Spinors in Five-Dimensional Contact Geometry
We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely 𝐺₂ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined onl...
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| description | We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely 𝐺₂ 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 𝐺₂ contact structure on the configuration space of a flying saucer (always a five-dimensional contact manifold).
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 031, 19 pages
Spinors in Five-Dimensional Contact Geometry
Michael EASTWOOD a and Timothy MOY b
a) School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
E-mail: meastwoo@gmail.com
b) Clare College, University of Cambridge, CB2 1TL, England, UK
E-mail: tjahm2@cam.ac.uk
Received January 31, 2022, in final form April 13, 2022; Published online April 16, 2022
https://doi.org/10.3842/SIGMA.2022.031
Abstract. We use classical (Penrose) two-component spinors to set up the differential
geometry of two parabolic contact structures in five dimensions, namely G2 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 G2 contact structure on the configuration space of
a flying saucer (always a five-dimensional contact manifold).
Key words: spinors; contact geometry; parabolic geometry
2020 Mathematics Subject Classification: 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
SL(2,C) −→ SO↑(3, 1),
where SO↑(3, 1) is the identity-connected component of the Lorentz group. Similarly, in three
dimensions, the 2–1 covering
SL(2,R) −→ SO↑(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
SL(2,R) ↪→ Sp(4,R),
where Sp(4,R) is the subgroup of SL(4,R) consisting of matrices that preserve a fixed non-
degenerate skew form on R4. There are two such inclusions:
� by writing R4 =
⊙3R2, where
⊙
denotes symmetric tensor product,
� by writing R4 = R2 ⊕ R2,
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
mailto:meastwoo@gmail.com
mailto:tjahm2@cam.ac.uk
https://doi.org/10.3842/SIGMA.2022.031
https://www.emis.de/journals/SIGMA/Penrose.html
2 M. Eastwood and T. Moy
giving rise to G2 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 mani-
fold M as extra structure on the contact distribution H ⊂ TM . A contact form α is a 1-form so
that H = kerα. It gives rise to a nondegenerate skew form, the Levi form, namely dα|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 α. 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 G2 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 G2 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 ωa ∈ ∧1 instead of ωa ∈ Γ
(
∧1
)
or even ωa ∈ Γ
(
M,∧1
)
,
for a 1-form ωa. 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 in-
dependently 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)])
ωb 7−→ ∇[aωb].
For the Lie derivative on covariant 2-tensors, we have
ϕbc 7−→ (LXϕ)bc ≡ Xa∇aϕbc + (∇bX
a)ϕac + (∇cX
a)ϕba, (2.1)
for a vector fieldXa and, irrespective of the usual interpretation of LX in terms of the flow ofXa,
it easy to check that this expression does not depend on the choice of ∇a. It is convenient to
regard the right hand side of (2.1), let’s say for a symmetric tensor ϕab, as an invariantly defined
differential pairing
TM ×
⊙2∧1 −→
⊙2∧1.
Spinors in Five-Dimensional Contact Geometry 3
In particular, if gab is a semi-Riemannian metric, that is to say a nondegenerate symmetric
form, then the Lie derivative of gab, with a convenient factor of 1
2 thrown in, can be regarded as
a canonically defined linear differential operator
TM ∋ Xa 7−→ 1
2LXgbc ∈
⊙2∧1. (2.2)
Of course, the tensor gab also defines an isomorphism TM ∼= ∧1 by Xa 7→ gabX
b, and so we
have obtained an invariantly defined linear differential operator
∧1 ∼= TM −→
⊙2∧1. (2.3)
In combination with the exterior derivative d: ∧1 → ∧2, we have obtained
∧1 −→
⊙2∧1 ⊕∧2 = ∧1 ⊗∧1 (2.4)
and we claim that this is the Levi-Civita connection defined by gab. This is easy to check: if we
use the Levi-Civita connection in (2.1), then
1
2LXgab +∇[aXb] =
1
2
(
Xc∇cgab + (∇aX
c)gcb + (∇bX
c)gac
)
+∇[aXb]
= ∇(aXb) +∇[aXb] = ∇aXb,
as required. But we can play the moves (2.2), (2.3), and (2.4) to define the Levi-Civita con-
nection. 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 gab, we are given only
a conformal class of metrics. A convenient way of expressing this is to say that we are given ηab,
a nondegenerate section of
⊙2∧1 ⊗ L2 for some line bundle L. Thus, if σ is a non-vanishing
section of L, then gab ≡ σ−2ηab is a genuine metric. We shall refer to σ as a scale and, if we
choose a different section σ̂ = Ω−1σ for some nowhere vanishing function Ω, then we encounter
a rescaled metric ĝab ≡ σ̂−2ηab = Ω2gab. For a given conformal structure ηab, let us define ηab,
a section of
⊙2TM ⊗L−2, by ηacη
bc = δa
b, where δa
b is the invariant (Kronecker delta) pairing
TM ⊗∧1 → ∧0.
On an n-dimensional oriented conformal manifold, we may attempt to normalise a section
ϵab...d of the bundle ∧n, compatible with the orientation, by insisting that
ϵab...dϵef...hη
aeηbf · · · ηdh ≡ 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 ϵab...d take values in ∧n ⊗ Ln. The
normalisation (3.1) now makes good sense and ϵab...d is uniquely determined. Equivalently, we
may view ϵab...d as providing a canonical identification
L−n = ∧n given by ϕ 7→ ϕϵab...d.
It it usual to write L ≡ ∧0[1] and refer to sections of Lw ≡ ∧0[w] as conformal densities of
weight w. In the presence of a scale σ, defining a metric gab = σ−2ηab, the normalisation (3.1)
reads
(σnϵab...d)(σ
nϵef...h)g
aegbf · · · gdh = n!
4 M. Eastwood and T. Moy
so σnϵab...d is the usual volume form for the metric gab. This is consistent with the scale σ
trivialising all the density bundles ∧0[w]. To summarise, in the presence of a scale σ, a conformal
density of weight w can be regarded as an ordinary function but if we change to a new scale
σ̂ = Ω−1σ, then the same density is represented by a new function f̂ = Ωwf .
Once we have decided that ∧0[1] = L = (∧n)−1/n, the Lie derivative LX : ∧1 → ∧1 in-
duces invariantly defined differential pairings TM ×B → B for all conformally weighted tensor
bundles B and, in particular,
TM ×
⊙2∧1[2] →
⊙2∧1[2].
Recall that a conformal structure is a nondegenerate section ηab of this bundle
⊙2∧1[2] and
hence we obtain a conformally invariant first-order linear differential operator
TM ∋ Xa 7−→ LXηab ∈
⊙2∧1[2]. (3.2)
But recall that ϵab...d is the canonical section of ∧n[n] = ∧0 corresponding to the constant
function 1. It follows that LXϵab...d = 0 for any vector field Xa and therefore, from (3.1), that
ηabLXηab = 0. In addition, we may use the conformal metric ηab 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
∧1[2] −→
⊙2
◦∧
1[2],
where ◦ denotes the trace-free part (a manifestly conformally invariant notion). If ϕb...d is an
(n− 1)-form then
ϵab...dη
be · · · ηdgϕe...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 d: ∧n−1 → ∧n may be viewed as an invariant differential operator
∧1[2− n] → ∧0[−n].
Together with the exterior derivative d: ∧1 → ∧2, we now have three conformally invariant
linear differential operators defined on variously weighted 1-forms, namely
∧1[2] →
⊙2
◦∧
1[2], ∧1[2− n] → ∧0[−n], ∧1 → ∧2. (3.3)
As in Section 2, we are now in a position to define a preferred connection in the presence of
a scale σ ∈ Γ(∧0[1]). Specifically, we may drop all the weights in (3.3) and combine them to
obtain
∧1 ∇−→ ∧1 ⊗∧1 =
⊙2
◦∧
1 ⊕∧0 ⊕∧2.
The three bundles on the right constitute the orthogonal decomposition of ∧1 ⊗ ∧1 into irre-
ducibles with respect to the metric gab = σ−2ηab. Of course, it is easily verified that we have
found the Levi-Civita connection for gab 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 σ 7→ σ̂ = Ω−1σ. 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 σ ∈ Γ
(
∧0[1]
)
, is to require that ∇aσ = 0. This is consistent with the
Spinors in Five-Dimensional Contact Geometry 5
connections induced on the line bundles ∧0[w] in the sense that, for ϕ ∈ Γ
(
∧0[w]
)
, we have
∇ϕ = σwd(σ−wϕ). In particular, for a new scale σ̂ = Ω−1σ and ϕ ∈ Γ
(
∧0[w]
)
, we have
∇̂ϕ = σ̂wd
(
σ̂−wϕ
)
= Ω−wσwd
(
Ωwσ−wϕ
)
= σwd
(
σ−wϕ
)
+ wΩ−1(dΩ)ϕ = ∇ϕ+ wΥϕ,
where Υ ≡ Ω−1dΩ. It follows that, regarding the change in the Levi-Civita connection on
a 1-form ωb, we have
∇̂aωb = ∇aωb −Υaωb −Υbωa +Υcωcgab
since then, if ωb has conformal weight w, we deduce that
∇̂aωb = ∇aωb + (w − 1)Υaωb −Υbωa +Υcωcgab,
which is exactly so that the three operators (3.3), namely
ϕb 7→ ∇(aϕb) − 1
n(∇
cϕc)gab, ϕb 7→ ∇bϕb, ϕb 7→ ∇[aϕb],
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
× • •⟩
a b c
, where a ∈ R and b, c ∈ Z≥0
and for tensor bundles we restrict c to be even. (Roughly speaking, this indicates a (complex)
irreducible representation of R>0×Spin(5) with the real number a over the crossed node recording
the action of R>0.) The de Rham complex is
× • •⟩
0 0 0 → × • •⟩
−2 1 0 → × • •⟩
−3 0 2 → × • •⟩
−4 0 2 → × • •⟩
−5 1 0 → × • •⟩
−5 0 0
(3.4)
and the three conformally invariant first-order linear differential operators on weighted 1-forms
are
× • •⟩
0 1 0 → × • •⟩
−2 2 0
, × • •⟩
−5 1 0 → × • •⟩
−5 0 0
, × • •⟩
−2 1 0 → × • •⟩
−3 0 2
.
Notice that ∧0[w] = × • •⟩
w 0 0
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 T is a four-dimensional real vector space equipped with
a nondegenerate skew form ϵαβ. Let us write ϵ
αβ for its inverse so that ϵαβϵ
αγ = δα
γ . The vector
space ∧2T naturally splits:
∧2T =
{
Pαβ | ϵαβPαβ = 0
}
⊕ {Pαβ = λϵαβ} ≡ ∧2
⊥T⊕ R. (4.1)
Otherwise said, if we let Sp(4,R) denote the linear automorphisms of T ∼= R4 preserving the
symplectic form ϵαβ, then (4.1) is the decomposition of ∧2T into Sp(4,R)-irreducibles. The
5-dimensional vector space ∧2
⊥T is acquires a split signature metric
∥Pαβ∥2 ≡ PαβPγηϵ
αγϵβη.
6 M. Eastwood and T. Moy
Otherwise said, we have constructed the isomorphism Sp(4,R) ∼= Spin(3, 2). According to the
Plücker relations, the null vectors in ∧2
⊥T are the decomposable tensors.
With the conventions of [1], the bundle version of this discussion yields the splitting
∧2
(
× • •⟩
0 0 1 )
= × • •⟩
0 1 0 ⊕× • •⟩
1 0 0
= ∧1[2]⊕∧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 × • •⟩
0 0 1
and equipped with
� a nondegenerate section of ∧2
(
× • •⟩
0 0 1 )
with values in ∧0[−1],
� an identification ∧2
⊥
(
× • •⟩
0 0 1 )
= ∧1[2].
A scale σ ∈ Γ
(
∧0[1]
)
induces a nondegenerate section of ∧2
(
× • •⟩
0 0 1 )
(and, according to the
discussion in Section 3, a compatible nondegenerate (split signature) symmetric form on ∧1).
The first summand in the splitting
End
(
× • •⟩
0 0 1 )
= × • •⟩
−1 0 2 ⊕× • •⟩
−1 1 0 ⊕× • •⟩
0 0 0
(4.2)
captures the trace-free endomorphisms of × • •⟩
0 0 1
that preserve its conformal skew form
(
and,
simultaneously, the second summand in
End
(
∧1
)
= End
(
× • •⟩
−2 1 0 )
= × • •⟩
−2 2 0 ⊕× • •⟩
−1 0 2 ⊕× • •⟩
0 0 0
plays the same rôle with respect to the conformal metric on ∧1
)
. From this viewpoint, a scale
σ ∈ Γ
(
∧0[1]
)
gives rise to a connection on the spin bundle × • •⟩
0 0 1
as follows. We firstly insist
that this connection preserve its conformal skew form and also that the induced connection
on ∧0[1] annihilate σ. According to (4.2), the freedom in choosing such a connection lies in
∧1 ⊗× • •⟩
−1 0 2
= × • •⟩
−2 1 0 ⊗× • •⟩
−1 0 2
. (4.3)
On the other hand, the induced operator ∇ : ∧1 → ∧2 differs from the exterior derivative by
a homomorphism ∧1 → ∧2 (it is the torsion of the induced affine connection) and, according
to (3.4), lies in
Hom
(
∧1,∧2
)
= × • •⟩
0 1 0 ⊗× • •⟩
−3 0 2
. (4.4)
Comparing the bundles (4.3) and (4.4) and noting that they are canonically isomorphic suggests
that the homomorphism ∧1 → ∧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
⊙2S = ∧1[2] (cf. [15]). With the Dynkin diagram
notation from [1],
S = • ×⟨1 0 ,
⊙2 • ×⟨1 0 = • ×⟨2 0 , ∧1 = • ×⟨2 −2
, ∧0[1] = • ×⟨0 1 ,
and the de Rham complex is
∧0 = • ×⟨0 0 → • ×⟨2 −2→ • ×⟨2 −3→ • ×⟨0 −3
= ∧3.
Spinors in Five-Dimensional Contact Geometry 7
The conformal structure can now be characterised by decreeing that the simple spinors in • ×⟨2 0
are the null vectors in ∧1[2] = • ×⟨2 0 .
The first summand in
End
(
• ×⟨1 0 ) = • ×⟨2 −1⊕ • ×⟨0 0
captures the trace-free endomorphisms of • ×⟨1 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
(
• ×⟨1 0 ) = • ×⟨0 1 , lies in
∧1 ⊗ • ×⟨2 −1
= • ×⟨2 −2⊗ • ×⟨2 −1
.
On the other hand the torsion of the induced affine connection lies in the canonically isomorphic
bundle
Hom
(
∧1,∧2
)
= • ×⟨2 0 ⊗ • ×⟨2 −3
.
It is easy to check that the freedom in choice of connection on • ×⟨1 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 σ ∈ Γ
(
• ×⟨0 1 ), there is a unique
connection on • ×⟨1 0 so that
� the induced connection on ∧2
(
• ×⟨1 0 ) = • ×⟨0 1 annihilates σ,
� the torsion of the induced connection on ∧1 = • ×⟨2 −2
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
∇ : • ×⟨1 0 −→ • ×⟨2 −2⊗ • ×⟨1 0 as ϕC 7−→ ∇ABϕC .
Proposition 5.2. Let us change scale σ ∈ Γ
(
∧0[1]
)
by σ̂ = Ω−1σ. Then, for ϕC ∈ Γ
(
• ×⟨1 0 ),
∇̂ABϕC = ∇ABϕC +ΥABϕC −ΥC(AϕB), where ΥAB ≡ Ω−1∇ABΩ. (5.1)
Proof. Given ∇AB we use (5.1) to define ∇̂AB and then verify that it has the characterising
properties from Proposition 5.1 for the scale σ̂. Firstly, if σCD ∈ Γ
(
∧2
(
• ×⟨1 0 )), then
∇̂ABσCD = ∇ABσCD + 2ΥABσCD −ΥC(AσB)D +ΥD(AσB)C = ∇ABσCD +ΥABσCD.(5.2)
Therefore ∇̂ABσ̂CD = ∇AB
(
Ω−1σCD
)
+ΥAB
(
Ω−1σCD
)
= Ω−1∇ABσCD. Thus, if ∇ABσCD = 0,
then ∇̂ABσ̂CD = 0, which accounts for the first condition in Proposition 5.1. Now equation (5.2)
may be abbreviated as ∇̂ABσ = ∇ABσ +ΥABσ for σ ∈ Γ
(
• ×⟨0 1 ) and it follows that
∇̂ABρ = ∇ABρ+ wΥABρ, for ρ ∈ Γ
(
• ×⟨0 w)
.
Combining this with (5.1), it follows that ∇̂ABϕC = ∇ABϕC − ΥC(AϕB), for ϕC ∈ Γ
(
• ×⟨1 −1)
and hence that
∇̂ABωCD = ∇ABωCD −ΥC(AωB)D −ΥD(AωB)C ,
for ωCD ∈ Γ
(
• ×⟨2 −2)
= Γ
(
∧1
)
. It follows that
∇̂ABωCD − ∇̂CDωAB = ∇ABωCD −∇CDωAB,
which accounts for the second condition in Proposition 5.1. ■
8 M. Eastwood and T. Moy
6 The Rumin complex
Contact geometry is the geometry of amaximally non-integrable corank one subbundleH ⊂ 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 α such that α ∧ (dα)n is non-vanishing.
Alternatively, writing ∧1
H 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 → L→ ∧1 → ∧1
H → 0
and, therefore, short exact sequences
0 → ∧k−1
H ⊗ L→ ∧k → ∧k
H → 0 (6.1)
for k = 1, . . . , 2n. If we now consider the exterior derivative
∧1
H ∧2
H
↑ ↑
∧1 d−−→ ∧2
↑ ↑
L ∧1
H ⊗ L,
then, by the Leibniz rule, the composition L → ∧1 d−→ ∧2 → ∧2
H 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 dimM = 5 in which case the diagram to chase is
∧1
H ∧2
H ∧3
H ∧4
H
↑ ↑ ↑ ↑
∧0 d−−→ ∧1 d−−→ ∧2 d−−→ ∧3 d−−→ ∧4 d−−→ ∧5
↑ ↑ ↑ ↑
L ∧1
H ⊗ L ∧2
H ⊗ L ∧3
H ⊗ L,
and, writing ∧2
H = ∧2
H⊥⊕L, where ∧2
H⊥ comprises 2-forms on H that are trace-free with respect
to the Levi form, one obtains the invariantly defined complex
∧0 d⊥−−−→ ∧1
H
d⊥−−−→ ∧2
H⊥
d
(2)
⊥−−−→ ∧2
H⊥ ⊗ L
d⊥−−−→ ∧3
H ⊗ L
d⊥−−−→ ∧5.
A difference to the de Rham complex is that one obtains d
(2)
⊥ : ∧2
H⊥ → ∧2
H⊥ ⊗ L, which is
a second-order differential operator.
7 Spinors in G2 contact geometry
AG2 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→ ∧2
H and the contact distributionH thereby inherits a nondegenerate skew
form defined up to scale. This is just what is needed to talk about Legendrean varieties [10] in
Spinors in Five-Dimensional Contact Geometry 9
the projective bundle P(H) →M . A G2 contact structure on M is a field of Legendrean twisted
cubics in P(H). Precisely, this means that, for all m ∈M , there is a twisted cubic Cm ⊂ P(Hm),
varying smoothly with m ∈ M , such that the 2-planes in Hm covering the tangent lines to the
cubic are null for the Levi form. Equivalently, such a G2 contact structure may be viewed as
a rank two ‘spin bundle’ S equipped with a ‘Levi-compatible’ identification
⊙3S = ∧1
H [2], where
∧5 = ∧0[−3]. Levi-compatibility means that the Levi form
L[4] → ∧2
H [4] = ∧2
(⊙3S
)
=
(⊙4S ⊗∧2S
)
⊕
(
∧2S
)3
has its range in the second summand and thus provides an identification L[4] =
(
∧2S
)3
. In
these circumstances, notice that
∧4
H [8] = ∧4
(⊙3S
)
=
(
∧2S
)6 ⇒ ∧0[9] = ∧5[12] = ∧4
H [8]⊗ L[4] =
(
∧2S
)9
and, therefore, we find canonical identifications ∧2S = ∧0[1] and L = ∧0[−1]. In any case,
the G2 contact structure can now be characterised by decreeing that the simple spinors in
H =
⊙3S[−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 ‘H3-structures’ on the tangent bundle in four
dimensions [2].
The reason for the name ‘G2 contact structure’ is that this geometric data defines a parabolic
geometry of type (G2, P ) where G2 is the simply-connected exceptional Lie group of split type G2
and P is a particular parabolic subgroup such that G2/P is a contact manifold: see, for exam-
ple, [4, Section 4.2.8] (and, in particular, it is explained in [6] that this particular realisation
of the Lie algebra of G2 goes back to Engel [8]). With the Dynkin diagram notation from [1],
this motivates our writing
⊙kS[w] = • ×⟨k w
(just another way of organising the irreducible
representations of GL+(2,R)) so that
S = • ×⟨1 0 ,
⊙3 • ×⟨1 0 = • ×⟨3 0 , ∧1
H = • ×⟨3 −2
, H = • ×⟨3 −1
, ∧0[1] = • ×⟨0 1 ,
and the Rumin complex is
• ×⟨0 0 d⊥−−→ • ×⟨3 −2 d⊥−−→ • ×⟨4 −3 d
(2)
⊥−−−→ • ×⟨4 −4 d⊥−−→ • ×⟨3 −4→ • ×⟨0 −3
(consistent with the basic BGG complex from [1]). Notice that
∧2
H = ∧2
(
• ×⟨3 −2)
= • ×⟨4 −3⊕ • ×⟨0 −1
so that the Levi form L = • ×⟨0 −1
↪→ ∧2
(
∧1
H
)
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
∇H : E → ∧1
H ⊗ E
satisfying a partial Leibniz rule ∇H(fs) = f∇Hs+d⊥f⊗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 • ×⟨1 0 in the presence of a ‘scale’ σ ∈ Γ
(
• ×⟨0 1 ).
10 M. Eastwood and T. Moy
The construction of this preferred partial connection follows the same route save for a minor
yet crucial distinction. For any contact manifold of dimension ≥ 5, a partial connection on ∧1
H
gives rise to a linear differential operator ∇⊥ : ∧1
H → ∧2
H⊥ defined as the composition
∧1
H
∇H−−−→ ∧1
H ⊗∧1
H
∧−→ ∧2
H → ∧2
H⊥
with the same symbol as the invariantly defined Rumin operator. It follows that the difference
∇⊥ − d⊥ : ∧1
H → ∧2
H⊥
is actually a homomorphism of vector bundles. By definition, this is the partial torsion of
a partial connection ∇H : ∧1
H → ∧1
H ⊗∧1
H .
In the case of a G2 contact structure, bearing in mind that ∧2
(
• ×⟨1 0 ) = • ×⟨0 1 , a partial
connection on the spin bundle • ×⟨1 0 induces partial connections on all spin bundles • ×⟨k w
and,
in particular, on ∧1
H = • ×⟨3 −2
. Thus, we may ask about its partial torsion, which lies in
Hom
(
∧1
H ,∧
2
H⊥
)
= Hom
(
• ×⟨3 −2
, • ×⟨4 −3)
= • ×⟨7 −4⊕ • ×⟨5 −3⊕ • ×⟨3 −2⊕ • ×⟨1 −1
. (7.1)
This decomposition is crucial in characterising preferred spin connections as follows.
Proposition 7.1. Given a scale, i.e., a nowhere vanishing σ ∈ Γ
(
• ×⟨0 1 ), there is a unique
partial connection on • ×⟨1 0 so that
� the induced partial connection on ∧2
(
• ×⟨1 0 ) = • ×⟨0 1 annihilates σ,
� the partial torsion of the induced partial connection on ∧1
H = • ×⟨3 −2
lies in • ×⟨7 −4
.
Proof. The first summand in
End
(
• ×⟨1 0 ) = • ×⟨2 −1⊕ • ×⟨0 0
captures the trace-free endomorphisms of • ×⟨1 0 and it follows that the freedom in choosing
a partial connection on this spin bundle annihilating σ lies in
∧1
H ⊗ • ×⟨2 −1
= • ×⟨3 −2⊗ • ×⟨2 −1
= • ×⟨5 −3⊕ • ×⟨3 −2⊕ • ×⟨1 −1
. (7.2)
Comparison with (7.1) certainly suggests that all but the piece in • ×⟨7 −4
can be uniquely elim-
inated. We may verify this using spinors. With the familiar conventions of [12], let us write
ϵAB ∈ Γ
(
∧2
(
• ×⟨1 0 )) rather than σ ∈ Γ
(
• ×⟨0 1 ). Then, choosing a partial connection
ϕD
∇H−−−→ ∇ABCϕD on • ×⟨1 0 ,
the general partial connection on • ×⟨1 0 with ∇ABCϵDE = 0 has the form
∇ABCϕD + ΓABCD
EϕE ,
where ΓABCDE = Γ(ABC)(DE) (i.e., lying in • ×⟨3 −2⊗ • ×⟨2 −1
, as in (7.2)). By the Leibniz rule, the
induced operator • ×⟨3 −2→ • ×⟨4 −3
is
ωDEF 7−→ ∇(AB
EωCD)E + 2Γ(AB
E
C
GωD)EG − ΓE(AB
EGωCD)G. (7.3)
Therefore, according to the decomposition (7.2), we should now write
ΓABC
DE = λABC
DE + µ(AB
(DδC)
E) + ν(AδB
DδC)
E , (7.4)
Spinors in Five-Dimensional Contact Geometry 11
where λABCDE and µABC are symmetric spinors and compute
2Γ(AB
E
C
GωD)EG − ΓE(AB
EGωCD)G
for each term on the right hand side of (7.4). Clearly, this entails computing
Γ(AB
(E
C)
G) and ΓEAB
EG. (7.5)
Firstly, if ΓABC
DE = λABC
DE , where λABCDE = λ(ABCDE), then the second term in (7.5)
vanishes so for (7.3) we end up with
ωDEF 7−→ ∇(AB
EωCD)E + 2λ(ABC
EGωD)EG,
which is perfect for eliminating the • ×⟨5 −3
-component of partial torsion.
Secondly, if ΓABC
DE = µ(AB
(DδC)
E), where µABC = µ(ABC), then straightforward spinor
computations show that
Γ(AB
(E
C)
G) = 1
6µ(AB
(EδC)
G) and ΓEAB
EG = 5
6µAB
G
so for (7.3) we end up with
ωDEF 7→ ∇(AB
EωCD)E + 1
3µ(AB
EωCD)E − 5
6µ(AB
GωCD)G
= ∇(AB
EωCD)E − 1
2µ(AB
EωCD)E ,
which is perfect for eliminating the • ×⟨3 −2
-component of partial torsion.
Thirdly, if ΓABC
DE = ν(AδB
DδC)
E , then straightforward spinor calculations yield
Γ(AB
(E
C)
G) = −1
3ν(AδB
EδC)
G and ΓEAB
EG = 4
3ν(AδB)
G
so for (7.3) we end up with
ωDEF 7→ ∇(AB
EωCD)E − 2
3ν(AωBCD) − 4
3ν(AωBCD) = ∇(AB
EωCD)E − 2ν(AωBCD),
which is perfect for eliminating the • ×⟨1 −1
-component of partial torsion. ■
Several remarks are in order. Firstly, the preferred connection of Proposition 7.1 is con-
structed by eliminating all but the • ×⟨7 −4
-component of the partial torsion of the induced partial
connection on ∧1
H , decomposed according to (7.1). In fact, it is clear from the proof that the
component lying in • ×⟨7 −4
is the same for any choice of partial connection on • ×⟨1 0 and is,
therefore, an invariant of the structure. It is called the torsion of our G2 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 G2/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 σ of • ×⟨0 1 , has a nice geomet-
ric interpretation. Since • ×⟨0 −1
= L ↪→ ∧1 is the bundle of contact forms, we can interpret σ−1
as a choice of contact form. In other words, the preferred partial connection on the spin bundle
S = • ×⟨1 0 is obtained in the presence of a contact form.
The transformation law for preferred partial connections in G2 contact geometry is obtained
by analogy with Proposition 5.2. Its proof will therefore be omitted.
Proposition 7.2. Let us change scale σ ∈ Γ
(
∧0[1]
)
by σ̂ = Ω−1σ. Then, for ϕD ∈ Γ
(
• ×⟨1 0 ),
∇̂ABCϕD = ∇ABCϕD +ΥABCϕD −ΥD(ABϕC), where ΥABC ≡ Ω−1∇ABCΩ. (7.6)
12 M. Eastwood and T. Moy
As an immediate consequence of (7.6), if ϕD ∈ Γ
(
• ×⟨1 0 ), then ∇̂(ABCϕD) = ∇(ABCϕD).
Furthermore, if ϕD ∈ Γ
(
• ×⟨1 −4/3)
, then
∇̂ABCϕD = ∇ABCϕD − 1
3ΥABCϕD −ΥD(ABϕC),
whence ∇̂AB
CϕC = ∇AB
CϕC . We have found two invariant operators
• ×⟨1 0 −→ • ×⟨4 −3
and • ×⟨1 −4/3−→ • ×⟨2 −7/3
, (7.7)
defined only in terms of the G2 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 • ×⟨1 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
• ×⟨1 0 −→ • ×⟨3 −2
and • ×⟨1 −3/2−→ • ×⟨1 −5/2
,
given by ϕC 7→ ∇(ABϕC) and ϕC 7→ ∇ABϕ
B. This suggests that we should refer to the opera-
tors (7.7) as the ‘twistor’ or ‘tractor’ operator and ‘Dirac’ operator, respectively, on a G2 contact
manifold. Sure enough, this tractor operator is overdetermined and, in the flat case, has a 7-
dimensional kernel corresponding to the embedding G2 ↪→ SO↑(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 ⊕ F
each of which is null with respect to the Levi form (so that the Levi form reduces to a perfect
pairing E ⊗ F → 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
× • × in the notation of [1]. Then
H =
E
⊕
F
=
× • ×1 1 −1
⊕
× • ×−1 1 1 .
The general irreducible bundle has the form × • ×
u k v
for k ∈ Z≥0 and u, v,∈ R. It is convenient
to take S ≡ × • ×−1 1 −1
as our basic ‘spin’ bundle. If we also let ∧0[u, v] ≡ × • ×
u 0 v
, then
L = ∧0[−1,−1] and the general irreducible bundle is × • ×
u k v
=
⊙kS[u+ k, v+ k]. Notice that
E ⊗ F = × • ×1 1 −1 ⊗ × • ×−1 1 1 = × • ×0 2 0 ⊕× • ×1 0 1
Spinors in Five-Dimensional Contact Geometry 13
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⊗F → L∗ and the canonical
identification E∗ = E ⊗ detE∗ that
E ⊗ (detE∗)1/2 =
(
E ⊗ (detE∗)1/2
)∗
= F ⊗ (detF ∗)1/2
and we may take S ≡ E ⊗ (detE∗)1/2 ⊗ L1/2 ≡ F ⊗ (detF ∗)1/2 ⊗ L1/2.
)
In order to avoid confusion, by default we shall write a section of × • ×
u k v
with lower spinor
indices
ϕAB . . . C︸ ︷︷ ︸
k
= ϕ(AB...C) ∈ Γ
(
× • ×
u k v )
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∗ ⊗ detS = S∗ ⊗ 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
d⊥ : ∧0 → ∧1
H in the Rumin complex as
f 7−→ d⊥f ≡
[
∇Af
∇̄Af
]
∈
× • ×−2 1 0
⊕
× • ×0 1 −2
=
E∗
⊕
F ∗
= ∧1
H , (8.1)
where ∇A, respectively ∇̄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 ∇Af and ∇̄Af 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 σ ∈ Γ
(
× • ×1 0 1 ). As with all parabolic contact
structures [4, Section 4.2], we may interpret the section σ−1 of × • ×−1 0 −1
= L ↪→ ∧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 ϵAB by dint of
σ−1 ∈ Γ
(
× • ×−1 0 −1)
= Γ
(
∧2
(
× • ×−1 1 −1))
= Γ
(
∧2S
)
∋ ϵAB, (8.2)
which we may use to raise and lower spinor indices with the familiar conventions of [12]. In
particular, for ϕA ∈ Γ
(
× • ×−1 1 −1)
and ψA ∈ Γ
(
× • ×0 1 0 ), we may write
σ−1ϕ[AψB] =
1
2ϵABψ
CϕC to define ψCϕC ∈ Γ
(
∧0
)
,
independent of choice of σ and identifying
(
× • ×−1 1 −1)∗
= × • ×0 1 0 , 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
× • ×0 0 0
× • ×−2 1 0
× • ×0 1 −2
× • ×−3 0 1
× • ×−2 2 −2
× • ×1 0 −3
× • ×−4 0 0
× • ×−3 2 −3
× • ×0 0 −4
× • ×−4 1 −2
× • ×−2 1 −4
× • ×−3 0 −3���*
H
HHj
���*
H
HHj
���*
H
HHj
-
-
-
S
S
SSw
S
S
SSw�
�
��7
S
S
SSw�
�
��7
HHHj
���*
H
HHj
���*
H
HHj
���*
second-order operators
6
first-order operators
6
6
first-order operators
6
6
(8.3)
14 M. Eastwood and T. Moy
In this diagram we have omitted arrows that correspond to homomorphisms. For example one
can check that the part of the Rumin complex × • ×−2 1 0 → × • ×1 0 −3
is actually a homomorphism,
and its vanishing is equivalent to the integrability of F . As already noted, the operator
∇A : × • ×0 0 0 → × • ×−2 1 0
is just the directional derivative on functions in the E direction: f 7→ df |E . More generally, the
partial connections we aim to construct naturally split into a part that differentiates along E
(which we shall denote by ∇A) and a part that differentiates along F (denoted by ∇̄A). In
particular, amongst the invariant operators in (8.3) we find ∇A : × • ×0 0 −4 → × • ×−2 1 −4
and,
therefore, by insisting on the Leibniz rule, invariantly defined derivatives in the E direction
∇A : × • ×0 0 v → × • ×−2 1 0 ⊗× • ×0 0 v
, for all v ∈ R.
Now suppose we are given a nowhere vanishing scale σ ∈ Γ
(
× • ×1 0 1 ). Then we can define
∇A : × • ×
u 0 v → × • ×−2 1 0 ⊗× • ×
u 0 v
, for all u, v ∈ R (8.4)
by ∇A(fσ
u) ≡ (∇Af)σ
u for smooth sections f of × • ×0 0 v−u
. We may compute how this operator
changes under a change of scale σ̂ = Ω−1σ, for some nowhere vanishing smooth function Ω.
Firstly, note that ∇A(Ω
−uf) = Ω−u(∇Af − uΥAf), where ΥA ≡ Ω−1∇AΩ. Hence,
∇̂A(fσ̂
u) ≡ (∇Af)σ̂
u = Ω−u(∇Af)σ
u
= ∇A(Ω
−uf)σu + uΥAfσ̂
u = ∇A(Ω
−ufσu) + uΥAfσ̂
u
and, writing s = fσ̂u ∈ Γ
(
× • ×
u 0 v )
, we obtain
∇̂As = ∇As+ uΥAs.
In particular, this transformation law records the invariance of ∇A when u = 0. Similarly,
starting with ∇̄A : × • ×−4 0 0 → × • ×−4 1 −2
from (8.3), in the presence of a scale σ, we obtain
∇̄A : × • ×
u 0 v → × • ×0 1 −2⊗× • ×
u 0 v
, for all u, v ∈ R
and, under change of scale σ̂ = Ω−1σ, we find that̂̄∇As = ∇̄As+ vῩAs,
where ῩA ≡ Ω−1∇̄AΩ.
Referring back to the Rumin complex (8.3) we also have canonical differential operators (in
the E direction)
× • ×0 1 −2→ × • ×−2 2 −2
and × • ×−2 1 0 → × • ×−3 0 1 , (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
∇A : × • ×
u 1 v → × • ×u−2 2 v ⊕ × • ×u−1 0 v+1
= × • ×−2 1 0 ⊗× • ×
u 1 v
.
To express the transformation of this operator under change of scale, we may split it as
ϕB 7→ ∇AϕB = ∇(AϕB) +∇[AϕB],
and recall that operator (8.4) on densities s ∈ Γ
(
× • ×
u 0 v )
transforms as
∇̂As = ∇A + uΥAs, where ΥA ≡ Ω−1∇AΩ. (8.6)
Spinors in Five-Dimensional Contact Geometry 15
Proposition 8.1. Suppose we change scale σ ∈ Γ
(
× • ×1 0 1 ) by σ̂ = Ω−1σ. Then, for ϕB
a section of × • ×
u 1 v
we have
∇̂AϕB = ∇AϕB + (u+ 1)ΥAϕB −ΥBϕA. (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
ϕB 7−→ ∇(AϕB) and ϕB 7−→ ∇[AϕB]
and also with (8.6) on densities. ■
Similarly, from the canonical operators
∇̄ : × • ×−2 1 0 → × • ×−2 2 −2
and ∇̄ : × • ×0 1 −2→ × • ×1 0 −3
,
from (8.3) we may construct, in the presence of a scale,
Γ
(
× • ×
u 1 v ) ∋ ϕB 7−→ ∇̄AϕB
differentiating in the F direction and transforming by
̂̄∇AϕB = ∇̄AϕB + (v + 1)ῩAϕB − ῩBϕA, where ῩA ≡ Ω−1∇̄AΩ. (8.8)
Finally, we may combine ∇A and ∇̄A to define, in the presence of a scale σ ∈ Γ
(
× • ×1 0 1 ),
partial connections
Γ(× • ×
u 1 v
) ∋ ϕB →
[
∇AϕB
∇̄AϕB
]
∈
× • ×−2 1 0
⊕
× • ×0 1 −2
⊗× • ×
u 1 v
= ∧1
H ⊗× • ×
u 1 v
and, indeed by the Leibniz rule, on all weighted spinor bundles. These partial connections are
generated by ∧1
H → ∧1
H ⊗∧1
H and this basic one is characterised as follows.
Proposition 8.2. Let σ be a nowhere vanishing section of × • ×1 0 1 , equivalently a choice of
contact form. Then, there is a unique partial connection ∇H : ∧1
H → ∧1
H⊗∧1
H so that the induced
partial connection on ∧4
H = × • ×−2 0 −2
annihilates σ−2 and so that ∇H has minimal partial torsion
in the sense that the induced operator ∧1
H → ∧2
H⊥ agrees with the Rumin operator d⊥ 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 d⊥ : ∧1
H → ∧2
H⊥, where the two operators
× • ×0 0 −4→ × • ×−2 1 −4
and × • ×−4 0 0 → × • ×−4 1 −2
coming from further along the Rumin complex. However, it is straightforward to check that, for
example,
× • ×0 0 −4
= ∧5 ⊗
(
× • ×−3 0 1 )∗ −→ ∧5 ⊗
(
× • ×−2 1 0 )∗ = × • ×−2 1 −4
is the adjoint of × • ×−2 1 0 → × • ×−3 1 0 so these two hidden ingredients are also secretly carried
by d⊥ : ∧1
H → ∧2
H⊥. ■
16 M. Eastwood and T. Moy
In fact, this proposition also follows from the general theory [4], or by a more explicit spinor
calculation [11]. The prolongation of × • ×0 1 0 ∋ ϕB 7−→
(
∇(AϕB), ∇̄(AϕB)
)
gives an especially
convenient tractor bundle and its (partial) connection (cf. [4, 11]).
A more familiar way [12] of saying that ∇A annihilates the scale σ ∈ Γ
(
× • ×1 0 1 ) and hence
σ−1 ∈ Γ
(
× • ×−1 0 −1)
, is to say that ∇AϵBC = 0 for the corresponding skew form ϵBC under (8.2).
As a final consistency check, we may verify that this constraint is invariant under (8.7), as
follows. For ϕB a section of × • ×−1 1 −1
, the transformation (8.7) reads
∇̂AϕB = ∇AϕB −ΥBϕA
(cf. conformal spin geometry in four dimensions [12, equation (5.6.15)]) and, therefore, if ϕBC
is a section of × • ×−1 1 −1 ⊗ × • ×−1 1 −1
, then the Leibniz rule implies that
∇̂AϕBC = ∇AϕBC −ΥBϕAC −ΥCϕBA.
Hence, if ϕBC is skew, then ΥAϕBC = ΥBϕAC +ΥCϕBA and it follows that
∇̂AϕBC = ∇AϕBC −ΥAϕBC .
Finally, when σ̂ = Ω−1σ, we find that ϵ̂AB = ΩϵAB (cf. [12, equation (5.6.2)]) and, hence, that
∇̂Aϵ̂BC = ∇̂A(ΩϵBC) = Ω
(
∇̂AϵBC +ΥAϵBC
)
= Ω∇AϵBC ,
as required.
9 Flying saucers via spinors
Distilling the construction in [7] down to its key ingredients, we will explain how to construct
a G2 contact structure starting from a Legendrean contact structure plus some additional data,
a choice of appropriately weighted sections of the Legendrean subbundles. We then will calculate
the torsion of the resulting G2 contact structure in terms of the input data and the preferred
partial connection in the previous section.
Using the Levi form we may identify F ∗ = E ⊗ L = E[−1,−1] and so can write
∧1
H ∋ ωa =
[
ωA
ω̄A
]
∈
E∗
⊕
E[−1,−1]
.
In what follows we will fix a scale σ ∈ × • ×1 0 1 , or equivalently a contact form. The expression
for the torsion will turn out to be independent of this choice, but to write it down we will need
the distinguished connection from the previous section. The distinguished partial connection
annihilating the scale can be written on E∗ as
E∗ ∋ ωA 7→
[
∇AωB
∇̄AωB
]
∈
E∗
⊕
E[−1,−1]
⊗ E∗.
Recall from Section 8 that a choice of scale gives rise to a skew spinor ϵAB, covariant constant
under the distinguished partial connection, together with its inverse ϵAB (such that ϵABϵ
AC =
δB
C), which we may use to raise and lower indices as per [12]. Covariant constancy implies that
we may equate ∇Aσ
B = ∇A
(
σCϵ
BC
)
and (∇AσC)ϵ
BC , as in the usual spinor calculus.
Spinors in Five-Dimensional Contact Geometry 17
We decompose ∧2
H⊥ = ∧2E∗ ⊕ (E∗ ⊗ E)◦[−1,−1]⊕∧2E[−2,−2], and Proposition 8.2 then
means we can write the Rumin operator as
d⊥
[
σA
τA
]
=
−∇AσA +ΠAτ
A(
∇Aτ
B − ∇̄BσA
)
◦
∇̄Aτ
A +ΣAσA
,
where ΣA and ΠA are the obstructions to integrability and
(
ΞA
B
)
◦ = ΞA
B − 1
2δA
BΞC
C for
ΞA
B ∈ E ⊗ E∗.
Now suppose we are given sections oA ∈ E∗[2,−1] and ιA ∈ E[−2, 1] such that oAι
A = 1,
classically known as a spin-frame [12, pp. 110–115]. Then we may define a G2 contact geometry
as follows. These sections determine an isomorphism
∧0[0, 3]⊕∧0[1, 2]⊕∧0[2, 1]⊕∧0[3, 0] ∼= E∗[2, 2]⊕ E[1, 1]
given by
(x, y, z, w) 7−→
(
xoA − 1√
3
yιA, wι
A − 1√
3
zoA
)
. (9.1)
If we set S = ∧0[0, 1]⊕∧0[1, 0], we have an isomorphism⊙3S ∼= E∗[2, 2]⊕ E[1, 1] = ∧1
H [2, 2]
as required in the definition of a G2 contact structure. The peculiar factors here are chosen so
that the isomorphism also satisfies the additional Levi-compatibility condition.
Starting by considering the general connection on S that annihilates the scale, we can cal-
culate the torsion of the G2 contact structure by calculating the obstruction to the differential
operator Λ1
H → Λ2
H⊥ induced by the defining isomorphism being equal to the Rumin operator.
We can write any partial connection on S as
∇a
[
y
z
]
=
[
∇ay + κay + λaz
∇az + µay + νaz
]
for appropriately weighted sections κa, λa, µa, νa, where we will take ∇a to be the partial
connection distinguished by the contact form. The connection on S will be compatible with
the contact form when κa = −νa. Given this, the induced partial connection on
⊙3S ∼=
∧0[0, 3]⊕∧0[1, 2]⊕∧0[2, 1]⊕∧0[3, 0] is
∇a
x
y
z
w
=
∇ax
∇ay
∇az
∇aw
+
3κa λa 0 0
3µa κa 2λa 0
0 2µa −κa 3λa
0 0 µa −3κa
x
y
z
w
∈ ∧1
H ⊗
⊙3S.
Rewriting the right hand side using the defining isomorphism (9.1) yields the differential operator⊙3S → ∧1
H ⊗ (E∗[2, 2]⊕ E[1, 1]) given by
∇a
x
y
z
w
=
[
(∇ax+ 3xκa + yλa)oA − 1√
3
(∇ay + 3xµa + yκa + 2zλa)ιA
− 1√
3
(∇az + 2yµa − zκa + 3wλa)o
A + (∇aw + zµa − 3wκa)ι
A
]
.
To calculate the induced operator ∧1
H → ∧2
H⊥ we may use the canonical identification
∧1
H = E∗ ⊕ E[−1,−1] and then project the right hand side above onto the direct sum
∧2E∗[2, 2]⊕ (E∗ ⊗ E)◦[1, 1]⊕∧2E.
18 M. Eastwood and T. Moy
We write κa =
(
κA, κ̄
A
)
∈ E∗ ⊕ E[−1, 1] and so on to denote the projections. The induced
operator (pulled back via the isomorphism (9.1)) is therefore⊙3S[−2,−2] → ∧2E∗ ⊕ (E∗ ⊗ E)◦[−1,−1]⊕∧2E[−2,−2]
given by
x
y
z
w
7→
(
∇Ax+ 3xκA + yλA
)
oA − 1√
3
(
∇Ay + 3xµA + yκA + 2zλA
)
ιA(
− 1√
3
(
∇Az + 2yµA − zκA + 3wλA
)
oB +
(
∇Aw + zµA − 3wκA
)
ιB
−
(
∇̄Bx+ 3xκ̄B + yλ̄B
)
oA + 1√
3
(
∇̄By + 3xµ̄B + yκ̄B + 2zλ̄B
)
ιA
)
◦
1√
3
(
∇̄Az + 2yµ̄A − zκ̄A + 3wλ̄A
)
oA −
(
∇̄Aw + zµ̄A − 3wκ̄A
)
ιA
.
This is the operator that we should compare to the Rumin operator (pulled back via the iso-
morphism (9.1)), which can be written
d⊥
x
y
z
w
=
−∇A(xoA) +
1√
3
∇A(yιA) + ΠA
(
wιA − 1√
3
zoA
)
(
∇A
(
wιB − 1√
3
zoB
)
− ∇̄B
(
xoA − 1√
3
yιA
))
◦
∇̄A
(
wιA
)
− 1√
3
∇̄A
(
zoA
)
+ΣA
(
xoA − 1√
3
yιA
)
.
Insisting that these differential operators are equal (and hence that the torsion vanishes) we
obtain a system of twelve spinor equations
∇AoA = 3κAoA −
√
3µAιA, ∇AιA = −
√
3λAoA + κAιA,
ΠAo
A = −2λAιA ΠAι
A = 0,(
∇̄BoA
)
◦ =
(
−
√
3µ̄BιA + 3κ̄BoA
)
◦,(
∇̄BιA
)
◦ =
(
−2µAo
B −
√
3λ̄BoA + κ̄BιA
)
◦,(
∇Ao
B
)
◦ =
(
−κAoB −
√
3µAι
B − 2λ̄BιA
)
◦,(
∇Aι
B
)
◦ =
(
−
√
3λAo
B − 3κAι
B
)
◦,
∇̄Ao
A = −κ̄AoA −
√
3µ̄Aι
A, ∇̄Aι
A = −
√
3λ̄Ao
A − 3κ̄Aι
A,
ΣAιA = 2µ̄Ao
A, ΣAoA = 0.
Contracting the middle four equations above with combinations of oA, ι
B and their counterparts
with indices raised and lowered, respectively, produces (together with the other eight) a system
of twenty independent linear equations over R in twelve unknowns κAoA, µ
AιA, . . . (owing to the
trace-free condition, the middle four equations above yield three independent equations each).
This system is consistent if and only if the following eight obstructions vanish:
ψ0 = ΠAι
A,
ψ1 = ΠAo
A − 2√
3
ιA
(
∇Aι
B
)
ιB,
ψ2 = ιA
(
∇̄AoB
)
oB + oA
(
∇̄AoB
)
ιB − ∇̄Ao
A + 2oA
(
∇̄AιB
)
oB,
ψ3 = ιA
(
∇̄AιB
)
oB + oA
(
∇̄AιB
)
ιB − 1
3∇̄Aι
A + 2
3 ιA
(
∇̄AoB
)
ιB + 2√
3
oA
(
∇Ao
B
)
oB,
ψ4 = oA
(
∇Ao
B
)
ιB + ιA
(
∇Ao
B
)
oB − 1
3∇
AoA + 2
3o
A
(
∇Aι
B
)
oB + 2√
3
ιA
(
∇̄AιB
)
ιB,
ψ5 = oA
(
∇Aι
B
)
ιB + ιA
(
∇Aι
B
)
oB −∇AιA + 2ιA
(
∇Ao
B
)
ιB,
ψ6 = ΣAιA + 2√
3
oA
(
∇̄AoB
)
oB,
ψ7 = ΣAoA.
Spinors in Five-Dimensional Contact Geometry 19
The above eight functions vanish with the invariant torsion of the G2 contact structure. One
can check using the formulæ (8.7) and (8.8) that these expressions are invariant under change
of scale, as they should be.
This construction and resulting formulæ apply in some generality (locally all G2 contact
structures arise this way [11]). In particular, they generalise [7, equation (32)] in case that the
spin-frame oA, ι
A arises from flying saucer data [7, equation (24)] (precisely, with the notation
from [7], this means that ιA = π!ψ and oA = Θ−1π!ϕ).
Acknowledgements
We would like to thank all staff at SIGMA in Kyiv for their extraordinary courage, continuing
their work despite the shocking Russian invasion and unconscionable aggression.
We would also like to thank the referees for their careful reading of our manuscript and for
their valuable suggestions and corrections.
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https://doi.org/10.1017/CBO9780511524486
https://doi.org/10.1063/1.524351
1 Introduction
2 The Levi-Civita connection
3 Conformal differential geometry
4 Five-dimensional conformal spin geometry
5 Three-dimensional conformal spin geometry
6 The Rumin complex
7 Spinors in G_2 contact geometry
8 Legendrean contact geometry in five dimensions
9 Flying saucers via spinors
References
|
| id | nasplib_isofts_kiev_ua-123456789-211637 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T02:56:19Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Eastwood, Michael Moy, Timothy 2026-01-07T13:42:47Z 2022 Spinors in Five-Dimensional Contact Geometry. Michael Eastwood and Timothy Moy. SIGMA 18 (2022), 031, 19 pages 1815-0659 2020 Mathematics Subject Classification: 53B05; 53D10; 58J10 arXiv:2201.13048 https://nasplib.isofts.kiev.ua/handle/123456789/211637 https://doi.org/10.3842/SIGMA.2022.031 We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely 𝐺₂ 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 𝐺₂ contact structure on the configuration space of a flying saucer (always a five-dimensional contact manifold). We would like to thank all staff at SIGMA in Kyiv for their extraordinary courage, continuing their work despite the shocking Russian invasion and unconscionable aggression. We would also like to thank the referees for their careful reading of our manuscript and for their valuable suggestions and corrections. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spinors in Five-Dimensional Contact Geometry Article published earlier |
| spellingShingle | Spinors in Five-Dimensional Contact Geometry Eastwood, Michael Moy, Timothy |
| title | Spinors in Five-Dimensional Contact Geometry |
| title_full | Spinors in Five-Dimensional Contact Geometry |
| title_fullStr | Spinors in Five-Dimensional Contact Geometry |
| title_full_unstemmed | Spinors in Five-Dimensional Contact Geometry |
| title_short | Spinors in Five-Dimensional Contact Geometry |
| title_sort | spinors in five-dimensional contact geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211637 |
| work_keys_str_mv | AT eastwoodmichael spinorsinfivedimensionalcontactgeometry AT moytimothy spinorsinfivedimensionalcontactgeometry |