Twistor Geometry of Null Foliations in Complex Euclidean Space
We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface Qⁿ of dimension n≥3, and its twistor space PT, defined to be the space of all linear subspaces of maximal dimension of Qⁿ. Viewing complex Euclidean space CEⁿ as a dense open subset of...
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irk-123456789-1485602019-02-19T01:25:41Z Twistor Geometry of Null Foliations in Complex Euclidean Space Taghavi-Chabert, A. We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface Qⁿ of dimension n≥3, and its twistor space PT, defined to be the space of all linear subspaces of maximal dimension of Qⁿ. Viewing complex Euclidean space CEⁿ as a dense open subset of Qⁿ, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on CEⁿ can be constructed in terms of complex submanifolds of PT. The construction is illustrated by means of two examples, one involving conformal Killing spinors, the other, conformal Killing-Yano 2-forms. We focus on the odd-dimensional case, and we treat the even-dimensional case only tangentially for comparison. 2017 Article Twistor Geometry of Null Foliations in Complex Euclidean Space / A. Taghavi-Chabert // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 32L25; 53C28; 53C12 DOI:10.3842/SIGMA.2017.005 http://dspace.nbuv.gov.ua/handle/123456789/148560 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface Qⁿ of dimension n≥3, and its twistor space PT, defined to be the space of all linear subspaces of maximal dimension of Qⁿ. Viewing complex Euclidean space CEⁿ as a dense open subset of Qⁿ, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on CEⁿ can be constructed in terms of complex submanifolds of PT. The construction is illustrated by means of two examples, one involving conformal Killing spinors, the other, conformal Killing-Yano 2-forms. We focus on the odd-dimensional case, and we treat the even-dimensional case only tangentially for comparison. |
| format |
Article |
| author |
Taghavi-Chabert, A. |
| spellingShingle |
Taghavi-Chabert, A. Twistor Geometry of Null Foliations in Complex Euclidean Space Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Taghavi-Chabert, A. |
| author_sort |
Taghavi-Chabert, A. |
| title |
Twistor Geometry of Null Foliations in Complex Euclidean Space |
| title_short |
Twistor Geometry of Null Foliations in Complex Euclidean Space |
| title_full |
Twistor Geometry of Null Foliations in Complex Euclidean Space |
| title_fullStr |
Twistor Geometry of Null Foliations in Complex Euclidean Space |
| title_full_unstemmed |
Twistor Geometry of Null Foliations in Complex Euclidean Space |
| title_sort |
twistor geometry of null foliations in complex euclidean space |
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Інститут математики НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/148560 |
| citation_txt |
Twistor Geometry of Null Foliations in Complex Euclidean Space / A. Taghavi-Chabert // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 37 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT taghavichaberta twistorgeometryofnullfoliationsincomplexeuclideanspace |
| first_indexed |
2025-07-12T18:55:46Z |
| last_indexed |
2025-07-12T18:55:46Z |
| _version_ |
1837468540915941376 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 005, 42 pages
Twistor Geometry of Null Foliations
in Complex Euclidean Space
Arman TAGHAVI-CHABERT
Università di Torino, Dipartimento di Matematica “G. Peano”,
Via Carlo Alberto, 10 - 10123, Torino, Italy
E-mail: ataghavi@unito.it
Received April 01, 2016, in final form January 14, 2017; Published online January 23, 2017
https://doi.org/10.3842/SIGMA.2017.005
Abstract. We give a detailed account of the geometric correspondence between a smooth
complex projective quadric hypersurface Qn of dimension n ≥ 3, and its twistor space PT,
defined to be the space of all linear subspaces of maximal dimension of Qn. Viewing complex
Euclidean space CEn as a dense open subset of Qn, we show how local foliations tangent
to certain integrable holomorphic totally null distributions of maximal rank on CEn can
be constructed in terms of complex submanifolds of PT. The construction is illustrated by
means of two examples, one involving conformal Killing spinors, the other, conformal Killing–
Yano 2-forms. We focus on the odd-dimensional case, and we treat the even-dimensional
case only tangentially for comparison.
Key words: twistor geometry; complex variables; foliations; spinors
2010 Mathematics Subject Classification: 32L25; 53C28; 53C12
1 Introduction
The twistor space PT of a smooth complex projective quadric hypersurface Qn of dimension n =
2m+1 ≥ 3, is defined to be the space of all γ-planes, i.e., m-dimensional linear subspaces of Qn.
This is a complex projective variety of dimension 1
2(m + 1)(m + 2) equipped with a canonical
holomorphic distribution D of rank m+1, and maximally non-integrable, i.e., TPT = [D,D]+D.
Here, TPT denotes the holomorphic tangent bundle of PT. Noting that a smooth quadric can be
identified with a complexified n-sphere and is naturally equipped with a holomorphic conformal
structure, we shall view complex Euclidean space CEn as a dense open subset of Qn. In this
context, we shall prove the following new results holding locally:
• totally geodetic integrable holomorphic γ-plane distributions on CEn arise from (m+ 1)-
dimensional complex submanifolds of PT – Theorem 3.5;
• totally geodetic integrable holomorphic γ-plane distributions on CEn with integrable or-
thogonal complements arise from (m+1)-dimensional complex submanifolds of PT foliated
by holomorphic curves tangent to D – Theorem 3.6;
• totally geodetic integrable holomorphic γ-plane distributions on CEn with totally geodetic
integrable orthogonal complements arise from m-dimensional complex submanifolds of a 1-
dimensional reduction of a subset of PT known as mini-twistor space MT – Theorem 3.8.
Conversely, any such distributions arise in the ways thus described. These findings may be
viewed as odd-dimensional counterparts of the work of [20], where it is shown that local foliations
of a 2m-dimensional smooth quadric Q2m by α-planes, i.e., totally null self-dual m-planes, are in
one-to-one correspondence with certain m-dimensional complex submanifolds of twistor space,
here defined as the space of all α-planes in Q2m.
mailto:ataghavi@unito.it
https://doi.org/10.3842/SIGMA.2017.005
2 A. Taghavi-Chabert
The first two of the above results are conformally invariant, and to arrive at them, we shall
first describe the geometrical correspondence between Qn and PT in a manifestly conformally
invariant manner, by exploiting the vector and spinor representations of the complex conformal
group SO(n+ 2,C) and of its double-covering Spin(n+ 2,C). Such a tractor or twistor calculus,
as it is known, builds on Penrose’s twistor calculus in four dimensions [29]. The more ‘standard’,
local and Poincaré-invariant approach to twistor geometry will also be introduced to describe
non-conformally invariant mini-twistor space MT. In fact, a fairly detailed description of twistor
geometry in odd dimensions will make up the bulk of this article, and should, we hope, have
a wider range of applications than the one presented here. Once our calculus is all set up, our
main results will follow almost immediately. The effectiveness of the tractor calculus will be
exemplified by the construction of algebraic subvarieties of PT, which describe the null foliations
of Qn arising from certain solutions of conformally invariant differential operators.
Another aim of the present article is to distil the complex geometry contained in a number of
geometrical results on real Euclidean space and Minkowski space in dimensions three and four.
In fact, our work is motivated by the findings of [27] and [2]. In the former reference, the author
recasts the problem of finding pairs of analytic conjugate functions on En as a problem of finding
closed null complex-valued 1-forms, and arrives at a description of the solutions in terms of real
hypersurfaces of Cn−1. The case n = 3 is of particular interest, and is the focus of the article [2]:
the kernel of a null complex 1-form on E3 consists of a complex line distribution T(1,0)E3 and
the span of a real unit vector u. This complex 2-plane distribution is in fact the orthogonal
complement
(
T(1,0)E3
)⊥
of T(1,0)E3, and we can think of T(1,0)E3 as a CR-structure compatible
with the conformal structure on E3 viewed as an open dense subset of S3. The condition that(
T(1,0)E3
)⊥
be integrable is equivalent to u being tangent to a conformal foliation, otherwise
known as a shearfree congruence of curves. To find such congruences, the authors construct the
S2-bundle of unit vectors over S3, which turns out to be a CR hypersurface in CP3. A section
of this S2-bundle defines a congruence of curves, and this congruence is shearfree if and only if
the section is a 3-dimensional CR submanifold.
There are three antecedents for this result:
1) there is a one-to-one correspondence between local self-dual Hermitian structures on E4⊂S4
and holomorphic sections of the S2-bundle CP3 → S4 known as the twistor bundle – this
is a well-known result, see, e.g., [2, 4, 14, 20, 32];
2) there is a one-to-one correspondence between local analytic shearfree congruences of null
geodesics in Minkowski space M and certain complex hypersurfaces of its twistor space,
an auxilliary space isomorphic to CP3 – this is known as the Kerr theorem [11, 29, 31];
3) there is a one-to-one correspondence between local shearfree congruences of geodesics in E3
and certain holomorphic curves in its mini-twistor space, the holomorphic tangent bundle
of CP1 ∼= S2 – such congruences can also be equivalently described by harmonic morphisms
[3, 36, 37].
Statements (1) and (2) are essentially the same result once they are cast in the complexification
of E4 and M.
The analogy between statement (1) and the result of [2] can be understood in the following
terms: in the former case, the integrable complex null 2-plane distribution T(1,0)E4 defining the
Hermitian structure is totally geodetic, i.e., ∇XY ∈ Γ
(
T(1,0)E4
)
for all X,Y ∈ Γ
(
T(1,0)E4
)
.
In the latter case, the condition that u be tangent to a shearfree congruence is also equivalent
to the complex null line distribution T(1,0)E3 being (totally) geodetic. One could also think of
the integrability of both T(1,0)E3 (trivially) and
(
T(1,0)E3
)⊥
as an analogue of the integrability
of T(1,0)E4.
Twistor Geometry of Null Foliations in Complex Euclidean Space 3
Finally, statement (3), unlike (1) and (2), breaks conformal invariance, and the additional
data fixing a metric on E3 induces a reduction of the S2-bundle constructed in [2] to mini-
twistor space TS2 of (3). Correspondingly, for u to be tangent to a shearfree congruence of null
geodesics, both T(1,0)E3 and
(
T(1,0)E3
)⊥
must be totally geodetic, which is not a conformally
invariant condition.
The structure of the paper is as follows. Section 2 deals with the twistor geometry of a smooth
quadric Qn focussing mostly on the case n = 2m+1. In particular, we give an algebraic descrip-
tion of the canonical distribution on its twistor space. The geometric correspondence betweenQn
and PT is made explicit. Propositions 2.12 and 2.13, and Corollary 2.14 give a twistorial artic-
ulation of incidence relations between γ-planes in Qn. The mini-twistor space MT of complex
Euclidean space CEn is introduced in Section 2.4. Points in CEn correspond to embedded
complex submanifolds of PT and MT, and their normal bundles are described in Section 2.5.
The main results, Theorems 3.5, 3.6 and 3.8, as outlined above, are given in Section 3. In
each case, a purely geometrical explanation precedes a computational proof. In Section 4, we
give two examples on how to relate null foliations in Qn to complex varieties in PT, based on
certain solutions to the twistor equation, in Propositions 4.2 and 4.3, and the conformal Killing–
Yano equation, in Proposition 4.7. We wrap up the article with Appendix A, which contains
a description of standard open covers of twistor space and correspondence space.
2 Twistor geometry
We describe each of the three main protagonists involved in this article in turn: a smooth
quadric hypersurface in projective space, its twistor space and a correspondence space fibered
over them. The projective variety approach is very much along the line of [19, 31], while the
reader should consult [5, 9] for the corresponding homogeneous space description.
Throughout V will denote an (n+2)-dimensional complex vector space. We shall make use of
the following abstract index notation: elements of V and its dual V∗ will carry upstairs and down-
stairs calligraphic upper case Roman indices respectively, i.e., V A ∈ V and αA ∈ V∗. Symmetri-
sation and skew-symmetrisation will be denoted by round and square brackets respectively, i.e.,
α(AB) = 1
2(αAB +αBA) and α[AB] = 1
2(αAB −αBA). These conventions will apply to other types
of indices used throughout this article. We shall also use Einstein’s summation convention, e.g.,
V AαA will denote the natural pairing of elements of V and V∗. We equip V with a non-degenerate
symmetric bilinear form hAB, by means of which V ∼= V∗: indices will be lowered and raised
by hAB and its inverse hAB respectively. We also choose a complex orientation on V, i.e., a com-
plex volume element εA1...An+2 in ∧n+2V. We shall denote by G the complex spin group Spin(n+
2,C), the two-fold cover of the complex Lie group SO(n+ 2,C) preserving hAB and εA1...An+2 .
Turning now to the spinor representations of G, we distinguish the odd- and even-dimensional
cases:
• n = 2m+1: denote by S the 2m+1-dimensional irreducible spinor representation of G. Ele-
ments of S will carry upstairs bold lower case Greek indices, e.g., Sα ∈ S, and dual elements,
downstairs indices. The Clifford algebra C`(V, hAB) is linearly isomorphic to the exterior
algebra ∧•V, and, identifying ∧kV with ∧2m+3−kV by Hodge duality for k = 0, . . . ,m+ 1,
it is also isomorphic, as a matrix algebra, to the space End(S) of endomorphisms of S. It
is generated by matrices, denoted Γ γ
Aα , which satisfy the Clifford identity
Γ γ
(Aα Γ β
B)γ = −hABδβα. (2.1)
Here δβα is the identity element on S. There is a spin-invariant inner product on S denoted
Γ
(0)
δβ : S × S → C, yielding the isomorphism End(S) ∼= S ⊗ S. The resulting isomorphisms
4 A. Taghavi-Chabert
C`(V, hAB) ∼= ∧•V ∼= S⊗S will be realised by means of the bilinear forms on S with values
in ∧kV∗, for k = 1, . . . , n+ 2:
Γ
(k)
A1...Akαβ := Γ
γ1
[A1α
· · ·Γ δk
Ak]γk−1
Γ
(0)
δkβ
. (2.2)
These are symmetric in their spinor indices when k ≡ m + 1,m + 2 (mod 4) and skew-
symmetric otherwise.
• n = 2m: G has two 2m-dimensional irreducible chiral spinor representations, which we
shall denote S and S′. Elements of S and S′ will carry upstairs unprimed and primed lower
case bold Greek indices respectively, i.e., Aα ∈ S and Bα′ ∈ S′. Dual elements will carry
downstairs indices. The Clifford algebra C`(V, hAB) is isomorphic to End(S⊕S′) as a matrix
algebra, and, linearly, to ∧•V. We can write its generators in terms of matrices Γ γ′
Aα
and Γ γ
Aα′ satisfying
Γ γ′
(Aα Γ β
B)γ′ = −hABδβα, Γ γ
(Aα′ Γ β′
B)γ = −hABδ
β′
α′ ,
where δβα and δβ
′
α′ are the identity elements on S and S′ respectively. There are spin-
invariant bilinear forms on S ⊕ S′ inducing isomorphisms S∗ ∼= S′, (S′)∗ ∼= S when m is
even, and S∗ ∼= S and (S′)∗ ∼= S′ when m is odd, and denoted Γ
(0)
αβ′
, Γ
(0)
α′β, and Γ
(0)
αβ, Γ
(0)
α′β′
respectively. The resulting isomorphisms C`(V, hAB) ∼= ∧•V ∼= (S ⊕ S′) ⊗ (S ⊕ S′) are
realised by ∧kV-valued bilinear forms Γ
(k)
αβ, for k ≡ m + 1 (mod 2), and Γ
(k)
αβ′
, for k ≡ m
(mod 2) and so on.
We work in the holomorphic category throughout.
2.1 Smooth quadric hypersurface
Let us denote by XA the position vector in V, which can be viewed as standard Cartesian
coordinates on Cn+2. The equivalence class of non-zero vectors in V that projects down to the
same point in the projective space PV ∼= CPn+1 will be denoted [·], and thus [XA] will represent
homogeneous coordinates on PV.
The zero set of the quadratic form associated to hAB on V defines a null cone C in V, and
the projectivisation of C defines a smooth quadric hypersurface Qn in PV, i.e.,
Qn =
{[
XA
]
∈ PV : hABX
AXB = 0
}
.
By taking a suitable cross-section of C, one can identify Qn with the complexification CSn of
the standard n-sphere Sn in Euclidean space En+1. Using the affine structure on V, hAB can be
viewed as a field of bilinear forms on V and thus on C. We can then pull back hAB to Qn along
any section of C → Qn to a (holomorphic) metric on Qn. Different sections yield conformally
related metrics on Qn, i.e., a (holomorphic) conformal structure on Qn. The projective tangent
space at a point p of Qn with homogeneous coordinate
[
PA
]
is the linear subspace
TpQn :=
{[
XA
]
∈ Qn : hABX
APB = 0
}
,
which can be seen to be the closure of the (holomorphic) tangent space TpQn at p ∈ Qn in the
usual sense. The intersection of TpQn and Qn is a cone through p, and any point lying in this
cone is connected to its vertex by a line that is null with respect to the conformal structure.
To obtain the Kleinian model of Qn, we fix a null vector X̊A in V, and denote by P the
stabiliser of the line spanned by X̊A in G. The transitive action of G on V descends to a transitive
action on Qn, and since P stabilises a point in Qn, we obtain the identification G/P ∼= Qn. The
Twistor Geometry of Null Foliations in Complex Euclidean Space 5
subgroup P is a parabolic subgroup of G, and its Lie algebra p admits a Levi decomposition,
that is, a splitting p = p0 ⊕ p1, where p0 is the reductive Lie algebra so(n,C) ⊕ C, and p1
is a nilpotent part, here isomorphic to (Cn)∗. We choose a complement p−1 of p in g, dual
to p1 via the Killing form on g, so that g = p−1 ⊕ p. There is a unique element spanning the
centre z(p0) ∼= C of p0, which acts diagonally on p0, p1 and p−1 with eigenvalues 0, 1 and −1
respectively. For this reason, we refer to this element as the grading element of the splitting
g = p−1 ⊕ p0 ⊕ p1. This splitting is compatible with the Lie bracket [·, ·] : g × g → g on g in
the sense that [pi, pj ] ⊂ pi+j , with the convention that pi = {0} for |i| > 1. In particular, it is
invariant under p0, but not under p. However, the filtration p1 ⊂ p0 ⊂ p−1 := g, where p1 := p1
and p0 := p0 ⊕ p1, is a filtration of p-modules on g, and each of the p-modules p−1/p0, p0/p1
and p1 is linearly isomorphic to the p0-modules p−1, p0 and p1 respectively. These properties
are most easily verified by realising g in matrix form, i.e.,
p0 p1 p1 p1 0
p−1 p0 p0 p0 p1
p−1 p0 0 p0 p1
p−1 p0 p0 p0 p0
0 p−1 p−1 p0 p0
}1
}m
}1
}m
}1
p0 p1 p1 0
p−1 p0 p0 p1
p−1 p0 p0 p0
0 p−1 p0 p0
}1
}m
}m
}1
when n = 2m+ 1 and n = 2m respectively.
Given a vector representation V of P , one can construct the holomorphic homogeneous vector
bundle G ×P V over G/P : this is the orbit space of a point in G × V under the right action
of G. In particular, the tangent bundle of Qn can be described as T(G/P ) ∼= G ×P (g/p), and
the tangent space at any point of Qn is isomorphic to p−1
∼= g/p – for a proof, see, e.g., [9].
Similarly, denoting by P0 the reductive subgroup of P with Lie algebra p0, we can construct
holomorphic homogeneous vector bundles from representations of P0.
2.1.1 The tractor bundle
An important homogeneous vector bundle over Qn is the one constructed from the standard
representation V of G. It leads to a conformal invariant calculus, known as tractor calculus. The
reader should consult, e.g., [1, 12] for further details.
Definition 2.1. The (complex) standard tractor bundle over Qn ∼= G/P is the rank-(n + 2)
vector bundle T := G×P V ∼= G/P × V.
The symmetric bilinear form hAB on V induces a non-degenerate holomorphic section of
�2T ∗ → Qn on T , called the tractor metric, also denoted by hAB. Further, the affine structure
on V induces a unique tractor connection on T preserving hAB.
The vector space V admits a filtration of P -modules V =: V−1 ⊃ V0 ⊃ V1, where V1 =
〈
X̊A
〉
and V0 is the orthogonal complement of V1. These P -modules and their quotients Vi/Vi+1 give
rise to P -invariant vector bundles as explained above. For convenience, we choose a splitting
V = V−1 ⊕ V0 ⊕ V1, (2.3)
where V1 := V−1, V−1 is a null line in V complementary to V0 ⊂ V−1, and V0 is the n-
dimensional vector subspace orthogonal to both V−1 and V1. We note the linear isomorphisms
V−1
∼= V−1/V0 and V0
∼= V0/V1.
Let us introduce some abstract index notation. Elements of V0 and its dual (V0)∗ will be
adorned with upstairs and downstairs lower-case Roman indices respectively, e.g., V a ∈ V0 and
αa ∈ (V0)∗. We fix a null vector Y̊ A spanning V−1 such that X̊AY̊A = 1. We also introduce
6 A. Taghavi-Chabert
the injector Z̊Aa : V0 → V. Then, hAB restricts to a non-degenerate symmetric bilinear form
gab := Z̊Aa Z̊
B
b hAB on V0. Indices can be raised or lowered by means of hAB, gab and their inverses.
A geometric interpretation of T → G/P can be found in [12] in a real setting. Here, we
note that the line subbundle G ×P V1 of T can be identified with the pull-back O[−1] of the
tautological line bundle O(−1) on PV to Qn. The bundle G×P
(
V−1/V0
)
is isomorphic to the
dual of O[−1], i.e., to the pullback O[1] of the hyperplane bundle O(1) on PV. Finally, since
p−1 ⊗ V1
∼= V0, we have the identification G×P
(
V0/V1
) ∼= TQn ⊗O[−1].
The structure sheaf of Qn will be denoted O, and the sheaf of germs of holomorphic functions
onQn homogeneous of degree w byO[w]. We shall writeOa for the sheaf of germs of holomorphic
sections of TQn, and extend this notation in the obvious way to tensor products, e.g., OAab[w] :=
OA ⊗ Oab ⊗ O[w], and so on. In particular, the sheaf of germs of holomorphic sections of the
tractor bundle T reads
OA = O[1] +Oa[−1] +O[−1]. (2.4)
The line bundle O[1] has the geometric interpretation of the bundle of conformal scales, and
the conformal structure on Qn can be equivalently encoded in terms of a distinguished global
section gab of O(ab)[2] called the conformal metric. For any non-vanishing local section σ of O[1],
gab = σ−2gab is a metric in the conformal class. A choice of metric in the conformal class is
essentially equivalently to a splitting of (2.4), i.e., a choice of section Y A of OA[−1] such that
Y AYA = 0 and XAYA = 1, where we view XA ∈ OA[1] as the Euler vector field on C ⊂ V. We
can then choose a section ZAa of OAa [1] satisfying ZAa ZbA = gab and ZAa XA = ZAa YA = 0, so
that the tractor metric takes the form hAB = 2X(AYB) +ZaAZ
b
Bgab – see, e.g., [15]. A section ΣA
of OA can be expressed as
ΣA = σY A + ϕaZAa + ρXA, where (σ, ϕa, ρ) ∈ O[1]⊕Oa[−1]⊕O[−1]. (2.5)
We shall denote both the tractor connection and the Levi-Civita connection of a metric in the
conformal class by ∇a. The explicit formula for the tractor connection on a section (2.5) of OA
in terms of a splitting of (2.4) can then be recovered from the Leibniz rule and the formulae
∇aXA = ZAa , ∇aZAb = −PabX
A − gabY
A, ∇aY A = P b
a Z
A
b , (2.6)
where Pab is the Schouten tensor of ∇a defined by the relation 2∇[a∇b]V c = 2Pc[aVb] −
2V dPd[aδ
c
b].
Complex Euclidean space. Most of this paper will be concerned with the geometry on
n-dimensional complex Euclidean space CEn viewed as a dense open subset of Qn, i.e., CEn =
Qn \ {∞} where ∞ is a point at ‘infinity’ on CSn ∼= Qn. We choose a conformal scale σ ∈ O[1]
so that gab is the flat metric, i.e., Pab = 0. To realise σ geometrically, we use the splitting (2.3).
Then, CEn arises as the intersection of the affine hyperplane H := {XA ∈ V : XAY̊A = 1} with
C: V1 = 〈X̊A〉 descends to the origin on CEn, and V−1 = 〈Y̊ A〉 represents ∞ on Qn. The flat
metric gab is obtained by pulling back hAB along the local section C∩H of C → Qn. Letting {xa}
be flat coordinates on CEn so that ∇a = ∂
∂xa , we can integrate (2.6) explicitly to get
Y A = Y̊ A, ZAa = Z̊Aa − gabxbY̊ A, XA = X̊A + xaZ̊Aa − 1
2gabx
axbY̊ A. (2.7)
This description is also consistent with the identification of CEn with the tangent space at the
‘origin’ of Qn. In this case, the coordinates {xa} arise from p−1
∼= V−1⊗V0 via the exponential
map, which provides an embedding of CEn into Qn, xa 7→ [XA] where XA is given by (2.7).
The embedding can in fact be extended to a conformal embedding
CEn → C → Qn,
xa 7→ ΩXA = ΩX̊A + xaΩZ̊Aa − 1
2
(
Ω2gabx
axb
)
Ω−1Y̊ A 7→
[
XA
]
,
Twistor Geometry of Null Foliations in Complex Euclidean Space 7
obtained by intersecting C with the affine hypersurface HΩ := {XA ∈ V : XAY̊A = Ω}, where Ω
is a non-vanishing holomorphic function on V.
2.1.2 The tractor spinor bundle
We can play the same game by considering bundles over Qn arising from the spinor representa-
tions of G = Spin(n+ 2,C). Again, we distinguish the odd- and even-dimensional cases.
Odd dimensions. Assume n = 2m+ 1.
Definition 2.2. The tractor spinor bundle and dual tractor spinor bundle over Qn ∼= G/P are
the holomorphic homogeneous vector bundles S := G×P S and S∗ := G×P S∗ respectively.
The generators Γ β
Aα of the Clifford algebra (V, hAB) induce holomorphic sections of T ∗ ⊗
S∗ ⊗ S on Qn, which we shall also denote by Γ β
Aα . The tractor connection on Qn extends to
a tractor spinor connection on S preserving Γ β
Aα , and thus hAB.
There is a filtration of P -submodules S =: S−
1
2 ⊃ S
1
2 . These P -modules and their quotients
give rise to P -invariant vector bundles on Qn in the standard way. The splitting (2.3) of V
induces a splitting
S ∼= S− 1
2
⊕ S 1
2
, (2.8)
where S 1
2
∼= V1 ⊗ S− 1
2
, and we can identify S− 1
2
, and thus S 1
2
, as the spinor representation for
(V0, gab). Similar considerations apply to S∗. See, e.g., [16, 17] for details.
Elements of S± 1
2
will carry bold upper case Roman indices, e.g., ξA ∈ S± 1
2
. The Clifford
algebra generators γ B
aA satisfy γ C
(aA γ B
b)C = −gabδBA, where δBA is the identity on S± 1
2
. There
is a spin-invariant bilinear form γ
(0)
AB on S± 1
2
, by means of which we can define bilinear forms
γ
(k)
a1...akAB := γ C1
[a1A
· · · γ Ck
ak]Ck−1
γ
(0)
CkB
,
from S± 1
2
× S± 1
2
to ∧kV0 for k = 1, . . . , n. We introduce projectors O̊A
α : S → S− 1
2
and
I̊Aα : S→ S 1
2
, and injectors I̊αA : S− 1
2
→ S and O̊α
A : S 1
2
→ S, which satisfy O̊B
α I̊
α
A = δBA and
O̊A
α I̊
β
A + I̊Aα O̊
β
A = δβα. Then one can check that the relation between Γ β
Aα and γ B
aA is given by
Γ β
Aα = Z̊aA
(
O̊A
α I̊
β
Bγ
B
aA − I̊Aα O̊
β
Bγ
B
aA
)
+
√
2Y̊AO̊
A
α O̊
β
A −
√
2X̊AI̊
A
α I̊
β
A. (2.9)
Sheaves of germs of holomorphic sections of G ×P
(
S−
1
2 /S−
1
2
)
will be denoted OA, and we
shall write OA[−1] := OA ⊗ O[−1], and similarly for dual bundles in the obvious way. In
particular, the sheaves of germs of holomorphic sections of S and its dual are given by
Oα = OA +OA[−1], Oα = OA[1] +OA, (2.10)
respectively. The splitting of (2.10) can be realised by means of injectors/projectors OA
α ∈
OA
α , IAα ∈ OA
α [−1], Oα
A ∈ Oα
A[1] and IαA ∈ Oα
A, such that OA
α I
α
B = δAB , IAα O
α
B = δAB , and
OA
α I
β
A + IAα O
β
A = δβα, while all the other pairings are zero. In particular, we shall express
a section of Oα as
Ξα = IαAξ
A +Oα
Aζ
A, where
(
ξA, ζA
)
∈ OA +OA[−1],
and similarly for dual tractor spinors.
8 A. Taghavi-Chabert
By abuse of notation, the connection on S and the spin connection associated to a metric in
the conformal class will both be denoted ∇a. They satisfy
∇aOA
α = − 1√
2
γ A
aB IBα , ∇aIAα = − 1√
2
Pabγ
b A
B OB
α ,
∇aOα
A =
1√
2
γ B
aA IαB, ∇aIαA =
1√
2
Pabγ
b B
A Oα
B, (2.11)
where γ B
aA ∈ O B
aA [1] satisfy γ C
(aA γ B
b)C = −gabδ
B
A. The bundle analogue of (2.9) is
Γ β
Aα = ZaA
(
OA
α I
β
Bγ
B
aA − IAα O
β
Bγ
B
aA
)
+
√
2YAO
A
αO
β
A −
√
2XAI
A
α I
β
A.
With a choice of conformal scale σ ∈ O[1] for which gab = σ−2gab is flat, i.e., Pab = 0,
equations (2.11) can be integrated explicitly to give
IαA = I̊αA, Oα
A = O̊α
A +
1√
2
xaγ B
aA I̊αB, IAα = I̊Aα , OA
α = O̊A
α −
1√
2
xaγ B
aA I̊Bα ,
where γ B
aA = σ−1γ B
aA .
Even dimensions. When n = 2m, the story is similar, except that, by virtue of the two
chiral spinor representations, we have an unprimed tractor spinor bundle and a primed tractor
spinor bundle, defined as S := G×P S and S ′ := G×P S′ respectively. We shall view the genera-
tors Γ β′
Aα and Γ β
Aα′ as holomorphic sections of T ∗ ⊗ S∗ ⊗ S ′ and T ∗ ⊗ (S ′)∗ ⊗ S respectively
on Qn, both of which are preserved by the extension of the tractor connection to S ⊕ S ′.
The spinor spaces S and S′ admit filtrations of P -submodules S =: S−
1
2 ⊃ S
1
2 and S′ =:
S′−
1
2 ⊃ S′
1
2 . These P -modules and their quotients give rise to P -invariant vector bundles on Qn
in the standard way. The splitting (2.3) on V induces a splitting of these filtrations
S ∼= S− 1
2
⊕ S 1
2
, S′ ∼= S′− 1
2
⊕ S′1
2
, (2.12)
where S′1
2
∼= V1 ⊗ S− 1
2
and S 1
2
∼= V1 ⊗ S′− 1
2
, and we can identify S− 1
2
and S′− 1
2
, and thus S′1
2
and S 1
2
, as the chiral spinor representations of (V0, gab). Elements of S− 1
2
and S 1
2
will carry
unprimed and primed upper case Roman indices respectively, e.g., ηA ∈ S 1
2
and ξA
′ ∈ S− 1
2
.
The generators of the Clifford algebra are matrices denoted γ B′
aA and γ A
aB′ , satisfying the
Clifford identities γ C′
(aA γ B
b)C′ = −gabδBA and γ C
(aA′ γ B′
b)C = −gabδB
′
A′ , where δB
′
A′ and δBA are
the identity elements on S− 1
2
and S 1
2
respectively. We also obtain spin invariant bilinear forms
γ
(k)
A′B′ , γ
(k)
AB and γ
(k)
AB′ . The story for S′ is similar.
We introduce projectors O̊A
α , I̊A
′
α and injectors I̊αA and O̊α
A′ for the splitting (2.12), normalised
in the obvious way. The relation between the generators of the Clifford algebra C`(V, hAB) and
those of C`(V0, gab) is then given by
Γ β′
Aα = Z̊aA
(
O̊A
α I̊
β′
B′γ
B′
aA − I̊A′α O̊β′
B γ
B
aA′
)
+
√
2Y̊AO̊
A
α O̊
β′
A −
√
2X̊AI̊
A′
α I̊β
′
A′ ,
and similar for Γ β
Aα′ by interchanging primed and unprimed indices.
These algebraic objects extend to weighted tensor or spinor fields just as in odd dimensions
in the obvious way and notation. In particular, we have composition series of the unprimed and
primed tractor spinor bundles:
Oα = OA +OA′ [−1], Oα′ = OA′ +OA[−1],
Oα = OA′ [1] +OA, Oα′ = OA[1] +OA′ .
Twistor Geometry of Null Foliations in Complex Euclidean Space 9
2.2 Twistor space
The linear subspaces of Qn can be described in terms of representations of G = Spin(n+ 2,C).
We shall be interested in those of maximal dimension, arising from maximal totally null vector
subspaces of (V, hAB). In even dimensions, the complex orientation on V determines the duality
of the corresponding linear subspaces, via Hodge duality, which are then described as either
self-dual or anti-self-dual.
Definition 2.3. An m-dimensional linear subspace of Q2m+1 is called a γ-plane. A self-dual,
respectively, anti-self-dual, m-dimensional linear subspace of Q2m is called an α-plane, respec-
tively, a β-plane.
We call the space of all γ-planes in Q2m+1 the twistor space of Q2m+1, and denote it
by PT(2m+1). The space of all α-planes, respectively, β-planes in Q2m will be called the twistor
space PT(2m), respectively, the primed twistor space PT′(2m).
A point in PT will be referred to as a twistor.
We shall often write PT and PT′ for PT(2m+1) or PT(2m), and PT′(2m) respectively. We now
distinguish the odd- and even-dimensional cases.
2.2.1 Odd dimensions
Assume n = 2m+ 1. Let Zα be a non-zero spinor in S, and define the linear map
Zα
A := Γ α
Aβ Zβ : V→ S. (2.13)
By (2.1), the kernel of (2.13) is a totally null vector subspace of V, and if it is non-trivial,
descends to a linear subspace of Qn.
Definition 2.4. We say that a non-zero spinor Zα in S is pure if the kernel of Zα
A := Γ α
Aβ Zβ
has maximal dimension m+ 1.
The (m + 1)-dimensional totally null subspace of V associated in this way to a pure spinor
descends to a γ-plane in Qn. Clearly, any two pure spinors differing by a factor give rise to the
same γ-plane. Further, one can show that any γ-plane in Qn arises from a pure spinor up to
scale. Hence,
Proposition 2.5 ([10]). The twistor space PT of Q2m+1 is isomorphic to the projectivisation
of the space of all pure spinors in S.
Every non-zero spinor in S is pure when m = 1. Let us recall that the Γ
(k)
αβ in the next
theorem denote the spin bilinear forms defined by (2.2).
Theorem 2.6 ([10]). When m > 1, a non-zero spinor Zα in S is pure if and only if it satisfies
Γ
(k)
αβZ
αZβ = 0, for all k < m+ 1, k ≡ m+ 2,m+ 1 (mod 4), (2.14)
and Γ
(m+1)
αβ ZαZβ 6= 0.
Alternatively, the quadratic relations (2.14) can be expressed more succinctly by [35]
ZAαZβ
A + ZαZβ = 0. (2.15)
In analogy with the description of the quadric, we shall view Zα as a position vector or
coordinates on S. The twistor space of Qn can then be described as a complex projective variety
10 A. Taghavi-Chabert
of the projectivisation PS of S with homogeneous coordinates [Zα] satisfying (2.14) or (2.15)
when m > 1. For Q3, we have PT(3)
∼= CP3.
We shall adopt the following notation: if Z is a point in PT, with homogeneous coordina-
tes [Zα], then the corresponding γ-plane in Qn will be denoted Ž, i.e.,
Ž :=
{[
XA
]
∈ Qn : XAZα
A = 0
}
.
Let Ξ be a twistor with homogeneous coordinates [Ξα] and associated γ-plane Ξ̌ in Qn. The
projective tangent space of PT at Ξ is the linear subspace of PS defined by
TΞPT :=
{
[Zα] ∈ PS : Γ
(k)
αβZ
αΞβ = 0, for all k < m− 1
}
. (2.16)
This is the closure of the holomorphic tangent space TΞPT at Ξ, and contains the linear subspace
DΞ :=
{
[Zα] ∈ PS : Γ
(k)
αβZ
αΞβ = 0, for all k < m
}
. (2.17)
This is the closure of a subspace DΞ of TΞPT. The smooth assignment of every point Ξ of PT
of DΞ yields a distribution that we shall denote D. Another convenient way of expressing the
locus in (2.17) is [35]
0 = ZAαΞβ
A + 2ZβΞα − ZαΞβ, (2.18)
where Zα
A := Γ α
Aβ Zβ and Ξα
A := Γ α
Aβ Ξβ.
To understand PT more fully, we realise it as a Kleinian geometry. Let us fix a pure spinor Ξα,
and denote by R the stabiliser of its span in G. This is a parabolic subgroup of G. Then, PT
is isomorphic to G/R. One could equivalently realise PT as the quotient of SO(n+ 2,C) by the
stabiliser of the corresponding γ-plane Ξ̌ in Qn. The Lie algebra r of R induces a |2|-grading on
g, i.e., g = r−2 ⊕ r−1 ⊕ r0 ⊕ r1 ⊕ r2, where r = r0 ⊕ r1 ⊕ r2, with r0 ∼= gl(m+ 1,C), r−1
∼= Cm+1
and r−2
∼= ∧2Cm+1, and r−1
∼= (r1)∗, r−2
∼= (r2)∗. In matrix form, this reads as
r0 r0 r1 r2 0
r0 r0 r1 r2 r2
r−1 r−1 0 r1 r1
r−2 r−2 r−1 r0 r0
0 r−2 r−1 r0 r0
}1
}m
}1
}m
}1
These r0-modules satisfy the commutation relations [ri, rj ] ⊂ ri+j where ri = {0} for |i| > 2.
Further, g is equipped with a filtration of r-modules g := r−2 ⊃ r−1 ⊃ r0 ⊃ r1 ⊃ r2 where
ri := ri⊕ ri+1 satisfy [ri, rj ] ⊂ ri+j . In particular, g/r is not an irreducible r-module, but admits
a splitting into irreducible r-submodules r−1/r and r−2/r−1. Since the tangent space at any
point of G/R can be identified with the quotient g/r, i.e., T(G/R) ∼= G ×R (g/r), the tangent
bundle of PT admits a filtration of R-invariant subbundles TPT = T−2PT ⊃ T−1PT, where the
rank-(m+ 1) distribution
T−1PT := G×R
(
r−1/r
)
(2.19)
is maximally non-integrable by virtue of the commutation relations among the various graded
pieces of g, i.e., at every point Z ∈ PT, T−1
Z PT ∼= r−1 and [T−1
Z PT,T−1
Z PT]+T−1
Z PT ∼= r−1⊕ r−2.
We shall presently show that the distributions D defined in terms of (2.17) and T−1PT
defined by (2.19) are the same. We first note that any spinor Zα ∈ S can be expressed as
Zα = Z(0)Ξ
α +
[(m+1)/2]∑
k=1
(
−1
4
)k 1
k!
(
Z(−2k) · Ξ
)α
+
i
2
[(m+1)/2]∑
k=0
(
−1
4
)k 1
k!
(
Z(−2k−1) · Ξ
)α
, (2.20)
Twistor Geometry of Null Foliations in Complex Euclidean Space 11
where Z(−i) ∈ ∧ir−1
∼= ∧iCm+1, and
[
m+1
2
]
is m+1
2 when m + 1 is even, m
2 when m + 1 is odd.
Here, the · denotes the Clifford action, i.e., (Φ · Ξ)α = ΦAΓ α
Aβ Ξβ, and so on as extended to the
action of ∧•V on S. The factors have been chosen for convenience. The representation (2.20) is
sometimes referred to as the Fock representation [7], and is already used implicitly in Cartan’s
work [10], where the Z(−i) are viewed as the components of a spinor.
Now, using (2.1) and (2.2), together with (2.14) applied to Ξα, we compute Γ
(m+2)
αβ ZαΞα =
Z(0)Γ
(m+2)
αβ ΞαΞα and, for k ≥ 0,
Γ
(m−2k+1)
A1...Am−2k+1αβ
ZαΞα =
(
−1
4
)k 1
k!
ZB1...B2k(−2k) Γ
(m+1)
B1...B2kA1...Am−2k+1αβ
ΞαΞα
+
i
2
(
−1
4
)k 1
k!
Z
B1...B2k+1
(−2k−1) Γ
(m+2)
B1...B2k+1A1...Am−2k+1αβ
ΞαΞα,
Γ
(m−2k)
A1...Am−2kαβ
ZαΞα =
i
2
(
−1
4
)k 1
k!
Z
B1...B2k+1
(−2k−1) Γ
(m+1)
B1...B2k+1A1...Am−2kαβ
ΞαΞα
+
(
−1
4
)k+1 1
(k + 1)!
Z
B1...B2k+2
(−2k−2) Γ
(m+2)
B1...B2k+2A1...Am−2kαβ
ΞαΞα. (2.21)
Here, we have added tractor indices to the Z(−i). We can immediately conclude
Lemma 2.7. The conditions that [Zα] ∈ PS lies in TΞPT and DΞ respectively are equivalent to
Zα = Z(0)Ξ
α + i
2
(
Z(−1) · Ξ
)α − 1
4
(
Z(−2) · Ξ
)α
, (2.22)
Zα = Z(0)Ξ
α + i
2
(
Z(−1) · Ξ
)α
, (2.23)
respectively, up to overall factors. When Z(0) is non-zero, [Zα] given by (2.22) and (2.23) lies
in TΞPT and DΞ respectively. In particular, D ∼= T−1PT.
Proof. Equations (2.22) and (2.23) follow from definitions (2.16) and (2.17) using (2.21). Equa-
tion (2.23) with Z(0) = 1 coincides with the exponential of an element of r−1 and thus describes
a point in T−1
Ξ PT. The story for (2.22) is similar. �
On the other hand, using (2.14) or referring to [10], the condition that Zα be pure is that
Z(0)Z(−2k−1) = Z(−1) ∧ Z(−2k),
Z(0)Z(−2k) = Z(−2) ∧ Z(−2k+2), k = 1, . . . , [(m+ 1)/2]. (2.24)
A dense open subset of PT containing [Ξα] can be obtained by intersecting the locus (2.24) with
the affine subspace Z(0) = 1 in S. Summarising,
Proposition 2.8. The twistor space PT of a (2m+ 1)-dimensional smooth quadric Q2m+1 has
dimension 1
2(m+ 1)(m+ 2), and is equipped with a maximally non-integrable distribution D of
rank m + 1, i.e., TPT = D + [D,D], where, for any Ξ ∈ PT, DΞ is a dense open subset of DΞ
as defined by (2.17).
Further, for any Ξ ∈ PT, the projective tangent space TΞPT intersects PT in a (2m + 1)-
dimensional linear subspace of PT, and DΞ is an (m+ 1)-dimensional linear subspace of PT.
Proof. The first part has already been explained and stems from the general theory of [9].
For the second part, we fix a pure spinor Ξα, and let [Zα] be an element of the projective
tangent space TΞPT so that Zα takes the form (2.22). If [Zα] also lies in PT, then, with
reference to (2.24), Z(−1) ∧ Z(−2) = 0 and Z(−2) ∧ Z(−2) = 0. Generically, Z(−1) is non-zero, so
Z(−2) = Z(−1) ∧ Φ(−1) for some Φ(−1) ∈ r−1. The form of Z(−2) remains invariant under the
12 A. Taghavi-Chabert
transformation Φ(−1) 7→ Φ(−1) +aZ(−1) for any a ∈ C. The choice of Z(0) is cancelled out by the
freedom in the choice of scale of (2.22). Thus, dim (TΞPT ∩ PT) = 2 × (m + 1) − 1 = 2m + 1.
If [Zα] lies in DΞPT , then it takes the form (2.23). In this case, the purity conditions (2.24) do
not yield any further constraints, and thus [Zα] must also lie in PT. �
Definition 2.9. The rank-(m+1) distribution D will be referred to as the canonical distribution
of PT.
When m = 1, the twistor space of Q3 is simply CP3 and the canonical distribution D is the
rank-2 contact distribution annihilated by the contact 1-form α := Γ
(0)
αβZ
αdZβ. The appropriate
generalisation of this contact 1-form to dimension 2m+ 1 is then the set of 1-forms
ααβ := ZAαdZβ
A + 2ZβdZα − ZαdZβ, (2.25)
annihilating the canonical distribution D. Here, the homogeneous coordinates [Zα] are assumed
to satisfy (2.14) or (2.15).
The following lemma follows directly from the exponential map from a given complement of r
in g to a dense open subset of PT.
Lemma 2.10. Let Ξ be a point in PT, and let r be its stabiliser in g. Then DΞ is foliated by
a family of distinguished curves passing through Ξ parametrised by the points of the (m + 1)-
dimensional module r−1, for any decomposition r = r−2 ⊕ r−1 ⊕ r0 ⊕ r1 ⊕ r2.
Geometric correspondences. The bilinear forms (2.2) can also be used to characterise
the intersections of γ-planes in terms of their corresponding pure spinors.
Theorem 2.11 ([10, 17]). Let Z and W be two twistors with homogeneous coordinates [Zα]
and [Wα], and corresponding γ-planes Ž and W̌ in Qn respectively. Then
dim
(
Ž ∩ W̌
)
≥ k ⇐⇒ Γ
(`)
αβZ
αWβ = 0, for all ` ≤ k.
Further, dim(Ž ∩ W̌ ) = k if and only if in addition Γ
(k+1)
αβ ZαWβ 6= 0.
A direct application leads to
Proposition 2.12. Let Ξ and Z be two twistors with corresponding γ-planes Ξ̌ and Ž respec-
tively. Then
1. dim(Ξ̌ ∩ Ž) ≥ m − 3 if and only if there exists W ∈ PT such that W ∈ DΞ ∩ TZPT or
W ∈ DZ ∩TΞPT.
2. dim(Ξ̌∩ Ž) ≥ m− 2 if and only if Ξ ∈ TZPT if and only if Z ∈ TΞPT if and only if there
exists W ∈ PT such that Z,Ξ ∈ DW , or equivalenly W ∈ DZ ∩DΞ.
3. dim(Ξ̌ ∩ Ž) ≥ m− 1 if and only if Z ∈ DΞ if and only if Ξ ∈ DZ .
Proof. We fix Ξα and we assume that Zα is given by (2.20) with components Z(−i) satis-
fying (2.24). In each case, we apply Theorem 2.11 and compute Γ
(`)
αβZ
αWβ = 0 to derive
conditions on Z(−i). With no loss of generality, we may assume Z(0) = 1.
1. We have Z(−i) for all i ≥ 4 and Z(−2) ∧ Z(−2) = 0, i.e., Z(−2) = Φ(−1) ∧ Ψ(−1) for some
Φ(−1),Ψ(−1) ∈ r−1, and Z(−3) = Z(−1)∧Z(−2) = Z(−1)∧Φ(−1)∧Ψ(−1). A suitable W ∈ DΞ∩TZPT
is given by Wα = Ξα + i
2(Z(−1) · Ξ)α and Wα = Zα + 1
4((Φ(−1) ∧Ψ(−1)) ·Z)α, and similarly for
a suitable W ∈ DZ ∩TΞPT.
2. The first two equivalences follow immediately from Proposition 2.8 and Theorem 2.11. For
the last equivalence, we have Z(−i) for all i ≥ 3, so that Z(−2)∧Z(−2) = 0 and Z(−1)∧Z(−2) = 0,
i.e., Z(−2) = Z(−1) ∧ Φ(−1) for some Φ(−1) ∈ r−1. A suitable W ∈ DZ ∩ DΞ is given by
Wα = Ξα + i
2(Z(−1) · Ξ)α and Wα = Zα − i
2(Φ(−1) · Z)α.
3. This follows immediately from Proposition 2.8 and Theorem 2.11. �
Twistor Geometry of Null Foliations in Complex Euclidean Space 13
In a similar vein, we obtain
Proposition 2.13. Fix a twistor Ξ in PT and let Ξ̌ be its corresponding γ-plane in Qn. Let Z
and W be two twistors in TΞPT, corresponding to γ-planes Ž and W̌ . Then dim(Ž∩W̌ ) ≥ m−4.
Further, if Z and W take the respective forms
Zα = Z(0)Ξ
α + i
2
(
Z(−1) · Ξ
)α − 1
4
(
Z(−2) · Ξ
)α
,
Wα = W(0)Ξ
α + i
2
(
W(−1) · Ξ
)α − 1
4
(
W(−2) · Ξ
)α
,
where Z(0)Z(−2) = Z(−1) ∧ Z(−1), Z(−2) ∧ Z(−2) = 0, W(0)W(−2) = W(−1) ∧W(−1) and W(−2) ∧
W(−2) = 0, then
dim
(
Ž ∩ W̌
)
≥ m− 3 ⇐⇒ Z(−2) ∧W(−2) = 0, (2.26a)
dim
(
Ž ∩ W̌
)
≥ m− 2 ⇐⇒ Z(−1) ∧W(−2) +W(−1) ∧ Z(−2) = 0, (2.26b)
dim
(
Ž ∩ W̌
)
≥ m− 1 ⇐⇒ W(−2) − Z(−2) −W(−1) ∧ Z(−1) = 0. (2.26c)
Proof. Let us rewrite
Wα = Zα + i
2
(
Φ(−1) · Z
)α − 1
4
(
Φ(−2) · Z
)α
− i
8
((
Φ(−1) ∧ Φ(−2)
)
· Z
)α
+ 1
32
((
Φ(−2) ∧ Φ(−2)
)
· Z
)α
,
where Φ−1 := W−1 − Z−1 and Φ−2 := W−2 − Z−2 − W−1 ∧ Z−1. It suffices to compute
Γ
(m−k)
αβ ZαWβ = 0 for all k ≥ 4, and
1) Γ
(m−k)
αβ ZαWβ = 0 for all k ≥ 3 if and only if Φ(−2) ∧ Φ(−2) = 0;
2) Γ
(m−k)
αβ ZαWβ = 0 for all k ≥ 2 if and only if Φ(−1) ∧ Φ(−2) = 0;
3) Γ
(m−k)
αβ ZαWβ = 0 for all k ≥ 1 if and only if Φ(−2) = 0.
Equivalences (2.26a), (2.26b) and (2.26c) now follow from the definitions of Φ(−1) and Φ(−2). �
A special case of this proposition is given below.
Corollary 2.14. Fix a twistor Ξ in PT and let Ξ̌ be its corresponding γ-plane in Qn. Let Z
and W be two twistors in DΞ, corresponding to γ-planes Ž and W̌ . Then dim(Ž ∩ W̌ ) ≥ m− 2.
Further, Z and W belong to the same distinguished curve in DΞ, as defined in Lemma 2.10,
if and only if dim(Ž ∩ W̌ ) ≥ m− 1.
Proof. This is a direct consequence of Proposition 2.13 with Z(−2) = W(−2) = 0, and Lem-
ma 2.10. �
2.2.2 Even dimensions
Assume n = 2m. Any non-zero chiral spinor Zα defines a linear map Zα′
A := Γ α′
Aβ Zβ : V→ S,
and similarly for primed spinors. Again, any non-trivial kernel of this map descends to a linear
subspace of Qn. A non-zero chiral spinor Zα is pure if the kernel of Zα
A has maximal dimension
m+ 1, and similarly for primed spinors.
Proposition 2.15 ([10]). The twistor space PT and the primed twistor space PT′ of Q2m are
isomorphic to the projectivations of the spaces of all pure spinors in S and S′ respectively.
14 A. Taghavi-Chabert
When m = 2, all spinors in S and S′ are pure. When m > 2, the analogue of the purity
condition (2.14) is now [10]
Γ
(k)
αβZ
αZβ = 0, for all k < m+ 1, k ≡ m+ 1 (mod 4), (2.27)
or alternatively, [20, 34], ZAα
′
Zβ′
A = 0. Again, we will think of PT and PT′ as complex projective
varieties of PS and PS′ respectively, when m > 2, while for Q4, we have PT(4)
∼= CP3.
The Kleinian model is again a homogeneous space G/R, where R is parabolic. But its
parabolic Lie algebra r this time induces a |1|-grading g = r−1 ⊕ r0 ⊕ r1 on g, where r0 ∼=
gl(m+ 1,C), r−1
∼= ∧2Cm+1 and r1 ∼= ∧2(Cm+1)∗, and r = r0 ⊕ r1, as given in matrix form by
r0 r0 r1 0
r0 r0 r1 r1
r−1 r−1 r0 r0
0 r−1 r0 r0
}1
}m
}m
}1
Again, the one-dimensional center of r0 is spanned by a unique grading element with eigenva-
lues i on ri. In this case, the tangent space of any point of G/R is irreducible and linearly
isomorphic to r−1.
Unlike in odd dimensions, the twistor space of Q2m is not equipped with any canonical
rank-m distribution. As we shall see in Section 2.2.3, one requires an additional structure to
endow PT(2m) with one.
Proposition 2.16. The twistor space PT of a 2m-dimensional smooth quadric Q2m has dimen-
sion 1
2m(m+ 1). Further, for any Z of PT, the projective tangent space TZPT intersects PT in
a (2m− 1)-dimensional linear subspace of PT.
Arguments similar to those used in odd dimensions lead to the following proposition.
Proposition 2.17. Let Z and W be two twistors corresponding to α-planes Ž and W̌ . Then
dim(Ž ∩ W̌ ) ∈ {m− 2,m} if and only if Z ∈ TWPT, or equivalently, W ∈ TZPT. Further, if Z
and W lie in TΞPT for some twistor Ξ in PT, then dim(Ž ∩ W̌ ) ∈ {m− 4,m− 2,m}.
2.2.3 From even to odd dimensions
We note that as 1
2(m + 1)(m + 2)-dimensional projective complex varieties of CP2m+1−1, the
respective twistor spaces PT := PT(2m+1) and P̃T := PT(2m+2) of Q2m+1 and Q2m+2 are iso-
morphic. The only geometric structure that distinguishes the former from the latter is the
rank-(m+ 1) canonical distribution. It is shown in [13] how P̃T can be viewed as a ‘Fefferman
space’ over PT – in fact, this reference deals with a more general, curved, setting. Here, we
explain how the canonical distribution on PT arises as one ‘descends’ from P̃T to PT.
Let Ṽ be a (2m + 4)-dimensional oriented complex vector space equipped with a non-
degenerate symmetric bilinear form h̃AB. Denote by XA the standard coordinates on Ṽ. As
before, we realise Q2m+2 as a smooth quadric of PṼ with twistor spaces P̃T and P̃T
′
induced
from the irreducible spinor representations S̃ and S̃′ of (Ṽ, h̃AB). Now, fix a unit vector UA in Ṽ,
so that Ṽ = U⊕V, where U := 〈UA〉, and V := U⊥ is its orthogonal complement in Ṽ. Then V
is equipped with a non-degenerate symmetric bilinear form hAB := h̃AB − UAUB, and we can
realise Q2m+1 as a smooth quadric of PV with twistor space PT induced from the irreducible
spinor representation S of (V, hAB).
Twistor Geometry of Null Foliations in Complex Euclidean Space 15
Observe that UA defines two invertible linear maps,
Uβ
α′ := UAΓ̃ β
Aα′ : S̃′ → S̃, Uβ′
α := UAΓ̃ β′
Aα : S̃→ S̃′,
where Γ̃ β
Aα′ and Γ̃ β′
Aα generate the Clifford algebra C`(Ṽ, h̃AB). These maps allow us to identi-
fy S̃ with S̃′, and thus P̃T with P̃T
′
. Further, using the Clifford property, it is straightforward
to check that Γ β
Aα := hBAΓ̃ γ′
Bα Uβ
γ′ = −hBAU
γ′
α Γ̃ β
Bγ′ = UBΓ̃ β
ABα generate the Clifford algebra
C`(V, hAB). More generally, the relation between the spanning elements of C`(V, hAB) and those
of C`(Ṽ, h̃AB) is given by
Γ
(k)
A1...Akαβ = hB1A1
· · ·hBkAk Γ̃
(k)
B1...Bkαβ, k ≡ m+ 2 (mod 2), (2.28)
Γ
(k)
A1...Akαβ = UBΓ̃
(k)
A1···AkBαβ = (−1)khB1A1
· · ·hBkAkU
γ′
α Γ̃
(k)
B1···Bkγ′β, k ≡ m+ 1 (mod 2).
If we now introduce homogeneous coordinates [Zα] on PS̃, we can identify the twistor space PT
equipped with its canonical distribution with the twistor space P̃T, as can be seen by inspection
of (2.14) and (2.27). Note that we could have played the same game with P̃T
′
.
Let us interpret this more geometrically. Clearly, the embedding of Q2m+1 into Q2m+2 arises
as the intersection of the hyperplane UAX
A = 0 in PṼ with the cone over Q2m+2. A γ-plane
of Q2m+1 then arises as the intersection of an α-plane of Q2m+2 with Q2m+1, and similarly for β-
planes. An α-plane Ž and a β-plane W̌ define the same γ-plane if and only if their corresponding
twistors satisfy Zα = Uα
β′
Wβ′ . In particular, such a pair must intersect maximally, i.e., in an
m-plane in Q2m+2. This much is already outlined in the appendix of [31].
Finally, we can see how the canonical distribution D on PT arises geometrically from P̃T
and P̃T
′
. Fix a point [Ξα] in P̃T. This represents an α-plane Ξ̌ in Q2m+2, and so a γ-plane
in Q2m+1, which also corresponds to the unique β-plane with associated primed twistor [Uα′
β Ξβ]
in P̃T
′
. We claim that the β-planes intersecting Ξ̌ maximally are in one-to-one correspondence
with the points of DΞ. To see this, let [Zα] be a point in TΞP̃T ⊂ PS̃ so that
Γ̃
(k)
αβZ
αΞβ = 0, for all k < m, k ≡ m (mod 2).
We can then conclude [Zα] ∈ TΞPT by virtue of (2.16) and (2.28) as expected. Now, consider the
set of all β-planes intersecting Ξ̌ maximally: these correspond to all primed twistors [Wα′ ] ∈ P̃T
′
satisfying
Γ̃
(k)
α′βW
α′Ξβ = 0, for all k < m+ 1, k ≡ m+ 1 (mod 2).
Identifying β-planes and α-planes on Q2m+1, i.e., setting Zα = Uα
β′
Wβ′ , and using (2.28) again
precisely yield that [Zα] ∈ DΞ by virtue of (2.17).
2.3 Correspondence space
We now formalise the correspondence between Qn and PT.
2.3.1 Odd dimensions
Assume n = 2m+ 1.
Definition 2.18. The correspondence space F of Qn and PT is the projective complex subvariety
of Qn × PT defined as the set of points ([XA], [Zα]) satisfying the incidence relation
XAZβ
A = 0, (2.29)
where Zβ
A := Γ β
Aα Zα.
16 A. Taghavi-Chabert
The usual way of understanding the twistor correspondence is by means of the double
fibration
F
µ
ν
~~
Qn PT,
where µ and ν denote the usual projections of maximal rank.
Clearly, since, by definition, a twistor [Zα] in PT corresponds to a γ-plane of Qn, namely the
set of points [XA] in Qn satisfying (2.29), we see that each fiber of µ is isomorphic to CPm.
Now, a point x of Qn is sent to a compact complex submanifold x̂ of PT isomorphic to the
fiber Fx of F over x, and similarly, a subset U of Qn will correspond to a subset Û of PT swept
out by those complex submanifolds {x̂} parametrised by the points x ∈ U , i.e.,
x ∈ Qn 7→ Fx := ν−1(x) 7→ x̂ := µ(Fx), U ⊂ Qn 7→ FU :=
⋃
x∈U
ν−1(x) 7→ Û :=
⋃
x∈U
µ(Fx).
To describe x̂, it is enough to describe the fiber Fx. By definition, this is the set of all γ-planes
incident on x. If Ž is a γ-plane incident on x, the intersection Ž ∩ TxQn is an m-dimensional
subspace totally null with respect to the bilinear form on TxQn ∼= CEn, which we shall also refer
to a γ-plane. This descends to a γ-plane in Q2m−1 viewed as the projectivisation of the null cone
through x. Thus, x̂ ∼= Fx is isomorphic to the 1
2m(m + 1)-dimensional twistor space PT(2m−1)
of Q2m−1.
We can get a little more information about F by viewing it as the homogeneous space G/Q
where Q := P ∩R is the intersection of P , the stabiliser of a null line in V, and R the stabiliser
of a totally null (m+ 1)-plane containing that line. The Lie algebra q of Q induces a |3|-grading
on g, i.e., g = q−3⊕ q−2⊕ q−1⊕ q0⊕ q1⊕ q2⊕ q3, where q = q0⊕ q1⊕ q2⊕ q3. For convenience,
we split q±1 and q±2 further as q±1 = qE±1 ⊕ qF±1 and q±2 = qE±2 ⊕ qF±2. Also, q0
∼= gl(m,C)⊕C,
qE−1
∼= Cm, qF−1
∼= (Cm)∗, qE−2
∼= C, qF−2
∼= ∧2Cm and q−3
∼= (Cm)∗ with (qi)
∗ ∼= q−i. In matrix
form, g reads as
q0 qE1 qE2 q3 0
qE−1 q0 qF1 qF2 q3
qE−2 qF−1 0 qF1 qE2
q−3 qF−2 qF−1 q0 qE1
0 q−3 qE−2 qE−1 q0
}1
}m
}1
}m
}1
These modules satisfy the commutation relations [qi, qj ] ⊂ qi+j where qi = {0} for |i| > 3. More
precisely, the action of q1 on these modules, carefully distinguishing qE1 and qF1 , can be recorded
in the form of a diagram:
qE−1
qE−2
qE1
!!
qF1 ==
q−3
qE1
!!
qF1 ==
qF−1
qF−2
qF1 ==
p−1
��
r−2
r1 //
""
r−1
��
Twistor Geometry of Null Foliations in Complex Euclidean Space 17
where the dotted arrows give the relations between q0-modules, and p0- and r0-modules. Invari-
ance follows from the inclusions qE1 ⊂ r0, qF1 ⊂ p0, qE1 ⊂ p1 and qF1 ⊂ r1.
Beside the filtration of vector subbundles of TF determined by the grading on g, we distin-
guish three Q-invariant distributions of interest on F:
• the rank-1
2m(m + 1) distribution T−2
F F corresponding to qF−2 ⊕ qF−1. It is integrable and
tangent to the fibers of ν : G/Q→ G/P , each isomorphic to the homogeneous space P/Q.
This follows from the relations [qF−1, q
F
−1] ⊂ qF−2, [qF−1, q
F
−2] = 0, and [qF−2, q
F
−2] = 0, and the
fact that the kernel of the projection g/q→ g/p is precisely qF−2⊕qF−1
∼= p/q. In fact, since
[qF−1, q
F
−1] ⊂ qF−2, each fiber is itself equipped with the canonical distribution on PT(2m−1).
• the rank-m distribution T−1
E F corresponding to qE−1. It is integrable and tangent to the
fibers of µ : G/Q → G/R, each isomorphic to the homogeneous space R/Q. This follows
from the relations [qE−1, q
E
−1] = 0 and the fact that the kernel of the projection g/q→ g/r
is precisely qE−1
∼= r/q.
• the rank-(2m+1) distribution T−2
E F corresponding to qE−2⊕qF−1⊕qE−1. It is non-integrable
and bracket generates TF since we have [qE−1, q
F
−1] ⊂ qE−2, [qE−1, q
E
−2] = 0, [qE−1, q
F
−2] ⊂ q−3,
[qF−1, q
E
−2] ⊂ q−3. Further, the quotient T−2
E F/T−1
E F descends to the canonical distribution
T−1PT.
The twistor space and correspondence space of CE2m+1. At this stage, we introduce
a splitting (2.3) of V, and as before denote by X̊A, Y̊ A and Z̊Aa vectors in V1, V−1 and V0 respec-
tively. There is an induced splitting (2.8) of S, and we shall accordingly split the homogeneous
twistor coordinates as Zα = (ωA, πA), or, using the injectors, as
Zα = I̊αAω
A + O̊α
Aπ
A. (2.30)
Needless to say that Cartan’s theory of spinors applies to S− 1
2
and S 1
2
in the obvious way and
notation, as we have done in Section 2.2. In particular, a spinor πA is pure if and only if the
kernel of the map πAa := πBγ A
aB is of maximal dimension m, and so on. The purity condition
on Zα can then be re-expressed as follows.
Lemma 2.19. Let Zα = (ωA, πA) be a non-zero spinor in S ∼= S− 1
2
⊕ S 1
2
. Then Zα is pure,
i.e., satisfies (2.14), if and only if ωA and πA satisfy
γ
(k)
ABπ
AπB = 0, for all k < m, k ≡ m+ 1,m (mod 4), (2.31a)
γ
(k)
ABω
AωB = 0, for all k < m, k ≡ m+ 1,m (mod 4), (2.31b)
γ
(k)
ABω
AπB = 0, for all k < m− 1. (2.31c)
Proof. This is a direct computation using (2.15), (2.9) and (2.30). Writing πAa := πBγ A
aB and
ωA
a := ωBγ A
aB , we find
πaAπBa + πAπB = 0, ωaAωB
a + ωAωB = 0, πaAωB
a − πAωB + 2ωAπB = 0,
which are equivalent to (2.31a), (2.31b) and (2.31c) respectively [35]. �
By Cartan’s theory of spinors, condition (2.31a) is equivalent to πA being pure provided it
is non-zero, and similarly for condition (2.31b) and ωA. Condition (2.31c) is equivalent to the
γ-planes of πA and ωA intersecting in an m- or (m−1)-plane in V0 provided these are non-zero.
18 A. Taghavi-Chabert
Remark 2.20. The annihilator (2.25) of the canonical distribution of PT can be re-expressed
as
αAB
(ω,ω) := ωaAdωB
a + 2ωBdωA − ωAdωB,
αAB
(π,π) := πaAdπBa + 2πBdπA − πAdπB,
αAB
(ω,π) := ωaAdπBa + ωAdπB + 4π[AdωB],
αAB
(π,ω) := πaAdωB
a + πAdωB + 4ω[AdπB],
(2.32)
where we have used (2.30) and (2.9), and it is understood that ωA and πA satisfy (2.31).
The twistor correspondence associates to the point∞ inQn, with coordinates [Y̊ A], a complex
submanifold ∞̂ of PT defined by the locus Y̊ AZα
A = 0 in PT, i.e.,
∞ ∈ Qn 7→ F∞ := ν−1(∞) 7→ ∞̂ := µ(F∞) = µ ◦ ν−1(∞).
Points of ∞̂ are parametrised by [ωA, 0]. Since removing ∞ from Qn yields complex Euclidean
space CEn, we accordingly remove ∞̂ to obtain the twistor space PT\{∞̂} = µ◦ν−1(CEn) of CEn.
This will be denoted by PT\∞̂. This region of twistor space is parametrised by {[ωA, πA] :
πA 6= 0}.
The correspondence space of CEn will be denoted FCEn , and is parametrised by the coordi-
nates (xa, [πA]), where {xa} are the flat standard coordinates on CEn and [πA] are homogeneous
pure spinor coordinates on the fibers of F. These parametrise the γ-planes of the tangent space
TxCEn at a point x in CEn, and are related to [ωA, πB] by means of the incidence relation (2.29)
ωA = 1√
2
xaπAa , (2.33)
which can be obtained from (2.7), (2.9) and (2.30). Indeed, the γ-plane defined by [πA] through
the origin is given by the locus 1√
2
xaπAa , so that the γ-plane defined by [πA] through any other
point x̊a is given by (2.33) with ωA = 1√
2
x̊aπAa .
Remark 2.21. By (2.33) and (2.31a), for a holomorphic function f on F to descend to PT, it
must be annihilated by the differential operator π[AπaB]∇a.
2.3.2 Even dimensions
The double fibration picture in dimension n = 2m is very similar to the odd-dimensional case,
and we only summarise the discussion here.
We realise F as a homogeneous space G/Q. Here, the Lie algebra q of Q induces a |2|-grading
g = q−2⊕q−1⊕q0⊕q1⊕q2 on g, where q = q0⊕q1⊕q2. We split q±1 further as q±1 = qE±1⊕qF±1,
and we have q0
∼= gl(m,C)⊕C, qE−1
∼= Cm, qF−1
∼= ∧2Cm and q−2
∼= (Cm)∗ with (qi)
∗ ∼= q−i. The
action of q1 on these q0-modules is recorded below together with the matrix form of the splitting:
q0 qE1 q2 0
qE−1 q0 qF1 q2
q−2 qF−1 q0 qE1
0 q−2 qE−1 q0
}1
}m
}m
}1
qE−1
q−2
qE1
""
qF1 <<
qF−1
p−1
��
r−1
""
Twistor Geometry of Null Foliations in Complex Euclidean Space 19
The modules qF−1 and qE−1 give rise to two integrable Q-invariant distributions T−1
F F and
T−1
E F on F of rank 1
2m(m − 1) and m respectively, and tangent to the fibers of G/Q → G/P
and G/Q→ G/R respectively.
The twistor space and correspondence space of CE2m. The even-dimensional analogue
of Lemma 2.19 is recorded below.
Lemma 2.22. Let Zα = (ωA, πA
′
) be a spinor in S ∼= S− 1
2
⊕ S′1
2
. Then Zα is pure if and only
if ωA and πA
′
satisfy
γ
(k)
A′B′π
A′πB
′
= 0, for all k < m, k ≡ m (mod 4), (2.34a)
γ
(k)
ABω
AωB = 0, for all k < m, k ≡ m (mod 4), (2.34b)
γ
(k)
AB′ω
AπB
′
= 0, for all k < m− 1, k ≡ m− 1 (mod 2). (2.34c)
Conditions (2.34a), (2.34b) and (2.34c) can equivalently be expressed as
πaAπBa = 0, ωaA
′
ωB′
a = 0, πaAωB′
a + 2ωAπB
′
= 0,
respectively. By Cartan’s theory of spinors, condition (2.34a) is equivalent to πA
′
being pure
provided it is non-zero, and similarly for conditions (2.34b) for ωA. Condition (2.34c) is equiva-
lent to the α-plane of πA
′
and the β-plane of ωA intersecting in an (m−1)-plane in V0 provided
these are non-zero.
Just as in the odd-dimensional case, the twistor space of CE2m is obtained by removing the
1
2m(m− 1)-dimensional complex submanifold ∞̂ corresponding to ∞ on Q2m from PT. We can
use [πA
′
] as homogeneous coordinates on the fibers of FCEn , and the incidence relation (2.29)
can be expressed as ωA = 1√
2
xaπAa .
2.4 Co-γ-planes and mini-twistor space
In odd dimensions, there is an additional geometric object of interest.
Definition 2.23. A co-γ-plane is an (m+ 1)-dimensional affine subspace of CE2m+1 with the
property that the orthogonal complement of its tangent space at any of its point is totally null
with respect to the metric.
The space of all co-γ-planes in CE2m+1 is called the mini-twistor space of CE2m+1, and is
denoted MT.
Viewed as a vector subspace of TxCEn ∼= CEn, a co-γ-plane through a point x in CEn is the
orthogonal complement of a γ-plane through x. Consider a co-γ-plane through the origin, and
let [πA] be a projective pure spinor associated to the γ-plane orthogonal to it. Then, it is easy
to check that this co-γ-plane consists of the set of points xa satisfying tπA = 1√
2
xaπAa where
t ∈ C with xaxa = −2t2. Shifting the origin to x̊a say, a point in a co-γ-plane containing x̊a now
satisfies ωA + πAt = 1√
2
xaπAa for some t ∈ C, and where ωA := 1√
2
x̊aπAa . Thus, a co-γ-plane
through x̊a consists of the set of points satisfying the incidence relation
ω[AπB] = 1√
2
xaπ[A
a πB], (2.35)
where [πC] is a projective pure spinor and ωA := 1√
2
x̊aπAa . In particular, a co-γ-plane consists
of a 1-parameter family of γ-planes, and thus corresponds to the curve
C 3 t 7→
[
ωA + πAt, πA
]
∈ PT\∞̂. (2.36)
20 A. Taghavi-Chabert
The relation between MT and PT\∞̂ can be made precise by involving our choice of ‘infini-
ty’ [Y̊ A] to define CEn. Let us write (Y̊ · Z)α := Y̊ AΓ α
Aβ Zβ. We can then define the vector
field
Y := − i
2
(
Y̊ · Z
)α ∂
∂Zα
=
i√
2
πA
∂
∂ωA
, (2.37)
on PT\∞̂, the factors having been added for later convenience. It is now pretty clear that the
curve (2.36) is an integral curve of the vector field (2.37) passing through the point [ωA, πA].
We therefore conclude
Lemma 2.24. Mini-twistor space MT is the quotient of PT\∞̂ by the flow of Y defined by (2.37).
An alternative geometric interpretation can be obtained by introducing weighted homoge-
neous coordinates on MT as follows. Since πA is pure, we can view [πA] as homogeneous
coordinates on PT(2m−1). Let ωa1...am−1
be an (m− 1)-form satisfying
πa1Aωa1a2...am−1
= 0, m > 1. (2.38)
Write [ωa1...am−1
, πA]2,1 for the equivalence class of pairs (ωa1...am−1
, πA) defined by the relation(
ωa1...am−1
, πA
)
∼
(
λ2ωa1...am−1
, λπA
)
for some λ ∈ C∗.
Then [ωa1...am−1
, πA]2,1 constitute weighted homogeneous coordinates on MT. To see this, we
note that for any choice of representative, the condition (2.38) is equivalent to
ωa1...am−1
= γ
(m−1)
a1...am−1ABπ
AωB (2.39)
for some pure spinor ωA satisfying (2.31b). Then, the projection of any [ωA, πA] in PT\∞̂ to
[ωa1...am−1
, πA]2,1 is independent of the choice of representative of [ωA, πA], and further, since πA
is pure, i.e., satisfies (2.31a), sending ωA to ωA + tπA for any t ∈ C leaves (2.39) unchanged.
With these coordinates, we can rewrite the incidence relation (2.35) as
ωa1...am−1
= 1√
2
xaγ
(m)
aa1...am−1ABπ
AπB. (2.40)
Now, turning to the geometrical interpretation, we fix a point π in PT(2m−1) with a choice
of pure spinor πA. Since T−1
π PT(2m−1) is a dense open subset of an m-dimensional linear
subspace of PT(2m−1) containing π, we can identify a vector in T−1
π PT(2m−1) with a point in this
subspace, which can be represented by a pure spinor ωA satisfying (2.31b). At this stage, this
identification is valid provided the scale of πA is fixed. Clearly the origin in T−1
π PT(2m−1) is πA
itself, so that (ωA, πA) maps injectively to (ωa1a2...am−1
, πA). That this map is also surjective
follows immediately from (2.39). Hence, we can conclude
Proposition 2.25. The mini-twistor space MT of CE2m+1 is a 1
2m(m+3)-dimensional complex
manifold isomorphic to the total space of the canonical rank-m distribution T−1PT(2m−1) of the
twistor space PT(2m−1) of Q2m−1.
For clarity, we represent MT by means of an extended double fibration
FCEn
µ
##
η
��
ν
{{
CEn PT\∞̂
τ
��
MT
Twistor Geometry of Null Foliations in Complex Euclidean Space 21
where µ, ν, τ and η are the usual projections. We shall introduce the following notation for
submanifolds of MT corresponding to points in CEn:
x ∈ CEn 7→ Fx := ν−1(x) 7→ x̂ := τ(x̂) = η(Fx),
U ⊂ CEn 7→ FU :=
⋃
x∈U
ν−1(x) 7→ Û := τ
(
Û
)
= η(FU ).
Remark 2.26. For a holomorphic function on F to descend to MT, it must be annihilated by
the differential operator πaA∇a.
2.5 Normal bundles
It will also be convenient to think of the correspondence space as an analytic family {x̂} of
compact complex submanifolds of twistor space parametrised by the points x of Qn. The way
each x̂ is embedded in PT is described by its (holomorphic) normal bundle Nx̂ in PT, which is
the rank-(m+ 1) vector bundle defined by the short exact sequence
0→ Tx̂→ T PT|x̂ → Nx̂→ 0.
As we shall see there are some crucial difference between the odd- and even-dimensional cases.
2.5.1 Odd dimensions
Assume n = 2m+ 1. We first note that the canonical distribution D on PT defines a subbundle
D|x̂ + Tx̂ of TPT|x̂ containing Tx̂. How much of this subbundle descends to Nx̂ is answered by
the following lemma.
Lemma 2.27. Let x be a point in Q2m+1. Then, for any Z ∈ x̂ ⊂ PT, the intersection of DZ
and TZ x̂ has dimension m. In particular, x̂ is equipped with a maximally non-integrable rank-m
distribution T−1x̂ := D|x̂ ∩ Tx̂. Further, there is a distinguished line subbundle of the normal
bundle Nx̂ of x̂ given by N−1x̂ := (D|x̂ + Tx̂)/Tx̂.
Proof. Denote by [XA] the homogeneous coordinates of x ∈ Q2m+1, and let Ξ ∈ x̂ ⊂ PT so
that XAΞα
A = 0. Then, by Lemma 2.7 and Proposition 2.8, a vector tangent to DΞ can be
identified with a point Zα = Ξα + i
2(Z(−1) · Ξ)α of a dense open subset of DΞ ⊂ PT. Here,
Z(−1) ∈ r−1
∼= Cm+1 lies in a complement of the stabiliser r of Ξ as explained in Section 2.2.
The condition that this vector is also tangent to x̂ is equivalent to 0 = XAZα
A = −2XAZ
A
(−1)Ξ
α,
by (2.1), i.e., XAZ
A
(−1) = 0. This gives a single additional algebraic condition on ZA(−1), and thus
the intersection of DΞ and TΞx̂ is m-dimensional (for a description in affine coordinates, see
the end of Appendix A.1). This defines a rank-m distribution T−1x̂ := D|x̂ ∩Tx̂ on x̂. Since D
is maximally non-integrable, so must be T−1x̂. That the subbundle N−1x̂ := (D|x̂ + Tx̂)/Tx̂
of Nx̂ is of rank 1 follows from the isomorphism D|x̂/(D|x̂ ∩ Tx̂) ∼= (D|x̂ + Tx̂)/Tx̂. �
That x̂ is endowed with a canonical rank-m distribution comes as no surprise since each x̂ is
isomorphic to the generalised flag manifold P/Q ∼= PT(2m−1).
As explained in [24], the tangent space at a point x of Q2m+1 injects into H0(x̂,O(Nx̂)),
the space of global holomorphic sections of Nx̂. If V a is a vector in TxQ2m+1 and y the point
infinitesimally separated from x by V a, then the corresponding section of H0(x̂,O(Nx̂)) can
be identified with ŷ. Let us fix x to be the origin in CE2m+1 ⊂ Q2m+1. Then V a can be
identified with ya. We view πA as coordinates on x̂ given by the locus ωA = 0. The infinitesimal
displacement of x̂ along V a at the origin is V A := V a∇aωA, i.e., V A = 1√
2
V aπAa . This represents
a global holomorphic section V̂x̂ of Nx̂, and can be identified with the complex submanifold ŷ
given by ωA = 1√
2
yaπAa .
Before describing such sections, we shall need the following two lemmata.
22 A. Taghavi-Chabert
Lemma 2.28. Let V a be a non-zero vector in CE2m+1, and let V B
A := V aγ B
aA be the corre-
sponding spin endomorphism. Then V a is null if and only if V B
A has a zero eigenvalue. Further,
• if V a is null, V B
A has a single zero eigenvalue of algebraic multiplicity 2m, and its eigenspace
is isomorphic to the 2m−1-dimensional spinor space of CE2m−1,
• if V a is non-null, V B
A has a pair of eigenvalues ±i
√
V aVa, each of algebraic multiplici-
ty 2m−1, and their respective eigenspaces are isomorphic to the 2m−1-dimensional chiral
spinor spaces of CE2m.
Proof. By the Clifford property, we have V C
A V B
C = −V aVaδ
B
A, and it follows that any eigenvalue
of V B
A must be equal to ±i
√
V aVa. Hence V a is null if and only if it has a zero eigenvalue. This
zero eigenvalue must be of algebraic multiplicity 2m since in this case V B
A is nilpotent. One can
check that the kernel of V B
A can be identified with the 2m−1-dimensional spinor space of CE2m−1
as the orthogonal complement of V a in CE2m+1 quotiented by 〈V a〉.
If V a is non-null, the square of V B
A is proportional to the identity, and thus, each of the
eigenvalues ±i
√
V aVa must have algebraic multiplicity 2m−1. Each of the eigenspaces can be
identified with each of the chiral spinor spaces of CE2m as the orthogonal complement of 〈V a〉
in CE2m+1 – see, e.g., [31]. �
Lemma 2.29. Let x and y be two points in Q2m+1 infinitesimally separated by a non-null
vector V a. Then, for every Z ∈ x̂ ⊂ PT such that V a is tangent to the co-γ-plane Ž⊥ ⊂
TxQ2m+1, DZ intersects ŷ in a unique point W , say, such that the corresponding γ-planes Ž
and W̌ intersect maximally.
Proof. With no loss of generality, we may assume that x is the origin in CE2m+1 ⊂ Q2m+1.
We then have V a = ya. Since V a is non-null, it must lie on some co-γ-plane of some twistor Z.
Following the discussion of Section 2.4, it can be represented by a 1-parameter family of γ-
planes. In particular, y must lie on one such γ-plane. If Z is a point on x̂, then [Zα] = [0, πA]
for some πA. The condition that y lies on the co-γ-plane Z⊥ is that πA is an eigenspinor of V a
with eigenvalue t or −t where t := i
√
V aVa. For definiteness, let us assume that the eigenvalue
is t. With reference to (2.36), the point ya lies in the γ-plane W given by [Wα] = [tπA, πA].
Re-expressing this twistor as Wα = Ξα − t
2 Y̊
AΓ α
Aβ Ξα, we see, by Lemma 2.7, that W lies in
the intersection of DZ and ŷ. In fact, one can see that the connecting vector from Z to W is
given by
√
V aVaY , where Y is given by (2.37). Finally, by Proposition 2.12, Z and W must
intersect maximally. �
Proposition 2.30. Let x be a point in Q2m+1 with corresponding submanifold x̂ in PT. Let V
be a tangent vector at x, and V̂x̂ its corresponding global holomorphic section of Nx̂.
• Suppose V is null. When m = 1, V̂x̂ vanishes at a single point on x̂, which corresponds
to the unique γ-plane (i.e., null line) to which V is tangent. When m > 1, there is
a 1
2m(m − 1)-dimensional algebraic subset of x̂ biholomorphic to PT(2m−3) on which V̂x̂
vanishes. Each point of this subset corresponds to a γ-plane to which V is tangent.
• Suppose V is non-null. When m = 1, there are precisely two points, Z± say, on x̂, at which
V̂x̂(Z±) ∈ N−1
Z±
x̂. Further, V is tangent to the two co-γ-planes determined by Z±. When
m > 1, there are two disjoint 1
2m(m−1)-dimensional algebraic subsets of x̂, biholomorphic
to PT(2m−2) and PT′(2m−2), over which V̂x̂ is a section of N−1x̂. Each point of these subsets
corresponds to a co-γ-plane to which V is tangent.
Conversely, if V̂x̂ vanishes at a point, then V must be null, and if V̂x̂(Z) ∈ N−1
Z x̂ for some Z ∈ x̂,
then V must be non-null.
Twistor Geometry of Null Foliations in Complex Euclidean Space 23
Proof. Again, let us assume that x is the origin in CE2m+1 ⊂ Q2m+1, and set V B
A := V aγ B
aA .
If V a is null, the vanishing of V̂x̂ of a point πA of x̂ is simply equivalent to V B
A π
A = 0, i.e.,
πA is a pure eigenspinor of V B
A . By Lemma 2.28, we can immediately conclude that V̂x̂ vanishes
at a point when m = 1, and on a subset of x̂ biholomorphic to PT(2m−3) when m > 1. Clearly,
each point of this subset corresponds to a γ-plane to which V a is tangent.
If V a is non-null, we know by Lemma 2.28 that V B
A has eigenvalues ±i
√
V aVa. In particular,
the pure eigenspinors up to scale determine two distinct points on x̂ when m = 1, and two
disjoint subsets of x̂ biholomorphic to the twistor spaces PT(2m−2) and PT′(2m−2) when m > 1.
A point Z on any of these sets corresponds to a co-γ-plane Ž⊥ to which V a is tangent. By
Lemma 2.29, the corresponding submanifold ŷ intersects DZ at a point W . The connecting
vector from Z to W clearly lies in DZ , but is not tangent to x̂. In particular, it descends to an
element of N−1
Z x̂. Thus, the restriction of V̂x̂ to these subsets is a section of N−1x̂.
Finally, if V̂x̂ vanishes at a point Z say, then V is tangent to the γ-plane Ž, and so must be
null. The non-null case is similar. �
2.5.2 Mini-twistor space
For any point x of Qn, the normal bundle Nx̂ of x̂ in MT is given by 0→Tx̂→T MT|x̂→Nx̂→0.
In this case, Nx̂ can be identified with T−1x̂, i.e., mini-twistor space itself, as follows from the
description of Section 2.4: taking x in CEn to be the origin, then the complex submanifold x̂
in MT is defined by ωa1...am−1
= 0, πA will be coordinates on x̂, and we shall view ωa1...am−1
as
coordinates off x̂.
Again, for any x ∈ CEn, TxCEn injects into H0(x̂,O(Nx̂)). If x is the origin and V ∈ TxCEn
be the vector connecting x to a point y, we can identify the global holomorphic section V̂x̂ of Nx̂
as in the previous section. If πA are coordinates on x̂ given by the locus ωA = 0, V̂x̂ can be
identified with the complex submanifold ŷ given by ωa1...am−1
= 1√
2
yaγ
(m)
aa1...am−1ABπ
AπB, where
ya = V a.
Proposition 2.31. Let x be a point in CE2m+1 with corresponding submanifold x̂ in MT. Let V
be a tangent vector at x, and V̂x̂ its corresponding global holomorphic section of Nx̂.
• Suppose V is null. When m = 1, V̂x̂ has a double zero, which corresponds to the γ-plane to
which V is tangent. When m > 1, V̂x̂ vanishes on a 1
2m(m−1)-dimensional algebraic subset
of x̂ biholomorphic to PT(2m−1) of multiplicity 2m. Each point of this subset corresponds
to a γ-plane to which V is tangent.
• Suppose V is non-null. When m = 1, V̂x̂ has two simple zeros, each of which determines
a co-γ-plane to which V is tangent. When m > 1, V̂x̂ vanishes on two disjoint 1
2m(m−1)-
dimensional algebraic subsets of x̂ bihomolomorphic to PT(2m−2) and PT′(2m−2), each of
multiplicity 2m−1. Each point of these subsets corresponds to a co-γ-plane to which V is
tangent.
Proof. With no loss, we assume that x is the origin in CE2m+1 ⊂ Q2m+1, and set V B
A :=
V aγ B
aA . To determine the zero set of V̂ x̂, we simply remark that V aγ
(m)
aa1...am−1ABπ
AπB = 0 is
equivalent to the eigenspinor equation πCV
[A
C πB] = 0. We can then proceed as in the proof of
Proposition 2.30 according to whether V a is null or non-null, and obtain the required zero sets
of the section V̂x̂ in each case, the multiplicities being given by the algebraic multiplicities of
the eigenvalues of V A
C . In particular, when m = 1, the solution set is defined by the vanishing
of a single homogeneous polynomial of degree 2, which has two distinct roots generically, but
a single root of multiplicity two when V a is null – see, e.g., [21]. �
24 A. Taghavi-Chabert
2.5.3 Even dimensions
The analysis when n = 2m is very similar to the odd-dimensional case without the added
complication of the canonical distribution. Again, for any x of Q2m, TxCE2m injects into
H0(x̂,O(Nx̂)). A null vector in V a is TxCE2m defines a global section V̂x̂ of Nx̂, which vanishes
at a single point when m = 2, and on a 1
2(m − 1)(m − 2)-dimensional algebraic subset of x̂,
isomorphic to PT(2m−2), when m > 2. Each point of this subset corresponds to an α-plane to
which V a is tangent.
2.5.4 Kodaira’s theorem and completeness
Let us now turn to the question of whether TxQn maps to H0(x̂,O(Nx̂)) bijectively, and not
merely injectively, for any x ∈ Qn. By Kodaira’s theorem [24], TxQn ∼= H0(x̂,O(Nx̂)) ∼= Cn if
and only if the family {x̂} in PT is complete, i.e., any infinitesimal deformation of x̂ arises from
an element of TxQn. As we have seen in Section 2.2.3, the twistor space PT of Q2m+1 and the
twistor space P̃T of Q2m+2 are both 1
2(m+ 1)(m+ 2)-dimensional complex projective varieties
in CP2m+1−1, and it is the embedding Q2m+1 ⊂ Q2m+2 that induces the canonical distribution D
on PT. The issue here is that Kodaira’s theorem is only concerned with the holomorphic structure
of the underlying manifolds, and does not depend on the additional distribution on PT.
Now, by the twistor correspondences, any point x inQ2m+1 andQ2m+2 gives rise to a 1
2m(m+
1)-dimensional complex submanifold x̂ of PT and P̃T respectively. This means that the ana-
lytic family {x̂} parametrised by the points {x} of Q2m+1 can be completed to a larger fa-
mily parametrised by the points {x} of Q2m+2 via the embedding Q2m+1 ⊂ Q2m+2. Further,
a complex submanifold x̂ corresponds to a point x in Q2m+1 if and only if x̂ is tangent to an
m-dimensional subspace of DZ at every point Z ∈ x̂.
We also need to check whether the family of x̂ is complete when x ∈ Q2m+2. If it were
not, one would be able to find a group of biholomorphic automorphisms of PT larger than
Spin(2m + 4,C) and a parabolic subgroup such that the quotient models PT. But the work of
[13, 28] tells us that there is no such group. The same applies to each x̂, and since these are
biholomorphic to flag varieties, the normal bundle Nx̂ can be identified with a rank-(m + 1)
holomorphic homogeneous vector bundle over x̂. In the notation of [5], we find that for a point x
in Q2m+1 or Q2m+2, the normal bundle Nx̂ in PT ∼= P̃T is given by
m = 1 m > 1
×× ×× 11 • • •
×
•
• • •
×
•
• • •
×
•
• • •
×
•
• • •
×
•
1 0 0
0
0︸ ︷︷ ︸
m + 1 nodes
Here, the mutilated Dynkin diagram corresponds to the parabolic subalgebra underlying the flag
variety x̂, and the coefficients over the nodes to the irreducible representation that determines
the vector bundle. When m = 1, i.e., for Q3 and Q4, we recover the well-known result Nx̂
∼=
Ox̂(1) ⊕ Ox̂(1), where Ox̂(1) is the hyperplane bundle over x̂ ∼= CP1. We can compute the
cohomology using the Bott–Borel–Weil theorem, and verify that indeed H0(x̂,O(Nx̂)) ∼= C2m+2
and H1(x̂,O(Nx̂)) = 0 – this latter condition tells us that there is no obstruction for the existence
of our family.
We can play the same game with the family of compact complex submanifolds {x̂} in MT
parametrised by the points x of CE2m+1. But in this case, for any x of CE2m+1, the normal
bundle Nx̂ is essentially the total space of T−1x̂→ x̂, and is described, in the notation of [5], as
Twistor Geometry of Null Foliations in Complex Euclidean Space 25
the rank-m holomorphic homogeneous vector bundle
m = 1 m > 1
×2 • • • ו • • ו • • ו • • ×>1 0 0 0︸ ︷︷ ︸
m nodes
When m = 1, i.e., Q3, x̂ ∼= CP1, and we recover the well-known result O(Nx̂) ∼= Ox̂(2) :=
⊗2Ox̂(1). Again, the Bott–Borel–Weil theorem confirms that H0(x̂,O(Nx̂)) ∼= C2m+1 and
H1(x̂,O(Nx̂)) = 0.
Remark 2.32. When n = 3, this analysis was already exploited in [25] in the curved setting,
where the twistor space of a three-dimensional holomorphic conformal structure is identified
with the space of null geodesics. See also [18].
3 Null foliations
As before, we work in the holomorphic category throughout, i.e., vector fields and distributions
will be assumed to be holomorphic.
Definition 3.1. An almost null structure is a holomorphic totally null m-plane distribution
on Qn, where n = 2m or 2m+ 1.
In other words, an almost null structure is a γ-plane, α-plane or β-plane distribution. From
the discussion of Section 2.3, an almost null structure, self-dual when n = 2m, can be viewed as
a holomorphic section of F→ Qn, or equivalently as a projective pure spinor field on Qn, that
is a spinor field defined up to scale, and which is pure at every point. The geometric properties
of an almost null structure on a general spin complex Riemannian manifold can be expressed in
terms of the differential properties of its corresponding projective pure spinor field as described
in [34, 35].
The question we now wish to address is the following one: given an almost null structure,
how can we encode its geometric properties in twistor space PT?
3.1 Odd dimensions
When n = 2m + 1, an almost null structure is more adequately expressed as an inclusion of
distributions N ⊂ N⊥ where N is a holomorphic totally null m-plane distribution and N⊥ is
its orthogonal complement. One can then investigate the geometric properties of N and N⊥
independently. In the following, Γ(U ,O(N)) denotes the space of holomorphic sections of N
over an open subset U of Qn, and similarly for N⊥.
Definition 3.2. Let N ⊂ N⊥ be an almost null structure on some open subset U of Qn. We
say that N is
• integrable if [X,Y ] ∈ Γ(U ,O(N)) for all X,Y ∈ Γ(U ,O(N)),
• totally geodetic if ∇YX ∈ Γ(U ,O(N)) for all X,Y ∈ Γ(U ,O(N)),
• co-integrable if [X,Y ] ∈ Γ(U ,O(N⊥)) for all X,Y ∈ Γ(U ,O(N⊥)),
• totally co-geodetic if ∇YX ∈ Γ(U ,O(N⊥)) for all X,Y ∈ Γ(U ,O(N⊥)).
An integrable almost null structure will be referred to as a null structure.
There is however some dependency regarding the geometric properties of N and N⊥.
26 A. Taghavi-Chabert
Lemma 3.3 ([35]). Let N be an almost null structure. Then
• if N is totally co-geodetic, it is also integrable and co-integrable,
• if N is integrable and co-integrable, it is also geodetic,
• if N is totally geodetic, it is also integrable.
Another important point is the conformal invariance of the above properties. All with the
exception of the totally co-geodetic property are conformal invariant – see [35].
3.1.1 Local description
The next theorems will be local in nature. That means that we shall work on CEn viewed
as a dense open subset of Qn. For their proofs, we shall make use of the local coordinates
on CEn, FCEn and PT\∞̂ given in Appendix A.1. Let N be an almost null structure on some
open subset U of CEn = {zA, zA, u}, and view N as a local holomorphic section of F → CEn,
i.e., a holomorphic projective pure spinor field [ξA]. We may assume that locally, [ξA] defines
a complex submanifold of U×U0, where (U0, (π
A, πAB)) is a coordinate chart on the fibers of FU ,
given by the graph
Γξ :=
{
(x, π) ∈ U × U0 : πAB = ξAB(x), πA = ξA(x)
}
, (3.1)
for some 1
2m(m − 1) and m holomorphic functions ξAB = ξ[AB] and ξA respectively on U . In
this case, the distribution N is spanned by the m holomorphic vector fields
ZA = ∂A +
(
ξAD − 1
2ξ
AξD
)
∂D + ξA∂, (3.2)
while its orthogonal complement N⊥ by the m+ 1 holomorphic vector fields
ZA = ∂A +
(
ξAD − 1
2ξ
AξD
)
∂D + ξA∂, U = ∂ − ξD∂D, (3.3)
where ∂A := ∂
∂zA
, ∂A := ∂
∂zA
and ∂ := ∂
∂u . Here, we shall make a slight abuse of notation by
denoting the vector fields spanning N and N⊥, and their lifts to FU , both by (3.2) and (3.3).
Remark 3.4. It will be understood that when m = 1 there are no coordinates πAB. This does
not affect the veracity of the following results in this case – see however Remark 3.7.
3.1.2 Totally geodetic null structures
Let W be an (m + 1)-dimensional complex submanifold of PT and let U be an open subset
of Q2m+1. Suppose that for every point x of U , x̂ ∈ Û intersects W transversely in a point.
Then each point of W ∩ x̂ determines a point in the fiber Fx, and thus a γ-plane through x.
Smooth variations of the point x in U thus define a holomorphic section of FU → U and an
(m + 1)-dimensional analytic family of γ-planes, each of which being the totally geodetic leaf
of an integrable almost null structure. Conversely, consider a local foliation by totally null and
totally geodetic m-dimensional leaves. Then, each leaf must be some affine subset of a γ-plane.
The (m+ 1)-dimensional leaf space of the foliation constitutes an (m+ 1)-dimensional analytic
family of γ-planes, and thus defines an (m+ 1)-dimensional complex submanifold of PT.
Theorem 3.5. A totally geodetic null structure on some open subset U of Q2m+1 gives rise to
an (m+1)-dimensional complex submanifold of Û ⊂ PT intersecting x̂ ⊂ Û transversely for each
x ∈ U . Conversely, any totally geodetic null structure locally arises in this way.
Twistor Geometry of Null Foliations in Complex Euclidean Space 27
Proof. Let N be an almost null structure as described in Section 3.1.1. The condition that N
be totally geodetic is g(∇ZAZB,ZC) = g(∇ZAZB,U) = 0, i.e.,(
∂A +
(
ξAD − 1
2ξ
AξD
)
∂D + ξA∂
)
ξBC = 0,(
∂A +
(
ξAD − 1
2ξ
AξD
)
∂D + ξA∂
)
ξB = 0.
(3.4)
We re-express the system (3.4) of holomorphic partial differential equations as
ρABC +
(
πAD − 1
2π
AπD
)
ρBCD + πAρBC = 0,
σAB +
(
πAD − 1
2π
AπD
)
σBD + πAσB = 0,
(3.5)
where ρABC := ∂AπBC , ρBCA := ∂Aπ
BC , ρAB := ∂πAB, σAB := ∂AπB, σBA := ∂Aπ
B, σA := ∂πA.
In the language of jets, the locus (3.5) defines a complex submanifold of the first jet space
J 1(CEn,U0), of which the prolongation of the section Γξ is a submanifold. Now, the distribution
T−1
E F = 〈ZA〉 tangent to the fibers of F → PT is annihilated by the 1-forms dπA, dπAB, θA
and θ0 as defined in Appendix A.1, which can be pulled back to J 1(CEn,U0). The 1-forms
defined by
φA := dπA − σACθC −
(
σA − σACπC
)
θ0,
φAB := dπAB − ρABC θC −
(
ρAB − ρABC πC
)
θ0,
(3.6)
vanish on the locus (3.5), and this implies in particular that, for generic ρABC , ρBC , ρAC , ρC ,
the section Γξ must be constant along the fibers of T−1
E F, i.e., the functions (ξA, ξAB) depend
only on the coordinates (ω0, ωA, πA, πAB) of the chart V0 of PT. Thus, quotienting Γξ along the
fibers of F→ PT yields an (m+ 1)-dimensional complex submanifold of PT intersecting each x̂
transversely in a point.
The converse is also true: we start with an (m + 1)-dimensional complex submanifold W,
say, of PT, which can be locally represented by the vanishing of 1
2m(m + 1) holomorphic func-
tions (FAB, FA) on the chart (V0, (ω
0, ωA, πA, πAB)). Then (dFAB,dFA) are a set of 1-forms
vanishing on W. We shall assume that for each x ∈ U , the submanifold x̂ ⊂ Û intersects W
transversely in a point. This singles out a local holomorphic section [ξA] of U ×U0 ⊂ F→ U . By
the implicit function theorem, we may assume with no loss of generality that this is the graph Γξ
given by (3.1). The pullbacks of (dFAB,dFA) to F vanish on Γξ and give the restriction(
QAC QACD
QABC QABCD
)(
dπC
dπCD
)
+
(
Y FA XCF
A
Y FAB XCF
AB
)(
θ0
θC
)
=
(
0
0
)
, (3.7)
where
(
QAC QACD
QABC QABCD
)
:=
(
∂
∂πC
+
1
2
u
∂
∂ωC
− zC
∂
∂ω0
)
FA
(
∂
∂πCD
+ z[C
∂
∂ωD]
)
FA(
∂
∂πC
+
1
2
u
∂
∂ωC
− zC
∂
∂ω0
)
FAB
(
∂
∂πCD
+ z[C
∂
∂ωD]
)
FAB
. (3.8)
At generic points, the matrix (3.8) is invertible, and equations (3.7) can immediately be seen to
be equivalent to the vanishing of the forms (3.6). In particular, πAB = ξAB(x) and πA = ξA(x)
satisfy (3.4), i.e., the distribution associated to the graph Γξ is integrable and totally geo-
detic. �
28 A. Taghavi-Chabert
3.1.3 Co-integrable null structures
Let us now suppose that our almost null structure N is integrable and co-integrable on U . We
then have two foliations of U , one for N and the other for N⊥. By Lemma 3.3, we know that each
leaf of N is totally geodetic and therefore a γ-plane. Since N ⊂ N⊥, each (m+ 1)-dimensional
leaf of N⊥ contains a one-parameter holomorphic family {Žt} of γ-planes, i.e., of leaves of N .
Thus each leaf of N⊥ descends to a holomorphic curve on the leaf space of N . In particular,
by Theorem 3.5, we can identify the leaf space of N with an (m + 1)-dimensional complex
submanifold W of PT foliated by curves, each of which being a one-parameter of twistors {Zt}
and, as we shall show, tangent to the canonical distribution D of PT.
We start by the remark that at any point Z ofW, any submanifold x̂ intersectsW transversely,
i.e., TZPT = TZ x̂ ⊕ TZW. Hence, by Lemma 2.27 the intersection of DZ with TZW can only
be at most one-dimensional. Now, let Z0 and Zt be two points on W corresponding to two
infinitesimally separated γ-planes, Ž0 and Žt in {Žt}, contained in the co-γ-plane Ž⊥0 . Let x
and y be points on Ž0 and Žt respectively, so that their corresponding complex submanifolds x̂
and ŷ of Û intersect W in Z0 and Zt respectively. The vector V a in TxU tangent to Ž⊥0
connecting x to y is non-null, and we know by Lemma 2.29 that the vector connecting Z0 to Zt
must lie in DZ0 . This is clearly independent of the choice of points x and y on Ž0 and Žt.
Assigning a vector tangent to DZt at every point of {Zt} yields a curve corresponding to a leaf
of N⊥. Proceeding in this way for each leaf of N⊥ gives rise to a foliation by holomorphic curves
tangent to D on W. Conversely, any such foliation by curves on a given (m + 1)-submanifold
of PT gives rise to an integrable and co-integrable almost null structure.
Theorem 3.6. An integrable and co-integrable almost null structure on some open subset U
of Q2m+1 gives rise to an (m + 1)-dimensional complex submanifold of Û ⊂ PT foliated by
holomorphic curves tangent to D and intersecting x̂ ⊂ Û transversely for each x ∈ U . Conversely,
any integrable and co-integrable almost null structure locally arises in this way.
Proof. We recycle the setting and notation of the proof of Theorem 3.5. In particular, we
take N and N⊥ to be spanned by the vector fields (3.2) and (3.3). The assumption that N
be integrable and co-integrable, i.e., g(∇ZAZB,ZC) = g(∇ZAZB,U) = g(∇UZB,ZC) = 0,
gives (3.4) and in addition,(
∂ − ξD∂D
)
ξBC +
((
∂ − ξD∂D
)
ξ[B
)
ξC] = 0. (3.9)
Thus, the system {(3.4), (3.9)} can be encoded as the complex submanifold of J 1(CEn,U0)
arising from the intersection of the locus (3.5) and the locus
ρBC − πDρBCD + σ[BπC] − πDσ[B
D π
C] = 0, (3.10)
and the prolongation of Γξ must lie in this intersection. Now, let us define ψAB := φAB−π[AφB],
where φA and φAB are the 1-forms (3.6). From the proof of Theorem 3.5, the 1-forms ψAB
and φA vanish on the locus (3.5). On the other hand, on restriction to the locus (3.10), we have
ψAB = αAB −
(
ρABC − π[Aσ
B]
C
)
θC , where 〈αAB,θA〉 annihilate the rank-(2m+ 1) distribution
T−2
E F =
〈
U ,WA,Z
A
〉
. One can further check that 〈ψAB,φA〉 annihilate the m+ 1 vector fields
U +
(
σA − σABπB
)
WA and ZA. These span a rank-(m + 1) subdistribution L, say, of T−2
E F
tangent to Γξ. By Theorem 3.5, Γξ descends to an (m+1)-dimensional complex submanifoldW
of PT. The quotient L/T−1
E F is a rank-1 subbundle of T−2
E F/T−1
E F, which also descends to
a rank-1 subdistribution of D = T−1PT tangent toW. This proves the first part of the theorem.
Conversely, consider a complex submanifold W of PT, transverse to every x̂ in Û , given
by the vanishing of holomorphic functions (FAB, FA) on the chart (V0, (ω
0, ωA, πA, πAB)). By
Theorem 3.5, we can associate to W a local section [ξA] of U × U0 ⊂ F with graph Γξ, so that
Twistor Geometry of Null Foliations in Complex Euclidean Space 29
equations (3.5) hold. Assume further that the intersection of TW and D|W is one-dimensional at
every point. Then the pullbacks of (dFAB,dFA) to U×U0 ⊂ F must vanish on Γξ and annihilate
both T−1
E F and a rank-(m + 1) subbundle of T−2
E F ⊃ T−1
E F. Thus, there exists a vector field
V = U + V AWA, for some holomorphic functions V A on Γξ, annihilating the 1-forms (3.6). It
is then straightforward to check that this gives us precisely the additional restrictions (3.10). In
particular, πAB = ξAB(x) and πA = ξA(x) satisfy (3.4) and (3.9), i.e., the distribution associated
to the graph Γξ is integrable and co-integrable. �
Remark 3.7. When n = 3, Theorems 3.5 and 3.6 are equivalent: since PT is 3-dimensional
and D has rank 2, any 2-dimensional complex submanifold of PT satisfying the transversality
property of the theorems must have non-trivial intersection with D.
3.1.4 Totally co-geodetic null structures
Finally, we consider a totally co-geodetic null structure N . The key point here is that this
stronger requirement is not conformally invariant, and for this reason, the appropriate arena
is the mini-twistor space MT of CE2m+1. In this case, each leaf of the foliation of N⊥ is
totally geodetic, and must therefore be a co-γ-plane. The m-dimensional leaf space can then be
identified as an m-dimensional complex submanifold W of MT.
Alternatively, we can recycle the setting of Theorems 3.5 and 3.6: since N is in particular inte-
grable and co-integrable, its leaf space is an (m+1)-dimensional complex submanifoldW of PT\∞̂
foliated by curves. However, these curves are very particular since they correspond to totally
geodetic leaves of N⊥. Breaking of the conformal invariance can be translated into these curves
being the integral curves of the vector field Y induced by the point ∞ on Qn. Quotienting the
submanifold W by the flow of Y thus yields an m-dimensional complex submanifold W of MT.
Theorem 3.8. A totally co-geodetic null structure on some open subset U of CE2m+1 gives rise
to an m-dimensional complex submanifold of Û ⊂ MT intersecting each x̂ ⊂ Û transversely for
each x ∈ U . Conversely, any totally co-geodetic null structure locally arises in this way.
Proof. Suppose N and N⊥ are both integrable as in the previous section. As already pointed
out the integral manifolds of N are totally geodetic. We now impose the further assump-
tion that the integral manifolds of N⊥ are also totally geodetic on U , i.e., g(∇ZAZB,ZC) =
g(∇ZAZB,U) = g(∇UZB,ZC) = g(∇UZA,U) = 0. Then, in addition to (3.4), we have(
∂ − ξD∂D
)
ξAB = 0,
(
∂ − ξD∂D
)
ξA = 0, (3.11)
which can be seen to imply (3.9). As before, using the same notation as in the proof of Theo-
rem 3.5, we express the system (3.4), (3.11) as a complex submanifold of J 1(CEn,U0) defined
by (3.5) and
ρAB − πDρABD = 0, σA − πDσAD = 0. (3.12)
In particular, the 1-forms dπAB − ρABC θC and dπA− σACθC vanish on the locus (3.5) and (3.12),
and this implies in particular that, for generic ρABC , ρBC , ρAC , ρC , the section Γξ must be constant
along the fibers of F → MT, i.e., the functions (ξA, ξAB) depend only on the coordinates
(ωA, πA, πAB) on the chart V0 of MT. Thus, quotienting Γξ along the fibers of F→ MT yields
an m-dimensional complex submanifold of MT.
For the converse, we simply run the argument backwards as in the proof of Theorem 3.5. �
30 A. Taghavi-Chabert
3.2 Even dimensions
The even-dimensional case is somewhat more tractable than the odd-dimensional case. For one,
the orthogonal complement of an α-plane or β-plane distribution N is N itself, i.e., N⊥ = N .
Definition 3.2 still applies albeit with much redundancy. In particular, N is integrable if and
only if it is co-integrable. The question now reduces to whether N is integrable or not, and if
so, whether it is totally geodetic. But it turns out that these two questions are equivalent.
Lemma 3.9. An almost null structure is integrable if and only if it is totally geodetic.
For a proof, see for instance [33, 34]. The argument leading up to Theorem 3.5 equally
applies to the even-dimensional case – simply substitute γ-plane for α-plane. For the sake of
completeness, we restate the theorem, which was first used in four dimensions in [23], reformu-
lated in twistor language in [29], and generalised to higher even dimensions in [20]. The proof
of Theorem 3.5 can be recycled entirely by ‘switching off’ the coordinates u, ω0, πA, and so on.
Theorem 3.10 ([20]). A self-dual null structure on some open subset U of Q2m gives rise to an
m-dimensional complex submanifold of Û ⊂ PT intersecting x̂ in Û transversely for each x ∈ U .
Conversely, any self-dual null structure locally arises in this way.
4 Examples
We now give two examples of co-integrable null structures that will illustrate the mechanism of
Theorems 3.6 and 3.10. These arise in connections with conformal Killing spinors and conformal
Killing–Yano 2-forms, and are more transparently constructed in the language of tractor bundles
reviewed in Section 2.1.1. As before, we work in the holomorphic category.
4.1 Conformal Killing spinors
For definiteness, let us stick to odd dimensions, i.e., n = 2m+ 1. The even-dimensional case is
similar. A (holomorphic) conformal Killing spinor on Qn is a section ξA of OA that satisfies
∇aξA + 1√
2
γ A
aB ζB = 0, (4.1)
where ζA =
√
2
n γa A
B ∇aξB is a section of OA[−1].
The prolongation of equation (4.1) is given by (see for instance [6] and references therein)
∇aξA + 1√
2
γ A
aB ζB = 0, ∇aζA + 1√
2
Pabγ
b A
B ξB = 0. (4.2)
These equations are equivalent to the tractor spinor Ξα = (ξA, ζA) being parallel with respect
to the tractor spinor connection, i.e., ∇aΞα = 0. In a conformal scale for which the metric is
flat, integration of (4.2) yields
ξA = ξ̊A − 1√
2
xaγ A
aB ζ̊B, ζA = ζ̊A, (4.3)
where ξ̊A and ζ̊A denote the constants of integrations at the origin.
A pure conformal Killing spinor ξA defines an almost null structure. The following propo-
sition combines results from [34, 35] recast in the language of tractors using Lemmata 2.19
and 2.22. It is valid on any conformal manifold of any dimension.
Proposition 4.1 ([34, 35]). The almost null structure of a pure conformal Killing spinor is
locally integrable and co-integrable if and only if its associated tractor spinor is pure.
By Theorems 3.6 and 3.10 one can associate to any such conformal Killing spinor on Qn
a complex submanifold in PT. These are described in the next two propositions.
Twistor Geometry of Null Foliations in Complex Euclidean Space 31
4.1.1 Odd dimensions
Proposition 4.2. Let Ξα = (ξA, ζA) be a constant pure tractor spinor on Q2m+1, Ξ its as-
sociated twistor in PT, Ξ̌ its corresponding γ-plane in Q2m+1, and U := Q2m+1 \ Ξ̌. Then ξA
is a pure conformal Killing spinor on Q2m+1 with zero set Ξ̌, and its associated integrable and
co-integrable almost null structure Nξ on U arises from the submanifold DΞ \ {Ξ} in Û ⊂ PT,
where DΞ is given by (2.17). In particular, each leaf of Nξ consists of a γ-plane intersecting Ξ̌
in an (m− 1)-plane. Each leaf of N⊥ξ consists of a 1-parameter family of γ-planes intersecting
in an (m − 1)-plane. Any two γ-planes contained in two distinct leaves of N⊥ξ intersect in an
(m− 2)-plane.
Proof. The line spanned by Ξα descends to a point Ξ (i.e., [Ξα]) in PT, and thus singles out
a γ-plane Ξ̌ in Qn, which by (4.3) can be immediately identified with the zero set of ξA. Off
that set, Proposition 4.1 tells us that Nξ is integrable and co-integrable. Correspondingly, the
conformal Killing spinor ξA gives rise to a section [ξA] of F, which we can re-express as
Γξ =
{
([XA], [Zα]) ∈ U × PT : Zα = XAΞα
A
}
⊂ F.
Clearly, a point on Γξ descends to a twistor Z on Dξ \ {Ξ} with γ-plane Ž tangent to Nξ.
Thus, for each Z on DΞ \ {Ξ} in Û ⊂ PT, Ž is precisely a leaf of Nξ. The point Ξ itself must be
excluded from DΞ since the foliation becomes singular there in the sense the leaves intersect in Ξ̌.
The geometric interpretation of the leaves of Nξ and N⊥ξ follows directly from Theorem 2.11
and Corollary 2.14. In particular, each distinguished curve on DΞ can be identified with a leaf
of N⊥ξ . �
Local form. Let us re-express the (m + 1)-plane DΞ as (2.18). We work in a conformal
scale for which gab is the flat metric. Since Ξα is constant, we can substitute the fields for their
constants of integration at the origin. Using (2.30) and Ξα = IαAξ̊
A +Oα
Aζ̊
A, we obtain, in the
obvious notation,
ωaAξ̊Ba + 2ξ̊AωB − ωAξ̊B = 0,
πaAζ̊ B
a + 2ζ̊AπB − πAζ̊B = 0,
ωaAζ̊Ba + ωAζ̊B + 4π[Aξ̊B] = 0,
πaAξ̊Ba + πAξ̊B + 4ω[Aζ̊B] = 0.
(4.4)
Evaluating at ωA = 1√
2
xaγ A
aB πB, using the second and third of (4.4) together with the purity
of Ξα, we find that πA must be proportional to ξA = ξ̊A − 1√
2
xaγ A
aB ζ̊B as expected. This
solution then satisfies the first and fourth equations.
Let us now work in the coordinate chart (V0, (ω
0, ωA, πA, πAB)) as defined in Section A.1,
and write
ξ̊A = ξ̊0oA + i
1
2
ξ̊AδAA −
1
4
ξ̊ABδAAB + · · · ,
ζ̊A =
1√
2
(
iζ̊0oA + ζ̊AδAA −
i
4ξ̊0
(
ξ̊AB ζ̊0 − 2ξ̊Aζ̊B
)
δAAB + · · ·
)
,
(4.5)
where the remaining components of ζ̊A and ξ̊A depend only on ζ̊0, ζ̊A, ξ̊A and ξ̊AB by the
purity of Ξα, and where we have assumed ξ̊0 6= 0. Substituting (A.7) and (4.5) into the last of
equations (4.4) yields
ξ̊0πA − ξ̊A + ζ̊0ωA − ω0ζ̊A = 0, ξ̊0πAB − ξ̊AB + 2ω[Aζ̊B] = 0,
32 A. Taghavi-Chabert
while the remaining equations do not yield any new information. Now, at every point Z of DΞ,
the 1-forms
βA := ξ̊0dπA + ζ̊0dωA − ζ̊Adω0, βAB := ξ̊0dπAB + 2dω[Aζ̊B],
annihilate the vectors tangent to DΞ at Z and the line in DZ spanned by
V := V 0Y + V AYA, (4.6)
where V 0 := ξ̊0 + 1
2 ζ̊
0ω0 and V A := ζ̊A + 1
2 ζ̊
0πA. This corroborates the claims of Theorem 3.6
and Proposition 4.2. Note that the vector field V vanishes at the point [Ξα] of DΞ. With no
loss, we can set ζ̊0 = −2. The integral curve, with complex parameter t, of (4.6) passing through
the point(
ω0, ωA, πA, πAB
)
=
(
ξ̊0 + α,−1
2
(
ξ̊A + αζ̊A − ξ̊0αA
)
, ζ̊A + αA,
1
ξ̊0
(
ξ̊AB + ξ̊[Aζ̊B]
)
− α[Aζ̊B]
)
,
for some α, αA, is given by(
ω0(t), ωA(t), πA(t), πAB(t)
)
=
(
ξ̊0,−1
2
ξ̊A, ζ̊A,
1
ξ̊0
(
ξ̊AB + ξ̊[Aζ̊B]
))
+
(
α,−1
2
(
αζ̊A − ξ̊0αA
)
, αA,−α[Aζ̊B]
)
e−t,
Writing AA = aY A +AaZAa + bXA and Aa = AAδaA +AAδ
aA +A0ua with
α = −a− 1
2
A0ξ̊0 +
1
2
AC ξ̊
C ,
αA =
1
2ξ̊0
(
AC ξ̊
C ζ̊A −A0ξ̊0ζ̊A − 2ξ̊ABAB −A0ξ̊A − 2ξ̊0AA
)
,
b =
1
ξ̊0
(
A0 − ζ̊CAC
)
,
one can recast this integral curve tractorially as Zα(t) = i√
2
(
Ξ̊α + i
2e−tÅAΞ̊α
A
)
, which is one of
the distinguished curves of Lemma 2.10 as expected.
4.1.2 Even dimensions
In even dimensions, the story is entirely analogous except for the choice of chirality of the tractor
spinor. We leave the details to the reader.
Proposition 4.3. Let Ξα′ = (ξA
′
, ζA) be a constant pure tractor spinor on Q2m, and let
U := Q2m \ Ξ̌ where Ξ̌ is the β-plane defined by Ξα
′
. Then ξA
′
is a pure conformal Killing
spinor on Q2m, and its associated null structure Nξ on U arises from the submanifold in Û ⊂ PT
defined by
Γ
(k)
αβ′
ZαΞβ′ = 0, for k < m, k ≡ m (mod 2). (4.7)
Each leaf of Nξ consists of an α-plane intersecting Ξ̌ in an (m− 1)-plane.
Remark 4.4. In four dimensions, tractor-spinors are always pure, and so almost null structures
associated to conformal Killing spinors are always integrable. In this case, the submanifold (4.7)
is a complex projective hyperplane in PT ∼= CP3 given by ΞαZ
α = 0 where we have used the
canonical isomorphism PT∗ ∼= PT′. This example was highly instrumental in the genesis of
twistor theory [29]. The null structure arising from the intersection of this submanifold with
real twistor space generates a shearfree congruence of null geodesics in Minkowski space known
as the Robinson congruence.
Twistor Geometry of Null Foliations in Complex Euclidean Space 33
4.2 Conformal Killing–Yano 2-forms
A (holomorphic) conformal Killing–Yano (CKY) 2-form on Qn is a section σab of O[ab][3] that
satisfies
∇aσbc − µabc − 2ga[bϕc] = 0, (4.8)
where µabc = ∇[aσbc] and ϕa = 1
n−2∇
bσba. The CKY 2-form equation (4.8) is prolonged to the
following system
∇aσbc − µabc − 2ga[bϕc] = 0,
∇aµbcd + 3ga[bρcd] + 3Pa[bσcd] = 0,
∇aϕb − ρab + P c
a σcb = 0,
∇aρbc − P d
a µdbc + 2Pa[bϕc] = 0.
(4.9)
This system can be seen to be equivalent to the existence of a parallel tractor 3-form, i.e.,
∇aΣABC = 0, (4.10)
where ΣABC := (σab, µabc, ϕa, ρab) ∈ O[ABC] ∼= O[ab][3] + (O[abc][3] ⊕ Oa[1]) + O[ab][1]. For an
arbitrary conformal manifold, equation (4.10) no longer holds in general, and necessitates the
addition of a ‘deformation’ term as explained in [15].
In flat space, i.e., with Pab = 0, we can integrate equations (4.9) to obtain
σab = σ̊ab + 2x[aϕ̊b] + µ̊abcx
c − 2
(
x[aρ̊b]cx
c + 1
4
(
xcxc
)
ρ̊ab
)
,
µabc = µ̊abc − 3x[aρ̊bc],
ϕa = ϕ̊a − ρ̊abxb,
ρab = ρ̊ab,
(4.11)
for some constants σ̊ab, µ̊abc, ϕ̊a and ρ̊ab.
Remark 4.5. In three dimensions, conformal Killing–Yano 2-forms are Hodge dual to conformal
Killing vector fields. These latter are in one-to-one correspondence with parallel sections of
tractor 2-forms.
In four dimensions, a 2-form σab is a CKY 2-form if and only if its self-dual part σ+
ab and
its anti-self-dual part σ−ab are CKY 2-forms, with, in the obvious notation, µ±abc = (∗ϕ±)abc.
Self-duality obviously carries over to tractor 3-forms.
4.2.1 Eigenspinors of a 2-form
Let us first assume n = 2m + 1. We recall that an eigenspinor ξA of a 2-form σab is a spinor
satisfying
σabγ
ab [A
C ξB]ξC = 0, (4.12)
i.e., σabγ
ab A
C ξC = λξA for some function λ. Here, γab A
C := γ
[a B
C γ
b] A
B . When ξA is pure,
another convenient way to express the eigenspinor equation (4.12) is given by
σabγ
(m+1)
abc3...cm+1ABξ
AξB = 0.
Therefore, to any 2-form σab, we can associate a complex submanifold of F given by the graph
Γσ :=
{(
xa,
[
πA
])
∈ CEn × PT(2m−1) : σabγ
(m+1)
abc3...cm+1ABπ
AπB = 0
}
. (4.13)
34 A. Taghavi-Chabert
For σab generic, this submanifold will have many connected components, each of which corre-
sponding to a local section of F→ Q2m+1, i.e., a projective pure spinor field that is an eigenspinor
of σab. To be precise, in 2m+1 dimensions, a generic 2-form σab viewed as an endomorphism σ b
a
of the tangent bundle, always has m distinct pairs of non-zero eigenvalues opposite to each other,
i.e., (λ,−λ), and a zero eigenvalue. In this case, a generic 2-form viewed as an element of the
Clifford algebra has 2m distinct eigenvalues, and thus 2m distinct eigenspaces, all of whose
elements are pure [26].
When n = 2m, the analysis is very similar: the pure eigenspinor equation is now
σabγ
(m)
abc3...cmA′B′ξ
A′ξB
′
= 0,
and similarity for spinors of the opposite chirality. Such a 2-form generically has m distinct
pairs of non-zero eigenvalues opposite to each other, and as an element of the Clifford algebra,
has 2m eigenspaces that split into two sets of 2m−1 eigenspaces according to the chirality of
the eigenspinors. The eigenspinor equation lifts to a submanifold Γσ := {(xa, [πA′ ]) ∈ CEn ×
PT(2m−2) : σabγ
(m)
abc3...cmA′B′π
A′πB
′
= 0} of F, whose connected components correspond to the
distinct primed spinor eigenspaces of σab.
4.2.2 The null structures of a conformal Killing–Yano 2-forms
The next question to address is when the almost null structure of an eigenspinor of a 2-form is
integrable and co-integrable.
Proposition 4.6 ([26]). Let σab be a generic conformal Killing 2-form on Qn (or any complex
Riemannian manifold). Let µabc := ∇[aσbc]. Let N be the almost null structure of some eigen-
spinor of σab, and suppose that µabcX
aY bZc = 0 for any sections Xa, Y a, Za of N⊥. Then N
is integrable and, when n is odd, co-integrable too.
In the light of Theorems 3.6 and 3.10, the foliations arising from the eigenspinors of a CKY
2-form σab can be encoded as complex submanifolds of the twistor space PT of Qn. As we shall
see in a moment, these submanifolds can be constructed from the corresponding tractor ΣABC .
The additional condition on µabc in Proposition 4.6 can also be understood in terms of the
graph of a connected component of Γσ defined by (4.13). For such a graph to descend to a com-
plex submanifold of PT, its defining equations should be annihilated by the vectors tangent to
F→ PT. Such a condition, in odd dimensions, can be expressed as 0 = π[CπcD]∇c(σabπaAπbB),
and using (4.8) gives µabcπ
aAπbBπbC = 0. Thus, we shall be interested in the local sections of
F→ Qn defined by
Γσ,µ :=
{(
xa,
[
πA
])
∈ CEn × PT(2m−1) :
σabγ
(m+1)
abc3...cm+1ABπ
AπB = 0, µabcγ
(m+1)
abcd4...dm+1ABπ
AπB = 0
}
. (4.14)
In even dimensions, this is entirely analogous except that (4.14) is now
Γσ,µ :=
{(
xa,
[
πA
′]) ∈ CEn × PT(2m−2) :
σabγ
(m)
abc3...cmA′B′π
A′πB
′
= 0, µabcγ
(m)
abcd4...dmA′B′π
A′πB
′
= 0
}
.
Proposition 4.7. Set n = 2m + ε, where ε ∈ {0, 1}. Let σab be a generic conformal Killing–
Yano 2-form on some open subset U of Qn, with associated tractor 3-form ΣABC. Then if the
almost null structure associated to some eigenspinor of σab is integrable and co-integrable, it
must arise from the submanifold in Û ⊂ PT defined by
ΣABCΓ
(m+1+ε)
ABCD4...Dm+1+εαβ
ZαZβ = 0. (4.15)
Twistor Geometry of Null Foliations in Complex Euclidean Space 35
Proof. We focus on the odd-dimensional case only, and leave the even-dimensional case to the
reader. Let us write
ΣABC = 3Y[AZ
b
BZ
c
C]σbc +
(
ZaAZ
b
BZ
c
Cµabc + 6X[AYBZ
c
C]ϕc
)
+ 3X[AZ
b
BZ
c
C]ρbc.
Since ΣABC is constant, we can substitute the fields for their constants of integration at the
origin, σ̊ab, µ̊abc, ϕ̊a and ρ̊ab, so that using (2.30) we can re-express (4.15) as
0 = −3
√
2σ̊abγ
(m+1)
abd4...dm+2ABπ
AπB + 2µ̊abcγ
(m+2)
abcd4...dm+2ABω
AπB − 12ϕ̊aγ
(m)
ad4...dm+2ABω
AπB
+ 3
√
2ρ̊abγ
(m+1)
abd4...dm+2ABω
AωB,
0 =
√
2µ̊abcγ
(m+1)
abcd4...dm+1ABπ
AπB − 6ρ̊abγ
(m)
abd4...dm+1ABω
AπB,
0 = −
√
2µ̊abcγ
(m+1)
abcd4...dm+1ABω
AωB + 6σ̊abγ
(m)
abd4...dm+1ABω
AπB,
0 = 2µ̊abcγ
(m)
abcd4...dmABω
AπB.
Evaluating this system of equations on the intersection of (4.15) and Û amounts to setting
ωA = 1√
2
xaγ A
aB πB, and we find, after some algebraic manipulations,
0 = −3
√
2
(
σabγ
(m+1)
abd4...dm+2ABπ
AπB
)
+
√
2(m− 1)
(
x[d4|µ
abcγ
(m+1)
abc|d5...dm+2]ABπ
AπB
)
,
0 =
√
2µabcγ
(m+1)
abcd4...dm+1ABπ
AπB,
0 = −(xexe)√
2
µabcγ
(m+1)
abcd4...dm+1ABπ
AπB + 3
√
2σabxcγ
(m+1)
abcd4...dm+1ABπ
BπB
+
√
2(m− 2)x[d4|µ
abcxfγ
(m+1)
abcf |d5...dm+1]ABπ
BπB,
0 =
√
2µabcxdγ
(m+1)
abcde5...em+1ABπ
AπB,
where we have made use of (4.11) and the identity
1
4
(
xcγ A
cC
)(
ρ̊abγ
ab B
A
)(
xdγ D
dB
)
=
(
xaρ̊bcx
c + 1
4(xcxc)ρ̊ab
)
γab D
C .
In particular, we immediately recover, that on the intersection of the twistor submanifold (4.15)
with Û ,
σabγ
(m+1)
abc3...cm+1ABπ
AπB = 0, µabcγ
(m+1)
abcd4...dm+1ABπ
AπB = 0.
But these are precisely the zero set (4.14) corresponding to the eigenspinors of σab. �
Remark 4.8. In three dimensions, the twistor submanifold is simply a smooth quadric in
PT ∼= CP3.
In four dimensions, the submanifold (4.15) restricts to an anti-self-dual tractor 3-form Σ−ABC
corresponding to a self-dual CKY 2-form σab. Setting Σ−αβ := Σ−ABCΓ
ABC
αβ, we recover the
quadratic polynomial Σ−αβZ
αZβ = 0 given in [31]. Under appropriate reality conditions, this
submanifold produces a shearfree congruence of null geodesics in Minkowski space known as
the Kerr congruence. A suitable perturbation of Minkowski space by the generator of such
a congruence leads to the solution of Einstein’s equations known as the Kerr metric [22, 23].
A Euclidean analogue is also given in [32].
In six dimensions, we have a splitting of µabc = µ+
abc + µ−abc into a self-dual part and an
anti-self-dual part. Since ξaAξbBξcCµ̊+
abc = 0 for any ξA
′
, the obstruction to the integrability of
a positive eigenspinor of a generic CKY 2-form σab is the anti-self-dual part µ−abc of µabc.
36 A. Taghavi-Chabert
5 Curved spaces
Let M be a complex manifold equipped with a holomorphic non-degenerate symmetric bili-
near form gab. The pair (M, gab) will be referred to as a complex Riemannian manifold. We
assume that M is equipped with a holomorphic complex orientation and a holomorphic spin
structure. We may also assume that one merely has a holomorphic conformal structure rather
than a metric one. For definiteness, we set n = 2m + 1 as the dimension of M. The analogue
of the correspondence space F is the projective pure spinor bundle ν : F →M: for any x ∈M,
a point p in a fiber ν−1(x) is a totally null m-plane in TxM, and sections of F are almost
null structures on M. To define the twistor space of (M, gab), one must replace the notion of
γ-plane by that of γ-surface, i.e., an m-dimensional complex submanifold ofM such that at any
point of such a surface, its tangent space is totally null with respect to the metric and totally
geodetic with respect to the metric connection. The integrability condition for the existence of
a γ-surface N through a point x is [35]
CabcdX
aY bZcW d = 0, for all Xa, Y a, Zc ∈ TxN , W a ∈ TxN . (5.1)
If we define the twistor space of (M, gab) to be the 1
2(m+1)(m+2)-dimensional complex manifold
parametrising the γ-surfaces of (M, gab), we must have a 1
2m(m + 1)-parameter family of γ-
surfaces through each point ofM. From the integrability condition (5.1), we must conclude that
for the twistor space of (M, gab) to exist, (M, gab) must be conformally flat in odd dimensions
greater than three. In even dimensions the story is similar: one replaces the notion of α-plane
by that of an α-surface in the obvious way. We then find that for (M, gab) to admit a twistor
space, it must be conformally flat in even dimensions greater than four, and anti-self-dual in
dimension four.
Curved twistor theory in dimensions three and four is pretty well-known. In dimension four,
we have the Penrose correspondence, whereby twistor space is a three-dimensional complex
manifold containing a complete analytic family of rational curves with normal bundleO(1)⊕O(1)
parameterised by the points of an anti-self-dual complex Riemannian manifold [30]. In dimension
three, the LeBrun correspondence can be seen as a special case of the Penrose correspondence: if
we endow twistor space with a holomorphic ‘twisted’ contact structure, then a three-dimensional
conformal manifold arises as the umbilic conformal infinity of an Einstein anti-self-dual four-
dimensional complex Riemannian manifold [25]. Finally, the mini-twistor space in the Hitchin
correspondence is a two-dimensional complex manifold containing a complete analytic family
of rational curves with normal bundle O(2) parameterised by the points of an Einstein–Weyl
space [18, 21].
Theorems 3.5 (or 3.6), 3.8 and 3.10 can be adapted to the curved setting by interpreting
the leaf space of a totally geodetic null foliation as a complex submanifold of twistor space.
See [8] for an application of a ‘curved’ Theorem 3.8 in the investigation of three-dimensional
Einstein–Weyl spaces.
A Coordinate charts on twistor space and correspondence space
In this appendix, we construct atlases of coordinates charts covering PT and F. We refer to the
setup of Section 2 throughout. In particular, we work with the splittings (2.3), (2.8) and (2.12).
A.1 Odd dimensions
Let us introduce a splitting of V0 as
V0
∼= W⊕W∗ ⊕ U, (A.1)
Twistor Geometry of Null Foliations in Complex Euclidean Space 37
where W ∼= Cm is a totally null m-plane of (V0, gab), and U ∼= C is the one-dimensional comple-
ment of W ⊕W∗ in V0. Elements of W and W∗ will carry upstairs and downstairs upper-case
Roman indices respectively, i.e., V A ∈W, and WA ∈W∗. The vector subspace U will be spanned
by a unit vector ua. Denote by δaA the injector from W∗ to V0, and δaA the injector from W
to V0 satisfying δAa δ
a
B = δAB, where δAB is the identity on W and W∗. We shall think of {δaA}
as a basis for W with dual basis {δaA} for W∗. The splitting (A.1) allows us to identify the two
copies S± 1
2
of the spinor space of (V0, gab) with its Fock representation, i.e.,
S± 1
2
∼= ∧mW⊕ ∧m−1W⊕ · · · ⊕W⊕ C.
This is essentially the strategy adopted in Section 2.2 for the spinors of Spin(2m + 3,C). To
realise it explicitly, we proceed as follows: let oA be a (pure) spinor annihilating W so that oA
is a spanning element of ∧mW. A (Fock) basis for S± 1
2
can then be produced by acting on oA
by basis elements of ∧•W∗, i.e.,
S± 1
2
=
〈
oA, δAA1
, δAA1A2
, . . .
〉
, (A.2)
where
δAA1...Ak
:= δa1[A1
· · · δakAk]o
A0γ A1
a1A0
· · · γ A
akAk−1
,
for each k = 1, . . . ,m. With this notation, the Clifford multiplication of V0 ⊂ C`(V0, gab) on S− 1
2
is given explicitly by
δaAγ C
aB δBB1...Bp = −2pδC[B1...Bp−1
δABp], δaAγ
C
aB δBB1...Bp = δCB1...BpA,
uaγ C
aB oB = ioC, uaγ C
aB δBB1...Bp = (−1)piδCB1...Bp . (A.3)
An arbitrary spinor πA in S 1
2
can then be expressed in the Fock basis (A.2) as
πA = π0oA +
[m/2]∑
k=1
(
−1
4
)k 1
k!
πA1...A2kδAA1...A2k
+
i
2
[m/2]∑
k=0
(
−1
4
)k 1
k!
πA1...A2k+1δAA1...A2k+1
, m > 1,
πA = π0oA + i
2π
AδAA , m = 1, (A.4)
where
[
m
2
]
is m
2 when m is even, m−1
2 when m is odd, and π0 and πA1A2...Ak = π[A1A2...Ak] are
the components of πA. Let us now assume that πA is pure, i.e., satisfies (2.31a). When m = 1
and 2, there are no algebraic constraints, and the space of projective pure spinors is isomorphic
to CP1 and CP3 respectively. When m > 2, the pure spinor variety is then given by the complete
intersections of the quadric hypersurfaces
π0πA1A2...A2k+1 = π[A1πA2...A2k+1], k = 1, . . . , [m/2],
π0πA1A2A3...A2k = π[A1A2πA3...A2k], k = 1, . . . , [m/2], (A.5)
in CP2m−1. We can therefore cover a fibre of F with 2m open subsets U0, UA1...Ak , where π0 6= 0
on U0 and πA1...Ak 6= 0 on UA1...Ak , and thus obtain 2m coordinate charts in the obvious way.
This induces an atlas of charts on FCEn given by the open subsets CEn × U0, CEn × UA1...Ak .
38 A. Taghavi-Chabert
Let us now write the spinor ωA in S− 1
2
in the Fock basis as
ωA =
i√
2
ω0oA +
1√
2
ωAδAA , m = 1,
ωA =
i√
2
ω0oA +
i
2
√
2
[m/2]∑
k=1
(
−1
4
)k−1 1
(k − 1)!
ωA1...A2kδAA1A2...A2k
+
1√
2
[m/2]∑
k=0
(
−1
4
)k 1
k!
ωA1...A2k+1δAA1...A2k+1
, m > 1, (A.6)
where ω0 and ωA1A2...Ak = ω[A1A2...Ak] are the components of ωA. The condition for Zα =
(ωA, πA) to be pure, so that (2.31) hold, is that the relations
π0ωA1...A2k−1A2k = π[A1...A2k−1ωA2k] − 1
2kπ
A1...A2kω0,
π0ωA1...A2kA2k+1 = π[A1...A2kωA2k+1],
hold for k ≥ 1 when m > 1, and that (A.5) hold too when m > 2. Hence, we can cover PT\∞̂
with 2m open subsets V0, where π0 6= 0, and VA1...Ak where πA1...Ak 6= 0 in the obvious way.
Coordinates on the complement ∞̂ parametrised by [ωA, 0] satisfy the conditions
ω0ωA1...A2kA2k+1 = −2kω[A1...A2kωA2k+1], ω[A1...A2k−1ωA2k] = 0.
Let (zA, zA, u) be null coordinates on CEn in the sense that xa = zAδaA + zAδ
aA + uua so
that the flat metric on CEn takes the form g = 2dzA � dzA + du ⊗ du. Then the incidence
relation (2.33) reads
ω0 = π0u− πBzB,
ωA = π0zA + πABzB + 1
2π
Au,
ωA1...A2k−1A2k = π[A1...A2k−1zA2k] + 4k+2
4k πA1...A2k−1A2kA2k+1zA2k+1
− 1
2kπ
A1...A2ku,
ωA1...A2kA2k+1 = π[A1...A2kzA2k+1] + πA1...A2kA2k+1A2k+2zA2k+2
+ 1
2π
A1...A2k+1u.
We now work in the chart U0, and since π0 6= 0 there, we can set with no loss of gene-
rality π0 = 1. Let (x, π) be a point in FCEn and let (U0, (π
A, πAB)) be a coordinate chart
containing π ∈ Fx. Let (ω, π) be the image of (x, π) under the projection µ : F → PT so
that (V0, (ω
0, ωA, πA, πAB)) is a coordinate chart containing (ω, π). Then, in these charts,
(A.6) and (A.4) reduce to
ωA = 1√
2
(
iω0oA + ωAδAA − i
4
(
πABω0 − 2πAωB
)
δAAB + · · ·
)
, (A.7a)
πA = oA + i
2π
AδAA − 1
4π
ABδAAB + · · · . (A.7b)
More succinctly, πA = exp(−1
4π
abγ A
abB )oB, where πab = πABδaAδ
b
B + 2πAδ
[a
Au
b] belongs to the
complement of the stabiliser of oA in so(V0, gab), i.e., (πA, πAB) are coordinates on a dense open
subset of the homogeneous space P/Q. We can also rewrite ωA more compactly in the two
alternative forms
ωA = 1√
2
(
ωAδaA + 1
2ω
0ua
)
πAa + i
2
√
2
ω0πA,
ωA = 1√
2
ωaπAa , where ωa :=
(
ωA − 1
2ω
0πA
)
δaA + ω0ua,
from which it is easy to check that πA and ωA indeed satisfy the conditions given in Lemma 2.19.
Twistor Geometry of Null Foliations in Complex Euclidean Space 39
Finally, in the coordinate chart (CEn × U0, (z
A, zA, u;πA, πAB)), we have
xaπAa = i
(
u− πBzB
)
oA +
(
zB + πBCzC + 1
2uπ
B
)
δAB + · · · ,
so that the incidence relation (2.33) reduces to
ωA = zA + πABzB + 1
2π
Au, ω0 = u− πBzB. (A.8)
Tangent and cotangent spaces. Let us introduce the short-hand notation
∂A :=
∂
∂zA
= δaA∇a, ∂A :=
∂
∂zA
= δaA∇a, ∂ :=
∂
∂u
= ua∇a,
so that T(x,π)Qn ∼= p−1 = 〈∂A, ∂A, ∂〉, and define 1-forms
αA := dωA + 1
2π
Adω0 − 1
2ω
0dπA, αAB := dπAB − π[AdπB], (A.9)
and vectors
XA :=
∂
∂ωA
, XAB :=
∂
∂πAB
,
Y :=
∂
∂ω0
− 1
2
πC
∂
∂ωC
, YA :=
∂
∂πA
− πB ∂
∂πAB
+
1
2
ω0 ∂
∂ωA
. (A.10)
Then bases for the cotangent and tangent spaces of PT at (ω, π) are given by
T∗(ω,π)PT ∼= r∗1 ⊕ r∗2 =
〈
dω0,dπA
〉
⊕
〈
αA,αAB
〉
,
T(ω,π)PT ∼= r−2 ⊕ r−1 =
〈
XAXAB
〉
⊕
〈
Y ,YA
〉
,
respectively.
Remark A.1. Using (A.7), one can check that the expressions for the set (A.9) of 1
2m(m+ 1)
1-forms are none other than the 1-forms (2.32), and thus (2.25). These forms annihilate the
rank-(m + 1) canonical distribution D on PT spanned by Y and YA. Further, the vector Y
clearly coincides with (2.37) to describe mini-twistor space – this can be checked by using
transformations (A.7).
Now, define the 1-forms and vectors
θA := dzA +
(
πAD − 1
2π
AπD
)
dzD + πAdu, θ0 := du− πCdzC ,
ZA := ∂A +
(
πAD − 1
2π
AπD
)
∂D + πA∂, U := ∂ − πD∂D,
WA :=
∂
∂πA
− πB ∂
∂πAB
.
Then bases for the cotangent and tangent spaces of F at (x, π) are given by
T∗(x,π)F ∼= q∗1
E ⊕ q∗1
F ⊕ q∗2
E ⊕ q∗2
F ⊕ q∗3 = 〈dzA〉 ⊕
〈
dπA
〉
⊕ 〈θ0〉 ⊕
〈
αAB
〉
⊕
〈
θA
〉
,
T(x,π)F ∼= q−3 ⊕ qF−2 ⊕ qE−2 ⊕ qF−1 ⊕ qE−1 = 〈∂A〉 ⊕ 〈XAB〉 ⊕ 〈U〉 ⊕ 〈WA〉 ⊕
〈
ZA
〉
,
respectively.
We note that the coordinates (ω0, ωA, πA, πAB) on V0 are indeed annihilated by the vec-
tors ZA tangent to the fibres of F → PT. Further, the pullback of αA to F is given by
µ∗(αA) = αABzB +θA, i.e., the annihilator of D = T−1PT pulls back to the annihilator of T−2
E F
corresponding to qE−2 ⊕ qF−1 ⊕ qE−1.
40 A. Taghavi-Chabert
Mini-twistor space. By Lemma 2.24, the mini-twistor space MT of CEn is the leaf space
of the vector field Y defined by (2.37), given in (A.10) in the coordinate chart (V0, (ω
0, ωA, πA,
πAB)). Accordingly, we have a local coordinate chart (V0, (ω
A, πAB, πA)) on MT where
ωA = ωA + 1
2π
Aω0,
which can be seen to be annihilated by Y . The incidence relation (2.35) or (2.40) can then be
expressed as
ωA = zA +
(
πAB − 1
2π
AπB
)
zB + πAu,
which are indeed annihilated by ZA and U . The tangent space of MT at a point (ω, π) in V0 is
clearly
T(ω,π)MT = 〈XA,XAB,WA〉, where XA :=
∂
∂ωA
.
Normal bundle of x̂ in PT\∞̂. Let x be a point in CEn. In the chart (V0, (ω
0, ωA, πA, πAB)),
the corresponding x̂ is given by (A.8). In particular, the 1-forms
βA(x) := dωA − dπABzB − 1
2dπAu, β0(x) := dω0 + dπBzB,
vanish on x̂, and the tangent space of x̂ at (ω, π) is spanned by the vectors YA − zAY and
XAB − z[AXB]. This distinguishes the m-dimensional subspace 〈YA − zAY 〉 tangent to both x̂
and the canonical distribution D at (ω, π).
A.2 Even dimensions
The local description of F and PT in even dimensions can be easily derived from the one above.
We split V0 as V0
∼= W⊕W∗ where W ∼= Cm is a totally null m-plane of (V0, gab), with adapted
basis {δaA, δaA}. The Fock representations of the irreducible spinor spaces S− 1
2
and S′− 1
2
on V0
are given by
S 1
2
∼= S′− 1
2
∼= ∧mW⊕ ∧m−2W⊕ · · · , S′1
2
∼= S− 1
2
∼= ∧m−1W⊕ ∧m−3W⊕ · · · .
Let oA
′
be a (pure) spinor annihilating W. Then bases for S 1
2
and S− 1
2
can then be produced
by acting on oA
′
by basis elements of ∧2kW∗ and of ∧2k−1W∗. Explicitly,
S 1
2
=
〈
oA
′
, δA
′
A1A2
, . . .
〉
, S− 1
2
=
〈
δAA1
, δAA1A2A3
, . . .
〉
,
where
δA
′
A1...A2k
:= δa1[A1
· · · δa2kA2k]o
A′0γ A1
a1A′0
· · · γ A′
a2kA2k−1
,
δAA1...A2k−1
:= δa1[A1
· · · δa2k−1
A2k−1]o
A′0γ A1
a1A′0
· · · γ A
a2k−1A
′
2k−2
.
The Clifford action of V0 ⊂ C`(V0, gab) on S± 1
2
follows the same lines as (A.3) with appropriate
priming of spinor indices.
Coordinate charts in even dimensions can be obtained from the odd-dimensional case by
switching off πA1...Ak for all odd k, and ωA1...Ak for all even k. We therefore have a covering of
each fibre of F by 2m−1 open subsets U0, UA1...A2k
, and a covering of PT\∞̂ by 2m−1 open subsets
V0, VA1...A2k
in the obvious way. In particular, in (V0, (ω
A, πAB)), the homogeneous coordinates
[ωA, πA
′
] are given by
ωA = 1√
2
(
ωAδAA − 1
4ω
AπBCδAABC + · · ·
)
, πA
′
= oA
′ − 1
4π
ABδA
′
AB + · · · .
Twistor Geometry of Null Foliations in Complex Euclidean Space 41
where the former can also be rewritten as ωA = 1√
2
ωaπAa with ωa := ωAδaA. Finally, the
even-dimensional version of the incidence relation (2.33) can be rewritten as ωA = zA + πABzB.
As for the tangent spaces of Q2m, its twistor space and their correspondence space, we find,
in the obvious notation, T(x,π)Qn ∼= p−1 = 〈∂A, ∂A, ∂〉, T(x,π)F ∼= q−2 ⊕ qF−1 ⊕ qE−1 = 〈∂A〉 ⊕
〈XAB〉 ⊕ 〈ZA〉, and T(ω,π)PT ∼= r−1 = 〈XA,XAB〉, where ZA := ∂A + πAB∂B, XAB := ∂
∂πAB
,
XA := ∂
∂ωA
, and so on.
Acknowledgements
The author would like to thank Boris Doubrov, Lionel Mason and Jan Slovák for helpful discus-
sions and comments, and the anonymous referees for their reports. He is also grateful to Lukáš
Vokř́ınek and Andreas Čap for clarifying some aspects of Section 2.5. This work was funded by
a GAČR (Czech Science Foundation) post-doctoral grant GP14-27885P.
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1 Introduction
2 Twistor geometry
2.1 Smooth quadric hypersurface
2.1.1 The tractor bundle
2.1.2 The tractor spinor bundle
2.2 Twistor space
2.2.1 Odd dimensions
2.2.2 Even dimensions
2.2.3 From even to odd dimensions
2.3 Correspondence space
2.3.1 Odd dimensions
2.3.2 Even dimensions
2.4 Co-gamma-planes and mini-twistor space
2.5 Normal bundles
2.5.1 Odd dimensions
2.5.2 Mini-twistor space
2.5.3 Even dimensions
2.5.4 Kodaira's theorem and completeness
3 Null foliations
3.1 Odd dimensions
3.1.1 Local description
3.1.2 Totally geodetic null structures
3.1.3 Co-integrable null structures
3.1.4 Totally co-geodetic null structures
3.2 Even dimensions
4 Examples
4.1 Conformal Killing spinors
4.1.1 Odd dimensions
4.1.2 Even dimensions
4.2 Conformal Killing–Yano 2-forms
4.2.1 Eigenspinors of a 2-form
4.2.2 The null structures of a conformal Killing–Yano 2-forms
5 Curved spaces
A Coordinate charts on twistor space and correspondence space
A.1 Odd dimensions
A.2 Even dimensions
References
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