Quaternionic Killing Spinors

W. Kramer^{∗}, U. Semmelmann^{†}, G. Weingart^{‡}

Mathematisches Institut der Universität Bonn Beringstraße 1, 53115 Bonn, Germany

                 kramer@math.uni-bonn.de, uwe@math.uni-bonn.de, gw@math.uni-bonn.de

March 19, 2003

Abstract

In [KSW97-1] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler manifolds. In the present article we study the limiting case, i. e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kähler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.

AMS Subject Classification: 53C25, 58G25

Contents

1 Introduction 1

2 Spin Geometry of Quaternionic Kähler Manifolds 2 2.1 Preliminaries on Sp(n)–Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Spinor Bundle and Clifford Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Dirac and Twistor Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Weitzenböck Formulas 6

4 The Quaternionic Killing Equation 9

5 The Killing Curvature on Wolf Spaces 13 5.1 The Curvature Endomorphism on Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 The Eigenvalues of the Curvature Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 15

1 Introduction

Let (M^{4n}, g), n ≥ 2 be a compact quaternionic Kähler manifold of positive scalar curvature κ. By definition its holonomy group is contained in the subgroup Sp(n) · Sp(1) ⊂ SO(4n). M is spin if and only if the quaternionic dimension n is even or M = ℍP^{n}. In [KSW97-1] we proved the following lower bound for the spectrum of the Dirac operator D on M:

λ^{2} ≥ \frac{κ}{4} \frac{n + 3}{n + 2} .

∗Supported by the SFB 256 ‘Nichtlineare partielle Differentialgleichungen’

†Supported by the Max–Planck–Institut für Mathematik

‡Supported by the SFB 256 ‘Nichtlineare partielle Differentialgleichungen’

Note that κ is constant on M, since any quaternionic Kähler manifold is automatically Einstein. This was first shown by D. V. Alekseevskii in [Ale68-1] and [Ale68-2] (see also [Ish74]). The estimate is sharp since the lower bound is the first eigenvalue of D^{2} on the quaternionic projective space, as follows from the computation of the spectrum done in [Mil92].

The natural task is then to study the limiting case and to find all manifolds where \frac{κ}{4} \frac{n+3}{n+2} is in the spectrum of D^{2}. In this article we rule out all Wolf spaces besides the quaternionic projective spaces, thus settling the question for all compact symmetric quaternionic Kähler manifolds. Up to now, no other examples of compact quaternionic Kähler manifolds of positive scalar curvature are known, and a common conjecture, proved by Y. S. Poon and S. Salamon [PSa91] in quaternionic dimension n = 2 and by C. LeBrun and S. Salamon [LSa94] under additional conditions on the Betti numbers in dimensions n = 3, 4, says that there are none.

The principal result shows that the existence of an eigenspinor with the minimal eigenvalue is equivalent to the existence of a solution for a suitable quaternionic Killing equation, i. e. a section of a suitable vector bundle which is parallel with respect to a modified connection. The curvature of this Killing connection is precisely the hyperkähler or Weyl part of the curvature tensor. Explicit calculation then shows that no Wolf space besides the quaternionic projective space allows a parallel section for this connection.

A peculiar feature of the quaternionic Killing connection is that unlike its Riemannian or Kählerian counterpart it is not defined on (a subbundle of) the spinor bundle, but involves a non–spinor bundle naturally. These “hidden parameters” account for the fact that the dimension of the space of eigenspinors with minimal eigenvalue on the quaternionic projective space exceeds the dimension of S_{0}(ℍP^{n})⊕S_{1}(ℍP^{n}), see Corollary 4.1. As the geometric significance of the additional bundle is obscure it seems difficult to describe the Killing connection in purely geometric terms without using representation theory of Sp(n)·Sp(1).

The authors would like to thank D. V. Alekseevskii and C. Bär for helpful discussions. In particular, we would like to thank W. Ballmann for encouragement and support.

Note: Recently we have been able to translate the quaternionic Killing equation into the equation for a parallel spinor on a hyperkähler manifold naturally associated to M in the spirit of previous work by C. Bär [Bär93] and A. Moroianu [Mor95]. Their holonomy argument goes through to show that the existence of a non–trivial quaternionic Killing spinor forces M to be the quaternionic projective space. This stronger result is presented in [KSW97-2].

2 Spin Geometry of Quaternionic Kähler Manifolds

Let (M^{4n}, g) be a quaternionic Kähler manifold, i. e. the Levi–Civita connection on M is already defined on a Sp(n) · Sp(1)–reduction P of the SO(4n)–bundle of orthonormal frames. Any representation V of Sp(n) × Sp(1) locally gives a vector bundle V associated to (a local two–fold covering of) P. This bundle exists globally if and only if the representation factors through Sp(n) · Sp(1).

The representation theory of Sp(1) and Sp(n) is governed by the defining representations H := ℍ ≅ ℂ^{2} and E := ℍ^{n} ≅ ℂ^{2n} respectively. More precisely, any irreducible Sp(n) × Sp(1)–representation can be realized as a subspace of H^{⊗p} ⊗ E^{⊗q} for some p and q; those with p + q even factor through Sp(n) · Sp(1). Hence, any vector bundle on M associated to P can be expressed in terms of two local bundles H and E. For example, the complexified tangent bundle is defined by the representation H ⊗ E, i. e.

T M^{ℂ} = H ⊗ E .

In this section we will recall some of the definitions and results given in [KSW97-1]. Besides elementary properties of the representations Λ^{s}E and Sym^{r}H we also need the explicit description of the curvature tensor given in [KSW97-1] as well as the definition of Dirac and twistor operators.

2.1 Preliminaries on Sp(n)–Representations

Let H and E be the defining complex representations of Sp(1) and Sp(n) with their invariant symplectic forms σ_{H} ∈ Λ^{2}H^{∗} and σ_{E} ∈ Λ^{2}E^{∗} and their compatible positive quaternionic structures J, i. e.

\begin{aligned}J^{2} & = & −1 \\ σ_{E}(Je_{1}, Je_{2}) & = & \overline{σ_{E}(e_{1}, e_{2})} \\ σ_{E}(e, Je) & > & 0 for e ≠ 0 . \end{aligned}

The symplectic form σ_{E} defines an isomorphism ♯ : E → E^{∗}, e ↦ e^{♯} := σ_{E}(e, .) with inverse ♭ : E^{∗} → E.

Of particular importance is the representation Sym^{2}E. Its real subspace is canonically isomorphic to 𝔰𝔭(n). Thus Sym^{2}E acts on every complex representation of Sp(n), e. g. its action on E is given by (e_{1}e_{2})e := σ_{E}(e_{1}, e)e_{2} + σ_{E}(e_{2}, e)e_{1}. Analogous statements are true for H.

Let {e_{i}} and {de_{i}} with de_{i}(e_{j}) = δ_{ij} be a dual pair of bases for E, E^{∗} respectively. In terms of this bases the symplectic form and its canonical bivector — associated by Λ^{2}E ≅ Λ^{2}E^{∗} — can be written as

σ_{E} = \frac{1}{2} ∑ de_{i} ∧ e_{i}^{♯} ∈ Λ^{2}E^{∗} L_{E} = \frac{1}{2} ∑ de_{i}^{♭} ∧ e_{i} ∈ Λ^{2}E .

Wedging with L_{E} determines a homomorphism L : Λ^{s−2}E ⟶ Λ^{s}E whereas contracting with σ_{E} defines its adjoint Λ := L^{∗} : Λ^{s}E ⟶ Λ^{s−2}E. The operators L, Λ and H := [Λ, L] with H|_{Λ^{s}E} = (n − s)id satisfy the commutator algebra of the Lie algebra 𝔰𝔩_{2}ℂ. Therefore, Λ^{s}E = im(L)⊕ker(Λ) splits as Sp(n)–representation and Λ_{∘}^{s}E := ker(Λ), the primitive space, turns out to be irreducible. Inductively, the complete decomposition of Λ^{s}E is proved to be:

Λ^{s}E = ⨁_{k=0}^{⌊ \frac{s}{2} ⌋} Λ_{∘}^{s−2k}E, 0 ≤ s ≤ n .

The primitive space is stable under contraction with elements of E^{∗} but it is not preserved by the wedge product. Therefore it is necessary to describe the projection e ∧_{∘} ω of e ∧ ω onto Λ_{∘}^{s}E:

Lemma 2.1 If ω ∈ Λ_{∘}^{s}E then e^{♯} ⌟ ω ∈ Λ_{∘}^{s−1}E. Furthermore

e ∧_{∘} ω = e ∧ ω − \frac{1}{n−s+1} L_{E} ∧ (e^{♯} ⌟ ω) .

Summarizing the properties of contraction and modified exterior multiplication we have:

Lemma 2.2 On Λ_{∘}^{s}E modified exterior multiplication and contraction operators satisfy the following anti– commutator relations

\{η_{1}⌟ , η_{2}⌟ \} = 0  \{e_{1}∧_{∘}, e_{2}∧_{∘}\} = 0  \{η ⌟ , e ∧_{∘} \} = η(e) + \frac{1}{n−s+1}η^{♭} ∧_{∘} e^{♯}⌟

for arbitrary η, η_{i} ∈ E^{∗} and e, e_{i} ∈ E. In addition the following variants of number operators are defined:

∑ e_{i} ∧_{∘} de_{i} ⌟ = s id ∑ de_{i} ⌟ e_{i} ∧_{∘} = \frac{(2n−s+2)(n−s)}{n−s+1} id .

On H, there are similar equations which relate contraction and symmetric product with h ∈ H. However, it is convenient to modify contraction. For α ∈ H^{∗} we define α ⌟_{∘} : Sym^{r}H → Sym^{r−1}H by α ⌟_{∘} := \frac{1}{r} α ⌟.

Lemma 2.3 On Sym^{r}H symmetric multiplication and modified contraction operators satisfy the following commutator relations

\begin{aligned}[h_{1}·, h_{2}·] & = & 0 & [α⌟_{∘}, h·] & = & −\frac{1}{r+1}α^{♭} · h^{♯}⌟_{∘} \\ [α_{1}⌟_{∘}, α_{2}⌟_{∘}] & = & 0 & α(h)id & = & h · α⌟_{∘} − α^{♭} · h^{♯}⌟_{∘} \end{aligned}

for arbitrary h, h_{i} ∈ H and α, α_{i} ∈ H^{∗}. In addition the following variants of number operators are defined:

∑ h_{i} · dh_{i}⌟_{∘} = id ∑ dh_{i}⌟_{∘}h_{i}· = \frac{r + 2}{r + 1} id .

2.2 The Curvature Tensor

For later use we need an explicit description of the curvature tensor which we take from [KSW97-1]. First we recall the definition of the following End(H ⊗ E)–valued 2–forms on H ⊗ E:

R_{h_{1}⊗e_{1},h_{2}⊗e_{2}}^{H} = σ_{E}(e_{1}, e_{2})(h_{1}h_{2} ⊗ id_{E})
R_{h_{1}⊗e_{1},h_{2}⊗e_{2}}^{E} = σ_{H}(h_{1}, h_{2})(id_{H} ⊗ e_{1}e_{2})
R_{h_{1}⊗e_{1},h_{2}⊗e_{2}}^{hyper} = σ_{H}(h_{1}, h_{2})(id_{H} ⊗ ℜ_{e_{1},e_{2}}),

where the endomorphisms ℜ_{e_{1},e_{2}} : e ↦ ℜ(e_{1}, e_{2}, e, .)^{♭} of E are induced by ℜ ∈ Γ(Sym^{4}E^{∗}).

Lemma 2.4 The curvature tensor of a quaternionic Kähler manifold M^{4n} is given by

R = − \frac{κ}{8n(n + 2)}(R^{H} + R^{E}) + R^{hyper} ,

where κ is the scalar curvature of M and the symmetric 4–form ℜ is necessarily the symmetrisation:

ℜ(e_{1}, e_{2}, e_{3}, e_{4}) = \frac{1}{24σ_{H}(h_{1}, h_{2})σ_{H}(h_{3}, h_{4})} ∑_{τ∈ S_{4}}⟨R_{h_{1}⊗e_{τ1},h_{2}⊗e_{τ2}}h_{3} ⊗ e_{τ3}, h_{4} ⊗ e_{τ4}⟩ ,

which is independent of the choice of the h_{i} as long as σ_{H}(h_{1}, h_{2})σ_{H}(h_{3}, h_{4}) ≠ 0.

2.3 Spinor Bundle and Clifford Multiplication

The spinor module considered as Sp(n) × Sp(1)–representation splits into a sum of n + 1 irreducible components. Hence, the spinor bundle of a 4n–dimensional quaternionic Kähler manifold decomposes into a sum of n + 1 subbundles which can be expressed using the locally defined bundles E and H.

Proposition 2.1 [BaS83], [HiM95], [Wan89] The spinor bundle S(M) of a quaternionic Kähler manifold M decomposes as

S(M) = ⨁_{r=0}^{n} S_{r}(M) := ⨁_{r=0}^{n} Sym^{r}H ⊗ Λ_{∘}^{n−r}E .

The rank of the subbundle S_{r}(M) is given by

rank (S_{r}(M)) = (r + 1)(\binom{2n}{n − r} − \binom{2n}{n − r − 2}) .

Note that the covariant derivative on S(M) induced by the Levi–Civita connection on (M, g) respects the decomposition given above. The following proposition presents the Clifford multiplication in terms of the E–H–formalism.

Proposition 2.2 [KSW97-1] For any tangent vector h ⊗ e ∈ H ⊗ E = T M^{ℂ}, the Clifford multiplication μ(h ⊗ e) : S(M) → S(M) is given by:

μ(h ⊗ e) = \sqrt{2} (h · ⊗ e^{♯} ⌟ + h^{♯} ⌟_{∘} ⊗ e ∧_{∘}) .

In particular, the Clifford multiplication maps the subbundle S_{r}(M) to the sum S_{r−1}(M) ⊕ S_{r+1}(M).

Thus, Clifford multiplication splits into two components:

μ_{−}^{+} : T M^{ℂ} ⊗ S_{r}(M) ⟶ S_{r+1}(M)  and  μ_{+}^{−} : T M^{ℂ} ⊗ S_{r}(M) ⟶ S_{r−1}(M) ,

where μ_{−}^{+}(h ⊗ e ⊗ ψ) = \sqrt{2} (h · ⊗ e^{♯} ⌟ ) ψ and μ_{+}^{−}(h ⊗ e ⊗ ψ) = \sqrt{2} (h^{♯} ⌟_{∘} ⊗ e ∧_{∘}) ψ. We note that this definition makes sense also for S_{r}(M) replaced by Sym^{r}H ⊗ Λ_{∘}^{s}E. In this spirit it is possible to define two operations similar to Clifford multiplication:

μ_{+}^{+} : T M^{ℂ} ⊗ (Sym^{r}H ⊗ Λ_{∘}^{s}E) ⟶ Sym^{r+1}H ⊗ Λ_{∘}^{s+1}E
h ⊗ e ⊗ ψ  ⟼ \sqrt{2} (h · ⊗ e ∧_{∘})ψ

and

μ_{−}^{−} : T M^{ℂ} ⊗ Sym^{r}(H ⊗ Λ_{∘}^{s}E) ⟶ Sym^{r−1}H ⊗ Λ_{∘}^{s−1}E
h ⊗ e ⊗ ψ  ⟼ \sqrt{2} (h^{♯} ⌟_{∘} ⊗ e^{♯} ⌟ )ψ .

Using the number operators of Lemmata 2.2 and 2.3 it is easy to prove the following useful formulas:

Lemma 2.5 The following relations are satisfied on Sym^{r}H ⊗ Λ_{∘}^{s}E:

\begin{aligned}∑ μ_{+}^{+}(X_{a})μ_{−}^{−}(X_{a}) & = & 2s \\ ∑ μ_{−}^{+}(X_{a})μ_{+}^{−}(X_{a}) & = & −2 \frac{(2n−s+2)(n−s)}{n−s+1} \\ ∑ μ_{+}^{−}(X_{a})μ_{−}^{+}(X_{a}) & = & −2s \frac{r+2}{r+1} \\ ∑ μ_{−}^{−}(X_{a})μ_{+}^{+}(X_{a}) & = & 2 \frac{(2n−s+2)(n−s)}{n−s+1} \frac{r+2}{r+1}, \end{aligned}

where the sum is over a local orthonormal base {X_{a}} of T M. All other combinations of the partial Clifford multiplications vanish upon summation over {X_{a}}.

2.4 Dirac and Twistor Operators

In this section we recall the definition of quaternionic Dirac and twistor operators. For defining these operators we have to decompose T M^{ℂ} ⊗ S_{r}(M) into irreducible components and to project the covariant differential of a spinor onto the different summands. For r ≠ 0, n we have the following decomposition

T M^{ℂ} ⊗ S_{r}(M) ≅ (H ⊗ E) ⊗ (Sym^{r}H ⊗ Λ_{∘}^{n−r}E)  (2.1)
≅ S_{r+1}(M) ⊕ S_{r−1}(M) ⊕ ( S_{r}^{+} ⊕ S_{r}^{−} ⊕ V_{r}^{+} ⊕ V_{r}^{−}) .

Here we used the notation S_{r}^{±} = Sym^{r±1}H ⊗ Λ_{∘}^{n−r±1}E and V_{r}^{±} = Sym^{r±1}H ⊗ K^{n−r}E, where K^{n−r}E is the summand corresponding to the sum of the highest weights in the decomposition of E ⊗ Λ_{∘}^{n−r}E. In the case r = 0 and r = n four of the above summands vanish and we obtain:

(H ⊗ E) ⊗ Λ_{∘}^{n}E ≅ S_{1}(M) ⊕ V_{0}^{+}  and  (H ⊗ E) ⊗ Sym^{n}H ≅ S_{n−1}(M) ⊕ S_{n}^{+} .  (2.2)

The two components of the Clifford multiplication define natural projections onto the first two summands appearing in the decomposition (2.1). The remaining four summands constitute the kernel of the Clifford multiplication. The projections onto S_{r}^{+} resp. S_{r}^{−} are given by μ_{+}^{+} resp. μ_{−}^{−} and we denote the projections onto V^{±} by pr_{V^{±}}. Applying these projectors to the section ∇ψ ∈ Γ(T M ⊗S(M)) we get the two components of the Dirac operator

D_{−}^{+} := μ_{−}^{+} ∘ ∇ : S_{r}(M) ⟶ S_{r+1}(M)  D_{+}^{−} := μ_{+}^{−} ∘ ∇ : S_{r}(M) ⟶ S_{r−1}(M) ,  (2.3)

where D = D_{−}^{+} + D_{+}^{−} is the Dirac operator, and four twistor operators:

\begin{aligned}D_{+}^{+} & := & μ_{+}^{+} ∘ ∇ : S_{r}(M) ⟶ S_{r}^{+} & D_{−}^{−} & := & μ_{−}^{−} ∘ ∇ : S_{r}(M) ⟶ S_{r}^{−} \\ T^{+} & := & pr_{V^{+}} ∘ ∇ : S_{r}(M) ⟶ V^{+} & T^{−} & := & pr_{V^{−}} ∘ ∇ : S_{r}(M) ⟶ V^{−} . \end{aligned}  (2.4)

The square of the Dirac operator respects the splitting of the spinor bundle, i. e. D^{2} : S_{r}(M) ⟶ S_{r}(M). In particular, we have: D_{−}^{+}D_{−}^{+} = 0 = D_{+}^{−}D_{+}^{−}.

According to the definition of the Dirac and twistor operators by decomposition (2.1) it is possible to reconstruct the covariant differential of a spinor with the help of these operators. As this is a prerequisite for deriving Killing equations we state the final formula with the help of right inverses ι_{∓}^{∓} for the partial Clifford multiplications μ_{±}^{±}:

Lemma 2.6

∇ϕ = ι_{+}^{−}(D_{−}^{+}ϕ) + ι_{−}^{+}(D_{+}^{−}ϕ) + ι_{−}^{−}(D_{+}^{+}ϕ) + ι_{+}^{+}(D_{−}^{−}ϕ) + T^{+}ϕ + T^{−}ϕ

where the embeddings ι_{∓}^{∓} are defined as the right inverses of μ_{±}^{±}, given explicitly by

ι_{+}^{−} : S_{r+1}(M) ⟶ T M^{ℂ} ⊗ S_{r}(M)  ι_{−}^{−} : S_{r}^{+} ⟶ T M^{ℂ} ⊗ S_{r}(M)
ϕ  ⟼ −\frac{r+2}{2(n+r+3)(r+1)} ∑ X_{a} ⊗ μ_{+}^{−}(X_{a})ϕ  ϕ ⟼ \frac{1}{2(n−r+1)} ∑ X_{a} ⊗ μ_{−}^{−}(X_{a})ϕ
ι_{−}^{+} : S_{r−1}(M) ⟶ T M^{ℂ} ⊗ S_{r}(M)  ι_{+}^{+} : S_{r}^{−} ⟶ T M^{ℂ} ⊗ S_{r}(M)
ϕ  ⟼ −\frac{r}{2(n−r+1)(r+1)} ∑ X_{a} ⊗ μ_{−}^{+}(X_{a})ϕ  ϕ ⟼ \frac{r(r+2)}{2(n+r+3)(r+1)^{2}} ∑ X_{a} ⊗ μ_{+}^{+}(X_{a})ϕ

according to Lemma 2.5, where {X_{a}} is a local orthonormal base of T M.

3 Weitzenböck Formulas

The central result of [KSW97-1] is a Weitzenböck formula in matrix form which relates two sets of naturally defined 2nd order differential operators from the spinor bundle to itself. The idea is to replace the Lichnerowicz Weitzenböck formula of general holonomy by the linear space of all Weitzenböck formulas adapted to the holonomy Sp(1) · Sp(n), in order to cope with the abundance of natural 2nd order differential operators. In the case of quaternionic Kähler manifolds the final matrix equation is an identity of differential operators defined on sections ψ of the bundles Sym^{r}H ⊗ Λ_{∘}^{s}E depending on r, s and the quaternionic dimension n:

( \begin{matrix}−∇^{∗}∇ψ \\ \frac{κ}{4} \frac{r(r+2)}{n+2}ψ \\ \frac{κ}{4} \frac{s(2n−s+2)}{n(n+2)}ψ \\ Cψ \\ Lψ \\ 0 \end{matrix} ) = W(r, s) ( \begin{matrix}−\frac{1}{2}(D_{+}^{+})^{∗}D_{+}^{+}ψ \\ \frac{1}{2}D_{−}^{+}D_{+}^{−}ψ \\ \frac{1}{2}D_{+}^{−}D_{−}^{+}ψ \\ −\frac{1}{2}(D_{−}^{−})^{∗}D_{−}^{−}ψ \\ −(T^{+})^{∗}T^{+}ψ \\ (T^{−})^{∗}T^{−}ψ \end{matrix} ) ,  (3.5)

where W(r, s) = W_{H}(r) ⊗ W_{E}(s) is the Kronecker product of the two matrices

W_{H}(r) = ( \begin{matrix}1 & −\frac{r}{r+1} \\ r & \frac{r(r+2)}{r+1} \end{matrix} )  and  W_{E}(s) = ( \begin{matrix}\frac{1}{s+1} & −\frac{n−s+2}{(n−s+1)(2n−s+3)} & 1 \\ −\frac{s}{s+1} & \frac{(n−s+2)(2n−s+2)}{(n−s+1)(2n−s+3)} & 1 \\ −\frac{(n+1)s}{n(s+1)} & −\frac{(n+1)(n−s)(2n−s+2)}{n(n−s+1)(2n−s+3)} & \frac{n−s}{n} \end{matrix} ) .  (3.6)

The proof given in [KSW97-1] for s = n − r goes through without modification in the general case, only Lemma 4.4 of [KSW97-1] has to be reformulated. We remark that this formula simplifies in case r = 0 or s = 0, n, because some of the operators involved vanish by definition. Though this Weitzenböck formula is powerful enough to prove the eigenvalue estimate for the Dirac operator, it turns out to be insufficient to derive the quaternionic Killing equations of the next section.

For that purpose we need additional Weitzenböck formulas relating 2nd order differential operators between different vector bundles, which are not covered by (3.5). Nevertheless, the basic idea of [KSW97-1] can be applied to derive these additional formulas.

We consider for s ≥ 2 the isotypical Sym^{r}H ⊗ K^{s−1}E–component of the second covariant differential ∇^{2}ψ ∈ Γ(H ⊗ E ⊗ H ⊗ E ⊗ Sym^{r}H ⊗ Λ_{∘}^{s}E) of a section ψ of Sym^{r}H ⊗ Λ_{∘}^{s}E. As this isotypical component contains four copies unless r = 0 the resulting formula will in general relate two sets of four projectors.

As in [KSW97-1] the problem can be split into two parts dealing with Sp(1) and Sp(n)–representations only. Representation theory of Sp(1), however, is very simple and no arguments beyond [KSW97-1] are needed. For this reason we briefly recall that two copies of Sym^{r}H are contained in H ⊗ H ⊗ Sym^{r}H, but the projectors onto these two copies are not unique. A first pair of projectors is obtained by decomposing H ⊗ H into irreducibles, which then act as endomorphisms on Sym^{r}H:

pr_{ℂ} : H ⊗ H ⊗ Sym^{r}H ⟶ Sym^{r}H
h_{1} ⊗ h_{2} ⊗ s  ⟼ σ_{H}(h_{1}, h_{2})s,
pr_{Sym^{2}H} : H ⊗ H ⊗ Sym^{r}H ⟶ Sym^{r}H
h_{1} ⊗ h_{2} ⊗ s  ⟼ (h_{1}h_{2})(s).

We get a second pair of projectors through the operation of H on Sym^{r}H by the H–part of Clifford multiplication:

pr_{−+} : H ⊗ H ⊗ Sym^{r}H ⟶ H ⊗ Sym^{r+1}H ⟶ Sym^{r}H
h_{1} ⊗ h_{2} ⊗ s  ⟼ h_{1} ⊗ h_{2} · s ⟼ h_{1}^{♯}⌟_{∘}(h_{2} · s),
pr_{+−} : H ⊗ H ⊗ Sym^{r}H ⟶ H ⊗ Sym^{r−1}H ⟶ Sym^{r}H
h_{1} ⊗ h_{2} ⊗ s  ⟼ h_{1} ⊗ h_{2}^{♯}⌟_{∘}s ⟼ h_{1} · (h_{2}^{♯}⌟_{∘}s).

These two pairs of projectors are related by the matrix W_{H}(r):

( \begin{matrix}pr_{ℂ} \\ pr_{Sym^{2}H} \end{matrix} ) = ( \begin{matrix}1 & −\frac{r}{r+1} \\ r & \frac{r(r+2)}{r+1} \end{matrix} ) ( \begin{matrix}pr_{−+} \\ pr_{+−} \end{matrix} ) .  (3.7)

Turning now to the second part of the problem dealing with Sp(n)–representations, we have to look at the isotypical K^{s−1}E–component of E ⊗ E ⊗ Λ_{∘}^{s}E containing two copies. First recall that the projector p̃r_{K} : E ⊗ Λ_{∘}^{s}E → K^{s}E is given by:

p̃r_{K}(e ⊗ ϕ) := e ⊗ ϕ − \frac{1}{s+1} ∑_{i} e_{i} ⊗ de_{i}⌟ e ∧_{∘} ϕ − \frac{n−s+2}{(2n−s+3)(n−s+1)} ∑_{i} de_{i}^{♭} ⊗ e_{i} ∧_{∘} e^{♯}⌟ ϕ,

The decomposition E ⊗ E ≃ Sym^{2}E ⊕ Λ_{∘}^{2}E ⊕ ℂ immediately yields two projectors:

pr_{KSym^{2}E} : E ⊗ E ⊗ Λ_{∘}^{s}E ⟶ Sym^{2}E ⊗ Λ_{∘}^{s}E ⟶ K^{s−1}E
e_{1} ⊗ e_{2} ⊗ ϕ  ⟼  p̃r_{K}(e_{2} ⊗ e_{1}^{♯}⌟ ϕ + e_{1} ⊗ e_{2}^{♯}⌟ ϕ)
pr_{KΛ_{∘}^{2}E} : E ⊗ E ⊗ Λ_{∘}^{s}E ⟶ Λ_{∘}^{2}E ⊗ Λ_{∘}^{s}E ⟶ K^{s−1}E
e_{1} ⊗ e_{2} ⊗ ϕ  ⟼  p̃r_{K}(e_{2} ⊗ e_{1}^{♯}⌟ ϕ − e_{1} ⊗ e_{2}^{♯}⌟ ϕ).

On the other hand, we can first project E ⊗ Λ_{∘}^{s}E onto Λ_{∘}^{s−1}E resp. K^{s}E and then look what the second E–factor can do:

pr_{K−} : E ⊗ E ⊗ Λ_{∘}^{s}E ⟶ E ⊗ Λ_{∘}^{s−1}E ⟶ K^{s−1}E
e_{1} ⊗ e_{2} ⊗ ϕ  ⟼  p̃r_{K}(e_{1} ⊗ e_{2}^{♯}⌟ ϕ),
pr_{−K} : E ⊗ E ⊗ Λ_{∘}^{s}E ⟶ E ⊗ K^{s}E ⟶ K^{s−1}E
e_{1} ⊗ e_{2} ⊗ ϕ  ⟼  p̃r_{K}((id ⊗ e_{1}^{♯}⌟ )p̃r_{K}(e_{2} ⊗ ϕ)).

The projector pr_{−K} is not yet in its final form. To simplify its definition we use a suitable operator identity on Λ_{∘}^{s−1}E, which is obtained by applying the anticommutator rules of Lemma 2.2 twice:

e_{1}^{♯}⌟ e_{2}∧_{∘} = −e_{2} ∧_{∘} e_{1}^{♯}⌟ + σ_{E}(e_{1}, e_{2}) + \frac{1}{n−s+2}e_{1} ∧_{∘} e_{2}^{♯}⌟
= −\frac{(n−s+1)(n−s+3)}{(n−s+2)^{2}}e_{2} ∧_{∘} e_{1}^{♯}⌟ + \frac{n−s+1}{n−s+2}σ_{E}(e_{1}, e_{2}) − \frac{1}{n−s+2}e_{2}^{♯}⌟ e_{1} ∧_{∘} .

We remark that by definition the projector p̃r_{K} kills elements of the form ∑_{i} e_{i} ⊗de_{i}⌟ ϕ and ∑_{i} de_{i}^{♭} ⊗e_{i} ∧_{∘} ϕ. With this in mind the projector pr_{−K} can be made completely explicit:

pr_{−K}(e_{1} ⊗ e_{2} ⊗ ϕ)

= p̃r_{K}(e_{2} ⊗ e_{1}^{♯}⌟ ϕ − \frac{1}{s+1} ∑_{i} e_{i} ⊗ e_{1}^{♯}⌟ de_{i}⌟ e_{2} ∧_{∘} ϕ − \frac{n−s+2}{(2n−s+3)(n−s+1)} ∑_{i} de_{i}^{♭} ⊗ e_{1}^{♯}⌟ e_{i} ∧_{∘} e_{2}^{♯}⌟ ϕ)
= p̃r_{K}(e_{2} ⊗ e_{1}^{♯}⌟ ϕ)

−\frac{n−s+2}{(2n−s+3)(n−s+1)} p̃r_{K}(∑_{i} de_{i}^{♭} ⊗ (−\frac{(n−s+1)(n−s+3)}{(n−s+2)^{2}}e_{i} ∧_{∘} e_{1}^{♯}⌟ + \frac{n−s+1}{n−s+2}σ_{E}(e_{1}, e_{i})− \frac{1}{n−s+2}e_{i}^{♯}⌟ e_{1}∧_{∘})e_{2}^{♯}⌟ ϕ)

= p̃r_{K}(e_{2} ⊗ e_{1}^{♯}⌟ ϕ) − \frac{1}{2n−s+3} p̃r_{K}(e_{1} ⊗ e_{2}^{♯}⌟ ϕ).

With this simpler form of pr_{−K} the relations between the projectors become obvious:

( \begin{matrix}pr_{KSym^{2}E} \\ pr_{KΛ_{∘}^{2}E} \end{matrix} ) = ( \begin{matrix}\frac{2n−s+4}{2n−s+3} & 1 \\ −\frac{2n−s+2}{2n−s+3} & 1 \end{matrix} ) ( \begin{matrix}pr_{K−} \\ pr_{−K} \end{matrix} ) .  (3.8)

In a final step the differential operators associated to the projectors have to be identified. This is simple for the right–hand side projectors, because by definition the associated operators are products of 1st order differential operators. To write down the result, we have to define two 1st order differential operators, which appear naturally in this way:

θ^{±} : Γ(Sym^{r}H ⊗ K^{s}E) → Γ(Sym^{r±1}H ⊗ K^{s−1}E)

are the composition of the covariant differential ∇ : Γ(Sym^{r}H ⊗ K^{s}E) → Γ((H ⊗ E) ⊗ Sym^{r}H ⊗ K^{s}E) with linear maps

(H ⊗ E) ⊗ Sym^{r}H ⊗ K^{s}E ↪ (H ⊗ E) ⊗ Sym^{r}H ⊗ E ⊗ Λ_{∘}^{s}E → Sym^{r±1}H ⊗ K^{s−1}E
(h ⊗ e_{1}) ⊗ s ⊗ e_{2} ⊗ ϕ  ↦ \{ \begin{matrix}h · s \\ h^{♯}⌟_{∘}s \end{matrix} \} ⊗ p̃r_{K}(e_{2} ⊗ e_{1}^{♯}⌟ ϕ) .

Then the right–hand projectors define the following operator products:

\begin{aligned}pr_{−+} ⊗ pr_{K−}(∇^{2}ϕ) & = & \frac{1}{\sqrt{2}}T^{−}D_{−}^{+}ϕ \\ pr_{+−} ⊗ pr_{K−}(∇^{2}ϕ) & = & \frac{1}{\sqrt{2}}T^{+}D_{−}^{−}ϕ \\ pr_{−+} ⊗ pr_{−K}(∇^{2}ϕ) & = & θ^{−}T^{+}ϕ \\ pr_{+−} ⊗ pr_{−K}(∇^{2}ϕ) & = & θ^{+}T^{−}ϕ . \end{aligned}

The left–hand projectors provide two new 2nd order differential operators:

\begin{aligned}T_{C}ϕ & := & pr_{Sym^{2}H} ⊗ pr_{KSym^{2}E}(∇^{2}ϕ) \\ T_{L}ϕ & := & pr_{ℂ} ⊗ pr_{KΛ_{∘}^{2}E}(∇^{2}ϕ) , \end{aligned}

which are of no particular importance for the time being, and two linear operators depending only on curvature: (pr_{Sym^{2}H} ⊗ pr_{KΛ_{∘}^{2}E}) ∘ ∇^{2} and (pr_{ℂ} ⊗ pr_{KSym^{2}E}) ∘ ∇^{2}.

It is easy to see that (pr_{Sym^{2}H} ⊗ pr_{KΛ_{∘}^{2}E}) ∘ ∇^{2} is the zero operator, because the curvature tensor of a quaternionic Kähler manifold takes values only in the complement Sym^{2}H ⊗ ℂ of Sym^{2}H ⊗ Λ_{∘}^{2}E in Sym^{2}H ⊗ Λ^{2}E. We now claim that the operator (pr_{ℂ} ⊗ pr_{KSym^{2}E}) ∘ ∇^{2} is trivial, too.

Of the three summands of the curvature tensor of a quaternionic Kähler manifold according to Lemma 2.4, only R^{E} and R^{hyper} take values in ℂ⊗Sym^{2}E. Tracing down the definitions of (pr_{ℂ}⊗pr_{KSym^{2}E})∘∇^{2} the contribution of R^{hyper} can be written in terms of ℜ ∈ Sym^{4}E^{∗} as the following morphism from Sym^{r}H ⊗Λ_{∘}^{s}E to Sym^{r}H ⊗ K^{s−1}E:

(pr_{ℂ} ⊗ pr_{KSym^{2}E}) ∘ ∇^{2} (s ⊗ ϕ) = s ⊗ (∑_{ij} (p̃r_{K} ∘ (de_{i}^{♭} ⊗ de_{j}⌟ + de_{j}^{♭} ⊗ de_{i}⌟ ) ∘ ℜ_{e_{i},e_{j}}) ϕ) .

Expanding ℜ in fourth powers \frac{1}{24}α^{4}, α ∈ E^{∗} the contribution from R^{hyper} is seen to vanish because already without projecting α^{♭} ⊗ α⌟ α^{♭} ∧_{∘} α⌟ ϕ = 0 for all ϕ. The contribution of R^{E} to (pr_{ℂ} ⊗ pr_{KSym^{2}E}) ∘ ∇^{2} can be written down similarly. The resulting homomorphism is immediately seen to be Sp(n)·Sp(1)–equivariant and has to vanish, because there is no non–trivial way to map Sym^{r}H ⊗ Λ_{∘}^{s}E to Sym^{r}H ⊗ K^{s−1}E equivariantly.

We conclude this section in tensoring equations (3.7) and (3.8) to get the following twistor Weitzenböck formula:

( \begin{matrix}0 \\ T_{C} \\ T_{L} \\ 0 \end{matrix} ) = W_{twist} ( \begin{matrix}\frac{1}{\sqrt{2}}T^{−}D_{−}^{+} \\ \frac{1}{\sqrt{2}}T^{+}D_{−}^{−} \\ θ^{−}T^{+} \\ θ^{+}T^{−} \end{matrix} ) W_{twist} = ( \begin{matrix}\frac{2n−s+4}{2n−s+3} & −\frac{r}{r+1} \frac{2n−s+4}{2n−s+3} & 1 & −\frac{r}{r+1} \\ r \frac{2n−s+4}{2n−s+3} & \frac{r(r+2)}{r+1} \frac{2n−s+4}{2n−s+3} & r & \frac{r(r+2)}{r+1} \\ −\frac{2n−s+2}{2n−s+3} & \frac{r}{r+1} \frac{2n−s+2}{2n−s+3} & 1 & −\frac{r}{r+1} \\ −r \frac{2n−s+2}{2n−s+3} & −\frac{r(r+2)}{r+1} \frac{2n−s+2}{2n−s+3} & r & \frac{r(r+2)}{r+1} \end{matrix} ) .

Corollary 3.1 The following operator identity holds on sections of the bundle Sym^{r}H ⊗ Λ_{∘}^{s}E with s ≥ 2:

\frac{1}{\sqrt{2}} \frac{2n − s + 4}{2n − s + 3}(T^{−}D_{−}^{+} − \frac{r}{r + 1}T^{+}D_{−}^{−}) + (θ^{−}T^{+} − \frac{r}{r + 1}θ^{+}T^{−}) = 0.

This identity is trivially satisfied for s = 1, because T^{−}D_{−}^{+}, T^{+}D_{−}^{−}, θ^{−}T^{+} and θ^{+}T^{−} all vanish separately.

4 The Quaternionic Killing Equation

The matrix Weitzenböck formula (3.5) generates a linear space of operator identities of 2nd order differential operators from a bundle S_{r}(M) to itself by multiplying it from the left by an arbitrary row vector. A particularly important identity in this linear space leads to the following key identity of operator norms for any section ψ_{r} of S_{r}(M):

\frac{κ}{4} \frac{(r+2)(n+r+2)}{n+2} ∥ψ_{r}∥^{2} = − \frac{r+1}{n−r+1} ∥D_{+}^{+}ψ_{r}∥^{2} + (r+2) ∥D_{+}^{−}ψ_{r}∥^{2} + \frac{(r+2)(n+r+2)}{n+r+3} ∥D_{−}^{+}ψ_{r}∥^{2} − 2(r+1) ∥T^{+}ψ_{r}∥^{2}.

(4.9)

Since D^{2} respects the splitting of S(M) into the subbundles S_{r}(M) an eigenspinor of D can be assumed to be localized in S_{r}(M) ⊕ S_{r+1}(M). If ψ = ψ_{r} + ψ_{r+1} is such an eigenspinor with eigenvalue λ, then D_{−}^{+}ψ_{r} = λψ_{r+1}, D_{+}^{−}ψ_{r} = 0 and identity (4.9) implies for ψ_{r}:

\frac{κ}{4} \frac{n+r+3}{n+2}∥ψ_{r}∥^{2} = λ^{2}∥ψ_{r+1}∥^{2} − \frac{r+1}{r+2} \frac{n+r+3}{n+r+2}(\frac{1}{n−r+1}∥D_{+}^{+}ψ_{r}∥^{2} + 2∥T^{+}ψ_{r}∥^{2}).

In the same vein identity (4.9) implies for ψ_{r+1}:

\frac{κ}{4} \frac{n+r+3}{n+2}∥ψ_{r+1}∥^{2} = λ^{2}∥ψ_{r}∥^{2} − \frac{r+2}{r+3}(\frac{1}{n−r} ∥D_{+}^{+}ψ_{r+1}∥^{2} + 2∥T^{+}ψ_{r+1}∥^{2}).

These identities prove:

Theorem 4.1 Let (M^{4n}, g) be a compact quaternionic Kähler spin manifold of positive scalar curvature κ and let ψ = ψ_{r} + ψ_{r+1} ∈ Γ(S_{r}(M) ⊕ S_{r+1}(M)) be an eigenspinor for D with eigenvalue λ. Then

λ^{2} ≥ \frac{κ}{4} \frac{n + r + 3}{n + 2}

with equality if and only if ψ_{r} and ψ_{r+1} are both minimal in the sense that D_{+}^{+}ψ_{r} = 0 = D_{+}^{+}ψ_{r+1} and T^{+}ψ_{r} = 0 = T^{+}ψ_{r+1}.

Thus an eigenspinor ψ of D with smallest possible eigenvalue λ with λ^{2} = \frac{κ}{4} \frac{n+3}{n+2} has to be of the form ψ = ψ_{0} + ψ_{1} ∈ Γ(S_{0}(M) ⊕ S_{1}(M)). As ψ_{0} is a section of S_{0}(M) its covariant differential ∇ψ_{0} splits into only two pieces D_{−}^{+}ψ_{0} and T^{+}ψ_{0} according to equation (2.2). Minimality implies T^{+}ψ_{0} = 0 and Lemma 2.6 reconstructs ∇ψ_{0} from D_{−}^{+}ψ_{0} = λψ_{1}:

∇ψ_{0} = λ ι_{+}^{−}(ψ_{1}).

The covariant differential of ψ_{1} is more complicated as it splits into six pieces: D_{±}^{±}ψ_{1} and T^{±}ψ_{1}. Minimality implies D_{+}^{+}ψ_{1} = 0 and T^{+}ψ_{1} = 0, and as part of the Dirac operator D_{−}^{+}ψ_{1} = 0. Plugging this into the Weitzenböck formula (3.5) the second, third and sixth row read:

( \begin{matrix}\frac{κ}{4} \frac{3}{n+2}ψ_{1} \\ \frac{κ}{4} \frac{(n+3)(n−1)}{n(n+2)}ψ_{1} \\ 0 \end{matrix} ) = ( \begin{matrix}\frac{3}{2n} & −\frac{9}{4(n+4)} & \frac{3}{2} \\ \frac{n−1}{2n} & −\frac{3(n+3)}{4(n+4)} & −\frac{1}{2} \\ −\frac{3(n−1)(n+1)}{2n^{2}} & −\frac{3(n+3)(n+1)}{4n(n+4)} & \frac{3}{2n} \end{matrix} ) ( \begin{matrix}\frac{1}{2}D_{−}^{+}D_{+}^{−}ψ_{1} \\ −\frac{1}{2}(D_{−}^{−})^{∗}D_{−}^{−}ψ_{1} \\ (T^{−})^{∗}T^{−}ψ_{1} \end{matrix} ) .

With D_{−}^{+}D_{+}^{−}ψ_{1} = \frac{κ}{4} \frac{n+3}{n+2}ψ_{1} this system of equations may be solved for (D_{−}^{−})^{∗}D_{−}^{−}ψ_{1} and (T^{−})^{∗}T^{−}ψ_{1} to find:

(D_{−}^{−})^{∗}D_{−}^{−}ψ_{1} = \frac{κ}{2} \frac{(n+4)(n−1)}{n(n+2)}ψ_{1}  and  (T^{−})^{∗}T^{−}ψ_{1} = 0 .

But, since we are on a compact manifold this implies T^{−}ψ_{1} = 0. Independently, T^{−}ψ_{1} = 0 can be shown by applying Corollary 3.1 to ψ_{0}. With the help of Lemma 2.6 the covariant differential ∇ψ_{1} is reconstructed from D_{+}^{−}ψ_{1} = λψ_{0} and D_{−}^{−}ψ_{1}:

∇ψ_{1} = λ ι_{−}^{+}(ψ_{0}) + ι_{+}^{+}(D_{−}^{−}ψ_{1}).

This in turn provides a first version of the quaternionic Killing equations:

\begin{aligned}∇_{X}ψ_{0} & = & − \frac{λ}{n+3} μ_{+}^{−}(X) ψ_{1} \\ ∇_{X}ψ_{1} & = & − \frac{λ}{4n} μ_{−}^{+}(X) ψ_{0} + \frac{3}{8(n+4)} μ_{+}^{+}(X) D_{−}^{−}ψ_{1}. \end{aligned}

Unfortunately it is not possible to say much about the section D_{−}^{−}ψ_{1}. The idea to overcome this obstacle is to include this special section and to consider a quaternionic Killing equation for two spinors and an auxiliary section of the bundle Λ_{∘}^{n−2}E. This yields indeed a useful Killing equation due to the following proposition:

Proposition 4.1

∇_{X}(D_{−}^{−}ψ_{1}) = − \frac{κ}{4} \frac{n+4}{n(n+2)} μ_{−}^{−}(X) ψ_{1}.

Proof. Since ψ_{−} := D_{−}^{−}ψ_{1} is a section of Λ_{∘}^{n−2}E its covariant differential ∇ψ_{−} splits into three pieces: D_{±}^{+}ψ_{−} and T^{+}ψ_{−}. However, from Corollary 3.1 applied to ψ_{1} we conclude T^{+}ψ_{−} = T^{+}D_{−}^{−}ψ_{1} = 0. With D_{+}^{+} = −(D_{−}^{−})^{∗} we have in addition:

(D_{+}^{+})^{∗}D_{+}^{+}ψ_{−} = D_{−}^{−}(D_{−}^{−})^{∗}D_{−}^{−}ψ_{1} = \frac{κ}{2} \frac{(n+4)(n−1)}{n(n+2)}ψ_{−}.

Now the third row of the Weitzenböck formula (3.5) applied to ψ_{−} reads

\frac{κ}{4} \frac{(n+4)(n−2)}{n(n+2)}ψ_{−} = \frac{n−2}{2(n−1)}(D_{+}^{+})^{∗}D_{+}^{+}ψ_{−} + \frac{2(n+4)}{3(n+5)}D_{+}^{−}D_{−}^{+}ψ_{−} = \frac{κ}{4} \frac{(n+4)(n−2)}{n(n+2)}ψ_{−} + \frac{2(n+4)}{3(n+5)}(D_{−}^{+})^{∗}D_{−}^{+}ψ_{−}.

On the compact manifold M this implies D_{−}^{+}ψ_{−} = 0. Thus the covariant differential of ψ_{−} can be reconstructed from D_{+}^{+}ψ_{−} = −(D_{−}^{−})^{∗}D_{−}^{−}ψ_{1} = −\frac{κ}{2} \frac{(n+4)(n−1)}{n(n+2)}ψ_{1} in the spirit of Lemma 2.6:

∇ψ_{−} = −\frac{κ}{2} \frac{(n+4)(n−1)}{n(n+2)}ι_{−}^{−}(ψ_{1}).

The proposition follows. □

Changing slightly the notation we obtain Theorem 4.2 Let (M^{4n}, g) be a compact quaternionic Kähler spin manifold of quaternionic dimension n and with positive scalar curvature κ. Let ψ = ψ_{0} + ψ_{1} ∈ Γ(S_{0}(M) ⊕ S_{1}(M)) be an eigenspinor for the smallest possible eigenvalue λ, i. e. λ^{2} = \frac{κ}{4} \frac{n+3}{n+2}. Then ψ_{0}, ψ_{1} and ψ_{−} := \frac{1}{4λ} \frac{n+3}{n+4} D_{−}^{−}ψ_{1} ∈ Γ(Λ_{∘}^{n−2}E) solve the following quaternionic Killing equation for the parameter λ:

∇_{X}ψ_{0} =  − \frac{λ}{n+3} μ_{+}^{−}(X) ψ_{1}
∇_{X}ψ_{1} = − \frac{λ}{4n} μ_{−}^{+}(X) ψ_{0}  + \frac{3λ}{2(n+3)} μ_{+}^{+}(X) ψ_{−}
∇_{X}ψ_{−} =  − \frac{λ}{4n} μ_{−}^{−}(X) ψ_{1}

Conversely, if the triple ψ_{0}, ψ_{1}, ψ_{−} is a solution of these equations for any λ ≠ 0 then

D_{−}^{+}ψ_{0} = λ ψ_{1}  D_{+}^{−}ψ_{1} = λ ψ_{0}  D_{+}^{+}ψ_{−} = − λ \frac{n−1}{2n} ψ_{1}
T^{+}ψ_{0} = 0  D_{−}^{−}ψ_{1} = 4λ \frac{n+4}{n+3} ψ_{−}  D_{−}^{+}ψ_{−} = 0
D_{±}^{+}ψ_{1} = 0  T^{+}ψ_{−} = 0
T^{±}ψ_{1} = 0

In particular, ψ_{0} + ψ_{1} is an eigenspinor for the smallest possible eigenvalue.

We will call solutions of the quaternionic Killing equations quaternionic Killing spinors. These are sections ψ = (ψ_{0}, ψ_{1}, ψ_{−}) of the bundle

S^{Killing}(M) := S_{0}(M) ⊕ S_{1}(M) ⊕ Λ_{∘}^{n−2}E ≅ Λ_{∘}^{n}E ⊕ (H ⊗ Λ_{∘}^{n−1}E) ⊕ Λ_{∘}^{n−2}E.

As in the Riemannian or the Kähler case it is possible to define a Killing connection ∇^{Killing} for which quaternionic Killing spinors are parallel. On sections of the bundle S^{Killing}(M) this connection is given by ∇_{X}^{Killing} := ∇_{X} + A_{X} where A_{X} is the following matrix

A_{X} = ( \begin{matrix}0 & \frac{λ}{n+3} μ_{+}^{−}(X) & 0 \\ \frac{λ}{4n} μ_{−}^{+}(X) & 0 & − \frac{3λ}{2(n+3)} μ_{+}^{+}(X) \\ 0 & \frac{λ}{4n} μ_{−}^{−}(X) & 0 \end{matrix} ) .

Hence, quaternionic Killing spinors ψ = (ψ_{0}, ψ_{1}, ψ_{−}) are annihilated by the curvature of the Killing connection, i. e. R_{X,Y}^{Killing}ψ = 0 for any vector fields X, Y. The following proposition is crucial for further investigations of the Killing equations:

Proposition 4.2

R^{Killing} = R^{hyper}.

Proof. As the partial Clifford multiplications are Sp(n) · Sp(1)–equivariant, they are parallel with respect to the Levi–Civita connection, and so is the endomorphism–valued 1–form A. Defining for arbitrary endomorphism–valued 1-forms B and C the endomorphism–valued 2–form (B ∧C)_{X,Y} := B_{X} ∘C_{Y} −B_{Y} ∘C_{X} the usual formula relating the curvature of ∇ and ∇^{Killing} = ∇ + A reads

R^{Killing} = R + [A, A] = R + \frac{λ^{2}}{4n(n + 3)} ( \begin{matrix}μ_{+}^{−} ∧ μ_{−}^{+} & 0 & −\frac{6n}{n+3}μ_{+}^{−} ∧ μ_{+}^{+} \\ 0 & μ_{−}^{+} ∧ μ_{+}^{−} − \frac{3}{2}μ_{+}^{+} ∧ μ_{−}^{−} & 0 \\ \frac{n+3}{4n}μ_{−}^{−} ∧ μ_{−}^{+} & 0 & −\frac{3}{2}μ_{−}^{−} ∧ μ_{+}^{+} \end{matrix} ) .

(4.10)

The entries of this matrix are endomorphism–valued 2-forms, which can be simplified further using Lemmas 2.2 and 2.3. Considering tangent vectors of the form X = h_{1} ⊗ e_{1} and Y = h_{2} ⊗ e_{2} we get:

(μ_{−}^{−} ∧ μ_{−}^{+})_{X,Y} = 2σ_{H}(h_{1}, h_{2})(e_{1}^{♯}⌟ e_{2}^{♯}⌟ + e_{2}^{♯}⌟ e_{1}^{♯}⌟ ) = 0
(μ_{+}^{−} ∧ μ_{+}^{+})_{X,Y} = 2σ_{H}(h_{1}, h_{2})(e_{1} ∧_{∘} e_{2} ∧_{∘} + e_{2} ∧_{∘} e_{1}∧_{∘}) = 0

and

(μ_{+}^{−} ∧ μ_{−}^{+})_{X,Y} = 2σ_{H}(h_{1}, h_{2})(e_{1} ∧_{∘} e_{2}^{♯}⌟ + e_{2} ∧_{∘} e_{1}^{♯}⌟ ) = 2R_{X,Y}^{E}
(μ_{−}^{−} ∧ μ_{+}^{+})_{X,Y} = 2σ_{H}(h_{1}, h_{2})(e_{1}^{♯}⌟ e_{2} ∧_{∘} + e_{2}^{♯}⌟ e_{1}∧_{∘})
= −\frac{4}{3}σ_{H}(h_{1}, h_{2})(e_{1} ∧_{∘} e_{2}^{♯}⌟ + e_{2} ∧_{∘} e_{1}^{♯}⌟ ) = −\frac{4}{3}R_{X,Y}^{E}

with e_{1}^{♯}⌟ e_{2}∧_{∘} = −e_{2} ∧_{∘} e_{1}^{♯}⌟ + σ_{E}(e_{1}, e_{2}) + \frac{1}{3}e_{1} ∧_{∘} e_{2}^{♯}⌟ on Λ_{∘}^{n−2}E. Using the same argument the calculation of the last matrix entry reduces to

(μ_{−}^{+} ∧ μ_{+}^{−} − \frac{3}{2}μ_{+}^{+} ∧ μ_{−}^{−})_{X,Y}

= 2(h_{1} · h_{2}^{♯}⌟_{∘} ⊗ (−e_{2} ∧_{∘} e_{1}^{♯}⌟ +σ_{E}(e_{1}, e_{2})−e_{1} ∧_{∘} e_{2}^{♯}⌟ ) − h_{2} · h_{1}^{♯}⌟_{∘} ⊗ (−e_{1} ∧_{∘} e_{2}^{♯}⌟ +σ_{E}(e_{2}, e_{1})−e_{2} ∧_{∘} e_{1}^{♯}⌟ ))
= 2σ_{E}(e_{1}, e_{2})(h_{1} · h_{2}^{♯}⌟_{∘} + h_{2} · h_{1}^{♯}⌟_{∘}) ⊗ id + 2(h_{2} · h_{1}^{♯}⌟_{∘} − h_{1} · h_{2}^{♯}⌟_{∘}) ⊗ (e_{1} ∧_{∘} e_{2}^{♯}⌟_{∘} + e_{2} ∧_{∘} e_{1}^{♯}⌟_{∘})
= 2(R^{H} + R^{E})_{X,Y}

using h_{2} · h_{1}^{♯}⌟_{∘} − h_{1} · h_{2}^{♯}⌟_{∘} = σ_{H}(h_{1}, h_{2}) on H. From Lemma 2.4 we know the explicit form of the curvature tensor R. Combining this with the above computations of the matrix entries in formula (4.10) for R^{Killing} we have

R^{Killing} = R + \frac{κ}{16n(n+2)} ( \begin{matrix}2R^{E} & 0 & 0 \\ 0 & 2(R^{H} + R^{E}) & 0 \\ 0 & 0 & 2R^{E} \end{matrix} ) = R^{hyper}

because R^{H} annihilates Λ_{∘}^{n}E and Λ_{∘}^{n−2}E. □

This proposition shows that there exist quaternionic Killing spinors on the quaternionic projective space, in fact, R^{hyper} vanishes identically on ℍP^{n}, which as a quaternionic Kähler manifold is simply connected:

Corollary 4.1 The vector bundle S^{Killing}(ℍP^{n}) is trivial. Any parallel section projects to a nontrivial eigenspinor on ℍP^{n} with minimal eigenvalue.

Let ψ = (ψ_{0}, ψ_{1}, ψ_{−}) be a quaternionic Killing spinor. We will now show how to construct eigenfunctions of the Laplace operator as linear combinations of the length functions of the three components. Similar constructions were already considered in the Riemannian and the Kähler case. The main tool is the following formula which holds for any section ψ of an hermitean vector bundle with hermitean connection ∇:

Δ |ψ|^{2} = 2 Re (∇^{∗}∇ ψ, ψ) − 2|∇ ψ|^{2}.

From this formula and the quaternionic Killing equations it is easy to derive

Δ ( \begin{matrix}|ψ_{0}|^{2} \\ |ψ_{1}|^{2} \\ |ψ_{−}|^{2} \end{matrix} ) = \frac{2λ^{2}}{n+3} ( \begin{matrix}1 & − 1 & 0 \\ − \frac{n+3}{4n} & 1 & −\frac{6(n+4)}{n+3} \\ 0 & − \frac{(n+3)(n−1)}{8n^{2}} & \frac{n+4}{n} \end{matrix} ) ( \begin{matrix}|ψ_{0}|^{2} \\ |ψ_{1}|^{2} \\ |ψ_{−}|^{2} \end{matrix} ) .

Diagonalizing the above matrix leads to the definition of the following functions

\begin{aligned}f_{0} & := & \frac{n+3}{4n} |ψ_{0}|^{2} + |ψ_{1}|^{2} + \frac{6n}{n+3} |ψ_{−}|^{2} \\ f_{1} & := & − \frac{n+3}{4n} |ψ_{0}|^{2} + \frac{1}{n} |ψ_{1}|^{2} + \frac{2(n+4)}{n+3} |ψ_{−}|^{2} \\ f_{2} & := & \frac{n+3}{4n} |ψ_{0}|^{2} − \frac{n+3}{n} |ψ_{1}|^{2} + \frac{6(n+4)}{n−1} |ψ_{−}|^{2}. \end{aligned}

Proposition 4.3 If they do not vanish identically, the functions f_{0}, f_{1}, f_{2} are eigenfunctions of the Laplace operator and the eigenvalues are the first three eigenvalues of the Laplace operator on the quaternionic projective space. More precisely, f_{0} is a constant and

Δ f_{1} = \frac{κ}{2n} \frac{n+1}{n+2} f_{1} and Δ f_{2} = \frac{κ}{2n} \frac{2n+3}{n+2} f_{2}.

That f_{0} is constant suggests that ∇^{Killing} is an hermitean connection with respect to a modified scalar product on the bundle S^{Killing}(M) defined for two sections ψ = (ψ_{0}, ψ_{1}, ψ_{−}) and ϕ = (ϕ_{0}, ϕ_{1}, ϕ_{−}) by

⟨ψ, ϕ⟩ = \frac{n+3}{4n} (ψ_{0}, ϕ_{0}) + (ψ_{1}, ϕ_{1}) + \frac{6n}{n+3} (ψ_{−}, ϕ_{−}).

Proposition 4.4 The connection ∇^{Killing} is hermitean with respect to ⟨, ⟩.

Proof. We have ∇_{X}^{Killing} = ∇_{X} + A_{X}. Since ∇ is an hermitean connection on the different components of S^{Killing}(M) we only have to prove that A_{X} is a skew–hermitean endomorphism on S^{Killing}(M) with respect to ⟨, ⟩ for all real tangent vectors X or equivalently Re ⟨A_{X}ψ, ψ⟩ = 0 for all ψ. This is straightforward use of the general formulas (cf. [KSW97-1]):

(μ_{+}^{−}(X) ψ_{1}, ψ_{0}) = − (ψ_{1}, μ_{−}^{+}(\overline{X}) ψ_{0})  and  (μ_{+}^{+}(X) ψ_{−}, ψ_{1}) = (ψ_{−}, μ_{−}^{−}(\overline{X}) ψ_{1}).  □

This proposition shows in particular that the usual construction of a Killing vector field X_{ψ} from a Killing spinor ψ in the Riemannian or Kähler case goes through in the quaternionic Kähler case, too. In fact, the formula

⟨Z, X_{ψ}⟩ = i⟨A_{Z}ψ, ψ⟩ = −i⟨∇_{Z}ψ, ψ⟩

defines a real vector field X_{ψ} such that

⟨Z, ∇_{Y} X_{ψ}⟩ = Y ⟨Z, X_{ψ}⟩ − ⟨∇_{Y} Z, X_{ψ}⟩
= i⟨A_{Z}∇_{Y} ψ, ψ⟩ + i⟨A_{Z}ψ, ∇\overline{Y} ψ⟩
= i⟨A_{Y} ψ, A\overline{Z}ψ⟩ − i⟨A_{Z}ψ, A\overline{Y} ψ⟩

is skew in Y and Z. It is a well–known fact that the first non–zero eigenvalue of the Laplace operator on a compact quaternionic Kähler manifold of scalar curvature κ > 0 is greater or equal to \frac{κ}{2n} \frac{n+1}{n+2} and that equality is attained if and only if the manifold is isometric to the quaternionic projective space (cf. [AlMa95] or [LeB95]). Hence, the above construction yields the following proposition:

Proposition 4.5 Let (M, g) be a compact quaternionic Kähler spin manifold which admits a quaternionic Killing spinor such that the associated function f_{1} is not identically zero. Then M is isometric to the quaternionic projective space.

5 The Killing Curvature on Wolf Spaces

In this section we determine the eigenvalues of the hyperkähler part of the curvature tensor considered as a symmetric bilinear form on 𝔰𝔭(n) for the Wolf spaces, i. e. the symmetric compact quaternionic Kähler manifolds. In particular we show that this symmetric bilinear form is always non–degenerate except for the quaternionic projective space and conclude:

Theorem 5.1 Let M be a Wolf space other than ℍP^{n}, then the values of R^{hyper} span 𝔰𝔭(n) at every point p ∈ M in the following sense

span{R_{X,Y}^{hyper} with X, Y ∈ T_{p}M} = 𝔰𝔭(n).

Thus there are no parallel sections of S^{Killing}(M), because if n > 2 there are no 𝔰𝔭(n)–invariant elements in the representation Λ_{∘}^{n}E ⊕ (H ⊗ Λ_{∘}^{n−1}E) ⊕ Λ_{∘}^{n−2}E. In case n = 2 there is up to scalar a unique 𝔰𝔭(n)- invariant element and in consequence the curvature R^{hyper} vanishes on sections of the trivial line bundle Λ_{∘}^{0}E ⊂ S^{Killing}(M). Nevertheless no nontrivial section of this line bundle is parallel with respect to the Killing connection. This shows that there are no quaternionic Killing spinors on Wolf spaces besides ℍP^{n}.

5.1 The Curvature Endomorphism on Symmetric Spaces

Let G/K be a symmetric space without euclidean factors and 𝔤 = 𝔨 ⊕ 𝔭 the corresponding decomposition of the semisimple Lie algebra 𝔤 of G into eigenspaces of the Cartan–involution. The Killing–form B of 𝔤 is non– degenerate on 𝔤 and assumed to be either negative or positive definite on 𝔭. The Riemannian metric on G/K is defined accordingly by g = ∓B, the upper sign corresponding to the negative definite or compact case, the lower to the positive definite or non–compact case. Despite the choice of metric the inverse isomorphisms ♯ : 𝔭 → 𝔭^{∗} and ♭ : 𝔭^{∗} → 𝔭 are always taken with respect to the Killing–form.

The decomposition of 𝔤 = 𝔨 ⊕ 𝔭 allows to define two partial Killing–forms B_{𝔨} and B_{𝔭} for X, Y ∈ 𝔤:

B_{𝔨}(X, Y) = tr_{𝔨}(ad_{X} ∘ ad_{Y}) B_{𝔭}(X, Y) = tr_{𝔭}(ad_{X} ∘ ad_{Y})

with B = B_{𝔨} + B_{𝔭} by definition. It is well known and easy to prove that B_{𝔨} and B_{𝔭} are symmetric, vanish on 𝔨 × 𝔭 and agree on 𝔭 × 𝔭, in particular B_{𝔨} = B_{𝔭} = \frac{1}{2}B on 𝔭 × 𝔭. However, no such simple relation is true on 𝔨 × 𝔨. In fact, if

𝔨 = ⨁_{i=1}^{r} 𝔨_{i}

is the orthogonal decomposition of 𝔨 into (center and) simple ideals, by Schur’s Lemma there exist constants l_{i} ∈ ℝ such that

B_{𝔨}|_{𝔨_{i}×𝔨_{i}} = B_{𝔨_{i}} = l_{i} B|_{𝔨_{i}×𝔨_{i}}.

Let L be the endomorphism of 𝔨 defined by L|_{𝔨_{i}} = l_{i} id , then the relation between the partial Killing–forms on 𝔨 × 𝔨 can be expressed as follows:

B_{𝔨}(K_{1}, K_{2}) = B(LK_{1}, K_{2}) B_{𝔭}(K_{1}, K_{2}) = B((id − L)K_{1}, K_{2}).

Next we define the cobracket Δ : 𝔨 → Λ^{2}𝔭 by

B(ΔK, X ∧ Y) := B(K, [X, Y]) = B([K, X], Y) = B(X^{♯}⌟ ΔK, Y)

for all X, Y ∈ 𝔭, in particular Δ is a Lie algebra homomorphism of 𝔨 into 𝔰𝔬 𝔭 ≅ Λ^{2}𝔭. The cobracket may be

used to define an endomorphism of 𝔨 by 𝔨 ⟶^{Δ} Λ^{2}𝔭 ⟶^{[,]} 𝔨.

Lemma 5.1 The endomorphism above equals \frac{L − id}{2} , compare [WZi93].

To prove this lemma one needs to expand the cobracket in terms of a dual pair of bases {E_{i}} and {dE_{i}} of 𝔭 and 𝔭^{∗}, namely ΔK = ∑_{i<j} B([K, E_{i}], E_{j})dE_{i}^{♭} ∧ dE_{j}^{♭}:

B([ΔK_{1}] , K_{2}) = B(ΔK_{1}, ΔK_{2}) = \frac{1}{2} ∑_{i,j} B([K_{1}, E_{i}], E_{j})B([K_{2}, dE_{i}^{♭}], dE_{j}^{♭})

= −\frac{1}{2} ∑_{i} B([K_{2}, [K_{1}, E_{i}]], dE_{i}^{♭}) = B(\frac{L − id}{2} K_{1}, K_{2}).

The extensions of the metric g = ∓B and the Killing–form agree on Λ^{2}𝔭. Thus we may define the curvature endomorphism ρ : Λ^{2}𝔭 → Λ^{2}𝔭 by

B(ρ(X ∧ Y), Z ∧ W) = B(R_{X,Y} Z, W) = −B([X, Y], [Z, W]) = −B(Δ[X, Y], Z ∧ W)

i. e. ρ = −Δ ∘ [,]. Together with [,] ∘ Δ = \frac{L−id}{2} it follows that ρ is diagonalizable and may be identified with \frac{id −L}{2} on the image Δ𝔨 of 𝔨, extended by zero on the orthogonal complement. Note that due to B(R_{X,Y} Z, W) = ±g(R_{Y,X}Z, W) the definition of ρ is the standard one only in the compact case. With this in mind the scalar curvature κ of G/K becomes:

±κ = 2 tr_{Λ^{2}𝔭}ρ = tr_{𝔨}(id − L).

Combined with the well known formula for the Ricci curvature

Ric(X, Y) = ∑_{i} B(R_{E_{i},X}Y, dE_{i}^{♭}) = − ∑_{i} B(dE_{i}^{♭}, [Y, [X, E_{i}]])

= −B_{𝔭}(Y, X) = ±\frac{1}{2}g(X, Y)

implying κ = ±\frac{dim 𝔭}{2} this formula relates the dimensions of 𝔭 and of the 𝔨_{i} with the eigenvalues of L:

Lemma 5.2

±κ = \frac{dim 𝔭}{2} = tr_{𝔨}(id − L).

5.2 The Eigenvalues of the Curvature Endomorphisms

Lemma 5.3 Let 𝔤 = 𝔨 ⊕ 𝔭 with 𝔨 = 𝔰𝔭(1) ⊕ 𝔨^{0} be the Lie algebra decomposition of a Wolf space G/K of quaternionic dimension n = \frac{1}{4}dim 𝔭. Then the Killing–forms B of 𝔤 restricted to 𝔰𝔭(1) and B_{𝔰𝔭(1)} of 𝔰𝔭(1) are related by

B_{𝔰𝔭(1)} = \frac{2}{n + 2}B,

i. e. the constant l_{𝔰𝔭(1)} is always \frac{2}{n+2}.

Proof. From Wolf’s construction there exists a base I, J, K of 𝔰𝔭(1) satisfying [I, J] = 2K etc. such that ad_{I} is a complex structure on 𝔭. As 𝔨^{0} centralizes 𝔰𝔭(1) the trace of ad_{I} ∘ ad_{I} on 𝔤 is easily computed:

B(I, I) = trace on 𝔰𝔭(1) + trace on 𝔭 = −8 − 4n.

On the other hand B_{𝔰𝔭(1)}(I, I) = −8 = \frac{2}{n+2}B(I, I). □

This result together with Lemma 5.2 will be used in the sequel to calculate the curvature endomorphisms ρ of all Wolf–spaces:

Sp(n + 1)/Sp(1) × Sp(n): On the quaternionic projective spaces 𝔨 decomposes into 𝔰𝔭(1) ⊕ 𝔰𝔭(n). As 𝔰𝔭(n) is the compact form of 𝔰𝔭 ℂ^{2n} the Killing–form is (2n + 2)tr_{ℂ^{2n}}. Now 𝔰𝔭(n) is embedded in 𝔰𝔭(n + 1) via the embedding of the defining representations ℂ^{2n} ↪ ℂ^{2n+2}. In consequence, the Killing–forms are related by the constant l_{𝔰𝔭(n)} = \frac{2n+2}{2n+4} = \frac{n+1}{n+2}. Thus the curvature endomorphism of the quaternionic projective space reads:

ρ = \frac{n}{2(n + 2)}proj_{𝔰𝔭(1)} + \frac{1}{2(n + 2)}proj_{𝔰𝔭(n)}.

SU(n + 2)/S(U(2) × U(n)): On the complex Grassmannian of 2-planes 𝔨 decomposes into 𝔰𝔭(1)⊕𝔰𝔲(n)⊕ℝ. The constant l_{ℝ} is certainly 0. As 𝔰𝔲(n) is the compact form of 𝔰𝔩(n) the Killing–form is 2n tr_{ℂ^{n}} and one concludes that l_{𝔰𝔲(n)} = \frac{2n}{2n+4} = \frac{n}{n+2}. The curvature endomorphism of the complex Grassmannian is thus:

ρ = \frac{n}{2(n + 2)}proj_{𝔰𝔭(1)} + \frac{1}{n + 2}proj_{𝔰𝔲(n)} + \frac{1}{2}proj_{ℝ}.

SO(n + 4)/S(O(4) × O(n)): On the real Grassmannian of 4-planes 𝔨 decomposes into 𝔰𝔭(1)⊕𝔰̃𝔭(1)⊕𝔰𝔬(n). The Killing–form of 𝔰𝔬(n) is (n − 2)tr_{ℂ^{n}}. Arguing in the same way as before one finds l_{𝔰𝔬(n)} = \frac{n−2}{n+2}. The last constant is then calculated with the help of Lemma 5.2 and agrees with l_{𝔰𝔭(1)} = l_{𝔰̃𝔭(1)} = \frac{2}{n+2}, because either of the two subalgebras could be used to define the quaternionic structure. The curvature endomorphism is:

ρ = \frac{n}{2(n + 2)}proj_{𝔰𝔭(1)} + \frac{n}{2(n + 2)}proj_{𝔰̃𝔭(1)} + \frac{2}{n + 2}proj_{𝔰𝔬(n)}.

G_{2}/SO(4): In this case dim 𝔭 = 8 or n = 2 and 𝔨 = 𝔰𝔭(1)⊕𝔰̃𝔭(1). However 𝔰𝔭(1) and 𝔰̃𝔭(1) are definitely distinct, 𝔰̃𝔭(1) does not define a proper quaternionic structure. In fact their constants differ, because Lemma 5.2 implies l_{𝔰̃𝔭(1)} = \frac{1}{6}.

ρ = \frac{1}{4}proj_{𝔰𝔭(1)} + \frac{5}{12}proj_{𝔰̃𝔭(1)}.

F_{4}/Sp(1)Sp(3): This space is not spin, however Theorem 5.1 may be of independent interest. With dim 𝔭 = 28 the quaternionic dimension is n = 7 and 𝔨 = 𝔰𝔭(1) ⊕ 𝔰𝔭(3). Using Lemma 5.2 one finds l_{𝔰𝔭(3)} = \frac{4}{9} and

ρ = \frac{7}{18}proj_{𝔰𝔭(1)} + \frac{5}{18}proj_{𝔰𝔭(3)}.

E_{6}/Sp(1)SU(6): With dim 𝔭 = 40 the quaternionic dimension is n = 10, 𝔨 = 𝔰𝔭(1) ⊕ 𝔰𝔲(6), l_{𝔰𝔲(6)} = \frac{1}{2} and

ρ = \frac{5}{12}proj_{𝔰𝔭(1)} + \frac{1}{4}proj_{𝔰𝔲(6)}.

E_{7}/Sp(1)Spin(12): With dim 𝔭 = 64 the quaternionic dimension is n = 16, 𝔨 = 𝔰𝔭(1) ⊕ 𝔰𝔬(12), l_{𝔰𝔬(12)} = \frac{5}{9} and

ρ = \frac{4}{9}proj_{𝔰𝔭(1)} + \frac{2}{9}proj_{𝔰𝔬(12)}.

E_{8}/Sp(1)E_{7}: With dim 𝔭 = 112 the quaternionic dimension is n = 28 and 𝔨 = 𝔰𝔭(1) ⊕ 𝔢_{7}, where dim 𝔢_{7} = 133. Applying Lemma 5.2 one finds l_{𝔢_{7}} = \frac{3}{5} and thus

ρ = \frac{7}{15}proj_{𝔰𝔭(1)} + \frac{1}{5}proj_{𝔢_{7}}.

To present the argument leading to Theorem 5.1 we recall that for any Wolf space we have a K-invariant subalgebra 𝔰𝔭(n) ⊂ 𝔰𝔬 𝔭 defined as the centralizer of the subalgebra 𝔰𝔭(1) ⊂ 𝔰𝔬 𝔭 defining the quaternionic structure and by definition 𝔨^{0} ⊂ 𝔰𝔭(n). In particular, the curvature endomorphism ρ^{ℍP^{n}} of the quaternionic projective space is well defined on any Wolf space. Bridging the gap between Lie algebra– and E–H– formalism, it is even possible to identify the “curvature” endomorphisms corresponding to R^{H} and R^{E} of Lemma 2.4 with:

ρ^{H} = −2n proj_{𝔰𝔭(1)} and ρ^{E} = −2 proj_{𝔰𝔭(n)}.

The scalar curvature of a compact Wolf space is κ = 2n and Lemma 2.4 implies for the curvature tensor:

ρ = −\frac{κ}{8n(n + 2)}(ρ^{H} + ρ^{E}) + ρ^{hyper} = \frac{n}{2(n + 2)}proj_{𝔰𝔭(1)} + \frac{1}{2(n + 2)}proj_{𝔰𝔭(n)} + ρ^{hyper}.

We conclude that the kernel of the hyperkähler part ρ^{hyper} of the curvature endomorphism ρ restricted to 𝔰𝔭(n) ⊂ 𝔰𝔬 𝔭 is the eigenspace of ρ with eigenvalue \frac{1}{2(n+2)} in 𝔰𝔭(n). Looking at the list of curvature endomorphisms above one verifies that this eigenvalue does only occur for the curvature endomorphism of the quaternionic projective space itself. □

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