The Geometry of Generalised Spinʳ Spinors on Projective Spaces
In this paper, we adapt the characterisation of the spin representation via exterior forms to the generalised spinʳ context. We find new invariant spinʳ spinors on the projective spaces ℂℙⁿ, ℍℙⁿ, and the Cayley plane ℙ² for all their homogeneous realisations. Specifically, for each of these realisat...
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| Cite this: | The Geometry of Generalised Spinʳ Spinors on Projective Spaces. Diego Artacho and Jordan Hofmann. SIGMA 21 (2025), 017, 32 pages |
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| citation_txt | The Geometry of Generalised Spinʳ Spinors on Projective Spaces. Diego Artacho and Jordan Hofmann. SIGMA 21 (2025), 017, 32 pages |
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| description | In this paper, we adapt the characterisation of the spin representation via exterior forms to the generalised spinʳ context. We find new invariant spinʳ spinors on the projective spaces ℂℙⁿ, ℍℙⁿ, and the Cayley plane ℙ² for all their homogeneous realisations. Specifically, for each of these realisations, we provide a complete description of the space of invariant spinʳ spinors for the minimum value of for which this space is non-zero. Additionally, we demonstrate some geometric implications of the existence of special spinʳ spinors on these spaces.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 017, 32 pages
The Geometry of Generalised Spinr Spinors
on Projective Spaces
Diego ARTACHO a and Jordan HOFMANN b
a) Imperial College London, London SW7 2AZ, UK
E-mail: d.artacho21@imperial.ac.uk
b) King’s College London, London WC2R 2LS, UK
E-mail: jordan.2.hofmann@kcl.ac.uk
Received July 01, 2024, in final form March 01, 2025; Published online March 11, 2025
https://doi.org/10.3842/SIGMA.2025.017
Abstract. In this paper, we adapt the characterisation of the spin representation via ex-
terior forms to the generalised spinr context. We find new invariant spinr spinors on the
projective spaces CPn, HPn, and the Cayley planeOP2 for all their homogeneous realisations.
Specifically, for each of these realisations, we provide a complete description of the space
of invariant spinr spinors for the minimum value of r for which this space is non-zero.
Additionally, we demonstrate some geometric implications of the existence of special spinr
spinors on these spaces.
Key words: special spinors; projective spaces; generalized spin structures; spinc; spinh
2020 Mathematics Subject Classification: 53C27; 15A66; 57R15
1 Introduction
A topic of major interest in differential geometry is the existence or non-existence of special
G-structures on a given smooth manifold M ; classical examples include Riemannian, complex,
symplectic, and spin structures. Spin geometry, in particular, gives a way of accessing global
geometric information about Riemannian spin manifolds via sections of a certain naturally
defined vector bundle called the spinor bundle. Indeed, for a Riemannian spin manifold M ,
there are a number of results of the form:
M carries a spinor satisfying equation E =⇒ M has geometric property P.
For example, it is well known that a manifold carrying a non-zero parallel spinor is Ricci-
flat, and, more generally, that the existence of a non-zero real (resp. purely imaginary) Killing
spinor implies that the metric is Einstein with positive (resp. negative) scalar curvature [9, 15].
Other notable examples include the bijection between generalised Killing spinors in dimension 5
(resp. 6, resp. 7) and hypo SU(2)-structures (resp. half-flat SU(3)-structures, resp. co-calibrated
G2-structures) [1, 12], and the spinorial description of isometric immersions into Riemannian
space forms [10, 16, 30].
However, not every manifold can be endowed with a spin structure; the question then natu-
rally arises as to whether one can apply the tools of spin geometry to non-spin manifolds. The
answer is affirmative, and there are several possible approaches. The unifying idea is to consider
suitable enlargements of the spin groups, i.e., Lie groups Ln equipped with homomorphisms
Spin(n)
ιn−→ Ln
pn−→ SO(n)
such that pn ◦ ιn is the usual two-sheeted covering Spin(n) → SO(n). Hence, an oriented
Riemannian n-manifold admitting a lift of the structure group to Ln is a weaker condition than
mailto:d.artacho21@imperial.ac.uk
mailto:jordan.2.hofmann@kcl.ac.uk
https://doi.org/10.3842/SIGMA.2025.017
2 D. Artacho and J. Hofmann
being spin. Following ideas by Friedrich and Trautman [20], the so-called spinorial Lipschitz
structures have garnered much attention in recent years [33, 34, 35]. These naturally arise
by following the inverse approach: starting with a suitable generalisation of the concept of
spinor bundle, one investigates the enlargement Ln of Spin(n) to which this bundle corresponds.
These Ln are called Lipschitz groups.
Another choice of Ln was introduced by Espinosa and Herrera [14], who had the idea of
spinorially twisting the spin group. In our setting, this corresponds to taking, for r ∈ N, the
groups
Lrn = Spinr(n) := (Spin(n)× Spin(r))/⟨(−1,−1)⟩
with the obvious homomorphisms. We say that an oriented Riemannian n-manifold is spinr if
it admits a lift of the structure group to Spinr(n). The case r = 1 is the classical spin case, and
the cases r = 2, 3 give rise to spinC and spinH geometry respectively, which have been a fruitful
field of study over the past decades – see [17, 37] for spinC and [25, 31] and references therein
for spinH.
These structures have been characterised topologically by Albanese and Milivojević in [4],
where they show that a manifold is spinr if, and only if, it can be embedded into a spin manifold
with codimension r. In [5], Lawn and the first author focused on the study of spinr structures
on homogeneous spaces, establishing a bijection between G-invariant spinr structures on G/H
and certain representation-theoretical data – see Theorem 2.6.
Analogously to the usual spin case, from a given spinr structure one can construct, for each
odd m ∈ N, the so-called m-twisted spinr spinor bundle – see Definition 2.8. Its sections are
called m-twisted spinr spinors or simply spinr spinors, and, as in the classical case, they encode
geometric information: for a spinr manifold M , there are results of the form:
M carries a spinr spinor satisfying equation E =⇒ M has property P.
Some of these results can be found in [14, 26], for example (see Theorem 2.16 for more explicit
results):
� The existence of a generalised Killing spinr spinor ensures a certain decomposition of the
Ricci tensor [14, Theorem 3.3].
� The existence of a parallel pure spin2
(
resp. spin3
)
spinor implies that the manifold is
Kähler (resp. quaternionic Kähler) [26, Corollaries 4.10 and 4.12].
In this paper, we illustrate how invariant twisted spinr spinors on the projective spaces CPn,
HPn and the Cayley plane OP2 encode different geometric properties of these manifolds. To this
end, we proceed as follows:
(1) Consider a homogeneous realisation M = G/H of the corresponding space, equipped with
a generic G-invariant metric.
(2) Find the minimum value of r ∈ N such that M has a G-invariant spinr structure carrying
non-trivial invariant m-twisted spinr spinors, for some odd m ∈ N, and describe the space
of such spinors.
(3) Study the geometric properties ofM encoded by those invariant spinr spinors which satisfy
additional algebraic properties.
The realm of projective spaces provides a fruitful ground for study. In particular, we prove
that Friedrich’s construction of generalised Killing spinors on CP3 [9, p. 146] cannot be gener-
alised to higher complex dimensions, showing that this is the only dimension for which CP2k+1
has an Sp(k + 1)-invariant metric carrying non-trivial invariant generalised Killing spinors. We
The Geometry of Generalised Spinr Spinors on Projective Spaces 3
M n G r m dim(Σm∗,r)inv Special spinors Geometry
CPn
k SU(k + 1) 2 1 2 pure, parallel Kähler–Einstein
2k + 1 Sp(k + 1)
2, if k even 1 2 pure, parallel Kähler–Einstein
1, if k odd 1 2 generalised Killing Einstein, nearly Kähler (n = 3 (†))
HPn
2k + 1 Sp(2k + 2) 3 2k + 1 1 pure, parallel quaternionic Kähler
2k Sp(2k + 1) – – – – –
OP2 – F4 9 3 4 – –
Table 1. For each compact, simple and simply connected Lie group G acting transitively on M : the
minimum values of r, m such that M admits a G-invariant spinr structure carrying a non-zero invariant
m-twisted spinr spinor, and the geometric significance of these. For †, see [9, p. 146].
also find the spinH spinor on HPn inducing the standard quaternionic Kähler structure (see [26,
Corollary 4.12]) with new representation-theoretical methods, and we show that this is, up to
scaling, the only Sp(n+ 1)−invariant spinH spinor on this space. Finally, we find that the min-
imum values of r and m such that OP2 carries non-trivial F4-invariant m-twisted spinr spinors
are r = 9 and m = 3, and the space of such spinors is four-dimensional.
The computations carried out in this paper illustrate an extension of the differential forms
approach to the spin representation (see, e.g., [2]) to the context of spinr structures. These
techniques allow us to express and manipulate complicated twisted spinors in an easy and
readable way, finding new examples of special spinr spinors. The main contribution of this
paper is, then, the fusion of the differential forms approach with the power of spinr geometry
to encode geometric properties of manifolds which are not necessarily spin. Our results are
summarised as follows.
Theorem. Let G be a compact, simple and simply connected Lie group acting transitively on
the projective space M = CPn,HPn or OP2. Then, the minimum values of r,m ∈ N (with
m odd) such that M admits a G-invariant spinr structure carrying a non-zero invariant m-
twisted spinr spinor are shown in Table 1, together with the geometric information such spinors
encode.
2 Preliminaries
We begin by introducing the necessary background definitions and results concerning spin and
spinr geometry within the context of homogeneous spaces. For an introduction to spin geometry
we refer the reader to [9, 32], for spinr manifolds to [4, 5, 14], and for homogeneous spaces to [6].
2.1 Invariant metrics on reductive homogeneous spaces
Let G/H be a reductive homogeneous space with reductive decomposition g = h ⊕ m, where g
is the Lie algebra of G, h is the Lie algebra of H and m is an AdH -invariant complement of h
in g. Suppose that the adjoint representation of H on m – which, under the usual identifications,
corresponds to the isotropy representation of the homogeneous space – decomposes as a direct
sum of irreducible components m = m1 ⊕ · · · ⊕mk. We would like to find all the AdH -invariant
inner products on m (which correspond to G-invariant metrics on the homogeneous space G/H,
see, e.g., [6]). Of course, such metrics need not exist. However, ifH is compact, using Weyl’s trick
one readily sees that they do exist. We would like to show that invariant inner products on each
irreducible component are unique up to positive scaling, and that any invariant inner product
on m is a positive linear combination of invariant inner products on the irreducible components,
4 D. Artacho and J. Hofmann
yielding a k-parameter family of invariant metrics. This is of course false in general (consider
two copies of the same irreducible representation admitting an invariant metric). However, this
is essentially the only obstruction, as we shall see now.
This is a very important result which appears to be well known, but it is surprisingly hard
to find in the literature. We include it here with a full proof.
Proposition 2.1. Let g be a finite-dimensional real Lie algebra and ρ : g → EndR(V ) a finite-
dimensional irreducible real representation of g. Suppose that there exists a ρ-invariant inner
product on V . Then, it is unique up to positive scaling.
Proof. Let B1, B2 : V ×V → R be two ρ-invariant inner products on V , i.e., two inner products
on V satisfying
∀X ∈ g, ∀v, w ∈ V : Bi(ρ(X)v, w) +Bi(v, ρ(X)w) = 0, i = 1, 2.
Then, for each i = 1, 2, Bi defines an isomorphism of representations φi between ρ and its dual
representation ρ∗ : g → EndR(V
∗):
φi : V → V ∗, v 7→ Bi(v,−).
In particular, ρ is self-dual. Now consider the endomorphism of representations given by φ−1
1 ◦
φ2 : V → V . By Schur’s lemma, the endomorphism ring of an irreducible representation (over
any ground field) is a division ring. In particular, the endomorphism ring of ρ is a finite-
dimensional associative division algebra over R. By the Frobenius theorem, these are, up to
isomorphism, R, C and H. So we need to consider these three cases separately.
(1) If End(ρ) ∼= R, then there exists λ ∈ R such that φ−1
1 ◦ φ2 = λ IdV , which implies
that φ2 = λφ1, which in turn means that B2 = λB1. As both B1, B2 are inner products, we
must have that λ > 0. This completes the proof for the case End(ρ) ∼= R.
(2) If End(ρ) ∼= C, then there exists J ∈ End(ρ), with J2 = − IdV . And any φ ∈ End(ρ) is of
the form φ = a IdV +bJ , for some a, b ∈ R. In particular, φ2 = aφ1 + bφ1 ◦ J . But this implies
that B2(−,−) = aB1(−,−)+bB1(J−,−). Now we claim that, for every v, w ∈ V , B1(Jv, Jw) =
B1(v, w). Indeed, define B̃ : V × V → R as B̃(v, w) = B1(Jv, Jw). As J2 = − IdV , B̃ is non-
degenerate. And, as J and B1 are ρ-invariant and B1 is symmetric, B̃ is ρ-invariant and symmet-
ric. Now define φ̃ : V → V ∗ by φ̃(v) = B̃(v,−). Consider the endomorphism of representations
φ−1
1 ◦ φ̃. Then, there exist c, d ∈ R such that φ−1
1 ◦ φ̃ = c IdV +dJ . Hence, for every v, w ∈ V ,
B̃(v, w) = cB1(v, w) + dB1(Jv,w). As B̃, B1 are symmetric, we have that, for every v, w ∈ V ,
dB1(Jv,w) = dB1(v, Jw). Suppose d ̸= 0. Then, for every v, w ∈ V , B1(Jv,w) = B1(v, Jw). In
particular, if v ̸= 0, we would have that B1(Jv, Jv) = B1
(
v, J2v
)
= −B1(v, v) < 0, which con-
tradicts positive-definiteness of B1. Hence, B̃ = cB1, for some c ∈ R. By positive-definiteness
and non-degeneracy, c > 0. Moreover, if v ̸= 0, B1(v, v) = B1
(
J2v, J2v
)
= c2B1(v, v). Hence,
c = 1. This completes the proof of the fact that, for every v, w ∈ V , B1(Jv, Jw) = B1(v, w).
Equivalently, B1(J−,−) is skew-symmetric.
Now, back to our previous situation. As B1 and B2 are symmetric and B1(J−,−) is skew-
symmetric, b = 0. And now positive-definiteness implies that a > 0, finishing the proof for the
case End(ρ) ∼= C.
(3) Finally, suppose End(ρ) ∼= H. Then, there exist J1, J2, J3 ∈ End(ρ) linearly independent,
with J2
i = − IdV for i = 1, 2, 3 and J1J2J3 = − IdV , such that any element φ ∈ End(ρ) is of
the form a IdV +b1J1 + b2J2 + b3J3, for some a, b1, b2, b3 ∈ R. Hence, in particular, φ−1
1 ◦ φ2 is
of this form, and thus B2(−,−) = aB1(−,−) + b1B1(J1−,−) + b2B1(J2−,−) + b3B1(J3−,−),
for some a, b1, b2, b3 ∈ R. As in the previous case, one shows that, for every i ∈ {1, 2, 3},
B1(Ji−,−) is skew-symmetric. For completeness, let us show it for J1. Define B̃ : V × V → R
as B̃(v, w) = B1(J1v, J1w). As J2
1 = − IdV , B̃ is non-degenerate. And, as J1 and B1 are
The Geometry of Generalised Spinr Spinors on Projective Spaces 5
ρ-invariant and B1 is symmetric, B̃ is ρ-invariant and symmetric. Now define φ̃ : V → V ∗ by
φ̃(v) = B̃(v,−). Consider the endomorphism of representations φ−1
1 ◦ φ̃. Then, there exist
c, d1, d2, d3 ∈ R such that φ−1
1 ◦ φ̃ = c IdV +d1J1 + d2J2 + d3J3. Hence, for every v, w ∈ V ,
B̃(v, w) = cB1(v, w)+d1B1(J1v, w)+d2B1(J2v, w)+d3B1(J3v, w). As B̃ and B1 are symmetric,
d1B1(J1−,−) + d2B1(J2−,−) + d3B1(J3−,−) is symmetric. Define Q = d1J1 + d2J2 + d3J3,
so that B1(Q−,−) is symmetric. Note that Q2 = −
(
d21 + d22 + d23
)
IdV . Now, for 0 ̸= v ∈ V ,
B1(Qv,Qv) = B1
(
v,Q2v
)
= −
(
d21 + d22 + d23
)
B1(v, v) ≤ 0. By positive-definiteness of B1, we
must have d21+d
2
2+d
2
3 = 0, and hence B̃(v, w) = cB1(v, w). Again by positive definiteness of B1,
c > 0. And, for 0 ̸= v ∈ V ,
B1(v, v) = B1
(
J2
1v, J
2
1v
)
= B̃(J1v, J1v) = cB1(J1v, J1v) = cB̃(v, v) = c2B1(v, v),
so c = 1. This shows that B1(J1−, J1−) = B1(−,−) or, equivalently, that B1(J1−,−) is
skew-symmetric. One can repeat this reasoning with J2 and J3 to show that B1(J2−,−)
and B1(J3−,−) are skew-symmetric. Going back to our previous situation, we have that
b1B1(J1−,−) + b2B1(J2−,−) + b3B1(J3−,−) is skew-symmetric. And, as B1, B2 are sym-
metric, this bilinear form is also symmetric. Hence, it is 0. Therefore, by positive-definiteness
of B1, b1J1 + b2J2 + b3J3 = 0, showing that b1 = b2 = b3 = 0, and hence that B2 = aB1. And,
by positive-definiteness, a > 0.
This concludes the proof. ■
Proposition 2.2. Let g be a finite-dimensional Lie algebra and (ρ1, V1), (ρ2, V2) two non-
isomorphic self-dual irreducible real representations of g. Then, V1 ⊥ V2 with respect to any
(ρ1 ⊕ ρ2)-invariant inner product on V1 ⊕ V2.
Proof. Let B be a (ρ1 ⊕ ρ2)-invariant inner product on V1 ⊕ V2. If V1 is not B-orthogonal
to V2, then we get a non-zero morphism of representations V1 → V ∗
2 , namely v 7→ B(v,−)|V2 .
By Schur’s lemma, this would be an isomorphism, which is a contradiction since V2 is self-dual
and V1 is not isomorphic to V2. ■
Finally, we have the result we were after.
Theorem 2.3. Let G/H be a reductive homogeneous space with reductive decomposition g =
h ⊕ m, where g is the Lie algebra of G, h is the Lie algebra of H and m is an AdH-invariant
complement of h in g. Suppose that the adjoint (isotropy) representation of H on m decomposes
as a direct sum of pairwise non-isomorphic irreducible components m = m1 ⊕ · · · ⊕mk. Suppose
that m admits an AdH-invariant inner product. Then, AdH-invariant inner products on each
irreducible component exist and are unique up to positive scaling, and any AdH-invariant inner
product on m is a positive linear combination of AdH-invariant inner products on the irreducible
components.
Proof. An AdH -invariant inner product on m restricts to an AdH -invariant inner product on
each irreducible component mi. Hence, mi is self-dual. Now, apply Propositions 2.1 and 2.2. ■
2.2 Some notation
(1) If V is a representation of a Lie group G, and H ⊆ G is a subgroup, then we shall denote
the restricted representation by V |H . We shall use analogous notation for restrictions of
Lie algebra representations to Lie subalgebras.
(2) Throughout the computations carried out in this paper, we will repeatedly use some matrix
notation and identities, taken from [2, p. 9]. We will denote by E
(n)
i,j (resp. F
(n)
i,j ) the
6 D. Artacho and J. Hofmann
elementary n× n skew-symmetric (resp. symmetric) matrix given by
E
(n)
i,j =
i j
...
i −1 . . .
j . . . 1
...
, F
(n)
i,j =
i j
...
i 1 . . .
j . . . 1
...
.
By convention, the matrix F
(n)
i,i has all the entries equal to zero except for the (i, i) entry,
which is 1. We will denote by B0 the bilinear form on the space of matrices of appropriate
size given by
B0(X,Y ) := −Re(tr(XY )),
where Re(z) denotes the real part of z and tr(A) is the trace of the matrix A. Finally,
if {ei}i is an orthonormal basis for some vector space V with respect to an inner prod-
uct B, we shall denote by ei,j := ei ∧ ej the standard basis elements for so(V,B) ∼= Λ2V ,
sending ei 7→ ej and ej 7→ −ei.
2.3 Invariant spinr structures
Denote by SO(n) the special orthogonal group, and let λn : Spin(n) → SO(n) be the standard
two-sheeted covering. This map induces an isomorphism at the level of Lie algebras, and its
inverse ρ : so(n) ∼= Λ2Rn → spin(n) ⊆ Cl(n) is given by 2ei ∧ ej 7→ ei · ej . If f is any map with
codomain so(n), we will refer to f̃ := ρ ◦ f as the spin lift of f .
For r ∈ N, we define the group
Spinr(n) := (Spin(n)× Spin(r))/Z2,
where Z2 = ⟨(−1,−1)⟩ ⊆ Spin(n)×Spin(r). Note that Spin1(n) = Spin(n), Spin2(n) = SpinC(n)
and Spin3(n) = SpinH(n), and that there are natural homomorphisms
λrn : Spinr(n) → SO(n), [µ, ν] 7→ λn(µ),
ξrn : Spinr(n) → SO(r), [µ, ν] 7→ λr(ν).
We recall a topological result that we will use multiple times throughout the text.
Proposition 2.4 ([5]). The map φr,n : Spinr(n) → SO(n)× SO(r) defined by λrn × ξrn is a two-
sheeted covering. Moreover,
(1) φ2,2
♯
(
π1
(
Spin2(2)
))
= ⟨(1,±1)⟩ ⊆ Z× Z ∼= π1(SO(2)× SO(2)),
(2) for n ≥ 3, φ2,n
♯
(
π1
(
Spin2(n)
))
= ⟨(1, 1)⟩ ⊆ Z2 × Z ∼= π1(SO(n)× SO(2)),
(3) for r, n ≥ 3, φr,n♯ (π1(Spin
r(n))) = ⟨(1, 1)⟩ ⊆ Z2 × Z2
∼= π1(SO(n)× SO(r)),
where we always take the identifications π1(SO(n)× SO(r)) ∼= π1(SO(n))× π1(SO(r)).
These enlargements or twistings of the spin group give rise to the main players in this paper.
Definition 2.5. Let M be an oriented Riemannian n-manifold with principal SO(n)-bundle of
positively oriented orthonormal frames FM . A spinr structure onM is a reduction of the struc-
ture group of FM along the homomorphism λrn. In other words, it is a pair (P,Φ) consisting of
The Geometry of Generalised Spinr Spinors on Projective Spaces 7
� a principal Spinr(n)-bundle P over M , and
� a Spinr(n)-equivariant bundle homomorphism Φ: P → FM , where Spinr(n) acts on FM
via λrn.
If there is no risk of confusion and Φ is clear from the context, we shall simply denote such
a structure by P . The principal SO(r)-bundle associated to P along ξrn is called the auxiliary
bundle of the spinr structure, and it is denoted by P̂ .
If (P1,Φ1) and (P2,Φ2) are spinr structures on M , an equivalence of spinr structures from
(P1,Φ1) to (P2,Φ2) is a Spinr(n)-equivariant diffeomorphism f : P1 → P2 such that Φ1 = Φ2 ◦f .
If, moreover, M = G/H is a Riemannian homogeneous space, we say that a spinr struc-
ture (P,Φ) on M is G-invariant if G acts smoothly on P by Spinr(n)-bundle homomorphisms
and Φ is G-equivariant.
G-invariant spinr structures on G/H are in one-to-one correspondence with representation-
theoretical data.
Theorem 2.6 ([5]). Let G/H be an n-dimensional oriented Riemannian homogeneous space
with H connected and isotropy representation σ : H → SO(n). Then, there is a bijective corre-
spondence between
� G-invariant spinr structures on G/H modulo G-equivariant equivalence of spinr structures,
and
� Lie group homomorphisms φ : H → SO(r) such that σ × φ : H → SO(n) × SO(r) lifts to
a homomorphism ϕ : H → Spinr(n) along λrn, modulo conjugation by an element of SO(r).
Explicitly, to such a φ corresponds the spinr structure (P,Φ) with P = G ×ϕ Spinr(n) and
Φ: P → FM ∼= G×σ SO(n) given by [g, x] 7→ [g, λrn(x)].
Definition 2.7. For an oriented Riemannian homogeneous space M = G/H, its G-invariant
spin type Σ(M,G) is defined by
Σ(M,G) := min{r ∈ N |M admits a G-invariant spinr structure}.
2.4 Exterior forms approach to the spin representation
It is well known – see, e.g., [32] – that, for n ∈ N, the complexification Cl(n) of the real Clifford
algebra Cl(n) satisfies
Cl(n) ∼=
{
M2k(C) if n = 2k,
M2k(C)⊕M2k(C) if n = 2k + 1.
(2.1)
For n = 2k or 2k + 1, define
Σn =
(
C2
)⊗k
,
and let sk : M2k(C) → EndC(Σn) be the standard representation of M2k(C). The spin represen-
tation ∆n : Cl(n) → EndC(Σn) is defined by
∆n :=
{
sk if n = 2k,
sk ◦ prj if n = 2k + 1,
where prj is the projection onto the j-th factor. Note that, for odd n, there are two non-
isomorphic irreducible representations of Cl(n), and we are choosing one of them (we shall
8 D. Artacho and J. Hofmann
specify which one below). We will also denote by ∆n its restriction to Spin(n) when there is no
risk of confusion. This restriction is independent of the choice of representation for odd n.
It is useful to have an explicit description of the spin representation which does not use the
isomorphism (2.1). We will describe one here, which we refer to as the exterior forms approach
to the spin representation. Similar realizations have appeared, e.g., in [23, 32] and in the early
supergravity literature (see, e.g., [22]). More details and examples, using our precise conventions,
can be found, e.g., in [2, 28].
Suppose n = 2k + 1, and let (e0, . . . , e2k) be the standard basis of Rn. Its complexification
decomposes as C0 ⊕ L⊕ L′, where C0 = spanC{u0 := ie0} and
L := spanC
{
xj :=
1√
2
(e2j−1 − ie2j)
}k
j=1
, L′ := spanC
{
yj :=
1√
2
(e2j−1 + ie2j)
}k
j=1
.
Note that dimC(Λ
•L′) = 2k, and Cl(n) acts on Λ•L′ by extending
xj · η := i
√
2xj⌟η, yj · η := i
√
2yj ∧ η, u0 · η := −ηeven + ηodd,
where ηeven and ηodd are, respectively, the even and odd parts of η ∈ Λ•L′. Hence,
e2j−1 · η := i(xj⌟η + yj ∧ η), e2j · η := yj ∧ η − xj⌟η, e0 · η := iηeven − iηodd. (2.2)
This representation is isomorphic to ∆n for n = 2k + 1 (the other possible choice of irreducible
representation of Cl(n) corresponds to letting e0 act by the negative of what is established
in (2.2)). To obtain it for n = 2k, repeat all the above ignoring everything with a zero subscript.
These representations have an invariant Hermitian product, which we shall denote by ⟨·, ·⟩, and
an associated norm |·|, for which the basis
{yj1,...,jk := yj1 ∧ · · · ∧ yjk | 0 ≤ k ≤ n, 1 ≤ j1 < · · · < jk ≤ n}
is orthonormal.1
2.5 Invariant spinr spinors
A classical spin structure allows us to build a spinor bundle. Similarly, a spinr structure naturally
induces a family of complex vector bundles as follows:
Definition 2.8. Let M be an n-dimensional oriented Riemannian manifold admitting a spinr
structure (P,Φ). For m ∈ N odd, its m-twisted spinr spinor bundle is defined by
Σmn,rM := P ×∆m
n,r
Σmn,r,
with the natural projection to M induced by that of P , where
∆m
n,r := ∆n ⊗∆⊗m
r , Σmn,r := Σn ⊗ Σ⊗m
r ,
where ∆m
n,r is viewed as a representation of Spinr(n).
The requirement that m be odd in Definition 2.8 comes from the fact that ∆m
n,r, which is
a representation of Spin(n)× Spin(r), descends to a representation of Spinr(n) if and only if m
is odd.
1Here we use the convention that the empty wedge product (i.e., the case k = 0) is equal to 1.
The Geometry of Generalised Spinr Spinors on Projective Spaces 9
If, moreover, M = G/H is a homogeneous space and (P,Φ) is G-invariant, then there ex-
ists a homomorphism φ : H → SO(r) such that σ × φ lifts to a map ϕ : H → Spinr(n) (see
Theorem 2.6), and this bundle takes the form
Σmn,rM = G×∆m
n,r◦ϕ Σ
m
n,r.
Sections of Σmn,rM are called m-twisted spinr spinors – if there is no risk of confusion, we will
just refer to them as spinr spinors or simply spinors. They are identified with H-equivariant
maps ψ : G → Σmn,r, and G acts on the space of spinors by (g · ψ)(g′) := ψ
(
g−1g′
)
. G-invariant
spinors correspond, then, to constant H-equivariant maps G → Σmn,r, which in turn correspond
to elements of Σmn,r which are stabilised by H. If H is connected, these are just elements of Σmn,r
which are annihilated by the differential action of the Lie algebra of H. We denote the space of
invariant m-twisted spinr spinors by (Σmn,r)inv.
Remark 2.9. It should be noted that the exterior forms approach to the spin representation
described above does not in general give an identification of (classical) spinors with globally de-
fined differential forms; the isotropy action on Σn is not generally equivalent (as representations)
to the isotropy action on Λ0,•m. Rather, this realisation of spinors via exterior forms is purely
algebraic, and is often non-canonical (i.e., it depends on the choice of basis for m). Notable ex-
ceptions exist in the presence of certain special geometric structures – see, e.g., [2, Remark 3.9]
and the proof of [28, Theorem 5.10]. On the other hand, there are other various constructions
(so-called squaring constructions) which associate (real) differential forms to spinors (see, e.g.,
[13, 32, 41]). One common such construction is to associate to a spinor ψ the k-form
ω(k)(X1, . . . , Xk) := Re⟨(X1 ∧ . . . Xk) · ψ,ψ⟩ for all X1, . . . , Xk ∈ TM,
and it is well known that if ψ is a Killing spinor then ω(1) (or, more precisely, its dual vector
field) is a Killing vector field (see, e.g., [9, Section 1.5]). It should be noted that one obtains
quite often ω(k) = 0, even for non-vanishing spinors ψ (see, e.g., [2, Table 6]). In particular,
the differential forms associated to an invariant spinor in this manner do not seem to be heavily
influenced by its realisation in the exterior form model of the spin representation (which itself
may be non-canonical).
In a similar spirit to Definition 2.7, we make the following definition.
Definition 2.10. For an oriented Riemannian homogeneous space M = G/H, the G-invariant
spinor type of M is defined by
σ(M,G) := min
{
r ∈ N
∣∣∣∣M admits a G-invariant spinr structure
with (Σmn,r)inv ̸= 0 for some odd m
}
.
Remark 2.11. The G-invariant spinor type σ(Mn, G) is well defined, and it satisfies 1 ≤
σ(M,G) ≤ n. This is because the G-invariant Spinn structure on M determined by tak-
ing φ : H → SO(n) to be equal to the isotropy representation always carries a non-zero invariant
1-twisted spinn spinor – see [14, Proposition 3.3].
The requirement that r be minimal in the definition of the G-invariant spinor type is moti-
vated by the next proposition, which shows that passing from a spinr structure to any spinr
′
structure (r′ > r) induced by it via the obvious inclusion Spinr(n) ↪→ Spinr
′
(n) leads to redun-
dancies. Before stating the proposition, we introduce some terminology which will be useful in
describing the relationship between the structures:
10 D. Artacho and J. Hofmann
Definition 2.12. Let Mn = G/H be an oriented Riemannian homogeneous space. We say
that a spinr structures Pr and a spinr
′
structure Pr′ (r ≤ r′) on M are in the same lineage
if Pr′ ∼= Pr ×ι Spin
r′(n), where ι : Spinr(n) ↪→ Spinr
′
(n) is the natural inclusion map induced by
the inclusion SO(r) ↪→ SO(r′) as the lower right-hand r × r block.
Proposition 2.13. Let Mn = G/H be an oriented Riemannian homogeneous space with con-
nected isotropy group H, equipped with a G-invariant spinr structure Pr. Furthermore, for
any r′ ≥ r consider the invariant spinr
′
structure Pr′ in the lineage of Pr. If ψ ∈ (Σmn,r)inv is
an invariant m-twisted spinr spinor, then it induces an invariant m′-twisted spinr
′
spinor for
any m′ ≥ m (m, m′ odd), i.e., there is an inclusion
(Σmn,r)inv ↪→
(
Σm
′
n,r′
)
inv
for all r′ ≥ r, m′ ≥ m (m, m′ odd).
Proof. It suffices to prove the result for r′ ∈ {r, r + 1}. Suppose first that r′ = r + 1,
and let φ : H → SO(r) be an auxiliary homomorphism corresponding to Pr in the sense of
Theorem 2.6. Denoting by σ : H → SO(n) the isotropy representation, we begin by observ-
ing that the invariant spinr+1 structure in the lineage of Pr is induced by the lift of the
homomorphism σ × φ′ : H → SO(n)× SO(r + 1) given by the composition of σ × φ : H →
SO(n) × SO(r) with the inclusion SO(n) × SO(r) ↪→ SO(n) × SO(r + 1). In particular, h acts
on Σm
′
n,r+1 = Σn ⊗ Σ⊗m′
r+1 by the (tensor product action associated to the) lift of σ∗ on the Σn
factor and the lift of φ∗ on the Σr+1 factors. We now split into two cases based on the parity
of r. Supposing first that r is even, we have Σr+1|spin(r)C ≃ Σr as spin(r)C representations, and
therefore Σm
′
n,r+1|hC ≃ Σm
′
n,r as h
C-representations. Since m, m′ are both odd we have m′−m = 2k
for some k ≥ 0, and therefore
Σ⊗m′
r |spin(r)C ≃ Σ⊗m
r ⊗ Σr ⊗ · · · ⊗ Σr︸ ︷︷ ︸
2k copies
.
But Σr is a self-dual representation of spin(r)C, hence also a self-dual representation of hC,
so Σ⊗2k
r contains a copy of the trivial h-representation. The corresponding H-representation
thus also contains a trivial subrepresentation since H is connected. In particular, there is
a copy of Σmn,r|H inside Σm
′
n,r|H and the result in this case follows. Suppose now that r is odd,
and denote by Σr+1 = Σ+
r+1 ⊕ Σ−
r+1 the splitting into positive and negative half-spinor spaces.
Then we have Σ+
r+1|spin(r)C ≃ Σr, hence Σm
′
n,r+1|hC contains a copy of Σm
′
n,r, and the result in this
case then follows by the same argument as in the even case. We have shown the result holds
for r′ = r + 1 (hence for all r′ > r), and all that remains is to consider the case r′ = r. The result
in this case follows by arguing exactly as above, using the fact that Σr is a self-dual spin(r)C
representation to find a copy of the trivial representation in Σ
⊗(m′−m)
r . ■
2.6 Special spinr spinors
In the classical spin setting, it is well known that spinors satisfying certain additional prop-
erties carry geometric information about the manifold. Some of the most widely studied ex-
amples are the so-called Riemannian Killing spinors, which are solutions of the differential
equation ∇g
Xψ = λX · ψ for all X ∈ TM (here ∇g denotes the spinorial connection induced by
the Levi-Civita connection, and λ ∈ R). We refer the reader to [8, 9, 19, 24], among others, for
a detailed exposition of their basic properties and relationship to geometric structures in low
dimensions. Another class of important special spinors are the pure spinors, which are defined
by the algebraic condition that their annihilator inside TCM (with respect to Clifford multipli-
cation) is a maximal isotropic subbundle. Such spinors correspond, uniquely up to scaling, with
orthogonal almost complex structures on the manifold – see [32, Chapter 9] for details.
As in the classical spin case, special spinr spinors also encode geometric properties. In analogy
with pure spinors, we define.
The Geometry of Generalised Spinr Spinors on Projective Spaces 11
Definition 2.14 ([26]). Let ψ ∈ Σmn,r, X,Y ∈ Rn and 1 ≤ k < l ≤ r, and let (ê1, . . . , êr) be the
standard basis of Rr. The real 2-form ηψkl and the skew-symmetric endomorphism η̂ψkl associated
to ψ are defined by
ηψkl(X,Y ) := Re⟨(X ∧ Y ) · (êk · êl) · ψ,ψ⟩, η̂ψkl(X) :=
(
ηψkl(X, ·)
)♯
,
where X ∧ Y = X · Y + ⟨X,Y ⟩ ∈ spin(n) and êk · êl ∈ spin(r). We say that ψ is pure if(
η̂ψkl
)2
= − IdRn and
(
ηψkl + 2êk · êl
)
· ψ = 0 (only for r ≥ 3),
for all 1 ≤ k < l ≤ r. An m-twisted spinr spinor on a manifold is pure if it is pure at every
point.
It is clear that an invariant spinr spinor on a homogeneous space is pure if, and only if, it is
pure at one point.
We are also interested in various differential equations that a spinr spinor might satisfy. Recall
that the Levi-Civita connection on a spin manifold naturally induces a connection on the spinor
bundle. Similarly, the Levi-Civita connection of a spinr manifold together with a connection θ
on the auxiliary bundle defines a connection ∇θ on each twisted spinr spinor bundle. There are
obvious analogues of the usual special spinorial field equations (including the classical Killing
spinor equation mentioned above) to the spinr setting.
Definition 2.15. Let ψ be a twisted spinr spinor on M and θ a connection on the auxiliary
bundle of the spinr structure.
(1) ψ is θ-parallel if ∇θψ = 0;
(2) ψ is θ-Killing if for all vector fields X one has that ∇θ
Xψ = λX ·ψ, for some constant λ ∈ R;
(3) ψ is θ-generalised Killing if there exists a symmetric endomorphism field A ∈ End(TM)
such that, for all vector fields X, one has ∇θ
Xψ = A(X) · ψ.
We collect here a number of results that relate the existence of special spinr spinors to
geometric properties of the manifold:
Theorem 2.16 ([14, 26]). Let M be an n-dimensional spinr manifold, and let θ be a connection
on its auxiliary bundle.
(1) If M carries a θ-parallel spinor ψ, then the Ricci tensor decomposes as
Ric =
1
|ψ|2
∑
k<l
Θ̂kl ◦ η̂ψkl,
where Θ̂kl is the skew-symmetric endomorphism associated to the 2-form on TM given
by Θkl(X,Y ) := ⟨Ω(X,Y )(êk), êl⟩, where Ω is the curvature 2-form of the connection θ on
the auxiliary bundle.
(2) If θ is flat and M carries a θ-Killing spinor, then M is Einstein.
(3) If M carries a θ-parallel m-twisted pure spinor for some m ∈ N, r ≥ 3, r ̸= 4, n ̸= 8,
n+ 4r − 16 ̸= 0 and n+ 8r − 16 ̸= 0, then M is Einstein.
(4) If r = 2 and M carries a θ-parallel pure spinor, then M is Kähler.
(5) If r = 3 and M admits a θ-parallel pure spinor, then M is quaternionic Kähler.
If M = G/H is a Riemannian homogeneous space, invariant connections on homogeneous
bundles over M are described by algebraic data [38] (see, e.g., [3] for a modern treatment).
12 D. Artacho and J. Hofmann
Proposition 2.17. Let G/H be a homogeneous space, and let ϕ : H → K be a Lie group homo-
morphism. There is a one-to-one correspondence between G-invariant connections on G×ϕ K
and linear maps Λ : g → k satisfying2
(1) Λ(X) = ϕ∗(X), X ∈ h;
(2) Λ ◦AdH(h) = AdK(ϕ(h)) ◦Λ, h ∈ H.
The map Λ corresponding to a connection is called the Nomizu map of said connection.
For the connections of interest in this article, the Nomizu maps are particularly easy to
describe:
Proposition 2.18 ([38, Theorem 13.1]). Let (G/H, g) be an n-dimensional oriented Rieman-
nian homogeneous space, where the metric g corresponds to an invariant inner product B on
a reductive complement m of h. The Nomizu map Λ : g → so(m) of the Levi-Civita connection
of g is given by
Λ(X)(Y ) =
1
2
[X,Y ]m + U(X,Y ), X ∈ g, Y ∈ m,
where U is defined by
B(U(X,Y ),W ) =
1
2
(B([W,X]m, Y ) +B(X, [W,Y ]m)).
The following proposition describes how the correspondence in Proposition 2.17 works in the
particular situation we are interested in.
Proposition 2.19. Let (G/H, g) be an n-dimensional Riemannian homogeneous space equipped
with a G-invariant spinr structure P . Let Λ : g → so(n) be the Nomizu map of the Levi-Civita
connection of g, and let Λ′ : g → so(r) be the Nomizu map of an invariant connection θ on the
associated bundle P̂ . Let Λ̃ be the spin lift of Λ to spin(n) and let Λ̃′ be the spin lift of Λ′
to spin(r). Then, Λ̃⊗
(
Λ̃′
)⊗m
is the Nomizu map of the invariant connection ∇θ on the m-
twisted spinr spinor bundle. Moreover, if ψ ∈ (Σmn,r)inv and X̂ is the fundamental vector field
on G/H defined by X ∈ m, then(
∇θ
X̂
ψ
)
eH
=
(
Λ̃⊗ Λ̃′⊗m
)
(X) · ψ.
In particular, an invariant m-twisted spinr spinor ψ is θ-parallel if, and only if, it satisfies the
equation ∀X ∈ m :
(
Λ̃⊗ Λ̃′⊗m
)
(X) · ψ = 0.
As we shall see later in the setting of spinC structures, the second condition in Proposition 2.17
is quite restrictive. Indeed, the auxiliary bundles of invariant spinC structures are principal
bundles of the abelian group SO(2). Hence, the second condition becomes Λ ◦ AdH(h) = Λ for
all h ∈ H. This will force the kernel of Λ|m to be quite large in most of our cases. The following
is a useful criterion.
Lemma 2.20. Let G/H be a homogeneous space with a reductive decomposition g = h⊕m, and
let ϕ : H → K be a Lie group homomorphism. Let H0 ⊆ H be the kernel of AdK ◦ϕ : H → GL(k).
If X ∈ spanR[h0,m], then Λ(X) = 0 for the Nomizu map Λ : g → k associated to any invariant
connection on G×ϕ K.
2Note that AdH in condition (2) refers to the restriction of the adjoint representation of G to H ⊆ G, whereas
AdK refers to the adjoint representation of K.
The Geometry of Generalised Spinr Spinors on Projective Spaces 13
Proof. By linearity of Λ, it suffices to consider X = [v, Y ] for some v ∈ h0 and Y ∈ m.
Let γ : R → H0 be a curve with γ(0) = eG and γ′(0) = v. By Proposition 2.17, the Nomizu
map Λ of any invariant connection on G×ϕ K satisfies Λ ◦AdH(γ(t)) = Λ, and hence
0 =
d
dt
∣∣
t=0
Λ(Y ) =
d
dt
∣∣
t=0
Λ(AdH(γ(t))Y ) = Λ([v, Y ]) = Λ(X). ■
Finally, we examine the differential equations satisfied by invariant spinr spinors on symmetric
spaces. The following proposition is analogous to the familiar fact in the spin setting that
invariant spinors on symmetric spaces are ∇g-parallel, since the Levi-Civita and the Ambrose–
Singer connections coincide.
Proposition 2.21. Let (M = G/H, g) be a Riemannian symmetric space admitting a G-
invariant spinr structure P . Let g = h ⊕ m be a reductive decomposition such that [m,m] ⊆ h.
Let E be the natural vector bundle associated to the auxiliary bundle P̂ , and let ∇E be the
unique G-invariant connection on E whose Nomizu map vanishes identically on m. Let ∇ :=
∇g ⊗
(
∇E
)⊗m
be the corresponding twisted connection on Σmn,rM .
If ψ ∈ Σmn,r is a G-invariant spinr spinor, then ∇ψ = 0.
Proof. With respect to the reductive decomposition g = h⊕m, the Nomizu map associated to
the Levi-Civita connection vanishes identically on the reductive complement m, i.e., Λg|m ≡ 0.
The Nomizu map of ∇ then vanishes identically on m, and the result follows. ■
This result will be useful for several of the cases in our classification, where the limited number
of low-dimensional representations of the isotropy groups will force the auxiliary bundles to be
isomorphic to familiar tensor (sub)bundles.
3 Projective spaces
Onǐsčik [39, p. 163] classified the compact, simple and simply connected Lie groups which act
transitively on the projective spaces CPn, HPn and OP2 – see also [44, p. 356]. We exhibit them
in Table 2.
Space G H
CPn SU(n+ 1) S(U(1)×U(n))
CP2n+1 Sp(n+ 1) U(1)× Sp(n)
HPn Sp(n+ 1) Sp(1)× Sp(n)
OP2 F4 Spin(9)
Table 2. Compact, simple and simply connected Lie groups G acting transitively with isotropy H on
projective spaces – see, e.g., [44, p. 356].
3.1 Hermitian complex projective space
In this section, we consider the complex projective space realised as the quotient
CPn ∼= SU(n+ 1)⧸S(U(1)×U(n)),
where
S(U(1)×U(n)) =
{(
z 0
0 B
)
∈ Mn+1(C) | z ∈ U(1), B ∈ U(n), z det(B) = 1
}
.
14 D. Artacho and J. Hofmann
In order to study SU(n + 1)-invariant spinr structures and spinors on this space, we need to
establish some notation and properties of the Lie algebras involved. Let us denote by h the Lie
algebra of H := S(U(1) × U(n)), and consider the copy of SU(n) included in SU(n + 1) as the
lower right-hand n× n block. Letting h′ := su(n) ⊆ su(n+ 1), we have the decomposition
h = Rξ ⊕ h′ (as Lie algebras),
where ξ := i
(
−nF (n+1)
1,1 +
∑n+1
l=2 F
(n+1)
l,l
)
and
h′ = su(n) = spanR
{
iF (n+1)
p,q , E(n+1)
p,q , i
(
F (n+1)
r,r − F
(n+1)
r+1,r+1
)}
2≤p<q≤n+1
r=2,...,n
.
The isotropy subalgebra h ⊆ su(n+ 1) has a reductive complement
m := (h)⊥B0 = spanR
{
iF
(n+1)
1,p , E
(n+1)
1,p
}
p=2,...,n+1
,
and the adjoint representation of h on m is irreducible. Hence, by Theorem 2.3, the SU(n+ 1)-
invariant metrics on CPn correspond to the inner products on m in the one-parameter fam-
ily ga := aB0|m×m, a > 0, and a ga-orthonormal basis of m is given by{
e2p−1 :=
i√
2a
F
(n+1)
1,p+1 , e2p :=
1√
2a
E
(n+1)
1,p+1
}
p=1,...,n
.
We take the orientation defined by the ordering (e1, e2, . . . , e2n−1, e2n).
3.1.1 Invariant spinr structures
We are now ready to determine the SU(n+ 1)-invariant spin type of CPn. By [27, p. 327], it is
clear that CPn admits an SU(n+1)-invariant spin structure if, and only if, n is odd. Moreover,
one has the following.
Theorem 3.1. The SU(n+ 1)-invariant spinC structures on CPn are given by
SU(n+ 1)×ϕs Spin
C(2n), s ∈ Z : n ̸≡ s mod 2,
where ϕs is the unique lift of σ×φs to SpinC(2n), σ : H → SO(2n) is the isotropy representation
and φs : H → SO(2) ∼= U(1) is given by(
z 0
0 B
)
7→ det(B)s.
In particular, the SU(n+ 1)-invariant spin type of CPn is
Σ(CPn, SU(n+ 1)) =
{
1, n odd,
2, n even.
Proof. Note that H ∼= U(n), and that every Lie group homomorphism U(n) → U(1) is of
the form B 7→ det(B)s, for some s ∈ Z. The loop α(t) = diag
(
e−2πit, 1, . . . , 1, e2πit
)
gener-
ates π1(H) ∼= Z, and
(σ × φs)♯(α) = (n− 1, s) ∈ π1(SO(2n))× π1(SO(2)).
This can be seen as follows: the image of α(t) under the isotropy representation σ is easily seen
to be
σ(α(t)) = diag
(
e2πit, . . . , e2πit, e4πit
)
∈ U(n) ⊆ SO(2n),
where e2πit appears n− 1 times. This can be seen using the realisation of σ as the action of H
on m by matrix conjugation.
Hence, by Proposition 2.4, σ × φs : H → SO(2n) × U(1) lifts to SpinC(2n) if, and only if,
n ̸≡ s mod 2. Finally, as U(1) is abelian, the representations φs are pairwise non-equivalent.
The result now follows from Theorem 2.6. ■
The Geometry of Generalised Spinr Spinors on Projective Spaces 15
3.1.2 Invariant spinr spinors
The classical spin case r = 1 does not yield any non-trivial invariant spinors, as we show in the
following theorem.
Theorem 3.2. For n odd, there are no non-trivial SU(n+ 1)-invariant spinors on CPn.
Proof. We need the explicit expression of the action of ξ ∈ h on m. Letting σ : H → SO(2n) be
the isotropy representation and σ̃ its lift to Spin(2n), and, e.g., using the commutation relations
in [2, p. 9], one can readily see that, for each p = 1, . . . , n,
ad(ξ)|m(e2p) = [ξ, e2p]m = (n+ 1)e2p−1, ad(ξ)|m(e2p−1) = [ξ, e2p−1]m = −(n+ 1)e2p.
Hence,
σ∗(ξ) = ad(ξ)|m = (n+ 1)
n∑
p=1
e2p ∧ e2p−1 ∈ so(2n),
and the spin lift is given by
σ̃∗(ξ) =
n+ 1
2
n∑
p=1
e2p · e2p−1 ∈ spin(2n) ⊆ Cl(2n).
A direct computation using (2.2) shows that, for each 1 ≤ k ≤ n and 1 ≤ j1 < · · · < jk ≤ n,
σ̃∗(ξ) · (yj1 ∧ · · · ∧ yjk) =
i(n+ 1)
2
(2k − n)yj1 ∧ · · · ∧ yjk .
From this we observe that, if n is odd, there are no non-trivial invariant spinors. ■
The fact that no non-trivial invariant spinors exist motivates the investigation of spinC
spinors.
Theorem 3.3. For n, s ∈ N with n ̸≡ s mod 2, the space of SU(n+1)-invariant 1-twisted spinC
spinors on CPn associated to the spinC structure SU(n+ 1)×ϕs Spin
C(2n) is given by
(
Σ1
2n,2
)
inv
=
spanC{1⊗ 1̂, (y1 ∧ · · · ∧ yn)⊗ ŷ1}, s = n+ 1,
spanC{(y1 ∧ · · · ∧ yn)⊗ 1̂, 1⊗ ŷ1}, s = −(n+ 1),
0, otherwise.
In particular, the SU(n+ 1)-invariant spinor type of CPn is σ(CPn, SU(n+ 1)) = 2.
Proof. Recall that h = Rξ ⊕ h′ as Lie algebras, and note that, for ψ ∈ Σ1
2n,2,
(∀X ∈ h′ : (ϕs)∗(X) · ψ = 0) ⇐⇒ ψ ∈ spanC{1, y1 ∧ · · · ∧ yn} ⊗ Σ2,
by [2, Theorem 3.7] and the definition of φs. Moreover,
(ϕs)∗(ξ) =
(
n+ 1
2
n∑
p=1
e2p · e2p−1,
sn
2
ê1 · ê2
)
∈ spin(2n)⊕ spin(2) ∼= spinC(2n).
Finally, an easy calculation shows that, for 0 ≤ k ≤ n and 1 ≤ j1 < · · · < jk ≤ n,
(ϕs)∗(ξ) ·
(
(yj1 ∧ · · · ∧ yjk)⊗ 1̂
)
=
i
2
((n+ 1)(2k − n) + sn)
(
(yj1 ∧ · · · ∧ yjk)⊗ 1̂
)
,
(ϕs)∗(ξ) · ((yj1 ∧ · · · ∧ yjk)⊗ ŷ1) =
i
2
((n+ 1)(2k − n)− sn)((yj1 ∧ · · · ∧ yjk)⊗ ŷ1).
From this it is straightforward to conclude the result. ■
16 D. Artacho and J. Hofmann
3.1.3 Special spinr spinors
The aim now is to show that the SU(n+1)-invariant spinC spinors on CPn found in Theorem 3.3
are pure and parallel with respect to a suitable connection on the auxiliary bundle.
Proposition 3.4. For s = n+1 (resp. s = −(n+1)), the SU(n+1)-invariant spinC spinors 1⊗ 1̂
and (y1∧· · ·∧yn)⊗ ŷ1
(
resp. (y1∧· · ·∧yn)⊗ 1̂ and 1⊗ ŷ1
)
on CPn are pure. Moreover, they are
parallel with respect to the invariant connection on the corresponding auxiliary bundle determined
by the Nomizu map Λ|m = 0.
Proof. We will only show the calculations for the spinor ψ = (y1 ∧ · · · ∧ yn) ⊗ 1̂, as the other
three are analogous. Since r = 2 < 3, we only need to show that
(
η̂ψ12
)2
= − Id. Indeed,
a straightforward calculation shows that
ηψ12(e2p, e2q) = Re⟨e2p · e2q · ê1 · ê2 · ψ,ψ⟩+ δp,q Re⟨ê1 · ê2 · ψ,ψ⟩
= Re⟨ie2p · e2q · ψ,ψ⟩+ δp,q Re⟨−iψ,ψ⟩ = 0,
ηψ12(e2p−1, e2q−1) = 0,
ηψ12(e2p, e2q−1) = Re⟨e2p · e2q−1 · ê1 · ê2 · ψ,ψ⟩ = Re⟨ie2p · e2q−1 · ψ,ψ⟩ = −δp,q.
Hence, η̂ψ12(e2p) = −e2p−1 and η̂ψ12(e2p−1) = e2p.
The last assertion of the proposition follows by noting that, as CPn ∼= SU(n + 1)/ S(U(1) ×
U(n)) is a symmetric space, the Levi-Civita connection coincides with the Ambrose–Singer
connection, whose Nomizu map satisfies Λg|m ≡ 0. ■
In light of the Ricci decomposition in [14, Theorem 3.1], the existence of parallel pure spinC
spinors encodes a very well-known fact – see, e.g., [44].
Theorem 3.5. The SU(n+ 1)-invariant metrics ga on CPn are Kähler–Einstein.
Proof. Take s = −(n+1). Consider the spinC structure defined by φs, and endow its auxiliary
bundle with the connection described in Proposition 3.4. We have seen that this spinC structure
carries a non-zero parallel pure spinC spinor ψ = (y1∧ · · ·∧yn)⊗ 1̂. This implies that the metric
is Kähler [26, Corollary 4.10] with respect to the invariant complex structure defined by η̂ψ12.
Now, by Theorem 2.16 (1), the Ricci tensor decomposes as
Ric =
1
|ψ|2
Θ̂12 ◦ η̂ψ12,
where Θ̂12 is the endomorphism associated to the 2-form on m Θ12(X,Y ) := ⟨Ω(X,Y )(ê1), ê2⟩,
X,Y ∈ m, where Ω is the curvature 2-form of the connection on the auxiliary bundle. Recall [3]
that, if Λ is the Nomizu map of the connection on the auxiliary bundle, then
∀X,Y ∈ m : Ω(X,Y ) = [Λ(X),Λ(Y )]so(2) −Λ([X,Y ]) = −Λ([X,Y ]).
It is now easy to see that, for all 1 ≤ p, q ≤ n,
Ω(e2p−1, e2q) = δp,q
s
a
ê1 ∧ ê2, Ω(e2p−1, e2q−1) = Ω(e2p, e2q) = 0.
Hence,
Θ̂12 =
s
a
n∑
p=1
e2p−1 ∧ e2p,
The Geometry of Generalised Spinr Spinors on Projective Spaces 17
and finally, using the expression of η̂ψ12 obtained in the proof of Proposition 3.4, we obtain
Ric =
1
|ψ|2
Θ̂12 ◦ η̂ψ12 =
n+ 1
a
Id . (3.1)
This proved the theorem. ■
Remark 3.6. Recall that the Fubini–Study metric gFS on CPn is SU(n+1)-invariant, and that
its Ricci constant is 2(n+ 1). From equation (3.1), we can deduce that gFS = g1/2.
3.2 Symplectic complex projective space
Consider, for n ≥ 1, the homogeneous realisation of odd-dimensional complex projective space
CP2n+1 ∼= Sp(n+ 1)⧸U(1)× Sp(n),
where H := U(1) × Sp(n) is realised as a subgroup of Sp(n + 1) by the upper left-hand 1 × 1
block for U(1) and the lower right-hand n×n block for Sp(n). Denote by h the Lie algebra of H
and h′ := sp(n) ⊆ sp(n+ 1). Then, h = Rξ1 ⊕ h′ (as Lie algebras), where ξ1 := iF
(n+1)
1,1 and
h′ = sp(n)
= spanR
{
iF (n+1)
p,p , jF (n+1)
p,p , kF (n+1)
p,p , iF (n+1)
r,s , jF (n+1)
r,s , kF (n+1)
r,s , E(n+1)
r,s
}
2≤r<s≤n+1
p=2,...,n+1
.
The Lie subalgebra h ⊆ sp(n+ 1) has a reductive complement m := (h)⊥B0 = V ⊕H, where
V := spanR
{
ξ2 := −kF
(n+1)
1,1 , ξ3 := jF
(n+1)
1,1
}
,
H := spanR
{
e4p := jF
(n+1)
1,p+1 , e4p+1 := kF
(n+1)
1,p+1 , e4p+2 := iF
(n+1)
1,p+1 , e4p+3 := E
(n+1)
1,p+1
}
p=1,...,n
,
and this is the decomposition of m into irreducible subrepresentations3 of the adjoint represen-
tation of h. We have, therefore, by Theorem 2.3, a two-parameter family of invariant metrics
ga,t := aB0|H×H + 2atB0|V×V , a, t > 0,
and a ga,t-orthonormal basis of m is given by{
ξa,t2 :=
1√
2ta
ξ2, ξ
a,t
3 :=
1√
2ta
ξ3, e
a,t
4p+ε :=
1√
2a
e4p+ε
}
ε=0,1,2,3
p=1,...,n
.
We take the orientation defined by the ordering
(
ξa,t2 , ξa,t3 , ea,t4 , . . . , ea,t4n+3
)
.
3.2.1 Invariant spinr structures
We begin by determining the Sp(n+1)-invariant spin type of CP2n+1. By [27, p. 327], it is clear
that CP2n+1 admits a unique spin structure, and this structure is Sp(n + 1)-invariant. Using
the algebraic characterisation in Theorem 2.6, we can explicitly obtain all Sp(n + 1)-invariant
spinC structures on CP2n+1:
Theorem 3.7. The Sp(n+ 1)-invariant spinC structures on CP2n+1 are given by
Sp(n+ 1)×ϕs Spin
C(4n+ 2), s ∈ 2Z,
where ϕs is the unique lift of σ × φs to SpinC(4n + 2), σ : H → SO(4n + 2) is the isotropy
representation and φs : H → SO(2) ∼= U(1) is defined by (z,A) 7→ zs.
3Note that this decomposition into V and H corresponds to the vertical and horizontal distributions of the
generalised Hopf fibration S2 ↪→ CP2n+1 → HPn.
18 D. Artacho and J. Hofmann
Proof. This follows from Theorem 2.6, together with the fact that Sp(n) is simple and that
all Lie group homomorphisms U(1) → U(1) are of the form z 7→ zs, for some s ∈ Z. Using
Proposition 2.4 as in the proof of Theorem 3.1, one sees that σ × φs lifts to SpinC(4n + 2) if,
and only if, s is even. As U(1) is abelian, the representations φs are pairwise non-equivalent.
Hence, these spinC structures are pairwise non-Sp(n+ 1)-equivariantly equivalent. ■
3.2.2 Invariant spinr spinors
First, we classify the Sp(n+ 1)-invariant spinors for the unique spin structure of CP2n+1.
Theorem 3.8. The space Σinv of Sp(n+ 1)-invariant spinors on CP2n+1 is given by
Σinv =
{
spanC
{
ψ+ := ω(n+1)/2, ψ− := y1 ∧ ω(n−1)/2
}
, n odd,
0, n even,
where ω :=
∑n
p=1 y2p ∧ y2p+1.
Proof. By [2, Theorem 4.11], the space of invariant spinors is quite restricted:
Σinv ⊆ spanC
{
ωk, y1 ∧ ωk
}
k=0,...,n
.
We only need to determine which of these are annihilated by ξ1. A computation analogous to
the one in the proof of Theorem 3.2 shows that, if σ̃ is the lift to Spin(4n + 2) of the isotropy
representation σ : H → SO(4n+ 2),
σ̃∗(ξ1) = ξa,t2 · ξa,t3 +
1
2
n∑
p=1
(
ea,t4p · ea,t4p+1 + ea,t4p+2 · e
a,t
4p+3
)
.
In particular,
σ̃∗(ξ1) · ωk = i(n+ 1− 2k)ωk, σ̃∗(ξ1) ·
(
y1 ∧ ωk
)
= i(n− 1− 2k)
(
y1 ∧ ωk
)
,
and the result follows. ■
We now turn to the study of spinC spinors. Using the same argument as in the Hermitian
case (see Theorem 3.3), one obtains.
Theorem 3.9. For n ∈ N and s = 2s′ ∈ 2Z, the space
(
Σ1
4n+2,2
)
inv
of Sp(n + 1)-invariant
1-twisted spinC spinors on CP2n+1 associated to the spinC structure Sp(n+1)×ϕs Spin
C(4n+2)
is given by
(
Σ1
4n+2,2
)
inv
=
spanC
{
ω(n+1+s′)/2, y1 ∧ ω(n−1+s′)/2
}
⊗ 1̂⊕
⊕ spanC
{
ω(n+1−s′)/2, y1 ∧ ω(n−1−s′)/2}⊗ ŷ1, n ̸≡ s′ mod 2,
0, otherwise,
where negative powers of ω are defined to be zero. In particular, the Sp(n+ 1)-invariant spinor
type of CP2n+1 satisfies
σ
(
CP2n+1,Sp(n+ 1)
)
=
{
1, n odd,
2, n even.
Remark 3.10. The spinC structure corresponding to s = 0 is the one induced by the usual
spin structure. Indeed, taking s = 0 in Theorem 3.9, one recovers the spinors in Theorem 3.8
tensored with Σ2.
The Geometry of Generalised Spinr Spinors on Projective Spaces 19
3.2.3 Special spinr spinors
In order to differentiate these spinors, one can see, using the formulas for the Nomizu map
from [9, p. 141], that the spin lift Λ̃a,t of the Nomizu map of the Levi-Civita connection of ga,t
is given by
Λ̃a,t
(
ξa,t2
)
=
1− t
2
√
2at
n∑
p=1
(
ea,t4p · ea,t4p+2 − ea,t4p+1 · e
a,t
4p+3
)
,
Λ̃a,t
(
ξa,t3
)
=
1− t
2
√
2at
n∑
p=1
(
ea,t4p · ea,t4p+3 + ea,t4p+1 · e
a,t
4p+2
)
,
Λ̃a,t
(
ea,t4p
)
=
1
2
√
t
2a
(
−ξa,t2 · ea,t4p+2 − ξa,t3 · ea,t4p+3
)
,
Λ̃a,t
(
ea,t4p+1
)
=
1
2
√
t
2a
(
ξa,t2 · ea,t4p+3 − ξa,t3 · ea,t4p+2
)
,
Λ̃a,t
(
ea,t4p+2
)
=
1
2
√
t
2a
(
ξa,t2 · ea,t4p + ξa,t3 · ea,t4p+1
)
,
Λ̃a,t
(
ea,t4p+3
)
=
1
2
√
t
2a
(
−ξa,t2 · ea,t4p+1 + ξa,t3 · ea,t4p
)
. (3.2)
Baum et al. proved in [9, p. 146] that CP3 admits non-trivial Sp(2)-invariant generalised
Killing spinors, given by ψ+ ± iψ−. For the Fubini–Study metric, the two eigenvalues of these
generalised Killing spinors coincide, yielding real Killing spinors which are related to the nearly
Kähler geometry of CP3. We now show that this does not occur in higher dimensions.
Theorem 3.11. The spaces CP2n+1 admit non-trivial Sp(n + 1)-invariant generalised Killing
spinors if, and only if, n = 1.
Proof. First, we recall from Theorem 3.8 that there are no invariant spinors when n is even,
so it remains only to consider the case where n is odd. Let n be odd, and suppose that n ≥ 3
so that ω(n±3)/2 ̸= 0. Using the above formulas (3.2) for the Nomizu map, we get, for α, β ∈ C,
Λ̃a,t
(
ea,t4p
)
· (αψ+ + βψ−) = −
√
t
2a
{
α
n+ 1
2
y1 ∧ y2p + βy2p+1
}
∧ ω(n−1)/2.
Writing a general element X ∈ m as a (real) linear combination
X = µ2ξ
a,t
2 + µ3ξ
a,t
3 +
n∑
p=1
(
µ4pe
a,t
4p + µ4p+1e
a,t
4p+1 + µ4p+2e
a,t
4p+2 + µ4p+3e
a,t
4p+3
)
of the basis vectors, we find that the Clifford product with an arbitrary invariant spinor is
given by
X · (αψ+ + βψ−)
= α
{
(iµ2 + µ3)y1 +
n∑
p=1
[(iµ4p + µ4p+1)y2p + (iµ4p+2 + µ4p+3)y2p+1]
}
∧ ω(n+1)/2
+ α
n+ 1
2
{
n∑
p=1
[(iµ4p − µ4p+1)y2p+1 + (−iµ4p+2 + µ4p+3)y2p]
}
∧ ω(n−1)/2
+ β{iµ2 − µ3}ω(n−1)/2
20 D. Artacho and J. Hofmann
+ βy1 ∧
{
n∑
p=1
[(−iµ4p − µ4p+1)y2p + (−iµ4p+2 − µ4p+3)y2p+1]
}
∧ ω(n−1)/2
+ β
n− 1
2
{
n∑
p=1
[(−iµ4p + µ4p+1) + (iµ4p+2 − µ4p+3)]
}
y1 ∧ ω(n−3)/2.
Hence, by equating coefficients in
Λ̃a,t
(
ea,t4p
)
· (αψ+ + βψ−) = X · (αψ+ + βψ−),
one easily concludes (using crucially that n ≥ 3) that the only possibility is α = β = 0, and the
result then follows from the preceding discussion about the case n = 1. ■
We now turn to the study of the Sp(n + 1)-invariant spinC spinors on CP2n+1 found in
Theorem 3.9. The aim is to show that, when s′ = s/2 = ±n±1 and t = 1, there is a pure spinor
which is parallel with respect to a suitable connection on the auxiliary bundle. This encodes
the fact that, for t = 1, the metric ga,t is Kähler.
Lemma 3.12. Let k ∈ N and ψ ∈
{
ωk⊗ 1̂,
(
y1∧ωk
)
⊗ 1̂, ωk⊗ ŷ1,
(
y1∧ωk
)
⊗ ŷ1
}
. Then, a scalar
multiple of ψ is pure if, and only if, k = 0 or k = n.
Proof. We will only prove it for ψ = ωk ⊗ 1̂, as the other cases are analogous. Since r = 2 < 3,
we only need to show that
(
η̂ψ12
)2
= − Id. Indeed, for all 1 ≤ p, q ≤ n and ε ∈ {0, 1, 2, 3}, one
calculates:
ηψ12
(
ξa,t2 , ξa,t3
)
= Re
〈
ξa,t2 · ξa,t3 · ê1 · ê2 · ψ,ψ
〉
= Re
〈
iξa,t2 · ξa,t3 · ψ,ψ
〉
= −Re
〈
ωk ⊗ 1̂, ωk ⊗ 1̂
〉
= −(k!)2
(
n
k
)
,
ηψ12
(
ea,t4p , e
a,t
4q+1
)
= Re
〈
ea,t4p · ea,t4q+1 · ê1 · ê2 · ψ,ψ
〉
= Re
〈
i · ea,t4p · ea,t4q+1 · ω
k, ωk
〉
= −δp,q
〈
ωk − 2ky2p ∧ y2p+1 ∧ ωk−1, ωk
〉
= −δp,q(k!)2
[(
n
k
)
− 2
(
n− 1
k − 1
)]
,
ηψ12
(
ea,t4p+2, e
a,t
4q+3
)
= −δp,q(k!)2
[(
n
k
)
− 2
(
n− 1
k − 1
)]
,
ηψ12
(
ξa,t2 , ea,t4p+ε
)
= ηψ12
(
ξa,t3 , ea,t4p+ε
)
= ηψ12
(
ea,t4p , e
a,t
4q+2
)
= ηψ12
(
ea,t4p , e
a,t
4q+3
)
= ηψ12
(
ea,t4p+1, e
a,t
4q+3
)
= 0,
where
(
n−1
k−1
)
is understood to be 0 if k = 0. Altogether, we have
η̂ψ12 = −(k!)2
[(
n
k
)
ξa,t2 ∧ ξa,t3 +
[(
n
k
)
− 2
(
n− 1
k − 1
)] n∑
p=1
(
ea,t4p ∧ ea,t4p+1 + ea,t4p+2 ∧ e
a,t
4p+3
)]
,
which is easily seen to square to a multiple of − Id if, and only if, k = 0 or k = n. ■
The preceding lemma, together with Theorem 3.9 describing the invariant spinC spinors,
implies that the Sp(n + 1)-invariant spinC structure corresponding to s = 2s′ ∈ 2Z admits
invariant pure spinC spinors if, and only if, s′ ∈ {n+ 1,−n− 1, n− 1,−n+ 1}, which are given
in each case by{
(y1 ∧ ωn)⊗ 1̂, 1⊗ ŷ1
}
, s′ = n+ 1,
{
1⊗ 1̂, (y1 ∧ ωn)⊗ ŷ1
}
, s′ = −n− 1,{
ωn ⊗ 1̂, y1 ⊗ ŷ1
}
, s′ = n− 1,
{
y1 ⊗ 1̂, ωn ⊗ ŷ1
}
, s′ = −n+ 1.
The Geometry of Generalised Spinr Spinors on Projective Spaces 21
In order to differentiate these spinors, one needs to fix a connection on the auxiliary bundle.
Applying the criterion in Lemma 2.20, one sees that the only Sp(n+1)-invariant connection on
the auxiliary bundle is the one with Nomizu map Λ|m = 0. This connection, together with the
Levi-Civita connection of the metric ga,t (with Nomizu map Λa,t), induces a connection ∇a,t on
the corresponding spinC spinor bundle. The following lemma is a straightforward calculation
using the expression of the spin lift of the Nomizu map (3.2):
Lemma 3.13. The invariant pure spinC spinor 1⊗ 1̂ is ∇a,t-parallel if, and only if, t = 1.
These spinC spinors encode some well-known geometric information of CP2n+1 – see, e.g., [44].
Theorem 3.14. The metric ga,1 on CP2n+1 is Kähler–Einstein, for all a > 0.
Proof. Let s = −2(n+ 1), and consider the spinC structure on CP2n+1 determined by φs. By
Lemmas 3.12 and 3.13, this structure carries a ∇a,1 parallel pure spinC spinor ψ = 1⊗ 1̂. Hence,
by [26, Corollary 4.10], the metric ga,1 is Kähler, for all a > 0.
Let us now see that these metrics are also Einstein. By Theorem 2.16 (1), the Ricci tensor
decomposes as
Ric =
1
|ψ|2
Θ̂12 ◦ η̂ψ12. (3.3)
By calculations similar to those in the proof of Theorem 3.5, one finds that, for 1 ≤ p, q ≤ n,
0 ≤ ε ≤ 3 and 2 ≤ l ≤ 3,
Ω
(
ξa,12 , ξa,13
)
= −Λ
([
ξa,12 , ξa,13
])
= −1
a
Λ(ξ1) = −1
a
(
φ−2(n+1)
)
∗(ξ1) =
2(n+ 1)
a
ê1 ∧ ê2,
Ω
(
ea,14p , e
a,1
4q+1
)
= Ω
(
ea,14p+2, e
a,1
4q+3
)
= δp,q
2(n+ 1)
a
ê1 ∧ ê2,
Ω
(
ea,14p , e
a,1
4q+2
)
= Ω
(
ea,14p , e
a,1
4q+3
)
= Ω
(
ea,14p+1, e
a,1
4q+2
)
= Ω
(
ea,14p+1, e
a,1
4q+3
)
= Ω
(
ξa,1l , ea,14p+ε
)
= 0.
Hence, using the definition of Θ̂12 in terms of Ω and taking k = 0 in the proof of Lemma 3.12,
Θ̂12 =
2(n+ 1)
a
(
ξa,12 ∧ ξa,13 +
n∑
p=1
(
ea,14p ∧ ea,14p+1 + ea,14p+2 ∧ ea,14p+3
))
,
η̂ψ12 = −
(
ξa,12 ∧ ξa,13 +
n∑
p=1
(
ea,14p ∧ ea,14p+1 + ea,14p+2 ∧ ea,14p+3
))
.
Finally, substituting everything into equation (3.3), we get Ric = 2(n+1)
a Id, which completes the
proof. ■
3.3 Quaternionic projective space
Consider the homogeneous realisation of quaternionic projective space given by
HPn ∼= Sp(n+ 1)⧸Sp(1)× Sp(n),
where H := Sp(1) × Sp(n) is realised as a subgroup of Sp(n + 1) by the upper left-hand 1 × 1
block for Sp(1) and the lower right-hand n × n block for Sp(n). Denote by h the Lie algebra
of H and h′ := sp(n) ⊆ h. Then, h = sp(1)⊕h′ (as Lie algebras), and explicit bases are given by
sp(1) = spanR
{
ξ1 := iF
(n+1)
1,1 , ξ2 := −kF
(n+1)
1,1 , ξ3 := jF
(n+1)
1,1
}
,
22 D. Artacho and J. Hofmann
h′ = sp(n)
= spanR
{
iF (n+1)
p,p , jF (n+1)
p,p , kF (n+1)
p,p , iF (n+1)
r,s , jF (n+1)
r,s , kF (n+1)
r,s , E(n+1)
r,s
}
2≤r<s≤n+1
p=2,...,n+1
.
The isotropy subalgebra h ⊆ sp(n+ 1) has a reductive complement
m := spanR
{
e4p := jF
(n+1)
1,p+1 , e4p+1 := kF
(n+1)
1,p+1 , e4p+2 := iF
(n+1)
1,p+1 , e4p+3 := E
(n+1)
1,p+1
}
p=1,...,n
,
and the adjoint representation of h on m is irreducible. Therefore, by Theorem 2.3, the invariant
metrics come in a one-parameter family ga := aB0|m×m, a > 0, and one easily verifies that the
above basis of m rescaled by 1/
√
2a is ga-orthonormal. Without virtually any loss of generality,
in order to simplify the notation we will only consider g := g1/2. We take the orientation defined
by the ordering (e4, e5, . . . , e4n+3).
3.3.1 Invariant spinr structures
As HP1 is just the sphere S4, we will suppose throughout this section that n > 1. By Theo-
rem 2.6, in order to understand Sp(n + 1)-invariant spinr structures on HPn, we need to find
all Lie group homomorphisms φ : H → SO(r) such that σ × φ lifts to Spinr(4n). Since H is
simply-connected, any such homomorphism lifts. Note also that, for r = 2, using simplicity
of Sp(1) and Sp(n), the only Lie group homomorphism Sp(1) × Sp(n) → SO(2) is the trivial
one. The corresponding Sp(n+ 1)-invariant spinC structure on HPn is naturally induced by its
unique spin structure. The first interesting case is r = 3, which corresponds to spinH structures.
In order to classify them, we need to describe all homomorphisms Sp(1) → SO(3):
Proposition 3.15. Up to conjugation by elements of SO(3) there are exactly two Lie group
homomorphisms Spin(3) → SO(3), namely the trivial homomorphism and the double covering λ3.
Proof. Let φ : Sp(1) → SO(3) be a non-trivial homomorphism, and recall that Sp(1) ∼= Spin(3).
As the Lie algebra so(3) is simple, the only non-trivial normal subgroups of Sp(1) are discrete.
Hence, φ has discrete kernel. As Sp(1) is compact, the image of φ is a closed subgroup of SO(3).
By the first isomorphism theorem for Lie groups, the image of φ is a 3-dimensional Lie subgroup
of SO(3), and hence it is open in SO(3). As SO(3) is connected, φ must be surjective.
In particular, the representation of Sp(1) on R3 induced by φ must be irreducible, since oth-
erwise the image of φ would be contained inside a subgroup isomorphic to {1}×SO(2). There is
only one real irreducible 3-dimensional representation of Sp(1) up to isomorphism [29, Proposi-
tion 11] (namely, the standard spin double-cover φ0), hence φ is conjugate to φ0 inside GL(3,R).
It follows that any two non-trivial homomorphisms φ1, φ2 : Sp(1) → SO(3) are conjugate to each
other inside GL(3,R), and it remains only to show that they are conjugate inside SO(3).
Fix T ∈ GL(3,R) such that, for all A ∈ Sp(1), we have T−1φ1(A)T = φ2(A). We claim that
there exists T̂ ∈ SO(3) such that T̂−1φ1T̂ = T−1φ1T . Indeed, let B := T−1φ1(A)T = φ2(A) ∈
SO(3). Then,
TT t = φ1(A)
−1TBT t = φ1(A)
−1TBBtT t
(
φ1(A)
−1
)t
= φ1(A)
−1TT tφ1(A).
As φ1 is surjective, TT t commutes with all elements of SO(3), hence it is a scalar multiple of
the identity. The result then follows by taking T̂ = det(T )−1/3T ∈ SO(3). ■
This allows us to classify invariant spinH structures on quaternionic projective spaces.
Theorem 3.16. For n > 1, the Sp(n+ 1)-invariant spinH structures on HPn are given by
Sp(n+ 1)×ϕi Spin
H(4n), i = 0, 1,
where σ : H → SO(4n) is the isotropy representation, φ0 is the trivial homomorphism Sp(1) ×
Sp(n) → SO(3), φ1(x, y) = λ3(x) and ϕi is the unique lift of σ × φi to SpinH(4n).
The Geometry of Generalised Spinr Spinors on Projective Spaces 23
The invariant spinH structure corresponding to φ0 is simply the one induced by the unique
spin structure, so for the rest of our discussion of HPn we fix the spinH structure corresponding
to φ1.
Remark 3.17. Observe that the auxiliary vector bundle of the spinH structure corresponding
to φ1 is Sp(n+1)-equivariantly isomorphic to the rank-3 vector subbundle of End(THPn) induced
by the standard quaternionic Kähler structure on HPn.
3.3.2 Invariant spinr spinors
To begin, it is easy to see that this homogeneous realisation carries no invariant spinors: as the
homogeneous realisation of HPn that we are considering is that of a symmetric space, invariant
spinors are parallel, and we know that HPn cannot have any non-trivial parallel spinor, since it
is not Ricci-flat. However, we shall see in the next proposition that there are always non-trivial
invariant spinH spinors, for sufficient twistings of the spinor bundle, when n is odd.
Proposition 3.18. The Sp(n+ 1)-invariant spinor type of HPn (n > 1) is
σ(HPn, Sp(n+ 1)) =
{
3, n odd,
> 3, n even.
Furthermore, for n odd, the number of twistings m ≥ 0 of the spinor bundle which realises this
is m = n.
Proof. The preceding discussion shows that there are no invariant spinors. As noted above,
the only invariant spinC structure is the one coming from the spin structure, and it is clear
that there are also no invariant spinC spinors in this case (since H acts trivially on Σ2 and
hence each Σm4n,2 is equivalent as H-modules to a direct sum of copies of Σ4n). This shows
that σ(HPn,Sp(n+ 1)) ≥ 3. Denote by Vt := V (tω1) the irreducible representation of sp(2,C) ∼=
sl(2,C) with highest weight tω1 (and dimension t+ 1). Arguing as in [2, Section 4.1.6], we con-
sider the structure of S := (Σ4n|hC)sp(2n,C) = spanC{ωk}nk=0 (ω :=
∑n
j=1 y2j ∧ y2j+1) as a module
for sp(2,C) ⊂ hC (here we adopt the usual convention sp(k)C ∼= sp(2k,C)). The action of sp(2,C)
on S follows from [28, Lemma 5.13] and is given explicitly by
ãd(ξ1)|m · ωk = i(n− 2k)ωk, (3.4)
ãd(ξ2)|m · ωk = k(n− k + 1)ωk−1 − ωk+1,
ãd(ξ3)|m · ωk = i
(
ωk+1 + k(n− k + 1)ωk−1
)
. (3.5)
The standard basis element for the Cartan subalgebra of sp(2,C) ∼= sl(2,C) is −iξ1 ∼ diag[1,−1],
and by (3.4) we see that the action of this element on S has highest eigenvalue n. In particular,
S contains a copy of Vn, and by reason of dimension we have S ≃ Vn as sp(2,C)-modules.
Note that this representation is self-dual (see, e.g., [40]). Thus, it suffices to show that the
smallest odd tensor power of Σ3 which contains a copy of Vn is Σ⊗n
3 . Recalling the well-known
decomposition
Vs ⊗ Vt ≃ Vs+t ⊕ Vs+t−2 ⊕ · · · ⊕ Vs−t, s ≥ t (3.6)
of sl(2,C)-representations (see, e.g., [21, Sections 11.1 and 11.2]), the result for the case where n
is odd follows by repeatedly using (3.6) to decompose tensor powers of Σ3 ≃ V1. For the case
where n is even, one sees from (3.6) that the decompositions of odd tensor powers of Σ3 ≃ V1
into irreducible representations contain only factors of the form Vt with t odd, and in particular
cannot contain a copy of Vn. ■
24 D. Artacho and J. Hofmann
Remark 3.19. The difference in behaviour between the even and odd cases in the preced-
ing proposition occurs as something of a technicality rather than a manifestation of any sig-
nificant geometric difference; indeed, the argument presented in the odd case also produces
representation-theoretic invariants in the even case if we allow even twistings of the spinor bun-
dle. The reason to exclude even twistings is that the twisted spinor module would then fail to be
well defined as a representation of Spinr(n), since [−1,−1] wouldn’t act by the identity map. In
order to obtain a notion of spinor bundles with even numbers of twistings, one needs to consider
instead the alternative structure groups Spin(n)× Spin(r) described in [26, Remark 2.3].
In the next proposition we describe explicitly the invariant n-twisted spinH spinors which re-
alise the equality σ(HPn,Sp(n + 1)) = 3 for n odd. Recall that, as noted in Remark 3.17,
the auxiliary vector bundle E of the spinH structure corresponding to φ1 can be seen as
a subbundle of the endomorphism bundle End(THPn). In particular, E inherits a natural
connection ∇E from the Levi-Civita connection ∇g on THPn, and the former induces a con-
nection ∇g,E := ∇g ⊗
(
∇E
)⊗m
on the spinor bundle Σm4n,3HPn for any odd m ≥ 1. Recall
that {ξ1, ξ2, ξ3} is a basis of sp(1) ⊂ sp(1) ⊕ sp(n) = h, and denote by (Φ1 := ad(ξ1)|m,
Φ2 := ad(ξ2)|m,Φ3 := ad(ξ3)|m) the standard basis of the invariant rank-3 subspace of End(m)
corresponding to E. The action of ξi on this subspace is given by
Φi 7→ 0, Φj 7→ 2Φk, Φk 7→ −2Φj ,
where (i, j, k) is an even permutation of (1, 2, 3), and the spin lift of this representation acts
on Σ3
∼= C2 by the standard basis matrices for su(2):
ρ(ξ1) :=
(
i 0
0 −i
)
, ρ(ξ2) :=
(
0 1
−1 0
)
, ρ(ξ3) :=
(
0 i
i 0
)
(these matrices are taken relative to the standard basis 1̂ := (1, 0), ŷ1 := (0, 1) for Σ3
∼= C2).
In order to relate these to the usual presentation of the Lie algebra sl(2,C) ∼= sp(2,C), we
introduce H := −iξ1, X := 1
2(ξ2 − iξ3), Y := −1
2(ξ2 + iξ3), so that
ρ(H) =
(
1 0
0 −1
)
, ρ(X) =
(
0 1
0 0
)
, ρ(Y ) =
(
0 0
1 0
)
(3.7)
act in the representation Σ3
∼= C2 by the usual operators. Using this setup, we obtain the
following.
Theorem 3.20. If n > 1 is odd, the space of Sp(n + 1)-invariant n-twisted spinH spinors is
spanned over C by
ψ :=
n∑
j=0
(−1)jωj ⊗
(
ρ(Y )n−j .1
)
,
where 1 := 1̂⊗ n· · · ⊗ 1̂.
Proof. As in the proof of Proposition 3.18, we note that S := (Σ4n|hC)sp(2n,C) ≃ Vn as mod-
ules for sl(2,C) ∼= sp(2,C) ⊂ hC; explicitly we have S = spanC
{
ωℓ
}n
ℓ=0
, with the action of
sp(2,C) = spanC{ξ1, ξ2, ξ3} by the formulas (3.4)–(3.5). It is clear from (3.7) that 1 ∈ Σ⊗n
3 is
a highest weight vector for sp(2,C), and that the sp(2,C)-submodule U(sp(2,C)).1 that it gen-
erates4 is isomorphic to Vn (since ρ(H) = diag[1,−1] acts on 1 by multiplication by n). There-
fore, we have U(sp(2,C)).1 = spanC
{
ρ(Y )k.1
}n
k=0
. On the other hand, we see from (3.4)–(3.5)
4Here, U(sp(2,C)) refers to the universal enveloping algebra of sp(2,C) and the . product refers to the action
via the representation.
The Geometry of Generalised Spinr Spinors on Projective Spaces 25
that 1 = ω0 ∈ S is a highest weight vector and Y.ωk = ωk+1 for all k = 0, . . . , n. In particular,
the isomorphism T : S → U(sp(2,C)).1 is given (up to rescaling) by T
(
ωk
)
= ρ(Y )k.1. We
have T ∈ Homsp(2,C)
(
S,Σ⊗n
3
)
≃ S∗ ⊗ Σ⊗n
3 , and the invariant spinH spinor we are seeking is the
corresponding element of S ⊗ Σ⊗n
3 obtained via the musical isomorphism ♯ : S∗ ≃ S associated
to the sp(2,C)-invariant symplectic form
Ω
(
ωj , ωk
)
=
{
(−1)j , j + k = n,
0, j + k ̸= n
on S. Defining ω̂j ∈ S∗ by ω̂j
(
ωk
)
= δj,k, one sees that
(
ω̂j
)♯
= (−1)j+1ωn−j , and the result then
follows by noting that T =
∑n
j=0 ω̂
j ⊗
(
ρ(Y )j .1
)
. ■
Remark 3.21. The spinor in the statement of Theorem 3.20 corresponds to the one in [26,
Section 3.4.1], which the authors show to be pure.
Finally, we give the differential equation satisfied by ψ. Recall that HPn = Sp(n+1)/Sp(1)×
Sp(n) is a symmetric space, and that the auxiliary bundle of the spinH structure under consid-
eration is isomorphic to the rank-3 bundle spanned by the (locally-defined) endomorphisms Φ1,
Φ2, Φ3. This bundle inherits a connection ∇E from the Levi-Civita connection, and its Nomizu
map vanishes identically when restricted to m. The following is an immediate consequence of
Proposition 2.21.
Corollary 3.22. For n > 1 odd, the invariant n-twisted spinH spinor ψ in Theorem 3.20 is
parallel with respect to the invariant connection ∇g,E := ∇g ⊗
(
∇E
)⊗n
.
This spinH spinor encodes, via [26, Corollary 4.12], a well-known geometric fact – see, e.g., [11].
Theorem 3.23. The metric ga on HPn is quaternionic Kähler.
3.4 Octonionic projective plane
Consider now the octonionic projective plane, realised as a homogeneous space via
OP2 ∼= F4⧸Spin(9).
A description of the isometric action of F4 can be found, e.g., in [7], and, importantly, the
isotropy representation is just the real spin representation of Spin(9) on R16:
Spin(9) ↪−→ Cl09,0
∼= Cl8,0 ∼= M16(R).
As in [42, Theorems 1.4.3 and 1.4.4, Proposition 1.4.5], at the level of Lie algebras this inclusion
is given by
spin(9) ↪−→ Cl09,0
ψ1∼= Cl8,0
ψ2∼= Cl0,6⊗Cl2,0
ψ3∼= Cl4,0⊗Cl0,2⊗Cl2,0
ψ4∼= Cl0,2⊗Cl2,0⊗Cl0,2⊗Cl2,0
ψ5∼= M2(R)⊗H⊗M2(R)⊗H
ψ6∼= M2(R)⊗M2(R)⊗H⊗H
ψ7∼= M4(R)⊗M4(R)
ψ8∼= M16(R). (3.8)
The algebra isomorphisms ψ1, ψ2, ψ3, ψ4, ψ5 are given explicitly in [42]; ψ6 is the obvious
permutation of the second and third factors; ψ7 is the tensor product of the Kronecker product
of the first two factors and the isomorphism
H⊗R H → M4(R), q1 ⊗ q2 7→ (x 7→ q1 · x · q2),
26 D. Artacho and J. Hofmann
where x = (x1, x2, x3, x4) ∈ R4 is thought of as the quaternion x1 + ix2 + jx3 + kx4; and ψ8 is
the Kronecker product. Letting {e0, e1, . . . , e8} be the canonical basis of R9, a basis of spin(9)
is given by
spin(9) = spanR{ei · ej}0≤i<j≤8.
With a slight abuse of notation, we will denote by e1, . . . , en the elements of the canonical basis
of Rn, for n = 2, 4, 6, 8, 16. Now we give the images in M16(R) of each of the elements of our
basis of spin(9). For the first basis vector e0 · e1, one computes, following the chain of maps
in (3.8)
e0 · e1 7→ e0 · e1 7→ e1 7→ e1 ⊗ (e1 · e2) 7→ e1 ⊗ (e1 · e2)⊗ (e1 · e2)
7→ e1 ⊗ (e1 · e2)⊗ (e1 · e2)⊗ (e1 · e2) 7→
(
1 0
0 −1
)
⊗ k ⊗
(
0 1
−1 0
)
⊗ k
7→
(
1 0
0 −1
)
⊗
(
0 1
−1 0
)
⊗ k ⊗ k 7→
0 1 0 0
−1 0 0 0
0 0 0 −1
0 0 1 0
⊗
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 1
7→ −E(16)
1,5 + E
(16)
2,6 + E
(16)
3,7 − E
(16)
4,8 + E
(16)
9,13 − E
(16)
10,14 − E
(16)
11,15 + E
(16)
12,16. (3.9)
The others are computed similarly, giving
e0 · e2 7→ −E(16)
1,13 + E
(16)
2,14 + E
(16)
3,15 − E
(16)
4,16 + E
(16)
5,9 − E
(16)
6,10 − E
(16)
7,11 + E
(16)
8,12 ,
e0 · e3 7→ −E(16)
1,7 − E
(16)
2,8 − E
(16)
3,5 − E
(16)
4,6 − E
(16)
9,15 − E
(16)
10,16 − E
(16)
11,13 − E
(16)
12,14,
e0 · e4 7→ E
(16)
1,6 + E
(16)
2,5 − E
(16)
3,8 − E
(16)
4,7 + E
(16)
9,14 + E
(16)
10,13 − E
(16)
11,16 − E
(16)
12,15,
e0 · e5 7→ −E(16)
1,4 + E
(16)
2,3 + E
(16)
5,8 − E
(16)
6,7 − E
(16)
9,12 + E
(16)
10,11 + E
(16)
13,16 − E
(16)
14,15,
e0 · e6 7→ −E(16)
1,8 + E
(16)
2,7 − E
(16)
3,6 + E
(16)
4,5 − E
(16)
9,16 + E
(16)
10,15 − E
(16)
11,14 + E
(16)
12,13,
e0 · e7 7→ −E(16)
1,2 + E
(16)
3,4 − E
(16)
5,6 + E
(16)
7,8 − E
(16)
9,10 + E
(16)
11,12 − E
(16)
13,14 + E
(16)
15,16,
e0 · e8 7→ −E(16)
1,3 − E
(16)
2,4 − E
(16)
5,7 − E
(16)
6,8 − E
(16)
9,11 − E
(16)
10,12 − E
(16)
13,15 − E
(16)
14,16. (3.10)
The images of the other basis vectors ei · ej (1 ≤ i ≤ 8) for spin(9) are then determined
by taking products of the above generators inside Cl09,0 using the Clifford algebra identities
ei · ej = (e0 · ei) · (e0 · ej).
3.4.1 Invariant spinr structures
By Theorem 2.6, in order to understand F4-invariant spin
r structures on OP2, we need to find
all Lie group homomorphisms φ : Spin(9) → SO(r) such that σ×φ lifts to Spinr(16) (of course,
as the group Spin(9) is simply connected, the lifting condition is automatically satisfied). As the
Lie algebra spin(9) ∼= so(9) is simple, for 1 ≤ r ≤ 8 the only homomorphism Spin(9) → SO(r) is
the trivial one. The corresponding F4-invariant spin
r structures are just the ones in the lineage
of the invariant spin structure.
The first non-trivial case is r = 9, where we have the covering homomorphism λ9 : Spin(9) →
SO(9). By essentially the same argument as in Proposition 3.15, there are only two Lie group
homomorphisms Spin(9) → SO(9) up to conjugation by elements of SO(9), namely the trivial
one φ0 and the double covering φ1 := λ9, leading to two possible invariant spin9 structures up
to equivalence.
Theorem 3.24. The F4-invariant spin9 structures on OP2 are given by F4 ×ϕi Spin
9(16),
i = 0, 1, where ϕi is the unique lift of σ × φi to Spin9(16) and σ : Spin(9) → SO(16) is the
isotropy representation.
The Geometry of Generalised Spinr Spinors on Projective Spaces 27
3.4.2 Invariant spinr spinors
As in the case of HPn, the octonionic projective plane OP2 does not admit any invariant spinors
(cf. the discussion before Proposition 3.18), so from this point forward we consider the non-trivial
invariant spin9 structure (i.e., the one corresponding to i = 1 in the preceding theorem). In order
to describe its twisted spin9 spinors we first need a small lemma describing the decomposition
of the spin lift of the isotropy representation as a direct sum of highest weight modules. This
result may be found, written in a slightly different form and without proof, in [18, Section 7];
we include a sketch of the proof here as the notation and formulas will be useful for subsequent
discussion.
Lemma 3.25 ([18]). As modules for Spin(9)C, the spin lift σ̃ of the isotropy representation
decomposes as
Σ16 ≃ V (ω1 + ω4)︸ ︷︷ ︸
Σ−
16
⊕V (ω3)⊕ V (2ω1)︸ ︷︷ ︸
Σ+
16
, (3.11)
where V (µ) denotes the irreducible representation of highest weight µ and ωi, i = 1, 2, 3, 4 denote
the fundamental weights of spin(9)C ∼= so(9,C).
Proof. In order to take advantage of the explicit operators calculated in (3.9)–(3.10), we
view spin(9)C ∼= so(9,C) as the set of 9 × 9 skew-symmetric matrices in gl(9,C). We take the
(real form of the) Cartan subalgebra spanned by hj := −iE
(9)
2j−1,2j , j = 1, 2, 3, 4, together with
the (positive) re-scaling of the Killing form such that the hj ’s are orthogonal and unit length.
Letting vj := h♭j , we have the simple roots
α1 = v1 − v2, α2 = v2 − v3, α3 = v3 − v4, α4 = v4,
and the corresponding fundamental weights ωj :=
2αj
||αj ||2 are given by
ω1 = v1, ω2 = v1 + v2, ω3 = v1 + v2 + v3, ω4 =
1
2
(v1 + v2 + v3 + v4).
The root vectors Xi := Xαi associated to the simple roots αi are
X1 = E
(9)
1,3 + E
(9)
2,4 + i
(
−E(9)
2,3 + E
(9)
1,4
)
, X2 = E
(9)
3,5 + E
(9)
4,6 + i
(
−E(9)
4,5 + E
(9)
3,6
)
,
X3 = E
(9)
5,7 + E
(9)
6,8 + i
(
−E(9)
6,7 + E
(9)
5,8
)
, X4 = E
(9)
7,9 − iE
(9)
8,9 ,
and the root vectors associated to the roots −αi, i = 1, 2, 3, 4 are given by Yi := Yαi := Xi.
We note that this setup is slightly unusual5 and can be found, e.g., in [43]. Using the explicit
formulas for σ from (3.9)–(3.10), we find that the Cartan subalgebra generators hi and simple
root vectors Xi act in the complexified isotropy representation mC ≃ Σ9 by the operators
σ(h1) = − i
2
(−e1,5 + e2,6 + e3,7 − e4,8 + e9,13 − e10,14 − e11,15 + e12,16),
σ(h2) = − i
2
(e1,11 − e2,12 − e3,9 + e4,10 + e5,15 − e6,16 − e7,13 + e8,14),
σ(h3) = − i
2
(e1,7 − e2,8 − e3,5 + e4,6 + e9,15 − e10,16 − e11,13 + e12,14),
σ(h4) = − i
2
(−e1,7 − e2,8 + e3,5 + e4,6 − e9,15 − e10,16 + e11,13 + e12,14),
5One usually chooses a different realization of the Lie algebra so(9,C) in order to make the elements of the
Cartan subalgebra diagonal matrices, but that realization is less convenient for our purposes here.
28 D. Artacho and J. Hofmann
and
2σ(X1) = e1,3 − ie1,7 − ie1,9 − e1,13 − e2,4 − ie2,8 − ie2,10 + e2,14 − ie3,5 − ie3,11
+ e3,15 − ie4,6 − ie4,12 − e4,16 + e5,7 + e5,9 − ie5,13 − e6,8 − e6,10 − ie6,14
− e7,11 − ie7,15 + e8,12 − ie8,16 − e9,11 − ie9,15 + e10,12 − ie10,16 − ie11,13
− ie12,14 − e13,15 + e14,16,
2σ(X2) = −ie1,4 + e1,6 − e1,10 + ie1,16 − ie2,3 − e2,5 + e2,9 + ie2,15 + e3,8 − e3,12
− ie3,14 − e4,7 + e4,11 − ie4,13 − ie5,8 + ie5,12 − e5,14 − ie6,7 + ie6,11 + e6,13
− ie7,10 − e7,16 − ie8,9 + e8,15 − ie9,12 + e9,14 − ie10,11 − e10,13 + e11,16
− e12,15 − ie13,16 − ie14,15,
2σ(X3) = −2e1,3 − 2ie1,5 − 2ie3,7 + 2e5,7 − 2e9,11 − 2ie9,13 − 2ie11,15 + 2e13,15,
2σ(X4) = ie1,4 + e1,6 − ie2,3 − e2,5 − e3,8 + e4,7 + ie5,8 − ie6,7 + ie9,12 + e9,14
− ie10,11 − e10,13 − e11,16 + e12,15 + ie13,16 − ie14,15.
Considering the action of the lifts σ̃(hi), σ̃(Xi) ∈ spin(16)C in the spin representation, a straight-
forward calculation using computer algebra software yields three linearly independent joint
eigenvectors for the σ̃(hi) which are simultaneously annihilated by the action of each σ̃(Xi)
(i.e., highest weight vectors). The corresponding weights are
1
2
(3v1 + v2 + v3 + v4) = ω1 + ω4, v1 + v2 + v3 = ω3, 2v1 = 2ω1,
and the assertion that Σ−
16 ≃ V (ω1 + ω4) and Σ+
16 ≃ V (ω3)⊕ V (2ω1) may be deduced from [18,
Section 7]. ■
Note that the preceding lemma immediately recovers the fact that the invariant spin struc-
ture carries no invariant spinors. It also allows us to readily describe the smallest twisting for
which OP2 admits invariant twisted spin9 spinors.
Theorem 3.26. The F4-invariant spinor type of OP2 is σ
(
OP2, F4
)
= 9, and the twisting of
the spinor bundle which realises this is m = 3. Furthermore, the space of invariant 3-twisted
spin9 spinors has complex dimension 4.
Proof. First, we recall that every representation of so(9,C) is self-dual (see, e.g., [40]). There-
fore, using a similar argument as in the proof of Proposition 3.18, and in light of the preceding
lemma, it suffices to show that Σ⊗3
9 is the smallest odd tensor power of Σ9 ≃ V (ω4) which con-
tains a copy of V (ω1 + ω4), V (ω3), or V (2ω1). It is easily verified using, e.g., the LiE software
package [36] that
Σ⊗3
9 ≃ 5V (ω4)⊕ V (3ω4)⊕ 2V (ω3 + ω4)⊕ 3V (ω2 + ω4)⊕ 4V (ω1 + ω4). (3.12)
Finally, using self-duality, it follows from (3.11) and (3.12) that
dimC
(
Σ3
16,9
)
inv
= dimC
(
Σ16 ⊗ Σ⊗3
9
)Spin(9)
= dimCHomSpin(9)
(
Σ16,Σ
⊗3
9
)
= 4. ■
Now we examine more closely the invariant 3-twisted spin9 spinors from the preceding theo-
rem. This 4-dimensional space corresponds to the pairings of the 4 copies of V (ω1+ω4) in (3.12)
with the single copy in (3.11), so in order to obtain formulas for the spinors we first need to
clarify the algebraic structure of this representation. With all notation as above, one finds using
computer algebra software an explicit highest weight vector w0 (unique up to scaling) gener-
ating Σ−
16 ≃ V (ω1 + ω4) ⊆ Σ16, and one may verify furthermore that any other weight vector
The Geometry of Generalised Spinr Spinors on Projective Spaces 29
can be obtained from w0 by applying at most 18 lowering operators Yi, i = 1, 2, 3, 4. Writing
YI := Yi1 .Yi2 . . . Yik for a multi-index I = {i1, . . . , ik}, one possible minimal choice of multi-
indices
⋃18
k=0{Ik,ℓ}
µk
ℓ=1 generating V (ω1 + ω4) is given in Table 3, where µk denotes the number
of k-multi-indices in the generating set. In what follows we describe explicitly the invariant
spinors, using a more sophisticated version of the technique from the proof of Proposition 3.20.
Theorem 3.27. A basis for the space of F4-invariant 3-twisted spin9 spinors on OP2 is given by
ψp :=
18∑
k=0
µk∑
ℓ=1
( ̂YIk,ℓ .w0
)♯ ⊗ (YIk,ℓ .wp), p = 1, 2, 3, 4, (3.13)
where ♯ : Σ∗
16 → Σ16 is the musical isomorphism, wp (p = 1, 2, 3, 4) denote highest weight vectors
for the four copies of Σ−
16 inside Σ⊗3
9 , the indices Ik,ℓ are as in Table 3, and for any (YIk,ℓ .w0) ∈
Σ−
16 ⊆ Σ16 we denote by ̂YIk,ℓ .w0 ∈ Σ∗
16 the corresponding dual map sending YIk′,ℓ′ .w0 7→ δk,k′δℓ,ℓ′
and Σ+
16 7→ 0.
Proof. From the preceding discussion and Table 3, we have the highest weight vector w0
for Σ−
16 ⊆ Σ16, together with explicit sequences of lowering operators Yi generating this subrepre-
sentation. Altogether this gives four spin(9)C-module isomorphisms Tp : Σ
−
16 → Σ⊗3
9 , p = 1, 2, 3, 4
defined by
Tp : YIk,ℓ .w0 7→ YIk,ℓ .wp, k = 0, . . . , 18, ℓ = 1, . . . , µk,
where the Ik,ℓ are as in Table 3 and we use the convention Y∅ = Id. By abuse of notation, we
also denote by Tp the extensions to all of Σ16 by Σ+
16 7→ 0. The spinors ψp are the images of
the Tp under the spin(9)C-module isomorphism (Σ−
16)
∗ ⊗ Σ⊗3
9 ≃ Σ−
16 ⊗ Σ⊗3
9 , which are precisely
given by (3.13). ■
Finally, we give the differential equation satisfied by the spinors from Theorem 3.27. To
begin, we need to first specify a connection on the vector bundle A associated to the auxil-
iary SO(9)-bundle of the spin9 structure. Note that A is associated to the principal Spin(9)
bundle F4 → F4/ Spin(9) by the composition of the covering map λ9 : Spin(9) → SO(9) with
the standard representation ρstd : SO(9) → GL
(
R9
)
. Indeed, there is a natural choice of in-
variant connection defined on A as follows. The structure of m ≃ ΣR
9 as a Clifford module
for Cl9 gives 9 linearly independent endomorphisms, corresponding to Clifford multiplication by
an orthonormal set of basis vectors for R9. By slightly modifying the Clifford multiplication
(see [18, Section 2]), one obtains endomorphisms Ti : m → m, i = 1, . . . , 9 satisfying the mod-
ified Clifford relations Ti ◦ Tj + Tj ◦ Ti = 2δi,j Id, T
∗
i = Ti, trTi = 0 for i = 1, . . . , 9. In this
description, the isotropy image σ(Spin(9)) ⊆ SO(m) ⊆ End(m) coincides with the normaliser of
the 9-dimensional subspace T := spanR{T1, . . . , T9} ⊆ End(m) [18, p. 132]:
σ(Spin(9)) =
{
g ∈ SO(m) | gT g−1 = T
}
.
The 9-dimensional Spin(9)-representation T (action via conjugation) is isomorphic to ρstd ◦ λ9
(since there is only one irreducible real 9-dimensional representation of Spin(9) up to isomor-
phism), hence we have
A ∼= F4 ×Spin(9) T ⊆ F4 ×Spin(9) End(m) ∼= End
(
T
(
OP2
))
.
In particular, A naturally inherits a connection ∇End from the extension of the Levi-Civita
connection to the endomorphism bundle, whose Nomizu map vanishes on m. In light of Propo-
sition 2.21, we finally see that the invariant twisted spinors found above are parallel:
Theorem 3.28.The 4-dimensional space of F4-invariant 3-twisted spin9 spinors on OP2 is span-
ned by parallel spinors for the connection ∇g,End := ∇g ⊗
(
∇End
)⊗3
.
30 D. Artacho and J. Hofmann
k µk Ik,ℓ (ℓ = 1, . . . , µk)
0 1 ∅
1 2 {1}, {4}
2 3 {2, 1}, {1, 4}, {3, 4}
3 5 {3, 2, 1}, {2, 1, 4}, {1, 3, 4}, {2, 3, 4}, {4, 3, 4}
4 6 {4, 3, 2, 1}, {3, 2, 1, 4}, {2, 1, 3, 4}, {4, 1, 3, 4}, {1, 2, 3, 4}, {4, 2, 3, 4}
5 8 {3, 4, 3, 2, 1}, {4, 4, 3, 2, 1}, {2, 3, 2, 1, 4}, {4, 3, 2, 1, 4}, {1, 2, 1, 3, 4},
{4, 2, 1, 3, 4}, {4, 1, 2, 3, 4}, {3, 4, 2, 3, 4}
6 10 {2, 3, 4, 3, 2, 1}, {4, 3, 4, 3, 2, 1}, {3, 4, 4, 3, 2, 1}, {4, 4, 4, 3, 2, 1}, {1, 2, 3, 2, 1, 4},
{4, 2, 3, 2, 1, 4}, {4, 1, 2, 1, 3, 4}, {3, 4, 2, 1, 3, 4}, {3, 4, 1, 2, 3, 4}, {4, 3, 4, 2, 3, 4}
7 11 {1, 2, 3, 4, 3, 2, 1}, {4, 2, 3, 4, 3, 2, 1}, {4, 4, 3, 4, 3, 2, 1}, {2, 3, 4, 4, 3, 2, 1},
{4, 3, 4, 4, 3, 2, 1}, {4, 1, 2, 3, 2, 1, 4}, {3, 4, 2, 3, 2, 1, 4}, {3, 4, 1, 2, 1, 3, 4},
{4, 3, 4, 2, 1, 3, 4}, {2, 3, 4, 1, 2, 3, 4}, {4, 3, 4, 1, 2, 3, 4}
8 12 {4, 1, 2, 3, 4, 3, 2, 1}, {3, 4, 2, 3, 4, 3, 2, 1}, {4, 4, 2, 3, 4, 3, 2, 1}, {3, 4, 4, 3, 4, 3, 2, 1},
{4, 4, 4, 3, 4, 3, 2, 1}, {1, 2, 3, 4, 4, 3, 2, 1}, {4, 2, 3, 4, 4, 3, 2, 1}, {3, 4, 1, 2, 3, 2, 1, 4},
{4, 3, 4, 2, 3, 2, 1, 4}, {2, 3, 4, 1, 2, 1, 3, 4}, {4, 3, 4, 1, 2, 1, 3, 4}, {4, 2, 3, 4, 1, 2, 3, 4}
9 12 {3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 3, 4, 2, 3, 4, 3, 2, 1}, {3, 4, 4, 2, 3, 4, 3, 2, 1},
{4, 4, 4, 2, 3, 4, 3, 2, 1}, {2, 3, 4, 4, 3, 4, 3, 2, 1}, {4, 3, 4, 4, 3, 4, 3, 2, 1}, {4, 1, 2, 3, 4, 4, 3, 2, 1},
{2, 3, 4, 1, 2, 3, 2, 1, 4}, {4, 3, 4, 1, 2, 3, 2, 1, 4}, {4, 2, 3, 4, 1, 2, 1, 3, 4}, {3, 4, 2, 3, 4, 1, 2, 3, 4}
10 12 {2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {3, 4, 4, 1, 2, 3, 4, 3, 2, 1},
{4, 4, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 4, 3, 4, 2, 3, 4, 3, 2, 1}, {2, 3, 4, 4, 2, 3, 4, 3, 2, 1},
{4, 3, 4, 4, 2, 3, 4, 3, 2, 1}, {1, 2, 3, 4, 4, 3, 4, 3, 2, 1}, {4, 2, 3, 4, 4, 3, 4, 3, 2, 1},
{4, 2, 3, 4, 1, 2, 3, 2, 1, 4}, {3, 4, 2, 3, 4, 1, 2, 1, 3, 4}, {4, 3, 4, 2, 3, 4, 1, 2, 3, 4}
11 11 {4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {2, 3, 4, 4, 1, 2, 3, 4, 3, 2, 1},
{4, 3, 4, 4, 1, 2, 3, 4, 3, 2, 1}, {3, 4, 4, 3, 4, 2, 3, 4, 3, 2, 1}, {4, 4, 4, 3, 4, 2, 3, 4, 3, 2, 1},
{1, 2, 3, 4, 4, 2, 3, 4, 3, 2, 1}, {4, 2, 3, 4, 4, 2, 3, 4, 3, 2, 1}, {4, 1, 2, 3, 4, 4, 3, 4, 3, 2, 1},
{3, 4, 2, 3, 4, 1, 2, 3, 2, 1, 4}, {4, 3, 4, 2, 3, 4, 1, 2, 1, 3, 4}
12 10 {3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {3, 4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1},
{4, 4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {1, 2, 3, 4, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 2, 3, 4, 4, 1, 2, 3, 4, 3, 2, 1},
{2, 3, 4, 4, 3, 4, 2, 3, 4, 3, 2, 1}, {4, 3, 4, 4, 3, 4, 2, 3, 4, 3, 2, 1}, {4, 1, 2, 3, 4, 4, 2, 3, 4, 3, 2, 1},
{4, 3, 4, 2, 3, 4, 1, 2, 3, 2, 1, 4}
13 8 {4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {3, 4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1},
{2, 3, 4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 3, 4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 1, 2, 3, 4, 4, 1, 2, 3, 4, 3, 2, 1},
{1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 3, 2, 1}, {4, 2, 3, 4, 4, 3, 4, 2, 3, 4, 3, 2, 1}
14 6 {4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {2, 3, 4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1},
{4, 3, 4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {1, 2, 3, 4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1},
{4, 2, 3, 4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 3, 2, 1}
15 5 {3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1},
{1, 2, 3, 4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 2, 3, 4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1},
{4, 1, 2, 3, 4, 4, 3, 4, 1, 2, 3, 4, 3, 2, 1}
16 3 {2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1},
{4, 1, 2, 3, 4, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}
17 2 {1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}, {4, 2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}
18 1 {4, 1, 2, 3, 4, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 3, 2, 1}
Table 3. Ordered sequences of lowering operators generating V (ω1 + ω4).
Acknowledgements
The authors are grateful to Travis Schedler for his contributions to the representation-theoretical
aspect of the paper, and to Marie-Amélie Lawn for her comments and fruitful discussions. We
The Geometry of Generalised Spinr Spinors on Projective Spaces 31
are grateful to the referees for their helpful comments. D. Artacho is funded by the UK Engi-
neering and Physical Sciences Research Council (EPSRC), grant EP/W5238721. J. Hofmann
was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1,
EP/W522429/1]; the EPSRC Centre for Doctoral Training in Geometry and Number Theory
(The London School of Geometry and Number Theory: University College London, King’s
College London, and Imperial College London); and a DAAD Short Term Research Grant for
a research stay at Philipps-Universität Marburg.
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https://arxiv.org/abs/dg-ga/9610005
https://doi.org/10.3842/SIGMA.2023.012
https://arxiv.org/abs/2301.09683
https://doi.org/10.1016/j.difgeo.2018.08.006
https://doi.org/10.1016/j.difgeo.2018.08.006
https://arxiv.org/abs/1711.07765
https://doi.org/10.4310/AJM.2019.v23.n5.a3
https://arxiv.org/abs/1606.07894
https://doi.org/10.1016/j.difgeo.2022.101849
https://doi.org/10.1016/j.difgeo.2022.101849
https://arxiv.org/abs/2022.10184
http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/
https://doi.org/10.1007/s002200050142
https://doi.org/10.2307/2372398
https://doi.org/10.1090/trans2/055
https://doi.org/10.1007/978-1-4613-9014-5
https://doi.org/10.1007/BF00137402
https://arxiv.org/abs/1911.09766
https://www2.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf
https://www2.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf
https://doi.org/10.1007/BF01456947
1 Introduction
2 Preliminaries
2.1 Invariant metrics on reductive homogeneous spaces
2.2 Some notation
2.3 Invariant spin^r structures
2.4 Exterior forms approach to the spin representation
2.5 Invariant spin^r spinors
2.6 Special spin^r spinors
3 Projective spaces
3.1 Hermitian complex projective space
3.1.1 Invariant spin^r structures
3.1.2 Invariant spin^r spinors
3.1.3 Special spin^r spinors
3.2 Symplectic complex projective space
3.2.1 Invariant spin^r structures
3.2.2 Invariant spin^r spinors
3.2.3 Special spin^r spinors
3.3 Quaternionic projective space
3.3.1 Invariant spin^r structures
3.3.2 Invariant spin^r spinors
3.4 Octonionic projective plane
3.4.1 Invariant spin^r structures
3.4.2 Invariant spin^r spinors
References
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| id | nasplib_isofts_kiev_ua-123456789-212874 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:30:54Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Artacho, Diego Hofmann, Jordan 2026-02-13T13:49:09Z 2025 The Geometry of Generalised Spinʳ Spinors on Projective Spaces. Diego Artacho and Jordan Hofmann. SIGMA 21 (2025), 017, 32 pages 1815-0659 2020 Mathematics Subject Classification: 53C27; 15A66; 57R15 arXiv:2406.18337 https://nasplib.isofts.kiev.ua/handle/123456789/212874 https://doi.org/10.3842/SIGMA.2025.017 In this paper, we adapt the characterisation of the spin representation via exterior forms to the generalised spinʳ context. We find new invariant spinʳ spinors on the projective spaces ℂℙⁿ, ℍℙⁿ, and the Cayley plane ℙ² for all their homogeneous realisations. Specifically, for each of these realisations, we provide a complete description of the space of invariant spinʳ spinors for the minimum value of for which this space is non-zero. Additionally, we demonstrate some geometric implications of the existence of special spinʳ spinors on these spaces. The authors are grateful to Travis Schedler for his contributions to the representation-theoretical aspect of the paper, and to Marie-Am´elie Lawn for her comments and fruitful discussions. We are grateful to the referees for their helpful comments. D. Artacho is funded by the UK Engineering and Physical Sciences Research Council (EPSRC), grant EP/W5238721. J. Hofmann was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1, EP/W522429/1]; the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory: University College London, King’s College London, and Imperial College London); and a DAAD Short Term Research Grant for a research stay at Philipps-Universität Marburg. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Geometry of Generalised Spinʳ Spinors on Projective Spaces Article published earlier |
| spellingShingle | The Geometry of Generalised Spinʳ Spinors on Projective Spaces Artacho, Diego Hofmann, Jordan |
| title | The Geometry of Generalised Spinʳ Spinors on Projective Spaces |
| title_full | The Geometry of Generalised Spinʳ Spinors on Projective Spaces |
| title_fullStr | The Geometry of Generalised Spinʳ Spinors on Projective Spaces |
| title_full_unstemmed | The Geometry of Generalised Spinʳ Spinors on Projective Spaces |
| title_short | The Geometry of Generalised Spinʳ Spinors on Projective Spaces |
| title_sort | geometry of generalised spinʳ spinors on projective spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212874 |
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