Spinors in Five-Dimensional Contact Geometry

Spinors in Five-Dimensional Contact Geometry††This paper is a contribution to
the Special Issue on Twistors from Geometry to Physics in honour of Roger
Penrose. The full collection is available at
https://www.emis.de/journals/SIGMA/Penrose.html

Michael EASTWOOD a and Timothy MOY b

M. Eastwood and T. Moy

a) School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
<EMAIL_ADDRESS>
b) Clare College, University of Cambridge, CB2 1TL, England, UK
<EMAIL_ADDRESS>
Received January 31, 2022, in final form April 13, 2022; Published online
April 16, 2022

We use classical (Penrose) two-component spinors to set up the differential
geometry of two parabolic contact structures in five dimensions, namely
$G_{2}$ contact geometry and Legendrean contact geometry. The key players in
these two geometries are invariantly defined directional derivatives defined
only in the contact directions. We explain how to define them and their usage
in constructing basic invariants such as the harmonic curvature, the
obstruction to being locally flat from the parabolic viewpoint. As an
application, we calculate the invariant torsion of the $G_{2}$ contact
structure on the configuration space of a flying saucer (always a five-
dimensional contact manifold).

spinors; contact geometry; parabolic geometry

53B05; 53D10; 58J10

Dedicated to Roger Penrose
on the occasion of his 90th birthday

##  1 Introduction

Two-component spinors are widely used in four-dimensional Lorentzian geometry.
The seminal books ‘Spinors and space-time’ [12, 13] are devoted to such usage.
At a very basic level, two-component spinors arise via the 2–1 covering of Lie
groups

${\mathrm{SL}}(2,{\mathbb{C}})\longrightarrow{\mathrm{SO}}^{\uparrow}(3,1),$

where ${\mathrm{SO}}^{\uparrow}(3,1)$ is the identity-connected component of
the Lorentz group. Similarly, in three dimensions, the 2–1 covering

${\mathrm{SL}}(2,{\mathbb{R}})\longrightarrow{\mathrm{SO}}^{\uparrow}(2,1)$

is responsible for the utility of two-component spinors in three dimensions
(as in Section 5).

At a very basic level, the two-component spinors in this article arise via
inclusions

${\mathrm{SL}}(2,{\mathbb{R}})\hookrightarrow{\mathrm{Sp}}(4,{\mathbb{R}}),$

where ${\mathrm{Sp}}(4,{\mathbb{R}})$ is the subgroup of
${\mathrm{SL}}(4,{\mathbb{R}})$ consisting of matrices that preserve a fixed
nondegenerate skew form on ${\mathbb{R}}^{4}$. There are two such inclusions:

  * •

by writing ${\mathbb{R}}^{4}=\bigodot^{3}{\mathbb{R}}^{2}$, where $\bigodot$
denotes symmetric tensor product,

  * •

by writing ${\mathbb{R}}^{4}={\mathbb{R}}^{2}\oplus{\mathbb{R}}^{2}$,

giving rise to $G_{2}$ contact geometry (as in Section 7) and Legendrean
contact geometry (as in Section 8), respectively. These two geometries are
defined on a five-dimensional contact manifold $M$ as extra structure on the
contact distribution $H\subset TM$. A contact form $\alpha$ is a $1$-form so
that $H=\ker\alpha$. It gives rise to a nondegenerate skew form, the Levi
form, namely ${\rm d}\alpha|_{H}$. The extra spin structures in Sections 7 and
8 are required to be compatible with the Levi form.

Another important theme in [12, 13] is conformal geometry. It is concerned
with what happens if the (Lorentzian) metric is replaced by a smooth positive
multiple of itself. The resulting formulæ, for example (5.1), fit very well
with spinors. For the contact geometries in this article, the corresponding
freedom is in choosing a contact form $\alpha$. The resulting formulæ, for
example (7.6), (8.7), (8.8), also fit very well with spinors. These
‘conformal’ structures are examples of parabolic geometries [4]. In
particular, invariant differential operators play a key part in parabolic
constructions.

Ideally, one would like to approach the natural differential geometric
calculus on these various geometries via invariant differential operators.
More specifically, for a chosen ‘scale’ (a metric in the conformal class or a
choice of contact form) one expects a canonical (partial) connection on all
the natural irreducible vector bundles. This expectation follows from the Čap-
Slovák theory of Weyl structures and scales in parabolic geometry (see [3] or
[4, Section 5.1]). In conformal geometry, it is just the Levi-Civita
connection: see Sections 2 and 3 for details. For parabolic contact
structures, one expects partial connections defined only in the contact
directions. We shall see in Sections 7 and 8 that a suitable collection of
invariant operators is, indeed, sufficient for these purposes. For the five-
dimensional Legendrean contact structures in Section 8, these operators can
easily be found (within the Rumin complex, explained in Section 6). The two
key invariant operators (7.7) in $G_{2}$ contact geometry remain somewhat
mysterious: it is shown in Section 7 that these operators are, indeed,
invariant but we have not been able to find a suitable shortcut to their
construction. Nevertheless, we are able to construct, in Section 9, the
general $G_{2}$ contact geometry from a suitable Legendrean contact structure
and, thereby, calculate its basic spinor invariant (a certain septic).

Through this article we shall use Penrose’s abstract index notation and other
conventions from [12]. Also, to ease the notational burden, we shall not
carefully distinguish between bundles and sections thereof, for example
writing $\omega_{a}\in{}^{1}$ instead of
$\omega_{a}\in\Gamma\big{(}{}^{1}\big{)}$ or even
$\omega_{a}\in\Gamma\big{(}M,{}^{1}\big{)}$, for a $1$-form $\omega_{a}$.
Especially as this article is concerned with local differential geometry, this
should cause no confusion.

##  2 The Levi-Civita connection

On a general smooth manifold, the exterior derivative and the Lie derivative
are defined independently of local coördinates (and there is little else with
this property [9]). Both of these operations can be defined in terms of an
arbitrary torsion-free affine connection. For the exterior derivative on
$1$-forms, we have (following the conventions of [12, equation (4.3.14)])

$\omega_{b}\longmapsto\nabla_{[a}\omega_{b]}.$

For the Lie derivative on covariant $2$-tensors, we have

$\phi_{bc}\longmapsto({\mathcal{L}}_{X}\phi)_{bc}\equiv
X^{a}\nabla_{a}\phi_{bc}+(\nabla_{b}X^{a})\phi_{ac}+(\nabla_{c}X^{a})\phi_{ba},$
(2.1)

for a vector field $X^{a}$ and, irrespective of the usual interpretation of
${\mathcal{L}}_{X}$ in terms of the flow of $X^{a}$, it easy to check that
this expression does not depend on the choice of $\nabla_{a}$. It is
convenient to regard the right hand side of (2.1), let’s say for a symmetric
tensor $\phi_{ab}$, as an invariantly defined differential pairing

$\textstyle
TM\times\bigodot^{2}\\!{}^{1}\longrightarrow\bigodot^{2}\\!{}^{1}.$

In particular, if $g_{ab}$ is a semi-Riemannian metric, that is to say a
nondegenerate symmetric form, then the Lie derivative of $g_{ab}$, with a
convenient factor of $\frac{1}{2}$ thrown in, can be regarded as a canonically
defined linear differential operator

$\textstyle TM\ni
X^{a}\longmapsto\frac{1}{2}{\mathcal{L}}_{X}g_{bc}\in\bigodot^{2}\\!{}^{1}.$
(2.2)

Of course, the tensor $g_{ab}$ also defines an isomorphism $TM\cong{}^{1}$ by
$X^{a}\mapsto g_{ab}X^{b}$, and so we have obtained an invariantly defined
linear differential operator

$\textstyle{}^{1}\cong TM\longrightarrow\bigodot^{2}\\!{}^{1}.$ (2.3)

In combination with the exterior derivative ${\rm d}\colon{}^{1}\to{}^{2}$, we
have obtained

$\textstyle{}^{1}\longrightarrow\bigodot^{2}\\!{}^{1}\oplus{}^{2}={}^{1}\otimes{}^{1}$
(2.4)

and we claim that this is the Levi-Civita connection defined by $g_{ab}$. This
is easy to check: if we use the Levi-Civita connection in (2.1), then

$\displaystyle\tfrac{1}{2}{\mathcal{L}}_{X}g_{ab}+\nabla_{[a}X_{b]}$
$\displaystyle=\tfrac{1}{2}\big{(}X^{c}\nabla_{c}g_{ab}+(\nabla_{a}X^{c})g_{cb}+(\nabla_{b}X^{c})g_{ac}\big{)}+\nabla_{[a}X_{b]}$
$\displaystyle=\nabla_{(a}X_{b)}+\nabla_{[a}X_{b]}=\nabla_{a}X_{b},$

as required. But we can play the moves (2.2), (2.3), and (2.4) to define the
Levi-Civita connection. An advantage of this viewpoint is that it is easily
modified to define other connections (and partial connections), as we shall
soon see.

##  3 Conformal differential geometry

In semi-Riemannian conformal differential geometry, instead of a metric
$g_{ab}$, we are given only a conformal class of metrics. A convenient way of
expressing this is to say that we are given $\eta_{ab}$, a nondegenerate
section of $\bigodot^{2}\\!{}^{1}\otimes L^{2}$ for some line bundle $L$.
Thus, if $\sigma$ is a non-vanishing section of $L$, then
$g_{ab}\equiv\sigma^{-2}\eta_{ab}$ is a genuine metric. We shall refer to
$\sigma$ as a scale and, if we choose a different section
$\widehat{\sigma}=\Omega^{-1}\sigma$ for some nowhere vanishing function
$\Omega$, then we encounter a rescaled metric
$\widehat{g}_{ab}\equiv\widehat{\sigma}^{-2}\eta_{ab}=\Omega^{2}g_{ab}$. For a
given conformal structure $\eta_{ab}$, let us define $\eta^{ab}$, a section of
$\bigodot^{2}\\!TM\otimes L^{-2}$, by $\eta_{ac}\eta^{bc}=\delta_{a}{}^{b}$,
where $\delta_{a}{}^{b}$ is the invariant (Kronecker delta) pairing
$TM\otimes{}^{1}\to{}^{0}$.

On an $n$-dimensional oriented conformal manifold, we may attempt to normalise
a section $\epsilon_{ab\dots d}$ of the bundle n, compatible with the
orientation, by insisting that

$\epsilon_{ab\dots d}\epsilon_{ef\dots
h}\eta^{ae}\eta^{bf}\cdots\eta^{dh}\equiv n!,$ (3.1)

just as we may do on an $n$-dimensional oriented Riemannian manifold to
normalise and define the volume form. The problem with (3.1) is that the left
hand side takes values in the line bundle $L^{-2n}$. To remedy this problem,
we may insist that $\epsilon_{ab\dots d}$ take values in ${}^{n}\otimes
L^{n}$. The normalisation (3.1) now makes good sense and $\epsilon_{ab\dots
d}$ is uniquely determined. Equivalently, we may view $\epsilon_{ab\dots d}$
as providing a canonical identification

$L^{-n}={}^{n}\qquad\mbox{given by}\quad\phi\mapsto\phi\epsilon_{ab\dots d}.$

It it usual to write $L\equiv{}^{0}[1]$ and refer to sections of
$L^{w}\equiv{}^{0}[w]$ as conformal densities of weight $w$. In the presence
of a scale $\sigma$, defining a metric $g_{ab}=\sigma^{-2}\eta_{ab}$, the
normalisation (3.1) reads

$(\sigma^{n}\epsilon_{ab\dots d})(\sigma^{n}\epsilon_{ef\dots
h})g^{ae}g^{bf}\cdots g^{dh}=n!$

so $\sigma^{n}\epsilon_{ab\dots d}$ is the usual volume form for the metric
$g_{ab}$. This is consistent with the scale $\sigma$ trivialising all the
density bundles ${}^{0}[w]$. To summarise, in the presence of a scale
$\sigma$, a conformal density of weight $w$ can be regarded as an ordinary
function but if we change to a new scale $\widehat{\sigma}=\Omega^{-1}\sigma$,
then the same density is represented by a new function
$\widehat{f}=\Omega^{w}f$.

Once we have decided that ${}^{0}[1]=L=({}^{n})^{-1/n}$, the Lie derivative
${\mathcal{L}}_{X}\colon{}^{1}\to{}^{1}$ induces invariantly defined
differential pairings $TM\times B\to B$ for all conformally weighted tensor
bundles $B$ and, in particular,

$\textstyle TM\times\bigodot^{2}\\!{}^{1}[2]\to\bigodot^{2}\\!{}^{1}[2].$

Recall that a conformal structure is a nondegenerate section $\eta_{ab}$ of
this bundle $\bigodot^{2}\\!{}^{1}[2]$ and hence we obtain a conformally
invariant first-order linear differential operator

$\textstyle TM\ni
X^{a}\longmapsto{\mathcal{L}}_{X}\eta_{ab}\in\bigodot^{2}\\!{}^{1}[2].$ (3.2)

But recall that $\epsilon_{ab\dots d}$ is the canonical section of
${}^{n}[n]={}^{0}$ corresponding to the constant function $1$. It follows that
${\mathcal{L}}_{X}\epsilon_{ab\dots d}=0$ for any vector field $X^{a}$ and
therefore, from (3.1), that $\eta^{ab}{\mathcal{L}}_{X}\eta_{ab}=0$. In
addition, we may use the conformal metric $\eta_{ab}$ to lower indices, at the
expense of a conformal weight so that $TM={}^{1}[2]$. The Lie derivative in
(3.2) thus yields a conformal invariant differential operator

$\textstyle{}^{1}[2]\longrightarrow\bigodot_{\circ}^{2}\\!{}^{1}[2],$

where $\circ$ denotes the trace-free part (a manifestly conformally invariant
notion). If $\phi_{b\dots d}$ is an $(n-1)$-form then

$\epsilon_{ab\dots d}\eta^{be}\cdots\eta^{dg}\phi_{e\dots g}$

is a $1$-form of conformal weight $n-2(n-1)=2-n$. In other words, the bundles
n-1 and ${}^{1}[2-n]$ are canonically isomorphic. Similarly, we have
${}^{n}={}^{0}[-n]$ and so the exterior derivative ${\rm
d}\colon{}^{n-1}\to{}^{n}$ may be viewed as an invariant differential operator

${}^{1}[2-n]\to{}^{0}[-n].$

Together with the exterior derivative ${\rm d}\colon{}^{1}\to{}^{2}$, we now
have three conformally invariant linear differential operators defined on
variously weighted $1$-forms, namely

$\textstyle{}^{1}[2]\to\bigodot_{\circ}^{2}\\!{}^{1}[2],\qquad{}^{1}[2-n]\to{}^{0}[-n],\qquad{}^{1}\to{}^{2}.$
(3.3)

As in Section 2, we are now in a position to define a preferred connection in
the presence of a scale $\sigma\in\Gamma({}^{0}[1])$. Specifically, we may
drop all the weights in (3.3) and combine them to obtain

$\textstyle{}^{1}\xrightarrow{\,\nabla\,}{}^{1}\otimes{}^{1}=\bigodot_{\circ}^{2}\\!{}^{1}\oplus{}^{0}\oplus{}^{2}.$

The three bundles on the right constitute the orthogonal decomposition of
${}^{1}\otimes{}^{1}$ into irreducibles with respect to the metric
$g_{ab}=\sigma^{-2}\eta_{ab}$. Of course, it is easily verified that we have
found the Levi-Civita connection for $g_{ab}$ but there are two advantages of
this particular route. One is that we are able to read off the change in the
Levi-Civita connection under conformal rescaling
$\sigma\mapsto\widehat{\sigma}=\Omega^{-1}\sigma$. The other is that this
route may be employed in other parabolic geometries. Regarding the first
advantage, another way of pinning down the connection defined by (3.3) and a
scale $\sigma\in\Gamma\big{(}{}^{0}[1]\big{)}$, is to require that
$\nabla_{a}\sigma=0$. This is consistent with the connections induced on the
line bundles ${}^{0}[w]$ in the sense that, for
$\phi\in\Gamma\big{(}{}^{0}[w]\big{)}$, we have $\nabla\phi=\sigma^{w}{\rm
d}(\sigma^{-w}\phi)$. In particular, for a new scale
$\widehat{\sigma}=\Omega^{-1}\sigma$ and
$\phi\in\Gamma\big{(}{}^{0}[w]\big{)}$, we have

$\widehat{\nabla}\phi=\widehat{\sigma}^{w}{\rm
d}\big{(}\widehat{\sigma}^{-w}\phi\big{)}=\Omega^{-w}\sigma^{w}{\rm
d}\big{(}\Omega^{w}\sigma^{-w}\phi\big{)}=\sigma^{w}{\rm
d}\big{(}\sigma^{-w}\phi\big{)}+w\Omega^{-1}({\rm
d}\Omega)\phi=\nabla\phi+w\Upsilon\phi,$

where $\Upsilon\equiv\Omega^{-1}{\rm d}\Omega$. It follows that, regarding the
change in the Levi-Civita connection on a $1$-form $\omega_{b}$, we have

$\widehat{\nabla}_{a}\omega_{b}=\nabla_{a}\omega_{b}-\Upsilon_{a}\omega_{b}-\Upsilon_{b}\omega_{a}+\Upsilon^{c}\omega_{c}g_{ab}$

since then, if $\omega_{b}$ has conformal weight $w$, we deduce that

$\widehat{\nabla}_{a}\omega_{b}=\nabla_{a}\omega_{b}+(w-1)\Upsilon_{a}\omega_{b}-\Upsilon_{b}\omega_{a}+\Upsilon^{c}\omega_{c}g_{ab},$

which is exactly so that the three operators (3.3), namely

$\textstyle\phi_{b}\mapsto\nabla_{(a}\phi_{b)}-\frac{1}{n}(\nabla^{c}\phi_{c})g_{ab},\qquad\phi_{b}\mapsto\nabla^{b}\phi_{b},\qquad\phi_{b}\mapsto\nabla_{[a}\phi_{b]},$

are invariantly defined.

Regarding the second advantage, it is convenient to adopt a universal notation
for the natural irreducible vector bundles available on a parabolic geometry,
as detailed in [1, 4]. Let’s see how this works in $5$-dimensional conformal
geometry. The general irreducible spin bundle is

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and for tensor bundles we restrict $c$ to be even. (Roughly speaking, this
indicates a (complex) irreducible representation of
${\mathbb{R}}_{>0}\times{\mathrm{Spin(5)}}$ with the real number $a$ over the
crossed node recording the action of ${\mathbb{R}}_{>0}$.) The de Rham complex
is

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\end{picture}$ (3.4)

and the three conformally invariant first-order linear differential operators
on weighted $1$-forms are

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\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-2$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}\to\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-3$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 2$}}
\end{picture}.$

Notice that
${}^{0}[w]=\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle w$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}$ and that the identification ${}^{5}={}^{0}[-5]$ is built into
the notation. There are many other advantages of this seemingly arcane
notation: its utility in $5$-dimensional spin geometry is the subject of the
following section.

##  4 Five-dimensional conformal spin geometry

In this section we discuss five-dimensional conformal geometry by means of
spinors (that is to say four-spinors in this dimension (also known as
twistors)). For simplicity, we shall suppose that the conformal metric has
split signature. There is some elementary linear algebra behind this
discussion as follows. Suppose that ${\mathbb{T}}$ is a four-dimensional real
vector space equipped with a nondegenerate skew form $\epsilon_{\alpha\beta}$.
Let us write $\epsilon^{\alpha\beta}$ for its inverse so that
$\epsilon_{\alpha\beta}\epsilon^{\alpha\gamma}=\delta_{\alpha}{}^{\gamma}$.
The vector space ${}^{2}{\mathbb{T}}$ naturally splits:

${}^{2}{\mathbb{T}}=\big{\\{}P_{\alpha\beta}\,|\,\epsilon^{\alpha\beta}P_{\alpha\beta}=0\big{\\}}\oplus\\{P_{\alpha\beta}=\lambda\epsilon_{\alpha\beta}\\}\equiv{}_{\perp}^{2}{\mathbb{T}}\oplus{\mathbb{R}}.$
(4.1)

Otherwise said, if we let ${\mathrm{Sp}}(4,{\mathbb{R}})$ denote the linear
automorphisms of ${\mathbb{T}}\cong{\mathbb{R}}^{4}$ preserving the symplectic
form $\epsilon_{\alpha\beta}$, then (4.1) is the decomposition of
${}^{2}{\mathbb{T}}$ into ${\mathrm{Sp}}(4,{\mathbb{R}})$-irreducibles. The
$5$-dimensional vector space ${}_{\perp}^{2}{\mathbb{T}}$ is acquires a split
signature metric

$\|P_{\alpha\beta}\|^{2}\equiv
P_{\alpha\beta}P_{\gamma\eta}\epsilon^{\alpha\gamma}\epsilon^{\beta\eta}.$

Otherwise said, we have constructed the isomorphism
${\mathrm{Sp}}(4,{\mathbb{R}})\cong{\mathrm{Spin}}(3,2)$. According to the
Plücker relations, the null vectors in ${}_{\perp}^{2}{\mathbb{T}}$ are the
decomposable tensors.

With the conventions of [1], the bundle version of this discussion yields the
splitting

${}^{2}\big{(}\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\end{picture}\big{)}=\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}\oplus\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}={}^{1}[2]\oplus{}^{0}[1].$

More precisely, we may view a split signature conformal spin structure on a
five-dimensional split signature conformal manifold as a rank $4$ bundle,
denoted by $\times$$\bullet$$\bullet$$\rangle$$\vphantom{(}\scriptstyle
0$$\vphantom{(}\scriptstyle 0$$\vphantom{(}\scriptstyle 1$ and equipped with

  * •

a nondegenerate section of
${}^{2}\big{(}\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\end{picture}\big{)}$ with values in ${}^{0}[-1]$,

  * •

an identification
${}_{\perp}^{2}\big{(}\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\end{picture}\big{)}={}^{1}[2]$.

A scale $\sigma\in\Gamma\big{(}{}^{0}[1]\big{)}$ induces a nondegenerate
section of
${}^{2}\big{(}\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\end{picture}\big{)}$ (and, according to the discussion in Section 3, a
compatible nondegenerate (split signature) symmetric form on 1). The first
summand in the splitting

$\operatorname{End}\big{(}\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\end{picture}\big{)}=\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-1$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 2$}}
\end{picture}\oplus\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-1$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}\oplus\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}$ (4.2)

captures the trace-free endomorphisms of
$\times$$\bullet$$\bullet$$\rangle$$\vphantom{(}\scriptstyle
0$$\vphantom{(}\scriptstyle 0$$\vphantom{(}\scriptstyle 1$ that preserve its
conformal skew form (and, simultaneously, the second summand in

$\operatorname{End}\big{(}{}^{1}\big{)}=\operatorname{End}\big{(}\,\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-2$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}\big{)}=\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-2$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 2$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}\oplus\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-1$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 2$}}
\end{picture}\oplus\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}$

plays the same rôle with respect to the conformal metric on 1). From this
viewpoint, a scale $\sigma\in\Gamma\big{(}{}^{0}[1]\big{)}$ gives rise to a
connection on the spin bundle
$\times$$\bullet$$\bullet$$\rangle$$\vphantom{(}\scriptstyle
0$$\vphantom{(}\scriptstyle 0$$\vphantom{(}\scriptstyle 1$ as follows. We
firstly insist that this connection preserve its conformal skew form and also
that the induced connection on ${}^{0}[1]$ annihilate $\sigma$. According to
(4.2), the freedom in choosing such a connection lies in

${}^{1}\otimes\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-1$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 2$}}
\end{picture}=\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-2$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}\otimes\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-1$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 2$}}
\end{picture}.$ (4.3)

On the other hand, the induced operator $\nabla\colon{}^{1}\to{}^{2}$ differs
from the exterior derivative by a homomorphism ${}^{1}\to{}^{2}$ (it is the
torsion of the induced affine connection) and, according to (3.4), lies in

$\operatorname{Hom}\big{(}{}^{1},{}^{2}\big{)}=\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 1$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\end{picture}\otimes\begin{picture}(48.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(44.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}} \put(4.0,1.5){\line(1,0){20.0}}
\put(23.0,2.5){\line(1,0){20.0}} \put(23.0,0.5){\line(1,0){20.0}}
\put(34.0,1.5){\makebox(0.0,0.0)[]{$\rangle$}}
\put(4.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle-3$}}
\put(24.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 0$}}
\put(44.0,10.0){\makebox(0.0,0.0)[]{$\vphantom{(}\scriptstyle 2$}}
\end{picture}.$ (4.4)

Comparing the bundles (4.3) and (4.4) and noting that they are canonically
isomorphic suggests that the homomorphism ${}^{1}\to{}^{2}$ may be precisely
eliminated by the allowed freedom. It is easy to check that this is, indeed,
the case.

##  5 Three-dimensional conformal spin geometry

A split signature three-dimensional conformal spin structure may be viewed as
a rank two ‘spin bundle’ $S$ equipped with an identification
$\bigodot^{2}\\!S={}^{1}[2]$ (cf. [15]). With the Dynkin diagram notation from
[1],

$\textstyle
S=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture},\qquad\bigodot^{2}\\!\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture},\qquad{}^{1}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture},\qquad{}^{0}[1]=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture},$

and the de Rham complex is

${}^{0}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\to\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\to\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\end{picture}\to\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}} \end{picture}={}^{3}.$

The conformal structure can now be characterised by decreeing that the simple
spinors in $\bullet$$\times$$\langle$$\scriptstyle 2$$\scriptstyle 0$ are the
null vectors in
${}^{1}[2]=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}$.

The first summand in

$\operatorname{End}\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\oplus\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}$

captures the trace-free endomorphisms of
$\bullet$$\times$$\langle$$\scriptstyle 1$$\scriptstyle 0$ and it follows that
the freedom in choosing a connection on this spin bundle annihilating a scale,
i.e., a nowhere vanishing section of
${}^{2}\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}$, lies in

${}^{1}\otimes\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\otimes\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}.$

On the other hand the torsion of the induced affine connection lies in the
canonically isomorphic bundle

$\operatorname{Hom}\big{(}{}^{1},{}^{2}\big{)}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\otimes\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}} \end{picture}.$

It is easy to check that the freedom in choice of connection on
$\bullet$$\times$$\langle$$\scriptstyle 1$$\scriptstyle 0$ may be used exactly
to eliminate this torsion. More precisely, we have the following:

######  Proposition 5.1.

Given a scale, i.e., a nowhere vanishing
$\sigma\in\Gamma\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$,
there is a unique connection on $\bullet$$\times$$\langle$$\scriptstyle
1$$\scriptstyle 0$ so that

  * •

the induced connection on
${}^{2}\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}$
annihilates $\sigma$,

  * •

the torsion of the induced connection on
${}^{1}=\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}} \end{picture}$ vanishes.

It is straightforward to figure out how this preferred connection changes
under change of scale. To do this, let us adapt the classical two-spinor
notation of [12] to write

$\nabla\colon\
\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\longrightarrow\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\otimes\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\quad\mbox{as}\ \phi_{C}\longmapsto\nabla_{AB}\phi_{C}.$

######  Proposition 5.2.

Let us change scale $\sigma\in\Gamma\big{(}{}^{0}[1]\big{)}$ by
$\widehat{\sigma}=\Omega^{-1}\sigma$. Then, for
$\phi_{C}\in\Gamma\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}\big{)}$,

$\widehat{\nabla}_{AB}\phi_{C}=\nabla_{AB}\phi_{C}+\Upsilon_{AB}\phi_{C}-\Upsilon_{C(A}\phi_{B)},\qquad\mbox{where}\
\Upsilon_{AB}\equiv\Omega^{-1}\nabla_{AB}\Omega.$ (5.1)

###### Proof.

Given $\nabla_{AB}$ we use (5.1) to define $\widehat{\nabla}_{AB}$ and then
verify that it has the characterising properties from Proposition 5.1 for the
scale $\widehat{\sigma}$. Firstly, if
$\sigma_{CD}\in\Gamma\big{(}{}^{2}\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}\big{)}$, then

$\displaystyle\widehat{\nabla}_{AB}\sigma_{CD}=\nabla_{AB}\sigma_{CD}+2\Upsilon_{AB}\sigma_{CD}-\Upsilon_{C(A}\sigma_{B)D}+\Upsilon_{D(A}\sigma_{B)C}=\nabla_{AB}\sigma_{CD}+\Upsilon_{AB}\sigma_{CD}.\\!\\!\\!\\!\\!$
(5.2)

Therefore
$\widehat{\nabla}_{AB}\widehat{\sigma}_{CD}=\nabla_{AB}\big{(}\Omega^{-1}\sigma_{CD}\big{)}+\Upsilon_{AB}\big{(}\Omega^{-1}\sigma_{CD}\big{)}=\Omega^{-1}\nabla_{AB}\sigma_{CD}$.
Thus, if $\nabla_{AB}\sigma_{CD}=0$, then
$\widehat{\nabla}_{AB}\widehat{\sigma}_{CD}=0$, which accounts for the first
condition in Proposition 5.1. Now equation (5.2) may be abbreviated as
$\widehat{\nabla}_{AB}\sigma=\nabla_{AB}\sigma+\Upsilon_{AB}\sigma$ for
$\sigma\in\Gamma\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$
and it follows that

$\widehat{\nabla}_{AB}\rho=\nabla_{AB}\rho+w\Upsilon_{AB}\rho,\qquad\mbox{for}\
\rho\in\Gamma\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle w$}} \end{picture}\big{)}.$

Combining this with (5.1), it follows that
$\widehat{\nabla}_{AB}\phi_{C}=\nabla_{AB}\phi_{C}-\Upsilon_{C(A}\phi_{B)}$,
for
$\phi_{C}\in\Gamma\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}\,\big{)}$
and hence that

$\widehat{\nabla}_{AB}\omega_{CD}=\nabla_{AB}\omega_{CD}-\Upsilon_{C(A}\omega_{B)D}-\Upsilon_{D(A}\omega_{B)C},$

for
$\omega_{CD}\in\Gamma\big{(}\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\,\big{)}=\Gamma\big{(}{}^{1}\big{)}$. It follows that

$\widehat{\nabla}_{AB}\omega_{CD}-\widehat{\nabla}_{CD}\omega_{AB}=\nabla_{AB}\omega_{CD}-\nabla_{CD}\omega_{AB},$

which accounts for the second condition in Proposition 5.1. ∎

##  6 The Rumin complex

Contact geometry is the geometry of a maximally non-integrable corank one
subbundle $H\subset TM$, where $M$ is of dimension $2n+1$. Maximal non-
integrability is to say that locally $H$ is given as the kernel of a contact
form $\alpha$ such that $\alpha\wedge({\rm d}\alpha)^{n}$ is non-vanishing.

Alternatively, writing ${}_{H}^{1}$ for the bundle of one forms 1 restricted
naturally to $H$, and $L$ for the annihilator of $H$, we have a short exact
sequence

$0\to L\to{}^{1}\to{}_{H}^{1}\to 0$

and, therefore, short exact sequences

$0\to{}_{H}^{k-1}\otimes L\to{}^{k}\to{}_{H}^{k}\to 0$ (6.1)

for $k=1,\ldots,2n$. If we now consider the exterior derivative

$\begin{array}[]{ccc}{}_{H}^{1}&&{}_{H}^{2}\\\ \uparrow&&\uparrow\\\
{}^{1}&\xrightarrow{\ {\rm d}\ }&{}^{2}\\\ \uparrow&&\uparrow\\\
L&&{}_{H}^{1}\otimes L,\end{array}$

then, by the Leibniz rule, the composition $L\to{}^{1}\xrightarrow{\,{\rm
d}\,}{}^{2}\to{}_{H}^{2}$ is actually a vector bundle homomorphism known as
the Levi form. The maximal non-integrability condition is equivalent to the
Levi form being injective with image consisting of nondegenerate forms.

If one writes out the de Rham sequence along with the short exact sequences
(6.1), one can obtain by diagram chasing, the Rumin complex [14]. The Rumin
complex is a replacement for the de Rham complex on any contact manifold in
that it computes the de Rham cohomology, but it is in some sense more
efficient in that derivatives are only taken in contact directions. We are
concerned with the case $\dim M=5$ in which case the diagram to chase is

$\begin{array}[]{@{}ccccccccccc}&&{}_{H}^{1}&&{}_{H}^{2}&&{}_{H}^{3}&&{}_{H}^{4}&&\\\
&&\uparrow&&\uparrow&&\uparrow&&\uparrow&&\\\ {}^{0}&\hskip
8.53581pt\xrightarrow{\ {\rm d}\ }&{}^{1}&\hskip 8.53581pt\xrightarrow{\ {\rm
d}\ }&{}^{2}&\xrightarrow{\ {\rm d}\ }&{}^{3}&\xrightarrow{\ {\rm d}\
}&{}^{4}&\xrightarrow{\ {\rm d}\ }&\hskip 8.53581pt{}^{5}\\\
&&\uparrow&&\uparrow&&\uparrow&&\uparrow&&\\\ &&L&&{}_{H}^{1}\otimes
L&&{}_{H}^{2}\otimes L&&{}_{H}^{3}\otimes L,&&\end{array}$

and, writing ${}_{H}^{2}={}_{H\perp}^{2}\oplus L$, where ${}_{H\perp}^{2}$
comprises $2$-forms on $H$ that are trace-free with respect to the Levi form,
one obtains the invariantly defined complex

$\begin{array}[]{@{}ccccccccccc}{}^{0}&\hskip 5.69054pt\xrightarrow{\ {\rm
d}_{\perp}\ }&{}^{1}_{H}&\xrightarrow{\ {\rm d}_{\perp}\
}&{}^{2}_{H\perp}&\xrightarrow{\ {\rm d}_{\perp}^{(2)}\
}&{}_{H\perp}^{2}\otimes L&\xrightarrow{\ {\rm d}_{\perp}\ }&{}^{3}_{H}\otimes
L&\xrightarrow{\ {\rm d}_{\perp}\ }&\hskip 5.69054pt{}^{5}.\end{array}$

A difference to the de Rham complex is that one obtains ${\rm
d}_{\perp}^{(2)}\colon{}_{H\perp}^{2}\to{}_{H\perp}^{2}\otimes L$, which is a
second-order differential operator.

##  7 Spinors in $\boldsymbol{G_{2}}$ contact geometry

A $G_{2}$ contact geometry is an additional structure on the contact
distribution of a five-dimensional contact manifold. As observed in the
previous section, a contact geometry is naturally equipped with its Levi form
$L\to{}_{H}^{2}$ and the contact distribution $H$ thereby inherits a
nondegenerate skew form defined up to scale. This is just what is needed to
talk about Legendrean varieties [10] in the projective bundle
${\mathbb{P}}(H)\to M$. A $G_{2}$ contact structure on $M$ is a field of
Legendrean twisted cubics in ${\mathbb{P}}(H)$. Precisely, this means that,
for all $m\in M$, there is a twisted cubic $C_{m}\subset{\mathbb{P}}(H_{m})$,
varying smoothly with $m\in M$, such that the $2$-planes in $H_{m}$ covering
the tangent lines to the cubic are null for the Levi form. Equivalently, such
a $G_{2}$ contact structure may be viewed as a rank two ‘spin bundle’ S
equipped with a ‘Levi-compatible’ identification
$\bigodot^{3}\\!S={}_{H}^{1}[2]$, where ${}^{5}={}^{0}[-3]$. Levi-
compatibility means that the Levi form

$\textstyle
L[4]\to{}_{H}^{2}[4]={}^{2}\big{(}{\bigodot}^{3}\\!S\big{)}=\big{(}{\bigodot}^{4}\\!S\otimes{}^{2}S\big{)}\oplus\big{(}{}^{2}S\big{)}^{3}$

has its range in the second summand and thus provides an identification
$L[4]=\big{(}{}^{2}S\big{)}^{3}$. In these circumstances, notice that

$\textstyle{}_{H}^{4}[8]={}^{4}\big{(}{\bigodot}^{3}\\!S\big{)}=\big{(}{\wedge}^{2}S\big{)}^{6}\quad\Rightarrow\quad{}^{0}[9]={}^{5}[12]={}_{H}^{4}[8]\otimes
L[4]=\big{(}{}^{2}S\big{)}^{9}$

and, therefore, we find canonical identifications ${}^{2}S={}^{0}[1]$ and
$L={}^{0}[-1]$. In any case, the $G_{2}$ contact structure can now be
characterised by decreeing that the simple spinors in $H=\bigodot^{3}\\!S[-1]$
constitute the cone over the twisted cubic (and there is a clear analogy with
conformal spin geometry in dimension three). More detail can be found in [7,
Section 5] and the flat model is presented in [6, Section 4]. We should also
point out that the geometry of the rank four bundle $H$ follows that of
Bryant’s ‘$H_{3}$-structures’ on the tangent bundle in four dimensions [2].

The reason for the name ‘$G_{2}$ contact structure’ is that this geometric
data defines a parabolic geometry of type $(G_{2},P)$ where $G_{2}$ is the
simply-connected exceptional Lie group of split type $G_{2}$ and $P$ is a
particular parabolic subgroup such that $G_{2}/P$ is a contact manifold: see,
for example, [4, Section 4.2.8] (and, in particular, it is explained in [6]
that this particular realisation of the Lie algebra of $G_{2}$ goes back to
Engel [8]). With the Dynkin diagram notation from [1], this motivates our
writing
$\bigodot^{k}\\!S[w]=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle k$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle w$}} \end{picture}$ (just
another way of organising the irreducible representations of
${\mathrm{GL}}_{+}(2,{\mathbb{R}})$) so that

$\displaystyle\textstyle
S=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture},\qquad\bigodot^{3}\\!\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture},\qquad{}_{H}^{1}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}} \end{picture},\qquad
H=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture},\qquad{}^{0}[1]=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture},$

and the Rumin complex is

$\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\xrightarrow{\,{\rm
d}_{\perp}\,}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\xrightarrow{\,{\rm
d}_{\perp}\,}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 4$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\end{picture}\xrightarrow{\,{\rm
d}_{\perp}^{(2)}\,}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 4$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\end{picture}\xrightarrow{\,{\rm
d}_{\perp}\,}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\end{picture}\to\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}} \end{picture}$

(consistent with the basic BGG complex from [1]). Notice that

${}_{H}^{2}={}^{2}\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\,\big{)}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 4$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\end{picture}\oplus\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}$

so that the Levi form
$L=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\hookrightarrow{}^{2}\big{(}{}_{H}^{1}\big{)}$ is built into the
notation.

Regarding calculus on a contact manifold, it is natural to consider partial
connections, rather than connections, on vector bundles in which directional
derivatives are defined, in the first instance, only in the contact
directions. More precisely, a partial connection on a vector bundle $E$ is a
linear differential operator

$\nabla_{H}\colon\ E\to{}_{H}^{1}\otimes E$

satisfying a partial Leibniz rule $\nabla_{H}(fs)=f\nabla_{H}s+{\rm
d}_{\perp}f\otimes s$. (In fact, a partial connection can be uniquely promoted
[5, Proposition 3.5] to a full connection but we shall not need this trick.)
In analogy with three-dimensional spin geometry, we may construct a preferred
partial connection on $\bullet$$\times$$\langle$$\scriptstyle 1$$\scriptstyle
0$ in the presence of a ‘scale’
$\sigma\in\Gamma\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$.

The construction of this preferred partial connection follows the same route
save for a minor yet crucial distinction. For any contact manifold of
dimension $\geq 5$, a partial connection on ${}_{H}^{1}$ gives rise to a
linear differential operator
$\nabla_{\perp}\colon{}_{H}^{1}\to{}_{H\perp}^{2}$ defined as the composition

${}_{H}^{1}\xrightarrow{\,\nabla_{H}\,}{}_{H}^{1}\otimes{}_{H}^{1}\xrightarrow{\,\wedge\,}{}_{H}^{2}\to{}_{H\perp}^{2}$

with the same symbol as the invariantly defined Rumin operator. It follows
that the difference

$\nabla_{\perp}-{\rm d}_{\perp}\colon\ {}_{H}^{1}\to{}_{H\perp}^{2}$

is actually a homomorphism of vector bundles. By definition, this is the
partial torsion of a partial connection
$\nabla_{H}\colon{}_{H}^{1}\to{}_{H}^{1}\otimes{}_{H}^{1}$.

In the case of a $G_{2}$ contact structure, bearing in mind that
${}^{2}\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}$, a
partial connection on the spin bundle $\bullet$$\times$$\langle$$\scriptstyle
1$$\scriptstyle 0$ induces partial connections on all spin bundles
$\bullet$$\times$$\langle$$\scriptstyle k$$\scriptstyle w$ and, in particular,
on
${}_{H}^{1}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}} \end{picture}$. Thus, we
may ask about its partial torsion, which lies in

$\operatorname{Hom}\big{(}{}_{H}^{1},{}_{H\perp}^{2}\big{)}=\operatorname{Hom}\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture},\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 4$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\end{picture}\,\big{)}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 7$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\end{picture}\oplus\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 5$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\end{picture}\oplus\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\oplus\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}.$ (7.1)

This decomposition is crucial in characterising preferred spin connections as
follows.

######  Proposition 7.1.

Given a scale, i.e., a nowhere vanishing
$\sigma\in\Gamma\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$,
there is a unique partial connection on
$\bullet$$\times$$\langle$$\scriptstyle 1$$\scriptstyle 0$ so that

  * •

the induced partial connection on
${}^{2}\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}$
annihilates $\sigma$,

  * •

the partial torsion of the induced partial connection on
${}_{H}^{1}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}} \end{picture}$ lies in
$\bullet$$\times$$\langle$$\scriptstyle 7$$\scriptstyle-4$.

###### Proof.

The first summand in

$\operatorname{End}\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\oplus\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}$

captures the trace-free endomorphisms of
$\bullet$$\times$$\langle$$\scriptstyle 1$$\scriptstyle 0$ and it follows that
the freedom in choosing a partial connection on this spin bundle annihilating
$\sigma$ lies in

${}_{H}^{1}\otimes\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\otimes\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 5$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\end{picture}\oplus\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\oplus\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}.$ (7.2)

Comparison with (7.1) certainly suggests that all but the piece in
$\bullet$$\times$$\langle$$\scriptstyle 7$$\scriptstyle-4$ can be uniquely
eliminated. We may verify this using spinors. With the familiar conventions of
[12], let us write
$\epsilon_{AB}\in\Gamma\big{(}{}^{2}\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\big{)}\big{)}$ rather than
$\sigma\in\Gamma\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$.
Then, choosing a partial connection

$\phi_{D}\xrightarrow{\,\nabla_{H}\,}\nabla_{ABC}\phi_{D}\qquad\mbox{on}\quad\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture},$

the general partial connection on $\bullet$$\times$$\langle$$\scriptstyle
1$$\scriptstyle 0$ with $\nabla_{ABC}\epsilon_{DE}=0$ has the form

$\nabla_{ABC}\phi_{D}+\Gamma_{ABCD}{}^{E}\phi_{E},$

where $\Gamma_{ABCDE}=\Gamma_{(ABC)(DE)}$ (i.e., lying in
$\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\otimes\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}$, as in
(7.2)). By the Leibniz rule, the induced operator
$\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\to\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 4$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}} \end{picture}$ is

$\omega_{DEF}\longmapsto\nabla_{(AB}{}^{E}\omega_{CD)E}+2\Gamma_{(AB}{}^{E}{}_{C}{}^{G}\omega_{D)EG}-\Gamma_{E(AB}{}^{E}{}^{G}\omega_{CD)G}.$
(7.3)

Therefore, according to the decomposition (7.2), we should now write

$\Gamma_{ABC}{}^{DE}=\lambda_{ABC}{}^{DE}+\mu_{(AB}{}^{(D}\delta_{C)}{}^{E)}+\nu_{(A}\delta_{B}{}^{D}\delta_{C)}{}^{E},$
(7.4)

where $\lambda_{ABCDE}$ and $\mu_{ABC}$ are symmetric spinors and compute

$2\Gamma_{(AB}{}^{E}{}_{C}{}^{G}\omega_{D)EG}-\Gamma_{E(AB}{}^{E}{}^{G}\omega_{CD)G}$

for each term on the right hand side of (7.4). Clearly, this entails computing

$\Gamma_{(AB}{}^{(E}{}_{C)}{}^{G)}\qquad\mbox{and}\qquad\Gamma_{EAB}{}^{EG}.$
(7.5)

Firstly, if $\Gamma_{ABC}{}^{DE}=\lambda_{ABC}{}^{DE}$, where
$\lambda_{ABCDE}=\lambda_{(ABCDE)}$, then the second term in (7.5) vanishes so
for (7.3) we end up with

$\omega_{DEF}\longmapsto\nabla_{(AB}{}^{E}\omega_{CD)E}+2\lambda_{(ABC}{}^{EG}\omega_{D)EG},$

which is perfect for eliminating the $\bullet$$\times$$\langle$$\scriptstyle
5$$\scriptstyle-3$-component of partial torsion.

Secondly, if $\Gamma_{ABC}{}^{DE}=\mu_{(AB}{}^{(D}\delta_{C)}{}^{E)}$, where
$\mu_{ABC}=\mu_{(ABC)}$, then straightforward spinor computations show that

$\textstyle\Gamma_{(AB}{}^{(E}{}_{C)}{}^{G)}=\frac{1}{6}\mu_{(AB}{}^{(E}\delta_{C)}{}^{G)}\qquad\mbox{and}\qquad\Gamma_{EAB}{}^{EG}=\frac{5}{6}\mu_{AB}{}^{G}$

so for (7.3) we end up with

$\displaystyle\omega_{DEF}$
$\displaystyle\mapsto\nabla_{(AB}{}^{E}\omega_{CD)E}+\tfrac{1}{3}\mu_{(AB}{}^{E}\omega_{CD)E}-\tfrac{5}{6}\mu_{(AB}{}^{G}\omega_{CD)G}$
$\displaystyle=\nabla_{(AB}{}^{E}\omega_{CD)E}-\tfrac{1}{2}\mu_{(AB}{}^{E}\omega_{CD)E},$

which is perfect for eliminating the $\bullet$$\times$$\langle$$\scriptstyle
3$$\scriptstyle-2$-component of partial torsion.

Thirdly, if $\Gamma_{ABC}{}^{DE}=\nu_{(A}\delta_{B}{}^{D}\delta_{C)}{}^{E}$,
then straightforward spinor calculations yield

$\textstyle\Gamma_{(AB}{}^{(E}{}_{C)}{}^{G)}=-\frac{1}{3}\nu_{(A}\delta_{B}{}^{E}\delta_{C)}{}^{G}\qquad\mbox{and}\qquad\Gamma_{EAB}{}^{EG}=\frac{4}{3}\nu_{(A}\delta_{B)}{}^{G}$

so for (7.3) we end up with

$\textstyle\omega_{DEF}\mapsto\nabla_{(AB}{}^{E}\omega_{CD)E}-\frac{2}{3}\nu_{(A}\omega_{BCD)}-\frac{4}{3}\nu_{(A}\omega_{BCD)}=\nabla_{(AB}{}^{E}\omega_{CD)E}-2\nu_{(A}\omega_{BCD)},$

which is perfect for eliminating the $\bullet$$\times$$\langle$$\scriptstyle
1$$\scriptstyle-1$-component of partial torsion. ∎

Several remarks are in order. Firstly, the preferred connection of Proposition
7.1 is constructed by eliminating all but the
$\bullet$$\times$$\langle$$\scriptstyle 7$$\scriptstyle-4$-component of the
partial torsion of the induced partial connection on ${}_{H}^{1}$, decomposed
according to (7.1). In fact, it is clear from the proof that the component
lying in $\bullet$$\times$$\langle$$\scriptstyle 7$$\scriptstyle-4$ is the
same for any choice of partial connection on
$\bullet$$\times$$\langle$$\scriptstyle 1$$\scriptstyle 0$ and is, therefore,
an invariant of the structure. It is called the torsion of our $G_{2}$ contact
structure. In the general theory of parabolic geometry [4], this is the only
component of harmonic curvature and is therefore the only obstruction to local
flatness, i.e., to being locally isomorphic to the flat model $G_{2}/P$.
Secondly, we should point out that the spinor identities established by direct
calculation in our proof can be avoided by judicious use of Lie algebra
cohomology (as in done in [4]). Thirdly, we note that a scale, a nowhere
vanishing section $\sigma$ of $\bullet$$\times$$\langle$$\scriptstyle
0$$\scriptstyle 1$, has a nice geometric interpretation. Since
$\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}=L\hookrightarrow{}^{1}$ is the bundle of contact forms, we can
interpret $\sigma^{-1}$ as a choice of contact form. In other words, the
preferred partial connection on the spin bundle
$S=\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}$ is
obtained in the presence of a contact form.

The transformation law for preferred partial connections in $G_{2}$ contact
geometry is obtained by analogy with Proposition 5.2. Its proof will therefore
be omitted.

######  Proposition 7.2.

Let us change scale $\sigma\in\Gamma\big{(}{}^{0}[1]\big{)}$ by
$\widehat{\sigma}=\Omega^{-1}\sigma$. Then, for
$\phi_{D}\in\Gamma\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}\big{)}$,

$\widehat{\nabla}_{ABC}\phi_{D}=\nabla_{ABC}\phi_{D}+\Upsilon_{ABC}\phi_{D}-\Upsilon_{D(AB}\phi_{C)},\qquad\mbox{where}\quad\Upsilon_{ABC}\equiv\Omega^{-1}\nabla_{ABC}\Omega.$
(7.6)

As an immediate consequence of (7.6), if
$\phi_{D}\in\Gamma\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}\big{)}$,
then $\widehat{\nabla}_{(ABC}\phi_{D)}=\nabla_{(ABC}\phi_{D)}$. Furthermore,
if
$\phi_{D}\in\Gamma\big{(}\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4/3$}} \end{picture}\
\big{)}$, then

$\textstyle\widehat{\nabla}_{ABC}\phi_{D}=\nabla_{ABC}\phi_{D}-\frac{1}{3}\Upsilon_{ABC}\phi_{D}-\Upsilon_{D(AB}\phi_{C)},$

whence $\widehat{\nabla}_{AB}{}^{C}\phi_{C}=\nabla_{AB}{}^{C}\phi_{C}$. We
have found two invariant operators

$\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\longrightarrow\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 4$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\end{picture}\qquad\mbox{and}\qquad\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4/3$}}
\end{picture}\;\longrightarrow\begin{picture}(28.0,10.0)\put(4.0,1.1){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,3.5){\line(1,0){18.0}}
\put(4.0,1.5){\line(1,0){20.0}} \put(4.0,-0.5){\line(1,0){18.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-7/3$}} \end{picture}\ ,$
(7.7)

defined only in terms of the $G_{2}$ contact structure itself. Conversely, it
is easy to see that the existence of these two operators is sufficient to
define the preferred partial connection on
$\bullet$$\times$$\langle$$\scriptstyle 1$$\scriptstyle 0$ associated with a
scale and to capture the transformation law (7.6). Sadly, we have not been
able to manufacture either of the invariant operators (7.7) directly.

In three-dimensional conformal spin geometry, the transformation law (5.1)
leads to a pair of basic first-order invariant differential operators

$\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\longrightarrow\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 3$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\qquad\mbox{and}\qquad\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3/2$}}
\end{picture}\;\longrightarrow\begin{picture}(28.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(24.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(23.0,2.5){\line(-1,0){20.0}} \put(23.0,0.5){\line(-1,0){20.0}}
\put(14.0,1.5){\makebox(0.0,0.0)[]{$\langle$}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(24.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-5/2$}} \end{picture}\ ,$

given by $\phi_{C}\mapsto\nabla_{(AB}\phi_{C)}$ and
$\phi_{C}\mapsto\nabla_{AB}\phi^{B}$. This suggests that we should refer to
the operators (7.7) as the ‘twistor’ or ‘tractor’ operator and ‘Dirac’
operator, respectively, on a $G_{2}$ contact manifold. Sure enough, this
tractor operator is overdetermined and, in the flat case, has a
$7$-dimensional kernel corresponding to the embedding
$G_{2}\hookrightarrow{\mathrm{SO}}^{\uparrow}(4,3)$. The prolongation of this
tractor operator (leading to the standard tractor bundle) is detailed in [11].

##  8 Legendrean contact geometry in five dimensions

A Legendrean contact geometry in five dimensions is a $5$-dimensional contact
manifold equipped with a splitting of the contact distribution as two rank $2$
subbundles

$H=E\oplus F$

each of which is null with respect to the Levi form (so that the Levi form
reduces to a perfect pairing $E\otimes F\to L^{*}$). The flat model naturally
arises in [6, Section 2] as the moduli space of flying saucers in ‘attacking
mode’ and [6, Proposition 2.5] gives the $15$-dimensional symmetry algebra.

In the spirit of previous sections our aim will be, in the presence of a scale
(equivalently, a choice of contact form), to construct preferred partial
connections on all the natural irreducible bundles on such a geometry. As
before, these connections can be obtained by means of the canonical
differential operators present on this type of geometry. In fact, we shall
only need to examine the Rumin complex to find sufficiently many canonical
differential operators for these purposes.

A Legendrean contact geometry is a type of parabolic geometry [4, Section
4.2.3], specifically $\times$$\bullet$$\times$ in the notation of [1]. Then

$H=\begin{array}[]{c}E\\\ \oplus\\\
F\end{array}=\begin{array}[]{c}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}\\\
\oplus\\\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\end{picture}.\end{array}$

The general irreducible bundle has the form
$\times$$\bullet$$\times$$\scriptstyle u$$\scriptstyle k$$\scriptstyle v$ for
$k\in{\mathbb{Z}}_{\geq 0}$ and $u,v,\in{\mathbb{R}}$. It is convenient to
take
$S\equiv\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}$ as our
basic ‘spin’ bundle. If we also let
${}^{0}[u,v]\equiv\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}$, then
$L={}^{0}[-1,-1]$ and the general irreducible bundle is
$\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle k$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture}=\bigodot^{k}\\!S[u+k,v+k]$. Notice that

$E\otimes
F=\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\;\otimes\;\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\end{picture}=\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\oplus\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}$

and the Levi form is built into the notation as projection onto the second
summand. (Without the Dynkin diagram notation, it follows from the perfect
pairing $E\otimes F\to L^{*}$ and the canonical identification
$E^{*}=E\otimes\det E^{*}$ that

$E\otimes(\det E^{*})^{1/2}=\big{(}E\otimes(\det
E^{*})^{1/2}\big{)}^{*}=F\otimes(\det F^{*})^{1/2}$

and we may take $S\equiv E\otimes(\det E^{*})^{1/2}\otimes L^{1/2}\equiv
F\otimes(\det F^{*})^{1/2}\otimes L^{1/2}$.)

In order to avoid confusion, by default we shall write a section of
$\times$$\bullet$$\times$$\scriptstyle u$$\scriptstyle k$$\scriptstyle v$ with
lower spinor indices

$\phi_{\mbox{\scriptsize$\underbrace{AB\dots C}_{k}$}}=\phi_{(AB\dots
C)}\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle k$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}\big{)}$

with no special terminology to record the bundle of which it is a section (in
other words, we shall forgo any systematic notion of ‘weight’). Of course, we
may use the tautological identification $S=S^{*}\otimes\det S=S^{*}\otimes L$
to replace lower spinor indices by upper spinor indices (with an appropriate
change in ‘weight’ if we were to assign one) so there is no loss in using
lower indices by default. As an example of these conventions in action, we may
write the first operator ${\rm d}_{\perp}\colon{}^{0}\to{}_{H}^{1}$ in the
Rumin complex as

$f\longmapsto{\rm d}_{\perp}f\equiv\left[\\!\begin{array}[]{c}\nabla_{A}f\\\
\bar{\nabla}_{A}f\end{array}\\!\right]\in\begin{array}[]{c}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\\\\[-3.0pt] \oplus\\\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\end{array}=\begin{array}[]{c}E^{*}\\\\[-3.0pt] \oplus\\\
F^{*}\end{array}={}^{1}_{H},$ (8.1)

where $\nabla_{A}$, respectively $\bar{\nabla}_{A}$, is the directional
derivative in the $E$, respectively $F$, direction, both of which are
manifestly invariantly defined. (Although the notion of Legendrean contact
geometry pertains in any odd dimension, it is only in five dimensions that one
has the convenience of spinors and, in particular, that the bundles $E$ and
$F$ agree save for a line bundle factor.) Of course, the directional
derivatives $\nabla_{A}f$ and $\bar{\nabla}_{A}f$ end up as sections of
different bundles even though each of them has a single spinor index.

Soon (as with all parabolic geometries [3]), we shall find it convenient to
work in a particular scale, i.e., with a nowhere vanishing section
$\sigma\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$.
As with all parabolic contact structures [4, Section 4.2], we may interpret
the section $\sigma^{-1}$ of
$\,\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\,=L\hookrightarrow{}^{1}$ as a choice of contact form. For
$5$-dimensional Legendrean contact geometry, however, we may also interpret a
scale as a choice of skew spinor $\epsilon_{AB}$ by dint of

$\sigma^{-1}\in\Gamma\big{(}\,\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\,\big{)}=\Gamma\big{(}{}^{2}\big{(}\,\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\,\big{)}\big{)}=\Gamma\big{(}{}^{2}S\big{)}\ni\epsilon_{AB},$
(8.2)

which we may use to raise and lower spinor indices with the familiar
conventions of [12]. In particular, for
$\phi_{A}\in\Gamma\big{(}\,\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}} \end{picture}\,\big{)}$
and
$\psi_{A}\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}\big{)}$,
we may write

$\textstyle\sigma^{-1}\phi_{[A}\psi_{B]}=\frac{1}{2}\epsilon_{AB}\psi^{C}\phi_{C}\qquad\mbox{to
define }\psi^{C}\phi_{C}\in\Gamma\big{(}{}^{0}\big{)},$

independent of choice of $\sigma$ and identifying
$\big{(}\,\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-1$}}
\end{picture}\,\big{)}^{*}=\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}$, as
expected.

Continuing from (8.1), we may decompose the entire Rumin complex into its
constituent parts via the usual spin-bundle decompositions to obtain an array
of invariantly defined linear differential operators

$\times$$\bullet$$\times$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
0$$\times$$\bullet$$\times$$\scriptstyle-2$$\scriptstyle 1$$\scriptstyle
0$$\times$$\bullet$$\times$$\scriptstyle 0$$\scriptstyle
1$$\scriptstyle-2$$\times$$\bullet$$\times$$\scriptstyle-3$$\scriptstyle
0$$\scriptstyle 1$$\times$$\bullet$$\times$$\scriptstyle-2$$\scriptstyle
2$$\scriptstyle-2$$\times$$\bullet$$\times$$\scriptstyle 1$$\scriptstyle
0$$\scriptstyle-3$$\times$$\bullet$$\times$$\scriptstyle-4$$\scriptstyle
0$$\scriptstyle 0$$\times$$\bullet$$\times$$\scriptstyle-3$$\scriptstyle
2$$\scriptstyle-3$$\times$$\bullet$$\times$$\scriptstyle 0$$\scriptstyle
0$$\scriptstyle-4$$\times$$\bullet$$\times$$\scriptstyle-4$$\scriptstyle
1$$\scriptstyle-2$$\times$$\bullet$$\times$$\scriptstyle-2$$\scriptstyle
1$$\scriptstyle-4$$\times$$\bullet$$\times$$\scriptstyle-3$$\scriptstyle
0$$\scriptstyle-3$second-order operatorsfirst-order operatorsfirst-order
operators (8.3)

In this diagram we have omitted arrows that correspond to homomorphisms. For
example one can check that the part of the Rumin complex
$\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}} \end{picture}$ is
actually a homomorphism, and its vanishing is equivalent to the integrability
of $F$. As already noted, the operator

$\nabla_{A}\colon\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}} \end{picture}$

is just the directional derivative on functions in the $E$ direction:
$f\mapsto{\rm d}f|_{E}$. More generally, the partial connections we aim to
construct naturally split into a part that differentiates along $E$ (which we
shall denote by $\nabla_{A}$) and a part that differentiates along $F$
(denoted by $\bar{\nabla}_{A}$). In particular, amongst the invariant
operators in (8.3) we find
$\nabla_{A}\colon\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}} \end{picture}$ and,
therefore, by insisting on the Leibniz rule, invariantly defined derivatives
in the $E$ direction

$\nabla_{A}\colon\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\otimes\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture},\qquad\mbox{for all}\ v\in{\mathbb{R}}.$

Now suppose we are given a nowhere vanishing scale
$\sigma\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$.
Then we can define

$\displaystyle\nabla_{A}\colon\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\otimes\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture},\qquad\mbox{for all}\ u,v\in{\mathbb{R}}$ (8.4)

by $\nabla_{A}(f\sigma^{u})\equiv(\nabla_{A}f)\sigma^{u}$ for smooth sections
$f$ of $\times$$\bullet$$\times$$\scriptstyle 0$$\scriptstyle 0$$\scriptstyle
v-u$ . We may compute how this operator changes under a change of scale
$\widehat{\sigma}=\Omega^{-1}\sigma$, for some nowhere vanishing smooth
function $\Omega$. Firstly, note that
$\nabla_{A}(\Omega^{-u}f)=\Omega^{-u}(\nabla_{A}f-u\Upsilon_{A}f)$, where
$\Upsilon_{A}\equiv\Omega^{-1}\nabla_{A}\Omega$. Hence,

$\displaystyle\widehat{\nabla}_{A}(f\widehat{\sigma}^{u})$
$\displaystyle\equiv(\nabla_{A}f)\widehat{\sigma}^{u}=\Omega^{-u}(\nabla_{A}f)\sigma^{u}$
$\displaystyle=\nabla_{A}(\Omega^{-u}f)\sigma^{u}+u\Upsilon_{A}f\widehat{\sigma}^{u}=\nabla_{A}(\Omega^{-u}f\sigma^{u})+u\Upsilon_{A}f\widehat{\sigma}^{u}$

and, writing
$s=f\widehat{\sigma}^{u}\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}\big{)}$,
we obtain

$\widehat{\nabla}_{A}s=\nabla_{A}s+u\Upsilon_{A}s.$

In particular, this transformation law records the invariance of $\nabla_{A}$
when $u=0$. Similarly, starting with
$\bar{\nabla}_{A}\colon\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}} \end{picture}$ from
(8.3), in the presence of a scale $\sigma$, we obtain

$\bar{\nabla}_{A}\colon\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\otimes\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture},\qquad\mbox{for all}\ u,v\in{\mathbb{R}}$

and, under change of scale $\widehat{\sigma}=\Omega^{-1}\sigma$, we find that

$\widehat{\bar{\nabla}}_{A}s=\bar{\nabla}_{A}s+v\bar{\Upsilon}_{A}s,$

where $\bar{\Upsilon}_{A}\equiv\Omega^{-1}\bar{\nabla}_{A}\Omega$.

Referring back to the Rumin complex (8.3) we also have canonical differential
operators (in the $E$ direction)

$\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\qquad\mbox{and}\qquad\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture},$ (8.5)

which may be combined, via the Leibniz, rule with (8.4) to obtain, in the
presence of a scale, a first-order differential operator

$\nabla_{A}\colon\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}\to\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}\;\oplus\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u-1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v+1$}} \end{picture}\
=\;\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\otimes\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}.$

To express the transformation of this operator under change of scale, we may
split it as

$\phi_{B}\mapsto\nabla_{A}\phi_{B}=\nabla_{(A}\phi_{B)}+\nabla_{[A}\phi_{B]},$

and recall that operator (8.4) on densities
$s\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}\big{)}$
transforms as

$\widehat{\nabla}_{A}s=\nabla_{A}+u\Upsilon_{A}s,\qquad\mbox{where}\quad\Upsilon_{A}\equiv\Omega^{-1}\nabla_{A}\Omega.$
(8.6)

######  Proposition 8.1.

Suppose we change scale
$\sigma\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$ by
$\widehat{\sigma}=\Omega^{-1}\sigma$. Then, for $\phi_{B}$ a section of
$\times$$\bullet$$\times$$\scriptstyle u$$\scriptstyle 1$$\scriptstyle v$ we
have

$\widehat{\nabla}_{A}\phi_{B}=\nabla_{A}\phi_{B}+(u+1)\Upsilon_{A}\phi_{B}-\Upsilon_{B}\phi_{A}.$
(8.7)

###### Proof.

It suffices to note that this transformation law is consistent with the
invariance of the operators (8.5), which may be written as

$\phi_{B}\longmapsto\nabla_{(A}\phi_{B)}\qquad\mbox{and}\qquad\phi_{B}\longmapsto\nabla_{[A}\phi_{B]}$

and also with (8.6) on densities. ∎

Similarly, from the canonical operators

$\bar{\nabla}\colon\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 2$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\qquad\mbox{and}\qquad\bar{\nabla}\colon\
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-3$}} \end{picture},$

from (8.3) we may construct, in the presence of a scale,

$\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture}\big{)}\ni\phi_{B}\longmapsto\bar{\nabla}_{A}\phi_{B}$

differentiating in the $F$ direction and transforming by

$\widehat{\bar{\nabla}}_{A}\phi_{B}=\bar{\nabla}_{A}\phi_{B}+(v+1)\bar{\Upsilon}_{A}\phi_{B}-\bar{\Upsilon}_{B}\phi_{A},\qquad\mbox{where}\quad\bar{\Upsilon}_{A}\equiv\Omega^{-1}\bar{\nabla}_{A}\Omega.$
(8.8)

Finally, we may combine $\nabla_{A}$ and $\bar{\nabla}_{A}$ to define, in the
presence of a scale
$\sigma\in\Gamma\big{(}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}} \end{picture}\big{)}$,
partial connections

$\Gamma(\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture})\ni\phi_{B}\to\left[\\!\begin{array}[]{c}\nabla_{A}\phi_{B}\\\\[3.0pt]
\bar{\nabla}_{A}\phi_{B}\end{array}\\!\right]\in\begin{array}[]{c}\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\end{picture}\\\\[-2.0pt] \oplus\\\\[-1.0pt]
\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\end{picture}\end{array}\\!\\!\otimes\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}}
\end{picture}={}_{H}^{1}\otimes\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle u$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle v$}} \end{picture}$

and, indeed by the Leibniz rule, on all weighted spinor bundles. These partial
connections are generated by ${}_{H}^{1}\to{}_{H}^{1}\otimes{}_{H}^{1}$ and
this basic one is characterised as follows.

######  Proposition 8.2.

Let $\sigma$ be a nowhere vanishing section of
$\times$$\bullet$$\times$$\scriptstyle 1$$\scriptstyle 0$$\scriptstyle 1$,
equivalently a choice of contact form. Then, there is a unique partial
connection $\nabla_{H}\colon{}_{H}^{1}\to{}_{H}^{1}\otimes{}_{H}^{1}$ so that
the induced partial connection on
${}_{H}^{4}=\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}} \end{picture}$
annihilates $\sigma^{-2}$ and so that $\nabla_{H}$ has minimal partial torsion
in the sense that the induced operator ${}_{H}^{1}\to{}_{H\perp}^{2}$ agrees
with the Rumin operator ${\rm d}_{\perp}$ modulo the homomorphisms that are
the obstructions to integrability.

###### Proof.

Recall that we constructed this partial connection from the Rumin complex
(8.3) modulo the obstructions to integrability. The only ingredients in this
argument not immediately visible in ${\rm
d}_{\perp}\colon\wedge_{H}^{1}\to{}_{H\perp}^{2}$, where the two operators

$\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\end{picture}\to\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-2$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 1$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\end{picture}\qquad\mbox{and}\qquad\begin{picture}(40.0,11.0)\put(4.0,1.5){\makebox(0.0,0.0)[]{$\times$}}
\put(20.0,1.5){\makebox(0.0,0.0)[]{$\bullet$}}
\put(36.0,1.5){\makebox(0.0,0.0)[]{$\times$}} \put(4.0,1.5){\line(1,0){32.0}}
\put(4.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle-4$}}
\put(20.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}
\put(36.0,8.0){\makebox(0.0,0.0)[]{$\scriptstyle 0$}}