# Almost Robinson geometries

Anna Fino  Università di Torino, Dipartimento di Matematica “G. Peano", Via
Carlo Alberto, 10 - 10123, Torino, Italy<EMAIL_ADDRESS>,  Thomas
Leistner  School of Mathematical Sciences, University of Adelaide, SA 5005,
Australia<EMAIL_ADDRESS>and  Arman Taghavi-Chabert  Faculty
of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland
<EMAIL_ADDRESS>
(Date: August 28, 2024 at )

###### Abstract.

We investigate the geometry of almost Robinson manifolds, Lorentzian analogues
of almost Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian
manifolds of even dimension equipped with a totally null complex distribution
of maximal rank. Associated to such a structure, there is a congruence of null
curves, which, in dimension four, is geodesic and non-shearing if and only if
the complex distribution is involutive. Under suitable conditions, the
distribution gives rise to an almost Cauchy–Riemann structure on the leaf
space of the congruence.

We give a comprehensive classification of such manifolds on the basis of their
intrinsic torsion. This includes an investigation of the relation between an
almost Robinson structure and the geometric properties of the leaf space of
its congruence. We also obtain conformally invariant properties of such a
structure, and we finally study an analogue of so-called generalised optical
geometries as introduced by Robinson and Trautman.

###### Key words and phrases:

Lorentzian manifolds, almost Robinson structures, G-structure, intrinsic
torsion, congruences of null geodesics, conformal geometry, almost CR
structures

###### 2010 Mathematics Subject Classification:

Primary 53C50, 53C10; Secondary 53B30, 53C18

AF was supported by GNSAGA of INdAM and by PRIN 2017 “Real and Complex
Manifolds: Topology, Geometry and Holomorphic Dynamics". TL was supported by
the Australian Research Council (Discovery Program DP190102360). ATC & TL
declare that this work was partially supported by the grant 346300 for IMPAN
from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. ATC
was also supported by a long-term faculty development grant from the American
University of Beirut for his visit to IMPAN, Warsaw, in the summer 2018. He
has also received funding from the GAČR (Czech Science Foundation) grant
20-11473S. For the revision of this article, ATC is supported by the
fellowship “Conformal and CR methods in general relativity",
2020/37/K/ST1/02788, funded as part of the POLS NCN competition research
projects financed from the Norwegian Financial Mechanism for 2014-2021

###### Contents

  1. 1 Introduction
  2. 2 Algebraic description
    1. 2.1 Notation and conventions
    2. 2.2 Linear algebra
    3. 2.3 The stabiliser of a Robinson structure
    4. 2.4 One-dimensional representations of $Q$
    5. 2.5 Space $\mathbb{G}$ of algebraic intrinsic torsions
    6. 2.6 Isotypic $Q_{0}$-submodules of $\mathbb{G}$
    7. 2.7 The $Q$-submodules of $\mathbb{G}$
  3. 3 Almost Robinson manifolds
    1. 3.1 Almost Robinson structures
    2. 3.2 Pure spinors
    3. 3.3 Almost Robinson structures as $G$-structures
    4. 3.4 Intrinsic torsion
    5. 3.5 Congruences of null geodesics
    6. 3.6 Almost CR geometry
    7. 3.7 Nearly Robinson structures and almost CR structures
    8. 3.8 Conditions on the Robinson $3$-forms
    9. 3.9 Twist-induced almost Robinson structures
    10. 3.10 Almost Robinson structures as almost null structures
    11. 3.11 Analogies between almost Robinson geometry and almost Hermitian geometry
    12. 3.12 Almost Robinson manifolds of Kundt type
    13. 3.13 Almost Robinson manifolds of Robinson–Trautman type
    14. 3.14 Compatible linear connections
  4. 4 Conformal almost Robinson structures
    1. 4.1 Conformal invariants of an almost Robinson structure
    2. 4.2 Conformally parallel Robinson structures
    3. 4.3 Conformal lift of almost CR structures
    4. 4.4 Non-shearing twist-induced almost Robinson structures
    5. 4.5 The Mariot–Robinson theorem
    6. 4.6 The Kerr theorem
  5. 5 Generalised almost Robinson geometries
    1. 5.1 Generalised Robinson structures
    2. 5.2 Generalised Robinson geometries as $G$-structures
  6. 6 Generalisation to other metric signatures and odd dimensions
  7. A Projections
  8. B Generalised almost Robinson geometry — connections

##  1\. Introduction

In a recent article [FLTC20], the authors give a comprehensive review of the
notion of _optical structure_ on a Lorentzian manifold $(\mathcal{M},g)$,
simply understood as a null line distribution $K$ on $\mathcal{M}$. Many of
the geometric properties of this distribution and its orthogonal complement
are encoded in terms of its screen bundle $H_{K}=K^{\perp}/K$, which is
naturally equipped with a bundle metric $h$ inherited from $g$. One may
naturally wish to endow $H_{K}$ with further bundle structures. In the present
article, where we assume $\mathcal{M}$ to have dimension $2m+2$, we equip
$H_{K}$ with a bundle complex structure $J$ compatible with $h$. Such a
structure was introduced by Nurowski and Trautman in [NT02, Tra02, Tra02a],
where it is equivalently described in terms of a totally null complex
$(m+1)$-plane distribution $N$. The real span of the intersection
$N\cap\overline{N}$ then determines the line distribution $K$. Following their
terminology, we shall refer to the pair $(N,K)$ as an _(almost) Robinson
structure_. The structure group of the frame bundle is reduced to
$(\mathbf{R}_{>0}\times\mathbf{U}(m))\ltimes(\mathbf{R}^{2m})^{*}$, which is a
subgroup of the group $\mathbf{Sim}(2m)$, which characterises optical
structures, and as in [FLTC20], we shall describe the geometric properties of
an almost Robinson structure in terms of its _intrinsic torsion_.

In dimension four, as is well-known [PR86, NT02, Tra02, Tra02a, FLTC20], an
almost Robinson structure $(N,K)$ is essentially equivalent to an optical
structure. The key point, here, is that the involutivity of the totally null
complex $2$-plane distribution $N$ is equivalent to the congruence
$\mathcal{K}$ of null curves tangent to $K$ being geodesic and _non-shearing_
, that is, the conformal class of the bundle metric $h$ is preserved along the
geodesic curves of $\mathcal{K}$. What is more, the rank-one complex vector
bundle $N/{}^{\mathbf{C}}K$ descends to the leaf space
$\underline{\mathcal{M}}$ of $\mathcal{K}$, thereby endowing it with a
_Cauchy–Riemann (CR) structure_. This CR geometrical aspect of Robinson
manifolds was particularly emphasised by Robinson, Trautman and the ‘Warsaw’
school [Taf85, Taf86, RT86, Nur93, LN90, Nur97], and in parallel, by the
twistor school Mason1985,Mason1998. This property is useful when seeking
solutions to the Einstein field equations [LNT91, TNL91, NT92], a problem that
is in turn linked to analytic questions regarding the embeddability of CR
manifolds [Taf85, Taf86, LNT90, HLN08, SG19].

There are also three important theorems worthy of mention in the development
of mathematical relativity in the present context:

  * •

The _Mariot–Robinson theorem_ [Mar54, Rob61] gives a one-to-one correspondence
between analytic non-shearing congruences of null geodesics and _null_ or
_algebraically special_ electromagnetic fields in vacuum.

  * •

The _Goldberg-Sachs theorem_ [GS62, GS09] relates the existence of non-
shearing congruences of null geodesics to the algebraic degeneracy of the Weyl
tensor for Einstein spacetimes.

  * •

The _Kerr theorem_ , as formulated in [Pen67], tells us how such congruences
arise in Minkowski space from complex submanifolds of three-dimensional
complex projective space.

In higher (even) dimensions, the congruence of null curves of an involutive
almost Robinson structure $(N,K)$ is always geodesic, but shearing in general
[Tra02a]. The leaf space of $\mathcal{K}$ nevertheless still acquires a CR
structure [NT02, Tra02]. In addition, (almost) Robinson structures are
Lorentzian analogues of (almost) Hermitian structures, to borrow the
expression from Nurowski and Trautman [NT02]. In both cases, the underlying
geometric object is that of an _almost null structure_ , that is, a totally
null complex $(m+1)$-plane distribution. This perspective allows one to have a
unified approach to pseudo-Riemannian geometry in any signature. In dimension
four, the analogies between Lorentzian and Hermitian geometries were already
pointed out in [Nur93, Nur96] especially in connection with the aforementioned
theorems of mathematical relativity. For instance, the Kerr theorem finds an
articulation in Riemannian signature as follows: any local Hermitian structure
on four-dimensional Euclidean space corresponds to a holomorphic section of
its twistor bundle [ES85, SV09]. A Riemannian counterpart of the
Goldberg–Sachs theorem is given in [PB83, Nur93, AG97, GHN11]. In split
signature, one obtains analogous results – see e.g. [GS86, GHN11].

Almost null structures are also intimately connected with the notion of _pure
spinors_ , and thus hark back to Élie Cartan’s seminal work [Car67], which was
subsequently developed in [BT88, BT89, KT92, Kop97] among others. It is then
no surprise that in dimension four, the spinorial approach to general
relativity promoted by Penrose and his school [Wit59, Pen60, PR84, PR86] shed
much light on the _complex_ aspect of congruences of null geodesics, and was
influential in the development of twistor theory [Pen67]. These ideas were
later developed in higher even dimensions in [Hug90a, Hug90, Hug90b, Hug95,
Jef95], and most notably in the article [HM88] by Hughston and Mason, where
the Kerr and Robinson theorems are generalised in the context of involutive
almost null structures. These results were expanded by the third author of the
present article in [TC16, TC17, TC17a], where a comprehensive study of almost
null structures according to their intrinsic torsion is given in both even and
odd dimensions. The recent articles [Pap19, FO20, Pet19] also touch on related
topics on pseudo-Riemannian geometry.

Non-shearing congruences of null geodesics are ubiquitous in four-dimensional
mathematical relativity, see e.g. [Ste+03] and references therein. One
question that arises is, which of non-shearing congruences and (almost)
Robinson structures have most relevance in higher dimensions? On the one hand,
Robinson–Trautman and Kundt spacetimes, which are by definition characterised
by the existence of a _non-twisting_ non-shearing congruence of null
geodesics, have been well studied in arbitrary dimensions, see e.g. [PŽ09,
PŠ15]. On the other hand, the Kerr metric and its variants admit a pair of
_twisting_ congruences of null geodesics, which are non-shearing in dimension
four, but fail to be so in higher dimensions [PPO07]. Nonetheless, as was
first brought to light in [MT10], these metrics admit several Robinson
structures in any dimensions111These are not explicitly referred to as
Robinson structures there, but may be interpreted as such.. Almost Robinson
structures can also be defined in terms of a maximal totally null complex
distribution on odd-dimensional Lorentzian manifolds: the black ring in
dimension five is equipped with a pair of Robinson structures,222In [TC11],
these Robinson structures are referred to as optical structures in a sense
similar to [Nur96]. This terminology is now obsolete by virtue of [FLTC20].
but does not admit any non-shearing congruences of null geodesics [TC11] — see
also [TC12]. In dimension three, one can similarly obtain analogous results –
see e.g. [NTC15].

Another reason why non-shearing congruences of null geodesics in higher
dimensions are not as common as in dimension four has to do with curvature.
Let us review the various geometric interpretations of the following algebraic
condition on the Weyl tensor:

(1.1) $\displaystyle W(k,v,k,v)$ $\displaystyle=0\,,$ for any sections $k$ of
$K$ and $v$ of $K^{\perp}$,

where $K$ is an optical structure with congruence of null geodesics
$\mathcal{K}$. These are as follows:

  * •

In dimension _four_ , if $\mathcal{K}$ is geodesic and non-shearing
_regardless of whether $\mathcal{K}$ is non-twisting or not_, then condition
(1.1) holds — see e.g. [Ste+03, PR86].

  * •

In dimensions _greater_ than four, if $\mathcal{K}$ is non-shearing _and non-
twisting_ , then condition (1.1) holds — in fact, the Weyl tensor satisfies an
even stronger condition [OPP07].

  * •

In _odd_ dimensions greater than four, if $\mathcal{K}$ is twisting and
condition (1.1) holds, then $\mathcal{K}$ must also be shearing [OPP07].

  * •

In _even_ dimensions greater than four, if $\mathcal{K}$ is twisting and non-
shearing, and condition (1.1) holds, then the twist induces an almost Robinson
structure on $(\mathcal{M},g)$ [TC21].

In fact, in even dimensions greater than four, all the Einstein metrics
admitting a non-shearing twisting congruence of null geodesics satisfying a
mild algebraic condition on the Weyl tensor were found in [TC21]. These depend
on three real parameters, and include the Taub–NUT–(A)dS metric and the
Fefferman–Einstein metric — see also [Ale+21]. We thus see that the non-
shearing condition is rather restrictive in dimensions greater than four.

The structure of the paper is as follows. Section 2 focuses on the algebraic
description of the intrinsic torsion of an almost Robinson structure: we
classify the possible intrinsic torsion modules in Theorems 2.15 and 2.18.
This is then translated into the language of bundles in Section 3, where we
introduce almost Robinson geometries and investigate the geometric properties
of their associated congruences of null curves and leaf spaces, especially in
Sections 3.6 and 3.7 in connection with almost CR geometry. Their relation to
pure spinors fields is explained in Theorem 3.44, and is used in Section 3.11
to make contact with almost Hermitian manifolds. In Section 4, our definition
of almost Robinson manifold is extended to the conformal setting. In
particular, Theorem 4.4 gives all the conformally invariant classes of the
intrinsic torsion. Finally, we introduce the notion of generalised almost
Robinson geometry first considered in [RT85] in Section 5 by weakening the
conformal structure to a certain equivalence class $\mathbf{o}$ of Lorentzian
metrics on a smooth manifold. Of notable interest Theorem 5.4, which lists all
the possible $\mathbf{o}$-invariant classes of the intrinsic torsion. Section
6 discusses the possible generalisations to other metric signatures. We have
relegated to Appendices A and B a number of technical formulae that are used
in the main text.

We shall provide examples to illustrate some of the algebraic conditions that
the intrinsic torsion of an almost Robinson structure can satisfy, focussing
essentially on dimensions greater than four. Considering the rich range of
classes of almost Robinson structures, this article does not aim to cover
every possible case, but it leaves the construction of almost Robinson
structures with prescribed intrinsic torsion as open problems — see for
instance Remark 3.12. We do not touch on questions related to the curvature of
almost Robinson manifolds, these being dealt with in [TC14]. Finally, we point
out the recent articles [AGS18, Ale+21] whose content overlaps with some
aspects of the present article.

##  2\. Algebraic description

###  2.1. Notation and conventions

We set up the notation and conventions used throughout this article by
recalling some basic notions of algebra – see e.g. [Sal89, Che95] for further
details. The fields of real numbers and complex numbers will be denoted
$\mathbf{R}$ and $\mathbf{C}$ respectively, the imaginary unit by
$\mathrm{i}$, i.e. $\mathrm{i}^{2}=-1$.

Let ${\mathbb{V}}$ and $\mathbb{W}$ be two real or complex vector spaces with
respective duals ${\mathbb{V}}^{*}$ and ${\mathbb{W}}^{*}$. The annihilator of
a vector subspace $\mathbb{U}$ of ${\mathbb{V}}$ will be abbreviated to
$\mathrm{Ann}(\mathbb{U})$. The tensor product of ${\mathbb{V}}$ and
$\mathbb{W}$ will be denoted ${\mathbb{V}}\otimes\mathbb{W}$, the $p$-th
exterior power of ${\mathbb{V}}$ by $\scaleobj{1.2}{\wedge}^{p}{\mathbb{V}}$,
its $p$-th symmetric power by $\scaleobj{1.2}{\odot}^{p}{\mathbb{V}}$.

If $g$ is a non-degenerate symmetric bilinear form on ${\mathbb{V}}$, the
orthogonal complement of a subspace $\mathbb{U}$ of $\mathbb{V}$ with respect
to $g$ will be denoted $\mathbb{U}^{\perp}$. The subspace of
$\scaleobj{1.2}{\odot}^{p}{\mathbb{V}}$ consisting of elements that are
tracefree with respect to $g$ will be denoted by
$\scaleobj{1.2}{\odot}^{p}_{\circ}{\mathbb{V}}$.

Let us assume that ${\mathbb{V}}$ is complex and of dimension $2m$. Under the
Hodge duality operator
$\star:\scaleobj{1.2}{\wedge}^{p}{\mathbb{V}}^{*}\rightarrow\scaleobj{1.2}{\wedge}^{2m-p}{\mathbb{V}}^{*}$,
for $p=0,\ldots,2m$, the space $\scaleobj{1.2}{\wedge}^{m}{\mathbb{V}}^{*}$
splits into the space of self-dual $m$-forms
$\scaleobj{1.2}{\wedge}^{m}_{+}{\mathbb{V}}^{*}$ and the space of anti-self-
dual $m$-forms $\scaleobj{1.2}{\wedge}^{m}_{-}{\mathbb{V}}^{*}$, i.e.

$\displaystyle\scaleobj{1.2}{\wedge}^{m}{\mathbb{V}}^{*}=\scaleobj{1.2}{\wedge}^{m}_{+}{\mathbb{V}}^{*}\oplus\scaleobj{1.2}{\wedge}^{m}_{-}{\mathbb{V}}^{*}\,,$

where $\star\alpha=\pm(\mathrm{i})\alpha$ for any
$\alpha\in\scaleobj{1.2}{\wedge}^{m}_{\pm}{\mathbb{V}}^{*}$.

Suppose now that ${\mathbb{V}}$ is real. The complexification
$\mathbf{C}\otimes{\mathbb{V}}\cong{\mathbb{V}}\oplus\mathrm{i}{\mathbb{V}}$
of ${\mathbb{V}}$ will be denoted ${}^{\mathbf{C}}{\mathbb{V}}$. There is an
induced reality structure,
$\bar{}:{}^{\mathbf{C}}{\mathbb{V}}\rightarrow{}^{\mathbf{C}}{\mathbb{V}}$ on
${}^{\mathbf{C}}{\mathbb{V}}$, which preserves the elements of ${\mathbb{V}}$,
i.e. for $v\in{}^{\mathbf{C}}{\mathbb{V}}$, we have that $v\in{\mathbb{V}}$ if
and only if $\bar{v}=v$. If $\mathbb{A}$ is a vector subspace of
${}^{\mathbf{C}}{\mathbb{V}}$, its complex conjugate is defined by
$\overline{\mathbb{A}}:=\\{v\in{}^{\mathbf{C}}{\mathbb{V}}:\overline{v}\in\mathbb{A}\\}$.
We say that $\mathbb{A}$ is (totally) real if
$\overline{\mathbb{A}}=\mathbb{A}$.

Suppose now that ${\mathbb{V}}$ has dimension $2m$ and is equipped with a
complex structure $J$, that is an endomorphism of ${\mathbb{V}}$ that squares
to the identity on ${\mathbb{V}}$, i.e. $J\circ J=-\mathrm{Id}$. Then

$\displaystyle{}^{\mathbf{C}}{\mathbb{V}}$
$\displaystyle={\mathbb{V}}^{(1,0)}\oplus{\mathbb{V}}^{(0,1)}\,,$

where ${\mathbb{V}}^{(1,0)}$ and ${\mathbb{V}}^{(0,1)}$ are the
$+\mathrm{i}$\- and $-\mathrm{i}$-eigenspaces of $J$ respectively. These
$m$-dimensional complex vector subspaces are complex conjugate to each other,
i.e. ${\mathbb{V}}^{(1,0)}\cong\overline{{\mathbb{V}}^{(0,1)}}$. Similarly, we
have a splitting of the dual space

$\displaystyle{}^{\mathbf{C}}{\mathbb{V}}^{*}$
$\displaystyle=({\mathbb{V}}^{(1,0)})^{*}\oplus({\mathbb{V}}^{(0,1)})^{*}\,,$

and $({\mathbb{V}}^{(1,0)})^{*}=\mathrm{Ann}({\mathbb{V}}^{(0,1)})$ and
$({\mathbb{V}}^{(0,1)})^{*}=\mathrm{Ann}({\mathbb{V}}^{(1,0)})$. For any non-
negative integer $p$, $q$, the space of all $(p,q)$-forms on ${\mathbb{V}}$ is
defined to be

$\displaystyle\scaleobj{1.2}{\wedge}^{(p,q)}{\mathbb{V}}^{*}$
$\displaystyle:=\scaleobj{1.2}{\wedge}^{p}({\mathbb{V}}^{(1,0)})^{*}\otimes\scaleobj{1.2}{\wedge}^{q}({\mathbb{V}}^{(0,1)})^{*}\,.$

Similarly, we define the spaces

$\displaystyle\scaleobj{1.2}{\odot}{}^{(p,q)}{\mathbb{V}}^{*}$
$\displaystyle:=\scaleobj{1.2}{\odot}{}^{p}({\mathbb{V}}^{(1,0)})^{*}\otimes\scaleobj{1.2}{\odot}{}^{q}({\mathbb{V}}^{(0,1)})^{*}\,,$
(2.1)
$\displaystyle\raisebox{-1.72218pt}{\scaleobj{0.35}{\yng(2,1)}}({\mathbb{V}}^{*})$
$\displaystyle:=\left\\{\alpha\in\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}^{*}\otimes\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}^{*}:\pi^{(3,0)}(\alpha)=0\right\\}\,.$

where $\pi^{(3,0)}$ is the natural projection from
$\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}^{*}\otimes\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}^{*}$
to $\scaleobj{1.2}{\wedge}^{(3,0)}{\mathbb{V}}^{*}$. This notation reflects
the Young diagram symmetries of this irreducible
$\mathbf{GL}(m,\mathbf{C})$-module, where $\mathbf{GL}(m,\mathbf{C})$ is the
complex general linear group acting on
${\mathbb{V}}^{(1,0)}\cong\mathbf{C}^{m}$.

Since we are interested in real vector spaces, we also define, following the
notation in [Sal89],

(2.2) $\displaystyle\begin{aligned}
[\\![\scaleobj{1.2}{\wedge}^{(p,q)}{\mathbb{V}}^{*}]\\!]\otimes_{\mathbf{R}}\mathbf{C}&:=\scaleobj{1.2}{\wedge}^{(p,q)}{\mathbb{V}}^{*}\oplus\scaleobj{1.2}{\wedge}^{(q,p)}{\mathbb{V}}^{*}\,,&&p\neq
q\,,\\\
[\scaleobj{1.2}{\wedge}^{(p,p)}{\mathbb{V}}^{*}]\otimes_{\mathbf{R}}\mathbf{C}&:=\scaleobj{1.2}{\wedge}^{(p,p)}{\mathbb{V}}^{*}\,,\\\
\end{aligned}$

This notation will be extended in the obvious way to
$\scaleobj{1.2}{\odot}{}^{(p,q)}{\mathbb{V}}^{*}$ and
$\raisebox{-1.72218pt}{\scaleobj{0.35}{\yng(2,1)}}({\mathbb{V}}^{*})$.

Finally, we shall consider a Hermitian vector space $({\mathbb{V}},J,h)$ where
$J$ is a complex structure compatible with a positive-definite symmetric
bilinear form $h$, i.e. $J\circ h=-h\circ J$. Then
${\mathbb{V}}^{(1,0)}\cong({\mathbb{V}}^{(0,1)})^{*}$ and
${\mathbb{V}}^{(0,1)}\cong({\mathbb{V}}^{(1,0)})^{*}$ so that
${\mathbb{V}}^{(1,0)}$ and ${\mathbb{V}}^{(0,1)}$ are totally null with
respect to $h$. The Hermitian $2$-form on ${\mathbb{V}}$ is defined by
$\omega=h\circ J$. For $pq\neq 0$, the subspace of
$\scaleobj{1.2}{\wedge}^{(p,q)}{\mathbb{V}}^{*}$ and
$\scaleobj{1.2}{\odot}{}^{(p,q)}{\mathbb{V}}^{*}$ consisting of all
$(p,q)$-forms that are tracefree with respect to $\omega^{-1}$ will be denoted
$\scaleobj{1.2}{\wedge}^{(p,q)}_{\circ}{\mathbb{V}}^{*}$ and
$\scaleobj{1.2}{\odot}{}^{(p,q)}_{\circ}{\mathbb{V}}^{*}$ respectively. Note
that $\scaleobj{1.2}{\wedge}^{(1,1)}{\mathbb{V}}^{*}\cong\mathfrak{u}(m)$ and
$\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}^{*}\cong\mathfrak{su}(m)$,
where $\mathfrak{u}(m)$ and $\mathfrak{su}(m)$ are the Lie algebras of the
unitary group $\mathbf{U}(m)$ and special unitary group $\mathbf{SU}(m)$
respectively.

###  2.2. Linear algebra

####  2.2.1. Null structures

Let $\widetilde{{\mathbb{V}}}$ be a $(2m+2)$-dimensional oriented complex
vector space equipped with a non-degenerate symmetric bilinear form
$\widetilde{g}$. We introduce abstract indices following the convention of
[FLTC20]: minuscule Roman indices starting with the beginnning of the alphabet
$a,b,c,\ldots$ will refer to elements of $\widetilde{{\mathbb{V}}}$ and its
dual, and tensor products thereof, e.g. $v^{a}\in\widetilde{{\mathbb{V}}}$ and
$\alpha_{a}{}^{b}\in\widetilde{{\mathbb{V}}}^{*}\otimes\widetilde{{\mathbb{V}}}$.
Round brackets and squared brackets enclosing a group of indices will denote
symmetrisation and skew-symmetrisation respectively, e.g.

$\displaystyle T^{(ab)}$
$\displaystyle=\frac{1}{2}\left(T^{ab}+T^{ba}\right)\,,$
$\displaystyle\beta_{[abc]}$
$\displaystyle=\frac{1}{3!}\left(\beta_{abc}-\beta_{acb}+\beta_{bca}-\beta_{bac}+\beta_{cab}-\beta_{cba}\right)\,.$

In particular, the symmetric bilinear form satisfies
$\widetilde{g}_{ab}=\widetilde{g}_{(ab)}$, and together with its inverse
$\widetilde{g}^{ab}$, will be used to raise and lower indices. The tracefree
(symmetric) part of a tensor with respect to $\widetilde{g}$ will be adorned
with a small circle, e.g. either as $T_{(ab)_{\circ}}$ or
$\left(T_{ab}\right)_{\circ}$.

######  Definition 2.1.

[BT89, KT92, NT02, TC16] A _null structure_ on
$(\widetilde{{\mathbb{V}}},\widetilde{g})$ is a maximal totally null (MTN)
vector subspace of $\widetilde{{\mathbb{V}}}$, i.e.
$\mathbb{N}=\mathbb{N}^{\perp}$. In other words, for any $v,w\in\mathbb{N}$,
$\widetilde{g}(v,w)=0$ and $\mathbb{N}$ has dimension $m+1$.

A null structure $\mathbb{N}$ on $(\widetilde{{\mathbb{V}}},\widetilde{g})$
singles out the one-dimensional vector subspace
$\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})$ of
$\scaleobj{1.2}{\wedge}^{m+1}\widetilde{{\mathbb{V}}}^{*}$. Any element $\nu$
of $\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})$ is then totally
null, i.e. $\nu_{aa_{1}\ldots a_{m}}\nu^{a}{}_{b_{1}\ldots b_{m}}=0$. In
particular, $\nu$ satisfies the following properties:

  1. (1)

$\nu$ is simple, i.e. $\nu_{a_{1}\ldots a_{m}[a_{m+1}}\nu_{b_{1}\ldots
b_{m+1}]}=0$.

  2. (2)

$\nu$ is either self-dual or anti-self-dual, i.e. either
$\star\nu=(\mathrm{i})\nu$ or $\star\nu=-(\mathrm{i})\nu$.

Conversely, any self-dual or anti-self-dual simple $(m+1)$-form $\nu$ must be
totally null, and thus defines the MTN vector subspace

$\displaystyle\mathbb{N}$
$\displaystyle=\left\\{v\in\widetilde{{\mathbb{V}}}:v\makebox[7.0pt]{\rule{6.0pt}{0.3pt}\rule{0.3pt}{5.0pt}}\,\nu=0\right\\}\,.$

We shall therefore refer to a null structure $\mathbb{N}$ as either self-dual
or anti-self-dual depending on whether
$\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})\subset\scaleobj{1.2}{\wedge}^{m+1}_{+}\widetilde{{\mathbb{V}}}^{*}$
or
$\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})\subset\scaleobj{1.2}{\wedge}^{m+1}_{-}\widetilde{{\mathbb{V}}}^{*}$.

The space of all MTN vector subspaces of $\widetilde{{\mathbb{V}}}$, i.e. null
structures, is a complex homogeneous space of complex dimension
$\frac{1}{2}m(m+1)$, referred to as the _isotropic Grassmannian_
$\mathrm{Gr}_{m+1}(\widetilde{{\mathbb{V}}},\widetilde{g})$ of
$(\widetilde{{\mathbb{V}}},\widetilde{g})$. This space splits into two
disconnected components
$\mathrm{Gr}^{+}_{m+1}(\widetilde{{\mathbb{V}}},\widetilde{g})$ and
$\mathrm{Gr}^{-}_{m+1}(\widetilde{{\mathbb{V}}},\widetilde{g})$ according to
whether their elements are self-dual or anti-self-dual. The complex Lie group
$\mathbf{SO}(2m+2,\mathbf{C})$ acts transitively on each of these components.

######  Remark 2.2.

Any complement of $\mathbb{N}$ in $\widetilde{{\mathbb{V}}}$ must be dual to
$\mathbb{N}$ and totally null with respect to $\widetilde{g}$, and we shall
write $\widetilde{{\mathbb{V}}}=\mathbb{N}^{*}\oplus\mathbb{N}$, bearing in
mind that in general such a splitting is not canonical. For consistency with
the notation introduced subsequently, we shall assume with no loss of
generality that $\mathbb{N}$ is self-dual. In abstract index notation,
elements of $\mathrm{Ann}(\mathbb{N})$ will be adorned with lower Roman
majuscule indices, and elements of $\mathbb{N}^{*}$ with upper Roman majuscule
indices, e.g. $\alpha_{A}\in\mathrm{Ann}(\mathbb{N})\cong\mathbb{N}$ and
$v^{A}\in\mathbb{N}^{*}\cong\mathrm{Ann}(\mathbb{N}^{*})$. These indices will
be immovable. We also introduce splitting operators
$(\delta^{a}_{A},\delta^{aA})$, that is, projections
$\delta^{aA}:\widetilde{{\mathbb{V}}}^{*}\rightarrow\mathbb{N}^{*}$ and
$\delta^{a}_{A}:\widetilde{{\mathbb{V}}}^{*}\rightarrow\mathbb{N}$ that
satisfy $\delta^{a}_{A}\delta_{a}^{B}=\delta_{A}^{B}$. These will also be used
to inject elements of $\mathbb{N}$ and $\mathbb{N}^{*}$ into
$\widetilde{{\mathbb{V}}}$.

####  2.2.2. Robinson structures

Let ${\mathbb{V}}$ be a $(2m+2)$-dimensional real vector space equipped with a
non-degenerate symmetric bilinear form $g$ of (Lorentzian) signature
$(2m+1,1)$, i.e. $(+,+,\ldots,+,-)$. As is customary, we call
$({\mathbb{V}},g)$ Minkowski space. The abstract index notation introduced in
the previous section will equally apply to $({\mathbb{V}},g)$.

Denote by ${}^{\mathbf{C}}{\mathbb{V}}$ the complexification of
${\mathbb{V}}$, and extend $g$ to a non-degenerate complex-valued symmetric
bilinear form ${}^{\mathbf{C}}g$ on ${}^{\mathbf{C}}{\mathbb{V}}$. By abuse of
notation, we shall often denote ${}^{\mathbf{C}}g$ by $g$. The
complexification $({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$ of
$({\mathbb{V}},g)$ thus gives rise to the complex space
$(\widetilde{{\mathbb{V}}},\widetilde{g})$ considered in the previous section,
together with a reality condition. By extension, there is a well-defined
notion of null structure on $({\mathbb{V}},g)$ via
$({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$. To make this idea more
precise, we note that the complex conjugate $\overline{\mathbb{N}}$ of a MTN
vector subspace $\mathbb{N}$ on
$({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$ is also MTN.

######  Definition 2.3 ([KT92]).

The _real index_ of a null structure $\mathbb{N}$ on
$({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$ is the complex dimension of
the intersection of $\mathbb{N}$ and $\overline{\mathbb{N}}$.

######  Definition 2.4.

A _Robinson structure_ on Minkowski space $({\mathbb{V}},g)$ of dimension
$2m+2$ is a null structure $\mathbb{N}$ of real index one on
$({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$. We shall denote it by the
pair $(\mathbb{N},{\mathbb{K}})$ where

  1. (1)

$\mathbb{N}$ is an MTN vector subspace of ${}^{\mathbf{C}}{\mathbb{V}}$ of
real index one,

  2. (2)

${\mathbb{K}}$ is the real null line $\mathbb{N}\cap{\mathbb{V}}$.

With this second condition, we have that
${}^{\mathbf{C}}{\mathbb{K}}=\mathbb{N}\cap\overline{\mathbb{N}}$ and
$\mathbb{N}+\overline{\mathbb{N}}={}^{\mathbf{C}}{\mathbb{K}}^{\perp}$.

It turns out that the real Lie group $\mathbf{SO}(2m+1,1)$ also acts
transitively on each of the connected spaces of MTN vector spaces of
$({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$. In other words:

######  Lemma 2.5 ([KT92]).

Let $({\mathbb{V}},g)$ be Minkowski space of dimension $2m+2$. A null
structure on $({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$ always has real
index one, and hence is a Robinson structure.

######  Remark 2.6.

Note that $(\mathbb{N},{\mathbb{K}})$ and
$(\overline{\mathbb{N}},{\mathbb{K}})$ define the same Robinson structure.
Their Hodge duality is the same when $m$ is even, but opposite when $m$ is
odd. We shall say that a Robinson structure $(\mathbb{N},{\mathbb{K}})$ is
(anti-)self-dual if $\mathbb{N}$ is (anti-)self-dual. This entails of course a
preference of $\mathbb{N}$ over $\overline{\mathbb{N}}$ when $m$ is odd, but
there is no ambiguity when $m$ is even.

Any element $\nu$ of $\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})$
will be referred to as a _complex Robinson $(m+1)$-form_. In particular, when
$m$ is odd, if $\nu$ is a self-dual, so are its complex conjugate $\bar{\nu}$
and the real $(m+1)$-form $\nu+\overline{\nu}$.

####  2.2.3. Robinson structures and optical structures

It is clear that a Robinson structure $(\mathbb{N},{\mathbb{K}})$ on
$({\mathbb{V}},g)$ determines in particular an optical structure, namely
${\mathbb{K}}$, in the sense of [FLTC20]. We therefore have a filtration of
vector subspaces

(2.3)
$\displaystyle\\{0\\}\subset{\mathbb{K}}\subset{\mathbb{K}}^{\perp}\subset{\mathbb{V}}\,,$

and the screen space
$\mathbb{H}_{\mathbb{K}}={\mathbb{K}}^{\perp}/{\mathbb{K}}$ inherits a
positive-definite symmetric bilinear form $h$ given by

$\displaystyle h(v+{\mathbb{K}},w+{\mathbb{K}})$ $\displaystyle:=g(v,w)\,,$
for any $v,w\in\Gamma({\mathbb{K}}^{\perp})$.

Any element of ${\mathbb{K}}$ will be referred to as an _optical vector_ , and
any element of $\mathrm{Ann}({\mathbb{K}})$ as an _optical $1$-form_.

Now, define an endomorphism $J$ of $\mathbb{H}_{\mathbb{K}}$ and its
complexification ${}^{\mathbf{C}}\mathbb{H}_{\mathbb{K}}$ by

$\displaystyle J(v+{}^{\mathbf{C}}{\mathbb{K}})$
$\displaystyle=-\mathrm{i}v+{}^{\mathbf{C}}{\mathbb{K}}\,,$ for any
$v\in\mathbb{N}$, $\displaystyle J(v+{}^{\mathbf{C}}{\mathbb{K}})$
$\displaystyle=\mathrm{i}v+{}^{\mathbf{C}}{\mathbb{K}}\,,$ for any
$v\in\overline{\mathbb{N}}$.

Then $J$ is a complex structure on $\mathbb{H}_{\mathbb{K}}$, and
${}^{\mathbf{C}}\mathbb{H}_{\mathbb{K}}$ splits into the eigenspaces of $J$,
i.e.

(2.4)
$\displaystyle{}^{\mathbf{C}}\mathbb{H}_{\mathbb{K}}=\mathbb{H}_{\mathbb{K}}^{(1,0)}\oplus\mathbb{H}_{\mathbb{K}}^{(0,1)}\,,$

where
$\mathbb{H}_{\mathbb{K}}^{(1,0)}:=\overline{\mathbb{N}}/{}^{\mathbf{C}}{\mathbb{K}}$
and $\mathbb{H}_{\mathbb{K}}^{(0,1)}:=\mathbb{N}/{}^{\mathbf{C}}{\mathbb{K}}$.
These can be shown to be maximal totally null with respect to the bilinear
form $h$ on
${}^{\mathbf{C}}\mathbb{H}_{\mathbb{K}}=(\mathbb{N}+\overline{\mathbb{N}})/(\mathbb{N}\cap\overline{\mathbb{N}})$,
and thus $J$ is compatible with $h$, i.e. $J$ is Hermitian [NT02].

Conversely, suppose that $({\mathbb{V}},g)$ is equipped with an optical
structure ${\mathbb{K}}$ together with a complex structure $J$ on the screen
space $\mathbb{H}_{\mathbb{K}}$ compatible with $h$. Define

$\displaystyle\mathbb{N}=\left\\{v\in{}^{\mathbf{C}}{\mathbb{K}}^{\perp}:J(v+{}^{\mathbf{C}}{\mathbb{K}})=-\mathrm{i}v+{}^{\mathbf{C}}{\mathbb{K}}\right\\}\,.$

Then $\mathbb{N}$ has dimension $m+1$, and is totally null. Indeed, for any
$v,w\in\mathbb{N}$, we have

$\displaystyle g(v,w)$
$\displaystyle=h(v+{}^{\mathbf{C}}{\mathbb{K}},w+{}^{\mathbf{C}}{\mathbb{K}})$
$\displaystyle=h(\mathrm{i}J(v+{}^{\mathbf{C}}{\mathbb{K}}),\mathrm{i}J(w+{}^{\mathbf{C}}{\mathbb{K}}))$
$\displaystyle=-h(v+{}^{\mathbf{C}}{\mathbb{K}},w+{}^{\mathbf{C}}{\mathbb{K}})$
$\displaystyle=-g(v,w)\,,$

since $J\circ h=-h\circ J$, and thus $g(v,w)=0$. The complex conjugate
$\overline{\mathbb{N}}$ is defined analogously.

In abstract index notation, elements of $\mathbb{H}_{\mathbb{K}}$ and of its
dual, and tensor product thereof, will be adorned with minuscule Roman indices
starting from the middle of the alphabet $i,j,k,\ldots$. In particular, the
screen space inner product and its inverse will be expressed as $h_{ij}$ and
$h^{ij}$ respectively, and will be used to lower and raise this type of
indices. The complex structure and the Hermitian $2$-form will take the form
$J_{i}{}^{j}$ and $\omega_{ij}:=J_{i}{}^{k}h_{kj}$ respectively. As before,
symmetrisation and skew-symmetrisation will be denoted by round and squared
brackets around indices respectively, and the tracefree part of a tensor with
respect to $h_{ij}$ will be adorned by a small circle.

We shall use upper and lower Greek indices to denote elements of
$\mathbb{H}_{{\mathbb{K}}}^{(1,0)}$ and
$(\mathbb{H}_{{\mathbb{K}}}^{(1,0)})^{*}$ respectively, and upper and lower
overlined Greek indices to denote elements of
$\mathbb{H}_{{\mathbb{K}}}^{(0,1)}$ and
$(\mathbb{H}_{{\mathbb{K}}}^{(0,1)})^{*}$ respectively, e.g.
$v^{\alpha}\in\mathbb{H}_{{\mathbb{K}}}^{(1,0)}$ and
$\alpha_{\bar{\beta}}\in(\mathbb{H}_{{\mathbb{K}}}^{(0,1)})^{*}$. As usual,
symmetrisation and skew-symmetrisation will be denoted by round brackets and
squared brackets respectively. The Hermitian form on
${}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}}}$ will then be expressed as
$h_{\alpha\bar{\beta}}$ and its inverse by $h^{\alpha\bar{\beta}}$, which will
be used to convert indices, i.e.
$v_{\bar{\beta}}=v^{\alpha}h_{\alpha\bar{\beta}}$ for any
$v^{\alpha}\in\mathbb{H}_{{\mathbb{K}}}^{(1,0)}$. The totally tracefree part
of a mixed tensor $T_{\alpha\beta\bar{\gamma}}$, say, with respect to
$h^{\alpha\bar{\beta}}$ will be denoted
$\left(T_{\alpha\beta\bar{\gamma}}\right)_{\circ}$.

We shall also introduce for convenience splitting operators
$(\delta^{i}_{\alpha},\delta^{i}_{\bar{\alpha}})$ on
${}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}}}$, that is projections
$\delta^{i}_{\alpha}:{}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}}}^{*}\rightarrow(\mathbb{H}_{{\mathbb{K}}}^{(1,0)})^{*}$
and
$\delta^{i}_{\bar{\alpha}}:{}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}}}^{*}\rightarrow(\mathbb{H}_{{\mathbb{K}}}^{(0,1)})^{*}$
that satisfy

$\displaystyle\delta^{i}_{\alpha}J_{i}{}^{j}$
$\displaystyle=\mathrm{i}\delta^{j}_{\alpha}\,,$
$\displaystyle\delta^{i}_{\bar{\alpha}}J_{i}{}^{j}$
$\displaystyle=-\mathrm{i}\delta^{j}_{\bar{\alpha}}\,,$ $\displaystyle
h_{ij}\delta^{i}_{\alpha}\delta^{i}_{\bar{\beta}}$
$\displaystyle=h_{\alpha\bar{\beta}}\,,$ $\displaystyle
h_{ij}\delta^{i}_{\alpha}\delta^{i}_{\beta}$ $\displaystyle=0\,.$

These will also be used to inject $\mathbb{H}_{{\mathbb{K}}}^{(1,0)}$ into
${}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}}}$, and so on. Their dual versions
$(\delta_{i}^{\alpha},\delta_{i}^{\bar{\alpha}})$ can be obtained by raising
and lowering the indices with $h_{ij}$ and $h^{\alpha\bar{\beta}}$. Thus, in
particular, we can express $h_{ij}$ and $\omega_{ij}=J_{i}{}^{k}h_{kj}$ as

$\displaystyle\omega_{ij}$
$\displaystyle=2\mathrm{i}h_{\alpha\bar{\beta}}\delta_{[i}^{\alpha}\delta_{j]}^{\bar{\beta}}\,,$
$\displaystyle h_{ij}$
$\displaystyle=2h_{\alpha\bar{\beta}}\delta_{(i}^{\alpha}\delta_{j)}^{\bar{\beta}}\,.$

######  Remark 2.7.

From the discussion above, it is also conceptually useful to start with an
optical structure ${\mathbb{K}}$ on $({\mathbb{V}},g)$, and declare a null or
Robinson structure on $({\mathbb{V}},g)$ be _compatible_ with ${\mathbb{K}}$
if ${\mathbb{K}}={\mathbb{V}}\cap\mathbb{N}$. In dimension four, there is a
single Robinson structure (up to complex conjugation) compatible with an
optical structure, but this is not true in higher dimensions – see Remark
2.14.

####  2.2.4. Splitting

As before let $(\mathbb{N},{\mathbb{K}})$ be a Robinson structure on Minkowski
space $({\mathbb{V}},g)$. Any choice of splitting

(2.5) $\displaystyle{}^{\mathbf{C}}{\mathbb{V}}$
$\displaystyle=\mathbb{N}^{*}\oplus\mathbb{N}\,,$

for some choice of dual $\mathbb{N}^{*}$ induces a splitting of the filtration
(2.3) as

(2.6) $\displaystyle{\mathbb{V}}$
$\displaystyle=\mathbb{L}\oplus\mathbb{H}_{{\mathbb{K}},\mathbb{L}}\oplus{\mathbb{K}}\,,$

where

$\displaystyle{\mathbb{K}}$
$\displaystyle=\mathbb{N}\cap{\mathbb{V}}=\overline{\mathbb{N}}\cap{\mathbb{V}}\,,$
$\displaystyle\mathbb{L}$
$\displaystyle:=\mathbb{N}^{*}\cap{\mathbb{V}}=\overline{\mathbb{N}}^{*}\cap{\mathbb{V}}\,,$
$\displaystyle\mathbb{H}_{{\mathbb{K}},\mathbb{L}}$
$\displaystyle:={\mathbb{K}}^{\perp}\cap\mathbb{L}^{\perp}\,.$

In particular, $\mathbb{L}$ is a null line dual to ${\mathbb{K}}$. Note that
$\mathbb{H}_{{\mathbb{K}},\mathbb{L}}$ is isomorphic to the screen space
$\mathbb{H}_{\mathbb{K}}$, but this isomorphism depends on the choice of
$\mathbb{L}$. Further, in complete analogy with (2.4), we obtain the splitting

(2.7)
$\displaystyle{}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}},\mathbb{L}}=\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(1,0)}\oplus\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(0,1)}\,,$

where

$\displaystyle\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(1,0)}$
$\displaystyle:=\overline{\mathbb{N}}\cap{}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}},\mathbb{L}}=\overline{\mathbb{N}}\cap\mathbb{N}^{*}\,,$
$\displaystyle\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(0,1)}$
$\displaystyle:=\mathbb{N}\cap{}^{\mathbf{C}}\mathbb{H}_{{\mathbb{K}},\mathbb{L}}=\mathbb{N}\cap\overline{\mathbb{N}}^{*}\,.$

We note that
$\mathbb{H}_{\mathbb{K}}^{(1,0)}\cong\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(1,0)}$
and
$\mathbb{H}_{\mathbb{K}}^{(0,1)}\cong\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(0,1)}$,
but these isomorphisms depend on the choice of $\mathbb{N}^{*}$

We introduce splitting operators $(\ell^{a},\delta^{a}_{i},k^{a})$ with dual
$(\kappa_{a},\delta_{a}^{i},\lambda_{a})$ adapted to (2.6), where $k^{a}$ and
$\ell^{a}$ are elements in ${\mathbb{K}}$ and in $\mathbb{L}$ respectively
such that $g_{ab}k^{a}\ell^{b}=1$, and $\delta^{a}_{i}$ projects from
${\mathbb{V}}^{*}$ to $\mathbb{H}^{*}_{{\mathbb{K}},\mathbb{L}}$ and satisfies
$g_{ab}\delta^{a}_{i}\delta^{b}_{j}=h_{ij}$. Any change of splitting which
preserves $k^{a}$ induces the transformations

(2.8) $\displaystyle k^{a}$ $\displaystyle\mapsto k^{a}\,,$
$\displaystyle\delta^{a}_{i}$
$\displaystyle\mapsto\delta^{a}_{i}+\phi_{i}k^{a}\,,$ $\displaystyle\ell^{a}$
$\displaystyle\mapsto\ell^{a}-\phi^{i}\delta^{a}_{i}-\frac{1}{2}\phi^{i}\phi_{i}k^{a}\,,$

for some $\phi_{i}$ in $(\mathbf{R}^{2m})^{*}$, and similarly for their duals.

Finally, combining the splitting operators $(\ell^{a},\delta^{a}_{i},k^{a})$
and $(\delta^{i}_{\alpha},\delta^{i}_{\bar{\alpha}})$ yield new ones
$(\ell^{a},\delta^{a}_{\alpha},\delta^{a}_{\bar{\alpha}},k^{a})$ where
$\delta^{a}_{\alpha}:=\delta^{a}_{i}\delta^{i}_{\alpha}$ projects from
${}^{\mathbf{C}}{\mathbb{V}}^{*}$ to
$(\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(1,0)})^{*}$. We naturally obtain dual
spitting operators
$(\kappa_{a},\delta_{a}^{\alpha},\delta_{a}^{\bar{\alpha}},\lambda_{a})$. In
terms of the splitting operators $(\delta^{a}_{A},\delta_{a}^{A})$, we have
$\lambda_{a}\delta^{a}_{A}=\delta_{a}^{\bar{\alpha}}\delta^{a}_{A}=0$ and
$k^{a}\delta_{a}^{A}=\delta^{a}_{\bar{\alpha}}\delta_{a}^{A}=0$, and we may
define further splitting operators
$\delta^{A}_{\alpha}:=\delta^{A}_{a}\delta^{a}_{\alpha}$ projecting from
$\mathbb{N}$ to $(\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(1,0)})^{*}$, and
$\delta_{A}^{\alpha}:=\delta_{A}^{a}\delta_{a}^{\alpha}$ from $\mathbb{N}^{*}$
to $(\mathbb{H}_{{\mathbb{K}},\mathbb{L}}^{(1,0)})$. One may similarly define
$\delta^{A\bar{\alpha}}$ and $\delta_{A\bar{\alpha}}$.

####  2.2.5. Robinson $3$-forms

For any choice of splitting operators $(\kappa_{a},\delta^{i}_{a},\ell_{a})$,
and recalling that $\omega_{ij}$ is the Hermitian form on
$\mathbb{H}_{{\mathbb{K}}}$, we set
$\omega_{ab}=\omega_{ij}\delta^{i}_{a}\delta^{j}_{b}$ and define

(2.9) $\displaystyle\rho_{abc}$ $\displaystyle:=3\,\kappa_{[a}\omega_{bc]}\,.$

By Lemma 3.1 of [FLTC20], the definition of $\rho_{abc}$ depends only on the
choice of optical $1$-form $\kappa_{a}$, and not on the choice of
$\delta^{a}_{i}$ and $\lambda_{a}$. One can check that the $3$-form
$\rho_{abc}$ satisfies

(2.10) $\displaystyle\rho_{ab}\,{}^{e}\rho_{cde}$
$\displaystyle=-4\,\kappa_{[a}g_{b][c}\kappa_{d]}\,.$

We shall refer to such a $3$-form as a _Robinson $3$-form (associated to the
optical $1$-form $\kappa_{a}$)_.

Conversely, let $\rho_{abc}$ be a $3$-form that satisfies the algebraic
property (2.10) for some null $1$-form $\kappa_{a}$. Then, one can check that
$\kappa_{[a}\rho_{bcd]}=0$ and $k^{a}\rho_{abc}=0$. To prove
$\kappa_{[a}\rho_{bcd]}=0$, we skew (2.10) with $\kappa_{a}$ to get
$\kappa_{[a}\rho_{bc]}\,{}^{f}\rho_{def}=0$. Now contracting with
$\rho^{e}{}_{gh}$ and using $\eqref{eq:3-form-sq2}$ again yields
$\kappa_{[a}\rho_{bc][d}\kappa_{e]}=0$, and the result follows by skewing over
the first four indices. That $k^{a}\rho_{abc}=0$ can be proved in a similar
fashion. Hence, we can apply Lemma 3.1 of [FLTC20], and setting
$\omega_{ij}=\ell^{a}\delta^{b}_{i}\delta^{c}_{j}\rho_{abc}$ for any choice of
splitting operators $(\ell^{a},\delta^{a}_{i},k^{a})$ where
$\kappa_{a}\ell^{b}=1$, we see that $\omega_{ij}$ is the required Hermitian
form on $\mathbb{H}_{\mathbb{K}}$ up to sign. This sign can be fixed so that
if $k^{a}=g^{ab}\kappa_{b}$,

$\displaystyle v^{c}\rho_{c}{}^{ab}$
$\displaystyle=-2\mathrm{i}v^{[a}k^{b]}\,,$ for any $v^{a}\in\mathbb{N}$,
$\displaystyle w^{c}\rho_{c}{}^{ab}$
$\displaystyle=2\mathrm{i}w^{[a}k^{b]}\,,$ $w^{a}\in\overline{\mathbb{N}}$.

######  Remark 2.8.

In low dimensions, we note the following:

  * •

In dimension four, $\rho{{}_{bcd}}$ is the Hodge dual of $\kappa_{a}$, i.e.
$\kappa=\star\rho$. This reflects the fact that an optical structure is
equivalent to a Robinson structure.

  * •

In dimension six, the $3$-form $\rho_{abc}$ defined above can be either self-
dual or anti-self-dual under Hodge duality, consistent with the fact that the
complex conjugate $\overline{\mathbb{N}}$ of a MTN vector space $\mathbb{N}$
has the same Hodge duality as $\mathbb{N}$ in this case – see Remark 2.6.

####  2.2.6. Robinson spinors

We proceed to describe a Robinson structure in terms of spinors following the
treatment of [Car67, PR86, BT88, BT89, KT92, LM89, TC16].

We first consider a $(2m+2)$-dimensional complex vector space
$(\widetilde{{\mathbb{V}}},\widetilde{g})$ equipped with a non-degenerate
symmetric bilinear form. The double cover $\mathbf{Spin}(2m+2,\mathbf{C})$ of
$\mathbf{SO}(2m+2,\mathbf{C})$ allows us to define the spinor representation
$\mathbb{S}$ of $(\widetilde{{\mathbb{V}}},\widetilde{g})$, which splits into
a direct sum of two $2^{m}$-dimensional irreducible chiral spin spaces
$\mathbb{S}_{+}$ and $\mathbb{S}_{-}$, the spaces of spinors of positive and
negative chiralities respectively. Following the notation of [TC17a], elements
of $\mathbb{S}_{+}$ and $\mathbb{S}_{-}$ will be adorned with bold primed and
unprimed majuscule Roman indices respectively, e.g.
$\alpha^{\mathbf{A}^{\prime}}\in\mathbb{S}_{+}$ and
$\beta^{\mathbf{A}}\in\mathbb{S}_{-}$, and similarly for the dual spin spaces
$\mathbb{S}_{\pm}^{*}$ with lower indices. The spin space $\mathbb{S}$ is also
equipped with $\mathbf{Spin}(2m+2,\mathbf{C})$-invariant bilinear forms, which
allow the following identifications:

(2.13) $\displaystyle\begin{array}[]{|c|c|}\hline\cr\mbox{$m$ odd}&\mbox{$m$
even}\\\
\hline\cr\hline\cr\mathbb{S}_{\pm}\cong\mathbb{S}_{\pm}^{*}&\mathbb{S}_{\pm}\cong\mathbb{S}_{\mp}^{*}\\\
\hline\cr\end{array}$

The _Clifford action_ of $\widetilde{{\mathbb{V}}}$ on $\mathbb{S}$ is
effected by means of the _van der Waerden symbols_
$\gamma_{a\mathbf{A}}{}^{\mathbf{B}^{\prime}}$ and
$\gamma_{a\mathbf{A}^{\prime}}{}^{\mathbf{B}}$, injective maps from
${\mathbb{V}}$ to the space of homomorphisms
$\mathrm{Hom}(\mathbb{S}_{\pm},\mathbb{S}_{\mp})$. These satisfy the Clifford
property

(2.14)
$\displaystyle\gamma_{(a\mathbf{A}^{\prime}}{}^{\mathbf{C}}\gamma_{b)\mathbf{C}}{}^{\mathbf{B}^{\prime}}$
$\displaystyle=g_{ab}\delta_{\mathbf{A}^{\prime}}^{\mathbf{B}^{\prime}}\,,$
$\displaystyle\gamma_{(a_{\mathbf{A}}}{}^{\mathbf{C}^{\prime}}\gamma_{b)\mathbf{C}^{\prime}}{}^{\mathbf{B}}$
$\displaystyle=g_{ab}\delta_{\mathbf{A}}^{\mathbf{B}}\,,$

where $\delta_{\mathbf{A}^{\prime}}^{\mathbf{B}^{\prime}}$ and
$\delta_{\mathbf{A}}^{\mathbf{B}}$ denote the identity elements on
$\mathbb{S}_{+}$ and $\mathbb{S}_{-}$ respectively. Let
$\nu^{\mathbf{A}^{\prime}}$ be a spinor, and consider the linear map
$\nu_{a}^{\mathbf{A}}:=\nu^{\mathbf{B}^{\prime}}\gamma_{a\mathbf{B}^{\prime}}{}^{\mathbf{A}}:\widetilde{{\mathbb{V}}}\rightarrow\mathbb{S}_{-}$.
Denote by $\mathbb{N}$ the kernel of $\nu_{a}^{\mathbf{A}}$. By (2.14),
$\mathbb{N}$ must be totally null. We say that $\nu^{\mathbf{A}^{\prime}}$ is
_pure_ if $\mathbb{N}$ has maximal dimension $m+1$, i.e. $\mathbb{N}$ is a
null structure on $(\widetilde{{\mathbb{V}}},\widetilde{g})$. Any spinor
proportional to $\nu^{\mathbf{A}^{\prime}}$ defines the same null structure.
More generally, Cartan showed [Car67] that there is a one-to-one
correspondence between null structures on
$(\widetilde{{\mathbb{V}}},\widetilde{g})$ and pure spinors up to scale.
Further, self-dual null structures correspond to pure spinors of positive
chirality, and anti-self-dual null structures to pure spinors of negative
chirality. The components of a pure spinor are algebraically constrained
[Car67]. Indeed, a spinor $\nu^{\mathbf{A}^{\prime}}$ is pure if and only if
it satisfies the _purity condition_ [PR86, HM88, TC12]

(2.15) $\displaystyle\nu_{a}^{\mathbf{A}}\nu^{a\mathbf{B}}=0\,.$

When $m=0,1,2$, conditions (2.15) is vacuous, i.e. all spinors are pure. A
self-dual null structure $\mathbb{N}$ thus singles out a one-dimensional
vector subspace $\mathbb{S}_{+}^{\mathbb{N}}$ of $\mathbb{S}_{+}$, any element
$\nu^{\mathbf{A}^{\prime}}$ of which satisfies (2.15).

Note that the image of $\nu_{a}^{\mathbf{A}}$ is isomorphic to
$\widetilde{{\mathbb{V}}}/\mathbb{N}$, and thus to any choice of complement
$\mathbb{N}^{*}$ of $\mathbb{N}$ in $\widetilde{{\mathbb{V}}}$. More
precisely, we have injective linear maps
$\delta^{\mathbf{A}}_{A}:=\delta^{a}_{A}\nu^{\mathbf{A}}_{a}$ from
$\mathbb{N}^{*}$ to $\mathbb{S}_{-}$, and $\delta^{A}_{\mathbf{A}}$ from
$\mathbb{N}$ to $\mathbb{S}_{-}^{*}$ such that
$\delta^{\mathbf{C}}_{A}\delta_{\mathbf{C}}^{B}=\delta_{A}^{B}$. Hence, by
means of these, we can express a $1$-form $\alpha_{a}$ in
$\mathrm{Ann}(\mathbb{N})$ in the form
$\alpha_{a}=\nu_{a}^{\mathbf{A}}\alpha_{\mathbf{A}}$ where
$\alpha_{\mathbf{A}}=\delta^{B}_{\mathbf{A}}\delta^{c}_{B}\alpha_{c}$.

The van der Waerden symbols generate the _Clifford algebra_
$\mathcal{C}\ell(\widetilde{{\mathbb{V}}},\widetilde{g})$ of
$(\widetilde{{\mathbb{V}}},\widetilde{g})$, which, by virtue of (2.14), is
isomorphic to the exterior algebra
$\scaleobj{1.2}{\wedge}^{\bullet}\widetilde{{\mathbb{V}}}\cong\scaleobj{1.2}{\wedge}^{\bullet}\widetilde{{\mathbb{V}}}^{*}$
as a vector space. The Clifford algebra is also a matrix algebra isomorphic to
the space of endomorphisms of $\mathbb{S}$. These two properties allow us to
construct invariant bilinear forms on $\mathbb{S}$ with values in
$\scaleobj{1.2}{\wedge}^{k}\widetilde{{\mathbb{V}}}^{*}$ for $k=0,\ldots,m+1$.
The case $k=0$ yields the spin inner products implicitly used in (2.13).
Depending on the values of $m$ and $k$, these forms restrict to non-degenerate
forms on either $\mathbb{S}_{\pm}\times\mathbb{S}_{\mp}$ or
$\mathbb{S}_{\pm}\times\mathbb{S}_{\pm}$. Of relevance to the present article
are the cases $k=1,3,m+1$. For $k=1,3$, we have

(2.21) $\displaystyle\begin{array}[]{|c|c|}\hline\cr\mbox{$m$ odd}&\mbox{$m$
even}\\\
\hline\cr\hline\cr\gamma_{a\mathbf{A}^{\prime}\mathbf{B}}:\mathbb{S}_{+}\times\mathbb{S}_{-}\rightarrow\widetilde{{\mathbb{V}}}^{*}&\gamma_{a\mathbf{A}\mathbf{B}}:\mathbb{S}_{+}\times\mathbb{S}_{+}\rightarrow\widetilde{{\mathbb{V}}}^{*}\\\
\gamma_{a\mathbf{A}\mathbf{B}^{\prime}}:\mathbb{S}_{-}\times\mathbb{S}_{+}\rightarrow\widetilde{{\mathbb{V}}}^{*}&\gamma_{a\mathbf{A}^{\prime}\mathbf{B}^{\prime}}:\mathbb{S}_{-}\times\mathbb{S}_{-}\rightarrow\widetilde{{\mathbb{V}}}^{*}\\\
\hline\cr\gamma_{abc\mathbf{A}^{\prime}\mathbf{B}}:\mathbb{S}_{+}\times\mathbb{S}_{-}\rightarrow\scaleobj{1.2}{\wedge}^{3}\widetilde{{\mathbb{V}}}^{*}&\gamma_{abc\mathbf{A}^{\prime}\mathbf{B}^{\prime}}:\mathbb{S}_{+}\times\mathbb{S}_{+}\rightarrow\scaleobj{1.2}{\wedge}^{3}\widetilde{{\mathbb{V}}}^{*}\\\
\gamma_{abc\mathbf{A}\mathbf{B}^{\prime}}:\mathbb{S}_{+}\times\mathbb{S}_{-}\rightarrow\scaleobj{1.2}{\wedge}^{3}\widetilde{{\mathbb{V}}}^{*}&\gamma_{abc\mathbf{A}\mathbf{B}}:\mathbb{S}_{+}\times\mathbb{S}_{+}\rightarrow\scaleobj{1.2}{\wedge}^{3}\widetilde{{\mathbb{V}}}^{*}\\\
\hline\cr\end{array}$

For $k=m+1$, regardless of whether $m$ is odd or even, we have the following
two bilinear forms

(2.22) $\displaystyle\begin{aligned} \gamma_{a_{0}\ldots
a_{m}\mathbf{A}^{\prime}\mathbf{B}^{\prime}}&:\mathbb{S}_{+}\times\mathbb{S}_{+}\rightarrow\scaleobj{1.2}{\wedge}^{m+1}_{+}\widetilde{{\mathbb{V}}}^{*}\,,\\\
\gamma_{a_{0}\ldots
a_{m}\mathbf{A}\mathbf{B}}&:\mathbb{S}_{-}\times\mathbb{S}_{-}\rightarrow\scaleobj{1.2}{\wedge}^{m+1}_{-}\widetilde{{\mathbb{V}}}^{*}\,,\end{aligned}$

where we recall
$\scaleobj{1.2}{\wedge}^{m+1}_{\pm}\widetilde{{\mathbb{V}}}^{*}$ are the self-
dual and anti-self-dual parts of
$\scaleobj{1.2}{\wedge}^{m+1}\widetilde{{\mathbb{V}}}^{*}$.

For specificity, we assume that $\mathbb{N}$ is a self-dual null structure,
the argument being the same for anti-self-dual null structure. Then the
restriction of the first display of (2.22) to $\mathbb{S}_{+}^{\mathbb{N}}$
yields the isomorphism
$\mathbb{S}_{+}^{\mathbb{N}}\otimes\mathbb{S}_{+}^{\mathbb{N}}\cong\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})$.
We can thus think of $\mathbb{S}_{+}^{\mathbb{N}}$ as a square root of
$\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})$.

At this stage, we return to the real picture by consider $(2m+2)$-dimensional
Minkowski space $({\mathbb{V}},g)$. We can then apply all the facts outlined
above to the complexification $({}^{\mathbf{C}}{\mathbb{V}},{}^{\mathbf{C}}g)$
of $({\mathbb{V}},g)$. In addition, the real structure $\bar{}$ on
${}^{\mathbf{C}}{\mathbb{V}}$ preserving ${\mathbb{V}}$ induces an antilinear
map on $\mathbb{S}$, which interchanges the chiralities of spinors when $m$ is
odd, and preserve them when $m$ is even. The image of a spinor under this
antilinear map is referred to as the _charge conjugate_ of that spinor. Thus,
the charge conjugate of a spinor $\nu^{\mathbf{A}^{\prime}}\in\mathbb{S}_{+}$
will be denoted $\overline{\nu}^{\mathbf{A}}$ when $m$ is odd, and
$\overline{\nu}^{\mathbf{A}^{\prime}}\in\mathbb{S}_{+}$ when $m$ is
even.333Using the isomorphisms (2.13), the charge conjugate of a spinor
$\nu^{\mathbf{A}^{\prime}}\in\mathbb{S}_{+}$ can always be identified with
$\overline{\nu}_{\mathbf{A}}\in\mathbb{S}^{*}_{-}$ regardless of the parity of
$m$. Moreover, if $\nu^{\mathbf{A}^{\prime}}$ is pure so is its charge
conjugate. We define the _real index_ of a pure spinor to be the real index of
its associated null structure [KT92]. In particular, if $g$ has Lorentzian
signature, all pure spinors have real index one. A Robinson structure
$(\mathbb{N},{\mathbb{K}})$ can thus be defined by a pure spinor up to scale.
We shall refer to any such spinor as a _Robinson spinor_.

The pure spinor $\nu^{\mathbf{A}^{\prime}}$ and its charge conjugate can be
paired using the spinor bilinear forms to obtain invariants of the Robinson
structure as shown in [Car67, KT92]. These are listed below:

(2.28) $\displaystyle\begin{array}[]{|c|c|}\hline\cr\mbox{$m$ odd}&\mbox{$m$
even}\\\
\hline\cr\hline\cr\kappa_{a}=\gamma_{a\mathbf{A}^{\prime}\mathbf{B}}\nu^{\mathbf{A}^{\prime}}\overline{\nu}^{\mathbf{B}}&\kappa_{a}=\gamma_{a\mathbf{A}^{\prime}\mathbf{B}^{\prime}}\nu^{\mathbf{A}^{\prime}}\overline{\nu}^{\mathbf{B}^{\prime}}\\\
\rho_{abc}=\mathrm{i}\gamma_{abc\mathbf{A}^{\prime}\mathbf{B}}\nu^{\mathbf{A}^{\prime}}\overline{\nu}^{\mathbf{B}}&\rho_{abc}=\mathrm{i}\gamma_{abc\mathbf{A}^{\prime}\mathbf{B}^{\prime}}\nu^{\mathbf{A}^{\prime}}\overline{\nu}^{\mathbf{B}^{\prime}}\\\
\hline\cr\nu_{a_{0}\ldots a_{m}}=\gamma_{a_{0}\ldots
a_{m}\mathbf{A}^{\prime}\mathbf{B}^{\prime}}\nu^{\mathbf{A}^{\prime}}\nu^{\mathbf{B}^{\prime}}&\nu_{a_{0}\ldots
a_{m}}=\gamma_{a_{0}\ldots
a_{m}\mathbf{A}^{\prime}\mathbf{B}^{\prime}}\nu^{\mathbf{A}^{\prime}}\nu^{\mathbf{B}^{\prime}}\\\
\overline{\nu}_{a_{0}\ldots a_{m}}=\gamma_{a_{0}\ldots
a_{m}\mathbf{A}\mathbf{B}}\overline{\nu}^{\mathbf{A}}\overline{\nu}^{\mathbf{B}}&\overline{\nu}_{a_{0}\ldots
a_{m}}=\gamma_{a_{0}\ldots
a_{m}\mathbf{A}^{\prime}\mathbf{B}^{\prime}}\overline{\nu}^{\mathbf{A}^{\prime}}\overline{\nu}^{\mathbf{B}^{\prime}}\\\
\hline\cr\end{array}$

With these definitions, $\rho_{abc}$ is the Robinson $3$-form associated to
the optical $1$-form $\kappa_{a}$, i.e. these satisfy (2.10), and
$\nu_{a_{0}\ldots a_{m}}$ is a complex Robinson $(m+1)$-form. Forms of odd
degrees can be constructed in a similar way. Details can be found in [KT92].

######  Remark 2.9.

Clearly, any given optical $1$-form or Robinson $3$-form is defined by a
Robinson spinor up to a phase, and a given complex Robinson $(m+1)$-form by a
Robinson spinor up to a sign. To see this we note that, under the
transformation $\nu^{\mathbf{A}^{\prime}}\mapsto z\nu^{\mathbf{A}^{\prime}}$,
for any non-zero complex number $z$, the charge conjugate of
$\nu^{\mathbf{A}^{\prime}}$ gets multiplied by $\bar{z}$, and the forms
defined in (2.28) transform as

$\displaystyle\kappa_{a}$ $\displaystyle\mapsto r\kappa_{a}\,,$
$\displaystyle\rho_{abc}$ $\displaystyle\mapsto r\rho_{abc}\,,$
$\displaystyle\nu_{a_{0}\ldots a_{m}}$ $\displaystyle\mapsto
z^{2}\nu_{a_{0}\ldots a_{m}}\,,$

where $|z|=r\in\mathbf{R}_{>0}$.

######  Remark 2.10.

Any choice of splitting (2.5) of ${}^{\mathbf{C}}{\mathbb{V}}$ is equivalent
to choosing a one-dimensional subspace of $\mathbb{S}_{+}^{*}$ consisting of
pure spinors dual to $\mathbb{S}_{+}^{\mathbb{N}}$. Elements thereof
annihilate $\mathbb{N}^{*}$.

####  2.2.7. Characterisations of Robinson structures

We summarise the findings of the previous section in the following
proposition.

######  Proposition 2.11.

Let $({\mathbb{V}},g)$ be Minkowski space of dimension $2m+2$. The following
statements are equivalent.

  1. (1)

$({\mathbb{V}},g)$ is equipped with a Robinson structure
$(\mathbb{N},{\mathbb{K}})$.

  2. (2)

$({\mathbb{V}},g)$ is equipped with a totally null complex $(m+1)$-form.

  3. (3)

$({\mathbb{V}},g)$ is equipped with an optical structure ${\mathbb{K}}$ whose
screenspace $\mathbb{H}_{\mathbb{K}}={\mathbb{K}}^{\perp}/{\mathbb{K}}$ is
endowed with a complex structure $J_{i}{}^{j}$ compatible with the induced
metric $h_{ij}$.

  4. (4)

$({\mathbb{V}},g)$ admits a $1$-form $\kappa_{a}$ and $3$-form $\rho_{abc}$
satisfying

$\displaystyle\rho_{ab}\,{}^{e}\rho_{cde}$
$\displaystyle=-4\kappa_{[a}g_{b][c}\kappa_{d]}\,.$

  5. (5)

$({\mathbb{V}},g)$ admits a pure spinor of real index $1$.

######  Remark 2.12.

In Proposition 2.11, the $1$-form $\kappa_{a}$ and the $3$-form $\rho_{abc}$
are defined up to an overall real factor, while the pure spinor is defined up
to an overall complex factor as explained in Remark 2.9.

###  2.3. The stabiliser of a Robinson structure

There are two approaches to describe the stabiliser of a Robinson structure
$(\mathbb{N},{\mathbb{K}})$:

  1. (1)

From its definition, it suffices to consider the respective stabilisers $R$
and $\overline{R}$ of $\mathbb{N}$ and $\overline{\mathbb{N}}$ in
$\mathbf{SO}(2m+2,\mathbf{C})$. The stabiliser of a Robinson structure is then
the intersection $R\cap\overline{R}\cap\mathbf{SO}^{0}(2m+1,1)$.

  2. (2)

We characterise a Robinson structure as an optical structure together with a
Hermitian structure on the screen space to derive its stabiliser $Q$ as a
closed Lie subgroup of the stabiliser $P$ of ${\mathbb{K}}$ in
$G=\mathbf{SO}^{0}(2m+1,1)$.

For the present purpose, it will be more useful to use the second approach. We
shall assume that ${\mathbb{K}}$ is oriented so that the stabiliser of
${\mathbb{K}}$ together with its orientation is

$\displaystyle P$
$\displaystyle=\mathbf{Sim}^{0}(2m)=\mathbf{CO}^{0}(2m)\ltimes(\mathbf{R}^{2m})^{*}\
=\ (\mathbf{R}_{>0}\times\mathbf{SO}(2m))\ltimes(\mathbf{R}^{2m})^{*}\,.$

Note that $\mathbf{CO}^{0}(2m)$ acts on the screen space
$\mathbb{H}_{\mathbb{K}}={\mathbb{K}}^{\perp}/{\mathbb{K}}$ as
$\mathbf{SO}(2m)$ does, that is, $\mathbf{R}_{>0}$ acts trivially on
$\mathbb{H}_{\mathbb{K}}$. The nilpotent part of $P$ will be denoted $P_{+}$.
Choose a semi-null frame $(e_{0},e_{1},\ldots,e_{n},e_{n+1})$ and dual coframe
$(\theta^{0},\theta^{1},\ldots,\theta^{n},\theta^{n+1})$ where by convention
$K=\mathrm{span}(e_{n+1})$ and $K^{\perp}=\mathrm{Ann}(\theta^{0})$. Then,
since $Q\subset P$ is required to stabilise in addition a Hermitian structure
on $\mathbb{H}_{\mathbb{K}}$, we obtain that

$\displaystyle Q$
$\displaystyle=(\mathbf{R}_{>0}\times\mathbf{U}(m))\ltimes(\mathbf{R}^{2m})^{*}\,,$
$\displaystyle=\left\\{(\mathrm{e}^{\varphi},\psi,\phi):=\begin{pmatrix}\mathrm{e}^{-\varphi}&0&0\\\
-\mathrm{e}^{\varphi}\iota(\psi)\phi^{\top}&\iota(\psi)&0\\\
-\tfrac{\mathrm{e}^{\varphi}}{2}\phi\phi^{\top}&\phi&\mathrm{e}^{\varphi}\end{pmatrix}\mid\begin{array}[]{l}\varphi\in\mathbf{R}\,,\\\
\psi\in\mathbf{U}(m)\,,\\\
\phi\in(\mathbf{R}^{2m})^{*}\end{array}\right\\}\,,$

where we have used the standard embedding
$\iota:\mathbf{U}(m)\rightarrow\mathbf{SO}(2m)$. The reductive part
$\mathbf{R}_{>0}\times\mathbf{U}(m)$ of $Q$ will be denoted $Q_{0}$. Clearly,
$P_{+}$ is also the nilpotent part of $Q$.

To describe the Lie algebra $\mathfrak{q}$ of $Q$, we shall refer to the
notation already introduced in [FLTC20]. Setting
${\mathbb{V}}^{1}={\mathbb{V}}_{1}={\mathbb{K}}$,
${\mathbb{V}}^{0}={\mathbb{K}}^{\perp}$, we have a filtration of $P$-modules

(2.29)
$\displaystyle\\{0\\}=:{\mathbb{V}}^{2}\subset{\mathbb{V}}^{1}\subset{\mathbb{V}}^{0}\subset{\mathbb{V}}^{-1}:={\mathbb{V}}\,,$

which we shall conveniently split into a direct sum of $P_{0}$-modules,

(2.30) $\displaystyle{\mathbb{V}}$
$\displaystyle={\mathbb{V}}_{-1}\oplus{\mathbb{V}}_{0}\oplus{\mathbb{V}}_{1}\,,$

where $P_{0}=\mathbf{CO}^{0}(2m)$ is the reductive part of $P$. For each
$i=-1,0,1$, we have ${\mathbb{V}}_{i}\cong{\mathbb{V}}^{i}/{\mathbb{V}}^{i+1}$
as vector spaces. In terms of our earlier notation
${\mathbb{V}}_{-1}=\mathbb{L}$ and
${\mathbb{V}}_{0}=\mathbb{H}_{{\mathbb{K}},\mathbb{L}}$.

Recall from [FLTC20] that the Lie algebra
$\mathfrak{g}\cong\scaleobj{1.2}{\wedge}^{2}{\mathbb{V}}^{*}$ of
$G=\mathbf{SO}^{0}(n+1,1)$ can then be expressed as a direct sum of
$P_{0}$-modules

(2.31) $\displaystyle\mathfrak{g}$
$\displaystyle=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\,,$

where $\mathfrak{g}_{\pm 1}\cong{\mathbb{V}}_{\pm
1}^{*}\otimes{\mathbb{V}}_{0}^{*}$ and
$\mathfrak{g}_{0}\cong\left({\mathbb{V}}_{-1}^{*}\otimes{\mathbb{V}}_{1}^{*}\right)\oplus\scaleobj{1.2}{\wedge}^{2}{\mathbb{V}}_{0}^{*}$.
Note that ${\mathbb{V}}_{-1}^{*}\otimes{\mathbb{V}}_{1}^{*}$ is the one-
dimensional centre $\mathfrak{z}_{0}$ of $\mathfrak{g}_{0}$.

Now, the complex structure splits ${}^{\mathbf{C}}{\mathbb{V}}_{0}$ and its
dual as

(2.32) $\displaystyle{}^{\mathbf{C}}{\mathbb{V}}_{0}$
$\displaystyle={\mathbb{V}}^{(1,0)}_{0}\oplus{\mathbb{V}}^{(0,1)}_{0}\,,$
$\displaystyle{}^{\mathbf{C}}{\mathbb{V}}_{0}^{*}$
$\displaystyle=({\mathbb{V}}^{(1,0)}_{0})^{*}\oplus({\mathbb{V}}^{(0,1)}_{0})^{*}\,,$

where
${\mathbb{V}}_{0}^{(1,0)}:=\overline{\mathbb{N}}\cap{}^{\mathbf{C}}{\mathbb{V}}_{0}$
and ${\mathbb{V}}_{0}^{(0,1)}:=\mathbb{N}\cap{}^{\mathbf{C}}{\mathbb{V}}_{0}$.
Writing

(2.33) $\displaystyle{\mathbb{V}}^{*}$
$\displaystyle={\mathbb{V}}_{-1}^{*}\oplus[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]\oplus{\mathbb{V}}_{1}^{*}\,,$

we find that the summands in the $P_{0}$-invariant decomposition of
$\mathfrak{g}$ given in (2.31) split further into irreducible $Q_{0}$-modules:

(2.34) $\displaystyle\begin{aligned} \mathfrak{g}_{\pm 1}&={\mathbb{V}}_{\pm
1}^{*}\otimes[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]\,,\\\
\mathfrak{g}_{0}&=\mathfrak{z}_{0}\oplus\left(\mathrm{span}(\omega)\oplus[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}_{0}^{*}]\oplus[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]\right)\,,\end{aligned}$

where we recall that $\omega$ is the Hermitian $2$-form on ${\mathbb{V}}_{0}$.
We identify the Lie algebra $\mathfrak{q}_{0}$ of $Q_{0}$ as the Lie
subalgebra

$\displaystyle\mathfrak{q}_{0}=\mathfrak{z}_{0}\oplus\mathrm{span}(\omega)\oplus[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}_{0}^{*}]\,,$

which can be seen to be isomorphic to $\mathbf{R}\oplus\mathfrak{u}(m)$. In
addition, the Lie algebra $\mathfrak{q}$ of the stabiliser $Q$ of the Robinson
structure is given by

(2.35) $\displaystyle\mathfrak{q}$
$\displaystyle:=\mathfrak{z}_{0}\oplus\left(\mathrm{span}(\omega)\oplus[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}_{0}^{*}]\right)\oplus\left({\mathbb{V}}_{1}^{*}\otimes[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]\right)\,.$

As expected, $\mathfrak{q}_{0}=\mathfrak{g}_{0}\cap\mathfrak{q}$.

######  Remark 2.13.

In low dimensions, we note the following points:

  * •

In dimension four, we have that $\mathfrak{p}\cong\mathfrak{q}$, i.e. an
optical structure is a Robinson structure.

  * •

In dimension six, the semi-simple part
$\scaleobj{1.2}{\wedge}^{2}{\mathbb{V}}_{0}^{*}$ of $\mathfrak{g}_{0}$ splits
into a self-dual part and an anti-self-dual part. More explicitly,
$\mathfrak{su}^{+}(2)=\mathrm{span}(\omega)\oplus[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]$
and
$\mathfrak{su}^{-}(2)=[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}_{0}^{*}]$,
where $\mathfrak{su}^{\pm}(2)$ are isomorphic to two copies of
$\mathfrak{su}(2)$.

######  Remark 2.14.

Clearly, the space of all (oriented) self-dual Robinson structure on
$({\mathbb{V}},g)$ is isomorphic to $\mathbf{SO}^{0}(2m+1,1)/Q$. It
corresponds to the isotropic Grassmannian
$\mathrm{Gr}_{m+1}^{+}(\mathbb{V},g)$ of self-dual MTN planes in
$({\mathbb{V}},g)$ and has real dimension $m(m+1)$.

In addition, for a given optical structure ${\mathbb{K}}$ on
$({\mathbb{V}},g)$, the space of all (oriented) self-dual Robinson structures
compatible with ${\mathbb{K}}$ is isomorphic to
$P/Q\cong\mathbf{SO}(2m)/\mathbf{U}(m)$, which has real dimension $m(m-1)$. It
will be denoted $\mathrm{Gr}_{m+1}^{+}({\mathbb{V}},g,\mathbb{K})$.

###  2.4. One-dimensional representations of $Q$

For any $w\in\mathbf{R}$, we define the one-dimensional representations
$\mathbf{R}(w)$ and $\mathbf{C}(w,0)$ of $Q$ on $\mathbf{R}$ of weight $w$ and
on $\mathbf{C}$ of weight $(w,0)$ by

$\displaystyle(\mathrm{e}^{\varphi},\psi,\phi)\cdot r$
$\displaystyle=\mathrm{e}^{w\varphi}r\,,$ for any $r\in\mathbf{R}$,
$\displaystyle(\mathrm{e}^{\varphi},\psi,\phi)\cdot z$
$\displaystyle=(\mathrm{e}^{\varphi}\det A)^{w}z\,,$ for any $z\in\mathbf{C}$.

We also define $\mathbf{C}(0,w):=\overline{\mathbf{C}(w,0)}$. One can check
that

$\displaystyle\mathbf{R}(-1)$ $\displaystyle\cong{\mathbb{K}}\,,$
$\displaystyle\mathbf{C}(-1,0)$
$\displaystyle\cong\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\mathbb{N})\,,$
$\displaystyle\mathbf{C}(0,-1)$
$\displaystyle\cong\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\overline{\mathbb{N}})\,.$

These leads to the one-dimensional representations
$\mathbf{C}(w,w^{\prime}):=\mathbf{C}(w,0)\otimes\mathbf{C}(0,w^{\prime})$ for
any real $w$, $w^{\prime}$. We also note
$\mathbf{C}\otimes\mathbf{R}(w)\cong\mathbf{C}(w,w)$ for any real $w$, and
$\mathbf{R}(w)\otimes\mathbf{R}(w^{\prime})\cong\mathbf{R}(w+w^{\prime})$ for
any real $w,w^{\prime}$, and similarly for the complex representations.

###  2.5. Space $\mathbb{G}$ of algebraic intrinsic torsions

Let us now consider the $Q$-module
$\mathbb{G}={\mathbb{V}}^{*}\otimes\left(\mathfrak{g}/\mathfrak{q}\right)$. We
treat only the case $m>1$ since for $m=1$, we have $Q=P$, which is already
dealt with [FLTC20].

######  Theorem 2.15.

Assume $m>1$. The $Q$-module
$\mathbb{G}={\mathbb{V}}^{*}\otimes\left(\mathfrak{g}/\mathfrak{q}\right)$
admits a filtration

(2.36)
$\displaystyle\mathbb{G}^{1}\subset\mathbb{G}^{0}\subset\mathbb{G}^{-1}\subset\mathbb{G}^{-2}$

of $Q$-modules
$\mathbb{G}^{i}:=({\mathbb{V}}^{*})^{i+1}\otimes\left(\mathfrak{g}/\mathfrak{q}\right)$
for $i=-2,-1,0$. The summands of its associated graded $Q$-module

$\displaystyle\mathrm{gr}(\mathbb{G})=\mathrm{gr}_{-2}(\mathbb{G})\oplus\mathrm{gr}_{-1}(\mathbb{G})\oplus\mathrm{gr}_{0}(\mathbb{G})\oplus\mathrm{gr}_{1}(\mathbb{G})\,,$

where, for each $i=-2,-1,0$,
$\mathrm{gr}_{i}(\mathbb{G})=\mathbb{G}^{i}/\mathbb{G}^{i+1}$, and
$\mathrm{gr}_{1}(\mathbb{G})=\mathbb{G}^{1}$, decompose into direct sums of
irreducible $Q$-modules $\mathrm{gr}_{i}^{j,k}(\mathbb{G})$ as follows:

$\displaystyle\mathrm{gr}_{-2}(\mathbb{G})$
$\displaystyle=\mathrm{gr}_{-2}^{0,0}(\mathbb{G})\,,$
$\displaystyle\mathrm{gr}_{-1}(\mathbb{G})$
$\displaystyle=\mathrm{gr}_{-1}^{0,0}(\mathbb{G})\oplus\left(\mathrm{gr}_{-1}^{1,0}(\mathbb{G})\oplus\mathrm{gr}_{-1}^{1,1}(\mathbb{G})\oplus\mathrm{gr}_{-1}^{1,2}(\mathbb{G})\right)\oplus\left(\mathrm{gr}_{-1}^{2,0}(\mathbb{G})\oplus\mathrm{gr}_{-1}^{2,1}(\mathbb{G})\right)\oplus\mathrm{gr}_{-1}^{3,0}\,,$
$\displaystyle\mathrm{gr}_{0}(\mathbb{G})$
$\displaystyle=\mathrm{gr}_{0}^{0,0}(\mathbb{G})\oplus\left(\mathrm{gr}_{0}^{1,0}(\mathbb{G})\oplus\mathrm{gr}_{0}^{1,1}(\mathbb{G})\oplus\mathrm{gr}_{0}^{1,2}(\mathbb{G})\oplus\mathrm{gr}_{0}^{1,3}(\mathbb{G})\right)\,,$
$\displaystyle\mathrm{gr}_{1}(\mathbb{G})$
$\displaystyle=\mathrm{gr}_{1}^{0,0}(\mathbb{G})\,,$

where, for each $i,j,k$, the $Q$-module $\mathrm{gr}_{i}^{j,k}(\mathbb{G})$ is
isomorphic to the $Q_{0}$-module $\mathbb{G}_{i}^{j,k}$ as given in Table 1.
Note that $\mathrm{gr}_{0}^{1,1}(\mathbb{G})$ and
$\mathrm{gr}_{0}^{1,3}(\mathbb{G})$ do not occur when $m=2$.

${\begin{array}[]{|c|c|c|c|}\hline\cr\text{$Q_{0}$-module}&\text{Description}&\text{Dimension}\\\
\hline\cr\hline\cr\mathbb{G}_{-2}^{0,0}&{\mathbb{V}}_{0}^{*}\otimes\mathbf{R}(2)&2m\\\
\hline\cr\hline\cr\mathbb{G}_{-1}^{0,0}&\mathbf{R}(1)&1\\\
\hline\cr\mathbb{G}_{-1}^{1,0}&\mathbf{R}(1)&1\\\
\mathbb{G}_{-1}^{1,1}&[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]\otimes\mathbf{R}(1)&m(m-1)\\\
\mathbb{G}_{-1}^{1,2}&[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}_{0}^{*}]\otimes\mathbf{R}(1)&(m+1)(m-1)\\\
\hline\cr\mathbb{G}_{-1}^{2,0}&[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}_{0}^{*}]\otimes\mathbf{R}(1)&(m+1)(m-1)\\\
\mathbb{G}_{-1}^{2,1}&[\\![\scaleobj{1.2}{\odot}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]\otimes\mathbf{R}(1)&m(m+1)\\\
\hline\cr\mathbb{G}_{-1}^{3,0}&[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]\otimes\mathbf{R}(1)&m(m-1)\\\
\hline\cr\hline\cr\mathbb{G}_{0}^{0,0}&{\mathbb{V}}_{0}^{*}&2m\\\
\hline\cr\mathbb{G}_{0}^{1,0}&[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]&2m\\\
\mathbb{G}_{0}^{1,1}&[\\![\scaleobj{1.2}{\wedge}^{(3,0)}{\mathbb{V}}_{0}^{*}]\\!]&\scriptstyle{\frac{1}{3}m(m-1)(m-2)}\\\
\mathbb{G}_{0}^{1,2}&[\\![\raisebox{-1.72218pt}{\scaleobj{0.35}{\yng(2,1)}}\,{\mathbb{V}}_{0}^{*}]\\!]&\scriptstyle{\frac{2}{3}m(m+1)(m-1)}\\\
\mathbb{G}_{0}^{1,3}&[\\![\scaleobj{1.2}{\wedge}^{(1,2)}_{\circ}{\mathbb{V}}_{0}^{*}]\\!]&\scriptstyle{m(m+1)(m-2)}\\\
\hline\cr\hline\cr\mathbb{G}_{1}^{0,0}&[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]\otimes\mathbf{R}(-1)&m(m-1)\\\
\hline\cr\end{array}}$ Table 1. Irreducible $Q_{0}$-submodules of $\mathbb{G}$

###### Proof.

We use the same strategy as in Proposition 3.3 of [FLTC20]: the filtration
(2.29) induces the filtration (2.36) of $Q$-modules on
$\mathbb{G}\subset{\mathbb{V}}^{*}\otimes\left(\mathfrak{g}^{-1}/\mathfrak{g}^{1}\right)$.
We proceed as before using the decompositions (2.33), (2.34) and (2.35) to
find

$\displaystyle\mathbb{G}$
$\displaystyle\cong\left({\mathbb{V}}^{*}_{-1}\oplus[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]\oplus{\mathbb{V}}^{*}_{1}\right)\otimes\left(\left({\mathbb{V}}_{-1}^{*}\otimes[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]\right)\oplus[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]\right)\,.$

The result follows by distributing this expression and splitting each summand
into irreducibles:

$\displaystyle{\mathbb{V}}^{*}_{-1}\otimes{\mathbb{V}}_{-1}^{*}\otimes[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]$
$\displaystyle=\mathbb{G}_{-2}^{0,0}\,,$
$\displaystyle{\mathbb{V}}^{*}_{-1}\otimes[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]$
$\displaystyle=\mathbb{G}_{-1}^{3,0}\,,$
$\displaystyle[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]\otimes{\mathbb{V}}_{-1}^{*}\otimes[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]$
$\displaystyle=\mathbb{G}_{-1}^{1,0}\oplus\left(\mathbb{G}_{-1}^{1,0}\oplus\mathbb{G}_{-1}^{1,1}\oplus\mathbb{G}_{-1}^{1,2}\right)\oplus\left(\mathbb{G}_{-1}^{2,0}\oplus\mathbb{G}_{-1}^{2,1}\right)\,,$
$\displaystyle[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]\otimes[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]$
$\displaystyle=\mathbb{G}_{0}^{1,0}\oplus\mathbb{G}_{0}^{1,1}\oplus\mathbb{G}_{0}^{1,2}\oplus\mathbb{G}_{0}^{1,3}\,,$
$\displaystyle{\mathbb{V}}^{*}_{1}\otimes{\mathbb{V}}_{-1}^{*}\otimes[\\![\scaleobj{1.2}{\wedge}^{(1,0)}{\mathbb{V}}_{0}^{*}]\\!]$
$\displaystyle=\mathbb{G}_{0}^{0,0}\,,$
$\displaystyle{\mathbb{V}}^{*}_{1}\otimes[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]$
$\displaystyle=\mathbb{G}_{1}^{0,0}\,,$

where, recalling that ${\mathbb{V}}_{\pm 1}^{*}\cong\mathbf{R}(\mp 1)$, the
modules are described in Table 1. ∎

######  Remark 2.16.

Observe that the $P$-module
${\mathbb{V}}^{*}\otimes\left(\mathfrak{g}/\mathfrak{p}\right)$ represents the
space of intrinsic torsions of the underlying optical structure. This can be
viewed as a $Q$-submodule of $\mathbb{G}$. In the next proposition, which is a
direct consequence of Theorem 2.15, we single out the irreducible
$P$-submodules of $\mathrm{gr}(\mathbb{G})$, thereby making contact with the
intrinsic torsion of an optical structure described in [FLTC20].

######  Proposition 2.17.

Assume $m>1$. Let
$\mathbb{G}={\mathbb{V}}^{*}\otimes\left(\mathfrak{g}/\mathfrak{q}\right)$ and
consider the graded module $\mathrm{gr}(\mathbb{G})$ given in Theorem 2.15.
Define

$\displaystyle\mathrm{gr}_{-2}^{0}(\mathbb{G})$
$\displaystyle:=\mathrm{gr}_{-2}^{0,0}(\mathbb{G})\,,$
$\displaystyle\mathrm{gr}_{-1}^{0}(\mathbb{G})$
$\displaystyle:=\mathrm{gr}_{-1}^{0,0}(\mathbb{G})\,,$
$\displaystyle\mathrm{gr}_{-1}^{1}(\mathbb{G})$
$\displaystyle:=\mathrm{gr}_{-1}^{1,0}(\mathbb{G})\oplus\mathrm{gr}_{-1}^{1,1}(\mathbb{G})\oplus\mathrm{gr}_{-1}^{1,2}(\mathbb{G})\,,$
$\displaystyle\mathrm{gr}_{-1}^{2}(\mathbb{G})$
$\displaystyle:=\mathrm{gr}_{-1}^{2,0}(\mathbb{G})\oplus\mathrm{gr}_{-1}^{2,1}(\mathbb{G})\,,$
$\displaystyle\mathrm{gr}_{0}^{0}(\mathbb{G})$
$\displaystyle:=\mathrm{gr}_{0}^{0,0}(\mathbb{G})\,.$

and in dimension six,

$\displaystyle\mathrm{gr}_{-1}^{1,+}(\mathbb{G})$
$\displaystyle:=\mathrm{gr}_{-1}^{1,0}(\mathbb{G})\oplus\mathrm{gr}_{-1}^{1,1}(\mathbb{G})\,,$
$\displaystyle\mathrm{gr}_{-1}^{1,-}(\mathbb{G})$
$\displaystyle:=\mathrm{gr}_{-1}^{1,2}(\mathbb{G})\,.$

Then, for each $i,j$, $\mathrm{gr}_{i}^{j}(\mathbb{G})$ is an irreducible
$P$-module except in dimension six, where $\mathrm{gr}_{-1}^{1}(\mathbb{G})$
is not irreducible, but $\mathrm{gr}_{-1}^{1,\pm}(\mathbb{G})$ are.

Moreover, we define

$\displaystyle\mathbb{G}_{-2}^{0}$ $\displaystyle:=\mathbb{G}_{-2}^{0,0}\,,$
$\displaystyle\mathbb{G}_{-1}^{0}$ $\displaystyle:=\mathbb{G}_{-1}^{0,0}\,,$
$\displaystyle\mathbb{G}_{-1}^{1}$
$\displaystyle:=\mathbb{G}_{-1}^{1,0}\oplus\mathbb{G}_{-1}^{1,1}\oplus\mathbb{G}_{-1}^{1,2}\,,$
$\displaystyle\mathbb{G}_{-1}^{2}$
$\displaystyle:=\mathbb{G}_{-1}^{2,0}\oplus\mathbb{G}_{-1}^{2,1}\,,$
$\displaystyle\mathbb{G}_{0}^{0}$ $\displaystyle:=\mathbb{G}_{0}^{0,0}\,,$

and in dimension six,

$\displaystyle\mathbb{G}_{-1}^{1,+}$
$\displaystyle:=\mathbb{G}_{-1}^{1,0}\oplus\mathbb{G}_{-1}^{1,1}\,,$
$\displaystyle\mathbb{G}_{-1}^{1,+}$ $\displaystyle:=\mathbb{G}_{-1}^{1,2}\,.$

Then, for each $i,j$, $\mathbb{G}_{i}^{j}$ is an irreducible $P_{0}$-module
except in dimension six, where $\mathbb{G}_{-1}^{1}$ is not irreducible, but
$\mathbb{G}_{-1}^{1,\pm}$ are.

###  2.6. Isotypic $Q_{0}$-submodules of $\mathbb{G}$

Let us fix a splitting of $\mathbb{G}$ into $Q_{0}$-modules. Observe that the
modules in each of the pairs $(\mathbb{G}_{-1}^{0,0},\mathbb{G}_{-1}^{1,0})$,
$(\mathbb{G}_{-1}^{1,2},\mathbb{G}_{-1}^{2,0})$,
$(\mathbb{G}_{-1}^{1,1},\mathbb{G}_{-1}^{3,0})$ and
$(\mathbb{G}_{-1}^{1,2},\mathbb{G}_{-1}^{2,0})$ are isotypic, i.e. they have
the same dimensions. This means that one can construct further irreducible
$Q_{0}$-modules by assigning some algebraic relations among these in terms of
parameters. To this end, we need to describe them by means of the projections

(2.37) $\displaystyle\begin{aligned} (\Pi_{-1}^{0\times
1})_{[x:y]}&:{\mathbb{V}}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{G}_{-1}^{0,0}\oplus\mathbb{G}_{-1}^{1,0}\,,&[x:y]&\in\mathbf{R}\mathbb{P}^{1}\,,\\\
(\Pi_{-1}^{1\times
2})_{[x:y]}&:{\mathbb{V}}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{G}_{-1}^{1,2}\oplus\mathbb{G}_{-1}^{2,0}\,,&[x:y]&\in\mathbf{R}\mathbb{P}^{1}\,,\\\
(\Pi_{-1}^{1\times
3})_{[z:w]}&:{\mathbb{V}}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{G}_{-1}^{1,1}\oplus\mathbb{G}_{-1}^{3,0}\,,&[z:w]&\in\mathbf{C}\mathbb{P}^{1}\,,\\\
(\Pi_{0}^{0\times
1})_{[z:w]}&:{\mathbb{V}}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{G}_{0}^{0,0}\oplus\mathbb{G}_{0}^{1,0}\,,&[z:w]&\in\mathbf{C}\mathbb{P}^{1}\,,\end{aligned}$

whose precise definitions have been relegated to Appendix A for convenience.
Here, $\mathbf{R}\mathbb{P}^{1}$ and $\mathbf{C}\mathbb{P}^{1}$ are real and
complex projective lines, respectively. We can then define the following
additional $Q_{0}$-submodules of $\mathbb{G}$:

(2.38) $\displaystyle\begin{aligned} (\mathbb{G}_{-1}^{0\times
1})_{[x:y]}&:=\mathrm{im}(\Pi_{-1}^{0\times
1})_{[x:y]}\subset\mathbb{G}_{-1}^{0,0}\oplus\mathbb{G}_{-1}^{1,0}\,,&\mbox{for
$[x:y]\in\mathbf{R}\mathbb{P}^{1}$.}\\\ (\mathbb{G}_{-1}^{1\times
2})_{[x:y]}&:=\mathrm{im}(\Pi_{-1}^{1\times
2})_{[x:y]}\subset\mathbb{G}_{-1}^{1,2}\oplus\mathbb{G}_{-1}^{2,0}\,,&\mbox{for
$[x:y]\in\mathbf{R}\mathbb{P}^{1}$.}\\\ (\mathbb{G}_{-1}^{1\times
3})_{[z:w]}&:=\mathrm{im}(\Pi_{-1}^{1\times
3})_{[z:w]}\subset\mathbb{G}_{-1}^{1,1}\oplus\mathbb{G}_{-1}^{3,0}\,,&\mbox{for
$[z:w]\in\mathbf{C}\mathbb{P}^{1}$.}\\\ (\mathbb{G}_{0}^{0\times
1})_{[z:w]}&:=\mathrm{im}(\Pi_{-1}^{1\times
2})_{[z:w]}\subset\mathbb{G}_{-1}^{1,2}\oplus\mathbb{G}_{-1}^{2,0}\,,&\mbox{for
$[z:w]\in\mathbf{C}\mathbb{P}^{1}$.}\end{aligned}$

Their descriptions and dimensions are given in Table 2.

${\begin{array}[]{|c|c|c|c|}\hline\cr\text{$Q_{0}$-module}&\text{Description}&\text{Dimension}\\\
\hline\cr\hline\cr(\mathbb{G}_{-1}^{0\times 1})_{[x:y]}&\mathbf{R}(1)&1\\\
(\mathbb{G}_{-1}^{1\times
2})_{[x:y]}&[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}{\mathbb{V}}_{0}^{*}]\otimes\mathbf{R}(1)&(m-1)(m+1)\\\
(\mathbb{G}_{-1}^{1\times
3})_{[z:w]}&[\\![\scaleobj{1.2}{\wedge}^{(2,0)}{\mathbb{V}}_{0}^{*}]\\!]\otimes\mathbf{R}(1)&m(m-1)\\\
\hline\cr\hline\cr(\mathbb{G}_{0}^{0\times
1})_{[z:w]}&{\mathbb{V}}_{0}^{*}&2m\\\ \hline\cr\end{array}}$ Table 2.
$Q_{0}$-submodules of $\mathbb{G}$ — here $[x:y]\in\mathbf{R}\mathbb{P}^{1}$
and $[z:w]\in\mathbf{C}\mathbb{P}^{1}$

Note that by definition,

$\displaystyle(\mathbb{G}_{-1}^{0\times 1})_{[1,0]}$
$\displaystyle=\mathbb{G}_{-1}^{0,0}\,,$
$\displaystyle(\mathbb{G}_{-1}^{1\times 2})_{[1,0]}$
$\displaystyle=\mathbb{G}_{-1}^{1,2}\,,$
$\displaystyle(\mathbb{G}_{-1}^{1\times 3})_{[1,0]}$
$\displaystyle=\mathbb{G}_{-1}^{1,1}\,,$
$\displaystyle(\mathbb{G}_{0}^{0\times 1})_{[1,0]}$
$\displaystyle=\mathbb{G}_{-1}^{1,2}\,,$
$\displaystyle(\mathbb{G}_{-1}^{0\times 1})_{[0,1]}$
$\displaystyle=\mathbb{G}_{-1}^{1,0}\,,$
$\displaystyle(\mathbb{G}_{-1}^{1\times 2})_{[0,1]}$
$\displaystyle=\mathbb{G}_{-1}^{2,0}\,,$
$\displaystyle(\mathbb{G}_{-1}^{1\times 3})_{[0,1]}$
$\displaystyle=\mathbb{G}_{-1}^{3,0}\,,$
$\displaystyle(\mathbb{G}_{0}^{0\times 1})_{[0,1]}$
$\displaystyle=\mathbb{G}_{-1}^{2,0}\,.$

###  2.7. The $Q$-submodules of $\mathbb{G}$

We are now in the position of determining all the $Q$-submodules of
$\mathbb{G}$. For this purpose, we shall appeal to the $Q_{0}$-module
epimorphisms
$\Pi_{i}^{j,k}:{\mathbb{V}}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{G}_{i}^{j,k}$,
$\Pi_{i}^{j}:{\mathbb{V}}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{G}_{i}^{j}$
and (2.37), all of which are described in Appendix A with respect to some
chosen splitting of $\mathbb{G}$. Any $Q$-submodule of $\mathbb{G}$ must be a
sum of the irreducible $Q_{0}$-submodules $\mathbb{G}_{i}^{j,k}$ given in
Table 1, together with the families of irreducibles $Q_{0}$-modules given in
Table 2. Not every such sum is a $Q$-module. To determine which
$Q_{0}$-submodules of $\mathbb{G}$ are also $Q$-submodules, we compute how a
change of splitting (2.8) transforms the maps $\Pi_{i}^{j,k}$. This will tell
us how the various modules $\mathbb{G}_{i}^{j,k}$ are related under the action
of $P_{+}$, the nilpotent part of $Q$. We will then be able to determine the
$Q$-submodules of $\mathbb{G}$ accordingly.

To facilitate the readability, we contract the projections (A.4) with suitable
combinations of $\delta^{i}_{\alpha}$, $\delta^{i}_{\bar{\alpha}}$ to define

(2.39) $\displaystyle\begin{aligned}
\gamma_{\alpha}&:=\Pi_{-2}^{0,0}(\Gamma)_{\alpha}\,,\\\
\epsilon&:=\Pi_{-1}^{0,0}(\Gamma)\,,\\\
\tau^{\omega}&:=\Pi_{-1}^{1,0}(\Gamma)\,,\\\
\tau_{\alpha\beta}&:=\Pi_{-1}^{1,1}(\Gamma)_{\alpha\beta}={\Pi^{\prime}}_{-1}^{1,1}(\Gamma)_{\alpha\beta}\,,&\tau^{\circ}_{\alpha\bar{\beta}}&:=\Pi_{-1}^{1,2}(\Gamma)_{\alpha\bar{\beta}}\,,\\\
\sigma_{\alpha\bar{\beta}}&:=\Pi_{-1}^{2,0}(\Gamma)_{\alpha\bar{\beta}}\,,&\sigma_{\alpha\beta}&:=\Pi_{-1}^{2,1}(\Gamma)_{\alpha\beta}\,,\\\
\zeta_{\alpha\beta}&:=\Pi_{-1}^{3,0}(\Gamma)_{\alpha\beta}\,,\\\
E_{\alpha}&:=\Pi_{0}^{0,0}(\Gamma)_{\alpha}={\Pi^{\prime}}_{0}^{0,0}(\Gamma)_{\alpha}\,,&\\\
G_{\alpha}&:=\Pi_{0}^{1,0}(\Gamma)_{\alpha}\,,&G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}&:=\Pi_{0}^{1,1}(\Gamma)_{\alpha\beta\gamma}\,,\\\
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{\alpha\beta\gamma}&:=\Pi_{0}^{1,2}(\Gamma)_{\alpha\beta\gamma}\,,&G^{\circ}_{\bar{\alpha}\beta\gamma}&:=\Pi_{0}^{1,3}(\Gamma)_{\bar{\alpha}\beta\gamma}\,,\\\
B_{\alpha\beta}&:=\Pi_{1}^{0,0}(\Gamma)_{\alpha\beta}\,.\end{aligned}$

Their complex conjugates are defined similarly. Note that with these
definitions,

$\displaystyle\sigma_{\alpha\beta}$ $\displaystyle=\sigma_{(\alpha\beta)}\,,$
$\displaystyle\tau_{\alpha\beta}$ $\displaystyle=\tau_{[\alpha\beta]}\,,$
$\displaystyle\sigma_{\alpha\bar{\beta}}h^{\alpha\bar{\beta}}$
$\displaystyle=0\,,$
$\displaystyle\tau^{\circ}_{\alpha\bar{\beta}}h^{\alpha\bar{\beta}}$
$\displaystyle=0\,,$ $\displaystyle\overline{\sigma_{\alpha\bar{\beta}}}$
$\displaystyle=\sigma_{\bar{\alpha}\beta}=\sigma_{\beta\bar{\alpha}}\,,$
$\displaystyle\overline{\tau^{\circ}_{\alpha\bar{\beta}}}$
$\displaystyle=\tau^{\circ}_{\bar{\alpha}\beta}=-\tau^{\circ}_{\beta\bar{\alpha}}\,,$
$\displaystyle\zeta_{\alpha\beta}$ $\displaystyle=-\zeta_{\beta\alpha}\,,$
$\displaystyle B_{\alpha\beta}$ $\displaystyle=-B_{\beta\alpha}\,,$
$\displaystyle
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{(\alpha\beta)\gamma}$
$\displaystyle=0\,,$ $\displaystyle
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{[\alpha\beta\gamma]}$
$\displaystyle=0\,,$ $\displaystyle
G^{\circ}_{\bar{\beta}\alpha\gamma}h^{\alpha\bar{\beta}}$ $\displaystyle=0\,.$

######  Theorem 2.18.

Define

$\displaystyle\not{\mathbb{G}}_{-2}^{0,0}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle\not{\mathbb{G}}_{-1}^{0,0}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{-1}^{0,0}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle\not{\mathbb{G}}_{-1}^{1,i}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{-1}^{1,i}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,\qquad
i=0,1$ $\displaystyle\not{\mathbb{G}}_{-1}^{2,i}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{-1}^{2,i}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,\qquad
i=0,1,2$ $\displaystyle\not{\mathbb{G}}_{-1}^{3,0}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{-1}^{3,0}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle\not{\mathbb{G}}_{0}^{0,0}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{0}^{0,0}(\Gamma)=\Pi_{-1}^{0}(\Gamma)=\Pi_{-1}^{1}(\Gamma)=\Pi_{-1}^{2}(\Gamma)=\Pi_{-2}^{0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle\not{\mathbb{G}}_{0}^{1,0}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{0}^{1,0}(\Gamma)=\Pi_{-1}^{0,0}(\Gamma)=\Pi_{-1}^{1,0}(\Gamma)=\Pi_{-1}^{1,2}(\Gamma)$
$\displaystyle\qquad\qquad\qquad\qquad=\Pi_{-1}^{2,0}(\Gamma)=\Pi_{-1}^{3,0}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle\not{\mathbb{G}}_{0}^{1,1}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{0}^{1,1}(\Gamma)=(\Pi_{-1}^{1\times
3})_{[-4\mathrm{i},1]}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle\not{\mathbb{G}}_{0}^{1,2}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{0}^{1,2}(\Gamma)=(\Pi_{-1}^{1\times
3})_{[2\mathrm{i},1]}(\Gamma)=\Pi_{-1}^{2,1}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle\not{\mathbb{G}}_{0}^{1,3}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{0}^{1,3}(\Gamma)=\Pi_{-1}^{1,2}(\Gamma)=\Pi_{-1}^{2,0}(\Gamma)=\Pi_{-1}^{3,0}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$

and,

$\displaystyle\not{\mathbb{G}}_{1}^{0,0}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{1}^{0,0}(\Gamma)=(\Pi_{0}^{0\times
1})_{[2(m-1)\mathrm{i},-1]}(\Gamma)=\Pi_{0}^{1,1}(\Gamma)=\Pi_{0}^{1,2}(\Gamma)=\Pi_{0}^{1,3}(\Gamma)$
$\displaystyle\qquad\qquad=\Pi_{-1}^{1,1}(\Gamma)=\Pi_{-1}^{1,2}(\Gamma)=\Pi_{-1}^{2}(\Gamma)$
$\displaystyle\qquad\qquad\qquad\qquad=\Pi_{-1}^{3,0}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,\qquad\mbox{when
$m>2$}\,,$ while $\displaystyle\not{\mathbb{G}}_{1}^{0,0}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:\Pi_{1}^{0,0}(\Gamma)=(\Pi_{0}^{0\times
1})_{[2\mathrm{i},-1]}(\Gamma)=\Pi_{0}^{1,1}(\Gamma)=\Pi_{0}^{1,2}(\Gamma)$
$\displaystyle\qquad\qquad=(\Pi_{-1}^{1\times
3})_{[2\mathrm{i},1]}(\Gamma)=\Pi_{-1}^{1,2}(\Gamma)=\Pi_{-1}^{2}(\Gamma)$
$\displaystyle\qquad\qquad\qquad\qquad=\Pi_{-1}^{3,0}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,\qquad\mbox{when
$m=2$}\,.$

In dimension six, i.e. $m=2$, we always have
$\not{\mathbb{G}}_{0}^{1,1}=\not{\mathbb{G}}_{0}^{1,3}=\mathbb{G}$.

Define further, for any $[x:y]\in\mathbf{R}\mathbb{P}^{1}$,
$[z:w]\in\mathbf{C}\mathbb{P}^{1}$,

$\displaystyle(\not{\mathbb{G}}_{-1}^{0\times 1})_{[x:y]}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:(\Pi_{-1}^{0\times
1})_{[x:y]}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle(\not{\mathbb{G}}_{-1}^{1\times 2})_{[x:y]}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:(\Pi_{-1}^{1\times
2})_{[x:y]}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle(\not{\mathbb{G}}_{-1}^{1\times 3})_{[z:w]}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:(\Pi_{-1}^{1\times
3})_{[z:w]}(\Gamma)=(z+4\mathrm{i}w)\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,,$
$\displaystyle(\not{\mathbb{G}}_{0}^{0\times 1})_{[z:w]}$
$\displaystyle:=\\{\Gamma\in{\mathbb{V}}^{*}\otimes\mathfrak{g}:(\Pi_{0}^{0\times
1})_{[z:w]}(\Gamma)=(2(m-1)\mathrm{i}w+z)\Pi_{-1}^{0,0}(\Gamma)$
$\displaystyle\qquad\qquad\qquad=(2(m-1)\mathrm{i}w+z)\Pi_{-1}^{1,0}(\Gamma)=(\Pi_{-1}^{1\times
3})_{[z:w]}(\Gamma)=(2w\,\mathrm{i}-z)\Pi_{-1}^{1,2}(\Gamma)$
$\displaystyle\qquad\qquad\qquad\qquad=(2w\,\mathrm{i}-z)\Pi_{-1}^{2,0}(\Gamma)=z\Pi_{-1}^{2,1}(\Gamma)=\Pi_{-2}^{0,0}(\Gamma)=0\\}/\left({\mathbb{V}}^{*}\otimes\mathfrak{q}\right)\,.$

Then, for each $i,j,k$, $\not{\mathbb{G}}_{i}^{j,k}$ is the largest
$Q$-submodule of $\mathbb{G}$ that does not contain $\mathbb{G}_{i}^{j,k}$,
and similarly for $(\not{\mathbb{G}}_{-1}^{0\times 1})_{[x:y]}$,
$(\not{\mathbb{G}}_{-1}^{1\times 2})_{[x:y]}$,
$(\not{\mathbb{G}}_{-1}^{1\times 3})_{[z:w]}$ and
$(\not{\mathbb{G}}_{0}^{0\times 1})_{[z:w]}$. In particular, any $Q$-submodule
of $\mathbb{G}$ arises as the intersection of any of the ones above.

In addition, there are inclusions of $Q$-submodules, which are denoted by
arrows in the diagrams below.

For $m>2$:

For $m=2$:

For any $m>1$,

For any $m>1$, any
$[z:w]\in\mathbf{C}\mathbb{P}^{1}\setminus\\{[-2(m-1)\mathrm{i},1],[2\mathrm{i},1]\\}$:

For $m>1$, any $[-4\mathrm{i}:1]\neq[z:w]\in\mathbf{C}\mathbb{P}^{1}$, and any
$[x:y]\in\mathbf{R}\mathbb{P}^{1}$:

######  Remark 2.19.

We note that

  1. (1)

There is some redundancy in the inclusions: e.g.

$\displaystyle\not{\mathbb{G}}_{0}^{1,1}\cap\not{\mathbb{G}}_{-1}^{1,1}\cap\not{\mathbb{G}}_{-1}^{3,0}=\not{\mathbb{G}}_{0}^{1,1}\cap\not{\mathbb{G}}_{-1}^{1,1}=\not{\mathbb{G}}_{0}^{1,1}\cap\not{\mathbb{G}}_{-1}^{3,0}\,,$

and, for $m>2$,

$\displaystyle\not{\mathbb{G}}_{1}^{0,0}\subset(\not{\mathbb{G}}_{0}^{0\times
1})_{[-2(m-1)\mathrm{i}:1]}\cap\not{\mathbb{G}}_{0}^{1,2}\subset(\not{\mathbb{G}}_{-1}^{1\times
3})_{[2\mathrm{i}:1]}\cap(\not{\mathbb{G}}_{-1}^{1\times
3})_{[2(m-1)\mathrm{i}:1]}=\not{\mathbb{G}}_{-1}^{1,0}\cap\not{\mathbb{G}}_{-1}^{3,0}\,,$

  2. (2)

The only $Q$-submodules of $\mathbb{G}$ that are not contained in
$\not{\mathbb{G}}_{-2}^{0,0}$ are
$\not{\mathbb{G}}_{0}^{1,1}\subset(\not{\mathbb{G}}_{-1}^{1\times
3})_{[4\mathrm{i}:-1]}$.

###### Proof.

The theorem is a direct consequence of the following transformation rules of
the quantities defined by (2.39) under the change (2.8):

$\displaystyle\gamma_{\alpha}$ $\displaystyle\mapsto\gamma_{\alpha}\,,$
$\displaystyle\epsilon$
$\displaystyle\mapsto\epsilon+\gamma_{\alpha}\phi^{\alpha}+\gamma^{\alpha}\phi_{\alpha}\,,$
$\displaystyle\tau^{\omega}$
$\displaystyle\mapsto\tau^{\omega}-\mathrm{i}\left(\gamma_{\alpha}\phi^{\alpha}-\gamma^{\alpha}\phi_{\alpha}\right)\,,$
$\displaystyle\tau_{\alpha\beta}$
$\displaystyle\mapsto\tau_{\alpha\beta}-\gamma_{[\alpha}\phi_{\beta]}\,,$
$\displaystyle\tau^{\circ}_{\alpha\bar{\beta}}$
$\displaystyle\mapsto\tau^{\circ}_{\alpha\bar{\beta}}+\left(-\frac{1}{2}\gamma_{\alpha}\phi_{\bar{\beta}}+\frac{1}{2}\gamma_{\bar{\beta}}\phi_{\alpha}\right)_{\circ}\,,$
$\displaystyle\sigma_{\alpha\bar{\beta}}$
$\displaystyle\mapsto\sigma_{\alpha\bar{\beta}}+\left(\frac{1}{2}\gamma_{\alpha}\phi_{\bar{\beta}}+\frac{1}{2}\gamma_{\bar{\beta}}\phi_{\alpha}\right)_{\circ}\,,$
$\displaystyle\sigma_{\alpha\beta}$
$\displaystyle\mapsto\sigma_{\alpha\beta}+\gamma_{(\alpha}\phi_{\beta)}\,,$
$\displaystyle\zeta_{\alpha\beta}$
$\displaystyle\mapsto\zeta_{\alpha\beta}-4\,\mathrm{i}\gamma_{[\alpha}\phi_{\beta]}\,,$
$\displaystyle E_{\alpha}$ $\displaystyle\mapsto
E_{\alpha}+\tau_{\alpha\beta}\phi^{\beta}-\sigma_{\alpha\beta}\phi^{\beta}+\tau^{\circ}_{\alpha\bar{\beta}}\phi^{\bar{\beta}}-\sigma_{\alpha\bar{\beta}}\phi^{\bar{\beta}}-\frac{\mathrm{i}}{2m}\tau^{\omega}\phi_{\alpha}-\frac{1}{2m}\epsilon\phi_{\alpha}-\gamma_{\alpha}\phi^{\beta}\phi_{\beta}\,,$
$\displaystyle G_{\alpha}$ $\displaystyle\mapsto
G_{\alpha}-2\,\mathrm{i}\tau^{\circ}_{\alpha\bar{\beta}}\phi^{\bar{\beta}}+2\,\mathrm{i}\sigma_{\alpha\bar{\beta}}\phi^{\bar{\beta}}+\frac{m-1}{m}\tau^{\omega}\phi_{\alpha}-\frac{m-1}{m}\,\mathrm{i}\epsilon\,\phi_{\alpha}$
$\displaystyle\qquad\qquad+\phi^{\beta}\zeta_{\beta\alpha}-2\,\mathrm{i}\phi^{\beta}\gamma_{\beta}\phi_{\alpha}+2\,\mathrm{i}\phi^{\beta}\phi_{\beta}\gamma_{\alpha}\,,$
$\displaystyle
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}$
$\displaystyle\mapsto
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}+\left(-4\,\mathrm{i}\tau_{[\alpha\beta}+\zeta_{[\alpha\beta}\right)\phi_{\gamma]}\,,$
$\displaystyle
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{(\alpha\beta)\gamma}$
$\displaystyle\mapsto
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{(\alpha\beta)\gamma}-\left(2\,\mathrm{i}\tau_{\gamma(\alpha}+\zeta_{\gamma(\alpha}\right)\phi_{\beta)}+2\,\mathrm{i}\sigma_{\gamma(\alpha}\phi_{\beta)}-2\,\mathrm{i}\sigma_{\alpha\beta}\phi_{\gamma}-2\,\mathrm{i}\phi_{(\alpha}\gamma_{\beta)}\phi_{\gamma}+2\,\mathrm{i}\phi_{\alpha}\phi_{\beta}\gamma_{\gamma}\,,$
$\displaystyle G^{\circ}_{\bar{\alpha}\beta\gamma}$ $\displaystyle\mapsto
G^{\circ}_{\bar{\alpha}\beta\gamma}+\left(4\,\mathrm{i}\tau^{\circ}_{[\beta|\bar{\alpha}}\phi_{|\gamma]}-4\,\mathrm{i}\sigma_{[\beta|\bar{\alpha}}\phi_{|\gamma]}+\phi_{\bar{\alpha}}\zeta_{\beta\gamma}-4\,\mathrm{i}\phi_{\bar{\alpha}}\gamma_{[\beta}\phi_{\gamma]}\right)_{\circ}\,,$
$\displaystyle B_{\alpha\beta}$ $\displaystyle\mapsto
B_{\alpha\beta}+\left(\frac{2}{m-1}G_{[\alpha}-4\,\mathrm{i}E_{[\alpha}\right)\phi_{\beta]}-\phi^{\gamma}G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\gamma\alpha\beta}-\phi^{\gamma}G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{\gamma\alpha\beta}-\phi^{\bar{\gamma}}G^{\circ}_{\bar{\gamma}\alpha\beta}$
$\displaystyle\qquad\qquad+4\mathrm{i}\phi^{\gamma}\tau_{\gamma[\alpha}\phi_{\beta]}-\phi^{\gamma}\phi_{\gamma}\zeta_{\alpha\beta}+4\,\mathrm{i}\phi^{\gamma}\sigma_{\gamma[\alpha}\phi_{\beta]}$
$\displaystyle\qquad\qquad\qquad\qquad-4\mathrm{i}\phi^{\bar{\gamma}}\tau^{\circ}_{[\alpha|\bar{\gamma}}\phi_{|\beta]}+4\,\mathrm{i}\phi^{\bar{\gamma}}\sigma_{[\alpha|\bar{\gamma}}\phi_{|\beta]}+4\mathrm{i}\phi^{\gamma}\phi_{\gamma}\gamma_{[\alpha}\phi_{\beta]}\,.$

From these, we immediately deduce, for any $[x:y]\in\mathbf{R}\mathbb{P}^{1}$,
$[z:w]\in\mathbf{C}\mathbb{P}^{1}$,

$\displaystyle x\epsilon+y\tau^{\omega}$ $\displaystyle\mapsto
x\epsilon+y\tau^{\omega}+(x-\mathrm{i}y)\gamma_{\alpha}\phi^{\alpha}+(x+\mathrm{i}y)\gamma^{\alpha}\phi_{\alpha}\,,$
$\displaystyle
x\sigma_{\alpha\bar{\beta}}-y\mathrm{i}\tau^{\circ}_{\alpha\bar{\beta}}$
$\displaystyle\mapsto
x\sigma_{\alpha\bar{\beta}}-y\mathrm{i}\tau^{\circ}_{\alpha\bar{\beta}}+\left(-(x+y\mathrm{i})\frac{1}{2}\gamma_{\alpha}\phi_{\bar{\beta}}+(x-y\mathrm{i})\frac{1}{2}\gamma_{\bar{\beta}}\phi_{\alpha}\right)_{\circ}\,,$
$\displaystyle z\tau_{\alpha\beta}+w\zeta_{\alpha\beta}$ $\displaystyle\mapsto
z\tau_{\alpha\beta}+w\zeta_{\alpha\beta}-(z+4\mathrm{i}w)\gamma_{[\alpha}\phi_{\beta]}\,,$
$\displaystyle z\,E_{\alpha}+w\,G_{\alpha}$ $\displaystyle\mapsto
z\,E_{\alpha}+w\,G_{\alpha}-z\,\phi^{\beta}\sigma_{\beta\alpha}+\left(-z\,\tau_{\beta\alpha}+w\,\zeta_{\beta\alpha}\right)\phi^{\beta}$
$\displaystyle\qquad-(2w\,\mathrm{i}-z)\tau^{\circ}_{\alpha\bar{\beta}}\phi^{\bar{\beta}}+(2w\,\mathrm{i}-z)\sigma_{\alpha\bar{\beta}}\phi^{\bar{\beta}}$
$\displaystyle\qquad\qquad-\frac{\mathrm{i}}{2m}(2(m-1)\mathrm{i}w+z)\tau^{\omega}\phi_{\alpha}-\frac{1}{2m}(2(m-1)\mathrm{i}w+z)\epsilon\,\phi_{\alpha}$
$\displaystyle\qquad\qquad\qquad-2w\,\mathrm{i}\phi^{\beta}\gamma_{\beta}\phi_{\alpha}+(2w\,\mathrm{i}-z)\phi^{\beta}\phi_{\beta}\gamma_{\alpha}\,.$

The result follows as in the proof of Proposition 3.4 of [FLTC20].

In dimension six, one has

$\displaystyle\phi^{\gamma}\phi_{\gamma}\tau_{\alpha\beta}$
$\displaystyle=-2\phi^{\gamma}\tau_{\gamma[\alpha}\phi_{\beta]}\,,$
$\displaystyle
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\gamma\alpha\beta}$
$\displaystyle=0\,,$ $\displaystyle G^{\circ}_{\bar{\gamma}\alpha\beta}$
$\displaystyle=0\,,$

so that

$\displaystyle B_{\alpha\beta}$ $\displaystyle\mapsto
B_{\alpha\beta}+\left(2G_{[\alpha}-4\,\mathrm{i}E_{[\alpha}\right)\phi_{\beta]}-\phi^{\gamma}G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{\gamma\alpha\beta}-\left(2\mathrm{i}\tau_{\alpha\beta}+\zeta_{\alpha\beta}\right)\phi^{\gamma}\phi_{\gamma}$
$\displaystyle\qquad+4\mathrm{i}\phi^{\gamma}\sigma_{\gamma[\alpha}\phi_{\beta]}-4\mathrm{i}\phi^{\bar{\gamma}}\tau^{\circ}_{[\alpha|\bar{\gamma}}\phi_{|\beta]}+4\mathrm{i}\phi^{\bar{\gamma}}\sigma_{[\alpha|\bar{\gamma}}\phi_{|\beta]}+4\mathrm{i}\phi^{\gamma}\phi_{\gamma}\gamma_{[\alpha}\phi_{\beta]}$

This completes the proof. Some of these computations were verified using the
symbolic computer algebra system cadabra [Pee07, Pee18]. ∎

Finally, for future use, and to make contact with the intrinsic torsion of an
optical structure given in [FLTC20], we also define

(2.40) $\displaystyle\begin{aligned}
\not{\mathbb{G}}_{-2}^{0}&:=\not{\mathbb{G}}_{-2}^{0,0}\,,\\\
\not{\mathbb{G}}_{-1}^{0}&:=\not{\mathbb{G}}_{-1}^{0,0}\,,&\not{\mathbb{G}}_{-1}^{1}&:=\not{\mathbb{G}}_{-1}^{1,0}\cap\not{\mathbb{G}}_{-1}^{1,1}\cap\not{\mathbb{G}}_{-1}^{1,2}\,,&\not{\mathbb{G}}_{-1}^{2}&:=\not{\mathbb{G}}_{-1}^{2,0}\cap\not{\mathbb{G}}_{-1}^{2,1}\,,\\\
\not{\mathbb{G}}_{0}^{0}&:=\not{\mathbb{G}}_{0}^{0,0}\,.\end{aligned}$

##  3\. Almost Robinson manifolds

###  3.1. Almost Robinson structures

Throughout we shall follow the notation and conventions of Section 2 now
translated into the bundle setting.

######  Definition 3.1.

[NT02, TC16] Let $(\mathcal{M},g)$ be an oriented pseudo-Riemannian manifold
of dimension $2m+2$. An _almost null structure_ on $(\mathcal{M},g)$ is a
complex distribution $N$ of rank $(m+1)$ and totally null with respect to
${}^{\mathbf{C}}g$. When $N$ is involutive, i.e. $[N,N]\subset N$, we call $N$
a _null structure_.

In other words, an almost null structure is a smooth assignement of a null
structure to the tangent space at each point, and according to Definition 2.3,
one may talk of the real index of an almost null structure at a point. When
$g$ is of Lorentzian signature, we make the following definition.

######  Definition 3.2.

Let $(\mathcal{M},g)$ be an oriented and time-oriented Lorentzian manifold of
dimension $2m+2$. An _almost Robinson structure_ on $(\mathcal{M},g)$ consists
of a pair $(N,K)$ where $N$ is a complex distribution of rank $m+1$ totally
null with respect to ${}^{\mathbf{C}}g$, and $K$ a real line distribution such
that ${}^{\mathbf{C}}K=N\cap\overline{N}$. We shall refer to the quadruple
$(\mathcal{M},g,N,K)$ as an _almost Robinson manifold_ or _almost Robinson
geometry_.

In addition, we call $(N,K)$

  * •

a _nearly Robinson structure_ when $[K,N]\subset N$, and

  * •

a _Robinson structure_ when $[N,N]\subset N$, i.e. $N$ is involutive.

We shall accordingly refer to $(\mathcal{M},g,N,K)$ as a _nearly Robinson
manifold_ or as a _Robinson manifold_.

Clearly, a Robinson manifold is a nearly Robinson manifold. Definitions
involving weaker assumptions on orientability are possible.

######  Remark 3.3.

Equivalently put, an almost Robinson structure is an almost null structure of
real index one. By Lemma 2.5, any almost null structure on a Lorentzian
manifold defines an almost Robinson manifold. The terminology ‘almost null
structure’ will nevertheless be preferred in the case when we wish to
emphasize the geometric aspects of the almost Robinson structure not
particularly tied to the geometry of the real null line distribution $K$, as
will be done in Section 3.10.

An almost Robinson structure $(N,K)$ induces an optical structure on
$(\mathcal{M},g)$ in the sense of [FLTC20], namely the filtration of vector
bundles

(3.1) $\displaystyle K\subset K^{\perp}\subset T\mathcal{M}\,.$

The orientation and time-orientation on $\mathcal{M}$ induce an orientation on
$K$, and the screen bundle $H_{K}:=K^{\perp}/K$ of $K$ inherits a positive-
definite bundle metric $h$ from $g$. Any section of $K$ will be referred to as
an _optical vector field_ , while any section of $\mathrm{Ann}(K^{\perp})$
will be referred to as an _optical $1$-form_.

In addition, there is a bundle complex structure $J$ on the screen bundle
$H_{K}$ compatible with $h$, which induces a splitting of its complexification

$\displaystyle{}^{\mathbf{C}}H_{K}$ $\displaystyle=H^{(1,0)}_{K}\oplus
H^{(0,1)}_{K}\,,$ $\displaystyle{}^{\mathbf{C}}H_{K}^{*}$
$\displaystyle=(H^{(1,0)}_{K})^{*}\oplus(H^{(0,1)}_{K})^{*}\,,$

where $H^{(1,0)}_{K}$ and $H^{(0,1)}_{K}$ denote the $+\mathrm{i}$\- and
$-\mathrm{i}$-eigenbundles of $J$ respectively. In abstract index notation, we
shall denote the bundle complex structure and the bundle Hermitian structure
on $H_{K}$ by $J_{i}{}^{j}$ and $\omega_{ij}$ respectively, so that
$\omega_{ij}=J_{i}{}^{k}h_{kj}$. Following the notation of Section 2.1, we
also define the bundles, for any non-negative integer $p$, $q$,

$\displaystyle\scaleobj{1.2}{\wedge}^{(p,q)}H_{K}^{*}$
$\displaystyle:=\scaleobj{1.2}{\wedge}^{p}(H^{(1,0)}_{K})^{*}\otimes\scaleobj{1.2}{\wedge}^{q}(H^{(0,1)}_{K})^{*}\,,$
$\displaystyle\scaleobj{1.2}{\odot}{}^{(p,q)}H_{K}^{*}$
$\displaystyle:=\scaleobj{1.2}{\odot}{}^{p}(H^{(1,0)}_{K})^{*}\otimes\scaleobj{1.2}{\odot}{}^{q}(H^{(0,1)}_{K})^{*}\,.$

For $pq\neq 0$, the subbundles of elements of
$\scaleobj{1.2}{\wedge}^{(p,q)}H_{K}^{*}$ and
$\scaleobj{1.2}{\odot}{}^{(p,q)}H_{K}^{*}$ that are tracefree with respect to
the bundle Hermitian structure will be denoted by
$\scaleobj{1.2}{\wedge}^{(p,q)}_{\circ}H_{K}^{*}$ and
$\scaleobj{1.2}{\odot}{}^{(p,q)}_{\circ}H_{K}^{*}$ respectively. Similarly, we
introduce the subbundle
$\raisebox{-1.72218pt}{\scaleobj{0.35}{\yng(2,1)}}H_{K}^{*}$ as a bundle
analogue of (2.1). The corresponding real spans of these complex bundles will
be enclosed between $[\\![\cdot]\\!]$ or $[\cdot]$ as described by (2.2).

As in Section 2, we split the complexified tangent bundle as

$\displaystyle{}^{\mathbf{C}}T\mathcal{M}=N^{*}\oplus N$

for some chosen complement $N^{*}$ of $N$ in ${}^{\mathbf{C}}T\mathcal{M}$,
dual to $N$ via ${}^{\mathbf{C}}g$. This splitting is not canonical in
general. This induces a splitting of the filtration (3.1)

(3.2) $\displaystyle T\mathcal{M}$ $\displaystyle=L\oplus H_{K,L}\oplus K\,,$

where

$\displaystyle L$ $\displaystyle:=N^{*}\cap T\mathcal{M}\,,$ $\displaystyle
H_{K,L}$ $\displaystyle:=K^{\perp}\cap L^{\perp}\,.$

Note that $N^{*}$ defines the almost Robinson structure $(N^{*},L)$ on
$(\mathcal{M},g)$, where $L$ is the real span of $N^{*}\cap\overline{N^{*}}$
and is dual to $K$. In addition,

$\displaystyle{}^{\mathbf{C}}H_{K,L}$ $\displaystyle=H^{(1,0)}_{K,L}\oplus
H^{(0,1)}_{K,L}\,,$ $\displaystyle{}^{\mathbf{C}}H^{*}_{K,L}$
$\displaystyle=(H^{(1,0)}_{K,L})^{*}\oplus(H^{(0,1)}_{K,L})^{*}\,.$

where

$\displaystyle H_{K,L}^{(1,0)}$ $\displaystyle=\overline{N}\cap N^{*}\,,$
$\displaystyle H_{K,L}^{(0,1)}$ $\displaystyle=N\cap\overline{N^{*}}\,.$

We also have isomorphisms of vector bundles $H_{K,L}\cong H_{K}$,
$H^{(1,0)}_{K,L}=H^{(1,0)}_{K}$, and so on, which depend on the choice of
$N^{*}$.

The splitting operators and their duals, introduced in Section 2, that is,

$\displaystyle(\delta^{a}_{A},\delta^{aA})\,,$
$\displaystyle(\ell^{a},\delta^{a}_{i},k^{a})\,,$
$\displaystyle(\delta^{i}_{\alpha},\delta^{i}_{\bar{\alpha}})\,,$
$\displaystyle(\ell^{a},\delta^{a}_{\alpha},\delta^{a}_{\bar{\alpha}},k^{a})\,,$
$\displaystyle(\delta_{a}^{A},\delta_{aA})\,,$
$\displaystyle(\kappa_{a},\delta_{a}^{i},\lambda_{a})\,,$
$\displaystyle(\delta_{i}^{\alpha},\delta_{i}^{\bar{\alpha}})\,,$
$\displaystyle(\kappa_{a},\delta_{a}^{\alpha},\delta_{a}^{\bar{\alpha}},\lambda_{a})\,,$

will be used throughout the article to convert index types, with the
convention that $k^{a}\alpha_{a}=\kappa_{a}\alpha^{a}=\alpha^{0}$ and
$\ell^{a}\alpha_{a}=\lambda_{a}\alpha^{a}=\alpha_{0}$ for any $1$-form
$\alpha_{a}$.

In order to avoid ambiguity when taking components of the covariant derivative
of some tensor $\alpha_{b\ldots d}$, we shall often write

$\displaystyle(\nabla\alpha)_{ab\ldots d}$
$\displaystyle:=\nabla_{a}\alpha_{b\ldots d}\,.$

For instance,

$\displaystyle(\nabla\alpha)^{0}{}_{\alpha 0i}$
$\displaystyle=k^{a}\delta^{b}_{\alpha}\ell^{c}\delta^{d}_{i}\left(\nabla_{a}\alpha_{bcd}\right)\,.$

The splitting operators will also be used as injectors. Thus, if $\omega_{ij}$
is the Hermitian $2$-form on $H_{K}$, we can set
$\omega_{ab}=\omega_{ij}\delta^{i}_{a}\delta^{j}_{b}$ for some chosen
splitting operators, and construct the _Robinson $3$-form_
$\rho_{abc}:=3\kappa_{[a}\omega_{bc]}$ associated to the optical $1$-form
$\kappa_{a}$.

###  3.2. Pure spinors

Whenever $(\mathcal{M},g)$ is assumed to be spin, we introduce a spin bundle
$S$ and translate the theory of spinors summarised in Section 2.2.6 to the
language of bundles. We shall denote the spinor bundle by $S$ and its
irreducible parts by $S_{+}$ and $S_{-}$. We shall not distinguish
notationally between the Levi-Civita connection $\nabla$ and the induced spin
connection. We now view the van der Waerden symbols
$\gamma_{a\mathbf{A}^{\prime}}{}^{\mathbf{B}}$ and
$\gamma_{a\mathbf{A}}{}^{\mathbf{B}^{\prime}}$, the bilinear forms (2.22) as
fields on $\mathcal{M}$ compatible with $\nabla$. There is also an antilinear
map on each fiber of $S$, induced from the reality structure on
$({}^{\mathbf{C}}T\mathcal{M},{}^{\mathbf{C}}g)$, and thus, a notion of charge
conjugate of a spinor field.

An almost Robinson structure $(N,K)$, where we assume for specificity that $N$
is self-dual, can therefore be expressed by a non-vanishing section
$\nu^{\mathbf{A}^{\prime}}$ of $S_{+}$ that is pure at every point, i.e. the
kernel of the map

$\displaystyle\nu_{a}^{\mathbf{A}}$
$\displaystyle:=\gamma_{a\mathbf{B}^{\prime}}{}^{\mathbf{A}}\nu^{\mathbf{B}^{\prime}}:\Gamma(T\mathcal{M})\rightarrow\Gamma(S^{-})$

is precisely $N$. Since the kernel $N$ of the map $\nu_{a}^{\mathbf{A}}$ is
invariant under rescaling of the spinor $\nu^{\mathbf{A}^{\prime}}$, the
almost Robinson structure is in fact equivalent to the existence of a complex
line subbundle $S_{+}^{N}$ of $S_{+}$, which is spanned by
$\nu^{\mathbf{A}^{\prime}}$. The bundle $S_{+}^{N}$ can be viewed as a square
root of the line bundle $\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(N)$. Indeed,
we have an isomorphism of bundles

(3.3) $\displaystyle\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(N)\cong
S_{+}^{N}\otimes S_{+}^{N}\,.$

Any spinor $\nu^{\mathbf{A}^{\prime}}$ annihilating the almost null structure
$N$ will be referred to as a _Robinson spinor_ , and any section of
$\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(N)$ a _complex Robinson
$(m+1)$-form_.

A completely parallel analysis can be carried out starting with the charge
conjugate of $\nu^{\mathbf{A}^{\prime}}$, which spans the complex conjugate
bundle $\overline{S_{+}^{N}}$. The invariants $1$-form $\kappa_{a}$, $3$-form
$\rho_{abc}$, $(m+1)$-forms $\nu_{a_{1}\ldots a_{m+1}}$ and
$\overline{\nu}_{a_{1}\ldots a_{m+1}}$ of the almost Robinson structure can
then be recovered from $\nu^{\mathbf{A}^{\prime}}$ and its charge conjugate
using (2.28).

We are now in the position of stating the direct translation of Proposition
2.11 into the language of manifolds:

######  Proposition 3.4.

Let $(\mathcal{M},g)$ be an oriented and time-oriented smooth Lorentzian
manifold of dimension $2m+2$. The following statements are equivalent.

  1. (1)

$(\mathcal{M},g)$ is endowed with an almost Robinson structure $(N,K)$.

  2. (2)

$(\mathcal{M},g)$ admits a simple totally null complex $(m+1)$-form.

  3. (3)

$(\mathcal{M},g)$ is endowed with an optical structure $K$ whose screen bundle
$(H_{K},h)$ is equipped with a bundle complex structure compatible with the
bundle metric $h$.

  4. (4)

$(\mathcal{M},g)$ admits a null $1$-form $\kappa_{a}$ and a $3$-form
$\rho_{abc}$ such that

(3.4) $\displaystyle\rho_{ab}\,{}^{e}\rho_{cde}$
$\displaystyle=-4\kappa_{[a}g_{b][c}\kappa_{d]}\,.$

  5. (5)

when $(\mathcal{M},g)$ is spin, it admits a pure spinor (of real index one).

######  Remark 3.5.

In general, if one assumes that $(\mathcal{M},g)$ is spin, any geometric
statement on a complex Robinson $(m+1)$-form can equivalently be expressed in
terms of a Robinson spinor.

###  3.3. Almost Robinson structures as $G$-structures

From the discussion of [FLTC20], we shall also view an almost Robinson
structure as a reduction of the frame bundle to the $Q$-bundle
$\mathcal{F}^{Q}$ where
$Q=(\mathbf{R}_{>0}\times\mathbf{U}(m))\ltimes(\mathbf{R}^{2m})^{*}$, and
given any $Q$-module $\mathbb{A}$, we will construct associated vector bundles
$\mathcal{F}^{Q}(\mathbb{A}):=\mathcal{F}^{Q}\times_{Q}\mathbb{A}$. Similarly,
a choice of splitting gives rise to the $Q_{0}$-invariant vector bundles where
$Q_{0}=\mathbf{R}_{>0}\times\mathbf{U}(m)$. It will often be more convenient
to deal with the reduced coframe bundle ${\mathcal{F}^{*}}^{Q}$. A section of
${\mathcal{F}^{*}}^{Q}$ will then consist of a null complex coframe
$(\kappa,\theta^{\alpha},\overline{\theta}{}^{\bar{\alpha}},\lambda)$ such
that

  1. (1)

$\kappa$ annihilates $K^{\perp}$, or equivalently, $\kappa=g(k,\cdot)$ for
some section $k$ of $K$;

  2. (2)

$(\kappa,\theta^{\alpha})$ annililate $N$, or equivalently,
$(\kappa,\overline{\theta}{}^{\bar{\alpha}})$ annililate $\overline{N}$;

  3. (3)

$(\theta^{\alpha})$ are unitary with respect to the screen bundle metric;

  4. (4)

the metric takes the form

$\displaystyle g$
$\displaystyle=2\kappa\lambda+2h_{\alpha\bar{\beta}}\theta^{\alpha}\overline{\theta}{}^{\bar{\beta}}\,.$

We shall refer to
$(\kappa,\theta^{\alpha},\overline{\theta}{}^{\bar{\alpha}},\lambda)$ as a
_Robinson coframe_.

Any two Robinson coframes
$(\kappa,\theta^{\alpha},\overline{\theta}{}^{\bar{\alpha}},\lambda)$ and
$(\widehat{\kappa},\widehat{\theta}^{\alpha},\overline{\widehat{\theta}}{}^{\bar{\alpha}},\widehat{\lambda})$
are related by a transformation of the form

(3.5) $\displaystyle\widehat{\kappa}$
$\displaystyle=\mathrm{e}^{\varphi}\kappa\,,$
$\displaystyle\widehat{\theta}^{\alpha}$
$\displaystyle=\psi_{\beta}{}^{\alpha}\theta^{\beta}+\phi^{\alpha}\kappa\,,$
$\displaystyle\widehat{\lambda}$
$\displaystyle=\mathrm{e}^{-\varphi}\left(\lambda-\psi_{\alpha}{}^{\beta}\phi_{\beta}\theta^{\alpha}-\psi_{\bar{\alpha}}{}^{\bar{\beta}}\phi_{\bar{\beta}}\theta^{\bar{\alpha}}-\phi_{\alpha}\phi^{\alpha}\kappa\right)\,,$

where $\varphi$ is a smooth real-valued function, and $\phi^{\alpha}$,
$\psi_{\beta}{}^{\alpha}$ are smooth complex-valued functions on $\mathcal{M}$
with $\psi_{\alpha}{}^{\beta}$ being a $\mathbf{U}(m)$-transformation at any
point, i.e.
$h_{\alpha\bar{\beta}}=h_{\gamma\bar{\delta}}\psi_{\alpha}{}^{\gamma}\psi_{\bar{\beta}}{}^{\bar{\delta}}$,
and
$\phi_{\alpha}\phi^{\alpha}=h_{\alpha\bar{\beta}}\phi^{\alpha}\phi^{\bar{\beta}}$.

Associated to the representations $\mathbf{R}(w)$ and
$\mathbf{C}(w,w^{\prime})$ defined in Section 2.4, where $w$ and $w^{\prime}$
are real, we define the bundle $\mathcal{E}(w)$ of _boost densities_ of weight
$w$, already introduced in [FLTC20], and the bundle
$\mathcal{E}(w,w^{\prime})$ of _boost-spin densities_ of weight
$(w,w^{\prime})$. In particular, we have the identifications

$\displaystyle K$ $\displaystyle\cong\mathcal{E}(-1)\,,$ $\displaystyle L$
$\displaystyle\cong\mathcal{E}(1)\,,$ (3.6) $\displaystyle\mathcal{E}(-1,0)$
$\displaystyle:=\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(N)\,,$
$\displaystyle\mathcal{E}(0,-1)$
$\displaystyle:=\overline{\mathcal{E}(-1,0)}=\scaleobj{1.2}{\wedge}^{m+1}\mathrm{Ann}(\overline{N})\,.$

If $(\mathcal{M},g)$ is assumed to be spin, we define the smooth complex line
bundles

(3.7) $\displaystyle\begin{aligned}
\mathcal{E}(\tfrac{1}{2},0)&:=\left(S_{+}^{N}\right)^{-1}\,,&\mathcal{E}(0,\tfrac{1}{2})&:=\left(\overline{S_{+}^{N}}\right)^{-1}\,,\\\
\mathcal{E}(-\tfrac{1}{2},0)&:=\left(\mathcal{E}(\tfrac{1}{2},0)\right)^{*}\,,&\mathcal{E}(0,-\tfrac{1}{2})&:=\left(\mathcal{E}(0,\tfrac{1}{2})\right)^{*}\,.\end{aligned}$

In particular, we recover (3.6) by virtue of (3.3). Our definitions are
consistent with those of the real line bundles $\mathcal{E}(w)$ in the sense
that

$\displaystyle\mathcal{E}(w)$
$\displaystyle=\mathcal{E}(\tfrac{w}{2},\tfrac{w}{2})/S^{1}\,,$ for any real
$w$.

This can be readily be checked using (2.28).

###  3.4. Intrinsic torsion

As explained in Section 2 of [FLTC20], the intrinsic torsion of an almost
Robinson structure is given by a section
$T^{*}\mathcal{M}\otimes\mathcal{F}^{Q}(\mathfrak{g}/\mathfrak{q})$, which we
identify with the $Q$-invariant subbundle
$\mathcal{G}:=\mathcal{F}^{Q}\times_{Q}\mathbb{G}$ where
$\mathbb{G}:={\mathbb{V}}^{*}\otimes\mathfrak{g}/\mathfrak{q}$. We shall
accordingly call $\mathcal{G}$ the bundle of intrinsic torsions of $(N,K)$.
Its $Q_{0}$-invariant subbundles and $Q$-invariant subbundles will presently
be defined with reference to Section 2, and in particular Theorem 2.15.

The filtration (2.36) on $\mathbb{G}$ induces a filtration of $Q$-invariant
subbundles

$\displaystyle\mathcal{G}$
$\displaystyle=:\mathcal{G}^{-2}\supset\mathcal{G}^{-1}\supset\mathcal{G}^{0}\supset\mathcal{G}^{-1}\,,$

where $\mathcal{G}^{i}:=\mathcal{F}^{Q}\times_{Q}\mathbb{G}^{i}$.
Correspondingly, the associated graded vector bundle

$\displaystyle\mathrm{gr}(\mathcal{G})=\mathrm{gr}_{-2}(\mathcal{G})\oplus\mathrm{gr}_{-1}(\mathcal{G})\oplus\mathrm{gr}_{0}(\mathcal{G})\oplus\mathrm{gr}_{1}(\mathcal{G})\,,$

where
$\mathrm{gr}_{i}(\mathcal{G}):=\mathcal{F}^{Q}\times_{Q}\mathrm{gr}_{i}(\mathbb{G})$,
splits into irreducible $Q$-invariant subbundles
$\mathrm{gr}_{i}^{j,k}(\mathcal{G}):=\mathcal{F}^{Q}\times_{Q_{0}}\mathrm{gr}_{i}^{j,k}(\mathbb{G})$.
For each choice of splitting, these are isomorphic to the $Q_{0}$-invariant
subbundles

$\displaystyle\mathcal{G}_{i}^{j,k}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q_{0}}\mathbb{G}_{i}^{j,k}\,,$

and we introduce further

$\displaystyle(\mathcal{G}_{-1}^{0\times 1})_{[x:y]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q_{0}}(\mathbb{G}_{-1}^{0\times
1})_{[x:y]}\,,$ for each $[x:y]\in\mathbf{R}\mathbb{P}^{1}$,
$\displaystyle(\mathcal{G}_{-1}^{1\times 2})_{[x:y]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q_{0}}(\mathbb{G}_{-1}^{1\times
2})_{[x:y]}\,,$ for each $[x:y]\in\mathbf{R}\mathbb{P}^{1}$,
$\displaystyle(\mathcal{G}_{-1}^{1\times 3})_{[z:w]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q_{0}}(\mathbb{G}_{-1}^{1\times
3})_{[z:w]}\,,$ for each $[z:w]\in\mathbf{C}\mathbb{P}^{1}$,
$\displaystyle(\mathcal{G}_{0}^{0\times 1})_{[z:w]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q_{0}}(\mathbb{G}_{0}^{0\times
1})_{[z:w]}\,,$ for each $[z:w]\in\mathbf{C}\mathbb{P}^{1}$,

with reference to Table 1 and (2.38).

We correspondingly define the $Q$-invariant subbundles

$\displaystyle\not{\mathcal{G}}_{i}^{j,k}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q}\not{\mathbb{G}}_{i}^{j,k}\,,$
$\displaystyle(\not{\mathcal{G}}_{-1}^{0\times 1})_{[x:y]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q}(\not{\mathbb{G}}_{-1}^{0\times
1})_{[x:y]}\,,$ for each $[x:y]\in\mathbf{R}\mathbb{P}^{1}$,
$\displaystyle(\not{\mathcal{G}}_{-1}^{1\times 2})_{[x:y]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q}(\not{\mathbb{G}}_{-1}^{1\times
2})_{[x:y]}\,,$ for each $[x:y]\in\mathbf{R}\mathbb{P}^{1}$,
$\displaystyle(\not{\mathcal{G}}_{-1}^{1\times 3})_{[z:w]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q}(\not{\mathbb{G}}_{-1}^{1\times
3})_{[z:w]}\,,$ for each $[z:w]\in\mathbf{C}\mathbb{P}^{1}$,
$\displaystyle(\not{\mathcal{G}}_{0}^{0\times 1})_{[z:w]}$
$\displaystyle:=\mathcal{F}^{Q}\times_{Q}(\not{\mathbb{G}}_{0}^{0\times
1})_{[z:w]}\,,$ for each $[z:w]\in\mathbf{C}\mathbb{P}^{1}$,

where the $Q$-modules are all defined in Theorem 2.18.

Finally, in order to make contact with the intrinsic torsion of an optical
geometry described in [FLTC20], we define, for each $i,j$,

$\displaystyle\mathcal{G}_{i}^{j}$
$\displaystyle:=\mathcal{F}^{P}\times_{P_{0}}\mathbb{G}_{i}^{j}\,,$
$\displaystyle\not{\mathcal{G}}_{i}^{j}$
$\displaystyle:=\mathcal{F}^{P}\times_{P}\not{\mathbb{G}}_{i}^{j}\,,$

where $\mathbb{G}_{i}^{j}$ are defined in Proposition 2.17, and
$\not{\mathbb{G}}_{i}^{j}$ are defined by (2.40).

To determine the algebraic properties of the intrinsic torsion of an almost
Robinson structure, we choose an optical $1$-form $\kappa_{a}$ and its
associated Robinson $3$-form $\rho_{abc}$, compute their respective covariant
derivatives $\nabla_{a}\kappa_{b}$ and $\nabla_{a}\rho_{bcd}$, and project
these tensors onto the various $Q_{0}$-invariant subbundles of $\mathcal{G}$
by means of splitting operators
$(\ell^{a},\delta^{a}_{\alpha},\delta^{a}_{\bar{\alpha}},k^{a})$. This is
achieved by extending the projections $\Pi_{i}^{j,k}$ defined in Appendix A to
bundle projections, where we identify the tensor $\Gamma_{ab}{}^{c}$ as a
connection $1$-form of $\nabla_{a}$ adapted to the splitting operators. We
shall make use of the notation already introduced in Section 2, mirroring the
definition (2.39). Let us set

(3.8) $\displaystyle\begin{aligned}
&&\gamma_{i}&:=(\nabla\kappa)^{0}{}_{i}\,,\\\
\epsilon&:=(\nabla\kappa)_{ij}h^{ij}\,,&\tau_{ij}&:=(\nabla\kappa)_{[ij]}\,,&\sigma_{ij}&:=(\nabla\kappa)_{(ij)_{\circ}}\,,\\\
&&E_{i}&:=(\nabla\kappa)_{0i}\,,\end{aligned}$

These components split further into irreducible $Q_{0}$-components. In
particular, the component $\tau_{\alpha\bar{\beta}}$ splits as

$\displaystyle\tau_{\alpha\bar{\beta}}$
$\displaystyle=\frac{\mathrm{i}}{2m}\tau^{\omega}h_{\alpha\bar{\beta}}+\tau^{\circ}_{\alpha\bar{\beta}}\,,$

where

$\displaystyle\tau^{\omega}$
$\displaystyle:=\omega^{ij}\tau_{ij}=-2\mathrm{i}\tau_{\alpha\bar{\beta}}h^{\alpha\bar{\beta}}\,,$
$\displaystyle\tau^{\circ}_{\alpha\bar{\beta}}$
$\displaystyle:=\left(\tau_{\alpha\bar{\beta}}\right)_{\circ}\,,$

In addition, the components of the covariant derivative of the Robinson
$3$-form $\rho_{abc}$ of interest are

(3.9) $\displaystyle\zeta_{\alpha\beta}$
$\displaystyle:=(\nabla\rho)^{0}{}_{\alpha\beta 0}\,,$ (3.10) $\displaystyle
G_{i\beta\gamma}$ $\displaystyle:=(\nabla\rho)_{i\beta\gamma 0}\,,$
$\displaystyle B_{\alpha\beta}$ $\displaystyle:=(\nabla\rho)_{0\alpha\beta
0}\,.$

the second of which splits further into the irreducible components

$\displaystyle
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}$
$\displaystyle:=G_{[\alpha\beta\gamma]}\,,$ $\displaystyle
G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{\alpha\beta\gamma}$
$\displaystyle:=\frac{2}{3}\left(G_{(\alpha\beta)\gamma}-G_{(\alpha\gamma)\beta}\right)\,,$
$\displaystyle G^{\circ}_{\bar{\gamma}\alpha\beta}$
$\displaystyle:=\left(G_{\bar{\gamma}\alpha\beta}\right)_{\circ}\,,$
$\displaystyle G_{\alpha}$
$\displaystyle:=h^{\beta\bar{\gamma}}G_{\bar{\gamma}\beta\alpha}\,,$

so that

$\displaystyle G_{\alpha\beta\gamma}$
$\displaystyle=G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}+G^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{\alpha\beta\gamma}\,,$
$\displaystyle G_{\bar{\alpha}\beta\gamma}$
$\displaystyle=G^{\circ}_{\bar{\alpha}\beta\gamma}-\frac{2}{m-1}G_{[\beta}h_{\gamma]\bar{\alpha}}\,,$
$\displaystyle G_{i\beta\gamma}$
$\displaystyle=\delta^{\alpha}_{i}G_{\alpha\beta\gamma}+\delta^{\bar{\alpha}}_{i}G_{\bar{\alpha}\beta\gamma}\,.$

The other relevant components of $\nabla_{a}\rho_{bcd}$ can be found below:

$\displaystyle(\nabla\rho)_{a}{}^{0}{}_{jk}$ $\displaystyle=0\,,$
$\displaystyle(\nabla\rho)_{a\alpha\beta\gamma}$ $\displaystyle=0\,,$
$\displaystyle(\nabla\rho)^{00}{}_{0\alpha}$
$\displaystyle=-\mathrm{i}\gamma_{\alpha}\,,$
$\displaystyle(\nabla\rho)^{0}{}_{\alpha\beta\bar{\gamma}}$
$\displaystyle=2\mathrm{i}\gamma_{[\alpha}h_{\beta]\bar{\gamma}}\,,$
$\displaystyle(\nabla\rho)_{\alpha\beta}{}^{0}{}_{0}$
$\displaystyle=-\mathrm{i}\tau_{\alpha\beta}-\mathrm{i}\sigma_{\alpha\beta}\,,$
$\displaystyle(\nabla\rho)_{\alpha\beta\gamma\bar{\delta}}$
$\displaystyle=2\mathrm{i}\left(\tau_{\alpha[\beta}+\sigma_{\alpha[\beta}\right)h_{\gamma]\bar{\delta}}\,,$
$\displaystyle(\nabla\rho)_{\bar{\alpha}\beta}{}^{0}{}_{0}$
$\displaystyle=-\mathrm{i}\frac{1}{2m}\epsilon
h_{\beta\bar{\alpha}}+\mathrm{i}\tau_{\beta\bar{\alpha}}-\mathrm{i}\sigma_{\beta\bar{\alpha}}\,,$
$\displaystyle(\nabla\rho)_{\bar{\alpha}\beta\gamma\bar{\delta}}$
$\displaystyle=2\mathrm{i}\left(\frac{1}{2m}\epsilon
h_{[\beta|\bar{\alpha}}-\tau_{[\beta|\bar{\alpha}}+\sigma_{[\beta|\bar{\alpha}}\right)h_{|\gamma]\bar{\delta}}\,,$
$\displaystyle(\nabla\rho)_{0\alpha}{}^{0}{}_{0}$
$\displaystyle=-\mathrm{i}E_{\alpha}\,,$
$\displaystyle(\nabla\rho)_{0\alpha\beta\bar{\gamma}}$
$\displaystyle=2\mathrm{i}E_{[\alpha}h_{\beta]\bar{\gamma}}\,,$
$\displaystyle(\nabla\rho)^{0}{}_{0\beta\bar{\gamma}}$
$\displaystyle=\mathrm{i}(\nabla\kappa)^{0}{}_{0}h_{\beta\bar{\gamma}}\,,$
$\displaystyle(\nabla\rho)_{i0\beta\bar{\gamma}}$
$\displaystyle=\mathrm{i}(\nabla\kappa)_{i0}h_{\beta\bar{\gamma}}\,,$
$\displaystyle(\nabla\rho)_{00\beta\bar{\gamma}}$
$\displaystyle=\mathrm{i}(\nabla\kappa)_{00}h_{\beta\bar{\gamma}}\,.$

The elements defined above should be regarded as trivialisations of sections
of $\mathcal{G}_{i}^{j,k}$. Such sections generally carry a boost weight.
These will be adorned with a breve accent. To be clear, we have collected them
in Tables 3 and 4 — the bracket notation $[\\![\cdot]\\!]$ and $[\cdot]$
therein should be understood as taking the real span of the quantities
enclosed therebetween. For instance,

$\displaystyle[\\![\breve{G}^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}]\\!]$
$\displaystyle\sim\breve{G}^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}\delta^{\alpha}_{i}\delta^{\beta}_{j}\delta^{\gamma}_{k}+\breve{G}^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\bar{\alpha}\bar{\beta}\bar{\gamma}}\delta^{\bar{\alpha}}_{i}\delta^{\bar{\beta}}_{j}\delta^{\bar{\gamma}}_{k}\,,$
and $\displaystyle[\breve{\tau}^{\circ}_{\alpha\bar{\beta}}]$
$\displaystyle\sim
2\breve{\tau}^{\circ}_{\alpha\bar{\beta}}\delta^{\alpha}_{[i}\delta^{\bar{\beta}}_{j]}\,,$

and so on.

${\begin{array}[]{|c|c|c|c|}\hline\cr\text{$P_{0}$-bundle}&\text{Description}&\text{Tensor}\\\
\hline\cr\hline\cr\mathcal{G}_{-2}^{0}&H^{*}_{K,L}\otimes\mathcal{E}(2)&\breve{\gamma}_{i}\\\
\hline\cr\hline\cr\mathcal{G}_{-1}^{0}&\mathcal{E}(1)&\breve{\epsilon}\\\
\hline\cr\mathcal{G}_{-1}^{1}&\scaleobj{1.2}{\wedge}^{2}H^{*}_{K,L}\otimes\mathcal{E}(1)&\breve{\tau}_{ij}\\\
\hline\cr\mathcal{G}_{-1}^{2}&\scaleobj{1.2}{\odot}^{2}_{\circ}H^{*}_{K,L}\otimes\mathcal{E}(1)&\breve{\sigma}_{ij}\\\
\hline\cr\hline\cr\mathcal{G}_{0}^{0}&H^{*}_{K,L}&\breve{E}_{i}\\\
\hline\cr\end{array}}$ Table 3. Irreducible $P_{0}$-submodules of
$\mathcal{G}$
${\begin{array}[]{|c|c|c|c|}\hline\cr\text{$Q_{0}$-bundle}&\text{Description}&\text{Tensor}\\\
\hline\cr\hline\cr\mathcal{G}_{-2}^{0,0}&H^{*}_{K,L}\otimes\mathcal{E}(2)&\breve{\gamma}_{i}\\\
\hline\cr\hline\cr\mathcal{G}_{-1}^{0,0}&\mathcal{E}(1)&\breve{\epsilon}\\\
\hline\cr\mathcal{G}_{-1}^{1,0}&\mathcal{E}(1)&\breve{\tau}^{\omega}\\\
\mathcal{G}_{-1}^{1,1}&[\\![\scaleobj{1.2}{\wedge}^{(2,0)}H^{*}_{K,L}]\\!]\otimes\mathcal{E}(1)&[\\![\breve{\tau}_{\alpha\beta}]\\!]\\\
\mathcal{G}_{-1}^{1,2}&[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}H^{*}_{K,L}]\otimes\mathcal{E}(1)&[\breve{\tau}^{\circ}_{\alpha\bar{\beta}}]\\\
\hline\cr\mathcal{G}_{-1}^{2,0}&[\scaleobj{1.2}{\wedge}^{(1,1)}_{\circ}H^{*}_{K,L}]\otimes\mathcal{E}(1)&[\breve{\sigma}_{\alpha\bar{\beta}}]\\\
\mathcal{G}_{-1}^{2,1}&[\\![\scaleobj{1.2}{\odot}^{(2,0)}H^{*}_{K,L}]\\!]\otimes\mathcal{E}(1)&[\\![\breve{\sigma}_{\alpha\beta}]\\!]\\\
\hline\cr\mathcal{G}_{-1}^{3,0}&[\\![\scaleobj{1.2}{\wedge}^{(2,0)}H^{*}_{K,L}]\\!]\otimes\mathcal{E}(1)&[\\![\breve{\zeta}_{\alpha\beta}]\\!]\\\
\hline\cr\hline\cr\mathcal{G}_{0}^{0,0}&H^{*}_{K,L}&\breve{E}_{i}\\\
\hline\cr\mathcal{G}_{0}^{1,0}&[\\![\scaleobj{1.2}{\wedge}^{(1,0)}H^{*}_{K,L}]\\!]&[\\![\breve{G}_{\alpha}]\\!]\\\
\mathcal{G}_{0}^{1,1}&[\\![\scaleobj{1.2}{\wedge}^{(3,0)}H^{*}_{K,L}]\\!]&[\\![\breve{G}^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(1,1,1)}}}}_{\alpha\beta\gamma}]\\!]\\\
\mathcal{G}_{0}^{1,2}&[\\![\raisebox{-1.72218pt}{\scaleobj{0.35}{\yng(2,1)}}H^{*}_{K,L}]\\!]&[\\![\breve{G}^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{\alpha\beta\gamma}]\\!]\\\
\mathcal{G}_{0}^{1,3}&[\\![\scaleobj{1.2}{\wedge}^{(1,2)}_{\circ}H^{*}_{K,L}]\\!]&[\\![\breve{G}^{\circ}_{\bar{\gamma}\alpha\beta}]\\!]\\\
\hline\cr\hline\cr\mathcal{G}_{1}^{0,0}&[\\![\scaleobj{1.2}{\wedge}^{(2,0)}H^{*}_{K,L}]\\!]\otimes\mathcal{E}(-1)&[\\![\breve{B}_{\alpha\beta}]\\!]\\\
\hline\cr\end{array}}$ Table 4. Irreducible $Q_{0}$-submodules of
$\mathcal{G}$

We shall also introduce the quantity

$\displaystyle\breve{\tau}_{\alpha\bar{\beta}}$
$\displaystyle=\frac{\mathrm{i}}{2m}\breve{\tau}^{\omega}h_{\alpha\bar{\beta}}+\breve{\tau}^{\circ}_{\alpha\bar{\beta}}\,,$

which we identify as a section of
$\mathcal{G}_{-1}^{1,0}\oplus\mathcal{G}_{-1}^{1,2}$.

Finally, to characterise sections of the bundles
$\not{\mathcal{G}}_{i}^{j,k}$, we simply apply the results of Theorem 2.18.
For instance, the intrinsic torsion is a section of
$\not{\mathcal{G}}_{0}^{1,2}$ if and only if its weighted components in any
splitting satisfy

$\displaystyle\breve{\gamma}_{i}=\breve{\sigma}_{\alpha\beta}=2\mathrm{i}\breve{\tau}^{\circ}_{\alpha\beta}+\breve{\zeta}_{\alpha\beta}=\breve{G}^{\tiny{\raisebox{-0.86108pt}{\scaleobj{0.35}{\yng(2,1)}}}}_{\alpha\beta\gamma}=0\,.$

######  Remark 3.6.

We can also express some of the properties of the almost Robinson structure in
terms of a Robinson spinor — this will be done in Section 3.10.

###  3.5. Congruences of null geodesics

Since an almost Robinson manifold $(\mathcal{M},g,N,K)$ defines in particular
an optical geometry, there is a congruence of null curves $\mathcal{K}$
associated to it. The algebraic properties of the intrinsic torsion of the
optical geometry $(\mathcal{M},g,K)$ can be related to the geometric
properties of $\mathcal{K}$ as reviewed in [FLTC20]. We summarise the
correspondence in Table 5.

${\begin{array}[]{|c|c|}\hline\cr\text{Intrinsic torsion}&\text{Congruence}\\\
\hline\cr\hline\cr\not{\mathcal{G}}_{-2}^{0}=\mathcal{G}^{-1}&\text{geodesic}\\\
\hline\cr\not{\mathcal{G}}_{-1}^{0}&\text{non-expanding}\\\
\not{\mathcal{G}}_{-1}^{1}&\text{non-twisting}\\\
\not{\mathcal{G}}_{-1}^{2}&\text{non-shearing}\\\
\hline\cr\not{\mathcal{G}}_{0}^{0}=\\{0\\}&\text{parallel}\\\
\hline\cr\end{array}}$ Table 5.  Geometric properties of $\mathcal{K}$ and
intrinsic torsion $\mathring{T}$

We also record the following lemma, whose proof is straightforward.

######  Lemma 3.7.

Let $(\mathcal{M},g,N,K)$ be an almost Robinson manifold with congruence of
null curves $\mathcal{K}$. Denote by $J_{i}{}^{j}$ the complex structure on
the screen bundle $H_{K}$. Suppose that the intrinsic torsion $\mathring{T}$
of $(N,K)$ is a section of $\not{\mathcal{G}}_{-2}^{0}$ so that $\mathcal{K}$
is geodesic with twist $\breve{\tau}_{ij}$ and shear $\breve{\sigma}_{ij}$.
Then $\mathring{T}$ is a section of

  * •

$\not{\mathcal{G}}_{-1}^{1,1}$, i.e. $\breve{\tau}_{\alpha\beta}=0$, if and
only if $\breve{\tau}_{ij}$ and $J_{i}{}^{j}$ commute, i.e.
$J_{[i}{}^{k}\breve{\tau}_{j]k}=0$;

  * •

$\not{\mathcal{G}}_{-1}^{2,1}$, i.e. $\breve{\sigma}_{\alpha\beta}=0$, if and
only if $\breve{\sigma}_{ij}$ and $J_{i}{}^{j}$ commute, i.e.
$J_{(i}{}^{k}\breve{\sigma}_{j)k}=0$;

  * •

$\not{\mathcal{G}}_{-1}^{1,0}\cap\not{\mathcal{G}}_{-1}^{1,2}$, i.e.
$\breve{\tau}_{\alpha\bar{\beta}}=0$, if and only if $\breve{\tau}_{ij}$ and
$J_{i}{}^{j}$ anti-commute, i.e. $J_{(i}{}^{k}\breve{\tau}_{j)k}=0$;

  * •

$\not{\mathcal{G}}_{-1}^{2,0}$, i.e. $\breve{\sigma}_{\alpha\bar{\beta}}=0$,
if and only if $\breve{\sigma}_{ij}$ and $J_{i}{}^{j}$ anti-commute, i.e.
$J_{[i}{}^{k}\breve{\sigma}_{j]k}=0$.

Locally, we shall identify an almost Robinson manifold $(\mathcal{M},g,N,K)$
of dimension $2m+2$ as a surjective submersion over a $(2m+1)$-dimensional
smooth manifold $\underline{\mathcal{M}}$, namely, the local leaf space of
$\mathcal{K}$. This leaf space will be endowed with various geometric
structures depending on the geometric properties of $\mathcal{K}$. In the next
sections, we shall examine the relation between the intrinsic torsion of
$(N,K)$ with the induced geometric structures on the leaf space. As a
notational rule followed in this article, tensor fields on
$\underline{\mathcal{M}}$ will be underlined to distinguish them from tensor
fields on $\mathcal{M}$.

######  Remark 3.8.

We briefly recall the results of [FLTC20] (see also [RT83]) that pertain to
the optical structure $K$ associated to an almost Robinson manifold
$(\mathcal{M},g,N,K)$. In the following, we shall denote the conformal class
of $g$ by $[g]$. If the congruence of null curves $\mathcal{K}$ tangent to $K$
is geodesic, there is a subclass $[g]_{n.e.}$ of metrics in $[g]$ for which
$\mathcal{K}$ is also non-expanding. By extension, $[g]$ and $[g]_{n.e}$
determine conformal subclasses of bundle metrics $[h]$ and $[h]_{n.e}$
respectively on the screen bundle $H_{K}$. We then have the following
relations between the geometric properties of $\mathcal{K}$ and its leaf
space:

  * •

If $\mathcal{K}$ is geodesic, $H_{K}$ descends to a rank-$2m$ distribution
$\underline{H}$ on $\underline{\mathcal{M}}$. In this case, there is a one-to-
one correspondence between optical vector fields $k$ such that
$\mathsterling_{k}\kappa=0$, where $\kappa=g(k,\cdot)$, and sections of
$\mathrm{Ann}(\underline{H})$.

  * •

If $\mathcal{K}$ is geodesic and non-twisting, the induced distribution
$\underline{H}$ on $\underline{\mathcal{M}}$ is involutive.

  * •