{_A^B} \, .
\end{align}
Thus, only skew-symmetrised products of $\gamma$-matrices count, and we shall make use of the notational short hand
\begin{align}\label{eq-skew-gamma}
 \gamma \ind{_{a_1 a_2 \ldots a_p} _A ^B} & := \gamma \ind{_{\lb{a_1}} _A ^{C_1}} \gamma \ind{_{a_2} _{C_1} ^{C_2}} \ldots \gamma \ind{_{\rb{a_p}} _{C_p} ^{B}} \, ,
\end{align}
for any $p$. These matrices give us an explicit realisation of the isomorphism $\Cl( \mfV, g ) \cong \bigwedge^\bullet \mfV$ as vector spaces.

The spinor space $\mfS$ and its dual $\mfS^*$ are equipped with non-degenerate bilinear forms, with which one can in effect raise or lower spinor indices. In particular, the $\gamma$-matrices \eqref{eq-skew-gamma} can now be viewed as bilinear maps from $\mfS \times \mfS$ to $\wedge^p \mfV$ some $p$. Depending on the values of $m$ and $k$, these bilinear forms can be either symmetric or skewsymmetric. Of interest is the case where $p=m$: the $\gamma$-matrices $\gamma \ind{_{a_1 a_2 \ldots a_m} ^{AB}}$ and $\gamma \ind{_{a_1 \ldots a_{m+1}}^{AB}}$ are always symmetric in their spinor indices.

Before we delve into the topic of pure spinors, we state without proof the following technical lemma that will be used later.
\begin{lem}\label{lem-technical}
 We have
\begin{multline*}
 \gamma \ind{_a_A^B} \gamma \ind{_{b_1 \ldots b_p}_{BD}} \gamma \ind{_c_C^D} 
= (-1)^m \left( \gamma \ind{_{a b_1 \ldots b_p c}_{AC}} - (p + 1 )\gamma \ind{_{\lb{a} b_1 \ldots b_{p-1}}_{AC}} g \ind{_{\rb{b_p} c}} \right. \\
\left. - p \, g \ind{_{a \lb{b_1}}} \gamma \ind{_{b_2 \ldots \rb{b_p}c}_{AC}} + p (p+1) g \ind{_{a \lb{b_1}}} \gamma \ind{_{b_2 \ldots b_{p-1}}_{AC}} g \ind{_{\rb{b_p}c}} \right) \, .
\end{multline*}
In particular,
\begin{align*}
 \gamma \ind{^a_A^B} \gamma \ind{_{b_1 \ldots b_p}_{BD}} \gamma \ind{_a_C^D} & = 
 (-1)^{m + p} ( 2p - 2m - 1 )\gamma \ind{_{b_1 \ldots b_p}_{AC}} \, .
\end{align*}
\end{lem}

\subsection{Pure spinors}
Let $\xi \ind{^A}$ be a non-zero spinor in $\mfS$, and consider the map
\begin{align*}
 \xi \ind{_a^A} := \xi \ind{^B} \gamma \ind{_a_B^A} : \mfV \rightarrow \mfS \, .
\end{align*}
By the Clifford property \eqref{eq-red_Clifford_property}, the kernel of $\xi \ind{_a^A} : \mfV \rightarrow \mfS$ must be \emph{totally null}, i.e. for any $X^a, Y^a \in \ker \xi \ind{_a^A} : \mfV \rightarrow \mfS$, $g_{ab} X^a Y^b = 0$.
\begin{defn}
 A non-zero spinor $\xi \ind{^A}$ is said to be \emph{pure} if the kernel of $\xi \ind{_a^A}: \mfV \rightarrow \mfS$ is $m$-dimensional.
\end{defn}
Clearly, the purity property is invariant under rescaling of $\xi \ind{^A}$. The span of a pure spinor $\xi^A$ will be denoted $\langle \xi^A \rangle$ and will be referred to as a \emph{projective pure spinor}.
\begin{defn}
 A \emph{null structure} or \emph{$\gamma$-plane} on $\mfV$ is a totally null $m$-dimensional vector subspace of $\mfV$.
\end{defn}

\begin{rem}\label{rem-spinor-form2}
By identifying the spinor module $\mfS$ with the $\Cl(\mfV, g)$-module $\wedge^\bullet \mfN$ where $\mfN \subset \mfV$ is a null structure as in Remark \ref{rem-spinor-form1}, it is clear that pure spinors exist and in fact span $\mfS$. There is thus a one-to-one correspondence between $\gamma$-planes in $\mfV$ and projective pure spinors.
\end{rem}

Henceforth, we shall focus on the geometry arising from a fixed pure spinor $\xi^A$. By definition, $\xi^A$ induces a filtration
\begin{align}\label{eq-filtration}
\{ 0 \} =: \mfV^2 \subset \mfV^1 \subset \mfV^0 \subset \mfV^{-1} \, ,
\end{align}
where
\begin{align*}
 \mfS^{\frac{m}{2}} & := \langle \xi^A \rangle \, , & \mfS^{\frac{m-2}{2}} & := \im \xi \ind{_a^A} : \mfV \rightarrow \mfS \, , \\
 \mfV^{-1} & := \mfV \, , & \mfV^1 & := \ker \xi \ind{_a^A} : \mfV \rightarrow \mfS \, , & \mfV^0 & := (\mfV^1)^\perp \, ,
\end{align*}
where $\perp$ denotes the orthogonal complement with respect to $g_{ab}$. Thus, $\xi \ind{_a^A}$ establishes an isomorphism between $\mfV^{-1}/ \mfV^1$ and $\mfS^{\frac{m-2}{2}}$, which we shall write as\footnote{The use of factor $\mfS^{\frac{m}{2}}$ on the LHS of \eqref{eq-iso} balances the `degrees' on each side of \eqref{eq-iso}, i.e. $-1 + \frac{m}{2} = \frac{m-2}{2}$.}
\begin{align}
 \left( \mfV^{-1}/ \mfV^1 \right) \otimes \mfS^{\frac{m}{2}} & \cong \mfS^{\frac{m-2}{2}} \, , \label{eq-iso}
 \end{align}
of vector spaces. We note that $\mfV^{-1}/\mfV^1$ is an $(m+1)$-dimensional subspaces of $\mfV$ that contains the $1$-dimensional subspace $\mfV^0/\mfV^1$, on which the metric is non-degenerate. In this case, we can find an element $u^a$ in $\mfV^0$ such that $u^a \gamma \ind{_a_A^B}$ is proportional to the identity on $\mfS$. In particular, the image of $\mfV^0/\mfV^1$ under $\xi \ind{_a^A}$ must be a $1$-dimensional subspace $\mfS^{\frac{m}{2}}$ of $\mfS^{\frac{m-2}{2}}$, i.e.
 \begin{align}
  ( \mfV^0/ \mfV^1 ) \otimes \mfS^{\frac{m}{2}} & \cong \mfS^{\frac{m}{2}} \, , \label{eq-iso2}  \\
 ( \mfV^{-1} / \mfV^0 ) \otimes \mfS^{\frac{m}{2}} & \cong \mfS^{\frac{m-2}{2}}/\mfS^{\frac{m}{2}} \, . \label{eq-iso3}  
\end{align}

Turning now to the dual picture, we set
\begin{align*}
 \mfS^{-\frac{m}{2}} & := \mfS^* \, , & \mfS^{-\frac{m-2}{2}} & := \ker \xi^A : \C \leftarrow \mfS^* \, , & \mfS^{-\frac{m-4}{2}} := \ker \xi \ind{_a ^A} : \mfV^* \leftarrow \mfS^* \, ,
\end{align*}
so that, using the Clifford identity \eqref{eq-red_Clifford_property}, we obtain the inclusion
\begin{align*}
\mfS^{-\frac{m-4}{2}}  \subset \mfS^{-\frac{m-2}{2}}  \subset \mfS^{-\frac{m}{2}} \, .
\end{align*}
Dual to \eqref{eq-iso}, \eqref{eq-iso2} and \eqref{eq-iso3}, we have
\begin{align}
\mfV^0 & \cong \mfS^{\frac{m}{2}} \otimes \left( \mfS^{-\frac{m}{2}} / \mfS^{-\frac{m-4}{2}} \right) \, , \label{eq-dual_iso2-0} \\
\mfV^0 / \mfV^1 & \cong \mfS^{\frac{m}{2}} \otimes \left( \mfS^{-\frac{m}{2}} / \mfS^{-\frac{m-2}{2}} \right) \, ,
\label{eq-dual_iso2-0/1} \\
\mfV^1 & \cong \mfS^{\frac{m}{2}} \otimes \left( \mfS^{-\frac{m-2}{2}} / \mfS^{-\frac{m-4}{2}} \right) \, , 
\label{eq-dual_iso2-1}
\end{align}
which can be expressed concretely as follows.
\begin{lem}\label{lem-vector_decomposition}
 Let $V^a$ be a non-zero vector in $\mfV$. Then
 \begin{itemize}
 \item $V^a$ is an element of $\mfV^0$ if and only if $V^a = \xi \ind{^a^A} v \ind{_A}$ for some non-zero $v_A \in \mfS^{-\frac{m}{2}} / \mfS^{-\frac{m-4}{2}}$;
 \item $V^a$ is an element of $\mfV^1$ if and only if $V^a = \xi \ind{^a^A} v \ind{_A}$ for some non-zero $v_A \in \mfS^{-\frac{m-2}{2}} / \mfS^{-\frac{m-4}{2}}$.
 \end{itemize}
\end{lem}
As a direct consequence, a pure spinor $\xi^A$ must satisfy $\xi \ind{^a^A} \xi \ind{_a^B} = \lambda \, \xi^A \xi^B$ for some $\lambda$. Contracting each side by $\xi \ind{^c^C} \gamma \ind{_c_A^D}$ and a little algebra then lead to $\lambda=-1$.
\begin{prop}\label{prop-purity_cond}
A spinor $\xi^A$ is pure if and only if it satisfies
\begin{align}\label{eq-purity_cond}
 \xi \ind{^a ^C} \xi \ind{_a ^D} & = - \xi^C \xi^D \, .
\end{align}
\end{prop}
By Lemma \ref{lem-technical}, we can express express \eqref{eq-purity_cond} equivalently as the more familiar algebraic characterisation of pure spinors due to Cartan \cite{Cartan1981}.
\begin{prop}[Cartan \cite{Cartan1981}]\label{prop-Cartan_char}
A non-zero spinor $\xi^A$ is pure if and only if it satisfies
\begin{align}\label{eq-purity_cond_Cartan}
\begin{aligned}
 \gamma \ind{_{a_1 \ldots a_p} _{A B}} \xi^A \xi^B & = 0 \, , & \mbox{for all $p<m$, $m-p \equiv 0,3 \pmod 4$,} \\
 \gamma \ind{_{a_1 \ldots a_m} _{A B}} \xi^A \xi^B & \neq 0 \, .
 \end{aligned}
\end{align}
\end{prop}
We shall refer to both equations \eqref{eq-purity_cond} and \eqref{eq-purity_cond_Cartan} as the \emph{purity conditions} of a spinor $\xi^A$. These are vacuous when $m\leq2$, i.e. all spinors are pure when $m \leq 2$.

The only non-vanishing irreducible component of the tensor product $\xi^A \xi^B$ is thus the $m$-form
$\phi \ind{_{a_1 \ldots a_m}} := \gamma \ind{_{a_1 \ldots a_m}_{AB}} \xi \ind{^A} \xi \ind{^B}$, which can be seen to annihilate $\mfV^0$. It is \emph{null} (or \emph{simple} or \emph{decomposable}) in the sense that
\begin{align*}
 \phi \ind{_{a_1 \ldots a_m}} & = \xi \ind{_{a_1}^{A_1}} \ldots \xi \ind{_{a_m}^{A_m}} \varepsilon \ind{_{A_1 \ldots A_m}} \quad \in \quad \wedge^m \mfV^1 \, ,
\end{align*}
for some $\varepsilon \ind{_{A_1 \ldots A_m}} \in \wedge^m \left(\mfS^{-\frac{m-2}{2}} / \mfS^{-\frac{m-4}{2}} \right)$. Its Hodge dual annihilates $\mfV^1$ and takes the form
\begin{align*}
 (* \phi ) \ind{_{a_1 \ldots a_{m+1}}} & = \xi \ind{_{a_1}^{A_1}} \ldots \xi \ind{_{a_{m+1}}^{A_{m+1}}} \varepsilon \ind{_{A_1 \ldots A_{m+1}}} \quad \in \quad \wedge^{m+1} \mfV^0 \, ,
\end{align*}
for some $\varepsilon \ind{_{A_1 \ldots A_{m+1}}} \in \wedge^{m+1} \left(\mfS^{-\frac{m}{2}} / \mfS^{-\frac{m-4}{2}} \right)$.

\paragraph{Intersection of $\gamma$-planes}
\begin{prop}[\cite{Cartan1981}]\label{prop-pure-intersect}
Let $\alpha^A$ and $\beta^A$ be two spinors not proportional to each other. Then the $\gamma$-planes associated to $\alpha^A$ and $\beta^A$ intersect in a totally null $(m-k)$-plane if and only if
 \begin{align*}
  \gamma \ind{_{a_1 \ldots a_p}_{AB}} \alpha^A \beta^B & = 0 \, , & \mbox{for all $p < m-k$,} \\
  \gamma \ind{_{a_1 \ldots a_{m-k}}_{AB}} \alpha^A \beta^B & \neq 0 \, .
 \end{align*}
for $k=1, \ldots m$.
\end{prop}
As a consequence of Lemma \ref{lem-technical}, we have, in the special case when $k=1,2$, the equivalent characterisations.
\begin{prop}\label{prop-pure-intersect2}
Let $\alpha^A$ and $\beta^A$ be two spinors not proportional to each other. Then 
\begin{itemize}
\item the $\gamma$-planes associated to $\alpha^A$ and $\beta^A$ intersect in a totally null $(m-1)$-plane if and only if
\begin{align}\label{eq-intersect-m-1-short}
  \alpha \ind{^a^A} \beta \ind{_a^B} & = \alpha^A \beta^B - 2 \, \beta^A \alpha^B = - \alpha^{(A} \beta^{B)} + 3 \, \alpha^{[A} \beta^{B]} \, ;
 \end{align}
\item the $\gamma$-planes associated to $\alpha^A$ and $\beta^A$ intersect in a totally null $(m-k)$-plane where $k=1$ or $2$, if and only if
\begin{align}\label{eq-intersect-m-2-short}
 \alpha \ind{^a^{(A}} \beta \ind{_a^{B)}} & = - \alpha \ind{^{(A}} \beta \ind{^{B)}} \, .
\end{align}
\end{itemize}
\end{prop}
Finally, in the context of our present notation, we conclude
\begin{cor}\label{cor-pure-intersect}
Let $\xi^A$ be a pure spinor in $(\mfV,g)$ and let $\mfS^{\frac{m-2}{2}} := \im \xi \ind{_a^A} : \mfV \rightarrow \mfS$ as before. Then
\begin{itemize}
\item Any spinor in $\mfS^{\frac{m-2}{2}}$ is pure.
\item The $\gamma$-planes associated to any two pure spinors in $\mfS^{\frac{m-2}{2}}$ intersect in a totally null $(m-k)$-plane where $k$ can be either $0$ or $1$ or $2$.
\end{itemize}
\end{cor}
We omit the proof which is essentially the same as in the even-dimensional case and consists in checking the veracity of the algebraic conditions \eqref{eq-purity_cond}, \eqref{eq-intersect-m-1-short} and \eqref{eq-intersect-m-2-short}


\begin{rem}
The last part of Corollary \ref{cor-pure-intersect} is a restatement of a standard result that the sum of two pure spinors is again pure if and only if their corresponding $\gamma$-planes intersect in an $(m-k)$-plane where $k$ is at most $2$.
\end{rem}

\paragraph{Splitting}
For clarity, we can split the filtration \eqref{eq-filtration} as a direct sum
\begin{align}\label{eq-null-grading}
 \mfV & = \mfV_{-1} \oplus \mfV_{0} \oplus \mfV_1 \, ,
\end{align}
for some choice of vector subspaces $\mfV_{-1} \subset \mfV^{-1}$ complementary to $\mfV^0$ and $\mfV_0 \subset \mfV^0$ complementary to $\mfV^1$, and where $\mfV_1 := \mfV^{-1}$. These are linearly isomorphic to $\mfV^{-1}/\mfV^0$ and $\mfV^0/\mfV^1$ respectively. Now, $\mfV_{-1}$ is a totally null $m$-plane dual to $\mfV^1$, with orthogonal complement $(\mfV_{-1})^\perp = \mfV_{-1} \oplus \mfV_0$. We can therefore associate to $\mfV_{-1}$ a pure spinor $\eta_A$, say, dual to $\xi^A$, i.e.\footnote{In fact, $\mfV_{-1}$ annihilates the line $\langle \eta_A \rangle =: \mfS_{-\frac{m}{2}} \subset \mfS^{-\frac{m}{2}}$ complementary to $\mfS^{-\frac{m-2}{2}}$.} 
\begin{align*}
\mfV_{-1} & := \ker \eta \ind{_a_A} : \mfV \rightarrow \mfS^* \, ,
\end{align*}
where $\eta \ind{_a_A} := \eta_B \gamma \ind{_a_A^B}$. Conversely, any choice of spinor dual to $\xi^A$ induces a splitting \eqref{eq-null-grading}.

With no loss, we choose $\xi^A$ and $\eta_A$ to satisfy $\xi^A \eta_A = -\frac{1}{2}$. To describe elements of $\mfV_{-1}$ and $\mfV_0$, we define
\begin{align*}
\mfS_{-\frac{m-2}{2}} & := \{ \im \eta \ind{_a_A} : \mfV \rightarrow \mfS^* \} \cap \{ \ker \xi^A : \C \leftarrow \mfS^* \} \, ,
\end{align*}
which is dual to $\mfS_{\frac{m-2}{2}}$, the complement of $\mfS^{\frac{m}{2}} = \langle \xi^A \rangle$ in $\mfS^{\frac{m-2}{2}}$. Any element of $\mfV_{-1}$ must be of the form $V^a = \eta \ind{^a_A} v^A$ for some $v^A$ in $\mfS_{\frac{m-2}{2}}$. On the other hand, $\mfV_0$ is spanned by the vector $u^a := 2 \, \eta \ind{^a_A} \xi^A$, which can be seen to square to $-1$ and satisfies $u^a \xi \ind{_a^A} = \xi^A$ and $u^a \eta \ind{_a_A} = \eta_A$.

To make the pairing between $\mfS_{\frac{m-2}{2}}$ and $\mfS_{-\frac{m-2}{2}}$ explicit, we introduce
the map
\begin{align}\label{eq-identity-map}
I \ind*{_A^B} & := \eta \ind{_a_A} \xi \ind{^a^B} + \eta \ind{_A} \xi \ind{^B}: \mfS \rightarrow \mfS \, .
\end{align}
which can be seen to be idempotent with trace $I \ind*{_A^A}=m$, and degenerate on $\mfS^{\frac{m}{2}}$, i.e. $\xi \ind{^A} I \ind*{_A^B}=0$. Thus, $I \ind*{_A^B}$ is the identity element on $\mfS_{\frac{m-2}{2}}$, or dually, on $\mfS_{-\frac{m-2}{2}}$. It is the spinorial counterpart of the symmetric bilinear form $h \ind{_{ab}} := g \ind{_{ab}} + u \ind{_a} u \ind{_b}$ on $\mfV$, which satisfies $h_{ab} u^a = 0$, $h \ind{_a^c} h \ind{_c^b} = h \ind{_a^b}$, and $h \ind{_a^a} = n-1$.


\subsection{Spinorial description of the Lie algebra $\so(2m+1,\C)$}
We can extend the above calculus in the classification of the Lie algebra $\g :=\so(2m+1,\C)$, or equivalently to the space of $2$-forms $\wedge^2 \mfV$.  Again, $\xi^A$ is a pure spinor, with induced filtration \eqref{eq-filtration} on $\mfV$. At this stage, we also assume $n>3$, i.e. $m>1$, the case $n=3$ being somewhat more special and treated briefly in section \ref{sec-3dim}.
\paragraph{Filtration}
The filtration \eqref{eq-filtration} induces a filtration of vector subspaces
there is a filtration 
\begin{align*}
 \{ 0 \} =: \g^3 \subset \g^2 \subset \g^1 \subset \g^0 \subset \g^{-1} \subset \g^{-2} := \g \, ,
\end{align*}
on $\g$, where
\begin{align*}
 \g^{-1} & := \{ \phi \ind{_{ab}} \in \g : \xi \ind{^a ^{\lb{A}}} \xi \ind{^b ^B} \phi \ind{_{a b}} \xi \ind{^{\rb{C}}} = 0 \} \, ,&
 \g^0 & := \{ \phi \ind{_{ab}} \in \g : \xi \ind{^a ^A} \xi \ind{^b ^B} \phi \ind{_{a b}} = 0 \} \, , \\
 \g^1 & := \{ \phi \ind{_{ab}} \in \g : \xi \ind{^a ^{\lb{A}}} \phi_{ab} \xi \ind{^{\rb{B}}} = 0 \} \, , &
 \g^2 & := \{ \phi \ind{_{ab}} \in \g : \xi \ind{^a ^A} \phi_{ab} = 0 \} \, .
\end{align*}
The Lie bracket $[ \cdot , \cdot ]:\g \times \g \rightarrow \g$ on $\g$ is compatible with this filtration, i.e. $[\g^i,\g^j] \subset \g^{i+j}$, with the convention that $\g^i = \{ 0 \}$ for $i \geq 3$, and $\g^i = \g$ for all $i \leq -2$, i.e. $\g$ is a filtered Lie algebra.


\begin{prop}
The Lie subalgebra $\prb := \g^0$ is the stabilizer of $\langle \xi^A \rangle$, i.e.
\begin{align*}
\phi \ind{_{ab}} \xi \ind{^{ab}^A} & \propto \xi \ind{^A} \, .
\end{align*}
\end{prop}

\begin{proof}
We only need to show $\phi \ind{_{ab}} \xi \ind{^a^A} \xi \ind{^b^B} = 0$ if and only if $\phi \ind{_{ab}} \xi \ind{^{ab}^{\lb{A}}} \xi \ind{^{\rb{B}}} = 0$.
To see this, we use the identities
\begin{align}
 \phi \ind{_{ab}} \xi \ind{^a^A} \xi \ind{^b^B} \gamma \ind{^c_A^C} \gamma \ind{_c_B^D} & = - \phi \ind{_{ab}} \left( \xi \ind{^a^C} \xi \ind{^b^D} + 4 \, \xi \ind{^{ab}^{\lb{C}}} \xi \ind{^{\rb{D}}} \right) \, , \\
 \phi \ind{_{ab}} \xi \ind{^{ab}^A} \gamma \ind{^c_A^C} \xi \ind{_c^D} & = - \phi \ind{_{ab}} \left( \xi \ind{^{ab}^C} \xi \ind{^D} + 4 \, \xi \ind{^a^C} \xi \ind{^b^D} \right) \, . \label{eq-useful-id}
\end{align}
The first of these gives $\Rightarrow$ immediately, while the second gives $\Leftarrow$ by rewriting the LHS as $\phi \ind{_{ab}} \xi \ind{^{ab}^A} = \phi \xi \ind{^A}$ for some $\phi$.
\end{proof}

As in even dimensions, the stabiliser $\prb$ of a projective pure spinor is a \emph{parabolic Lie subalgebra} of $\g$ \cites{Fulton1991,vCap2009}.


\paragraph{Splitting}
The splitting \eqref{eq-null-grading} extends to tensor products of $\mfV$ and $\mfV^*$, and in particular, the Lie algebra $\g$ admits the structure of $|2|$-graded Lie algebra
\begin{align*}
 \g & = \g_{-2} \oplus \g_{-1} \oplus \g_0 \oplus \g_1 \oplus \g_2 \, , & [ \g_i , \g_j ] \subset \g_{i+j} \, ,
\end{align*}
for all $i,j$, where $\g_i \subset \g^i$ is complementary to $\g^{i-1}$ in $\g^i$, with the convention that $\g_i = \{ 0 \}$ for all $|i|>2$. We can make the identification
\begin{align*}
 \g_{-2} & \cong \wedge^2 \mfV_{-1} \, , & \g_{-1} & \cong \mfV_0 \otimes \mfV_{-1} \, , & \g_0 & \cong \mfV_{-1} \otimes \mfV_1 \, , & \g_1 & \cong \mfV_0 \otimes \mfV_1 \, , & \g_2 & \cong \wedge^2 \mfV_1 \, .
\end{align*}
Now, if $\eta_A$ is dual to $\xi^A$, and so annihilates $\mfV_{-1}$, and $u^a = 2 \, \xi \ind{^a^A} \eta_A$, one can give a spinorial decomposition of any element $\phi_{ab}$ of $\g$ as 
\begin{align*}
 \phi \ind{_{ab}} & = \xi \ind{_a^A} \xi \ind{_b^B} \phi \ind{_{AB}} - 2 \, \xi \ind{_{\lb{a}}^A} u \ind{_{\rb{b}}} \phi \ind{_A} + 2 \, \xi \ind{_{\lb{a}}^A} \eta \ind{_{\rb{b}}_B} \phi \ind{_A^B} - 2 \, \eta \ind{_{\lb{a}}_A} u \ind{_{\rb{b}}} \phi \ind{^A} + \eta \ind{_a_A} \eta \ind{_b_B} \phi \ind{^{AB}} \\
& \in \quad \g_2 \oplus \g_1 \oplus \g_0 \oplus \g_{-1} \oplus \g_{-2} \, ,
\end{align*}
where $\phi \ind{_{AB}} = \phi \ind{_{[AB]}}$, $\phi \ind{^{AB}} = \phi \ind{^{[AB]}}$, $\phi \ind{_A^B}$, $\phi_A$ and $\phi^A$ are all elements of $\mfS_{\frac{m-2}{2}}$ and $\mfS_{-\frac{m-2}{2}}$, and tensor products thereof. Here, $\phi \ind{_{AB}}$, $\phi \ind{^{AB}}$ and $\phi \ind{_A^B}$ are independent quantities in the sense that indices should be treated as `immovable' $\g_0$-indices -- the same applies to $\phi_A$ and $\phi^A$.

The Lie subalgebra $\g_0$ is isomorphic to $\glie(m,\C)$, and thus splits further as $\g_0 = \mfz_0 \oplus \slie_0$ where $\mfz_0$ is the center $\g_0$ and $\slie_0 = \slie(m,\C)$ is the semisimple part of $\g_0$. The center is spanned by the \emph{grading element}
\begin{align*}
E \ind{_{ab}} &:= - 2 \, \xi \ind{_{\lb{a}}^A} \eta \ind{_{\rb{b}}_A} = - 2 \, \xi \ind{_{a}^A} \eta \ind{_{b}_A}  + g_{ab} \, ,
\end{align*}
and from the identity  $\xi \ind{^a^A} E \ind{_a^b} = \xi \ind{^A} u^b + \xi \ind{^b^B}$, one can easily deduce that $E \ind{_a^b}$ has eigenvalues $i$ on $\mfV_i$. Its image in the Clifford algebra is given by $E \ind{_A^B} := - \frac{1}{4} E \ind{_{ab}} \gamma \ind{^{ab}_A^B}$, and a straightforward computation yields $E \ind{_B^A} \xi^B = \frac{m}{2} \xi^A$, and similarly for the dual action.

For consistency with reference \cite{Taghavi-Chabert2012a}, we set $\omega \ind{_{ab}} := E \ind{_{ab}}$. Then an element of $\g_0$ can be expressed as
\begin{align*}
 \phi \ind{_{ab}} & = \phi \, \omega \ind{_{ab}} + 2 \, \xi \ind{_{\lb{a}}^A} \eta \ind{_{\rb{b}}_B} \phi \ind{_A^B} 
\end{align*}
for some complex $\phi$ and tracefree $\phi \ind{_A^B}$, i.e. $\phi \ind{_A^B} I \ind*{_B^A} = 0$ where $I_A^B = \xi \ind{^a^B} \eta \ind{_a_A} + \xi^B \eta_A$ is the identity element on $\mfS_{\frac{m-2}{2}}$ (see \eqref{eq-identity-map}).

\paragraph{Associated graded vector space}
We now introduce the associated graded $\prb$-module $\gr(\g) = \bigoplus_{i=-2}^2 \gr_i (\g)$ where $\gr_i (\g) = \g^i/\g^{i+1}$. Each $\gr_i(\g)$ is linear isomorphic to the $\g_0$-module $\g_i$, and we have a direct sum decomposition
\begin{align*}
\gr_0(\g) & = \g_0^0 \oplus \g_0^1 \, , & & \mbox{where} & \g_0^0 & := \left(\g^1 + \mfz_0 \right) / \g^1 \, , &  \g_0^1 & := \left(\g^1 + \slie_0 \right) / \g^1 \, .
\end{align*}
For convenience, we also set $\g_i^0 := \gr_i (\g)$ for $i=\pm1,\pm2$.

The additional information needed to describe $\g$ in terms of irreducible $\prb$-modules comes from the (non-trivial) action of the nilpotent part of $\prb$, or more simply $\g_1$ on the $\g_0$-modules $\g_i^j$. This can be encoded by means of the directed graph (a `Penrose diagram')
\begin{align*}
 \xymatrix@R=1em{ & & \g_0^0 \ar[dr] & & \\
\g_{2}^0 \ar[r] & \g_1^0 \ar[ur] \ar[dr] &  & \g_{-1}^0 \ar[r] & \g_{-2}^0 \\
& & \g_0^1 \ar[ur] & & }
\end{align*}
where an arrow from $\g_i^j$ to $\g_{i-1}^k$ simply means that $\breve{\g}_i^j \subset \g_1 \cdot \breve{\g}_{i-1}^k$,
for any irreducible $\g_0$-modules $\breve{\g}_i^j$ linearly isomorphic to $\g_i^j$.

\begin{rem}
An alternative interpretation of an arrow $\g_i^j \rightarrow \g_{i-1}^k$ is as inclusions of $\prb$-modules
\begin{align}\label{eq-inclusions-p-g}
 \g^2 \subset \g^1 \subset \g^1 + \mfz_0 \subset \g^0 \subset \g^{-1} \subset \g^{-2} \, , & & \g^2 \subset \g^1 \subset \g^1 + \slie_0 \subset \g^0 \subset \g^{-1} \subset \g^{-2} \, .
\end{align}
We can make this explicit by defining, for any $\phi_{ab} \in \g$,
\begin{align*}
{}^\g  _\xi \Pi^0_{-2} ( \phi ) & := \xi \ind{^a ^{\lb{A}}} \xi \ind{^b ^B} \phi \ind{_{a b}} \xi \ind{^{\rb{C}}} \, , &
{}^\g  _\xi \Pi^0_{-1} ( \phi )  & := \xi \ind{^a ^A} \xi \ind{^b ^B} \phi \ind{_{a b}} \, , \\
{}^\g  _\xi \Pi^0_0 ( \phi )  & := \xi \ind{^{ab} ^A} \phi_{ab} \, , &
 {}^\g _\xi \Pi^1_0 ( \phi )  & := \xi \ind{^a ^{\lb{A}}} \phi_{ab} \xi \ind{^{\rb{B}}} + \frac{1}{n-1} \xi \ind{^{ac} ^C} \phi _{ac} \gamma \ind{_b _C ^{\lb{A}}} \xi \ind{^{\rb{B}}} \, , \\
{}^\g  _\xi \Pi^0_1 ( \phi )  & := \xi \ind{^a ^A} \phi_{ab} \, ,
\end{align*}
where $\xi \ind{_{ab}^A} := \xi \ind{^B} \gamma \ind{_{ab}_B^A} : \wedge^2 \mfV \rightarrow \mfS$ and $\mfS^{\frac{m-4}{2}} := \im \xi \ind{_{ab}^A} : \wedge^2 \mfV \rightarrow \mfS$. Then the kernels of the maps ${}^\g  _\xi \Pi^0_{-1}$ correspond to the $\prb$-submodules
\begin{align*}
\g^{-1} & = \{ \phi_{ab} \in \g : {}^\g  _\xi \Pi^0_{-2} ( \phi ) = 0 \} \, , &
\g^0 & = \{ \phi_{ab} \in \g : {}^\g  _\xi \Pi^0_{-1} ( \phi ) = 0 \} \, , \\
\g^1 + \mfz_0 & = \{ \phi_{ab} \in \g : {}^\g  _\xi \Pi^1_0 ( \phi ) = 0 \} \, , &
\g^1 + \slie_0 & = \{ \phi_{ab} \in \g : {}^\g  _\xi \Pi^0_0 ( \phi ) = 0 \} \, , \\
\g^1 & = \{ \phi_{ab} \in \g : {}^\g  _\xi \Pi^0_0 ( \phi ) = {}^\g  _\xi \Pi^1_0 ( \phi ) = 0 \} \, , &
\g^2 & = \{ \phi_{ab} \in \g : {}^\g  _\xi \Pi^0_{-1} ( \phi ) = 0 \} \, ,
\end{align*}
in the filtrations \eqref{eq-inclusions-p-g}. Passing now to the associated graded module $\gr(\g)$, the irreducibles $\g_i^j$ can be expressed in terms of ${}^\g _\xi \Pi_i^j$, eg.
\begin{align}\label{eq-g-pmod}
\g_0^0 & := \left\{ \phi \ind{_{ab}} \in \g^0 : {}^\g _\xi \Pi_0^1 (\phi) = 0 \right\} / \g^1 \, , &
\g_0^1 & := \left\{ \phi \ind{_{ab}} \in \g^0 : {}^\g _\xi \Pi_0^0 (\phi) = 0 \right\} / \g^1 \, ,
\end{align}
and similarly for the remaining $\g_i^j$. The irreducibility of $\g_i^j$ follows from the fact that the maps ${}^\g _\xi \Pi_i^j$ are saturated with symmetries.
\end{rem}

\paragraph{Parabolic Lie subgroups}
We can translate the description of the Lie algebra $\g$ and its parabolic Lie subalgebra $\prb$ to the Lie group level. We shall take $G$ to be the complex special orthogonal group $\SO(2m+1,\C)$. Then the \emph{parabolic Lie subgroup $P$} of $G$ stabilising the pure spinor $\xi^A$ can simply be obtained by exponentiating $\prb$. In particular, it admits a Levy decomposition $P = G_0 \ltimes P_+$ where $G_0$ is isomorphic to the complex general linear group $\GL(m,\C)$ and $P_+$ is the nilpotent Lie group generated by $\g_1 \oplus \g_2$. The spinor calculus developed here is then manifestly $P$-invariant.

Denote the null (or isotropic) Grassmanian $\Gr_m (\mfV,g)$ of $\gamma$-planes in $\mfV$. By definition, $\Gr_m (\mfV,g)$ can also be described as the space of projective pure spinors of $(\mfV,g)$, i.e. a compact complex subvariety of $\Pp \mfS$ defined by the purity conditions \eqref{eq-purity_cond_Cartan}. Since the action of $\SO(2m+1,\C)$ is transitive on $\Gr_m (\mfV,g)$ and $P$ stabilises a point in $\Gr_m (\mfV,g)$, the space of projective pure spinors of $(\mfV,g)$ is simply the $\frac{1}{2}m(m+1)$-dimensional homogeneous space $G/P$. When $m=1,2$, this space is isomorphic to the complex projective space $\CP^{\frac{1}{2}m(m+1)}$. See \cites{Fulton1991,Harnad1995} for more details.

\begin{rem}
By Proposition \ref{prop-pure-intersect}, each (non-projective) pure spinor defines an $(m+1)$-dimensional vector subspace of non-projective spinors. Therefore, to each point in $\Gr_m(\mfV,g)$ corresponds an $m$-dimensional complex projective space $\CP^m$ in $\Gr_m (\mfV,g)$. This is trivial when $m=1$, and when $m=2$, this is simply the hyperplane dual to a point in $\CP^3$.
\end{rem}


\subsection{Generalisation to $\so(2m+1,\C)$-modules}
The filtration \eqref{eq-filtration} of $\prb$-modules defined by the projective pure spinor $\langle \xi \ind{^A} \rangle$ induces a filtration
\begin{align*}
\{ 0 \} = \mfR^{p+1} \subset \mfR^p \subset \mfR^{p-1} \subset \ldots \subset \mfR^{-p+1} \subset \mfR^{-p} = \mfR
\end{align*}
of $\prb$-modules on $\mfR := \bigotimes^p \mfV$ for any $p \geq 0$, where
\begin{align*}
\mfR^j := \sum_{i_1 + \ldots + i_p = j} \mfV^{i_1} \otimes \ldots \otimes \mfV^{i_p} \, .
\end{align*}
We can then say that $\xi^{A'}$ is a \emph{$k$-fold principal spinor} of a tensor $T \in \mfR$, if $T$ lies in $\mfR^{-p+k}$, but not in $\mfR^{-p+k+1}$. However, as in the even-dimensional, it is best to replace the concept of principal spinors with 
that of $P$-invariant filtrations on $\g$-modules. 

The filtration on the spinor module $\mfS$ can be described as follows. Given a (projective) pure spinor $\xi \ind{^A}$, we can define the maps
\begin{align*}
 \xi \ind{_{a_1 \ldots a_k}^A} & := \xi \ind{^B} \gamma \ind{_{a_1 \ldots a_{2k+1}}_B ^A} : \wedge^k \mfV \rightarrow \mfS \, .
\end{align*}
By Hodge duality, it is enough to consider values of $k$ ranging from $0$ to $m$, with the understanding that $\xi \ind{^A} : \wedge^0 \mfV \cong \C \rightarrow \mfS$. By the Clifford property \eqref{eq-red_Clifford_property}, we can conclude that the spinor module $\mfS \cong \mfS^*$ admits a $P$-invariant filtration
\begin{align*}
  \mfS^{\frac{m}{2}} \subset \mfS^{\frac{m-2}{2}} \subset \ldots \subset \mfS^{-\frac{m-2}{2}} \subset \mfS^{-\frac{m}{2}}  := \mfS \, ,
\end{align*}
where we have defined
\begin{align*}
 \mfS^{\frac{m-2k}{2}} & := \im \xi \ind{_{a_1 \ldots a_k}^A} : \wedge^k \mfV \rightarrow \mfS \, , &
 \mfS^{-\frac{m-2k}{2}} & = \ker \xi \ind{_{a_1 \ldots a_k} ^A} : \wedge^k \mfV^* \leftarrow \mfS^* \, .
\end{align*}
Further, we can choose subspaces $\mfS_i \subset \mfS^i$ such that $\mfS^i = \mfS_i \oplus \mfS^{i+1}$ such that
\begin{align*}
 \mfS & = \mfS_{\frac{m}{2}} \oplus \mfS_{\frac{m-2}{2}} \oplus \ldots \oplus \mfS_{-\frac{m-2}{2}} \oplus \mfS_{-\frac{m}{2}} \, .
\end{align*}
The grading element $E \ind{_{ab}}$ in $\mfz_0$ and the spanning element $u_a$ of $\mfV_0$ have eigenvalues $\frac{m-2k}{2}$ and $(-1)^k$ respecively on $\mfS_{\frac{m-2k}{2}}$.

\begin{rem}
 In the form description of the spinor representation $\mfS$ given in Remarks \ref{rem-spinor-form1} and  \ref{rem-spinor-form2}, we can make the identification $\mfS_{\frac{m-2i}{2}} \cong \wedge^{m-i} \mfN$.
\end{rem}

The generalisation of isomorphism \eqref{eq-dual_iso2-1} is given by
\begin{align*}
 \wedge^k \mfV^1 & \cong \mfS^{\frac{m}{2}} \otimes \left( \mfS^{-\frac{m-2k}{2}}/\mfS^{-\frac{m-2k-2}{2}} \right) \, , & k & = 0 \, , \ldots \, , m-1\, , \\
 \wedge^m \mfV^1 \cong \wedge^m \mfV^1 \otimes \mfV^0 & \cong \mfS^{\frac{m}{2}} \otimes \mfS^{\frac{m}{2}} \, .
\end{align*}
When $k=2$, one has in particular
\begin{align*}
\left( \g^{-2} / \g^{-1} \right) \otimes \mfS^{\frac{m}{2}} & \cong \mfS^{\frac{m-4}{2}} / \mfS^{\frac{m-2}{2}} \, , &
\g^2 & \cong \mfS^{\frac{m}{2}} \otimes \left( \mfS^{-\frac{m-4}{2}} / \mfS^{-\frac{m-6}{2}} \right) \, , \\
\left( \g^{-1} / \g^0 \right) \otimes \mfS^{\frac{m}{2}} & \cong \mfS^{\frac{m-2}{2}} / \mfS^{\frac{m}{2}} \, &
\g^{-1} & \cong \mfS^{\frac{m}{2}} \otimes \left( \mfS^{-\frac{m-2}{2}} / \mfS^{-\frac{m-4}{2}} \right) \, .
\end{align*}


\subsection{Real pure spinors}\label{sec-real}
As in even dimensions, the above calculus applies equally in the case where $(\mfV,g)$ is a real vector space equipped with a non-degenerate bilinear form of signature $(m,m+1)$ or $(m+1,m)$, and our choice of pure spinor is \emph{real}, so that its associated $\gamma$-plane is a real totally null subspace of $\mfV$. The description of its stabiliser $\prb$ as a real parabolic Lie subalgebra of $\so(m,m)$, and similarly for its Lie group $P$ as a subgroup of the connected identity component of $\SO(m,m)$.

If the bilinear form $g_{ab}$ is now taken to be positive definite, one obtains a metric-compatible CR structure, which is none other than an odd-dimensional analogue of a Hermitian structure.

If the bilinear form $g_{ab}$ is of Lorentzian signature, i.e. $(1,2m)$ or $(2m,1)$, the resulting geometry is that of a \emph{Robinson structure} \cites{Nurowski2002,Trautman2002} -- this is treated in more details in \cite{Taghavi-Chabertb} -- or a metric-compatible CR structure.



\section{Algebraic classifications of curvature tensors}\label{sec-curvature}
Throughout $\g := \so(2m+1,\C)$ and $\prb$ is the parabolic Lie subalgebra of $\g$ stabilising a (positive) projective pure spinor $\langle \xi^A \rangle$ as described in section \ref{sec-algebra}. As before, we assume $n>3$, i.e. $m>1$, relegating the case $n=3$ to appendix \ref{sec-3dim}. We now apply the ideas developed in the previous section to the classifications of the $\g$-modules $\mfF$, $\mfA$ and $\mfC$ of tracefree symmetric $2$-tensors, of tensors with Cotton-York symmetries and of tensors with Weyl symmetries respectively, i.e.
\begin{align*}
 \mfA & := \{ A_{abc} \in \otimes^3 \mfV : A_{abc} = A_{a[bc]} \, , A_{[abc]} = 0 \, , A \ind{^a_{ac}} = 0 \} \, , \\
 \mfF & := \{ \Phi_{ab} \in \otimes^2 \mfV^* : \Phi_{ab} = \Phi_{(ab)} \, , \Phi \ind{_c^c} = 0 \} \, , \\
 \mfC & := \{ C_{abcd} \in \otimes^4 \mfV : C_{abcd} = C_{[ab][cd]} \, , C_{[abc]d} = 0 \, , C \ind{^a_{bad}} = 0 \} \, .
\end{align*}
From a representation theoretic viewpoint, the $\prb$-invariant decomposition of tensors can be derived the same algorithm as in the even-dimensional case. This is explained in details in \cite{Taghavi-Chabert2012a} -- see also \cite{vCap2009}. One can also describe these irreducibles explictly in terms of kernels of certain projection maps, denoted ${}^\mfF _\xi \Pi_i^j$, ${}^\mfA _\xi \Pi_i^j$ and ${}^\mfC _\xi \Pi_i^j$ below, but compared to the even-dimensional case, these maps are somewhat a little more `baroque'.

\subsection{Classification of the tracefree Ricci tensor}
\begin{prop}\label{prop-TFRicci-classification}
The space $\mfF$ of tracefree symmetric $2$-tensors admits a filtration
\begin{align*}
\{ 0 \} =: \mfF^3 \subset \mfF^2 \subset \mfF^1 \subset \mfF^0 \subset \mfF^{-1} \subset \mfF^{-2} := \mfF \, ,
\end{align*}
of $\prb$-modules
\begin{align*}
 \mfF^i & := \{ \Phi \ind{_{ab}} \in \mfF : {}^\mfF _\xi \Pi_{i-1}^k (\Phi) = 0 \, , \, \, \mbox{for all $k$} \} \, , & i=-1,0,1,2,
\end{align*}
where the maps ${}^\mfF _\xi \Pi_i^j$ are defined in appendix \ref{sec-spinor-descript}.
 
The associated graded $\prb$-module $\gr(\mfF)=\bigoplus_{i=-2}^2 \gr_i(\mfF)$, where $\gr_i(\mfF) := \mfF^i/\mfF^{i+1}$, splits into a direct sum
\begin{align*}
 \gr_{\pm2} (\mfF) & = \mfF_{\pm2}^0 \, , & \gr_{\pm1} (\mfF) & = \mfF_{\pm1}^0 \, , &
 \gr_0 (\mfF) & = \mfF_0^0 \oplus \mfF_0^1 \, , 
\end{align*}
of irreducible $\prb$-modules $\mfF_i^j$ as described in Table \ref{table-F} of appendix \ref{sec-rep-theory}. Further,
\begin{align*}
 \mfF_0^j & := \{ \Phi \ind{_{ab}} \in \mfF^0 : {}^\mfF _\xi \Pi_0^k(\Phi) = 0 \, , \, \, \mbox{for $k \neq j$} \} / \mfF^1 \, .
\end{align*}


Finally, the $\prb$-module $\gr(\mfF)$ can be expressed by means of the directed graph
\begin{equation*}
 \xymatrix@R=1em{ & & \mfF_0^1 \ar[dr] & & \\
\mfF_2^0 \ar[r] & \mfF_1^0 \ar[ur] \ar[dr] &  & \mfF_{-1}^0 \ar[r] & \mfF_{-2}^0 \\
& & \mfF_0^0 \ar[ur] & & }
\end{equation*}
where an arrow from $\mfF_i^j$ to $\mfF_{i-1}^k$ for some $i,j,k$ implies that $\breve{\mfF}_i^j \subset \g_1 \cdot \breve{\mfF}_{i-1}^k$ for any choice of irreducible $\g_0$-modules $\breve{\mfF}_i^j$ and $\breve{\mfF}_{i-1}^k$ isomorphic to $\mfF_i^j$ and $\mfF_{i-1}^k$ respectively.
\end{prop}

\subsection{Classification of the Cotton-York tensor}
\begin{prop}\label{prop-Cotton-York-classification}
The space $\mfA$ of tensors with Cotton-York symmetries admits a filtration
\begin{align*}
\{ 0 \} =: \mfA^4 \subset \mfA^3 \subset \mfA^2 \subset \mfA^1 \subset \mfA^0 \subset \mfA^{-1} \subset \mfA^{-2} \subset \mfA^{-3} := \mfA \, ,
\end{align*}
of $\prb$-modules
\begin{align*}
 \mfA^i & = \{ A_{abc} \in \mfA : {}^\mfA _\xi \Pi_{i-1}^k(A) = 0 \, , \, \, \mbox{for all $k$} \} \, ,  & i & = -2,-1,0,1,2,3,
\end{align*}
where the maps ${}^\mfC _\xi \Pi_i^j$ are defined in appendix \ref{sec-spinor-descript}.

The associated graded $\prb$-module $\gr(\mfA)=\bigoplus_{i=-3}^3 \gr_i(\mfA)$, where $\gr_i(\mfA) := \mfA^i/\mfA^{i+1}$, splits into a direct sum
\begin{align*}
 \gr_{\pm3} (\mfA) & = \mfA_{\pm3}^0 \, , & 
 \gr_{\pm2} (\mfA) & = \mfA_{\pm2}^0 \oplus \mfA_{\pm2}^1 \, , \\
 \gr_{\pm1} (\mfA) & = \mfA_{\pm1}^0 \oplus \mfA_{\pm1}^1 \oplus \mfA_{\pm1}^2 \oplus \mfA_{\pm1}^3 \, , &
 \gr_0 (\mfA) & = \mfA_0^0 \oplus \mfA_0^1 \oplus \mfA_0^2 \, ,
\end{align*}
of irreducible $\prb$-modules $\mfA_i^j$ as described in Table \ref{table-A} of appendix \ref{sec-rep-theory}, with the proviso that when $m=2$,  $\mfA_{\pm1}^2$ does not occur.  Further,
\begin{align*}
 \mfA_i^j & = \{ A_{abc} \in \mfA^i : {}^\mfA _\xi \Pi_i^k(A) = 0 \, , \, \, \mbox{for all $k \neq j$} \} / \mfA^{i+1} \, ,  & \mbox{for $|i|\leq 2$.}
\end{align*}

Finally, the $\prb$-module $\gr(\mfA)$ can be expressed by means of the directed graph
\begin{equation*}
 \xymatrix@R=1em{
	    & & \mfA_1^3 \ar[dr] &  & \mfA_{-1}^3 \ar[dr] &    &  \\
&  \mfA_2^1 \ar[ur] \ar[dddr]   &  & \mfA_0^1 \oplus \mfA_0^2 \ar[ur] \ar[dddr] \ar[dr] & &  \mfA_{-2}^1 \ar[dr]        	& \\
\mfA_3^0 \ar[ur] \ar[dr]   &    &  \mfA_1^2 \ar[ur] & &  \mfA_{-1}^2 \ar[dr] &  	& \mfA_{-3}^0  \\
&  \mfA_2^0 \ar[dr] \ar[ur]  &  & \mfA_0^0 \ar[dr]  & & \mfA_{-2}^0 \ar[ur] 	& \\
	    & & \mfA_1^0 \oplus \mfA_1^1 \ar[ur] \ar[uuur] &  & \mfA_{-1}^0 \oplus \mfA_{-1}^1 \ar[ur] \ar[uuur] &        &  }
\end{equation*}
where an arrow from  $\mfA_i^j$ to $\mfA_{i-1}^k$ for some $i,j,k$ implies that $\breve{\mfA}_i^j \subset \g_1 \cdot \breve{\mfA}_{i-1}^k$ for any choice of irreducible $\g_0$-modules $\breve{\mfA}_i^j$ and $\breve{\mfA}_{i-1}^k$ isomorphic to $\mfA_i^j$ and $\mfA_{i-1}^k$ respectively.
\end{prop}

\begin{rem}
From the above diagram, one can define additional $\prb$-submodules of the isotopic pairs of $\prb$-modules $\{\mfA_{\pm1}^0,\mfA_{\pm1}^1\}$ and $\{\mfA_0^1,\mfA_0^2\}$ in the decomposition of $\gr(\mfA)$ in terms of the maps ${}^\mfA _\xi \Pi_i^j$. For instance, one has
\begin{align*}
\{ A_{abc} \in \mfA_1^0 \oplus \mfA_1^1 : {}^\mfA _\xi \Pi_2^0 (A) = 0 \} \, , & & \{ A_{abc} \in \mfA_1^0 \oplus \mfA_1^1 : {}^\mfA _\xi \Pi_2^1 (A) = 0 \} \, ,
\end{align*}
and in particular, it is certainly not true that $\ker {}^\mfA _\xi \Pi_2^1 \subset \ker {}^\mfA _\xi \Pi_1^0$ or $\ker {}^\mfA _\xi \Pi_2^1 \subset \ker {}^\mfA _\xi \Pi_1^1$. This equally applies to the other pairs.
\end{rem}


\subsection{Classification of the Weyl tensor}
\begin{prop}\label{prop-Weyl-classification}
The space $\mfC$ of tensors with Weyl symmetries admits a filtration
\begin{align*}
 \{ 0 \} =: \mfC^5 \subset \mfC^4 \subset \mfC^3 \subset \mfC^2 \subset \mfC^1 \subset \mfC^0 \subset \mfC^{-1} \subset \mfC^{-2} \subset \mfC^{-3} \subset \mfC^{-4} := \mfC \, ,
\end{align*}
of $\prb$-modules
\begin{align*}
 \mfC^i & = \{ C_{abcd} \in \mfC : {}^\mfC _\xi \Pi_i^k(C) = 0 \, , \, \, \mbox{for all $k$} \} \, , & i = -3,-2,-1,0,1,2,3,4,
\end{align*}
where the maps ${}^\mfC _\xi \Pi_i^j$ are defined in appendix \ref{sec-spinor-descript}.

The associated graded $\prb$-module $\gr(\mfC)=\bigoplus_{i=-4}^4 \gr_i(\mfC)$, where $\gr_i(\mfC) := \mfC^i/\mfC^{i+1}$, splits into a direct sum
\begin{align*}
 \gr_{\pm4} (\mfC) & = \mfC_{\pm4}^0 \, , &
 \gr_{\pm3} (\mfC) & = \mfC_{\pm3}^0 \, , & 
 \gr_{\pm2} (\mfC) & = \mfC_{\pm2}^0 \oplus \mfC_{\pm2}^1 \oplus \mfC_{\pm2}^2 \, , \\
 \gr_{\pm1} (\mfC) & = \mfC_{\pm1}^0 \oplus \mfC_{\pm1}^1 \oplus \mfC_{\pm1}^2 \, , &
 \gr_0 (\mfC) & = \mfC_0^0 \oplus \mfC_0^1 \oplus \mfC_0^2 \oplus \mfC_0^3 \oplus \mfC_0^4  \, ,
\end{align*}
of irreducible $\prb$-modules $\mfC_i^j$ as described in Table \ref{table-C} of appendix \ref{sec-rep-theory}, with the proviso that when $m=2$, the modules $\mfC_{\pm2}^2$, $\mfC_{\pm1}^1$, $\mfC_0^1$ and $\mfC_0^3$ do not occur, and when $m=3$, the module $\mfC_0^3$ does not occur. Further,
\begin{align*}
 \mfC_i^j & = \{ C_{abcd} \in \mfC^i : {}^\mfC _\xi \Pi_i^k(C) = 0 \, , \, \, \mbox{for all $k \neq j$} \} / \mfC^{i+1} \, , & \mbox{for $|i|\leq 3$.}
\end{align*}

Finally, the $\prb$-module $\gr(\mfC)$ can be expressed by means of the directed graph
\begin{equation*}
\xymatrix@R=1em{ & & & & & & & & \\
& & & & \mfC_0^4 \ar[dr]   & & & & \\
& & \mfC_2^2 \ar[ddr] \ar[r] & \mfC_1^2 \ar[dddr]  \ar[ur] & & \mfC_{-1}^2 \ar[r] \ar[ddr] & \mfC_{-2}^2 \ar[ddr] & & \\
 & &  &  & \mfC_0^3 \ar[dr] &  &  & & \\
\mfC_4^0 \ar[r] & \mfC_3^0 \ar[ddr] \ar[uur] \ar[r] & \mfC_2^1 \ar[uur] \ar[ddr] & \mfC_1^1 \ar[dr] \ar[ur] & & \mfC_{-1}^1 \ar[uur] \ar[ddr] & \mfC_{-2}^1 \ar[r] & \mfC_{-3}^0 \ar[r] & \mfC_{-4}^0 \\
 & &  &  & \mfC_0^1 \oplus \mfC_0^2 \ar[uuur] \ar[dr] \ar[ur]  &  &  & & \\
& & \mfC_2^0 \ar[r] \ar[uur] & \mfC_1^0 \ar[dr] \ar[ur] & & \mfC_{-1}^0 \ar[r] \ar[uur] & \mfC_{-2}^0 \ar[uur] & & \\
& & & & \mfC_0^0 \ar[ur]  & & & & \\
& & & & & & & &}
\end{equation*}
where an arrow from  $\mfC_i^j$ to $\mfC_{i-1}^k$ for some $i,j,k$ implies that $\breve{\mfC}_i^j \subset \g_1 \cdot \breve{\mfC}_{i-1}^k$ for any choice of irreducible $\g_0$-modules $\breve{\mfC}_i^j$ and $\breve{\mfC}_{i-1}^k$ isomorphic to $\mfC_i^j$ and $\mfC_{i-1}^k$ respectively.
\end{prop}

\begin{rem}
From the above diagram, one can define additional $\prb$-submodules of the isotopic pair of $\prb$-modules $\{\mfC_0^1,\mfC_0^2\}$ in the decomposition of $\gr(\mfC)$ in terms of the maps ${}^\mfC _\xi \Pi_i^j$. For instance, one has
\begin{align*}
\{ C_{abcd} \in \mfC_0^1 \oplus \mfC_0^2 : {}^\mfC _\xi \Pi_1^0 (C) = 0 \} \, , & & \{ C_{abcd} \in \mfC_0^1 \oplus \mfC_0^2 : {}^\mfC _\xi \Pi_1^1 (C) = 0 \} \, ,
\end{align*}
and so on. In particular, it is not true that $\ker {}^\mfC _\xi \Pi_1^1 \subset \ker {}^\mfA _\xi \Pi_0^1$ or $\ker {}^\mfC _\xi \Pi_1^1 \subset \ker {}^\mfC _\xi \Pi_0^2$. Proposition \ref{prop-recurrent-spinor} in section \ref{sec-geometry} will illustrate the point.
\end{rem}


\section{The $\prb$-invariant classification of $\mfV \otimes \left(\g / \prb\right)$}\label{sec-intrinsic-torsion}
In a vein similar to the previous section, we can decompose the $\prb$-module $\mfW := \mfV \otimes \left( \g / \prb \right)$ into irreducible parts -- here, we recall that $\g := \so(2m+1,\C)$, $\mfV$ its standard representation, and $\prb \subset \g$ stabilises a projective pure spinor $\langle \xi^A \rangle$. Again, we assume $m>2$, leaving the case $m=2$ to appendix \ref{sec-3-5dim}.  The $\prb$-invariant filtrations on $\mfV$ and $\g$ induces a $\prb$-invariant filtration on $\mfW$
\begin{align*}
 \mfW^0 \subset \mfW^{-1} \subset \mfW^{-2} \subset \mfW^{-3} \, ,
\end{align*}
where
\begin{align*}
 \mfW^{-3} & := \mfV^{-1} \otimes \left( \g^{-2} / \g^0 \right) \, , & \mfW^{-2} & := \left( \mfV^{-1} \otimes \left( \g^{-1} / \g^0 \right) \right) \oplus \left( \mfV^0 \otimes \left( \g^{-2} / \g^0 \right) \right) \, , \\
 \mfW^{-1} & := \left( \mfV^0 \otimes \left( \g^{-1} / \g^0 \right) \right) \oplus \left( \mfV^1 \otimes \left( \g^{-2} / \g^0 \right) \right) \, , & \mfW^0 & :=  \mfV^1 \otimes \left( \g^{-1} / \g^0 \right) \, .
\end{align*}
We can express an element of $\mfW$ as $\Gamma \ind{_{abc}} \xi^{bB} \xi^{cC}$ or $\Gamma \ind{_{abc}} \xi^{bcB} \pmod{ \alpha \ind{_a} \xi^{B}}$.
To assist the analysis, it is convenient to work in the splitting \eqref{eq-null-grading}, and use the spinors $\xi^A$ and $\eta_A$ satisfying $\xi^A \eta_A = -\frac{1}{2}$, $I \ind*{_A^B} := \xi \ind{^a^B} \eta \ind{_a_A} + \eta_A \xi^B$, and $u_a = 2 \, \xi \ind{^a^A} \eta_A$ as in section \ref{sec-algebra}. We then have
\begin{align}\label{eq-connection1form}
\begin{aligned}
 \Gamma \ind{_{abc}} \xi \ind{^{bB}} \xi \ind{^{cC}} & = \xi \ind{_a^A} \Gamma \ind{_A^{BC}} - u \ind{_a} \Gamma \ind{^{BC}} + \eta \ind{_a_A} \Gamma \ind{^{ABC}} + 2 \, \xi \ind{_a^A} \Gamma \ind{_{A:}^{[B}} \xi \ind{^{C]}} - 2 \, u \ind{_a} \Gamma \ind{^{[B}} \xi \ind{^{C]}} + 2 \, \eta \ind{_a_A} \Gamma \ind{^{A:[B}} \xi \ind{^{C]}} \, , \\
 \Gamma \ind{_{abc}} \xi \ind{^{bc}^D} 
&  = \left( \xi \ind{_a^A} \Gamma \ind{_A^{EC}} - u \ind{_a} \Gamma \ind{^{EC}} 
 + \eta \ind{_a_A} \Gamma \ind{^{AEC}} \right) \eta \ind{_c_C} \gamma \ind{^c_E^D} 
 - 2 \, \left( \xi \ind{_a^A} \Gamma \ind{_{A:}^D} - u \ind{_a} \Gamma \ind{^D} + \eta \ind{_a_A} \Gamma \ind{^{A:D}} \right) \\
 & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
 + 2 \, \left( \xi \ind{_a^A} \Gamma \ind{_{AB}^B} - u \ind{_a} \Gamma \ind{_{B}^B} + \eta \ind{_a_A} \Gamma \ind{^A_{B}^B} \right) \xi^D \, , \\
 \gamma \ind{^a_D^A}  \Gamma \ind{_{abc}} \xi \ind{^{bc}^D}  
& = 4 \, \Gamma \ind{_{C:}^C} \xi^A + 4 \, \Gamma \ind{_C^{CA}} - 2 \, \Gamma \ind{^A} 
+ \eta \ind{_a_B} \gamma \ind{^a_C^A} \left( \Gamma \ind{^{BC}} - \Gamma \ind{^{B:C}} + \Gamma \ind{^{C:B}} \right) + \eta \ind{_b_B} \eta \ind{_c_C} \Gamma \ind{^{DBC}} \gamma \ind{^{bc}_D^A} \\
&  \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 
- 2 \, \Gamma \ind{_{B}^B} \xi^A + 2 \, \Gamma \ind{^A_{B}^B} \, ,
\end{aligned}
\end{align}
where $\Gamma \ind{_A^{BC}} = \Gamma \ind{_A^{[BC]}}$, $\Gamma \ind{^{ABC}} = \Gamma \ind{^{A[BC]}}$, $\Gamma \ind{_A_B^C}$, $\Gamma \ind{^A_B^C}$, $\Gamma \ind{^A}$, $\Gamma \ind{_{A:}^B}$, $\Gamma \ind{_A^B}$, $\Gamma \ind{^{A:B}}$, $\Gamma \ind{^{AB}}=\Gamma \ind{^{[AB]}}$ are all elements of $\mfS_{\frac{m-2}{2}}$ and $\mfS_{-\frac{m-2}{2}}$ and tensor products thereof.

\begin{rem}
The indices on the above $\g_0$-tensors are to be treated as immovable, e.g. one cannot go from $\Gamma \ind{^{ABC}}$ to $\Gamma \ind{_A^{BC}}$ by lowering indices.

Further, the colon $:$ in $\Gamma \ind{_{A:}^B}$ and $\Gamma \ind{^{A:B}}$ does \emph{not} denote any differential operation, but is simply a marker separating the `$1$-form' index from the `Lie algebra' indices.
\end{rem}

The module $\mfW$ decomposes as a direct sum of $\g_0$-modules $\breve{\mfW}_i^j$, each of which determined by the components
\begin{align*}
 \Gamma \ind{^{[ABC]}} & \in  \breve{\mfW}_{-3}^0 \, , &  \Gamma \ind{^{(AB)C}} & \in \breve{\mfW}_{-3}^1 \, , \\
 \Gamma \ind{^{AB}} & \in  \breve{\mfW}_{-2}^0 \, , &  \Gamma \ind{^{[A:B]}} & \in \breve{\mfW}_{-2}^1 \, , &  \Gamma \ind{^{(A:B)}} & \in \breve{\mfW}_{-2}^2 \, , \\
 \Gamma \ind{_B^{BA}} & \in \breve{\mfW}_{-1}^0 \, , &
 \Gamma \ind{_A^{BC}} - \frac{2}{m-1} I \ind*{_A^{[B|}} \Gamma \ind{_D^{D|C]}} & \in \breve{\mfW}_{-1}^1 \, ,  & \Gamma \ind{^A} & \in \breve{\mfW}_{-1}^2 \, ,  \\
 \Gamma \ind{_{A:}^A} & \in \breve{\mfW}_0^0 \, , & \Gamma \ind{_{A:}^B} - \frac{1}{m} I \ind*{_A^B} \Gamma \ind{_{C:}^C} & \in \breve{\mfW}_0^1 \, .
\end{align*}
To make the analysis invariant, we define
\begin{align*}
 {}^\mfW _\xi \Pi_{-3}^0 (\Gamma) & := \Gamma \ind{_{abc}} \xi \ind{^{a[A}} \xi \ind{^{bB}} \xi \ind{^{cC}} \xi \ind{^{D]}} \, , \\
 {}^\mfW _\xi \Pi_{-3}^1 (\Gamma) & := \Gamma \ind{_{abc}} \xi \ind{^{[A}} \xi \ind{^{aB]}} \xi \ind{^{b[C}} \xi \ind{^{D]}} \xi \ind{^{c[E}} \xi \ind{^{F]}} + \Gamma \ind{_{abc}} \xi \ind{^{[C}} \xi \ind{^{aD]}} \xi \ind{^{b[A}} \xi \ind{^{B]}} \xi \ind{^{c[E}} \xi \ind{^{F]}} \, , \\
 {}^\mfW _\xi \Pi_{-2}^0 (\Gamma) & := \Gamma \ind{_{abc}} \xi^{aA} \xi^{b[B} \xi^{cC} \xi^{D]} \, , \\
 {}^\mfW _\xi \Pi_{-2}^1 (\Gamma) & := \Gamma \ind{_{abc}} \xi \ind{^{a[A}} \xi \ind{^{bcB}} \xi \ind{^{C]}} \, , \\
 {}^\mfW _\xi \Pi_{-2}^2 (\Gamma) & := \Gamma \ind{_{abc}} \xi \ind{^{[A}} \xi \ind{^{aB]}} \xi \ind{^{b[C}} \xi \ind{^{D]}} \xi \ind{^{cE}} + \Gamma \ind{_{abc}} \xi \ind{^{[C}} \xi \ind{^{aD]}} \xi \ind{^{b[A}} \xi \ind{^{B]}} \xi \ind{^{cE}} \, , \\
 {}^\mfW _\xi \Pi_{-1}^0 (\Gamma) & := 
 2 \, \gamma \ind{^a_D^A} \Gamma \ind{_{abc}} \xi \ind{^b^D} \xi \ind{^c^{[B}} \xi^{C]} - \Gamma \ind{_{abc}} \xi \ind{^a^A} \xi \ind{^{bc}^{[B}} \xi \ind{^{C]}} \, , \\
 {}^\mfW _\xi \Pi_{-1}^1 (\Gamma) & :=
 \Gamma \ind{_{abc}} \xi \ind{^{b[B}} \xi \ind{^{cC}} \xi \ind{^{D]}} + \frac{1}{2(m-1)} \gamma \ind{_a_E^{[B|}} \left(  2 \, \gamma \ind{^d_F^E} \Gamma \ind{_{dbc}} \xi \ind{^b^F} \xi \ind{^c^{|C}} - \Gamma \ind{_{dbc}} \xi \ind{^d^E} \xi \ind{^{bc}^{|C}} \right) \xi \ind{^{D]}} \, , \\
 {}^\mfW _\xi \Pi_{-1}^2 (\Gamma) & := \Gamma \ind{_{abc}} \xi \ind{^{aA}} \xi \ind{^{bc[B}} \xi \ind{^{C]}} \, , \\
 {}^\mfW _\xi \Pi_0^0 (\Gamma) & := \gamma \ind{^a_C^A} \Gamma \ind{_{abc}} \xi \ind{^{bcC}} \xi \ind{^{B}} - \xi \ind{^a^A} \Gamma \ind{_{abc}} \xi^{bcB} \, , \\
 {}^\mfW _\xi \Pi_0^1 (\Gamma) & := \Gamma \ind{_{abc}} \xi^{bc[B} \xi^{C]} + \frac{1}{2m} \left( \gamma \ind{^d_A^{[B|}} \Gamma \ind{_{dbc}} \xi \ind{^{bcA}} \xi \ind{_a^{|C]}} - \xi \ind{^d^{[B|}} \Gamma \ind{_{dbc}} \xi \ind{^{bcA}} \gamma \ind{_a_A^{|C]}} \right) \, , 
\end{align*}
where $\Gamma \ind{_{abc}} \in \mfV \otimes \g$. We have been careful to distinguish the cases $m>2$ and $m=2$. In the latter case, we have made use of the $\Spin(5,\C)$-invariant skewsymmetric bilinear forms $\varepsilon \ind{_{AB}}$ and $\varepsilon \ind{^{AB}}$.

\begin{prop}\label{prop-intorsion-class-alg}
 The associated graded $\prb$-module $\gr(\mfW)$ of $\mfW$ decomposes into a direct sum
\begin{align*}
 \gr_{-3}(\mfW) & \cong \mfW_{-3}^0 \oplus \mfW_{-3}^1 \, , & \gr_{-2}(\mfW) & \cong \mfW_{-2}^0 \oplus \mfW_{-2}^1 \oplus \mfW_{-2}^2 \, , \\
 \gr_{-1}(\mfW) & \cong \mfW_{-1}^0 \oplus \mfW_{-1}^1 \oplus \mfW_{-1}^2 \, , & \mfW^0 & \cong \mfW_0^0 \oplus \mfW_0^1 \, ,
\end{align*}
of irreducible $\prb$-modules $\mfW_i^j$ as described in Table \ref{table-W} of appendix \ref{sec-rep-theory}, with the proviso that $\mfW_{-1}^1$, $\mfW_{-3}^0$ occur only when $m>2$. Further,
\begin{align*}
\mfW_i^j & = \left\{ \Gamma_{abc} \xi^{bB} \xi^{cC} \in \mfW^i : {}^\mfW _\xi \Pi_i^k (\Gamma) = 0 \, , \mbox{for all $k \neq j$} \right\} / \mfW^{i+1} \, , & i & =-3,-2,-1,0 \, .
\end{align*}
Finally, the $\prb$-module $\gr(\mfW)$ can be expressed by means of the directed graph
 \begin{align*}
 \xymatrix{ & \mfW_{-1}^2 \ar[rdd] \ar[r] \ar[rdddd] & \mfW_{-2}^2 \ar[rd] & \\
 \mfW_0^1 \ar[ru] \ar[rd] \ar[rddd] & & & \mfW_{-3}^1 \\
 & \mfW_{-1}^1 \ar[rdd] & \mfW_{-2}^1 \ar[rd] \ar[ru] & \\
 \mfW_0^0 \ar[uuur] \ar[dr] & & & \mfW_{-3}^0 \\
 & \mfW_{-1}^0 \ar[r] & \mfW_{-2}^0 \ar[ru] \ar[ruuu] & }
\end{align*}
with the proviso that $\mfW_{-1}^1$, $\mfW_{-3}^0$ occur only when $m>2$. Here, an arrow from  $\mfW_i^j$ to $\mfW_{i-1}^k$ for some $i,j,k$ implies that $\breve{\mfW}_i^j \subset \g_1 \cdot \breve{\mfW}_{i-1}^k$ for any choice of irreducible $\g_0$-modules $\breve{\mfW}_i^j$ and $\breve{\mfW}_{i-1}^k$ isomorphic to $\mfW_i^j$ and $\mfW_{i-1}^k$ respectively.
\end{prop}

Conformal.tex
\section{Conformal structures}\label{sec-conformal}
In this appendix, we collect a few facts, notations, and conventions pertaining to conformal geometry. We roughly follow \cite{Bailey1994}, although our staggering of indices differs from theirs. We work in the holomorphic category as before.

A \emph{holomorphic conformal structure} on a complex manifold $\mcM$ is an equivalence class of holomorphic metrics $[g_{ab}]$ on $\mcM$, whereby two metrics $\hat{g}_{ab}$ and $g_{ab}$ belong to the same class if and only if
\begin{align}\label{eq-conformal_change_metric}
\hat{g} \ind{_{ab}} & = \Omega^2 g \ind{_{ab}}  \, ,
\end{align}
for some non-vanishing holomorphic function $\Omega$ on $\mcM$. The respective Levi-Civita connections $\nabla_a$ and $\hat{\nabla}_a$ of $g_{ab}$ and $\hat{g}_{ab}$ are then related by
\begin{align*}
 \hat{\nabla} \ind{_a} V \ind{^b} & = \nabla \ind{_a} V \ind{^b} + Q \ind{_{ac}^b} V \ind{^c} \, , &
Q \ind{_{abc}} := Q \ind{_{ab}^d} g \ind{_{dc}} = 2 \Upsilon \ind{_{\lp{a}}} g \ind{_{\rp{b}c}} - \Upsilon \ind{_c} g \ind{_{ab}} \, ,
\end{align*}
for any holomorphic vector field $V^a$, where $\Upsilon _a := \Omega^{-1} \nabla _a \Omega$.

\paragraph{Spinor bundles}
For a formal definition of the spinor bundle, $\mcS$ say, on a conformal manifold, see e.g. \cite{Baum2010}. For our purpose, we simply note that under a rescaling \eqref{eq-conformal_change_metric}, the $\gamma$-matrices can be chosen to transform as
\begin{align*}
 \gamma \ind{_a_A^B} & \mapsto \hat{\gamma} \ind{_a_A^B} = \Omega \gamma \ind{_a_A^B} \, ,
\end{align*}
where $\hat{\gamma} \ind{_a_A^B}$ denote the $\gamma$-matrices for metric $\hat{g}_{ab}$. In addition, we can choose the spin invariant bilinear forms $\varepsilon_{AB}$ on $\mcS$ to rescale with a conformal weight of $1$, and their dual with a conformal weight of $-1$. This means in particular that the quantities $\gamma \ind{_a^{AB}}$ and $\gamma \ind{^a_{AB}}$ have conformal weight $0$. Then the spin connection $\hat{\nabla}_a$ is related to $\nabla_a$ by
\begin{align}
\begin{aligned}\label{eq-conformal_spin_connection2}
 \hat{\nabla} \ind{_a} \xi \ind{^B} & = \nabla \ind{_a} \xi \ind{^B} - \frac{1}{2} \Upsilon \ind{_b} \gamma \ind{^b_a_C^B} \xi \ind{^C} + \frac{1}{2} \Upsilon \ind{_a} \xi \ind{^B} \\
& = \nabla \ind{_a} \xi \ind{^B} - \frac{1}{2} \Upsilon \ind{_b} \gamma \ind{^b_C^D} \gamma \ind{_a_D^B} \xi \ind{^C} \, ,  
\end{aligned}
\end{align}
for any holomorphic spinor field $\xi^{A'}$, and similarly for dual spinors. This connection preserves the hatted $\gamma$-matrices \emph{and} the hatted bilinear forms on $\mcS$, in agreement with the convention of \cite{Penrose1984}.

If we now assume that $\xi^A$ is a pure spinor field, we then obtain from \eqref{eq-conformal_spin_connection2}
\begin{align*}
 \left( \hat{\nabla} \ind{_a} \hat{\xi} \ind{^{bB}} \right) \hat{\xi} \ind{_b^C}
& = \left( \nabla \ind{_a} \xi \ind{^{bB}} \right) \xi \ind{_b^C} + \frac{1}{2} \Upsilon \ind{_b} \xi \ind{^b^D} \gamma \ind{_a_D^B} \xi \ind{^C} + 2 \, \Upsilon \ind{_b} \xi \ind{^b^{[B}} \xi \ind{_a^{C]}} \, ,  \\
 \left( \hat{\gamma} \ind{^a_B^A} \hat{\nabla} \ind{_a} \hat{\xi} \ind{^{bB}} \right) \hat{\xi} \ind{_b^C}
& = \Omega^{-1} \left( \left( \gamma \ind{^a_B^A} \nabla \ind{_a} \xi \ind{^{bB}} \right) \xi \ind{_b^C} + \frac{n-2}{2} \left( 2 \, \Upsilon \ind{_b} \xi \ind{^b^C} \xi \ind{^A} - \Upsilon \ind{_b} \xi \ind{^b^{A}} \xi \ind{^{C}} \right) \right) \, ,  
\\
\hat{\xi} \ind{^a^A} \hat{\nabla} \ind{_a} \xi \ind{^B}
& = \Omega^{-1} \left( \xi \ind{^a^A} \nabla \ind{_a} \xi \ind{^{B}}  - \frac{1}{2} \Upsilon \ind{_b} \xi \ind{^b^B} \xi \ind{^A} + \Upsilon \ind{_b} \xi \ind{^b^A} \xi \ind{^B} \right) \, ,  \\
 \left( \hat{\xi} \ind{^a^A} \hat{\nabla} \ind{_a} \hat{\xi} \ind{^{bB}} \right) \hat{\xi} \ind{_b^C}
& = \Omega^{-1} \left( \left( \xi \ind{^a^A} \nabla \ind{_a} \xi \ind{^{bB}} \right) \xi \ind{_b^C} - \frac{1}{2} \Upsilon \ind{_b} \xi \ind{^b^B} \xi \ind{^A} \xi \ind{^C} - 2 \Upsilon \ind{_b} \xi \ind{^b^{[A}} \xi \ind{^{C]}} \xi \ind{^B} \right) \, ,  
\\
 \left( \hat{\nabla} \ind{_a} \hat{\xi} \ind{^{aB}} \right) \xi \ind{^C} -  \hat{\xi} \ind{^a^B} \hat{\nabla} \ind{_a} \xi \ind{^{C}}  
& = \Omega^{-1} \left( \left( \nabla \ind{_a} \xi \ind{^{aB}} \right) \xi \ind{^C} -  \xi \ind{^a^B} \nabla \ind{_a} \xi \ind{^{C}} + \frac{n-2}{2} \Upsilon \ind{_b} \xi \ind{^b^B} \xi \ind{^C} + \frac{1}{2} \Upsilon \ind{_b} \xi \ind{^b^C} \xi \ind{^B} \right) \, ,  
\end{align*}
where we have set $\hat{\xi} \ind{^a^A} := \hat{\gamma} \ind{^a_B^A} \xi \ind{^B}$. In particular, from the first three expressions, we get
\begin{align*}
 \left( \hat{\nabla} \ind{_a} \hat{\xi} \ind{^{b[B}} \right) \hat{\xi} \ind{_b^C} \xi^{D]}
& = \left( \nabla \ind{_a} \xi \ind{^{b[B}} \right) \xi \ind{_b^C} \xi^{D]} + 2 \, \Upsilon \ind{_b} \xi \ind{^b^{[B}} \xi \ind{_a^{C}} \xi^{D]} , ,  \\
 \left( \hat{\gamma} \ind{^a_B^A} \hat{\nabla} \ind{_a} \hat{\xi} \ind{^{bB}} \right) \hat{\xi} \ind{_b^{[C}}  \xi^{D]}
& = \Omega^{-1} \left( \left( \gamma \ind{^a_B^A} \nabla \ind{_a} \xi \ind{^{bB}} \right) \xi \ind{_b^{[C}} \xi^{D]}  + (n-2) \, \xi \ind{^A} \Upsilon \ind{_b} \xi \ind{^b^{[C}}   \xi^{D]} \right) \, ,
\\
 \left( \hat{\xi} \ind{^a^A} \hat{\nabla} \ind{_a} \xi \ind{^{[B}} \right) \xi \ind{^{C]}}
& = \Omega^{-1} \left(\left(  \xi \ind{^a^A} \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi \ind{^{C]}}  - \frac{1}{2} \xi \ind{^A} \Upsilon \ind{_b} \xi \ind{^b^{[B}}  \xi \ind{^{C]}} \right) \, ,
\end{align*}
from which follows the conformal invariance of \eqref{eq-cond-11}.

\paragraph{Curvature}
The conformal properties of the Riemann curvature can be conveniently brought out by means of the decomposition
\begin{align} \label{eq-Riem_decomposition-conformal}
 R \ind{_{abcd}} & = C \ind{_{abcd}} - 4 g_{\lb{c}|\lb{a}} \Rho_{\rb{b}|\rb{d}} \, , &
 \Rho_{ab} & := \frac{1}{2-n} \Phi_{ab} - R \frac{1}{2n(n-1)} g_{ab} \, .
\end{align}
where the Weyl tensor $C \ind{_{abcd}}$ is the conformally invariant part of $R \ind{_{abcd}}$, and the \emph{Schouten} or \emph{Rho tensor} $\Rho_{ab}$, together with its  its trace $\Rho$, transform as
\begin{align}\label{eq-conf-transf-Rho}
 \hat{\Rho}_{ab} & = \Rho_{ab} - \nabla_a \Upsilon_b + \Upsilon_a \Upsilon_b - \frac{1}{2} \Upsilon_c \Upsilon^c g_{ab} \, , &
 \hat{\Rho} & = \Omega^{-2} \left( \Rho - \nabla^c \Upsilon_c - \frac{n-2}{2} \Upsilon^c \Upsilon_c \right) \, .
\end{align}
Finally, the Cotton-York tensor $A_{abc} := 2 \nabla \ind{_{\lb{b}}} \Rho \ind{_{\rb{c}a}} = -(n-3) \nabla^d C_{dabc}$, where the expression on the RHS follows from the contracted Bianchi identity, transforms as $\hat{A}_{abc} = A \ind{_{abc}} - \Upsilon^d C_{dabc}$.
Five_dim.tex
\section{Spinor calculus in three and five dimensions}\label{sec-3-5dim}
In this appendix, we give a brief description of spinor calculus in three and five dimensions emphasising its relation to the classification of tensors with respect to a `principal spinor'.

\subsection{Three dimensions} \label{sec-3dim}
Let $(\mfV,g)$ be a three-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form. The double cover of the special orthogonal groups $\SO(3,\C)$ is the complex special linear group $\SL(2,\C)$. Thus, the spin space $\mfS$ and its dual space are two-dimensional complex vector spaces equipped with non-degenerate symplectic forms $\varepsilon_{AB}$ and $\varepsilon^{AB}$ respectively. By the two-dimensionality of $\mfS$, it thus follows that, as in the case of the two-spinor calculus in four dimensions, any irreducible spinor representations must be totally symmetric since skew-symmetry can be eliminated by means of the volume forms $\varepsilon_{AB}$ and $\varepsilon^{AB}$.  We take the convention that $\varepsilon \ind{_{AC}} \varepsilon \ind{^{BC}} = \delta \ind*{_A^B}$ and indices are lowered and raised as follows
\begin{align*}
\xi^A \varepsilon _{A B} & = \xi_B \, , & \varepsilon ^{A B} \xi_B & = \xi^A \, .
\end{align*}
We can convert tensorial quantities into spinorial ones by means of the normalised $\gamma$-matrices $\frac{1}{\sqrt{2}} \gamma \ind{_a^{AB}}$, which are symmetric in their spinor indices, and satisfy the identity
\begin{align*}
	\gamma \ind{_a _A ^B} \gamma \ind{^a _C ^D} & = - \delta \ind*{_A ^D} \delta \ind*{_C ^B} + \varepsilon \ind{_{AC}} \varepsilon \ind{^{BD}} \, , & \mbox{i.e.} & &
	\gamma \ind{_a _{AB}} \gamma \ind{^a _{CD}} & = - 2 \, \varepsilon \ind{_{A\lp{C}}} \varepsilon \ind{_{\rp{D}B}}  \, ,
\end{align*}
from which we can see that all spinors are pure. In particular, we have that $\mfV \cong \wedge^2 \mfV \cong \odot^2 \mfS$ where we have made use of Hodge duality, so that two-forms are also represented by symmetric spinors of valence $2$. Further, since it is well known that there is no Weyl tensor in dimension three, the only irreducible curvature tensors that remain are the tracefree Ricci tensor $\Phi_{ABCD} = \Phi_{(ABCD)}$ and the Cotton-York tensor $A_{ABCD} = A_{(ABCD)}$. In particular, the spaces $\mfF$ and $\mfA$ of tracefree Ricci tensors and of Cotton-York tensors are isomorphic to one another.


\paragraph{Principal spinors}
Fixing a spinor $\xi^A$ induces a filtration $\mfS^{\frac{1}{2}} \subset \mfS^{-\frac{1}{2}}$ on $\mfS$, where
\begin{align*}
 \mfS^{-\frac{1}{2}} & := \mfS \, , & \mfS^{\frac{1}{2}} & = \langle \xi^A \rangle = \{ \alpha^A \in \mfS : \alpha^A \xi_A = 0 \} \, .
\end{align*}
The notion of principal spinors can defined in exactly the same way as in four dimensions: the spinor $\xi^A$ defines a filtration
\begin{align*}
 \mfS^{\frac{k}{2}} \subset \mfS^{\frac{k}{2}-1} \subset \ldots \subset \mfS^{-\frac{k}{2}+1} \subset \mfS^{-\frac{k}{2}} 
\end{align*}
on $\mfS^{-\frac{k}{2}} := \odot^k \mfS$, where $\mfS^{\frac{k-2\ell+2}{2}} := \left\{ \phi \ind{_{A_1 \ldots A_k}} \in \mfS^{-\frac{k}{2}} : \phi \ind{_{A_1 \ldots A_\ell A_{\ell+1} \ldots A_k}} \xi \ind{^{A_1}} \ldots \xi \ind{^{A_\ell}} \right\}$, and $\xi^A$ is said to be a principal spinor of $\phi \ind{_{A_1 \ldots A_k}}$ if it lies in $\mfS^{-\frac{k}{2}+1}$.

In the particular case $k=2$, we have a filtration\footnote{So, unlike $\so(2m+1,\C)$ for $m>1$, the Lie algebra $\so(3,\C)$ admits a $|1|$-grading, not $|2|$-grading.} $\g^1 \subset \g^0 \subset \g^{-1}$ on $\g := \so(3,\C)$, or equivalently on $\mfV \cong \wedge^2 \mfV \cong \g$, where
\begin{align*}
 \g^{-1} & := \g \, , & \g^0 & := \left\{ \phi \ind{_{AB}} \in \odot^2 \mfS : \phi \ind{_{AB}} \xi \ind{^A} \xi \ind{^B} = 0 \right\} \, , &
 \g^1 & := \left\{ \phi \ind{_{AB}} \in \odot^2 \mfS : \phi \ind{_{AB}} \xi \ind{^A} = 0 \right\} \, .
\end{align*}
Similarly, the case $k=4$ yields filtrations on $\mfF$ and $\mfA$, but it is worth noting that unlike in higher dimensions, the filtration on $\mfA$ is `shortened' by two vector subspaces, i.e.
\begin{align*}
\{ 0 \} =: \mfA^3 \subset \mfA^2 \subset \mfA^1 \subset \mfA^0 \subset \mfA^{-1} \subset \mfA^{-2} := \mfA \, .
\end{align*}


\paragraph{Foliating spinors and others}
The filtration on the $\prb$-module $\mfW := \mfV \otimes \left( \g/\prb \right)$, where $\prb := \g^0$ stabilises $\xi^A$, defining the classes of intrinsic torsion associated to the spinor field $\xi^A$ is also shortened to $\mfW^0 \subset \mfW^{-1} \subset \mfW^{-2}$. From a geometric point of view, this is reflected by the fact that on a $3$-dimensional complex Riemannian manifold, the null plane $\mcN$ associated to a projective spinor field $\xi^A$ is only $1$-dimensional, and is thus trivially integrable. The remaining differential conditions then reduce to only three as given by the following proposition.
\begin{prop}
 Let $\langle \xi^A \rangle$ be a holomorphic projective spinor field on a $3$-dimensional complex Riemannian manifold with associated $\gamma$-plane distribution $\mcN$ and orthogonal complement $\mcN^\perp$. Then the intrinsic torsion of $\langle \xi^A \rangle$
 \begin{itemize}
  \item lies in $\mfW^{-1}$ if and only if $\xi \ind{^A} \xi \ind{^B} \xi \ind{^C} \nabla \ind{_{AB}} \xi \ind{_C} = 0$
 if and only if $\mcN^\perp$ is integrable;
  \item lies in $\mfW^0$ if and only if $\xi \ind{^B} \xi \ind{^C} \nabla \ind{_{AB}} \xi \ind{_C} = 0$
 if and only if $\mcN^\perp$ is integrable and geodetic;
  \item vanishes if and only if $\xi \ind{^C} \nabla \ind{_{AB}} \xi \ind{_C} = 0$.
 \end{itemize}
\end{prop}

\begin{rem}
The above conditions are equivalent to the null vector field $k \ind{^{AB}} := \xi \ind{^A} \xi \ind{^B}$ being geodetic, dilation free and recurrent respectively.
\end{rem}

\subsection{Five dimensions} \label{sec-5dim}
Let $(\mfV,g)$ be a five-dimensional complex vector space equipped with a non-degenerate symmetric bilinear form. The double cover of the special orthogonal groups $\SO(5,\C)$ is the complex symplectic group $\Sp(4,\C)$. The spinor space $\mfS$ and its dual space are thus four-dimensional complex vector spaces equipped with non-degenerate symplectic forms $\varepsilon_{AB}$ and $\varepsilon^{AB}$ respectively. One can convert tensor indices into spinorial ones by means of the normalised skewsymmetric $\gamma$-matrices $\frac{\ii}{2} \gamma \ind{_a^{AB}}$, tracefree with respect to $\varepsilon \ind{_{AB}}$, which satisfy the identity
\begin{align}\label{eq-g-epsilon}
	\gamma \ind{_a _A ^B} \gamma \ind{^a _C ^D} & = \delta \ind*{_A ^B} \delta \ind*{_C ^D} -  2 \delta \ind*{_A ^D} \delta \ind*{_C ^B} - 2 \varepsilon \ind{_{AC}} \varepsilon \ind{^{BD}} \, ,
&
\text{i.e.} & & 
	\gamma \ind{_a _{AB}} \gamma \ind{^a _{CD}} & = \varepsilon \ind{_{AB}} \varepsilon \ind{_{CD}} + 4 \varepsilon \ind{_{A \lb{C}}} \varepsilon \ind{_{\rb{D} B}} \, .
\end{align}
Again, this identity tells us that in five dimensions, all spinors are pure. In particular, we have $\mfV \cong ( \wedge^2 \mfS )_\circ$ and $\wedge^2 \mfV \cong \odot^2 \mfS$.

The tracefree Ricci tensor, the Weyl tensor and the Cotton tensor admit the spinorial expressions
\begin{align*}
\Phi \ind{_{ABCD}} & = \Phi_{[AB][CD]} \, , & \Phi_{[ABC]D} & = 0 \, , \\ 
C \ind{_{ABCD}} & = C_{(ABCD)} \, , \\
 A_{A B C D} & = A_{[AB](CD)} \, , & A_{[A B C] D} & = 0 \, ,
\end{align*}
respectively, all of which are completely tracefree.


\paragraph{Principal spinors}
Following the notation of section \ref{sec-algebra}, a spinor $\xi^A$ induces a filtration $\mfS^1 \subset \mfS^0 \subset \mfS^{-1}$ on $\mfS$ where
\begin{align*}
 \mfS^{-1} & := \mfS \, , & \mfS^0 & := \{ \alpha^A \in \mfS : \alpha_A \xi^A = 0 \} \, , & \mfS^1 & := \langle \xi^A \rangle = \{ \alpha^A \in \mfS : \alpha_{[A} \xi_{B]} = 0 \} \, ,
\end{align*}
In particular, the concept of a principal spinor $\xi^A$ of a spinorial object of any valence can be defined by means of the wedge and contraction operations. Thus the $\prb$-modules of the induced filtration $\mfV^1 \subset \mfV^0 \subset \mfV^{-1}$ can be characterised as\footnote{We can define equivalently $\mfV^{-1} = \left\{ V \ind{^{AB}} \in \mfV : \xi_{[A} V_{BC]} = 0 \right\}$. This can be seen by means of a volume form $\varepsilon _{A B C D}$ so that
$\varepsilon_{A B C D} = 3 \, \varepsilon \ind{_{\lb{A}B}} \varepsilon \ind{_{C\rb{D}}} = \frac{3}{5} \, \gamma \ind{_a _{\lb{A}B}} \gamma \ind{^a _{C\rb{D}}}$,
which satisfies the normalisation so that $\varepsilon_{ABCD} \varepsilon^{EFGH} = 24 \, \delta \ind*{_{[A}^E} \delta \ind*{_B^F} \delta \ind*{_C^G} \delta \ind*{_{D]}^H}$. Also, a vector (or $1$-form) $V^a$ is null if and only if it is of the form $V^{AB} = \xi \ind{^{\lb{A}}} \alpha \ind{^{\rb{B}}}$ for some spinors $\xi_A$ and $\alpha_A$ with $\alpha_A \xi^A = 0$.}
\begin{align*}
 \mfV^{-1} & := \mfV \, , & 
 \mfV^0 & := \left\{ V \ind{^{AB}} \in \mfV : \xi_C V \ind{^{C\lb{A}}} \xi \ind{^{\rb{B}}} = 0 \right\} \, , &
 \mfV^1 & := \left\{ V \ind{^{AB}} \in \mfV : \xi_C V \ind{^{CA}} = 0 \right\} \, .
\end{align*}
Similarly, the Lie algebra $\g:=\so(5,\C)$ admits a $\prb$-invariant filtration $\{ 0 \} \subset \g^2 \subset \g^1 \subset \g^0 \subset \g^{-1} \subset \g^{-2} := \g$, with associated graded module $\gr(\g)$, which decomposes into  irreducible $\prb$-modules $\g_i^j$. Elements of $\g_i^j$ can be expressed in terms of kernels of the maps
\begin{align*}
 {}^\g _\xi \Pi^0_{-2} ( \phi ) & := \xi^A \xi^B \phi_{A B} \, , &
 {}^\g _\xi \Pi^0_{-1} ( \phi ) & := \xi^A \phi \ind{_A ^{\lb{B}}} \xi^{\rb{C}} \, , \\
 {}^\g _\xi \Pi^0_0 ( \phi ) & := \xi^A \phi \ind{_A ^B} \, , &
 {}^\g _\xi \Pi^1_0 ( \phi ) & := \xi_{\lb{A}} \phi \ind{_{\rb{B} \lb{C}}} \xi_{\rb{D}} \, , &
 {}^\g _\xi \Pi^0_1 ( \phi ) & := \phi \ind{_{A \lb{B}}} \xi_{\rb{C}} \, ,
\end{align*}
where $\phi \ind{_{AB}} = \phi \ind{_{(AB)}}$.

The explicit expressions for the maps ${}^\mfF _\xi \Pi_i^j$, ${}^\mfA _\xi \Pi_i^j$ and ${}^\mfC _\xi \Pi_i^j$ defined in section \ref{sec-curvature} can be significantly simplified.
For $\Phi \ind{_{ABCD}} \in \mfF$, we have
\begin{align*}
{}^\mfF _\xi \Pi^0_{-2} ( \Phi ) & := \xi_{\lb{F}} \Phi_{\rb{A} B C \lb{D}} \xi^B \xi^C \xi_{\rb{E}} \, , &
{}^\mfF _\xi \Pi^0_{-1} ( \Phi ) & := \Phi_{A B C \lb{D}} \xi^B \xi^C \xi_{\rb{E}} \, , \\
{}^\mfF _\xi \Pi^0_{0} ( \Phi ) & := \Phi_{A B C D} \xi^B \xi^C \, , &
{}^\mfF _\xi \Pi^1_{0} ( \Phi ) & := \xi \ind{_{[A}} \Phi \ind{_{B] E C D}} \xi^E + \varepsilon \ind{_{[A|[C}} \Phi \ind{_{D]EF|B]}} \xi \ind{^E} \xi \ind{^F} \, , \\
{}^\mfF _\xi \Pi^0_{1} ( \Phi ) & := \Phi_{A B C D} \xi^B \, .
\end{align*}
For $A \ind{_{ABCD}} \in \mfA$, we have
\begin{align*}
 {}^\mfA _\xi \Pi^0_{-3} ( A ) & := \xi_{\lb{E}} A_{\rb{A} B C D} \xi^B \xi^C \xi^D \, , \\
 {}^\mfA _\xi \Pi^0_{-2} ( A ) & := A \ind{_{A B C D}} \xi \ind{^B} \xi \ind{^C} \xi \ind{^D} \, , \qquad \qquad \qquad \qquad
 {}^\mfA _\xi \Pi^1_{-2} ( A ) := \xi \ind{_{[A}} A \ind{_{B] EF [C}} \xi \ind{^E} \xi \ind{^F} \xi \ind{_{D]}} + \xi \ind{_{[C}} A \ind{_{D] EF [A}} \xi \ind{^E} \xi \ind{^F} \xi \ind{_{B]}} \, , \\
 {}^\mfA _\xi \Pi^0_{-1} ( A ) & := 4 \, \xi \ind{_{[A}} A \ind{_{B]DEC}} \xi \ind{^D} \xi \ind{^E} - A \ind{_{ABDE}} \xi \ind{^D} \xi \ind{^E} \xi \ind{_C} \, , \qquad \qquad \qquad
 {}^\mfA _\xi \Pi^1_{-1} ( A ) := A_{A B C \lb{D}} \xi^B \xi^C \xi_{\rb{E}} \, , \\
 {}^\mfA _\xi \Pi^3_{-1} ( A ) & := \xi_{\lb{A}} A_{\rb{B} G \lb{C} | \lb{E}} \xi^G \xi_{\rb{F}} \xi_{|\rb{D}} + \xi_{\lb{C}} A_{\rb{D} G \lb{E} | \lb{A}} \xi^G \xi_{\rb{B}} \xi_{|\rb{F}} + \xi_{\lb{E}} A_{\rb{F} G \lb{A} | \lb{C}} \xi^G \xi_{\rb{D}} \xi_{|\rb{B}} \, ,  \\
 {}^\mfA _\xi \Pi^0_{0} ( A ) & := A_{A B C D} \xi^B \xi^C  \, , \qquad \qquad \qquad \qquad \qquad \qquad 
 {}^\mfA _\xi \Pi^1_{0} ( A ) := \xi \ind{^F} A \ind{_{A F [B| [D}} \xi \ind{_{E]}} \xi \ind{_{|C]}} \, , \\
 {}^\mfA _\xi \Pi^2_{0} ( A ) & := \xi \ind{_{[A}} \xi \ind{^F} A \ind{_{B] F E [C}} \xi \ind{_{D]}} + \xi \ind{_{[A}} \xi \ind{^F} A \ind{_{B] E F [C}} \xi \ind{_{D]}} \, , \\
 {}^\mfA _\xi \Pi^0_{1} ( A ) & := \xi \ind{_{[A}} \xi \ind{^E} A_{B] E C D} - \xi \ind{^E} A \ind{_{ABE(C}} \xi \ind{_{D)}} \, , \qquad \qquad \qquad \qquad \qquad \qquad 
 {}^\mfA _\xi \Pi^1_{1} ( A ) := \xi \ind{^E} A \ind{_{AEB[C}} \xi \ind{_{D]}} \\
 {}^\mfA _\xi \Pi^3_{1} ( A ) & := A_{A B \lb{C} | \lb{E}} \xi_{\rb{F}} \xi_{|\rb{D}} + A_{C D \lb{E} | \lb{A}} \xi_{\rb{B}} \xi_{|\rb{F}} + A_{E F \lb{A} | \lb{C}} \xi_{\rb{D}} \xi_{|\rb{B}} \, , \\
 {}^\mfA _\xi \Pi^0_{2} ( A ) & := \xi \ind{^D} A \ind{_{ADBC}} \, ,  \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 
 {}^\mfA _\xi \Pi^1_{2} ( A ) := A \ind{_{ABC[D}} \xi \ind{_{E]}} + A \ind{_{DEC[A}} \xi \ind{_{B]}} \, .
\end{align*}
Finally, for $C \ind{_{ABCD}} \in \mfC$, we have
\begin{align*}
 {}^\mfC _\xi \Pi^0_{-4} ( C ) & := C_{A B C D} \xi^A \xi^B \xi^C \xi^D \, , &
 {}^\mfC _\xi \Pi^0_{-3} ( C ) & := C_{A B C \lb{D}} \xi^A \xi^B \xi^C \xi_{\rb{E}} \, , \\
 {}^\mfC _\xi \Pi^0_{-2} ( C ) & := C_{A B C D} \xi^A \xi^B \xi^C \, , &
 {}^\mfC _\xi \Pi^2_{-2} ( C ) & := \xi_{\lb{F}} C_{\rb{A} B C \lb{D}} \xi^B \xi^C \xi_{\rb{E}} \, , \\
 {}^\mfC _\xi \Pi^0_{-1} ( C ) & := C_{A B C \lb{D}} \xi^B \xi^C \xi_{\rb{E}} \, , &
 {}^\mfC _\xi \Pi^1_{-1} ( C ) & := \xi_{\lb{F}} C_{\rb{A} B \lb{C} | \lb{D}} \xi^B \xi_{\rb{E}} \xi_{|\rb{F}} \, , \\
 {}^\mfC _\xi \Pi^0_0 ( C ) & := C_{A B C D} \xi^B \xi^C \, , &
 {}^\mfC _\xi \Pi^2_0 ( C ) & := \xi_{\lb{F}} C_{\rb{A} B C \lb{D}} \xi^B \xi_{\rb{F}} \, , \\
 {}^\mfC _\xi \Pi^4_0 ( C ) & := \xi_{\lb{G}} \xi_{|\lb{F}} C_{\rb{A} | \rb{B} \lb{C} | \lb{D}} \xi_{\rb{E}} \xi_{|\rb{F}} \, , \\
 {}^\mfC _\xi \Pi^0_1 ( C ) & := \xi_{\lb{F}} C_{\rb{A} B C D} \xi^B \, , &
 {}^\mfC _\xi \Pi^1_1 ( C ) & := \xi_{\lb{F}} C_{\rb{A} B \lb{C} | \lb{D}} \xi_{\rb{E}} \xi_{|\rb{F}} \, , \\
 {}^\mfC _\xi \Pi^0_2 ( C ) & := C_{A B C D} \xi^B \, , &
 {}^\mfC _\xi \Pi^2_2 ( C ) & := \xi_{\lb{F}} C_{\rb{A} B C \lb{D}} \xi_{\rb{E}} \, , \\
 {}^\mfC _\xi \Pi^0_3 ( C ) & := C_{A B C \lb{D}} \xi_{\rb{E}} \, .
\end{align*}

\subsubsection{Foliating and not so foliating spinors}
Let $\langle \xi^A \rangle$ be a projective spinor field on a five-dimensional complex Riemannian manifold $(\mcM,g)$. Denote by $\nabla_{AB}$ the Levi-Civita connection of $g$. Then the differential characterisations of the intrinsic torsion of $\langle \xi^A \rangle$ can be re-expressed as
\begin{align}
 \eqref{eq-cond-31} & & \Longleftrightarrow &  & \xi_{\lb{A}} \left( \xi^C \nabla_{\rb{B}C} \xi^D \right) \xi_D & = 0 \, , \label{eq-cond5d-31}  \\
\eqref{eq-cond-20} & & \Longleftrightarrow &  & \left( \xi^B \nabla_{AB} \xi^C \right) \xi_C & = 0 \, , \label{eq-cond5d-20} \\
 \eqref{eq-cond-21} & & \Longleftrightarrow &  &  \left( \xi^D \nabla \ind{_{D[A}} \xi \ind{_{B}} \right) \xi \ind{_{C]}} & = 0 \, , \label{eq-cond5d-21} \\
 \eqref{eq-cond-22} & & \Longleftrightarrow &  &  \xi_{\lb{A}} \left( \xi^E \nabla_{\rb{B}E} \xi_{\lb{C}} \right) \xi_{\rb{D}} + \xi_{\lb{C}} \left( \xi^E \nabla_{\rb{D}E} \xi_{\lb{A}} \right) \xi_{\rb{B}} & = 0 \, , \label{eq-cond5d-22} \\
\eqref{eq-cond-10} & & \Longleftrightarrow &  &  (\nabla_{A B} \xi^C ) \xi_C & = 0 \, , \label{eq-cond5d-10} \\
\eqref{eq-cond-12} & & \Longleftrightarrow &  &  \left( \xi^D \nabla_{A D} \xi_{\lb{B}} \right) \xi_{\rb{C}} & = 0 \, , \label{eq-cond5d-12} \\
 \eqref{eq-cond00} & & \Longleftrightarrow &  &  (\nabla_{A C} \xi^C ) \xi^B - (\xi^C \nabla_{A C}  \xi^B  )& = 0 \, , \label{eq-cond5d-00} \\
\eqref{eq-cond01} & & \Longleftrightarrow &  &  \left( \nabla_{A B} \xi_{\lb{C}} \right) \xi_{\rb{D}} + \xi_{[C} \varepsilon _{D][A} \nabla_{B]E} \xi^E + \varepsilon _{[C|[A} \xi^E  \nabla_{B]E} \xi_{|D]} & = 0 \, . \label{eq-cond5d-01}
 \end{align}

Finally, we remark that
\begin{itemize}
\item condition \eqref{eq-almost-foliating} for an almost foliating spinor reduces to \eqref{eq-cond5d-31};
\item condition \eqref{eq-Nperp-int} for a $\gamma^\perp$-foliating spinor can be expressed as 
\begin{align}\label{eq-Nperp-foliating-5d}
 \xi \ind{_{[A}} \left( \xi \ind{^E} \nabla \ind{_{B]E}} \xi \ind{_{[C}} \right) \xi \ind{_{D]}} + \xi \ind{_{[C}} \varepsilon \ind{_{D][A}} \left( \xi \ind{^E} \nabla \ind{_{B]E}} \xi \ind{_F} \right) \xi \ind{^F} & = 0 \, ;
\end{align}
\item condition \eqref{eq-foliating} for a foliating spinor can be expressed as
 \begin{align}\label{eq-foliating-5d}
\left( \xi^B \nabla_{A B} \xi^C \right) \xi_C & = 0 \, , &  \xi_{\lb{A}} \left( \xi^E \nabla_{\rb{B}E} \xi_{\lb{C}} \right) \xi_{\rb{D}} & = 0 \, .
 \end{align}
\end{itemize}
As an example, one can check that a solution $\xi^A$ of the twistor equation
\begin{align*}
 \nabla \ind{_{AB}} \xi \ind{_C} + \frac{1}{5} \varepsilon \ind{_{AB}} \zeta \ind{_C} + \frac{4}{5} \zeta \ind{_{\lb{A}}} \varepsilon \ind{_{\rb{B}C}} & = 0 \, , & \zeta \ind{^A} & = \nabla \ind{^{AB}} \xi \ind{_B} \, ,
\end{align*}
satisfies equations \eqref{eq-foliating-5d} if and only if $\xi^A \zeta_A = 0$ as claimed in Proposition \ref{prop-conformal-Killing-foliating}.
Geometry.tex
\section{Differential geometry of pure spinor fields}\label{sec-geometry}
Throughout this section, $(\mcM,g)$ will denote an $n$-dimensional oriented complex Riemannian manifold, where $n=2m+1$, i.e. a complex manifold equipped with a global holomorphic non-degenerate symmetric bilinear form $g_{ab}$. We assume for $n>3$ definiteness. The holomorphic tangent and cotangent bundles will be denoted by $\Tgt \mcM$ and $\Tgt^* \mcM$ respectively. Thus, $g_{ab}$ is a holomorphic section of $\odot^2 \Tgt^* \mcM$. The structure group of the frame bundle $\mcF \mcM$ of $\mcM$ is reduced to $\SO(2m+1,\C)$.

The complex Riemannian manifold $(\mcM,g)$ admits a unique torsion-free connection preserving $g_{ab}$, the holomorphic Levi-Civita connection, which, together with its associated covariant derivative, will be denoted $\nabla_a$. For convenience, we can choose an orthonormal frame $\{ \delta \ind*{^a_{\mbf{a}}} \}$ with dual $\{ \delta \ind*{_a^{\mbf{a}}} \}$, where bold lower case Roman indices run from $1$ to $n$. The connection $\nabla_a$ can then be represented by the $\so(2m+1,\C)$-valued $1$-form $\Gamma \ind{_{\mbf{ab}}^{\mbf{c}}} := ( \delta \ind*{^a_{\mbf{a}}} \nabla \ind{_a} \delta \ind*{_{\mbf{b}}^b} ) \delta \ind*{_b^{\mbf{c}}}$. Choose a connnection $\partial_a$ compatible with $g_{ab}$, $\{ \delta \ind*{^a_{\mbf{a}}} \}$ and $\{ \delta \ind*{_a^{\mbf{a}}} \}$. Then the connections $\nabla_a$ and $\partial_a$ are related via
\begin{align}\label{eq-conn1-form}
\nabla \ind{_a} V \ind{^b} & = \partial \ind{_a} V \ind{^b} + \Gamma \ind{_{ac}^b} V \ind{^c} \, ,
\end{align}
for any holomorphic vector field $V \ind{^a}$, and where $\Gamma \ind{_{ab}^c} := \delta \ind*{_a^{\mbf{a}}} \delta \ind*{_b^{\mbf{b}}} \Gamma \ind{_{\mbf{ab}}^{\mbf{c}}} \delta \ind*{^c_{\mbf{c}}}$. The Riemann curvature $R \ind{_{abcd}}$ of $\nabla_a$ is defined by
\begin{align*}
2 \, \nabla \ind{_{[a}} \nabla \ind{_{b]}} V \ind{^c} & = R \ind{_{abd}^c} V \ind{^d} \, ,
\end{align*}
for any holomorphic vector field $V^a$, and similarly for holomorphic sections of $\Tgt^* \mcM$, and tensor products.

The splitting of the Riemann tensor into irreducible $\SO(2m+1,\C)$-invariant parts, when $n>3$, is given by
\begin{align}\label{eq-Riem_decomposition}
R \ind{_{abcd}} & =
C \ind{_{abcd}} + \frac{4}{n-2} \Phi \ind{_{[a|[c}} g \ind{_{d]|b]}} + \frac{2}{n(n-1)} R \, g \ind{_{a[c}} g \ind{_{d]b}} \, , 
\end{align}
where $C \ind{_{abcd}}$ is the Weyl tensor, $\Phi \ind{_{ab}}$ is the tracefree part of the Ricci tensor $R \ind{_{ab}} := R \ind{_{acb}^c}$, and $R := R \ind{_a^a}$ is the Ricci scalar.

\begin{rem}
It is worth pointing out that in the special case $n=3$, excluded from the main discussion, the Weyl tensor vanishes identically, i.e. $R \ind{_{abcd}} = 4 \, \Phi \ind{_{[a|[c}} g \ind{_{d]|b]}} + \frac{1}{3} R \, g \ind{_{a[c}} g \ind{_{d]b}}$. Nonetheless, many of the statements made in this section still apply by setting $C \ind{_{abcd}} = 0$. A two-spinor calculus in dimension $3$ is briefly presented in appendix \ref{sec-3dim}.
\end{rem}

We shall assume that $(\mcM,g)$ is equipped with a spin structure -- see \cite{Lawson1989} for definitions and topological restrictions. The spinor bundle on $\mcM$ admits a canonical connection as a lift of the Levi-Civita connection, and for this reason, this spinor connection will also be denoted $\nabla_a$. It preserves the Clifford module structure of the spinor bundle in the sense that
\begin{align*}
\nabla  \ind{_a} \left( V \ind{^b} \gamma \ind{_b_C^B} \xi \ind{^C} \right) & = \left( \nabla  \ind{_a} V \ind{^b} \right) \gamma \ind{_b_C^A} \xi \ind{^C} + V \ind{^b} \gamma \ind{_b_C^A} \nabla  \ind{_a} \xi \ind{^C} \, ,
\end{align*}
for any holomorphic vector field $V \ind{^a}$ and holomorphic spinor field $\xi^A$, and similarly for dual spinor fields. The relation between the lifts of $\partial_a$ introduced in \eqref{eq-conn1-form} and $\nabla_a$ to the spinor bundle is given by
\begin{align}\label{eq-cov-der-spinor}
\nabla \ind{_a} \xi \ind{^A} & = \partial \ind{_a} \xi \ind{^A} - \frac{1}{4} \Gamma \ind{_{abc}} \gamma \ind{^{bc}_B^A} \xi \ind{^B} \, .
\end{align}
for any holomorphic spinor field $\xi^A$. Similarly, the curvature tensor of the spinor bundle can be expressed as a lift of the Riemann curvature tensor, i.e.
\begin{align*}
 2 \nabla \ind{_{\lb{a}}} \nabla \ind{_{\rb{b}}} \xi^A & = - \frac{1}{4} R \ind{_{abcd}} \gamma \ind{^{cd} _B^A} \xi^B \, .
\end{align*}
for any holomorphic spinor field $\xi \ind{^A}$, and similarly for dual spinor fields.

\begin{rem}[Notation]
As in the previous sections, we shall make use of the short-hand notation
\begin{align*}
 \xi \ind{_{a_1 \ldots a_k} ^A} := \xi \ind{^B} \gamma \ind{_{a_1 \ldots a_k} _B ^A}
\end{align*}
for any holomorphic spinor field $\xi^A$ and any $k>0$.
\end{rem}

\subsection{Projective pure spinor fields}
A spinor field $\xi^A$ is said to be \emph{pure} if at every point $p \in \mcM$, $\xi^A$ defines a totally null $m$-dimensional vector subspace
\begin{align*}
\mcN_ p & := \{ X \ind{^a} \in \Tgt_p \mcM : X \ind{^a} \xi \ind{_a^A} = 0 \} \, ,
\end{align*}
of $\Tgt_p \mcM$. Clearly, at every point, the spinor $\xi^A$ will satisfy the purity condition \eqref{eq-purity_cond} or \eqref{eq-purity_cond_Cartan}. Thus, a pure spinor field defines a $\gamma$-plane distribution $\mcN$, and conversely, any such distribution determines a pure spinor field up to scale. This motivates the following definition, which, it is worth pointing out, does not require the existence of a spin structure on $\mcM$.

\begin{defn}
The \emph{projective pure spinor bundle $\Gr_m (\Tgt \mcM,g)$} is the bundle over $\mcM$ whose fiber over a point $p$ of $\mcM$ is isomorphic to the $\frac{1}{2}m(m+1)$-dimensional family $\Gr_m (\Tgt_p \mcM , g)$ of $\gamma$-planes in $\Tgt_p \mcM$.
\end{defn}

Now, the existence of a holomorphic projective pure spinor field $\langle \xi^A \rangle$ on $\mcM$, i.e. a (global) holomorphic section of $\Gr_m (\Tgt \mcM,g)$, with associated holomorphic distribution $\mcN$, is equivalent to a reduction of the structure group of the frame bundle of $\mcM$ to $P$, the parabolic Lie subgroup of $G:= \SO(2m+1,\C)$ stabilising $\langle \xi^A \rangle$, as described in section \ref{sec-algebra}. It is a standard result, e.g. \cites{Salamon1989,vCap2009} that one can construct holomorphic vector bundles from representations of $P$ or its Lie algebra $\prb$. Thus, if $\mfC$ is the irreducible $G$-module of tensors with Weyl symmetries, its associated vector bundle $\mcC := \mcF \mcM \times_G \mfC$ admits a filtration of $P$-invariant subbundles $\mcC^i := \mcF \mcM \times_P \mfC^i$, and similarly, to the associated graded $\prb$-modules $\gr(\mfC)$, $\g_0$-modules $\mfC_i$, and their irreducibles correspond associated graded vector bundles, and so on. Consequently, one can give pointwise algebraic characterisation of tensors on $\mcM$ in terms of irreducible $\prb$-modules, such as the ones defined in sections \ref{sec-algebra}, \ref{sec-curvature} and \ref{sec-intrinsic-torsion}, and we shall make use of the maps ${}^\mfF _\xi \Pi_i^j$, ${}^\mfA _\xi \Pi_i^j$ and ${}^\mfC _\xi \Pi_i^j$ to describe the curvature tensors as local sections of some $\prb$-invariant subbundles in the obvious way and notation.

\begin{rem}
For convenience, we can choose a spinor field $\eta_A$ dual to $\xi^A$ as described in section \ref{sec-algebra}, and let $\{\delta \ind*{^a^{\mbf{A}}}\}$ and $\{\delta \ind*{^a_{\mbf{A}}}\}$ be bases of holomorphic vector fields annihilating $\xi^A$ and $\eta_A$ respectively, which must be dual to each other, i.e. $\delta \ind*{^a_{\mbf{A}}} \delta \ind*{_a^{\mbf{B}}} = \delta \ind*{_{\mbf{A}}^{\mbf{B}}}$. Here, bold upper cases Roman indices run from $1$ to $m$. In a null frame $\{\delta \ind*{^a_{\mbf{A}}} , \delta \ind{^a^{\mbf{A}}} , u^a := 2 \, \xi \ind*{^a^A} \eta \ind{_A} \}$, the tensors $\Gamma \ind{^{ABC}}$, $\Gamma \ind{^{A:}^B}$, etc... appearing in \eqref{eq-connection1form} can be thought of as the abstract versions of the components $\Gamma \ind{^{\mbf{A}\mbf{B}\mbf{C}}} = \left( \delta \ind*{^a^{\mbf{A}}} \nabla \ind*{_a} \delta \ind*{^b^{\mbf{B}}} \right) \delta \ind*{_b^{\mbf{C}}}$, $\Gamma \ind{_{\mbf{A}:}^{\mbf{B}}} = \left( \delta \ind*{^a_{\mbf{A}}} \nabla \ind*{_a} \delta \ind*{^b^{\mbf{B}}} \right) u \ind{_b}$, and so on, of the connection $1$-form $\Gamma \ind{_{ab}^c}$.

Now, choosing our connection $\partial _a$ to preserve $\xi^A$ and $\eta_A$, the covariant derivative \eqref{eq-cov-der-spinor}.of $\xi^A$ reduces to
\begin{align*}
 \nabla \ind{_a} \xi \ind{^D} 
& = - \frac{1}{4}  \Gamma \ind{_{abc}} \gamma \ind{^{bc}_A^D} \xi \ind{^A} \, .
\end{align*}
\end{rem}

\begin{rem}
This section applies equally to a $(2m+1)$-dimensional smooth manifold equipped with a smooth metric of split signature. In this case, the projective pure spinor field $\langle \xi^A \rangle$ is assumed to be \emph{real} in the sense that its associated null $m$-plane distribution is real.
\end{rem}


\subsubsection{The intrinsic torsion of a projective pure spinor field}
The obstruction to the existence of a torsion-free affine connection compatible with the $P$-structure associated to a holomorphic projective pure spinor field $\langle \xi^A \rangle$ on $(\mcM,g)$ is known as the \emph{intrinsic torsion} \cite{Salamon1989} of the $P$-structure. Such a notion can be more concretely expressed in terms of the Levi-Civita covariant derivative of $\langle \xi^A \rangle$. One can then define the \emph{intrinsic torsion} of a holomorphic projective pure spinor field $\langle \xi^A \rangle$ on $(\mcM,g)$ as the measure of the failure of the Levi-Civita connection to be preserve $\langle \xi^A \rangle$, i.e.
\begin{align}\label{eq-recurrent}
\nabla \ind{_a} \langle \xi \ind{^A} \rangle & = 0 \, , & \mbox{i.e.} & &  \nabla \ind{_a} \xi \ind{^A} & = \alpha \ind{_a} \xi \ind{^A} \, ,
\end{align}
for some $1$-form $\alpha_a$, or equivalently, the failure of the connection $1$-form to take values in $\prb$. Since $\nabla_a$ takes values in $\g$, this measure is precisely obtained by consideration of the module $\mfW := \mfV \otimes \left( \g / \prb \right)$, which we defined in section \ref{sec-intrinsic-torsion}. The intrinsic torsion of $\xi^A$ can then simply be identified with
\begin{align*}
\Gamma \ind{_{abc}} \xi \ind{^b^B} \xi \ind{^c^C} \qquad \in \qquad \mfV^{-1} \otimes \wedge^2 \mfS^{\frac{m-2}{2}} \, ,
\end{align*}
where $\Gamma \ind{_{abc}}$ is a choice of connection $1$-form associated to $\nabla_a$. We shall then be able to apply Proposition \ref{prop-intorsion-class-alg} and the maps ${}^\mfW _\xi \Pi_i^j$ describing the various $\prb$-submodules of $\mfW$ to characterise the intrinsic torsion of $\langle \xi^A \rangle$. Before that, we compute, from \eqref{eq-cov-der-spinor} and \eqref{eq-useful-id}, the formula
\begin{align*}
 \left( \nabla \ind{_a} \xi \ind{^b^B} \right) \xi \ind{_b^C} = - \left( \nabla \ind{_a} \xi \ind{^B} \right) \xi \ind{^C} + \Gamma \ind{_{abc}} \xi \ind{^b^B} \xi \ind{^c^C} \, 
\end{align*}
from which we deduce
\begin{align*}
\left( \nabla \ind{_a} \xi \ind{^b^{(B}} \right) \xi \ind{_b^{C)}} & = - \left( \nabla \ind{_a} \xi \ind{^{(B}} \right) \xi \ind{^{C)}} \, , &
 \left( \nabla \ind{_a} \xi \ind{^b^{[B}} \right) \xi \ind{_b^{C]}} & = - \left( \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi \ind{^{C]}} + \Gamma \ind{_{abc}} \xi \ind{^b^B} \xi \ind{^c^C} \, , \\
  \left( \nabla \ind{_a} \xi \ind{^b^{[B}} \right) \xi \ind{_b^C} \xi \ind{^{D]}} & = \Gamma \ind{_{abc}} \xi \ind{^b^{[B}} \xi \ind{^c^C} \xi \ind{^{D]}} \, , &
  \left( \nabla \ind{_a} \xi \ind{^b^{[A}} \right)  \xi \ind{^{B]}} \xi \ind{_b^{[C}} \xi \ind{^{D]}} & = \Gamma \ind{_{abc}} \xi \ind{^b^{[A}} \xi \ind{^{B]}} \xi \ind{^c^{[C}} \xi^{D]} \, .
\end{align*}
The first of these identities is trivially satisfied by virtue of the purity condition. We are now in position to give covariant expressions characterising the intrinsic torsion of a pure spinor field.

\begin{prop}\label{prop-intorsion-class}
 Let $\langle \xi^A \rangle$ be a holomorphic projective pure spinor field on $(\mcM,g)$, and let $\Gamma _{abc} \xi^{bB} \xi^{cC} \in \mfW$ be its associated intrinsic torsion. Then, pointwise,
 \begin{itemize}
  \item ${}^\mfW _\xi \Pi_{-3}^0 (\Gamma ) = 0$ if and only if ($m>2$ only)
  \begin{align}\label{eq-cond-30}
  \left( \xi \ind{^{a[A}} \nabla \ind{_a} \xi \ind{^b^{B}} \right) \xi \ind{_b^C} \xi^{D]} & = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_{-3}^1 (\Gamma ) = 0$ if and only if
  \begin{align}\label{eq-cond-31}
      \xi \ind{^{[A}} \left( \xi \ind{^{aB]}} \nabla \ind{_a} \xi \ind{^b^{[C}} \right) \xi^{D]} \xi \ind{_b^{[E}} \xi^{F]} + \xi \ind{^{[C}} \left( \xi \ind{^{aD]}} \nabla \ind{_a} \xi \ind{^b^{[A}} \right) \xi^{B]} \xi \ind{_b^{[E}} \xi^{F]} & = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_{-2}^0 (\Gamma ) = 0$ if and only if
    \begin{align}\label{eq-cond-20}
      \left( \xi \ind{^{aA}} \nabla \ind{_a} \xi \ind{^b^{[B}} \right) \xi \ind{_b^{C}} \xi^{D]} & = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_{-2}^1 (\Gamma ) = 0$ if and only if
  \begin{align}\label{eq-cond-21}
   \left( \xi \ind{^{a[A}} \nabla \ind{_a} \xi \ind{^{B}} \right) \xi^{C]} & = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_{-2}^2 (\Gamma ) = 0$ if and only if
    \begin{align}\label{eq-cond-22}
  \xi \ind{^{[A}} \left( \xi \ind{^{aB]}} \nabla \ind{_a} \xi \ind{_b^E}  \right) \xi \ind{^b^{[C}} \xi \ind{^{D]}} + \xi \ind{^{[C}} \left( \xi \ind{^{aD]}} \nabla \ind{_a} \xi \ind{_b^E} \right) \xi \ind{^b^{[A}} \xi^{B]} &  = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_{-1}^0 (\Gamma ) = 0$ if and only if
    \begin{align}
   \left( \gamma \ind{^a_D^A} \nabla \ind{_a} \xi \ind{^{bD}} \right) \xi \ind{_b^{[B}} \xi^{C]} + 2 \left( \xi \ind{^{aA}} \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi^{C]} & = 0 \, ; \label{eq-cond-10} 
  \end{align}
  \item ${}^\mfW _\xi \Pi_{-1}^1 (\Gamma ) = 0$ if and only if ($m>2$ only)
  \begin{align}\label{eq-cond-11}
      \left( \nabla \ind{_a} \xi \ind{^b^{[B}} \right) \xi \ind{_b^{C}} \xi^{D]} + \frac{1}{m-1} \gamma \ind{_a_E^{[B}} \left( \left( \gamma \ind{^c_F^E} \nabla \ind{_c} \xi \ind{^{bF}} \right) \xi \ind{_b^{|C}} + 2 \left( \xi \ind{^{bE}} \nabla \ind{_b} \xi \ind{^{|C}} \right) \right) \xi \ind{^{D]}} & = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_{-1}^2 (\Gamma ) = 0$ if and only if
  \begin{align}\label{eq-cond-12}
   \left( \xi \ind{^{aA}} \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi^{C]} & = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_0^0 (\Gamma ) = 0$ if and only if
  \begin{align}\label{eq-cond00}
   \left( \nabla \ind{_a} \xi \ind{^{aA}} \right) \xi \ind{^B} - \xi \ind{^{aA}} \nabla \ind{_a} \xi \ind{^B}  & = 0 \, ;
  \end{align}
  \item ${}^\mfW _\xi \Pi_0^1 (\Gamma ) = 0$ if and only if
  \begin{align}\label{eq-cond01}
 \left( \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi^{C]} - \frac{2}{m} \left( \left( \nabla \ind{_b} \xi \ind{^{b[B}} \right) \xi \ind{_a^{C]}} - \xi \ind{^{b[B}}  \nabla \ind{_b} \xi \ind{_a^{C]}} \right) & = 0 \, .
  \end{align}
 \end{itemize}
\end{prop}



\subsubsection{Geometric properties}
We come to one of the main motivations of this article, finding differential conditions on a projective pure spinor to describe the integrability properties of its associated $\gamma$-plane distribution and its orthogonal complement.

\begin{prop}\label{prop-foliating-spinor} Let $\langle \xi^A \rangle$ be a holomorphic projective pure spinor field on $(\mcM,g)$ with associated $\gamma$-plane distribution $\mcN$, and denote by $\mcN^\perp$ the orthogonal complement of $\mcN$. Then
 \begin{itemize}
  \item $[\Gamma(\mcN) , \Gamma(\mcN) ] \subset \Gamma(\mcN^\perp)$ if and only if
\begin{align}\label{eq-almost-foliating}
      \xi \ind{^{[A}} \left( \xi \ind{^{aB]}} \nabla \ind{_a} \xi \ind{^b^{[C}} \right)  \xi \ind{_b^{D}} \xi^{E]} & = 0 \, ;
\end{align}
  \item $\mcN$ is integrable, i.e. $[\Gamma(\mcN) , \Gamma(\mcN) ] \subset \Gamma(\mcN)$ if and only if
\begin{align}\tag{\ref{eq-cond-21}}
   \left( \xi \ind{^{a[A}} \nabla \ind{_a} \xi \ind{^{B}} \right) \xi^{C]} & = 0 \, ;
\end{align}
  \item $\mcN$ is integrable and geodetic, i.e. $\nabla_X Y \in \Gamma(\mcN)$ for all $X,Y \in \Gamma(\mcN)$ if and only if
   \begin{align}\label{eq-N-geodetic}
   \left( \xi \ind{^{[A}} \xi \ind{^{aB]}} \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi^{C]} & = 0 \, .
  \end{align}
  \item $\mcN^\perp$ is integrable, i.e. $[\Gamma(\mcN^\perp) , \Gamma(\mcN^\perp) ] \subset \Gamma(\mcN^\perp)$ if and only if
\begin{align}\label{eq-Nperp-int}
 \left( \xi \ind{^a^{[A}} \nabla \ind{_a} \xi \ind{^b^{B]}} \right) \xi \ind{_b^{[C}} \xi \ind{^{D]}} & = 0 \, ;
\end{align}
  \item $\mcN$ and $\mcN^\perp$ are integrable, i.e. $[\Gamma(\mcN) , \Gamma(\mcN) ] \subset \Gamma(\mcN)$ and $[\Gamma(\mcN^\perp) , \Gamma(\mcN^\perp) ] \subset \Gamma(\mcN^\perp)$ if and only if
  \begin{align}\label{eq-foliating}
      \left( \xi \ind{^{aA}} \nabla \ind{_a} \xi \ind{^b^{[B}} \right) \xi \ind{_b^{C}} \xi^{D]} & = 0 \, ;
&
      \xi \ind{^{[A}} \left( \xi \ind{^{aB]}} \nabla \ind{_a} \xi \ind{^{[C}} \right) \xi \ind{^{D]}} & = 0 \, .
  \end{align}
    \item $\mcN^\perp$ is integrable and geodetic, i.e. $\nabla_X Y \in \Gamma(\mcN^\perp)$ for all $X,Y \in \Gamma(\mcN^\perp)$ if and only if
   \begin{align}\tag{\ref{eq-cond-12}}
   \left( \xi \ind{^{aA}} \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi^{C]} & = 0 \, .
  \end{align}
   \end{itemize}
\end{prop}

\begin{rem}
By Proposition \ref{prop-intorsion-class}, conditions \eqref{eq-almost-foliating} and \eqref{eq-foliating} are equivalent to the intrinsic torsion of $\langle \xi^A \rangle$ lying in the module $\mfW^{-2}$ and $\mfW^{-1}$ respectively. Further, condition \eqref{eq-Nperp-int} implies that the intrinsic torsion belongs to the module $\mfW^{-2}$ and that ${}^\mfW _\xi \Pi_{-2}^2 (\Gamma) = 0$.
\end{rem}

\begin{proof}
It is a matter of applying the above Lie bracket relations on vector fields of the form $V \ind{^a} = \xi \ind{^a^A} v \ind{_A}$, where $v \ind{_A}$ lies in $\mfS_{-\frac{m}{2}}$ or $\mfS_{-\frac{m-2}{2}}$, and, with reference to \eqref{eq-connection1form}, we note that each of the above conditions is equivalent to 
\begin{itemize}
\item $\Gamma \ind{^{ABC}} =0$,
\item $\Gamma \ind{^{ABC}} = \Gamma \ind{^{[A:B]}} = 0$,
\item $\Gamma \ind{^{ABC}} = \Gamma \ind{^{A:B}} = 0$,
\item $\Gamma \ind{^{ABC}}=0$ and $\Gamma \ind{^{A:B}} = \Gamma \ind{^{AB}}$, (in particular, $\Gamma \ind{^{(A:B)}} = 0$),
\item $\Gamma \ind{^{ABC}} = \Gamma \ind{^{AB}} = \Gamma \ind{^{A:B}} = 0$,
\item $\Gamma \ind{^{ABC}} = \Gamma \ind{^{AB}} = \Gamma \ind{^{A:B}} = \Gamma \ind{^A} = 0$,
\end{itemize}
respectively.
\end{proof}

\begin{rem}
We note that unlike in even dimensions, the leaves of the foliation of an integrable $\gamma$-plane distribution $\mcN$ are not necessarily geodetic, and similarly for its orthogonal complement. Further, since \eqref{eq-cond-12} implies \eqref{eq-N-geodetic}, we see that if $\mcN^\perp$ is integrable \emph{and} geodetic, then so is $\mcN$.
\end{rem}

The Frobenius theorem tells us that the involutivity of a totally null $m$-plane distribution is equivalent to its integrability, i.e. the existence of a local foliation of some region of $(\mcM,g)$ by $m$-dimensional leaves tangent to the distribution, and similarly for its orthogonal complement. This motivates the following definitions.

\begin{defn}
 Let $\langle \xi^A \rangle$ be a holomorphic projective pure spinor field on $(\mcM,g)$ with associated $\gamma$-plane distribution $\mcN$, and denote by $\mcN^\perp$ the orthogonal complement of $\mcN$. We say that
 \begin{itemize}
 \item $\langle \xi^A \rangle$ is \emph{almost foliating} if $[\Gamma(\mcN),\Gamma(\mcN)] \subset \Gamma(\mcN^\perp)$, i.e. $\xi^A$ satisfies \eqref{eq-almost-foliating};
 \item $\langle \xi^A \rangle$ is \emph{$\gamma$-foliating}, resp. \emph{$\gamma^\perp$-foliating}, if $\mcN$, resp. $\mcN^\perp$, is integrable, i.e. $\xi^A$ satisfies \eqref{eq-cond-21}, resp. \eqref{eq-Nperp-int};
 \item $\langle \xi^A \rangle$ is \emph{foliating} if both $\mcN$ and $\mcN^\perp$ are integrable, i.e. $\xi^A$ satisfies \eqref{eq-foliating};
 \item $\langle \xi^A \rangle$ is \emph{geodetic}, resp. \emph{co-geodetic} if $\mcN$, resp. $\mcN^\perp$, is integrable and geodetic, i.e. $\xi^A$ satisfies \eqref{eq-N-geodetic}, resp. \eqref{eq-cond-12}.
 \end{itemize}
\end{defn}

\paragraph{Conformal invariance}
With reference to appendix \ref{sec-conformal}, we prove
\begin{prop}\label{prop-conformal}
 Conditions \eqref{eq-cond-30}, \eqref{eq-cond-31}, \eqref{eq-cond-20}, \eqref{eq-cond-21}, \eqref{eq-cond-22}, \eqref{eq-cond-11} and \eqref{eq-Nperp-int}, (and  thus \eqref{eq-almost-foliating}, \eqref{eq-foliating}) are conformally invariant.
 
 Suppose that $\langle \xi \ind{^A} \rangle$ satisfies \eqref{eq-cond-11} and 
     \begin{align*}
   \left( \gamma \ind{^a_D^A} \nabla \ind{_a} \xi \ind{^{bD}} \right) \xi \ind{_b^{[B}} \xi^{C]} + 2 \left( \xi \ind{^{aA}} \nabla \ind{_a} \xi \ind{^{[B}} \right) \xi^{C]} & = (n-3) \, \xi^A \xi^{[B} \xi \ind{^b^{C]}} \nabla_b f  \, ;
  \end{align*}
for some holomorphic function $f$. Then there exists a (local) conformal rescaling for which $\langle \xi \ind{^A} \rangle$ satisfies
    \begin{align}\label{eq-N-parallel}
      \left( \nabla \ind{_a} \xi \ind{^b^{[B}} \right) \xi \ind{_b^{C}} \xi^{D]} & = 0 \, ,
  \end{align}
i.e. $\nabla X \in \Gamma ( \mcN^\perp )$ for all $X \in \Gamma ( \mcN )$, where $\mcN$ is the $\gamma$-plane distribution annihilating $\langle \xi^A \rangle$.
\end{prop}

\begin{rem}
 The second part of the above proposition should be compared with its even-dimensional counterpart given in \cite{Taghavi-Chabert2012a}, which was argued to be a complex Riemannian analogue of Hermitian manifolds locally conformal to K\"{a}hler \cite{Falcitelli1994}, in the sense of `conformal to a complex Riemmannian manifold admitting a recurrent pure spinor field'. Proposition \ref{prop-conformal} shows that in odd dimensions this is not the case, and the appropriate generalisation is \eqref{eq-N-parallel}, not \eqref{eq-recurrent}.
\end{rem}



\paragraph{Curvature conditions}
The integrability conditions for these equations can easily be computed by differentiation a second time and commuting the covariant derivatives. 
\begin{prop}\label{prop-foliation-int-cond}
 Let $\xi^A$ be a geodetic or foliating spinor on $(\mcM,g)$, i.e. $\xi^A$ satisfies \eqref{eq-N-geodetic} or \eqref{eq-foliating}. Then
\begin{align*}
{}^\mfC _\xi \Pi_{-3}^0 (C) & = 0 \, , & & \mbox{i.e.} & \xi \ind{^{[A}} \xi \ind{^a ^B} \xi \ind{^b ^{C]}} \xi \ind{^c ^D} \xi \ind{^d ^E} C_{abcd} & = 0 \, .
\end{align*}
Suppose further that $\xi^A$ is co-geodetic, i.e. $\xi^A$ satisfies \eqref{eq-cond-12}. Then
\begin{align*}
\xi \ind{^a ^A} \xi \ind{^b ^B} \xi \ind{^c ^C} \xi \ind{^d ^D} R_{abcd} & = 0 \, ,
\end{align*}
and 
\begin{align*}
{}^\mfF _\xi \Pi_{-2}^0 (\Phi) & = 0 & \Longleftrightarrow & & {}^\mfC _\xi \Pi_{-2}^1 (C) & = 0 \, ,
\end{align*}
i.e. $\xi \ind{^{[A}} \xi \ind{^a^{B]}} \Phi \ind{_{ab}} \xi \ind{^b ^{[C}} \xi \ind{^{D]}} = 0$ if and only if $\xi \ind{^a ^A} \xi \ind{^b ^B} \xi \ind{^c ^C} \xi \ind{^d ^D} C_{abcd} = 0$.
\end{prop}

The integrability condition for the existence of a parallel projective pure spinor is given next - (see also \cite{Galaev2013} for other types of recurrent spinors).
\begin{prop}\label{prop-recurrent-spinor}
Let $\langle \xi^A \rangle$ be a parallel holomorphic projective pure spinor on $(\mcM,g)$, i.e. $\xi^A$ satisfies \eqref{eq-recurrent}. Then
\begin{align}
\xi \ind{^a ^A} \xi \ind{^b ^B} R_{abcd} & = 0 \, , \label{eq-int_cond_Riem} \\
\xi \ind{^a ^A} \xi \ind{^b ^B} R_{ab} & = 0 \, , \label{eq-int_cond_Ric}\\
\xi \ind{^a ^A} \xi \ind{^b ^{\lb{B}}} \xi \ind{^{\rb{C}}} \Phi_{ab} & = 0 \, , & & \mbox{i.e.} & {}^\mfF _\xi \Pi_{-1}^0 ( \Phi) & = 0 \, , \label{eq-int_cond_Phi} \\
\xi \ind{^a ^A} \xi \ind{^b ^B} \xi \ind{^c ^{\lb{C}}} \xi \ind{^{\rb{D}}} C_{abcd} & = 0 & & \mbox{i.e.} & {}^\mfC _\xi \Pi_{-1}^0 ( C ) = {}^\mfC _\xi \Pi_{-1}^1 ( C ) = {}^\mfC _\xi \Pi_{-1}^2 ( C ) & = 0 \, , \label{eq-int_cond_Weyl}
\end{align}
and in addition, when $m>2$,
\begin{align}
{}^\mfC _\xi \Pi_1^1 (C) & = 0 \, . \label{eq-m3d-cond}
\end{align}
Further,
\begin{align}
 R & = 0 & & \Longleftrightarrow & {}^\mfF _\xi \Pi_0^0 (\Phi) & = 0 & & \Longleftrightarrow & {}^\mfC _\xi \Pi_0^0 (C) & = 0  \, , \label{eq-1d-cond} \\
 {}^\mfF _\xi \Pi_0^1 (\Phi) & = 0 \, , & & \Longleftrightarrow & {}^\mfC _\xi \Pi_0^1 (C) & = 0 & & \Longleftrightarrow & {}^\mfC _\xi \Pi_0^2 (C) & = 0 \, , \label{eq-m2-1d-cond} \\
 {}^\mfF _\xi \Pi_1^0 (\Phi) & = 0 & & \Longleftrightarrow & {}^\mfC _\xi \Pi_1^0 (C) & = 0 \, . \label{eq-md-cond} 
\end{align}
\end{prop}

\begin{proof}
 Taking a second covariant derivative of \eqref{eq-recurrent}, and commuting yield $- \frac{1}{4} R \ind{_{abcd}} \xi \ind{^{cd} ^A} = \left(\nabla _{\lb{a}} \alpha _{\rb{b}} \right) \xi^A$, which is equivalent to equation \eqref{eq-int_cond_Riem}. Equations \eqref{eq-int_cond_Ric} and \eqref{eq-int_cond_Phi} are then obtained by contracting equation \eqref{eq-int_cond_Riem} with $\gamma \ind{^{cd}_A^B}$, and using the expression relating $R_{ab}$ and $\Phi_{ab}$ for which we find
\begin{align*}
 \Phi \ind{_{ab}} \xi \ind{^a^A} \xi \ind{^b^B} & = \frac{1}{n} R \, \xi \ind{^A} \xi \ind{^B} \, ,
\end{align*}
from which we conclude \eqref{eq-1d-cond}.
Next, the decomposition of the Riemann tensor yields equation \eqref{eq-int_cond_Weyl}
\begin{align*}
 \xi \ind{^c^C} C \ind{_{c[ab]d}} \xi \ind{^d^D} & = \frac{2}{n-2} \xi \ind{_{[a}^{[C}} \Phi \ind{_{b]d}} \xi \ind{^{dD]}} + \frac{1}{n(n-1)}  R \, \xi \ind{_{[a}^C} \xi \ind{_{b]}^D} \, , \\
 C \ind{_{bcad}} \xi \ind{^{bc}^B} \xi \ind{^d^D} & = - \frac{2(n-4)}{n-2} \xi \ind{^B} \xi \ind{^d^D} \Phi \ind{_{ad}} - \frac{2}{n-2} \xi \ind{^D} \xi \ind{^{dB}} \Phi \ind{_{ad}}  - \frac{2(n-2)}{n(n-1)} R \, \xi \ind{_a^D} \xi \ind{^B} + \frac{2}{n(n-1)(n-2)} R \, \xi \ind{_a^B} \xi \ind{^D} \, , \\
    C \ind{_{bcad}} \xi \ind{^{bc}^{(B}} \xi \ind{^d^{D)}} & = - \frac{2(n-3)}{n-2} \left( \xi \ind{^{(B}} \xi \ind{^d^{D)}} \Phi \ind{_{ad}} + \frac{1}{n} R \, \xi \ind{^{(B}} \xi \ind{_a^{D)}} \right) \, , \\
 C \ind{_{bcad}} \xi \ind{^{ad}^A} \xi \ind{^{bc}^B} & = - 2 \frac{n-3}{n-2} R \, \xi \ind{^A} \xi \ind{^B}  \, , \\
 C \ind{_{abcd}} \xi \ind{^b^B} \xi \ind{^c^C} \xi \ind{^d^D} & = - \frac{2}{n-2} \xi \ind{^B} \xi \ind{^{[C}} \Phi \ind{_{ad}} \xi \ind{^d^{D]}} + \frac{2}{n(n-1)(n-2)} R \, \xi \ind{^B} \xi \ind{^{[C}} \xi \ind{_a^{D]}} \, .
\end{align*}
A further contraction with $\xi^{aA}$ in the last line yields \eqref{eq-int_cond_Weyl}. To conclude the remaining conditions, we note that the dimensions of the irreducible $\prb$-invariant parts of the Weyl tensor must match those of the tracefree Ricci tensor. Thus, from the first line, we can conclude immediately \eqref{eq-m3d-cond} and \eqref{eq-md-cond}. From the invariant diagram of Proposition \ref{prop-Weyl-classification}, one sees that condition \eqref{eq-m3d-cond} imposes algebraic conditions on elements of the isotopic modules $\mfC_0^1$ and $\mfC_0^2$, which, by dimension counting must match $\mfF_0^1$. More explicitly, on referring to the maps ${}^\mfC _\xi \Pi_i^j$, we have
\begin{align*}
 \xi^A {}^\mfC _\xi \Pi_0^1 ( C ) \ind{^{BC}_a} & = - 2 \frac{n-5}{n-3} \xi \ind{^{(B}} {}^\mfF _\xi \Pi_0^1 ( \Phi ) \ind{_a^{C)A}} \, , &
 {}^\mfC _\xi \Pi_0^2 ( C ) \ind{^{A(BC)}_d} & = - \frac{1}{n-2} \xi \ind{^{(B}} {}^\mfF _\xi \Pi_0^1 ( \Phi ) \ind{_d^{C)A}} \pmod{ \xi^B \xi^C \alpha_a^D } \, , 
\end{align*}
where we have rewritten
\begin{align*}
{}^\mfF _\xi \Pi_0^1 ( \Phi ) & := \xi \ind{^A} \xi \ind{^b ^{B}} \Phi \ind{_{b a}} - \frac{1}{n-1}  \xi \ind{^b ^A} \xi \ind{^c ^C} \Phi \ind{_{bc}} \gamma \ind{_b _C ^B} \pmod{ \xi^A \xi^B \alpha_a} \, .
\end{align*}
Condition \eqref{eq-m2-1d-cond} now follows.
\end{proof}




\subsection{Spinorial differential equations}
\subsubsection{Parallel pure spinors}
For reference, we recall the standard fact concerning parallel spinors -- see for instance \cite{Bryant2000} -- here, specialised to pure spinor fields.
\begin{prop}
 Let $\xi^A$ be a parallel pure spinor field on $(\mcM,g)$, i.e. $\nabla_a \xi^A = 0$. Then
 \begin{align*}
  R \ind{_{abcd}} \xi \ind{^{cd}^D} & = 0 \, , \\
 \Phi \ind{_{ab}} \xi \ind{^b^B} & = 0 \, , & & \mbox{i.e.} & {}^\mfF _\xi \Pi_1^0 (\Phi) & = 0 \, , \\
  R & = 0 \, , \\
  C \ind{_{abcd}} \xi \ind{^{cd}^D} & = 0 \, , & & \mbox{i.e.} & {}^\mfC _\xi \Pi_2^0 (C) & = 0 \, .
 \end{align*}
\end{prop}

The covariant derivative of a spinor field splits into two irreducible parts, which we shall presently turn our attention to.
\subsubsection{Null zero-rest-mass fields}
The smaller irreducible part of the covariant derivative of a spinor field $\xi^A$ leads to the \emph{(Weyl-)Dirac equation} 
\begin{align}\label{eq-Weyl-Dirac}
 \gamma \ind{^a _A^B} \nabla_a \xi^A & = 0 \, ,
\end{align}
In contrast to even dimensions, this equation admits not one, but \emph{two} generalisations to irreducible spinor fields of higher valence.

\begin{defn}
Let $\phi^{A_1 A_2 \ldots A_k} = \phi^{(A_1A_2 \ldots A_k)}$ be a holomorphic spinor field on $(\mcM,g)$ irreducible in the sense that
\begin{align*}
\gamma \ind{^a_{A_1}^C} \gamma \ind{_a_{A_2}^D} \phi^{A_1 A_2 A_3 \ldots A_k} = - \phi^{C D A_3\ldots A_k} \, .
\end{align*}
We say that $\phi \ind{^{A_1 \ldots A_k}}$ is a \emph{zero-rest-mass (zrm) field} if it satisfies
\begin{align}
 \gamma \ind{^a _{B}^{\lp{A_1}}} \nabla_a \phi^{A_2 \ldots \rp{A_k}B} & = 0 \, , \label{eq-zrm}
 \end{align}
and a \emph{co-zero-rest-mass (co-zrm) field} if it satisfies
\begin{align}
 \gamma \ind{^a _{B}^{\lb{A_1}}} \nabla_a \phi^{\rb{A_2} A_3 \ldots A_k B} & = 0 \, . \label{eq-co-zrm}
\end{align}
\end{defn}

\begin{rem}
When $k=2$, an irreducible spinor field as above is simply an $m$-form. Equation \eqref{eq-zrm}, respectively, \eqref{eq-co-zrm} are then equivalent to this $m$-form to be closed, respectively, \emph{co-closed}, hence the use of terminology.
\end{rem}

Equations \eqref{eq-zrm} and \eqref{eq-co-zrm} are conformally invariant provided that $\phi^{A_1 \ldots A_k}$ is of conformal weight $-\frac{n+k-1}{2}$ in the former case, and $-\frac{n+2k-1}{2}$ in the latter case. In particular, a solution of both \eqref{eq-zrm} and \eqref{eq-co-zrm}, i.e.
\begin{align}
 \gamma \ind{^a _{B}^{A_1}} \nabla_a \phi^{A_2 \ldots A_k B} & = 0 \, , \label{eq-nc-zrm} 
\end{align}
is not conformally invariant. This equation in the case $k=2$ corresponds to a closed and co-closed $m$-form.


The integrability condition for the existence of solutions to equations \eqref{eq-zrm} and \eqref{eq-co-zrm} of valence greater than two is given by the following lemma.
\begin{prop}\label{prop-int-zrm}
Assume $k>2$.
 Let $\phi^{A_1 A_2 \ldots A_k}$ be a solution of \eqref{eq-zrm} or \eqref{eq-co-zrm} on $(\mcM,g)$. Then
 \begin{align}\label{eq-int-cond-zrm}
  \gamma \ind{^a_{C_1}^{A}} \gamma \ind{^b_{C_2}^{B}} C \ind{_{abcd}} \gamma \ind{^{cd}_D^{(C_3}} \phi \ind{^{C_4 \ldots C_k) C_1 C_2 D}} = 0 \, .
 \end{align}
 If $\phi^{A_1 A_2 \ldots A_k}$ is a solution of \eqref{eq-nc-zrm}, then we have in addition
 \begin{align}\label{eq-int-cond-nc-zrm}
\gamma \ind{^b_{C_2}^{[A|}} \Phi \ind{_{bd}} \gamma \ind{^d_D^{(C_3}} \phi \ind{^{C_4 \ldots C_k)C_2 D |B]}} = 0 \, .
 \end{align}
\end{prop}

\begin{proof}
Equations \eqref{eq-zrm}, \eqref{eq-co-zrm} and \eqref{eq-nc-zrm} can be rewritten as
\begin{align*}
 \gamma \ind{^a _{B}^{A_1}} \nabla_a \phi^{A_2 \ldots A_k B} & = \psi \ind{^{A_1 A_2 \ldots A_k}} \, ,
\end{align*}
where $\psi \ind{^{(A_1 A_2 \ldots A_k)}} =0$, $\psi \ind{^{[A_1 A_2] A_3\ldots A_k}} =0$, and $\psi \ind{^{A_1 A_2 \ldots A_k}} = 0$ respectively.
Taking a second covariant derivative and commuting lead to
 \begin{multline*}
(k-2) \,  \gamma \ind{^a_{C_1}^{A}} \gamma \ind{^b_{C_2}^{B}} C \ind{_{abcd}} \gamma \ind{^{cd}_D^{(C_3}} \phi \ind{^{C_4 \ldots C_k) C_1 C_2 D}} - 4 (k-2) \, \gamma \ind{^b_{C_2}^{[A|}} \Rho \ind{_{bd}} \gamma \ind{^d_D^{(C_3}} \phi \ind{^{C_4 \ldots C_k)C_2 D |B]}} \\
= 2 \, \gamma \ind{^a_D^{[A}} \nabla \ind{_a} \psi \ind{^{B] C_3 C_4 \ldots C_k D}} \, .
 \end{multline*}
By the conformal invariance of \eqref{eq-zrm} and \eqref{eq-co-zrm}, the first term on the LHS must vanish identically, while the second term on the LHS cancels the RHS, hence \eqref{eq-int-cond-zrm}. When \eqref{eq-nc-zrm} holds, conformal invariance is broken, and one has the additional constraint \eqref{eq-int-cond-nc-zrm}.
\end{proof}

A spinor field $\phi^{A_1 A_2 \ldots A_k}$ is referred to as \emph{null} if it takes the form
\begin{align*}
 \phi^{A_1 A_2 \ldots A_k} & = \ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k} \, .
\end{align*}
for some holomorphic pure spinor field $\xi^A$, and holomorphic function $\psi$. Specialising Proposition \ref{prop-int-zrm} yields
\begin{cor}
Assume $k>2$. Suppose that $\phi^{A_1 A_2 \ldots A_k} := \ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k}$ is a solution of \eqref{eq-zrm} or \eqref{eq-co-zrm} on $(\mcM,g)$. Then
 \begin{align}\label{eq-simple-zrm-int-cond}
 {}^\mfC _\xi \Pi_{-1}^0 (C) & = 0 \, , & & \mbox{i.e.} & \xi \ind{^a^A} \xi \ind{^a^B } C \ind{_{abcd}} \xi \ind{^{cd}^{C}} & = 0 \, , 
 \end{align}
Further, if $\phi^{A_1 A_2 \ldots A_k}$ is a solution of \eqref{eq-nc-zrm}, then we have in addition
 \begin{align}\label{eq-simple-zrm-int-cond-xtra}
 {}^\mfF _\xi \Pi_{-1}^0 (\Phi) & = 0 \, , & & \qquad \mbox{i.e.} & \xi \ind{^a^{A}} \xi \ind{^b^{[B}} \Phi \ind{_{ab}} \xi \ind{^{C]}} = 0 \, .
 \end{align}
\end{cor}

The relation between null solutions of the zrm-field equation and the existence of foliating spinors is known as the Robinson theorem \cite{Robinson1961} in four dimensions, and was later generalised to even dimensions in \cite{Hughston1988}. Here, we give odd-dimensional versions of the theorem.


\begin{thm}[Robinson theorem for zrm fields]\label{thm-Robinson-zrm}
 Let $\xi^A$ be a holomorphic pure spinor field on $(\mcM,g)$. Let $\psi$ a holomorphic function and suppose that $\phi^{A_1 A_2 \ldots A_k} := \ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k}$ satisfies the zrm field equation \eqref{eq-zrm}. Then locally
\begin{align}\label{eq-Nperp-int-zrm}
 \left( \xi \ind{^a^{[A}} \nabla \ind{_a} \xi \ind{^b^{B]}} \right) \xi \ind{_b^{[C}} \xi \ind{^{D]}} - \frac{k-2}{2k} \left(  \xi \ind{^{[A}} \left( \xi \ind{^a^{B]}} \nabla \ind{_a} \xi \ind{^b^C} \right) \xi \ind{_b^D} + \xi \ind{^{[A}} \left( \xi \ind{^a^{B]}} \nabla \ind{_a} \xi \ind{^{C}} \right) \xi \ind{^{D}} \right) & = 0 \, .
\end{align}
In particular, when $k=2$, $\xi^A$ locally satisfies \eqref{eq-Nperp-int}, i.e. $\xi^A$ is $\gamma^\perp$-foliating.

Suppose that $\xi^{A}$ satisfies \eqref{eq-Nperp-int}. Then locally there exists a holomorphic function $\psi$ such that the spinor field $\phi^{A B} := \ee^\psi \xi \ind{^A} \xi \ind{^B}$ satisfies \eqref{eq-zrm}. There is the freedom of adding to $\psi$ a holomorphic function constant along the co-null $(m+1)$-dimensional leaves of the foliation defined by $\xi^A$.
\end{thm}

\begin{proof}
 For any $\phi^{A_1 A_2 \ldots A_k} := \ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k}$, we have, in regions where $\phi^{A_1 A_2 \ldots A_k}$ does not vanish,
\begin{align}\label{eq-master}
\gamma \ind{^a_B^{A_1}}  \nabla \ind{_a} \phi \ind{^{A_2 \ldots A_k B}} & = \ee^\psi \left( \xi \ind{^{A_2}} \ldots \xi \ind{^{A_k}} \xi \ind{^a^{A_1}} \nabla \ind{_a} \psi + (k-1) \left( \xi \ind{^a^{A_1}} \nabla \ind{_a} \xi \ind{^{(A_2}} \right) \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k)}} +  \left( \nabla \ind{_a} \xi \ind{^a^{A_1}} \right) \xi \ind{^{A_2}} \ldots \xi \ind{^{A_k}} \right) \, .	
\end{align}
If $\phi \ind{^{A_1 \ldots A_k}}$ satisfies \eqref{eq-zrm}, then we have
\begin{align}\label{eq-master-zrm}
 0 & = \xi \ind{^{(A_2}} \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k}} \xi \ind{^a^{A_1)}} \nabla \ind{_a} \psi + (k-1) \left( \xi \ind{^a^{(A_1}} \nabla \ind{_a} \xi \ind{^{A_2}} \right) \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k)}} +  \left( \nabla \ind{_a} \xi \ind{^a^{(A_1}} \right) \xi \ind{^{A_2}} \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k)}} \, .
\end{align}
Tensoring with $\xi^B \xi^C$ and skewing over $A_1 B$ and $A_2 C$ lead to
\begin{align*}
\xi \ind{^{[A}} \left( \xi \ind{^a^{B]}} \nabla \ind{_a} \xi \ind{^{[C}} \right) \xi \ind{^{D]}} + \xi \ind{^{[C}} \left( \xi \ind{^a^{D]}} \nabla \ind{_a} \xi \ind{^{[A}} \right) \xi \ind{^{B]}} & = 0 \, ,
\end{align*}
i.e. $\Gamma \ind{^{(A:B)}} = 0$ in the notation of \eqref{eq-connection1form}.
To show that equation \eqref{eq-Nperp-int-zrm} also holds, we expand
\begin{multline*}
(k-1) \left( \xi \ind{^a^{(A_1}} \nabla \ind{_a} \xi \ind{^{A_2}} \right) \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k)}} +  \left( \nabla \ind{_a} \xi \ind{^a^{(A_1}} \right) \xi \ind{^{A_2}} \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k)}} \\
= \left( - \frac{1}{4} \left( k \, \Gamma \ind{^{BC}} - 2 \, \Gamma \ind{^{B:C}} \right) \eta \ind{_a_B} \gamma \ind{^a_C^{(A_1}} + \psi \ind{^{(A_1}} \right) \xi \ind{^{A_2}} \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k)}} \, ,
\end{multline*}
where $\psi \ind{^{A}} \in \mfS_{\frac{m-2}{2}}$. Since the first term on the RHS lies in $\mfS_{\frac{m-4}{2}}$, we must have $k \, \Gamma \ind{^{BC}} = 2 \, \Gamma \ind{^{B:C}}$, i.e. \eqref{eq-Nperp-int-zrm} holds. When $k=2$, \eqref{eq-Nperp-int-zrm} reduces to \eqref{eq-Nperp-int}.

For the converse in the case $k=2$, we follow the geometrical proof given in \cite{Eastwood1995}. Let $\mcN$ be the $\gamma$-plane distribution defined by $\xi^A$, and $\mcN^\perp$ its orthogonal complement. Suppose that $\xi^{A}$ satisfies \eqref{eq-Nperp-int}, i.e. $\mcN^\perp$ is integrable. Then, locally, $\mcM$ is fibered over the leaf space $\mcL$ of $\mcN^\perp$. Choose a holomorphic section $\phi_{a_1 \ldots a_m}$ of the tautological line bundle $\wedge^m \Tgt^* \mcL$ of $\mcL$. Then, $\phi_{a_1 \ldots a_m}$ is clearly closed. Its pull-back to $\mcM$ must be orthogonal to each leaf of the foliation, i.e. it must be of the form $\phi^{A B} := \phi \ind{_{a_1 \ldots a_m}} \gamma \ind{^{a_1 \ldots a_m}^{AB}} = \ee^\psi \xi \ind{^A} \xi \ind{^B}$ for some holomorphic function $\psi$. Further, since the exterior derivative commutes with the pull-back, $\phi_{a_1 \ldots a_m}$ is also closed, i.e. $\phi^{AB}$ satisfy \eqref{eq-zrm}. Finally, the arbitrariness in choosing $\phi_{a_1 \ldots a_m}$ on $\mcL$ is reflected by the freedom of adding to $\psi$ any holomorphic function constant along the leaves of the foliation.
\end{proof}

\begin{thm}[Robinson theorem for co-zrm fields]\label{thm-Robinson-co-zrm}
 Let $\xi^A$ be a holomorphic pure spinor field on $(\mcM,g)$. Let $\psi$ a holomorphic function and suppose that $\phi^{A_1 A_2 \ldots A_k} := \ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k}$ satisfies the co-zrm field equation \eqref{eq-co-zrm}. Then locally $\xi^A$ satisfies \eqref{eq-cond-21}, i.e. $\xi^A$ is $\gamma$-foliating. Further, when $k>2$, one has in addition
  \begin{align}\label{eq-co-zrm>2-int}
  \xi \ind{^{[A}} \left( \xi \ind{^a^{B]}} \nabla \ind{_a} \xi \ind{^{[C}} \right) \xi \ind{^{D]}} & = 0 \, .
  \end{align}

Suppose that $\xi^{A}$ satisfies \eqref{eq-cond-21}. Then locally there exists a holomorphic function $\psi$ such that the pure spinor field $\phi^{A B} = \ee^\psi \xi \ind{^A} \xi \ind{^B}$ satisfies \eqref{eq-co-zrm}. Further, if $\xi^{A}$ satisfies \eqref{eq-co-zrm>2-int} and the curvature condition \eqref{eq-simple-zrm-int-cond}, then locally, for every $k>2$, there exists a holomorphic function $\psi$ such that the spinor field $\phi^{A_1 A_2 \ldots A_k}=\ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k}$ satisfies \eqref{eq-co-zrm}. In both cases, there is the freedom of adding to $\psi$ a holomorphic function constant along the null $m$-dimensional leaves of the foliation defined by $\xi^A$.
\end{thm}

\begin{proof}
For $k \geq 2$, if $\phi \ind{^{A_1 \ldots A_k}}$ satisfies \eqref{eq-co-zrm}, then equation \eqref{eq-master} becomes
\begin{multline}\label{eq-master-co-zrm}
 0 = \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k}} \xi \ind{^{[A_2}}\xi \ind{^a^{A_1]}} \nabla \ind{_a} \psi + \left( \xi \ind{^a^{[A_1}} \nabla \ind{_a} \xi \ind{^{A_2]}} \right) \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k}} \\
+ (k-2) \left( \xi \ind{^{[A_2}} \xi \ind{^a^{A_1]}} \nabla \ind{_a} \xi \ind{^{(A_3}} \right) \xi \ind{^{A_4}} \ldots \xi \ind{^{A_k)}} +  \left( \nabla \ind{_a} \xi \ind{^a^{[A_1}} \right) \xi \ind{^{A_2]}} \xi \ind{^{A_3}} \ldots \xi \ind{^{A_k}} \, .
\end{multline}
Then, tensoring with $\xi \ind{^B}$ and skewing over $A_1 A_2 B$ yield \eqref{eq-cond-21}. When $k>2$, one can also tensor with $\xi \ind{^B}$ and skew over $A_3 B$, and conclude \eqref{eq-co-zrm>2-int}.

For the `converse' when $k>2$, condition \eqref{eq-co-zrm>2-int} implies that
\begin{align}\label{eq-Robinson-co-zrm-proof}
\begin{aligned}
\xi \ind{^a^A} \nabla \ind{_a} \xi \ind{^B} & = \xi \ind{^A} A \ind{^B} + \xi \ind{^A} B \ind{^B} + C \xi \ind{^A} \xi \ind{^B} + D \ind{^A} \xi \ind{^B} \, , \\
\nabla \ind{_a} \xi \ind{^a^A} & = E \xi \ind{^A} + F \ind{^A} - B \ind{^A} + A \ind{^B} + C \xi \ind{^A} + D \ind{^A} \, ,
\end{aligned}
\end{align}
for some functions $C$, $E$, spinors $B \ind{^A}$, $D \ind{^A}$, $F \ind{^A}$ in $\mfS^{\frac{m-2}{2}}$, and $A \ind{^A}$ in $\mfS^{\frac{m-4}{2}}$. We want to show that locally there exists a holomorphic function $\psi$ such that \eqref{eq-master-co-zrm} holds, i.e.
\begin{align}\label{eq-Robinson-co-zrm-psi}
\begin{aligned}
\xi \ind{^{[A}} \xi \ind{^a^{B]}} \nabla \ind{_a} \psi & = \xi \ind{^{a[A}} \nabla \ind{_a} \xi \ind{^{B]}} + \left( \nabla \ind{_a} \xi \ind{^a^{[A}} \right) \xi \ind{^{B]}} - 2 (k-2) \, \xi \ind{^{[A}} D \ind{^{B]}} \\
& = \xi \ind{^{[A}} \left( 2 \, B \ind{^{B]}} - k D \ind{^{B]}} - F \ind{^{B]}} \right) =: \xi \ind{^{[A}} \psi \ind{^{B]}} \, .
\end{aligned}
\end{align}
Differentiating the above equation with respect to $\xi^{[A} \xi \ind{^a^{B]}} \nabla_a$, i.e. along vector fields annihilating $\xi^A$, yields the integrability condition
\begin{align}\label{eq-Robinson-co-zrm-intcond}
\xi \ind{^{[A}} D \ind{^{B}} \psi \ind{^{C]}} & = \xi \ind{^{[A}} \xi \ind{^a^B} \nabla \ind{_a} \psi \ind{^{C]}} \, .
\end{align}
We expand the RHS of \eqref{eq-Robinson-co-zrm-intcond} using the expression \eqref{eq-Robinson-co-zrm-psi} for $\psi^A$:
\begin{multline*}
\xi \ind{^{[A}} \xi \ind{^a^B} \nabla \ind{_a} \psi \ind{^{C]}} = ( k - 3) \xi \ind{^a^{[A}} \nabla \ind{_a} \left( \xi \ind{^B} D \ind{^{C]}} \right)  - \left( \nabla \ind{_a} \xi \ind{^a^{[A}} \right) \xi \ind{^B} D \ind{^{C]}} \\
+ \left( \xi \ind{^a^{[A}} \nabla \ind{_a} \xi \ind{^B} \right) \left( \nabla \ind{_b} \xi \ind{^b^{C]}} \right) + \left( \xi \ind{^a^{[A}} \nabla \ind{_a} \xi \ind{^B} \right) \psi \ind{^{C]}} 
+ \frac{1}{8} \xi \ind{^a^{[A}} \xi \ind{^b^B} C \ind{_{abcd}} \xi \ind{^{cd}^{C]}} \, .
\end{multline*}
Using the identity
\begin{align*}
\xi \ind{^a^{[A}} \nabla \ind{_a} \left( \xi \ind{^B} D \ind{^{C]}} \right) \xi \ind{^D} & = - \xi \ind{^{[A}} D \ind{^{B}} \left( A \ind{^{C]}} + B \ind{^{C]}} \right) \xi \ind{^D} + \frac{1}{8} \xi \ind{^{[A}} \xi \ind{^a^{B}} \xi \ind{^b^{C]}} C \ind{_{abcd}} \xi \ind{^{cd}^D}
\end{align*}
together with \eqref{eq-Robinson-co-zrm-proof}, \eqref{eq-Robinson-co-zrm-psi} and \eqref{eq-simple-zrm-int-cond} shows that \eqref{eq-Robinson-co-zrm-intcond} is indeed satisfied.
Finally, adding to $\psi$ any holomorphic function annihilated by vector fields $\xi^{[A} \xi \ind{^a^{B]}} \nabla_a$ leaves \eqref{eq-master-co-zrm} unchanged.

The case $k=2$ can easily be done in the language of differential forms as in \cite{Eastwood1995}, and is similar to the proof of Theorem \ref{thm-Robinson-zrm}.
\end{proof}


Finally, we omit the proof of the following theorem, which follows roughly the one given in \cite{Hughston1988}.
\begin{thm}[Non-conformally invariant Robinson theorem]\label{thm-Robinson-nc-zrm}
 Let $\xi^A$ be a holomorphic pure spinor field on $(\mcM,g)$. Let $\psi$ be a holomorphic function and suppose that $\phi^{A_1 A_2 \ldots A_k} := \ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k}$ is both a zrm field and a co-zrm field, i.e. $\phi^{A_1 A_2 \ldots A_k}$ satisfies \eqref{eq-nc-zrm}. Then locally $\xi^A$ satisfies \eqref{eq-cond-12}, i.e. $\xi^A$ is co-geodetic.

Suppose that $\xi^{A}$ satisfies \eqref{eq-cond-12}. Then locally there exists a holomorphic function $\psi$ such that $\phi^{AB} := \ee^\psi \xi^A \xi^B$ satisfies \eqref{eq-nc-zrm}. Suppose further that $\xi^A$ satisfies the curvature conditions \eqref{eq-simple-zrm-int-cond} and \eqref{eq-simple-zrm-int-cond-xtra}. Then, for every $k>2$, there exists a holomorphic function $\psi$ such that $\phi^{A_1 A_2 \ldots A_k} := \ee^\psi \xi^{A_1} \xi^{A_2} \ldots \xi^{A_k}$ satisfies \eqref{eq-nc-zrm}. In both cases, there is the freedom of adding to $\psi$ a holomorphic function constant along the co-null $(m+1)$-dimensional leaves of the foliation defined by $\xi^A$.
\end{thm}

\subsubsection{Pure conformal Killing spinor}
Complementary to \eqref{eq-Weyl-Dirac}, one defines the \emph{twistor equation}
\begin{align}\label{eq-twistor-spinor}
 \nabla_a \xi^{A} + \frac{1}{n} \gamma \ind{_a _B^{A}} \zeta^B & = 0 \, ,
\end{align}
for any holomorphic spinor field $\xi^A$. Here, \eqref{eq-twistor-spinor} determines $\zeta^B = \gamma \ind{^a _{A}^B} \nabla_a \xi^{A}$. A solution $\xi^A$ will be referred to as a \emph{conformal Killing spinor} or \emph{twistor-spinor}. The spinor field $\zeta^A$ can be shown to satisfy
\begin{align}\label{eq-twistor-spinor2}
\nabla \ind{_a} \zeta \ind{^B} + \frac{n}{2} \Rho \ind{_{ab}} \gamma \ind{^b_A^B} \xi \ind{^A} & = 0 \, ,
\end{align}
where $\Rho_{ab} := \frac{1}{2-n} \Phi_{ab} - R \frac{1}{2n(n-1)} g_{ab}$ is the \emph{Rho} or \emph{Schouten} tensor, which we have defined in appendix \ref{sec-conformal}. Equations \eqref{eq-twistor-spinor} and \eqref{eq-twistor-spinor2} are conformally invariant provided that $\xi^A$ and $\zeta^A$ transform as
\begin{align}\label{eq-twistor-spinor-transform}
\xi \ind{^A} & \mapsto \hat{\xi} \ind{^A} = \xi \ind{^A} \, , & \zeta \ind{^A} & \mapsto \hat{\zeta} \ind{^A} = \Omega^{-1} \left( \zeta \ind{^A} + \frac{n}{2} \Upsilon \ind{_a} \xi \ind{^a^A} \right) \, .
\end{align}
The equivalence class of pairs of spinors $(\xi^A,\zeta^A) \sim (\hat{\xi}^A,\hat{\zeta}^A)$ related by \eqref{eq-twistor-spinor-transform} can be thought of as a section $(\xi^A,\zeta^A)$ of the \emph{local twistor bundle} \cites{Penrose1986,Bailey1994} or \emph{spin tractor bundle} \cite{Hammerl2011}, and we shall refer to such a section as a \emph{tractor-spinor}. These are spinors for the group $\Spin(2m+3,\C)$. Tracing \eqref{eq-twistor-spinor2} yields
\begin{align}\label{eq-twistor-spinor3}
\nabla \ind{_a} \zeta \ind{^a^B} & = - \frac{n}{4(n-1)} R \xi \ind{^B} \, .
\end{align}
The integrability condition for the existence of a conformal Killing spinor is well-known, see e.g. \cite{Baum2010}. Here, we restate it in the context of pure spinor fields.
\begin{prop} \label{prop-int-cond-twistor-spinor} Let $\xi^A$ be a pure conformal Killing spinor on $(\mcM,g)$ with $\zeta^B := \gamma \ind{^a _{A}^B} \nabla_a \xi^{A}$. Then
\begin{align}\label{eq-twistor-spinor-int_cond}
\begin{aligned}
  C \ind{_{abcd}} \xi \ind{^{cd}^D} & = 0 \, , \qquad \mbox{i.e.} \qquad {}^\mfC _\xi \Pi_2^0 (C) = 0 \, , \\
  C \ind{_{abcd}} \zeta \ind{^{bcC}} - 2n A \ind{_{cab}} \xi \ind{^c^E} & = 0 \, ,
\end{aligned}
\end{align}
where $A \ind{_{abc}} := 2 \nabla_{[b} \Rho_{c]a}$ is the \emph{Cotton-York tensor} (see appendix \ref{sec-conformal}).
\end{prop}

Generically, at a point, the spinor $\zeta^A$ lies in the $\prb$-module $\mfS^{\frac{m-4}{2}}$. The geometric properties of the associated $\gamma$-plane distribution essentially depends on which submodule of $\mfS^{\frac{m-4}{2}}$ the spinor $\zeta^A$ belongs to.
\begin{prop}\label{prop-conformal-Killing-foliating}
 Let $\xi^{A}$ be a pure conformal Killing spinor on $(\mcM,g)$ with $\zeta^B := \gamma \ind{^a _{A}^B} \nabla_a \xi^{A}$. Then $\xi^{A}$ satisfies 
 \begin{align}\tag{\ref{eq-cond-22}}
  \xi \ind{^{[A}} \left( \xi \ind{^{aB]}} \nabla \ind{_a} \xi \ind{_b^E}  \right) \xi \ind{^b^{[C}} \xi \ind{^{D]}} + \xi \ind{^{[C}} \left( \xi \ind{^{aD]}} \nabla \ind{_a} \xi \ind{_b^E} \right) \xi \ind{^b^{[A}} \xi^{B]} &  = 0 \, .
  \end{align}

Further, $\xi^A$ is foliating if and only if $\zeta^A$ is pure and the respective $\gamma$-planes distribution associated to $\xi^A$ and $\zeta^A$ intersect in a totally null $(m-k)$-plane distribution, where $k=0$ or $1$, i.e.
\begin{align}\label{eq-pure-twistor-spinor}
  \zeta \ind{^a^A} \zeta \ind{_a^B} & = - \zeta^A \zeta^B  \, , & \zeta \ind{^a^A} \xi \ind{_a^B} & = \zeta \ind{^A} \xi \ind{^B} - 2 \, \xi \ind{^A} \zeta \ind{^B} \, .
\end{align}

Suppose that $\xi^A$ is foliating so that $\zeta^A$ satisfies \eqref{eq-pure-twistor-spinor}. Then
\begin{align}\label{eq-twistor-spinor2-almost_foliating}
\left( \zeta \ind{^a^{[A}} \nabla \ind{_a} \zeta \ind{^b^B} \right) \zeta \ind{_b^C} \zeta \ind{^{D]}} & = 0 \, .
\end{align}
\end{prop}

\begin{proof}
To prove that $\xi^A$ satisfies \eqref{eq-cond-22}, it suffices to contract equation \eqref{eq-twistor-spinor} with $\xi \ind{^a^A}$ and $\gamma \ind{^b_D^C} \xi \ind{_b^A}$. We find
\begin{align*}
\left( \xi \ind{^a^A} \nabla \ind{_a} \xi \ind{^b^B} \right) \xi \ind{_b^C} + \frac{1}{n} \left( \xi \ind{^{aA}} \zeta \ind{_{ab}^B} \xi \ind{^b^C} + \zeta \ind{^B} \xi \ind{^A} \xi \ind{^C} \right) & = 0 \, .
\end{align*}
The second term is skew-symmetric in $AC$. Therefore, symmetrising over $AC$ yields \eqref{eq-cond-22}.

We now want to derive conditions on $\xi^A$ and $\zeta^A$ for $\xi^A$ to be foliating, i.e. in addition to satisfying \eqref{eq-cond-22}, $\xi^A$ satisfies \eqref{eq-cond-20} and \eqref{eq-cond-21}. We first consider
\begin{align}\tag{\ref{eq-cond-21}}
\left( \xi \ind{^{a[A}} \nabla \ind{_a} \xi \ind{^{B}} \right) \xi \ind{^{C]}} & = 0 \, .
\end{align}
Contracting \eqref{eq-twistor-spinor} with $\xi \ind{^a^A}$ yields immediately the condition $\xi \ind{^a^{[A}} \zeta \ind{_a ^{B}} \xi \ind{^{C]}} = 0$, which is equivalent to \eqref{eq-pure-twistor-spinor} -- the factors can be obtained by means of Proposition \ref{prop-pure-intersect2}.
Assuming now that $\xi^A$ and $\zeta^A$ satisfy \eqref{eq-pure-twistor-spinor}, we compute
\begin{align*}
 \xi \ind{^a^B} \zeta \ind{_a ^A} \gamma \ind{^b_A^C} \xi \ind{_b^E} & = \left( \zeta \ind{^A} \xi \ind{^B} - 2 \, \xi \ind{^A} \zeta \ind{^B} \right) \gamma \ind{^b_A^C} \xi \ind{_b^E}  = \zeta \ind{^C} \xi \ind{^E} \xi \ind{^B} - 2 \, \xi \ind{^C} \zeta \ind{^E} \xi \ind{^B} + 2 \, \xi \ind{^C} \xi \ind{^E} \zeta \ind{^B} \, ,
\end{align*}
from which we deduce the remaining condition \eqref{eq-cond-20}.

Finally, if $\xi^A$ is pure and in addition $\zeta^A$ satisfies \eqref{eq-pure-twistor-spinor}, contracting equation \eqref{eq-twistor-spinor2} with $\zeta \ind{^a^A}$ and $\gamma \ind{^b_D^C} \zeta \ind{_b^A}$ leads to
\begin{align*}
\left( \zeta \ind{^a^A} \nabla \ind{_a} \zeta \ind{^b^B} \right) \zeta \ind{_b^C} - \frac{1}{2} \zeta \ind{^a^A} \Rho \ind{_{ab}} \xi \ind{^b^B} \zeta \ind{^C} + 2 \, \zeta \ind{^a^A} \Rho \ind{_{ab}} \zeta \ind{^b^{[B}} \xi \ind{^{C]}} & = 0 \, ,
\end{align*}
and the result \eqref{eq-twistor-spinor2-almost_foliating} follows by symmetry considerations.
\end{proof}

\begin{rem}
The statements of Proposition \ref{prop-conformal-Killing-foliating} are conformally invariant. In fact, it is straightforward to check that conditions \eqref{eq-pure-twistor-spinor} are invariant under the transformation \eqref{eq-twistor-spinor-transform}.
Further, the condition that the conformal Killing spinor $\xi^A$ be pure and $\zeta^A$ satisfy \eqref{eq-pure-twistor-spinor} is equivalent to the corresponding tractor-spinor $(\xi^A, \zeta^A)$ being a pure section of the local twistor bundle, i.e. it is a pure spinor for $\Spin(2m + 3, \C)$.

Another way of looking at Proposition \ref{prop-conformal-Killing-foliating} is to say that the conformal Killing spinor $\xi^A$ is foliating if and only if $\zeta^A$ lies in the $\prb$-module $\mfS^{\frac{m-2}{2}}$.
\end{rem}

\subsubsection{Pure Killing spinors}
A holomorphic spinor field $\xi^A$ that is both a solution to the twistor equation \eqref{eq-twistor-spinor} and an eigenspinor of the Dirac operator, i.e.
\begin{align*}
  \gamma \ind{^a_B^C} \nabla_a \xi^B = \lambda \xi^C \, , 
\end{align*}
for some holomorphic function $\lambda$ on $\mcM$, is known as a \emph{Killing spinor}. Otherwise put, $\xi^A$ satisfies the \emph{Killing equation}
\begin{align}\label{eq-Killing_spinor}
 \nabla \ind{_a} \xi \ind{^A} + \lambda \frac{1}{n} \xi \ind{_a^A} & = 0 \, .
\end{align}
In the case where $\xi^A$ is pure, the Killing spinor equation can be seen as an extension of Proposition \ref{prop-conformal-Killing-foliating}: at any point of $\mcM$, $\zeta^A$ lies in $\mfS^{\frac{m-4}{2}}$. Then $\xi^A$ is foliating if and only if $\zeta^A$ lies in $\mfS^{\frac{m-2}{2}}$ locally. Finally, $\xi^A$ is a Killing spinor if and only if $\zeta^A$ belongs to $\mfS^{\frac{m}{2}}$. 

The Killing spinor equation \eqref{eq-Killing_spinor} is clearly not conformally invariant, and one will expect non-conformally invariant integrability conditions and other geometric properties.

\begin{prop}
 Let $\xi^A$ be a pure Killing spinor on $(\mcM,g)$. Then 
\begin{align*}
{}^\mfC _\xi \Pi_2^0 (C) & = 0 \, , & & \mbox{i.e.} &   C \ind{_{abcd}} \xi \ind{^{cd}^D} & = 0 \, , \\
{}^\mfA _\xi \Pi_2^0 (A) = {}^\mfA _\xi \Pi_2^1 (A) & = 0 & & \mbox{i.e.} & \xi \ind{^a^A} A \ind{_{abc}} & = 0 \, , \\
{}^\mfF _\xi \Pi_0^1 ( \Phi ) & = 0 \, .
\end{align*}
Further, its eigenvalue $\lambda$ satisfies
\begin{align*}
\xi \ind{^a^A} \nabla \ind{_a} \lambda & = - \left( \lambda^2 + \frac{n}{4(n-1)} R \right) \xi \ind{^A} \, ,
\end{align*}
and in particular, $\lambda$ is constant along the $\gamma$-plane distribution defined by $\xi^A$.
\end{prop}
 
\begin{proof}
This is a special case of \eqref{eq-twistor-spinor}, \eqref{eq-twistor-spinor2}, \eqref{eq-twistor-spinor3} and \eqref{eq-twistor-spinor-int_cond} with $\zeta \ind{^A} = \lambda \, \xi \ind{^A}$.
\end{proof} 
 
This equation is also valid in even dimensions, but must necessarily involve non-irreducible Dirac spinors. Since there is no chirality in odd dimensions, one is free to restrict the Killing equation to pure spinor fields. The relation to the foliation condition is given by the following lemma, whose proof is left for the reader.
\begin{prop}\label{prop-Killing-foliating}
 Let $\xi^A$ be a pure conformal Killing spinor on $(\mcM,g)$. Then $\xi^A$ is a Killing spinor if and only if
 \begin{align*}
 \left( \nabla \ind{_a} \xi \ind{^b^{[A}} \right) \xi \ind{_b^B} \xi \ind{^{C]}} & = 0 \, , &
 \left( \xi \ind{^a^C} \nabla \ind{_a} \xi \ind{^{\lb{A}}} \right) \xi \ind{^{\rb{B}}} & = 0 \, .
\end{align*}
In particular, $\xi^A$ satisfies \eqref{eq-cond-12}, i.e. $\xi^A$ is co-geodetic.
\end{prop}

\begin{rem}
Non-foliating conformal Killing spinors are pertinent to the theory of differential equations. Using the method of equivalence, Cartan \cite{Cartan1910} showed how to encode the invariance properties of certain ODEs of Monge type in terms of a $(2,3,5)$-distribution, i.e. a rank-$2$ distribution $\mcN$ on a five-dimensional smooth manifold, generating the tangent bundle, i.e. $[\mcN , \mcN]$ is a rank-$3$ distribution, and $[\mcN,[\mcN , \mcN]]$ is the full tangent bundle. This is more invariantly expressed as a $\G_2$-principle bundle equipped with a Cartan connection. To this $(2,3,5)$-distribution, Nurowski \cite{Nurowski2005} was then able to associate a five-dimensional split-signature conformal structure, with respect to which $\mcN$ is totally null, with orthogonal complement $[\mcN , \mcN]$. The general theory, expounded in the language of parabolic geometries, is given in \cites{vCap2009a,Hammerl2009,Hammerl2011}. In particular, the conformal holomony of such manifolds is characterised by being contained in $\G_2$, or equivalently, by the existence of a real conformal Killing spinor, \emph{generic} in the sense that $\xi^A \zeta_A \neq 0$.
\end{rem}

\paragraph{Relation to the Goldberg-Sachs theorem}
In \cite{Taghavi-Chabert2012}, a higher-dimensional version of the Goldberg-Sachs theorem is given, based on a coarser version of the present classifications of the Weyl tensor and Cotton-York tensor.

\begin{thm}[\cite{Taghavi-Chabert2012}]
Let $\langle \xi^A \rangle$ be a holomorphic projective pure spinor field on a $(2m+1)$-dimensional complex Riemannian manifold $(\mcM,g)$. Suppose the Weyl tensor and the Cotton-York tensor satisfies the respective algebraic degeneracy conditions
\begin{align}\label{eq-GS}
\begin{aligned}
{}^\mfC _\xi \Pi_{-1}^0 (C) = {}^\mfC _\xi \Pi_{-1}^1 (C) = {}^\mfC _\xi \Pi_{-1}^2 (C) & = 0 \, , \qquad \mbox{i.e.} \qquad \xi \ind{^a^A} \xi \ind{^b^B} \xi \ind{^c^{[C}} C \ind{_{abcd}} \xi \ind{^{D]}} = 0 \, , \\
{}^\mfA _\xi \Pi_{-2}^0 (A) = {}^\mfA _\xi \Pi_{-2}^1 (A) & = 0 \, , \qquad \mbox{i.e.} \qquad \xi \ind{^{[A}} \xi \ind{^a^{B]}} \xi \ind{^b^C} \xi \ind{^c^D} A \ind{_{abc}} = 0 \, .
\end{aligned}
\end{align}
Suppose further that the Weyl tensor is otherwise generic. Then $\xi^{A'}$ is (locally) foliating.
\end{thm}

As in the even-dimensional case \cite{Taghavi-Chabert2012a}, Proposition \ref{prop-conformal-Killing-foliating} provides an example of a non-foliating pure spinor field that satisfies \eqref{eq-GS}, but violates the genericity assumption by virtue of Proposition \ref{prop-int-cond-twistor-spinor}. However, while a pure conformal Killing spinor may not be foliating, its intrinsic torsion satisfies a weaker condition. This motivates the following conjecture improving the author's work \cite{Taghavi-Chabert2012}.
\begin{conjec}\label{conjec-GS}
Let $\xi^A$ be a holomorphic pure spinor field on a $(2m+1)$-dimensional Einstein complex Riemannian manifold $(\mcM,g)$ such that the Weyl tensor satisfies $\xi \ind{^a^A} \xi \ind{^b^B} \xi \ind{^c^{[C}} C \ind{_{abcd}} \xi \ind{^{D]}} = 0$. Then $\xi \ind{^A}$ satisfies
 \begin{align}\tag{\ref{eq-cond-22}}
  \xi \ind{^{[A}} \left( \xi \ind{^{aB]}} \nabla \ind{_a} \xi \ind{_b^E}  \right) \xi \ind{^b^{[C}} \xi \ind{^{D]}} + \xi \ind{^{[C}} \left( \xi \ind{^{aD]}} \nabla \ind{_a} \xi \ind{_b^E} \right) \xi \ind{^b^{[A}} \xi^{B]} &  = 0 \, .
  \end{align}
\end{conjec}
Weaker conditions such as \eqref{eq-cond-31} may well be possible too, but an investigation of the veracity of the above conjecture is beyond the scope of this article.
Rep_theory.tex
\section{Tables of irreducibles}\label{sec-rep-theory}
In this appendix, we have collected tables of the irreducible $\g$-modules and their $\prb$-module decompositions in terms of their dominant weights and dimensions. Much of this representation theoretic description is already explained in \cite{Taghavi-Chabert2012a}, and more thoroughly in \cites{Baston1989,vCap2009}. We have abbreviated $\g$-module, $\prb$-module and $\g_0$-module to $\g$-mod, $\prb$-mod and $\g_0$-mod respectively. The Cartan product (see e.g. \cite{Eastwood2004}) between two irreducible modules $\mfV$ and $\mfW$ will be denoted by $\mfV \circledcirc \mfW$. More details can be found in the aforementioned references .

\subsection{The Lie algebra $\so(2m+1,\C)$}
The Dynkin diagram for $\g := \so(2m+1, \C)$ is 
\begin{align*}
  \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\bullet}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
\endxy \qquad \, , & & \mbox{$m$ nodes.}
\end{align*}
In the case $m=2$, this coincides with the Dynkin diagram $\quad \xy
 @={(5,0),(0,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}};
 s0; s1 **\dir2{-};
  {s0+(2.5,0)}*=0{<};
 \endxy \quad$ 
  of $\spl(4,\C)$.

The irreducible $\g$-modules considered in this article can be represented in terms of their dominant weights and dimensions as given in the table below:
\begin{displaymath}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{||c|c|c||}
\hline
\text{$\g$-mod} & \text{Dominant weight} & \text{Dimension} \\
\hline
\mfV  & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\bullet}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{1}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy & 2m+1 \\[.5em]
\hline
\g  & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\bullet}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{1}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy & m(2m+1) \\ [.5em]
\hline
\mfF  & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\bullet}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{2}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy & m(2m+3) \\[.5em]
\hline
\mfA & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\bullet}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{1}}; {s1+(0,2)}*=0{\sk{1}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy  & \frac{1}{3}(2m-1)(2m+1)(2m+3) \\[.5em]
\hline
\mfC & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\bullet}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{2}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy & \frac{1}{3}(m-1)(m+1)(2m+1)(2m+3) \\[.5em]
\hline
\end{array}}
\end{displaymath}




\subsection{The parabolic Lie subalgebra $\prb$}
To the (standard) parabolic Lie subalgebra $\prb$ stabilising a projective pure spinor $\langle \xi^A \rangle$ or its associated $\gamma$-plane corresponds the mutilated Dynkin diagram
\begin{align*}
  \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
\endxy \quad .
\end{align*}
The Dynkin diagram for the reductive part $\g_0 \cong \glie(m,\C)$ can be recovered by eliminating the crossed node.

We have collected the irreducibles $\prb$-modules occuring in the $\prb$-invariant decompositions of the representations $\mfV$, $\g$, $\mfF$, $\mfA$, $\mfC$ and $\mfW$ in Tables \ref{table-V}, \ref{table-g}, \ref{table-F}, \ref{table-A}, \ref{table-C} and \ref{table-W} respectively. Care must be taken in five dimensions where the modules $\mfA_{\pm1}^2$, $\mfC_{\pm2}^2$, $\mfC_{\pm1}^1$, $\mfC_0^1$ and $\mfC_0^3$ do not occur. In dimension seven, the module $\mfC_0^3$ does not occur.

\begin{table}[!h]
\begin{center}
\begin{displaymath}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{||c|c|c||c|c|c||c||}
\hline
\text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dominant weight} & \text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dominant weight} & \text{Dimension} \\
\hline
\mfV_1 & \mfV_1 &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{1}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy & 
\mfV^{-1}/\mfV^0 & \mfV_{-1} &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{1}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy & m \\[.5em] 
\hline
\mfV^0/\mfV^1  & \mfV_0 &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy  & & & & 1 \\[.5em]
\hline
\end{array}}
\end{displaymath}
\end{center}
\caption{Irreducible $\prb$-modules of $\mfV$} \label{table-V}
\end{table}
\begin{table}[!h]
\begin{center}
\begin{displaymath}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{||c|c|c||c|c|c||c||}
\hline
\text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dominant weight} & \text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dominant weight} & \text{Dimension} \\
\hline
\g_2^0 & \g_2 &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{1}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy & 
\g_{-2}^0  & \g_{-2} &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{1}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy & \frac{1}{2}m(m-1) \\[.5em] 
\hline
\g_1^0 &  \g_1  &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{1}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy & 
\g_{-1}^0  & \g_{-1} &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{1}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy & m \\[.5em] 
\hline
\g_0^0   & \mfz_0 &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy  & & & & 1 \\[.5em]
\g_0^1   & \slie_0 &
\xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{1}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{1}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy  & & & & m^2-1 \\[.5em]
\hline
\end{array}}
\end{displaymath}
\end{center}
\caption{Irreducible $\prb$-modules of $\g$}\label{table-g}
\end{table}
\begin{table}[!h]
\begin{minipage}[b]{0.45\linewidth}\centering
\begin{displaymath}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{||c|c|c||}
\hline
\text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dimension} \\
\hline
\mfF_{\pm2}^0 & \mfV_{\pm1} \circledcirc \mfV_{\pm1} & \frac{1}{2}m(m+1)  \\
\hline
\mfF_{\pm1}^0 & \mfV_0 \circledcirc \mfV_{\pm1}  & m \\
\hline
\mfF_0^0 & \mfV_0 \circledcirc \mfV_0 & 1 \\
\mfF_0^1 & \mfV_{\pm1} \circledcirc \mfV_{\mp1} & m^2-1 \\
\hline
\end{array}}
\end{displaymath}
\caption{Irreducible $\prb$-modules of $\mfF$}\label{table-F}
\begin{displaymath}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{||c|c|c||}
\hline
\text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dimension} \\
\hline
\mfA_{\pm3}^0 & V_{\pm1} \circledcirc \g_{\pm2} & \frac{1}{3}m(m+1)(m-1) \\
\hline
\mfA_{\pm2}^0 & V_0 \circledcirc \g_{\pm2} & \frac{1}{2}m(m-1) \\
\mfA_{\pm2}^1 & V_{\pm1} \circledcirc \g_{\pm1} & \frac{1}{2}m(m+1) \\
\hline
\mfA_{\pm1}^0 & V_{\pm1} \circledcirc \mfz_0 & m \\
\mfA_{\pm1}^1 & V_0 \circledcirc \g_{\pm1} & m \\
\mfA_{\pm1}^2 & V_{\mp1} \circledcirc \g_{\pm2} & \frac{1}{2}m(m-2)(m+1) \\
\mfA_{\pm1}^3 & V_{\pm1} \circledcirc \slie_0 & \frac{1}{2}m(m+2)(m-1) \\
\hline
\mfA_0^0 & V_0 \circledcirc \mfz_0 & 1 \\
\mfA_0^1 & V_0 \circledcirc \slie_0 & (m-1)(m+1) \\
\mfA_0^2 & V_1 \circledcirc \g_{-1} & (m-1)(m+1) \\
\hline
\end{array}}
\end{displaymath}
\caption{Irreducible $\prb$-modules of $\mfA$}\label{table-A}
\end{minipage}
\raisebox{2cm}{
\begin{minipage}[b]{0.45\linewidth}\centering
\begin{displaymath}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{||c|c|c||}
\hline
\text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dimension} \\
\hline
\mfC_{\pm4}^0 & \g_{\pm2} \circledcirc \g_{\pm2} & \frac{1}{12}m^2 (m^2-1) \\
\hline
\mfC_{\pm3}^0 & \g_{\pm1} \circledcirc \g_{\pm2} & \frac{1}{3}m (m^2-1) \\
\hline
\mfC_{\pm2}^0 & \mfz_0 \circledcirc \g_{\pm2} & \frac{1}{2}m (m-1) \\
\mfC_{\pm2}^1 & \g_{\pm1} \circledcirc \g_{\pm1} & \frac{1}{2} m (m +1) \\
\mfC_{\pm2}^2 & \slie_0 \circledcirc \g_{\pm2} & \frac{1}{3} m^2 (m^2 - 4) \\
\hline
\mfC_{\pm1}^0 & \mfz_0 \circledcirc \g_{\pm1} & m \\
\mfC_{\pm1}^1 & \g_{\mp1} \circledcirc \g_{\pm2} & \frac{1}{2} m (m-2) (m + 1)\\
\mfC_{\pm1}^2 & \slie_0 \circledcirc \g_{\pm1} & \frac{1}{2} m (m+2) (m - 1) \\
\hline
\mfC_0^0 & \mfz_0 \circledcirc \mfz_0 & 1 \\
\mfC_0^1 & \slie_0 \circledcirc \mfz_0 & m^2 - 1 \\
\mfC_0^2 & \g_1 \circledcirc \g_{-1} & m^2 - 1 \\
\mfC_0^3 & \g_2 \circledcirc \g_{-2} & \frac{1}{4}m^2 (m+1) (m-3) \\
\mfC_0^4 & \slie_0 \circledcirc \slie_0 & \frac{1}{4}m^2 (m-1) (m+3) \\
\hline
\end{array}}
\end{displaymath}
\caption{Irreducible $\prb$-modules of $\mfC$.}\label{table-C}
\end{minipage}}
\end{table}
\begin{table}[!h]
\begin{center}
\begin{displaymath}
{\renewcommand{\arraystretch}{1.2}
\begin{array}{||c|c|c|c||c|c|c|c||}
\hline
\text{$\prb$-mod} & \text{$\g_0$-mod} & \text{Dominant weight} & \text{Dimension} \\
\hline
\mfW_{-3}^0 & \wedge^3 \mfV_{-1} &
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{1}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy
& \frac{1}{3!}m(m-1)(m-2) \\[.5em]
\mfW_{-3}^1 & \mfV_{-1} \circledcirc \g_{-2} &
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{1}}; {s5+(0,2)}*=0{\sk{1}}; {s6+(0,2)}*=0{\sk{-4}};
\endxy
& \frac{1}{3}m(m^2-1) \\[.5em]
\hline
\mfW_{-2}^0 & \mfV_0 \otimes \g_{-2} & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{1}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy
& \frac{1}{2}m(m-1) \\[.5em]
\mfW_{-2}^1 & \wedge^2 \mfV_{-1} \otimes \mfV_0 & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{1}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy
& \frac{1}{2}m(m-1) \\[.5em]
\mfW_{-2}^2 & \odot^2 \mfV_{-1} \otimes \mfV_0 & 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{2}}; {s6+(0,2)}*=0{\sk{-4}};
\endxy
& \frac{1}{2}m(m+1) \\[.5em]
\hline
\mfW_{-1}^0 & \mfz_0 \otimes \mfV_{-1} &
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{1}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy
& m \\[.5em]
\mfW_{-1}^1 & \slie_0 \otimes_\circ \mfV_{-1} &
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{1}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{1}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy
& \frac{1}{2}m(m+1)(m-2) \\[.5em]
\mfW_{-1}^2 & \mfV_0 \otimes \g_{-1}
& 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{1}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy
& m \\[.5em]
\hline
\mfW_0^0 & \mfz_0 \otimes \mfV_0
&  \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{0}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{0}}; {s6+(0,2)}*=0{\sk{0}};
\endxy
& 1 \\[.5em]
\mfW_0^1 & \slie_0 \otimes \mfV_0
& 
 \xy
@={(5,0),(0,0),(-5,0),(-10,0),(-15,0),(-20,0),(-25,0)} @@{s0*=0{\bullet}, s1*=0{\bullet}, s2*=0{\bullet}, s3*=0{\bullet}, s4*=0{\bullet}, s5*=0{\bullet},s6*=0{\times}};
s0; s1 **\dir{-}; s1; s2 **\dir{-}; s2; s3 **\dir{.}; s3; s4 **\dir{-}; s4; s5 **\dir{-}; s5; s6 **\dir2{-};
 {s5+(2.5,0)}*=0{>};
{s0+(0,2)}*=0{\sk{1}}; {s1+(0,2)}*=0{\sk{0}}; {s2+(0,2)}*=0{\sk{0}}; {s3+(0,2)}*=0{\sk{0}}; {s4+(0,2)}*=0{\sk{0}}; {s5+(0,2)}*=0{\sk{1}}; {s6+(0,2)}*=0{\sk{-2}};
\endxy
& m^2-1 \\[.5em]
\hline
\end{array}}
\end{displaymath}
\end{center}
\caption{Irreducible $\prb$-modules of $\mfW$}\label{table-W}
\end{table}
Spinor-curvature-odd.tex
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\begin{document}

\title{Pure spinors, intrinsic torsion and curvature in odd dimensions}
 \author{Arman Taghavi-Chabert\\
 {\small Masaryk University, Faculty of Science, Department of Mathematics and Statistics,}\\
  {\small Kotl\'{a}\v{r}sk\'{a} 2, 611 37 Brno, Czech Republic } }
\date{}

\maketitle

\begin{abstract}
We develop a spinor calculus for a $(2m+1)$-dimensional complex Riemannian manifold $(\mcM,g)$ equipped with a preferred holomorphic projective pure spinor field $\langle \xi \rangle$. Such a spinor defines  a holomorphic distribution $\mcN$ of $m$-planes, totally null with respect to $g$. The stabiliser of $\langle \xi \rangle$ is a parabolic Lie subgroup $P$ of $\SO(2m+1,\C)$, and using its algebraic properties, we give $P$-invariant classifications of the curvature tensors of the Levi-Civita connection. In particular, we generalise the Petrov-Penrose classification of the Weyl tensor to odd dimensions. From a spinorial point of view, this generalises the notion of principal spinors.
 
We also classify the intrinsic torsion of $\langle \xi \rangle$ in terms of $P$-irreducibles, which gives an algebraic measure of the failure of $\langle \xi \rangle$ to be parallel with respect to the Levi-Civita connection. In particular, we interpret the integrability properties of the associated null $m$-plane distribution $\mcN$ and its orthogonal complement $\mcN^\perp$ in terms of the algebraic types of the intrinsic torsion of $\langle \xi \rangle$. The conformal invariance of these classes are studied and we give curvature conditions for a number of cases.
 
We then investigate the geometric properties of a number of spinorial differential equations. Notably, we relate the integrability properties of a null $m$-plane distribution to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We also give necessary and sufficient conditions for a pure conformal Killing spinor to be foliating. This leads in particular to a conjecture refining the author's odd-dimensional generalisation of the Goldberg-Sachs theorem.
 
 This work can also be applied to a smooth real pseudo-Riemannian manifold of split signature equipped with a preferred projective real  pure spinor field.
\end{abstract}

\section{Introduction and motivation}
The present article is the `odd-dimensional counterpart' of the author's work presented in \cite{Taghavi-Chabert2012a}, which focused on the geometric properties of projective pure spinor fields in even dimensions. Both this article and \cite{Taghavi-Chabert2012a} work share the same motivations and goals, and the reader should refer to the latter work for further details.

The introduction of spinors in \cite{Witten1959} and \cite{Penrose1960} to formulate Petrov's classification of the Weyl tensor in Lorentzian signature \cite{Petrov2000} marks an important step in the understanding of the complex geometric properties of spacetimes. This approach impinges on the fact that in four dimensions, Lorentzian signature, the self-dual part and anti-self-dual part of the Weyl tensor are complex conjugate of each other. Thus, the Petrov classification of the Weyl tensor can be reduced to the classification of its complex self-dual part. Algebraically, this can be represented by an element of the fourth-symmetric-power $\odot^4 \mfS^+$ of the two-dimensional complex representation $\mfS^+$ of positive spinors. This element can be thought of as the components of a homogeneous polynomial of degree $4$, and the multiplicities of its roots give an invariant way of classifying the Weyl tensor. A root of this polynomial is essentially a positive spinor $\langle \xi \rangle$ up to scale, i.e. a point in $\CP^1$, and such a spinor is known as a \emph{gravitational principal spinor}. When the Weyl tensor (at a point) admits a repeated root, it is then said to be \emph{algebraically special}. This is known as the Petrov-Penrose type $\{2,1,1\}$ or type II.

In general relativity, algebraically special solutions to Einstein's field equations play an important geometric r\^{o}le: by the Goldberg-Sachs theorem \cite{Goldberg2009}, an Einstein spacetime is algebraically special if and only if it admits a \emph{shearfree congruence of null geodesics (SCNG)}. It is however more convenient to understand a SCNG from the perspective of spinors. A projective chiral spinor field defines a distribution $\mcN$, say, of complex $2$-planes, totally null with respect to the complexified metric. The integrability of $\mcN$ is then \emph{equivalent} to the existence of a SCNG, which can be seen to arise from the intersection of $\mcN$ and its complex conjugate $\bar{\mcN}$.

The advantage of such an interpretation is that it can easily be adapted to other metric signatures. If our underlying manifold is not Lorentzian, we then have two independent Petrov-Penrose classifications corresponding to the self-dual part and anti-self-dual part of the Weyl tensor. The Goldberg-Sachs theorem still holds, but each signature yields a different geometric interpretation of what a foliating spinor is. For instance, in Euclidean signature, a totally null complex $2$-plane distribution and its complex conjugate are simply the $+\ii$-eigenspace and $-\ii$-eigenspace of an almost Hermitian structure respectively so that algebraically special Einstein Riemannian manifolds locally admit complex structures \cites{Apostolov1997,Gover2010}. On the other hand, if the underlying manifold is a \emph{complex Riemannian manifold}, i.e. a complex manifold endowed with a holomorphic non-degenerate symmetric bilinear form, on which no particular reality structure is assumed, one only has a \emph{single} spinor field defining a totally null $2$-plane distribution. This is in a sense the `barest' and most transparent setting, and will be the focus of this paper.

One way to generalise this to complex Riemannian manifolds of dimension $2m+1$ is to consider a \emph{$\gamma$-plane distribution}, i.e. a holomorphic distribution $\mcN$ of $m$-planes that are totally null with respect to the metric, or equivalently, a holomorphic projective spinor field $\langle \xi \rangle$, say, annihilated by this distribution -- such a spinor is said to be \emph{pure} \cite{Cartan1981}. The existence of such a spinor field reduces the structure group of the frame bundle to its stabiliser $P$, say, a subgroup of the complex special orthogonal group $\SO(2m+1,\C)$. The subgroup $P$ is a well-known example of a \emph{parabolic Lie subgroup} \cites{Baston1989,Fulton1991,vCap2009}. This naturally leads to an algebraic classification of the Weyl tensor, and other irreducible curvature tensors, invariant under $P$: the space $\mfC$ of tensors with Weyl symmetries admits a filtration $\mfC^4 \subset \mfC^3 \subset \ldots \mfC^{-3}\subset \mfC^{-4} := \mfC$ of $P$-modules $\mfC^i$. One can then go a step further by decomposing each $P$-module $\mfC^i/\mfC^{i+1}$ into irreducibles, each being isomorphic to an irreducible $\GL(m,\C)$-module, the reductive part of $P$.

In fact, a partial higher-dimensional generalisation of the Goldberg-Sachs theorem was presented in \cite{Taghavi-Chabert2012}: therein a notion of algebraic special complex Riemannian manifolds based on the filtration $\{ \mfC^i \}$ is defined, which under further genericity assumptions on the Weyl tensor and degeneracy assumptions on the Cotton-York tensor, guarantees the integrability of both $\mcN$ \emph{and} its orthogonal complement.\footnote{This was motivated by the Lorentzian version given in \cite{Taghavi-Chabert2011}, where it was shown that such conditions naturally lead to the existence of a congruence of null \emph{geodesics}. Also, this property was shown to feature in a number of higher-dimensional solutions of Einstein's field equations - see also \cite{Mason2010}.}  Such a result however remains coarse on account of the genericity assumption, and as we shall see in section \ref{sec-geometry}, a pure conformal Killing spinor may not be foliating, yet its integrability condition implies the algebraic speciality of the Weyl tensor in the sense of \cite{Taghavi-Chabert2012}. One must therefore seek weaker differential conditions on the distribution $\mcN$ (or its associated projective spinor field $\langle \xi \rangle$), and for this purpose, we turn our attention to the \emph{intrinsic torsion} of the $P$-structure of the frame bundle \cite{Salamon1989}. This is a tensorial quantity identified with the Levi-Civita covariant derivative $\nabla \langle \xi \rangle$ of $\langle \xi \rangle$, and lies in the $P$-module $\mfW := \mfV \otimes \left( \g/\prb \right)$ -- here, $\g := \so(2m+1,\C)$ is the Lie algebra of $\SO(2m+1,\C)$, and $\mfV$ its standard representation. The decomposition of $\mfW$ into irreducibles $P$-modules then provides a natural scheme under which differential conditions on the distribution $\mcN$ and its orthogonal complement can be formulated.

The aim of this article is to refine the classification of \cite{Taghavi-Chabert2012} and to extend the work of \cite{Taghavi-Chabert2012a} to odd dimensions. In particular, given a projective pure spinor $\langle \xi \rangle$, we shall develop a spinor calculus in order to
\begin{itemize}
 \item formulate a spinorial classification of irreducible curvature tensors such as the Weyl tensor generalising the Petrov-Penrose classification;
 \item characterise some of the geometric properties of the $\gamma$-plane distribution of $\langle \xi \rangle$ in terms of its intrinsic torsion, i.e. algebraic conditions on $\nabla \langle \xi \rangle$;
  \item and apply this machinery to a number of spinorial differential equations such as the zero-rest-mass field equation and the twistor equation.
\end{itemize}

The differential properties of pure spinor fields appear to have first been investigated in six and higher even dimensions in \cites{Hughston1995,Jeffryes1995,Hughston1988}. Reference \cite{Jeffryes1995} in fact contains a prototypical spinorial classification of the Weyl tensor in six dimensions. In five dimensions, the author of \cite{De2005} put forward a spinorial classification of the Weyl tensor distinct from the one presented in the present article. It stems from the remark that in five dimensions, the Weyl tensor is a totally symmetric $4$-spinor, from which one can construct a homogeneous polynomial of degree $4$ in $\CP^3$. This was thoroughly investigated in \cite{Godazgar2010} in Lorentzian signature. A five-dimensional spinor calculus was also introduced in \cite{Garc'ia-ParradoG'omez-Lobo2009}.

The classifications of the intrinsic torsion and the curvature of almost Hermian manifolds were given in \cite{Gray1980} and \cites{Tricerri1981,Falcitelli1994} respectively, while the classification of the intrinsic torsion of almost contact metric manifolds in \cites{Alexiev1986,Chinea1990}. The application of this work and \cite{Taghavi-Chabert2012a} to the study of Lorentzian geometry will be presented in \cite{Taghavi-Chabertb}, and can be thought of as a refinement of the higher-dimensional Petrov-Weyl classification expounded in \cites{Coley2004,Milson2005}.

The structure of this article follows that of \cite{Taghavi-Chabert2012a}. In section \ref{sec-algebra}, we recall some background on Clifford algebras and its spinor representation. Fixing a preferred pure spinor, we build a spinor calculus emphasising its relation to the representation theory of its stabiliser, a parabolic Lie subalgebra $\prb$ of $\g = \so(2m+1,\C)$. This leads, in section \ref{sec-curvature}, to the $\prb$-invariant classifications of the tracefree Ricci tensor, the Cotton-York tensor and the Weyl tensor leading to Propositions \ref{prop-TFRicci-classification}, \ref{prop-Cotton-York-classification} and \ref{prop-Weyl-classification} respectively, which include `Penrose diagrams' for these tensors.

In section \ref{sec-intrinsic-torsion}, we classify the module $\mfW = \mfV \otimes \left( \g /\prb \right)$ in terms of irreducible $\prb$-modules as given by Proposition \ref{prop-intorsion-class-alg}. This paves the way for the classification of the intrinsic torsion of a projective pure spinor field given in the next section.

Section \ref{sec-geometry} focuses on the geometric applications of the algebraic classifications of the previous sections. Building on section \ref{sec-intrinsic-torsion}, we give algebraic characterisations of the intrinsic torsion of a projective pure spinor field $\langle \xi \rangle$ in Proposition \ref{prop-intorsion-class}. The integrability properties of the null $m$-plane distribution of $\langle \xi \rangle$ and its orthogonal complement are interpreted as conditions on the intrinsic torsion of $\langle \xi \rangle$ in Proposition \ref{prop-foliating-spinor}. The conformal invariance of the intrinsic torsion is examined in Proposition \ref{prop-conformal}, which also includes an odd-dimensional complex Riemannian of the `conformal to K\"{a}hler condition'. We also give integrability conditions for the existence of projective pure spinor fields that are foliating in Proposition \ref{prop-foliation-int-cond} and parallel in Proposition \ref{prop-recurrent-spinor}.

We end the section by investigating the relation between differential equations on pure spinor fields and the differential properties of their associated $\gamma$-plane distributions. More specifically, Theorems \ref{thm-Robinson-zrm}, \ref{thm-Robinson-co-zrm} and \ref{thm-Robinson-nc-zrm} give odd-dimensional versions of the Robinson theorem for three distinct types of zero-rest-mass fields. Further, Proposition \ref{prop-conformal-Killing-foliating} and \ref{prop-Killing-foliating} investigate the differential properties of the $\gamma$-plane distribution associated to a pure conformal Killing spinor. Finally, we give a conjecture \ref{conjec-GS} of the Goldberg-Sachs theorem, generalising that of \cite{Taghavi-Chabert2012}.

Appendix \ref{sec-spinor-descript} contains formulae, which may be use in the explicit description of the classifications of the curvature tensors given in Propositions \ref{prop-TFRicci-classification}, \ref{prop-Cotton-York-classification} and \ref{prop-Weyl-classification}. A brief discussion on the special three- and five-dimensional cases is given in appendix \ref{sec-3-5dim}. In appendix \ref{sec-conformal}, we collect some useful formulae regarding conformal geometry. Finally, we have collected tables of irreducibles and their dimensions in appendix \ref{sec-rep-theory}.

\input{Algebra}
\input{Geometry}

\paragraph{Acknowlegments}
I would like to thank Andreas \v{C}ap, Pawe\l Nurowski, Josef \v{S}ilhan and Dimitri Alekseevsky for useful discussions which led to the revision of this paper.

This work was funded by a SoMoPro (South Moravian Programme) Fellowship: it has received a financial contribution from the European
Union within the Seventh Framework Programme (FP/2007-2013) under Grant Agreement No. 229603, and is also co-financed by the South Moravian Region.

Revision of this paper was carried out while the author was on an Eduard \v{C}ech Institute postdoctoral fellowship GPB201/12/G028.
\appendix
\input{Spinor-description}
\input{Five_dim}
\input{Conformal}
\input{Rep_theory}


\bibliography{biblio}



\end{document}
Spinor-description.tex
\section{Spinorial descriptions of curvature tensors}\label{sec-spinor-descript}
The aim of this appendix is to describe in an explicit manner the irreducible $\prb$-modules treated in section \ref{sec-curvature}
– here $\prb$ is the stabiliser of a projective pure spinor $\langle \xi^A \rangle$. Our notation follows that of sections \ref{sec-algebra} and \ref{sec-curvature}.

\subsection{Projection maps}
We give explicit formulae describing the various irreducibles $\prb$-modules of the spaces $\mfF$, $\mfA$ and $\mfC$ as kernels of maps ${}^\mfF _\xi \Pi_i^j$, ${}^\mfA _\xi \Pi_i^j$ and ${}^\mfC _\xi \Pi_i^j$, which on restriction to the $\prb$-modules $\gr_i (\mfF)$, $\gr_i (\mfA)$ and $\gr_i (\mfC)$ project into the irreducible $\prb$-modules $\mfF_i^j$, $\mfA_i^j$ and $\mfC_i^j$.

\paragraph{The tracefree Ricci tensor}
For $\Phi \ind{_{ab}} \in \mfF$, define
\begin{align*}
{}^\mfF _\xi \Pi_{-2}^0 ( \Phi ) & := \xi \ind{^{\lb{A}}} \xi \ind{^a ^{\rb{B}}} \Phi \ind{_{a b}} \xi \ind{^b ^{\lb{C}}} \xi \ind{^{\rb{D}}} \, , &
{}^\mfF _\xi \Pi_{-1}^0 ( \Phi ) & := \xi \ind{^{\lb{A}}} \xi \ind{^a ^{\rb{B}}} \Phi \ind{_{a b}} \xi \ind{^b ^C} \, , \\
{}^\mfF _\xi \Pi_0^0 ( \Phi ) & := \xi \ind{^a ^A} \xi \ind{^b ^B} \Phi\ind{_{a b}} \, , &
{}^\mfF _\xi \Pi_0^1 ( \Phi ) & := \xi \ind{^{\lb{A}}} \xi \ind{^a ^{\rb{B}}} \Phi \ind{_{a b}} + \frac{1}{n-1} \gamma \ind{_b _C ^{\lb{A}}} \xi \ind{^c ^{\rb{B}}} \xi \ind{^d ^C} \Phi \ind{_{c d}} \, , \\
{}^\mfF _\xi \Pi_1^0 ( \Phi ) & := \xi \ind{^a ^A} \Phi \ind{_{a b}} \, , &
\end{align*}

\paragraph{The Cotton-York tensor}
For $A \ind{_{abc}} \in \mfA$, define
\begin{flalign*}
& {}^\mfA _\xi \Pi^0_{-3} ( A ) := \xi \ind{^{\lb{A}}} \xi \ind{^a^{\rb{B}}} \xi \ind{^b^{\lb{C}}} \xi \ind{^c^D} \xi \ind{^{\rb{E}}} A \ind{_{a b c}} \, , & \\
& {}^\mfA _\xi \Pi^0_{-2} ( A ) := \xi^{[A} \xi \ind{^a^{B}} \xi \ind{^b^{C]}} \xi \ind{^c^D} A \ind{_{a b c}}  \, , & 
& {}^\mfA _\xi \Pi^1_{-2} ( A ) := \xi^{[A} \xi \ind{^a^{B]}} \xi \ind{^b^{[C}} \xi^{D]} \xi \ind{^c^E} A \ind{_{a b c}} \qquad + \qquad \left( [AB] \leftrightarrow [CD] \right)  \, , &
\end{flalign*}
\begin{flalign*}
& {}^\mfA _\xi \Pi^0_{-1} ( A ) := \xi \ind{^{\lb{A}}} \xi \ind{^a^{\rb{B}}} \xi \ind{^{bc}^C} A \ind{_{a b c}} - \frac{1}{n-2} \xi \ind{^a^C} \xi \ind{^b^A} \xi \ind{^c^B} A \ind{_{abc}} \, ,  & \\
& {}^\mfA _\xi \Pi^1_{-1} ( A ) := \xi \ind{^a^A} \xi \ind{^b^B} \xi \ind{^c^C} A \ind{_{a b c}} \, , & \\
& {}^\mfA _\xi \Pi^2_{-1} ( A ) := \xi \ind{^{[A}} \xi \ind{^a^B} \xi \ind{^b^{C]}} A \ind{_{a b c}} + \frac{1}{2(n-3)} \xi \ind{^{[A}} \xi \ind{^a^{B|}} \xi \ind{^{bd}^D} A \ind{_{a b d}} \gamma \ind{_c_D^{|C]}} - \frac{1}{2(n-3)} \xi \ind{^a^D} \xi \ind{^b^{[A}} \xi \ind{^{d}^B} A \ind{_{a b d}} \gamma \ind{_c_D^{C]}}   \, , & \\
& {}^\mfA _\xi \Pi^3_{-1} ( A ) := \xi \ind{^{[A}} \xi \ind{^a^{B]}} \xi \ind{^b^{[C}} \xi^{D]} A \ind{_{a b c}} + \frac{3}{2(n+1)} \xi \ind{^{[A}} \xi \ind{^a^{B]}} \xi \ind{^{bd}^E} A \ind{_{a b d}} \gamma \ind{_c_E^{[C}} \xi^{D]} + \frac{1}{2(n+1)} \xi \ind{^a^E} \xi \ind{^b^C} \xi \ind{^{d}^D} A \ind{_{a b d}} \gamma \ind{_c_E^{[A}} \xi^{B]} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \qquad \qquad \left( [AB] \leftrightarrow [CD] \right)  \, , & 
\end{flalign*}
\begin{flalign*}
& {}^\mfA _\xi \Pi^0_0 ( A ) := \xi \ind{^a^A} \xi \ind{^{bc}^B} A \ind{_{a b c}} \, ,  & \\
& {}^\mfA _\xi \Pi^1_0 ( A ) := \xi \ind{^a^A} \xi \ind{^b^{[B}} A \ind{_{a b c}} \xi^{C]} + \frac{1}{n-1} \xi \ind{^a^A} \xi \ind{^{bd}^D} A \ind{_{a b d}} \gamma \ind{_c_D^{[B}} \xi^{C]} \, , \\
& {}^\mfA _\xi \Pi^2_0 ( A ) := \xi^{[A} \xi \ind{^a^{B]}} \xi \ind{^b^C} A \ind{_{a b c}} - \frac{1}{2} A \ind{_{c a b}} \xi \ind{^{ab}^{[A}} \xi^{B]} \xi \ind{^C} - \frac{1}{n-1} \xi \ind{^a^C} \xi \ind{^{bd}^D} A \ind{_{abd}} \gamma \ind{_c_D^{[A}} \xi^{B]} \, , & 
\end{flalign*}
\begin{flalign*}
& {}^\mfA _\xi \Pi^0_1 ( A ) := \xi \ind{^{bc}^C} A \ind{_{a b c}} \xi \ind{^D} + \frac{2}{n-2} \xi \ind{^b^B} \xi \ind{^b^C} A \ind{_{bca}} \, , &
& {}^\mfA _\xi \Pi^1_1 ( A ) := \xi \ind{^a^A} \xi \ind{^b^B} A \ind{_{a b c}}  \, ,  & 
\end{flalign*}
\begin{flalign*}
& {}^\mfA _\xi \Pi^2_1 ( A ) := A \ind{_{[a b] c}} \xi \ind{^c^{[C}} \xi \ind{^{D]}} + \frac{1}{2(n-3)} \gamma \ind{_{[a}_E^{[C}} \xi \ind{^{D]}} A \ind{_{b]cd}} \xi \ind{^{cd}^E}  + \frac{1}{n-3} \xi \ind{^c^E} \xi \ind{^d^{[C}} A \ind{_{cd[a}} \gamma \ind{_{b]}_E^{D]}}  \, , & \\
& {}^\mfA _\xi \Pi^3_1 ( A ) := A \ind{_{(a b) c}} \xi \ind{^c^{[C}} \xi \ind{^{D]}} - \frac{3}{2(n+1)} \gamma \ind{_{(a}_E^{[C}} \xi \ind{^{D]}} A \ind{_{b)cd}} \xi \ind{^{cd}^E}  - \frac{1}{n+1} \xi \ind{^c^E} \xi \ind{^d^{[C}} A \ind{_{cd(a}} \gamma \ind{_{b)}_E^{D]}} \, , & 
\end{flalign*}
\begin{flalign*}
& {}^\mfA _\xi \Pi^0_2 ( A ) := \xi \ind{^c^C} A \ind{_{[a b] c}} \, ,  & 
& {}^\mfA _\xi \Pi^1_2 ( A ) := \xi \ind{^c^C} A \ind{_{(a b) c}} \, .  & 
\end{flalign*}

\paragraph{The Weyl tensor}
For $C \ind{_{abcd}} \in \mfC$, define
\begin{flalign*}
 & {}^\mfC _\xi \Pi_{-4}^0 (C) := \xi \ind{^{\lb{A}}} \xi \ind{^a ^B} \xi \ind{^b ^{\rb{C}}} C \ind{_{abcd}} \xi \ind{^c ^{\lb{D}}} \xi \ind{^d ^E} \xi \ind{^{\rb{F}}} \, ,  & \\
 & {}^\mfC _\xi \Pi_{-3}^0 (C) := \xi \ind{^{\lb{A}}} \xi \ind{^a ^B} \xi \ind{^b ^{\rb{C}}} C \ind{_{abcd}} \xi \ind{^c ^D} \xi \ind{^d ^E} \, ,  &
\end{flalign*}
\begin{flalign*}
 & {}^\mfC _\xi \Pi_{-2}^0 (C) := \xi \ind{^{\lb{A}}} \xi \ind{^a ^B} \xi \ind{^b ^{\rb{C}}} C \ind{_{abcd}} \xi \ind{^{cd} ^D} \, ,  & 
 & {}^\mfC _\xi \Pi_{-2}^1 (C) := \xi \ind{^a ^A} \xi \ind{^b ^B} C \ind{_{abcd}} \xi \ind{^c ^C} \xi \ind{^d ^D} \, ,  & 
\end{flalign*}
\begin{multline*}
 {}^\mfC _\xi \Pi_{-2}^2 (C) := \xi \ind{^a ^A} \xi \ind{^b ^B} C \ind{_{abcd}} \xi \ind{^c ^{[C}} \xi \ind{^{D]}} + \frac{1}{n+1} \left( \xi \ind{^a^A} \xi \ind{^b^B} C \ind{_{abce}} \xi \ind{^{ce}^E} \gamma \ind{_d_E^{[C}} - \xi \ind{^a^{[C|}} \xi \ind{^b^{[A|}} C \ind{_{abce}} \xi \ind{^{ce}^E} \gamma \ind{_d_E^{|B]}} \right) \xi \ind{^{D]}} \\
 - \frac{1}{n-3} \gamma \ind{_d_E^{[A|}} \xi \ind{^a^E} \xi \ind{^b^{|B]}} C \ind{_{abde}} \xi \ind{^d^C} \xi \ind{^e^D} \pmod {\xi \ind{^{[A}} \alpha \ind{_d^{B][C}} \xi \ind{^{D]}} }\, ,
\end{multline*}
\begin{flalign*}
& {}^\mfC _\xi \Pi_{-1}^0 (C) := \xi \ind{^a ^A} \xi \ind{^b ^B} C \ind{_{abcd}} \xi \ind{^{cd} ^C} \, ,  & \\
& {}^\mfC _\xi \Pi_{-1}^1 (C) := \xi \ind{^a ^A} \xi \ind{^b ^{B}} C \ind{_{abcd}} \xi \ind{^d ^D} - \xi \ind{^{[A}} \xi \ind{^{ab} ^{B]}} C \ind{_{abcd}} \xi \ind{^d ^D} + \frac{1}{n-3} \xi \ind{^{ab}^D} C \ind{_{abed}}  \xi \ind{^e^E} \xi \ind{^d^{[A}} \gamma \ind{_c_E^{B]}} \, , & \\
& {}^\mfC _\xi \Pi_{-1}^2 (C) := \xi \ind{^a ^A} \xi \ind{^b ^{B}} C \ind{_{abcd}} \xi \ind{^d ^D} - \xi \ind{^{ab} ^D} C \ind{_{abcd}} \xi \ind{^d ^{[A}} \xi \ind{^{B]}} + \xi \ind{^{ab} ^{[A}} C \ind{_{abcd}} \xi \ind{^d ^{B]}} \xi \ind{^D} \\
& \qquad \qquad \qquad - \frac{1}{n+1} \left( \xi \ind{^{ab}^E} C \ind{_{abed}} \xi \ind{^e^A} \xi \ind{^d^B} \gamma \ind{_c_E^D} - \xi \ind{^{ab}^{[A}} C \ind{_{abed}} \xi \ind{^e^{B]}} \xi \ind{^d^E} \gamma \ind{_c_E^D} \right) \\
& \qquad \qquad \qquad \qquad \qquad - \frac{1}{n+1} \left( \xi \ind{^{ab}^E} C \ind{_{abde}} \xi \ind{^{de}^{[A}} \gamma \ind{_c_E^{B]}} \xi \ind{^D} - \xi \ind{^{ab}^E} C \ind{_{abde}} \xi \ind{^{de}^D} \gamma \ind{_c_E^{[A}} \xi \ind{^{B]}} \right) \pmod { \xi \ind{^D} \xi \ind{^{[A}} \alpha \ind{_d^{B]}} }  \, , & 
\end{flalign*}
\begin{flalign*}
& {}^\mfC _\xi \Pi_0^0 (C) := \xi \ind{^{ab} ^A} C \ind{_{abcd}} \xi \ind{^{cd} ^B} \, ,  \\
& {}^\mfC _\xi \Pi_0^1 (C) := \xi \ind{^{ab} ^{(A}} C \ind{_{abcd}} \xi \ind{^d ^{B)}} \xi \ind{^C} - \frac{1}{n-1} \xi \ind{^{ab} ^{E}}  \xi \ind{^{de} ^{(A}} C \ind{_{abde}} \gamma \ind{_c _E ^{B)}} \xi \ind{^C} - 2 \frac{n-1}{n-3} \xi \ind{^a ^C} \xi \ind{^b ^{(A}} C \ind{_{abcd}} \xi \ind{^d ^{B)}} \pmod{ \xi^A \xi^B \alpha \ind{_d^C} }\, ,   \\
& {}^\mfC _\xi \Pi_0^2 (C) := \xi \ind{^a ^A} \xi \ind{^b ^B} C \ind{_{abcd}} \xi \ind{^c ^C} \, ,  & 
\end{flalign*}
\begin{multline*}
{}^\mfC _\xi \Pi_0^3 (C) := \xi \ind{^a ^{[A}} C \ind{_{a[bc]d}} \xi \ind{^d ^B} \xi \ind{^{C]}} + \frac{1}{n-5} \left( \gamma \ind{_{[b}_E^{[A}} \xi \ind{^{dB}} \xi \ind{^{C]}} C \ind{_{c]dae}} \xi \ind{^{ae}^E} + \xi \ind{^a^{[A}} \xi \ind{^e^{B|}} C \ind{_{ae[b|f}} \xi \ind{^f^E} \gamma \ind{_{c]}_E^{|C]}} \right) \\
 - \frac{1}{2(n-3)(n-5)} \left( \xi \ind{^{aeE}} C \ind{_{aedf}} \xi \ind{^{df}^{[A}} \gamma \ind{_{[b}_E^B} \xi \ind{_{c]}^{C]}} - \xi \ind{^{aeE}} C \ind{_{aedf}} \xi \ind{^{df}^{[A}} \gamma \ind{_{bc}_E^B} \xi ^{C]} \right) \, ,
\end{multline*}
\begin{multline*}
 {}^\mfC _\xi \Pi_0^4 (C) := \xi^{[A} \xi \ind{^a ^{B]}} C \ind{_{a(bc)d}} \xi \ind{^d ^{[C}} \xi \ind{^{D]}} + \frac{1}{n+3} \left( \xi^{[A} \xi \ind{_{(b}^{B]}} C \ind{_{c)dae}} \xi ^{d[C} \xi \ind{^{ae}^{D]}} + \xi^{[A} \gamma \ind{_{(b}_E^{B]}} C \ind{_{c)dae}} \xi^{dE} \xi \ind{^{ae}^{[C}} \xi ^{D]} \right. \\
\left. + 2 \, \xi^{[A} \gamma \ind{_{(b}_E^{B]}} C \ind{_{c)dae}} \xi^{d[C} \xi^{D]} \xi \ind{^{ae}^E} - \xi^{[A} \gamma \ind{_{(b}_E^{B]}} C \ind{_{c)dae}} \xi^{dE} \xi \ind{^{aC}} \xi ^{eD}
   \right) \\
 - \frac{1}{(n+1)(n+3)} \left( \xi^{[A} \gamma \ind{_{(b}_E^{B]}} \gamma \ind{_{c)}_F^{[C}} \xi^{D]} \xi \ind{^{aeE}} C \ind{_{aedf}} \xi \ind{^{dfF}} + \frac{1}{2} \xi \ind{_b^{[A}} \xi \ind{^{aeB]}} C \ind{_{aedf}} \xi \ind{^{df[C}} \xi^{D]} \right. \\
\left.  + \xi^{[A} \gamma \ind{_{(b}_E^{B]}} \gamma \ind{_{c)}_F^{[C}} \xi^{aD]} \xi^{fF} \xi \ind{^{aeE}} C \ind{_{aedf}} - \gamma \ind{_{(b}_E^{[A}} \xi \ind{^{aeB]}} \xi \ind{_{c)}^{[C}} \xi^{dD]} \xi^{fE} C \ind{_{aedf}}  \right) \\
+ \left( [AB] \leftrightarrow [CD] \right)  \, , 
\end{multline*}
\begin{flalign*}
 & {}^\mfC _\xi \Pi_1^0 (C) := \xi \ind{^{ab} ^B} C \ind{_{abcd}} \xi \ind{^d ^C} \, , &
 \end{flalign*}
 \begin{multline*} 
 {}^\mfC _\xi \Pi_1^1 (C) := \xi \ind{^a ^A} C \ind{_{a[bc]d}} \xi \ind{^d ^D} + \frac{1}{2(n-3)} \left( \gamma \ind{_{[b|}_E^{[A|}} \xi \ind{^{ae}^E} C \ind{_{ae|c]d}} \xi \ind{^d^{|D]}} + \gamma \ind{_{[b|}_E^{[A}} \xi \ind{^{ae}^{D]}} C \ind{_{ae|c]d}} \xi \ind{^d^E} \right) \\
 - \frac{1}{2(n-1)(n-3)} \left( \xi \ind{^{ae}^E} C \ind{_{aefd}} \xi \ind{^{fd}^{[A}} \gamma \ind{_{bc}_E^{D]}} \right) \, ,
\end{multline*}
\begin{multline*} 
 {}^\mfC _\xi \Pi_1^2 (C) := \xi \ind{^a ^A} C \ind{_{a(bc)d}} \xi \ind{^d ^D} + \frac{3}{2(n+1)} \left( \gamma \ind{_{(b|}_E^{(A|}} \xi \ind{^{ae}^E} C \ind{_{ae|c)d}} \xi \ind{^d^{|D)}} + \gamma \ind{_{(b|}_E^{(A}} \xi \ind{^{ae}^{D)}} C \ind{_{ae|c)d}} \xi \ind{^d^E} \right) \\
  + \frac{3}{2(n-1)(n+1)} \left( \xi \ind{^{ae}^E} C \ind{_{aefd}} \xi \ind{^{fd}^F} \gamma \ind{_{(b}_E^A}\gamma \ind{_{c)}_F^D} \right) \pmod { \xi^A \xi^D C \ind{_{bc}} } \, , 
 \end{multline*}
\begin{flalign*}
 & {}^\mfC _\xi \Pi_2^0 (C) := \xi \ind{^{ab} ^A} C \ind{_{abcd}}  \, ,  & 
 & {}^\mfC _\xi \Pi_2^1 (C) := \xi \ind{^a ^A} C \ind{_{a(bc)d}} \xi \ind{^d ^B} \, ,  & 
\end{flalign*}
\begin{multline*}
 {}^\mfC _\xi \Pi_2^2 (C) := \xi \ind{^{[A}} \xi \ind{^a^{D]}} C \ind{_{abcd}}  - \frac{1}{n-3} \left( \xi \ind{^a^{[A}} C \ind{_{ab[c|e}} \gamma \ind{_{|d]E}^{D]}} \xi ^{eE} - \xi \ind{^a^{[A}} C \ind{_{a[c|be}} \gamma \ind{_{|d]E}^{D]}} \xi ^{eE}  \right) \\
 - \frac{1}{n+1}  \left( \xi \ind{^{ae}^E} C \ind{_{aecd}} \gamma \ind{_b_E^{[A}} \xi^{D]} - \xi \ind{^{ae}^E} C \ind{_{aeb[c}} \gamma \ind{_{d]}_E^{[A}} \xi^{D]} \right) \\
- \frac{3}{2(n+1)(n-3)} \left( \xi \ind{^{ae}^E} C \ind{_{aebf}} \xi \ind{^f^F} \gamma \ind{_{[c}_E^{[A}} \gamma \ind{_{d]}_F^{D]}} - \xi \ind{^{ae}^E} C \ind{_{ae[c|f}} \xi \ind{^f^F} \gamma \ind{_{|d]}_E^{[A}} \gamma \ind{_b_F^{D]}} \right) \\
+ \frac{1}{2(n+1)(n-3)} \left( \xi \ind{^{ae}^{[A|}} C \ind{_{aebf}} \xi \ind{^f^E} \gamma \ind{_{cd}_E^{|D]}} - \xi \ind{^{ae}^{[A|}} C \ind{_{ae[c|f}} \xi \ind{^f^E} \gamma \ind{_{|d]b}_E^{|D]}} \right) \\
- \frac{2}{(n+1)(n-3)} \left( \xi \ind{^{ae}^E} C \ind{_{aebf}} \xi \ind{^f^{[A}} \gamma \ind{_{cd}_E^{D]}} - \xi \ind{^{ae}^E} C \ind{_{ae[c|f}} \xi \ind{^f^{[A}} \gamma \ind{_{|d]b}_E^{D]}} \right) \\
+ \frac{2}{(n+1)(n-1)(n-3)} \xi \ind{^{ae}^E} C \ind{_{aefg}} \xi \ind{^{fg}^F} \left( \gamma \ind{_b_E^{[A}} \gamma \ind{_{cd}_F^{D]}} - \gamma \ind{_{[c}_E^{[A}} \gamma \ind{_{d]b}_F^{D]}} \right) \, ,
\end{multline*}
\begin{flalign*}
 & {}^\mfC _\xi \Pi_3^0 (C) := \xi \ind{^a ^A} C \ind{_{abcd}} \, ,  &
\end{flalign*}

\subsection{Spinorial representatives}
It is often more convenient to work with a particular choice of grading \eqref{eq-null-grading} adapted to the filtration \eqref{eq-filtration}. This is achieved by introducing a spinor $\eta_A$ dual to $\xi^A$ satisfying $\xi^A \eta_A = -\frac{1}{2}$. In particular, $\xi^A$ and $\eta_A$ annihilate $\mfV_1$ and $\mfV_{-1}$ respectively. We also set as before
\begin{align*}
u_a & := 2 \, \xi \ind{^a^A} \eta \ind{_A} \, , & h \ind{_{ab}} & := g \ind{_{ab}} + u \ind{_a} u \ind{_b} \, , & \omega \ind{_{ab}} & := - 2 \, \xi \ind{_{[a}^A} \eta \ind{_{b]}_A} \, .
\end{align*}
Further, any upstairs and downstairs upper case Roman index refers to the $\g_0$-modules $\mfS_{\frac{m-2}{2}}$ and $\mfS_{-\frac{m-2}{2}}$ respectively. Finally, an element
\begin{align*}
\sigma \ind{_{A \ldots C}^{D \ldots F}} \quad \in \quad \mfS_{-\frac{m-2}{2}} \otimes \ldots \otimes \mfS_{-\frac{m-2}{2}} \otimes \mfS_{\frac{m-2}{2}} \otimes \ldots \otimes \mfS_{\frac{m-2}{2}} 
\end{align*}
will be referred to as \emph{(totally) tracefree}, if the contraction of any pair of indices with the identity element $I \ind*{_A^B}$ defined by \eqref{eq-identity-map} vanishes, e.g. $\sigma \ind{_A^B} I \ind*{_B^A} = 0$.

In what follows, $\breve{\mfF}_i^j$, $\breve{\mfA}_i^j$ and $\breve{\mfC}_i^j$ will denote the irreducibles $\g_0$-modules induced from the grading \eqref{eq-null-grading}, and isomorphic to the $\prb$-modules $\mfF_i^j$, $\mfA_i^j$ and $\mfC_i^j$ of Propositions \ref{prop-TFRicci-classification}, \ref{prop-Cotton-York-classification} and \ref{prop-Weyl-classification} respectively. We shall then give explicit spinorial `representatives' for elements of those modules.

\paragraph{The tracefree Ricci tensor}
 Let $\Phi \ind{_{ab}} \in \mfF$. Then
\begin{itemize}
 \item $\Phi \ind{_{ab}} \in \breve{\mfF}_0^1$ if and only if $\Phi_{ab} = \xi \ind{_{\lp{a}} ^A} \eta \ind{_{\rp{b}} _B} \Phi \ind{_A^B}$
for some tracefree $\Phi \ind{_A^B}$;
\item $\Phi \ind{_{ab}} \in \breve{\mfF}_0^0$ if and only if $\Phi _{ab} = \Phi \left( u \ind{_a} u \ind{_b} + \frac{1}{n-1} h \ind{_{ab}} \right)$ for some complex $\Phi$;
\item $\Phi \ind{_{ab}} \in \breve{\mfF}_1^0$ if and only if $\Phi _{ab} = \xi \ind{_{(a} ^A}  \Phi_A u \ind{_{b)}}$
for some $\Phi_{A}$;
\item $\Phi \ind{_{ab}} \in \breve{\mfF}_2^0$ if and only if $\Phi _{ab} = \xi \ind{_a ^A} \xi \ind{_b ^B} \Phi_{AB}$ for some $\Phi_{AB}= \Phi_{(AB)}$, and similarly for an element of $\breve{\mfF}_{-2}^0$ by substituting $\xi^A$ for $\eta_A$ and changing the index structure in the obvious way.
\end{itemize}

\paragraph{The Cotton-York tensor}
 Let $A_{abc} \in \mfA$. Then
\begin{itemize}
\item $A_{abc} \in \breve{\mfA}_0^0$ if and only if $A \ind{_{abc}} = a \left( u \ind{_a} \omega \ind{_{bc}} - u \ind{_{\lb{b}}} \omega \ind{_{\rb{c}a}} \right)$ for some complex $a$;
\item $A_{abc} \in \breve{\mfA}_0^1$ if and only if $A \ind{_{abc}} = u \ind{_a} A \ind{_{bc}} - u \ind{_{\lb{b}}} A \ind{_{\rb{c}a}}$
where $A_{ab} = \xi \ind{_{\lb{a}}^A} \eta \ind{_{\rb{b}}_B} A \ind{_A^B}$ for some tracefree $A \ind{_A^B}$;
\item $A_{abc} \in \breve{\mfA}_0^2$ if and only if $A \ind{_{abc}} = A \ind{_{a\lb{b}}} u \ind{_{\rb{c}}}$ where $A_{ab} = \xi \ind{_{\lp{a}}^A} \eta \ind{_{\rp{b}}_B} A \ind{_A^B}$ for some tracefree $A \ind{_A^B}$;
 \item $A_{abc} \in \breve{\mfA}_1^0$ if and only if $A \ind{_{abc}} = A \ind{_a} \omega \ind{_{bc}} - A \ind{_{\lb{b}}} \omega \ind{_{\rb{c}a}} + \frac{3}{n-2} h \ind{_{a \lb{b}}} \omega \ind{_{\rb{c} d}} A \ind{^d}$ where $A_c = \xi \ind{_c^C} A_C$ for some $A_A$;
 \item $A_{abc} \in \breve{\mfA}_1^1$ if and only if $A \ind{_{abc}} = u \ind{_a} u \ind{_{\lb{b}}} A \ind{_{\rb{c}}} + \frac{1}{n-2} h \ind{_{a \lb{b}}} A \ind{_{\rb{c}}}$ where $A_a = \xi \ind{_a^A} A_A$ for some $A_A$;
 \item $A_{abc} \in \breve{\mfA}_1^2$ if and only if $A_{abc} =  \eta \ind{_a _C} \xi \ind{_b ^A} \xi \ind{_c ^B} A \ind{_{AB}^C}  - \xi \ind{_a ^A} \xi \ind{_{\lb{b}} ^B} \eta \ind{_{\rb{c}} _C} A \ind{_{AB}^C}$ for some tracefree $A \ind{_{AB}^C} = A \ind{_{[AB]}^C}$; 
 \item $A_{abc} \in \breve{\mfA}_1^3$ if and only if $A_{abc} = \xi \ind{_a ^A} \xi \ind{_{\lb{b}} ^B} \eta \ind{_{\rb{c}} _C} A \ind{_{AB}^C}$ for some tracefree $A \ind{_{AB}^C} = A \ind{_{(AB)}^C}$;
\item $A_{abc} \in \breve{\mfA}_2^0$ if and only if $A \ind{_{abc}} = u \ind{_a} A \ind{_{bc}} - u \ind{_{\lb{b}}} A \ind{_{\rb{c}a}}$
where $A_{ab} = \xi \ind{_a^A} \xi \ind{_b^B} A \ind{_{AB}}$ fo some $A \ind{_{AB}}=A \ind{_{[AB]}}$;
\item $A_{abc} \in \breve{\mfA}_2^1$ if and only if $A \ind{_{abc}} = A \ind{_{a\lb{b}}} u \ind{_{\rb{c}}}$ where $A_{ab} = \xi \ind{_a^A} \xi \ind{_b^B} A \ind{_{AB}}$ for some $A \ind{_{AB}}=A \ind{_{(AB)}}$;
 \item $A_{abc} \in \breve{\mfA}_3^0$ if and only if $A_{abc} = \xi \ind{_a ^A} \xi \ind{_b ^B} \xi \ind{_c ^C} A \ind{_{ABC}}$
for some $A_{ABC} = A_{A[BC]}$ satisfying $A_{[ABC]} = 0$.
\end{itemize}
Using the duality $(\breve{\mfA}_{-i}^j)^* \cong \breve{\mfA}_i^j$, spinorial decompositions of elements of $\breve{\mfA}_{-i}^j$ for $i=1,2,3$ can be obtained by interchanging $\xi^A$ and $\eta_A$ in the obvious way and notation.

\paragraph{The Weyl tensor}
Let $C_{abcd} \in \mfC$. Then
\begin{itemize}
 \item $C_{abcd} \in \breve{\mfC}_0^0$ if and only if
\begin{align*}
 C \ind{_{abcd}} & = c \left( 2 \, \omega \ind{_{ab}} \omega \ind{_{cd}} - 2 \, \omega \ind{_{a\lb{c}}} \omega \ind{_{\rb{d}b}} 
+ \frac{6}{n-2} \, h \ind{_{a \lb{c}}} h \ind{_{\rb{d}b}} \right)
\end{align*}
for some complex $c$;
 \item $C_{abcd} \in \breve{\mfC}_0^1$ if and only if
\begin{align*}
 C \ind{_{abcd}} & = \omega \ind{_{ab}} C \ind{_{cd}} + C \ind{_{ab}} \omega \ind{_{cd}} - 2 \, \omega \ind{_{\lb{a}|\lb{c}}} C \ind{_{\rb{d}|\rb{b}}}
- \frac{6}{n-3} \, \left( h \ind{_{\lb{a} | \lb{c}}} \omega \ind{_{\rb{d}}^e} C \ind{_{|\rb{b}e}} + h \ind{_{\lb{c} | \lb{a}}} \omega \ind{_{\rb{b}}^e} C \ind{_{|\rb{d}e}} \right) \, ,
\end{align*}
where $C \ind{_{cd}} := 2 \, \xi \ind{_{\lb{c}} ^C} \eta \ind{_{\rb{d}} _D} C \ind{_C^D}$ for some tracefree $C \ind{_C^D}$;

 \item $C_{abcd} \in \breve{\mfC}_0^2$ if and only if
\begin{align*}
 C_{abcd} = u \ind{_{\lb{a}}} C \ind{_{\rb{b}\lb{c}}} u \ind{_{\rb{d}}} 
- \frac{1}{n-3} \, h \ind{_{\lb{a} | \lb{c}}} C \ind{_{\rb{d}|\rb{b}}} \, ,
\end{align*}
where $C \ind{_{cd}} := 2 \, \xi \ind{_{\lp{c}} ^C} \eta \ind{_{\rp{d}} _D} C \ind{_C^D}$ for some tracefree $C \ind{_C^D}$;

\item $C_{abcd} \in \breve{\mfC}_0^3$ if and only if
\begin{align*}
  C_{abcd} = \xi \ind{_a ^A} \xi \ind{_b ^B} \eta \ind{_c _C} \eta \ind{_d _D} C \ind{_{A B}^{C D}} + \xi \ind{_c ^A} \xi \ind{_d ^B} \eta \ind{_a _C} \eta \ind{_b _D} C \ind{_{A B}^{C D}} - 2 \, \xi \ind{_{\lb{a}|} ^A} \xi \ind{_{\lb{c}} ^C} \eta \ind{_{\rb{d} |} _D} \eta \ind{_{\rb{b}} _B} C \ind{_{A C}^{D B}}
\end{align*}
for some tracefree  $C \ind{_{A C}^{D B}} = C \ind{_{[A C]}^{[D B]}}$;

 \item $C_{abcd} \in \breve{\mfC}_0^4$ if and only if $C_{abcd} =  \xi \ind{_{\lb{a}|} ^A} \xi \ind{_{\lb{c}} ^C} \eta \ind{_{\rb{d}|} _D} \eta \ind{_{\rb{b}} _B} C  \ind{_{A C}^{D B}}$ for some tracefree $C \ind{_{A C}^{D B}} = C \ind{_{(A C)}^{(D B)}}$;

\item $C_{abcd} \in \breve{\mfC}_1^0$ if and only if
\begin{multline*}
 C_{abcd} = \omega \ind{_{ab}} C \ind{_{\lb{c}}} u \ind{_{\rb{d}}} + \omega \ind{_{cd}} C \ind{_{\lb{a}}} u \ind{_{\rb{b}}} - \omega \ind{_{\lb{a}|\lb{c}}} C \ind{_{\rb{d}}} u \ind{_{|\rb{b}}} - \omega \ind{_{\lb{c}|\lb{a}}} C \ind{_{\rb{b}}} u \ind{_{|\rb{d}}}  \\
+ \frac{3}{n-2} \left( h \ind{_{\lb{a} | \lb{c}}} u \ind{_{\rb{d}}} \omega \ind{_{|\rb{b}}^e} C \ind{_e}  + h \ind{_{\lb{c} | \lb{a}}} u \ind{_{\rb{b}}} \omega \ind{_{|\rb{d}}^e} C \ind{_e}\right) \, ,
\end{multline*}
where $C \ind{_a} = \xi \ind{_a^A} C \ind{_A}$ for some $C \ind{_A}$;

\item $C_{abcd} \in \breve{\mfC}_1^1$ if and only if 
\begin{align*}
C_{abcd} = u \ind{_{\lb{a}}} C \ind{_{\rb{b}cd}} + u \ind{_{\lb{c}}} C \ind{_{\rb{d}ab}} \, ,
\end{align*}
where $C \ind{_{cab}} = \eta \ind{_c _C} \xi \ind{_a ^A} \xi \ind{_b ^B} C \ind{_{A B}^C} - \xi \ind{_c ^A} \xi \ind{_{\lb{a}} ^B} \eta \ind{_{\rb{b}} _C} C \ind{_{A B}^C}$ for some tracefree $C \ind{_{AB}^C} = C \ind{_{[AB]}^C}$;

\item $C_{abcd} \in \breve{\mfC}_1^2$ if and only if $C_{abcd} = u \ind{_{\lb{a}}} C \ind{_{\rb{b}cd}} + u \ind{_{\lb{c}}} C \ind{_{\rb{d}ab}}$, where $C \ind{_{cab}} = \xi \ind{_c ^A} \xi \ind{_{\lb{a}} ^B} \eta \ind{_{\rb{b}} _C} C \ind{_{A B}^C}$ for some tracefree $C \ind{_{AB}^C} = C \ind{_{(AB)}^C}$;

\item $C_{abcd} \in \breve{\mfC}_2^0$ if and only if
\begin{align*}
 C \ind{_{abcd}} & = \omega \ind{_{ab}} C \ind{_{cd}} + C \ind{_{ab}} \omega \ind{_{cd}} - 2 \, \omega \ind{_{\lb{a}|\lb{c}}} C \ind{_{\rb{d}|\rb{b}}} \, ,
\end{align*}
where $C_{ab} := \xi \ind{_a ^A} \xi \ind{_b ^B} C_{A B}$ for some $C _{AB} = C _{[AB]}$;

\item $C_{abcd} \in \breve{\mfC}_2^1$ if and only if
\begin{align*}
 C_{abcd} = u \ind{_{\lb{a}}} C \ind{_{\rb{b}\lb{c}}} u \ind{_{\rb{d}}} 
- \frac{1}{n-3} h \ind{_{\lb{a} | \lb{c}}} C \ind{_{\rb{d}|\rb{b}}} \, ,
\end{align*}
where $C \ind{_{cd}} := \xi \ind{_c ^C} \xi \ind{_d ^D} C \ind{_{CD}}$ for some $C _{AB} = C _{(AB)}$;

\item $C_{abcd} \in \breve{\mfC}_2^2$ if and only if
\begin{align*}
 C_{abcd} = \xi \ind{_a ^A} \xi \ind{_b ^B} \xi \ind{_{\lb{c}} ^C} \eta \ind{_{\rb{d}}_D} C \ind{_{A B C}^D} + \xi \ind{_c ^A} \xi \ind{_d ^B} \xi \ind{_{\lb{a}} ^C} \eta \ind{_{\rb{b}}_D} C \ind{_{A B C}^D} \, ,
\end{align*}
for some $C \ind{_{ABC}^D} = C \ind{_{[AB]C}^D}$ satisfying $C \ind{_{[ABC]}^D} = 0$;

\item $C_{abcd} \in \breve{\mfC}_3^0$ if and only if $C_{abcd} = u \ind{_{\lb{a}}} C \ind{_{\rb{b}cd}} + u \ind{_{\lb{c}}} C \ind{_{\rb{d}ab}}$, where $C \ind{_{abc}} = \xi \ind{_a ^A} \xi \ind{_b ^B} \xi \ind{_c ^C} C \ind{_{ABC}}$ for some $C _{ABC} = C _{[AB]C}$ satisfying $C _{[ABC]} = 0$;

\item $C_{abcd} \in \breve{\mfC}_4^0$ if and only if $C_{abcd} = \xi \ind{_a ^A} \xi \ind{_b ^B} \xi \ind{_c ^C} \xi \ind{_d ^D} C_{A B C D}$ for some $C _{ABCD} = C _{[AB][CD]}$ satisfying $C _{[ABC]D} = 0$.
\end{itemize}
Using the duality $(\breve{\mfC}_{-i}^j)^* \cong \breve{\mfC}_i^j$, spinorial decompositions of elements of $\breve{\mfC}_{-i}^j$ for $i=1,2,3$ can be obtained by interchanging $\xi^A$ and $\eta_A$ in the obvious way and notation.
macro.tex
% THEOREMS -------------------------------------------------------
\newtheorem{thm}{Theorem}[section]
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%\delta \ind*{_{#2} ^\rb{#4}}}
Spinor-curvature-odd.bbl
% \bib, bibdiv, biblist are defined by the amsrefs package.
\begin{bibdiv}
\begin{biblist}

\bib{Alexiev1986}{incollection}{
      author={Alexiev, Valentin~A.},
      author={Ganchev, Georgi~T.},
       title={On the classification of the almost contact metric manifolds},
        date={1986},
   booktitle={Mathematics and mathematical education ({B}ulgarian) ({S}unny
  {B}each ({S}l\cdprime nchev {B}ryag), 1986)},
   publisher={Publ. House Bulgar. Acad. Sci., Sofia},
       pages={155\ndash 161},
      review={\MR{872914 (88c:53035a)}},
}

\bib{Apostolov1997}{article}{
      author={Apostolov, V.},
      author={Gauduchon, P.},
       title={The {R}iemannian {G}oldberg-{S}achs theorem},
        date={1997},
     journal={Internat. J. Math.},
      volume={8},
      number={4},
       pages={421\ndash 439},
}

\bib{Baston1989}{book}{
      author={Baston, Robert~J.},
      author={Eastwood, Michael~G.},
       title={The {P}enrose transform},
      series={Oxford Mathematical Monographs},
   publisher={The Clarendon Press Oxford University Press},
     address={New York},
        date={1989},
        note={Its interaction with representation theory, Oxford Science
  Publications},
}

\bib{Bailey1994}{article}{
      author={Bailey, T.~N.},
      author={Eastwood, M.~G.},
      author={Gover, A.~R.},
       title={Thomas's structure bundle for conformal, projective and related
  structures},
        date={1994},
        ISSN={0035-7596},
     journal={Rocky Mountain J. Math.},
      volume={24},
      number={4},
       pages={1191\ndash 1217},
}

\bib{Baum2010}{book}{
      author={Baum, Helga},
      author={Juhl, Andreas},
       title={Conformal differential geometry},
      series={Oberwolfach Seminars},
   publisher={Birkh\"auser Verlag},
     address={Basel},
        date={2010},
      volume={40},
        ISBN={978-3-7643-9908-5},
         url={http://dx.doi.org/10.1007/978-3-7643-9909-2},
        note={$Q$-curvature and conformal holonomy},
      review={\MR{2598414 (2011d:53055)}},
}

\bib{Bryant2000}{incollection}{
      author={Bryant, Robert~L.},
       title={Pseudo-{R}iemannian metrics with parallel spinor fields and
  vanishing {R}icci tensor},
        date={2000},
   booktitle={Global analysis and harmonic analysis ({M}arseille-{L}uminy,
  1999)},
      series={S\'emin. Congr.},
      volume={4},
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}

\bib{Budinich1989}{article}{
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\bib{Cartan1910}{article}{
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\bib{Cartan1981}{book}{
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\bib{Chinea1990}{article}{
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\bib{vCap2009}{book}{
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\bib{Eastwood1995}{incollection}{
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\bib{Falcitelli1994}{article}{
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\bib{Taghavi-Chabertb}{article}{
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