The First Eigenvalue of the Dirac Operator on Quaternionic Kähler Manifolds

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

August 16, 2002

Abstract

In [KSW97a] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler manifolds. In the present article we show that the only manifolds in the limit case, i. e. the only manifolds where the lower bound is attained as an eigenvalue, are the quaternionic projective spaces. We use the equivalent formulation in terms of the quaternionic Killing equation introduced in [KSW97b] and show that a nontrivial solution defines a parallel spinor on the associated hyperkähler manifold.

AMS Subject Classification: 53C25, 58G25

Contents

1 Introduction 2 2 Semiquaternionic Vector Spaces and Representations 3 3 Principal Bundles on Quaternionic Kähler Manifolds 5 4 The Levi–Civitá Connection on S and M̂ 8 5 Quaternionic Killing Spinors 10 6 The Geometry of M̂ and Application to Spinors 13

6.1 The Hyperkähler Structure of M̂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2 Reinterpretation of the Quaternionic Killing Equation . . . . . . . . . . . . . . . . . . . . . . 15

7 Proof of the Theorem 16 A Spinors on Cones 17 B The Curvature Tensor of M̂ 17

∗Supported by the SFB 256 ‘Nichtlineare partielle Differentialgleichungen’ †Supported by the Max–Planck–Institut für Mathematik ‡Supported by the SFB 256 ‘Nichtlineare partielle Differentialgleichungen’

1 Introduction

The square of the first eigenvalue of the Dirac operator on the sphere S^{n} of scalar curvature κ is \frac{κ}{4} \frac{n}{n−1}. In [Fri80] Friedrich showed that this eigenvalue is a universal lower bound for all eigenvalues on an arbitrary compact Riemannian spin manifold (M^{n}, g) with positive scalar curvature κ in the following sense: all eigenvalues of the Dirac operator satisfy

λ^{2} ≥ \frac{min_{M}κ}{4} \frac{n}{n − 1} .

An eigenspinor realizing this lower bound is characterized by a special differential equation called the Killing equation. Conversely, on manifolds admitting Killing spinors, i. e. nontrivial solutions of this Killing equation, the lower bound is realized as an eigenvalue. These manifolds have been characterized by C. Bär [Bär93] translating the Killing equation on M into the equation of a parallel spinor on the cone M̂ = ℝ^{+} ×_{t^{2}} M. The existence problem of solutions of the Killing equation was thus reduced to the description of manifolds with parallel spinors by M. Wang [Wan89].

Despite the fact that Friedrich’s estimate is sharp, it is not optimal if M is assumed to have additional geometric structure, namely special holonomy. Due to a result of O. Hijazi and A. Lichnerowicz ([Hij84], [Lic87]), there is no solution of the Killing equation if M possesses a non–trivial parallel k–form, k ≠ 0, n. There are two canonical classes of such manifolds which in addition have positive scalar curvature: Kähler manifolds and quaternionic Kähler manifolds.

The eigenvalue estimate for Kähler manifolds has been improved by K.–D. Kirchberg in [Kir86] and [Kir90] (see also [Lic90] and [Hij94]). Again, this estimate is sharp: the lower bound is attained as first eigenvalue on the complex projective space ℂP^{m} resp. on its product with a flat 2–torus in odd resp. even complex dimensions. On Kähler manifolds of odd complex dimension, a spinor with smallest possible eigenvalue is characterized by a suitable modification of the Killing equation, the Kählerian Killing equation. A. Moroianu showed in [Mor95] that a Kählerian Killing spinor defines an ordinary Killing spinor on the canonical S^{1}– bundle over M. Hence, the holonomy argument of Bär’s work can be used to study the limit case. In even complex dimensions, the problem of characterizing the limit case is still open. Nevertheless, there are partial results by A. Moroianu [Mor97] and A. Lichnerowicz [Lic90].

A quaternionic Kähler manifold is an oriented 4n–dimensional Riemannian manifold with n ≥ 2 whose holonomy group is contained in the subgroup Sp(1) · Sp(n) ⊂ SO(4n). Equivalently they are characterized by the existence of a certain parallel 4–form Ω, the so–called fundamental or Kraines form (cf. [Bon67], [Kra66]). All quaternionic Kähler manifolds are Einstein with constant scalar curvature (cf. [Ale68b] or [Ish74]) and possess a unique spin structure if n is even, whereas for odd n only the quaternionic projective spaces are spin (cf. [Sal82]). In [KSW97a] we proved that on a compact quaternionic Kähler spin manifold (M^{4n}, g) of positive scalar curvature κ the eigenvalues λ of the Dirac operator satisfy

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

As in the Riemannian or Kähler case, this estimate is sharp, and the lower bound is attained as first eigenvalue on the quaternionic projective space (cf. [Mil92]).

The natural task is to study the limit case and to find all manifolds which have \frac{κ}{4} \frac{n+3}{n+2} in the spectrum of D^{2}. A first step in this direction was taken in [KSW97b], where we introduced the equation characterizing an eigenspinor with this particular eigenvalue. A new feature of this quaternionic Killing equation is that it involves not only the eigenspinor but also an auxiliary section of an additional bundle, which is not itself a spinor. We used it to show that no compact symmetric quaternionic Kähler manifolds besides the quaternionic projective spaces carry quaternionic Killing spinors. In the present article we prove the following more general

Theorem 1 Let M be a compact quaternionic Kähler manifold of quaternionic dimension n and positive scalar curvature κ. If there is an eigenspinor for the Dirac operator with eigenvalue λ satisfying

λ^{2} = \frac{κ}{4} \frac{n + 3}{n + 2} ,

then M is isometric to the quaternionic projective space.

For the proof we follow the approach of C. Bär and A. Moroianu. We consider the canonical SO(3)–bundle S associated with any quaternionic Kähler manifold. Introducing an appropriate metric on the total space S the warped product M̂:= ℝ^{+} ×_{t^{2}} S has a natural hyperkähler metric. Reformulating the Killing equation in terms of equivariant functions on the Sp(1) · Sp(n)–frame bundle of the quaternionic Kähler manifold we show that a quaternionic Killing spinor induces a Killing spinor on S and a parallel spinor on M̂. The result of M. Wang then implies that the hyperkähler manifold M̂ has to be locally isometric to ℍ^{n+1} forcing M to be isometric to the quaternionic projective space.

We would like to thank A. Swann for several hints and comments and W. Ballmann for encouragement and support.

2 Semiquaternionic Vector Spaces and Representations

The tangent space of a quaternionic Kähler manifold is not a priori a quaternionic left vector space because in general the three local complex structures are not defined globally. This ambivalence gives rise to the weaker notion of a semiquaternionic structure on a real vector space:

Definition. A semiquaternionic structure on a real vector space T is a subalgebra Q ⊂ End T with id_{T} ∈ Q and Q ≅ ℍ as ℝ–algebras. It is said to be adapted to a euclidean scalar product ⟨, ⟩ on T if

⟨ qt_{1}, t_{2} ⟩ = ⟨ t_{1}, \overline{q}t_{2} ⟩

for all t_{1}, t_{2} ∈ T and q ∈ Q, where \overline{q} denotes conjugation defined by Q = ℝ ⊕ Im Q := ℝ id_{T} ⊕ [Q, Q].

Thus, choosing an isomorphism from ℍ to Q makes T a quaternionic left vector space, however no particular isomorphism is preferred. Accordingly, the notion of quaternionic linear map has to be refined:

Definition. An ℝ–linear map f : T → T′ between vector spaces T, T′ with semiquaternionic structures Q, Q′ is semilinear, if there exists an isomorphism of ℝ–algebras f^{Q} : Q → Q′ such that f(qt) = f^{Q}(q)f(t) for all t ∈ T and q ∈ Q. If f is semilinear and not identically zero f^{Q} is uniquely defined, because id_{T′} ∈ Q′ and Q′ ≅ ℍ implies that every non–zero endomorphism in Q′ is invertible.

If T is a euclidean vector space with an adapted semiquaternionic structure, then the group of all semilinear isometries of T is isomorphic to Sp(1) · Sp(n) := Sp(1) × Sp(n)/ℤ_{2}. Choosing a particular isomorphism makes T a true representation of Sp(1) · Sp(n) and any two representations T, T′ defined this way are intertwined by a semilinear isometry, which is unique up to sign. For this reason we will call any such T the defining representation of Sp(1) · Sp(n) with a choice of isomorphism tacitly understood. It turns out that the defining representation T comes along with a preferred isomorphism ℍ → Q given by the infinitesimal action of i, j, k ∈ 𝔰𝔭(1) ≅ Im ℍ on T.

Similarly, one may construct the defining representation of the group Sp(n) of unitary quaternionic n × n–matrices. If E is a complex vector space of dimension 2n endowed with a symplectic form σ_{E} and an adapted positive quaternionic structure J, i. e. σ_{E}( Je_{1}, Je_{2} ) = \overline{σ_{E}( e_{1}, e_{2} )} for all e_{1}, e_{2} ∈ E and σ_{E}( e, Je) > 0 for all 0 ≠ e ∈ E, then the group of all ℂ–linear symplectic transformations of E commuting with J is isomorphic to Sp(n). Choosing a particular isomorphism makes E a true Sp(n)– representation and any two representations E, E′ defined this way are intertwined by a ℂ–linear symplectic map preserving the quaternionic structure, which is unique up to sign. For this reason we will call any such E the defining representation of Sp(n). Note that the Lie algebra of all infinitesimal symplectic transformations of E is canonically isomorphic to Sym^{2}E with e_{1}e_{2} acting as the endomorphism σ_{E}(e_{1}, ·)e_{2} + σ_{E}(e_{2}, ·)e_{1}, and elements of Sym^{2}E commute with J if and only if they are real with respect to the real structure Sym^{2}J. Hence, the defining representation comes along with a canonical real Sp(n)–equivariant isomorphism ℂ ⊗_{ℝ} 𝔰𝔭(n) → Sym^{2}E of Lie algebras. We will denote the defining representation of Sp(1) by H.

There are several possibilities to give explicit realizations of the representations introduced above. In calculations and proofs below we will use the following standard picture, differing somewhat from Salamon’s conventions (cf. [Sal82]). Consider the space of row vectors ℍ^{n} over the quaternions with complex and quaternionic structure given by multiplication with i and j from the left. The group of unitary quaternionic matrices Sp(n) := {A ∈ M_{n,n}ℍ : A^{H}A = 1} acts on ℍ^{n} from the left by multiplying with A^{H} from the right. Thus, it commutes with the complex and quaternionic structure and preserves the linear form

σ_{ℍ^{n}}( v_{1}, v_{2} ) := [ v_{1}v_{2}^{H}j]_{ℂ} ,

where [q]_{ℂ} := \frac{1}{2}(q −iqi) ∈ ℂ is the ℂ–part of q ∈ ℍ. The ℂ–part is obviously ℂ–bilinear, i. e. [xqy]_{ℂ} = xy[q]_{ℂ} for all x, y ∈ ℂ ⊂ ℍ, and satisfies

[\overline{q}]_{ℂ} = \overline{[q]}_{ℂ} = [−jqj]_{ℂ} .

Using these properties it is easily checked that σ_{ℍ^{n}} is indeed ℂ–bilinear symplectic and that the quaternionic structure is adapted and positive. In this way ℍ^{n} becomes the defining representation of Sp(n).

With a slight modification of the construction above we can make ℍ^{n} the defining representation of Sp(1) · Sp(n), too. The scalar multiplication with q ∈ ℍ from the left on the row vectors in ℍ^{n} determines a subspace Q ⊂ End ℍ^{n}, which obviously is a semiquaternionic structure adapted to the standard scalar product on ℍ^{n} given by

⟨ v_{1}, v_{2} ⟩ := Re v_{1}v_{2}^{H} = Re σ_{ℍ^{n}}( v_{1}, jv_{2} ) .

The group Sp(1) · Sp(n) = {z · A : z ∈ Sp(1) and A ∈ Sp(n)} acts on ℍ^{n} from the left through semilinear isometries by (z · A)v := zvA^{H}.

An important point is particularly obvious in this standard picture and in consequence true for every defining representation of Sp(1) · Sp(n). The infinitesimal action of i, j, k ∈ 𝔰𝔭(1) ⊂ 𝔰𝔭(1) ⊕ 𝔰𝔭(n) on T defines a canonical Sp(1) · Sp(n)–equivariant isomorphism of algebras ℍ → Q making T a quaternionic left vector space. Thus, the natural homomorphism Sp(1) · Sp(n) → Aut Q, z · A ↦ (z · A)^{Q} is trivial on the subgroup Sp(n) and descends to an isomorphism SO(3) := Sp(1)/ℤ_{2} ≅ Aut Q on the complementary subgroup Sp(1) sending z · 1 ∈ Sp(1) to z^{Q} := (z · 1)^{Q}.

This observation allows a construction which becomes fundamental for quaternionic Kähler geometry once it is “gauged”. Let T be the defining representation of Sp(1) · Sp(n) and T′ an arbitrary euclidean vector space with an adapted semiquaternionic structure Q′. The representations of Sp(1) · Sp(n) on T and SO(3) on Q define simply transitive right group actions on

P = {f : T → T′ semilinear isometry}

and

S = {f^{Q} : Q → Q′ isomorphism of algebras}

such that the natural projection f ↦ f^{Q} is Sp(1)–equivariant. Fixing a base point in P to identify P with Sp(1) · Sp(n) we get the following diagram:

[figure: P; S; {T′}; z · A; z^{Q}; 1]

The quaternionic structure on T can be used to construct two important relations between the defining representations H, E and T of Sp(1), Sp(n) and Sp(1) · Sp(n):

Lemma 2.1 Let T be the defining representation of Sp(1) · Sp(n). Define the complex and quaternionic structure on T by the infinitesimal action of i, j (and k) in 𝔰𝔭(1) ≅ Im ℍ. With the ℂ–bilinear symplectic form σ_{T} ( t_{1}, t_{2} ) = ⟨ jt_{1}, t_{2} ⟩ + i⟨ kt_{1}, t_{2} ⟩ the vector space T becomes the defining representation of Sp(n). Conversely, the complex and quaternionic structure of the defining representation E of Sp(n) generate a subalgebra Q in the ℝ–linear endomorphisms of E, which is a semiquaternionic structure adapted to the scalar product ⟨ ·, · ⟩ := Re σ_{E}( ·, J·) making E the defining representation of Sp(1) · Sp(n). In particular, there is up to sign a unique Sp(n)–equivariant, ℂ–linear symplectic isomorphism Ψ : T → E preserving the quaternionic structure. In the standard picture this isomorphism is simply the identity.

This isomorphism has the disadvantage of spoiling the Sp(1)–action on T. Consequently, it is impossible to use it directly on quaternionic Kähler manifolds. Nevertheless, we may use it to define a family of Sp(n)– equivariant isomorphisms ℂ ⊗_{ℝ} T → H ⊗_{ℂ} E depending on the choice of a canonical base p, q := Jp of H satisfying σ_{H}(p, q) = 1 by

Φ : x ⊗_{ℝ} t ⟼ \frac{1}{\sqrt{2}}(xp ⊗_{ℂ} Ψ(t) + xq ⊗_{ℂ} JΨ(t)) .  (2.2)

All these isomorphisms are real with respect to the real structure J ⊗_{ℂ} J on H ⊗_{ℂ} E and isometries from ⟨ , ⟩ to σ_{H} ⊗_{ℂ} σ_{E}:

\begin{aligned}σ_{H} ⊗ σ_{E}(Φ(1 ⊗_{ℝ} t_{1}), Φ(1 ⊗_{ℝ} t_{2})) & = & \frac{1}{2}(σ_{E}(Ψ(t_{1}), JΨ(t_{2})) − σ_{E}(JΨ(t_{1}), Ψ(t_{2}))) \\ & = & \frac{1}{2}(⟨ jt_{1}, jt_{2} ⟩ + i⟨ kt_{1}, jt_{2} ⟩ + ⟨ t_{1}, t_{2} ⟩ − i⟨ it_{1}, t_{2} ⟩) = ⟨ t_{1}, t_{2} ⟩ , \end{aligned}

since JΨ(t) = Ψ(jt) and σ_{E}(Ψ(t_{1}), Ψ(t_{2})) = ⟨ jt_{1}, t_{2} ⟩ + i⟨ kt_{1}, t_{2} ⟩ by construction. It turns out that there are exactly two canonical bases p, q for which the isomorphism above is not only Sp(n)– but already Sp(1) · Sp(n)–equivariant, thus defining a more fundamental isomorphism better suited for globalization:

Lemma 2.3 Up to sign there is a unique real Sp(1) · Sp(n)–equivariant isomorphism of complex vector spaces

Φ : ℂ ⊗_{ℝ} T ≅ H ⊗_{ℂ} E ,

which is an isometry from the the ℂ–bilinear extension of ⟨ , ⟩ to σ_{H} ⊗ σ_{E}.

Proof. It is sufficient to prove this lemma in the standard picture as it translates immediately to arbitrary realizations of the defining representations. In this picture the canonical base to choose is p = j and q = −1 (or p = −j and q = 1) leading to:

\begin{aligned}Φ : & ℂ ⊗_{ℝ} ℍ^{n} & ⟶ & ℍ ⊗_{ℂ} ℍ^{n} & Φ^{−1} : & ℍ ⊗_{ℂ} ℍ^{n} & ⟶ & ℂ ⊗_{ℝ} ℍ^{n} \\ & x ⊗_{ℝ} v & ⟼ & \frac{1}{\sqrt{2}}(xj ⊗_{ℂ} v − x ⊗_{ℂ} jv) & & q ⊗_{ℂ} v & ⟼ & \frac{1}{\sqrt{2}}(1 ⊗_{ℝ} \overline{q}jv + i ⊗_{ℝ} \overline{q}kv) . \end{aligned}

(2.4)

Note that Φ^{−1}(iq ⊗_{ℂ} v) = Φ^{−1}(q ⊗_{ℂ} iv) = iΦ^{−1}(q ⊗_{ℂ} v). Thus, Φ^{−1} is well defined and ℂ–linear. As Sp(1) · Sp(n)–equivariance of Φ^{−1} is obvious and Φ is an isometry it remains to show that Φ and Φ^{−1} are mutual inverses:

(Φ^{−1} ∘ Φ)(x ⊗_{ℝ} v) = \frac{1}{2}(1 ⊗_{ℝ} (−j\overline{x}jv + \overline{x}v) + i ⊗_{ℝ} (−j\overline{x}ijv + \overline{x}iv))
= \frac{1}{2}(1 ⊗_{ℝ} 2 Re(x)v + i ⊗_{ℝ} 2 Re(−xi)v)
= x ⊗_{ℝ} v .

A direct calculation of Φ ∘ Φ^{−1} = id is more tedious, but can be done. □

3 Principal Bundles on Quaternionic Kähler Manifolds

“Gauging” the pointwise constructions of the previous section leads to the definitions of the basic objects of quaternionic Kähler geometry. However, in strict analogy with Kähler geometry one has to impose an additional integrability condition:

Definition. A quaternionic Kähler manifold is a Riemannian manifold M of dimension 4n, n ≥ 2 with an adapted semiquaternionic structure Q_{x}M ⊂ End T_{x}M on every tangent space which is respected by the Levi–Civitá connection of M:

∇Γ(Q) ⊂ Γ(T^{∗}M ⊗ Q) .

Thus, parallel transport of endomorphisms along arbitrary curves γ induces isomorphisms of ℝ–algebras Q_{γ(0)}M → Q_{γ(τ)}M, and a fortiori parallel transport of tangent vectors defines semilinear isometries T_{γ(0)}M → T_{γ(τ)}M. In particular, the Levi–Civitá connection is tangent to the reduction of the frame bundle to the principal Sp(1) · Sp(n)–bundle of semilinear orthogonal frames

P := {f : T → T_{x}M semilinear isometry}  (3.5)

with projection π_{M} : P → M, f ↦ x. Additionally, P projects Sp(1)–equivariantly to

S := {f^{Q} : Q → Q_{x}M isomorphism of algebras}  (3.6)

via π_{S} : P → S, f ↦ f^{Q}, which is in turn a principal SO(3)–bundle over M with projection π : S → M, f^{Q} ↦ x. In this way P may be considered as a principal Sp(1) · Sp(n)–bundle over M or as a principal Sp(n)–bundle over S:

[figure: P; S; M; π_{S}; π_{M}; π; f:T →T_{x}M; f^{Q}:Q→Q_{x}M; x; π_{S}; π_{M}; π]

The tangent bundle T M of M is canonically isomorphic to the bundle associated to P by the defining representation T sending the class [f, t] ∈ P ×_{Sp(1)·Sp(n)} T to f(t). Alternatively, this isomorphism can be expressed by the soldering form θ_{M} := f^{−1} ∘ (π_{M})_{∗} ∈ Γ(T^{∗}P ⊗ T) of M. Considering the fundamental isomorphism Φ : ℂ ⊗_{ℝ} T → H ⊗ E one might try to define vector bundles H and E from the defining representations H and E of Sp(1) and Sp(n). In this generality, this is only possible on the quaternionic projective spaces, because only for these manifolds the bundle P can be covered by a principal Sp(1)×Sp(n)– bundle. Nevertheless, representations of Sp(1)×Sp(n) contained in some H^{⊗p}⊗E^{⊗q} with p+q even descend to Sp(1)·Sp(n) defining vector bundles on M associated to P. In this way M carries a multitude of naturally defined vector bundles besides bundles constructed out of the tangent bundle. The usefulness of these vector bundles has been shown by [Sal82] (see also [KSW97a]). In particular, the bundle H⊗E is globally defined and canonically isomorphic to T M^{ℂ} as expressed e. g. by the soldering form θ_{M}^{H⊗E} := Φ∘θ_{M} ∈ Γ(T^{∗}P ⊗(H ⊗E)).

By definition, the Levi–Civitá connection is tangent to P and determined by a connection 1–form ω_{M} with values in the direct sum 𝔰𝔭(1) ⊕ 𝔰𝔭(n); accordingly, ω_{M} splits into ω_{M}^{𝔰𝔭(1)} ⊕ ω_{M}^{𝔰𝔭(n)}. Additionally, the Levi–Civitá connection defines a connection on the principal bundle S. Recall that horizontal lifts X^{h} ∈ T_{f_{0}}P of tangent vectors X = \frac{d}{dτ} |_{0}x_{τ} ∈ T_{x_{0}}M can be represented by curves of semilinear orthogonal frames f_{τ} : T → T_{x_{τ}} M over x_{τ} satisfying \frac{∇}{dτ} |_{0}f_{τ} = 0. Likewise the connection on S is defined by representing horizontal lifts X^{h} ∈ T_{f_{0}^{Q}}S by curves f_{τ}^{Q} : Q → Q_{x_{τ}} M of algebra isomorphisms over x_{τ} satisfying \frac{∇}{dτ} |_{0}f_{τ}^{Q} = 0. More succinctly, the connection 1–form ω on S for arbitrary tangent vectors \frac{d}{dτ} |_{0}f_{τ}^{Q} is given by

ω(\frac{d}{dτ} |_{0}f_{τ}^{Q}) := (f_{0}^{Q})^{−1} \frac{∇}{dτ} |_{0}f_{τ}^{Q} .

Using the Leibniz rule for the covariant derivative along curves, it is immediately seen that π_{S} projects horizontal vectors \frac{d}{dτ} |_{0}f_{τ} on P to horizontal vectors \frac{d}{dτ} |_{0}f_{τ}^{Q} on S, because for arbitrary q ∈ Q, t ∈ T

(\frac{∇}{dτ} |_{0}f_{τ} )(qt) = (\frac{∇}{dτ} |_{0}f_{τ}^{Q})(q) f_{0}(t) + f_{0}^{Q}(q) (\frac{∇}{dτ} |_{0}f_{τ} )(t) ,

and consequently \frac{∇}{dτ} |_{0}f_{τ} = 0 implies \frac{∇}{dτ} |_{0}f_{τ}^{Q} = 0. Since the projection π_{S} : P → S is Sp(1)–equivariant, and vectors tangent to the Sp(n)–action on P are surely vertical with respect to the projection π_{S} we conclude:

Lemma 3.7 π_{S}^{∗}ω = ω_{M}^{𝔰𝔭(1)} .

Remarkably, the curvature of this connection on S depends only on the scalar curvature κ of M. In fact, according to the classification of 𝔰𝔭(1) ⊕ 𝔰𝔭(n)–curvature tensors due to Alekseevskii (cf. [Ale68a], [Sal82]) the curvature tensor of a quaternionic Kähler manifold can be expressed in terms of the scalar curvature κ and a section ℜ of Sym^{4}E^{∗}. We have

R = −\frac{κ}{8n(n + 2)}(R^{H} + R^{E}) + R^{hyper},  (3.8)

where R^{H}, R^{E} and R^{hyper} are Sym^{2}H– or Sym^{2}E–valued 2–forms defined on sections of H ⊗ E ≅ T M^{ℂ}:

R_{h_{1}⊗e_{1},h_{2}⊗e_{2}}^{H} = σ_{E}(e_{1}, e_{2})h_{1} · h_{2} ∈ Sym^{2}H
R_{h_{1}⊗e_{1},h_{2}⊗e_{2}}^{E} = σ_{H}(h_{1}, h_{2})e_{1} · e_{2} ∈ Sym^{2}E
R_{h_{1}⊗e_{1},h_{2}⊗e_{2}}^{hyper} = σ_{H}(h_{1}, h_{2})ℜ(e_{1}, e_{2}, ·, ·) ∈ Sym^{2}E^{∗} ≅ Sym^{2}E ,

(3.9)

acting as endomorphisms on H ⊗ E. Analyzing these terms leads to the following formula for the pull–back of the curvature 2–form Ω of the connection ω to P:

Lemma 3.10 The curvature 2–form of the connection ω on S pulled back to P is given by

π_{S}^{∗}Ω = \frac{κ}{16n(n + 2)}(⟨ θ_{M} ∧ iθ_{M} ⟩i + ⟨ θ_{M} ∧ jθ_{M} ⟩j + ⟨ θ_{M} ∧ kθ_{M} ⟩k) ,

where by definition ⟨ θ_{M} ∧ iθ_{M} ⟩(X, Y) := 2⟨ θ_{M}(X), iθ_{M}(Y) ⟩.

Proof. Instead of calculating the curvature on S directly we will calculate the curvature of the bundle Sym^{2}H, which can be associated to S or P inheriting the same connection due to π_{S}^{∗}ω = ω_{M}^{𝔰𝔭(1)}. Its curvature considered as an 𝔰𝔭(1)–valued 2–form on P is thus π_{S}^{∗}Ω. Obviously only R^{H} acts non–trivially on Sym^{2}H and neglecting for a moment that it is defined for sections of vector bundles we may consider it as a real Sp(1) · Sp(n)–equivariant morphism

ℂ ⊗_{ℝ} Λ^{2}T ⟶^{≅} Λ^{2}(H ⊗ E) ⟶^{R^{H}} Sym^{2}H ⟶^{≅} ℂ ⊗_{ℝ} 𝔰𝔭(1) ,

where the first isomorphism is the extension of ℂ⊗_{ℝ} T ≅ H ⊗E and the second is the canonical isomorphism ℂ ⊗_{ℝ} 𝔰𝔭(1) ≅ Sym^{2}H, which makes H the defining representation of Sp(1). To make this isomorphism explicit in the standard picture we choose the canonical base j, −1 of ℍ with σ_{ℍ}(j, −1) = 1 and find for the infinitesimal action of i, j and k ∈ Im ℍ:

\begin{aligned}i : & 1 ↦ −i = i(−1) & j : & 1 ↦ −j & k : & 1 ↦ −k = i(−j) \\ & j ↦ k = i(j) & & j ↦ 1 & & j ↦ −i = i(−1) . \end{aligned}

Hence, the isomorphism ℂ⊗_{ℝ}𝔰𝔭(1) ≅ Sym^{2}ℍ maps i to i(1 j), j to \frac{1}{2}(j^{2}+1^{2}) and k to \frac{i}{2}(j^{2}−1^{2}). Accordingly, the morphism R^{H} reads in the standard picture

R_{1⊗_{ℝ}v_{1},1⊗_{ℝ}v_{2}}^{H} = \frac{1}{2}(R_{j⊗v_{1},j⊗v_{2}}^{H} − R_{1⊗jv_{1},j⊗v_{2}}^{H} − R_{j⊗v_{1},1⊗jv_{2}}^{H} + R_{1⊗jv_{1},1⊗jv_{2}}^{H})
= \frac{1}{2}(σ_{ℍ^{n}}(v_{1}, v_{2})j^{2} − σ_{ℍ^{n}}(jv_{1}, v_{2})1 j − σ_{ℍ^{n}}(v_{1}, jv_{2})1 j + σ_{ℍ^{n}}(jv_{1}, jv_{2})1^{2})
= Re σ_{ℍ^{n}}(v_{1}, v_{2}) · \frac{1}{2}(j^{2} + 1^{2}) + Im σ_{ℍ^{n}}(v_{1}, v_{2}) · \frac{i}{2}(j^{2} − 1^{2}) − Im σ_{ℍ^{n}}(v_{1}, jv_{2}) · i(1 j)
= −⟨ v_{1}, jv_{2} ⟩j − ⟨ v_{1}, kv_{2} ⟩k − ⟨ v_{1}, iv_{2} ⟩i

for v_{1}, v_{2} ∈ ℍ^{n}. Being Sp(1) · Sp(n)–equivariant R^{H} can be made a ℂ ⊗_{ℝ} (P ×_{Ad} 𝔰𝔭(1)) ≅ Sym^{2}H–valued 2–form in a straightforward way and becomes the R^{H} defined above. Alternatively, −\frac{κ}{8n(n+2)}R^{H} can be thought of as the horizontal Sp(1) · Sp(n)–equivariant 𝔰𝔭(1)–valued curvature form π_{S}^{∗}Ω on P and is given by the stated formula. The additional factor \frac{1}{2} comes from the definition of the wedge product. □

4 The Levi–Civitá Connection on S and M̂

Let π : S → M be the canonical SO(3)–bundle over M defined in (3.6), with connection form ω and Riemannian metric

g_{S} = \frac{16n(n+2)}{κ} B(ω, ω) + π^{∗}g_{M} ,

where B is the standard metric on 𝔰𝔭(1) ≅ Im ℍ, and κ denotes the scalar curvature of M. This metric is Einstein, and if we rescale the metrics of M and S so that κ = 16n(n+2), then (S, g_{S}) has a natural Sasakian 3–structure. The structure group of S reduces to Sp(n), and we can embed the principal Sp(n)–bundle P into the frame bundle of S in such a way that the soldering form θ_{S} ∈ Γ(T^{∗}P ⊗ (𝔰𝔭(1) ⊕ T)) of S on P is given by

θ_{S} = \sqrt{\frac{16n(n+2)}{κ}} ω_{M}^{𝔰𝔭(1)} ⊕ θ_{M} .  (4.11)

In this way the Riemannian metric of S is associated to the standard metric B ⊕ ⟨ , ⟩ on 𝔰𝔭(1) ⊕ T. In terms of covariant derivatives the Levi–Civitá connection of g_{S} is easily computed, and we obtain:

Lemma 4.12 Let U, V be vertical vector fields given as fundamental vector fields induced by elements of the Lie algebra 𝔰𝔭(1). Further, let X^{h}, Y^{h} be the horizontal lifts of vector fields X, Y on M and {e_{ν}}_{ν=1,...,4n} a locally defined horizontal orthonormal frame on S. Then the only non–zero covariant derivatives are given by

\begin{aligned}∇_{U}V & = & \frac{1}{2} [U, V] \\ ∇_{X^{h}}Y^{h} & = & (∇_{X}Y)^{h} − \frac{1}{2} Ω(X^{h}, Y^{h}) \\ ∇_{X^{h}}U & = & ∇_{U}X^{h} = \frac{1}{2} ∑_{ν=1}^{4n} g_{S}(Ω(X^{h}, e_{ν}), U) e_{ν} . \end{aligned}

Here and in the sequel we identify elements of 𝔰𝔭(1) with their associated fundamental vector fields. The Levi–Civitá connection of S is determined by a 𝔰𝔬(𝔰𝔭(1) ⊕ T)–valued 1–form ω_{S} on the orthonormal frame bundle of S, but as the structure group reduces to Sp(n), it is sufficient to know its restriction to the Sp(n)–reduction P again denoted by ω_{S}.

Lemma 4.13 The connection form ω_{S} on P can be written as

ω_{S} = ω_{M}^{𝔰𝔭(n)} + \frac{1}{2} ad(ω_{M}^{𝔰𝔭(1)}) + \sqrt{\frac{κ}{16n(n+2)}} (i θ_{M} ∧ i + j θ_{M} ∧ j + k θ_{M} ∧ k) .

Proof. For the proof we identify vector fields on S with equivariant functions on P, i. e.

\begin{matrix}Γ(T S) & ≅ & C^{∞}(P, 𝔰𝔭(1) ⊕ T)^{Sp(n)} \\ A & ↦ & Â. \end{matrix}

With respect to this identification the covariant derivative translates as

{ \atop ∇_{A}B} = dB̂(Ã) + ω_{S}(Ã) B̂ ,

where Ã denotes an arbitrary lift of A to a vector field on P. We will use this formula and Lemma 4.12 to compute all non–zero terms ω_{S}(Ã) B̂, which then combine to give the stated expression for ω_{S}. The definition of the soldering form θ_{S} immediately implies that the function Â is given by θ_{S}(Ã). In particular, we have for a fundamental vector field U

Û = θ_{S}(Ũ) = \sqrt{\frac{16n(n+2)}{κ}} π_{S}^{∗}ω(Ũ) = \sqrt{\frac{16n(n+2)}{κ}} ω(U) = \sqrt{\frac{16n(n+2)}{κ}} U ∈ 𝔰𝔭(1) .  (4.14)

Let X^{h}, Y^{h} denote the horizontal lifts of the vector fields X, Y on M. Because of π ∘ π_{S} = π_{M} we can assume X̃^{h} = X̃. Then,

{ \atop ∇_{X^{h}}Y^{h}} = dŶ(X̃) + ω_{S}(X̃) Ŷ .

Using Lemma 4.12, Lemma 3.10 and equation (4.14) we find

ω_{S}(X̃) Ŷ = { \atop (∇_{X}Y)^{h}} − \frac{1}{2} { \atop Ω(X^{h}, Y^{h})} − dŶ(X̃)
= ω_{M}(X̃) Ŷ − \sqrt{\frac{κ}{16n(n+2)}} (⟨θ_{M}(X̃), i θ_{M}(Ỹ)⟩ i + ⟨θ_{M}(X̃), j θ_{M}(Ỹ)⟩ j
+ ⟨θ_{M}(X̃), k θ_{M}(Ỹ)⟩ k)
= ω_{M}(X̃) Ŷ + \sqrt{\frac{κ}{16n(n+2)}} (⟨i θ_{M}(X̃), Ŷ⟩ i + ⟨j θ_{M}(X̃), Ŷ⟩ j + ⟨k θ_{M}(X̃), Ŷ⟩ k)
= ω_{M}(X̃) Ŷ + \sqrt{\frac{κ}{16n(n+2)}} (i θ_{M}(X̃) ∧ i + j θ_{M}(X̃) ∧ j + k θ_{M}(X̃) ∧ k) Ŷ .

Let U, V be fundamental vector fields. Then V̂ is a constant function and dV̂(Ũ) = Ũ(V̂) = 0. Hence,

{ \atop ∇_{U}V} = ω_{S}(Ũ)V̂ = \frac{1}{2} { \atop [U, V]} = \frac{1}{2} [Û, V̂] = \frac{1}{2} ad(ω_{M}^{𝔰𝔭(1)}(Ũ)) V̂ .

Finally, let U be fundamental, X^{h} a horizontal lift of X and {e_{ν}}_{ν=1,...,4n} a horizontal orthonormal frame on S, with E_{ν} = θ_{S}(ẽ_{ν}) = ê_{ν}. Then dÛ(X̃) = X̃(Û) = 0 and we obtain

ω_{S}(X̃) Û = { \atop ∇_{X^{h}}U} = \frac{1}{2} ∑_{ν=1}^{4n} g_{S}(Ω(X^{h}, e_{ν}), U) E_{ν}
= \frac{κ}{16n(n+2)} ∑_{ν=1}^{4n} (⟨θ_{M}(X̃), i E_{ν}⟩ g_{S}(i, U) E_{ν} + ⟨θ_{M}(X̃), j E_{ν}⟩ g_{S}(j, U) E_{ν}
+ ⟨θ_{M}(X̃), k E_{ν}⟩ g_{S}(k, U) E_{ν})
= − ∑_{ν=1}^{4n} (⟨i X̂, E_{ν}⟩ B(i, U) E_{ν} + ⟨j X̂, E_{ν}⟩ B(j, U) E_{ν} + ⟨k X̂, E_{ν}⟩ B(k, U) E_{ν})
= − ∑_{ν=1}^{4n} ⟨(i B(i, U) + j B(j, U) + k B(k, U)) X̂, E_{ν}⟩ E_{ν}
= − ω_{M}^{𝔰𝔭(1)}(Ũ) X̂ .

Combining these three calculations leads to

ω_{S} = ω_{M} + \frac{1}{2} ad(ω_{M}^{𝔰𝔭(1)}) + \sqrt{\frac{κ}{16n(n+2)}} (i θ_{M} ∧ i + j θ_{M} ∧ j + k θ_{M} ∧ k) − ω_{M}^{𝔰𝔭(1)} .

In this formula the last summand −ω_{M}^{𝔰𝔭(1)} acts by the infinitesimal 𝔰𝔭(1)–action on T and thus cancels the action of ω_{M}^{𝔰𝔭(1)} as part of the first summand ω_{M}. So we end up with the stated formula for ω_{S}. □

Besides the manifold S we also need to consider the cone M̂ over S, i. e. the warped product M̂:= ℝ^{+}×_{t^{2}}S with metric

ĝ = \frac{16n(n+2)}{κ} dt^{2} + t^{2} g_{S}.

The structure group of M̂ reduces to Sp(n), and we can embed the principal Sp(n)–bundle PM̂ := ℝ^{+} × P into the bundle of orthonormal frames of M̂ in such a way that the ℝ ⊕ 𝔰𝔭(1) ⊕ T –valued soldering form on PM̂ is given by

θM̂ = \sqrt{\frac{16n(n+2)}{κ}} (−dt) ⊕ t θ_{S} = \sqrt{\frac{16n(n+2)}{κ}}( − dt ⊕ t ω_{M}^{𝔰𝔭(1)}) ⊕ t θ_{M} .

This convention makes the inward pointing vector field Ξ := −\sqrt{\frac{κ}{16n(n+2)}} \frac{∂}{∂t} correspond to 1 := θM̂(Ξ) ∈ ℝ.

This may be surprising at first but it turns out that only this orientation is compatible with a hyperkähler structure on M̂ introduced later. With this choice of soldering form the Riemannian metric ĝ is associated to the standard metric ⟨ , ⟩ ⊕ B ⊕ ⟨ , ⟩ on ℝ ⊕ 𝔰𝔭(1) ⊕ T.

Lemma 4.15 The restriction ωM̂ of the Levi–Civitá connection of M̂ to the reduction PM̂ of the bundle of orthonormal frames reads:

ωM̂ = ω_{S} + \sqrt{\frac{κ}{16n(n+2)}} θ_{S} ∧ 1
= ω_{M}^{𝔰𝔭(n)} + \frac{1}{2} ad(ω_{M}^{𝔰𝔭(1)}) + ω_{M}^{𝔰𝔭(1)} ∧ 1
+ \sqrt{\frac{κ}{16n(n+2)}} (θ_{M} ∧ 1 + i θ_{M} ∧ i + j θ_{M} ∧ j + k θ_{M} ∧ k) .

In particular, the connection form ωM̂ is the pull–back of a well defined 1–form on P.

Proof. The proof is similar to the proof of the corresponding formula for ω_{S}. Let ˆ∇ denote the covariant derivative for the Levi–Civitá connection of ĝ. The only non–vanishing terms are

ˆ∇_{X}Y = ∇_{X}^{S}Y + \sqrt{\frac{κ}{16n(n+2)}} \frac{1}{t} ĝ(X, Y) Ξ  and  ˆ∇_{X}Ξ = ˆ∇_{Ξ}X = −\sqrt{\frac{κ}{16n(n+2)}} \frac{1}{t} X .

Using the same notation as in the proof of Lemma 4.13 we obtain

ωM̂(X̃)Ŷ = ω_{S}(X̃)Ŷ + \sqrt{\frac{κ}{16n(n+2)}} \frac{1}{t} ⟨θM̂(X̃), θM̂(Ỹ)⟩ Ξ̂ = ω_{S}(X̃)Ŷ + \sqrt{\frac{κ}{16n(n+2)}} ⟨θ_{S}(X̃), Ŷ⟩ 1 .

Having this expression for horizontal vector fields X, Y we immediately derive

ωM̂ = ω_{S} + \sqrt{\frac{κ}{16n(n+2)}} θ_{S} ∧ 1 = ω_{S} + \sqrt{\frac{κ}{16n(n+2)}} θ_{M} ∧ 1 + ω_{M}^{𝔰𝔭(1)} ∧ 1 . □

An interesting application of the formulas given above relates the curvature tensor R̂ of the manifold (M̂, ĝ) to the hyperkähler part R^{hyper} of the curvature tensor of (M, g). The proof will be given in appendix B.

Proposition 4.16 The curvature tensor R̂ is horizontal with respect to π̂ : M̂ → M. Its only non–vanishing terms are

R̂(X, Y) = R^{hyper}(π̂_{∗}X, π̂_{∗}Y) .

The right–hand side acts on the orthogonal complement (T^{V} M̂)^{⊥} of the vertical tangent bundle, which is canonical isomorphic to π^{∗}T M.

The manifold M̂ has been previously studied by A. Swann using the notation U(M) [Swa91]. He constructs M̂ as the ℤ_{2}–quotient of the total space of the locally defined bundle H with zero section removed. The metric ĝ is a member of the family of hyperkähler metrics on M̂ introduced in [Swa91]. In particular, the proposition above is implicit in his work (see also [Swa97]).

5 Quaternionic Killing Spinors

In this chapter we recall the quaternionic Killing equation introduced in [KSW97b]. It will provide us with an equivalent formulation of the limit case. First, we have to collect some facts for the spinor bundle and Clifford multiplication on a quaternionic Kähler manifold (cf. [KSW97a]).

The spinor bundle of a 4n–dimensional quaternionic Kähler spin manifold decomposes into a sum of n + 1 subbundles, which can be expressed using the locally defined bundles E and H (cf. [BaS83], [HiM95] or [Wan89]). For this we have to introduce the bundles Λ_{∘}^{s}E. They are associated to the irreducible Sp(n)–representations on the spaces Λ_{∘}^{s}E of primitive s–vectors, which are the kernels of the contraction σ_{E}⌟ : Λ^{s}E ⟶ Λ^{s−2}E with the symplectic form σ_{E}. With this notation the spinor bundle can be written

S(M) = ⨁_{r=0}^{n} S_{r}(M) := ⨁_{r=0}^{n} Sym^{r}H ⊗ Λ_{∘}^{n−r}E .  (5.17)

In order to define the Clifford multiplication we have to fix notations for modified contraction and multiplication on Sym^{r}H and Λ_{∘}^{s}E. Contraction preserves the primitive spaces, i. e. if η is in Λ_{∘}^{s}E then e^{♯} ⌟ η ∈ Λ_{∘}^{s−1}E, where e^{♯} := σ_{E}(e, ·) ∈ E^{∗} denotes the dual of e ∈ E. However, this is not true for the wedge product and the projection e ∧_{∘} η of e ∧ η onto Λ_{∘}^{s+1}E is given by

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

where L_{E} is the canonical bivector associated to σ_{E} under the isomorphism Λ^{2}E ≅ Λ^{2}E^{∗}. Let h· denote the symmetric product with h ∈ H, and for h^{♯} := σ_{H}(h, ·) ∈ H^{∗} we define h^{♯} ⌟_{∘} : Sym^{r}H → Sym^{r−1}H by h^{♯} ⌟_{∘} := \frac{1}{r}h^{♯} ⌟ . Let h ⊗ e ∈ H ⊗ E = T M^{ℂ} be a tangent vector. Then, the Clifford multiplication μ(h ⊗ e) : S(M) → S(M) is given by

μ(h ⊗ e) = \sqrt{2} (h · ⊗ e^{♯} ⌟ + h^{♯} ⌟_{∘} ⊗ e ∧_{∘}) .  (5.18)

In particular, it maps the subbundle S_{r}(M) to the sum S_{r−1}(M) ⊕ S_{r+1}(M) and thus splits into two components

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

with μ_{−}^{+}(e ⊗ h) = \sqrt{2} (h · ⊗ e^{♯} ⌟ ) and μ_{+}^{−}(e ⊗ h) = \sqrt{2} (h^{♯} ⌟_{∘} ⊗ e ∧_{∘} ). There are two operations defined similar to Clifford multiplication

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

Using these notations the Dirac operator D can be written D = D_{−}^{+} + D_{+}^{−} with

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

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

A quaternionic Killing spinor is by definition (cf. [KSW97b]) a section ψ = (ψ_{0}, ψ_{1}, ψ_{−}) of the Killing 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 ,

satisfying the following quaternionic Killing equation for some parameter λ ≠ 0 and all tangent vectors X:

\begin{aligned}∇_{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} . \end{aligned}

We remark that if (ψ_{0}, ψ_{1}, ψ_{−}) is a solution for parameter λ, then (ψ_{0}, −ψ_{1}, ψ_{−}) is a solution for parameter −λ. In [KSW97b] we showed that for any solution ψ ≠ 0 the spinor ψ_{0} +ψ_{1} is an eigenspinor for the minimal eigenvalue

λ = ±\sqrt{\frac{κ}{4} \frac{n+3}{n+2}} .

In particular, only for two values of the parameter λ ≠ 0 there can possibly exist non–trivial solutions. Conversely, any eigenspinor for the minimal eigenvalue is of the form ψ_{0} + ψ_{1} ∈ Γ(S_{0}(M) ⊕ S_{1}(M)), and the augmented eigenspinor (ψ_{0}, ψ_{1}, ψ_{−}) with ψ_{−} := \frac{1}{4λ} \frac{n+3}{n+4} (μ_{−}^{−} ∘ ∇)ψ_{1} ∈ Γ(Λ_{∘}^{n−2}E) is a solution of the quaternionic Killing equation. Obviously, solutions are sections parallel with respect to a modified connection. Its curvature is precisely the hyperkähler part R^{hyper} of the curvature of M. Since this part vanishes on the quaternionic projective space the Killing bundle of ℍP^{n} is trivialized by augmented eigenspinors with minimal eigenvalue. Proposition 4.16 then motivates to lift a quaternionic Killing spinor to a parallel spinor on M̂.

For our purpose of characterizing the limit case it is more convenient to consider an equivalent version of the quaternionic Killing equation. Let ψ = (ψ_{0}, ψ_{1}, ψ_{−}) be a solution of the original equation, then the scaled section

ψ^{scal} := (ψ_{0}^{scal}, ψ_{1}^{scal}, ψ_{−}^{scal}) = (\sqrt{\frac{n+3}{4n}} ψ_{0}, ψ_{1}, − \sqrt{\frac{4n}{n+3}} ψ_{−})

is a solution of the equation

∇_{X} ψ^{scal} = −\sqrt{\frac{κ}{16n(n+2)}} A_{X} ψ^{scal} ,

where ψ^{scal} is considered as column vector with three entries and A_{X} denotes the matrix

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

For the remainder of this article, the index denoting the scaling will be omitted. The following lemma shows that A_{X} can be interpreted as part of an 𝔰𝔭(n + 1)–action.

Lemma 5.19 Let F = H⊕E be the defining representation of Sp(n+1) with symplectic form σ_{F} = σ_{H} +σ_{E}. Restricted to the subgroup Sp(1) × Sp(n) the Sp(n + 1)–representation Λ_{∘}^{s}F decomposes into

Λ_{∘}^{s}F ≅ Λ_{∘}^{s}E ⊕ (H ⊗ Λ_{∘}^{s−1}E) ⊕ Λ_{∘}^{s−2}E ,

which descends to a well defined representation of Sp(1) · Sp(n) if s is even. Explicitly, the isomorphism ι : Λ_{∘}^{s}E ⊕ (H ⊗ Λ_{∘}^{s−1}E) ⊕ Λ_{∘}^{s−2}E ⟶ Λ_{∘}^{s}F is given by

ι(ϕ_{0} ⊕ (h ⊗ ϕ_{1}) ⊕ ϕ_{−}) = ϕ_{0} + (h ∧ ϕ_{1}) + (L_{H} − \frac{1}{n−s+2}L_{E}) ∧ ϕ_{−} .

Similarly, Sym^{2}F ≅ ℂ ⊗_{ℝ} 𝔰𝔭(n + 1) decomposes into Sym^{2}H ⊕ (H ⊗ E) ⊕ Sym^{2}E. For s = n, the subspace H ⊗ E ⊂ Sym^{2}F acts on Λ_{∘}^{n}F via (h ⊗ e) ϕ = \frac{1}{\sqrt{2}} A_{h⊗e} ϕ.

Proof. It is clear that ι is an injective map to Λ^{s}F. It remains to show that its image is already contained in Λ_{∘}^{s}F. Since σ_{F} = σ_{H} + σ_{E} the statement follows from

σ_{F} ⌟ (L_{H} − \frac{1}{n−s+2} L_{E}) ∧ ϕ_{−} = [σ_{H} ⌟ , L_{H}∧] ϕ_{−} − \frac{1}{n−s+2} [σ_{E} ⌟ , L_{E}∧] ϕ_{−} = 0 ,

where we used the relations [σ_{H} ⌟ , L_{H}∧] ϕ_{−} = ϕ_{−} and [σ_{E} ⌟ , L_{E}∧] ϕ_{−} = (n − s + 2) ϕ_{−}. Comparing dimensions shows that ι is in addition surjective, hence it defines an isomorphism.

The action of an element f_{1} f_{2} ∈ Sym^{2}F on F is given by (f_{1} f_{2})(f) = σ_{F} (f_{1}, f)f_{2} + σ_{F} (f_{2}, f) f_{1} . It extends as derivation to Λ_{∘}^{∗}F and can be explicitly written as (f_{1} f_{2})(ω) = (f_{2} ∧f_{1}^{♯}⌟ + f_{1} ∧f_{2}^{♯}⌟ )(ω). Hence, for h ∈ H and e ∈ E considered as elements of F, the element h ⊗ e ∈ H ⊗ E is identified with h e ∈ Sym^{2}F, and the action on ι(ϕ) = ϕ_{0} ⊕ (a ∧ ϕ_{1}) ⊕ (L_{H} − \frac{1}{2}L_{E}) ∧ ϕ_{−} ∈ Λ_{∘}^{n}F is given by

(h ⊗ e) ι(ϕ_{0}) = h ∧ e^{♯}⌟ ϕ_{0}
= \frac{1}{\sqrt{2}} ι( μ_{−}^{+}(h ⊗ e)ϕ_{0} )
(h ⊗ e) ι(a ⊗ ϕ_{1}) = (h ∧ e^{♯}⌟ + e ∧ h^{♯}⌟ )(a ∧ ϕ_{1})
= − h ∧ a ∧ e^{♯}⌟ ϕ_{1} + σ_{H}(h, a) e ∧ ϕ_{1}
= −σ_{H}(h, a) (L_{H} − \frac{1}{2}L_{E}) ∧ e^{♯}⌟ ϕ_{1} + σ_{H}(h, a) e ∧_{∘} ϕ_{1}
= \frac{1}{\sqrt{2}} ι(μ_{+}^{−}(h ⊗ e) (a ⊗ ϕ_{1}) − μ_{−}^{−}(h ⊗ e) (a ⊗ ϕ_{1}))

and

(h ⊗ e) ι(ϕ_{−}) = − \frac{1}{2} h ∧ e^{♯}⌟ (L_{E} ∧ ϕ_{−}) + e ∧ h^{♯}⌟ (L_{H} ∧ ϕ_{−})
= \frac{1}{2} h ∧ e ∧ ϕ_{−} − \frac{1}{2} h ∧ L_{E} ∧ (e^{♯}⌟ ϕ_{−}) − e ∧ h ∧ ϕ_{−}
= \frac{3}{2} h ∧ e ∧_{∘} ϕ_{−}
= \frac{3}{2\sqrt{2}} ι( μ_{+}^{+}(h ⊗ e)ϕ_{−} ) .

Hence, we see that operation of (h ⊗ e) on ι(ϕ) is just application of the matrix \frac{1}{\sqrt{2}} A_{h⊗e} to the column vector ϕ. □

To stress the origin of the operation \frac{1}{\sqrt{2}}A_{X} from a group action, we introduce ⋆ : H ⊗ E ⊗ Λ_{∘}^{n}F → Λ_{∘}^{n}F for the infinitesimal action of H ⊗ E on Λ_{∘}^{n}F. With this notation the quaternionic Killing equation reads

∇_{X}ψ = − \sqrt{\frac{κ}{16n(n+2)}} A_{X}ψ = − \sqrt{\frac{κ}{8n(n+2)}} Φ(X) ⋆ ψ ,  (5.20)

where Φ is the isomorphism defined in Lemma 2.3.

6 The Geometry of M̂ and Application to Spinors

The aim of this section is to show that the quaternionic Killing equation, considered as a differential equation on equivariant functions on P can be interpreted in three different ways: first, of course, when we think of its solutions as sections of the Killing–bundle S^{Killing}(M) on M they are quaternionic Killing spinors. Interpreted as a section of the spinor bundle on S, the solutions are Killing spinors, and finally solutions pulled back to PM̂ = ℝ^{+} × P are parallel sections of the spinor bundle of M̂.

6.1 The Hyperkähler Structure of M̂

First we recall that the structure group of both S and M̂ reduce to Sp(n), i. e. the tangent bundles may be associated to the principal Sp(n)–bundles P and PM̂ through the Sp(n)–representations 𝔰𝔭(1) ⊕ T and ℝ⊕𝔰𝔭(1)⊕T respectively. However, as Sp(n)–representation T can be identified with E according to Lemma 2.1 reflecting the isomorphism

π^{∗}T M ≅ P ×_{Sp(n)} T ≅ P ×_{Sp(n)} E

of vector bundles on S. More important, this identification is respected by the Levi–Civitá connection, because the infinitesimal 𝔰𝔭(1)–action on T present in ω_{M} = ω_{M}^{𝔰𝔭(1)} ⊕ ω_{M}^{𝔰𝔭(n)} is canceled in the connection form ω_{S} of S. For this reason we will consequently identify T with E and consider E as a euclidean vector space with scalar product ⟨ ·, · ⟩ = Re σ_{E}(·, J·).

In the same vein we combine the obvious identification ℝ ⊕ 𝔰𝔭(1) ≅ ℝ ⊕ Im ℍ = ℍ with the isomorphism ℍ → H of defining representations of Sp(1) unique up to sign to get an isometry from the standard metric on ℝ ⊕ 𝔰𝔭(1) to H with scalar product Re σ_{H}(·, J·) sending 1 , i, j, k to 1, I, J, K ∈ H. In this way we get an isometry

ℝ ⊕ 𝔰𝔭(1) ⊕ T ⟶ F

with the defining representation F = H ⊕ E of Sp(n + 1) considered as a euclidean vector space with scalar product Re σ_{F} (·, J·). As this isometry is Sp(n)–equivariant by construction the tangent bundles of M̂ and S are associated to the Sp(n)–representations F and {1}^{⊥} =: (Im H) ⊕ E ⊂ F.

Though the structure group of M̂ reduces to Sp(n), the holonomy of M̂ does not. Nevertheless, we will show that it is a subgroup of Sp(n + 1), i. e. M̂ is hyperkähler. For this purpose we group the summands of the connection form ωM̂ of M̂ given in Lemma 4.15 as follows

\begin{aligned}ω𝔰𝔭(1) ; M̂ & := & \frac{1}{2} ad(ω_{M}^{𝔰𝔭(1)}) + ω_{M}^{𝔰𝔭(1)} ∧ 1 \\ ω𝔰𝔭(n) ; M̂ & := & ω_{M}^{𝔰𝔭(n)} \\ ωℍ^{n} ; M̂ & := & \sqrt{\frac{κ}{16n(n+2)}} (θ_{M} ∧ 1 + iθ_{M} ∧ I + jθ_{M} ∧ J + kθ_{M} ∧ K) . \end{aligned}

Thus, the Levi–Civitá connection can be written ωM̂ = ω𝔰𝔭(1) ; M̂ + ω𝔰𝔭(n) ; M̂ + ωℍ^{n} ; M̂.

Lemma 6.21 The actions of ω𝔰𝔭(1) ; M̂ and ω_{M}^{𝔰𝔭(1)} are the same, i. e.

ωM̂ = ω_{M}^{𝔰𝔭(1)} + ω_{M}^{𝔰𝔭(n)} + \sqrt{\frac{κ}{16n(n+2)}} (θ_{M} ∧ 1 + iθ_{M} ∧ I + jθ_{M} ∧ J + kθ_{M} ∧ K) .  (6.22)

In particular, the connection form ωM̂ takes values in 𝔰𝔭(n + 1).

Proof. Note that for q ∈ ℍ and imaginary z ∈ Im ℍ we have \frac{1}{2}(zq + qz) = (Re q)z − ⟨ q, z ⟩ = −(z ∧ 1)q. Using this algebraic identity the infinitesimal 𝔰𝔭(1)–action on ℍ in the standard picture can be written as

−qz = \frac{1}{2}ad(z)q + (z ∧ 1)q .

Hence, a particular merit of the identifications above is that the two summands \frac{1}{2}ad(ω_{M}^{𝔰𝔭(1)}) and ω_{M}^{𝔰𝔭(1)} ∧ 1 of the Levi–Civitá connection of M̂ acting on ℝ ⊕ 𝔰𝔭(1) combine into the infinitesimal action of ω_{M}^{sp(1)} on H. Consequently, the summands ω𝔰𝔭(1) ; M̂ and ω𝔰𝔭(n) ; M̂ take values in 𝔰𝔭(n + 1), i. e. in the infinitesimal quaternionic linear isometries of F. The same is true for ωℍ^{n} ; M̂ because of its ℍ–linearity. □

With the Levi–Civitá connection being 𝔰𝔭(n + 1)–valued the manifold M̂ is hyperkähler, and we may use the description of the spinor bundle for the more general quaternionic Kähler manifolds given in (5.17). Consider a complex vector space ℂ^{2} endowed with a symplectic form σ_{ℂ^{2}} and a positive quaternionic structure J. Choosing an isomorphism to the group of all symplectic transformations of ℂ^{2} commuting with J would make ℂ^{2} the defining representation of Sp(1). However, on a hyperkähler manifold this Sp(1)–symmetry is not a local “gauged” symmetry, but a purely global one. For this reason the trivial ℂ^{2}–bundle on M̂ plays the role of H on M. Accordingly, the spinor bundle of M̂ is associated to the Sp(n)–representation

Σ = ⨁_{r=0}^{n+1} Σ_{r} = ⨁_{r=0}^{n+1} Sym^{r}ℂ^{2} ⊗ Λ_{∘}^{n+1−r}F ,

where Sp(n) operates trivially on H ⊂ F. The Clifford multiplication with complex tangent vectors in ℂ^{2} ⊗ F is then given by the formula (5.18). To describe the Clifford multiplication with real tangent vectors however, we have to choose an isomorphism among the family of isometries Φ : ℂ ⊗_{ℝ} F → ℂ^{2} ⊗ F defined in equation (2.2). For quaternionic Kähler manifolds this isometry is uniquely fixed by the additional local Sp(1)–symmetry up to sign, but this is no longer true in the hyperkähler case. In fact, we get a family of Clifford multiplications depending on the choice of Φ, i. e. of a canonical base p, q of ℂ^{2} satisfying Jp = q and σ_{ℂ^{2}}( p, q) = 1. All these are intertwined by the global Sp(1)–symmetry acting on ℂ^{2}. In this sense, to define the Clifford multiplication with real tangent vectors f ∈ F, we first have to apply the isomorphism

Φ : 1 ⊗_{ℝ} f ⟼ \frac{1}{\sqrt{2}}(p ⊗ f − q ⊗ Jf) .  (6.23)

Note that this isomorphism points out the ambivalence of the vector bundle associated to the defining representation F of Sp(n+1). Normally, this vector bundle is the real tangent bundle, but in the description of the spinor bundle it plays a role strictly analogous to the isotropic subspace T^{0,1}M of the complexified tangent bundle on Kähler manifolds.

To describe the spinor bundle on S we recall that the spinor module of Spin(4n + 4) decomposes into the two half–spin modules Σ^{±}. When restricted to Sp(n + 1) the representations Σ^{±} decompose further into a sum of certain Σ_{r}. Since Clifford multiplication maps Σ_{r} to Σ_{r−1} ⊕ Σ_{r+1} and interchanges Σ^{+} with Σ^{−}, we conclude

Σ^{+} = ⨁_{{r=0 \atop r≡1 (2)}}^{n+1} Sym^{r}ℂ^{2} ⊗ Λ_{∘}^{n+1−r}F Σ^{−} = ⨁_{{r=0 \atop r≡0 (2)}}^{n+1} Sym^{r}ℂ^{2} ⊗ Λ_{∘}^{n+1−r}F ,

at least if n is even. Due to Lemma 5.19 Σ^{+} and Σ^{−} are equivalent as Sp(n)–representations, and the spinor module of S is associated to the Sp(n)–principal bundle P through either one, e. g. Σ^{+}. The Clifford multiplication with f ∈ {1}^{⊥} ⊂ F is then given by f ·_{S} := (f ∧ 1)· = f · 1·.

6.2 Reinterpretation of the Quaternionic Killing Equation

In this section we will translate the quaternionic Killing equation (5.20) on M into the equation for a parallel spinor on M̂. Let ψ be a quaternionic Killing spinor on M, which we will consider as an equivariant function on P, i. e. ψ ∈ C^{∞}(P, Λ_{∘}^{n}F)^{Sp(n)}. Then the quaternionic Killing equation reads

(d + ω_{M} + \sqrt{\frac{κ}{8n(n+2)}} θ_{M}^{H⊗E} ⋆ )ψ = 0 ,  (6.24)

where by definition θ_{M}^{H⊗E} := Φ ∘ θ_{M}. The subbundle S_{1}(M̂) of the spinor bundle of M̂ is associated to PM̂ by the representation ℂ^{2} ⊗ Λ_{∘}^{n}F. To construct a section of this bundle, we proceed as follows: we lift ψ to PM̂ by extending it constantly along the ℝ^{+}–direction and choose an arbitrary constant vector ξ ∈ ℂ^{2}. Then, the function ξ ⊗ ψ defines a section in S_{1}(M̂). As function on PM̂ it satisfies

(d + ω_{M} + \sqrt{\frac{κ}{8n(n+2)}} id ⊗ θ_{M}^{H⊗E} ⋆ )(ξ ⊗ ψ) = 0 ,

where the connection and soldering form are considered to be pulled back to PM̂. Obviously, they do not act on ξ but only on ψ. Using the crucial Lemma 6.25 below we can replace \sqrt{\frac{κ}{8n(n+2)}} id ⊗ θ_{M}^{H⊗E}⋆ by ωℍ^{n} ; M̂,

so that the resulting equation on PM̂ reads

(d + ω_{M}^{𝔰𝔭(1)} + ω_{M}^{𝔰𝔭(n)} + ωℍ^{n} ; M̂)(ξ ⊗ ψ) = (d + ωM̂)(ξ ⊗ ψ) = 0 .

Hence, ξ ⊗ ψ defines a parallel spinor on M̂.

We remark that the parallel spinor ξ ⊗ ψ also gives rise to a Killing spinor on S. This follows of course from the general equivalence between Killing spinors on a Riemannian manifold and parallel spinors on its cone, as described by C. Bär (cf. [Bär93]) and briefly summarized in appendix A. Nevertheless, we will include this construction since it is an immediate consequence of our approach. We have seen above that ξ ⊗ ψ, considered as function on PM̂, satisfies (d + ωM̂)(ξ ⊗ ψ) = 0 . If we substitute the expression ωM̂ given in Lemma 4.15 we obtain

(d + ω_{S} + \sqrt{\frac{κ}{16n(n+2)}} θ_{S} ∧ 1)(ξ ⊗ ψ) = 0 .

Interpreted as equation on P this is just the Killing equation, i.e.

(d + ω_{S}) (ξ ⊗ ψ) = − \frac{1}{2} \sqrt{\frac{κ}{16n(n+2)}} θ_{S} ·_{S}(ξ ⊗ ψ) .

The additional factor \frac{1}{2} is due to the action of the orthogonal Lie algebra on the spinor bundle. Before closing this section we formulate the lemma needed above.

Lemma 6.25

\sqrt{\frac{κ}{8n(n+2)}} id ⊗ θ_{M}^{H⊗E}⋆ = ωℍ^{n} ; M̂ .

The proof needs an additional proposition which is analogous to Proposition 2.3 in [KSW97a].

Proposition 6.26 Let p, q be a base of ℂ^{2} ≅ ℍ with σ( p, q) = 1 and f_{1}, f_{2} ∈ F. Then we have the following identity of operators on the spinor bundle:

(p ⊗ f_{1}) ∧ (q ⊗ f_{2}) − (q ⊗ f_{1}) ∧ (p ⊗ f_{2}) = id ⊗ f_{1} · f_{2} .

As element of 𝔰𝔬(4n + 4), the left hand side acts on the spinor module via the isomorphism 𝔰𝔬(4n + 4) ≅ 𝔰𝔭𝔦𝔫(4n + 4) sending e_{1} ∧ e_{2} to \frac{1}{2}( e_{1} e_{2} + ⟨ e_{1}, e_{2} ⟩).

We remark that the proof of these two technical Propositions amounts to prove the decomposition (5.17) of the spinor bundle. With the help of this proposition it is easy to prove Lemma 6.25.

Proof. Let p, q be the canonical base of ℂ^{2} used to define Clifford multiplication. We can then extend the isomorphism ℂ ⊗_{ℝ} F ≅ ℂ^{2} ⊗ F in equation (6.23) to ℂ ⊗_{ℝ} Λ^{2}F ≅ Λ^{2}(ℂ^{2} ⊗ F) to get

\begin{aligned}(θ_{M} ∧ 1) & = & \frac{1}{2} (p ⊗ θ_{M} + q ⊗ Jθ_{M}) ∧ (p ⊗ 1 + q ⊗ J) \\ (iθ_{M} ∧ I) & = & −\frac{1}{2} (p ⊗ θ_{M} − q ⊗ Jθ_{M}) ∧ (p ⊗ 1 − q ⊗ J) \\ (jθ_{M} ∧ J) & = & \frac{1}{2} (p ⊗ Jθ_{M} − q ⊗ θ_{M}) ∧ (p ⊗ J − q ⊗ 1) \\ (kθ_{M} ∧ K) & = & −\frac{1}{2} (p ⊗ Jθ_{M} + q ⊗ θ_{M}) ∧ (p ⊗ J + q ⊗ 1) . \end{aligned}

In the second and fourth line we use that J is conjugate linear. By summing up the four equations, some terms cancel, and we obtain

\begin{aligned}ωℍ^{n} ; M̂ & = & \sqrt{\frac{κ}{16n(n+2)}} ((p ⊗ θ_{M}) ∧ (q ⊗ J) − (q ⊗ θ_{M}) ∧ (p ⊗ J) \\ & & − (p ⊗ Jθ_{M}) ∧ (q ⊗ 1) + (q ⊗ Jθ_{M}) ∧ (p ⊗ 1)) . \end{aligned}

Applying the proposition above yields the following equivalence

ωℍ^{n} ; M̂ = \sqrt{\frac{κ}{16n(n+2)}} (id ⊗ θ_{M} · J − id ⊗ Jθ_{M} · 1)
= \sqrt{\frac{κ}{16n(n+2)}} id ⊗ (J ⊗ θ_{M} − 1 ⊗ Jθ_{M})⋆ = \sqrt{\frac{κ}{8n(n+2)}} id ⊗ θ_{M}^{H⊗E} ⋆ . □

7 Proof of the Theorem

The fact that a quaternionic Killing spinor on M translates into a parallel spinor on the hyperkähler manifold M̂ is crucial to the proof of the main theorem.

Theorem 1 Let M be a compact quaternionic Kähler manifold of quaternionic dimension n and positive scalar curvature κ > 0. If there is an eigenspinor for the Dirac operator with eigenvalue λ satisfying

λ^{2} = \frac{κ}{4} \frac{n + 3}{n + 2} ,

then M is isometric to the quaternionic projective space.

Proof. After the work done in the preceding sections the proof of this theorem reduces to a simple holonomy argument. The spinor bundle of the hyperkähler manifold M̂ is associated to the Sp(n)–representation

Σ = ⨁_{r=0}^{n+1} Σ_{r} = ⨁_{r=0}^{n+1} Sym^{r}ℂ^{2} ⊗ Λ_{∘}^{n+1−r}F ,

where F = H ⊕ E, and Sp(n) operates trivially on H. The holonomy of M̂ is contained in Sp(n + 1) and operates trivially on Sym^{r}ℂ^{2}. The subbundle S_{n+1}(M̂) of S(M̂) associated to Σ_{n+1} = Sym^{n+1}ℂ^{2} is consequently trivialized by n + 2 = dim (Sym^{n+1}ℂ^{2}) linearly independent parallel spinors.

If M admits a quaternionic Killing spinor, there are additional parallel spinors on M̂, which are sections of the subbundle S_{1}(M̂) associated to the representation ℂ^{2} ⊗ Λ_{∘}^{n}F.

Due to a result of Wang [Wan89], on a manifold with holonomy equal to Sp(n + 1) there are exactly n + 2 linearly independent parallel spinors, just those trivializing S_{n+1}(M̂). The additional parallel spinor constructed out of a quaternionic Killing spinor reduces the holonomy further. According to Berger’s list this can only happen if M̂ is reducible or locally symmetric. In the first case, as consequence of a theorem of Gallot [Gal79], M̂ has to be flat. But so it is in the second case, because it is hyperkähler, hence Ricci–flat, and in addition locally symmetric. Therefore M̂ is flat which forces M to be isometric to the quaternionic projective space. □

A Spinors on Cones

In this appendix we will describe how to lift spinors on a Riemannian spin manifold N to spinors on its cone N̂ := ℝ^{+} × N. In particular, we will show that Killing spinors on N translate into parallel spinors on N̂. This construction is originally due to C. Bär [Bär93].

Let (N, g_{N}) be a spin manifold of dimension n and let π : N̂ = ℝ^{+} ×N → N be the cone over N endowed with the warped product metric gN̂ = dt^{2} + e^{2λ}π^{∗}g_{N}. The soldering and connection form on the principal bundle P := P_{Spin(n)}N associated to the chosen spin structure are denoted by θ_{N} and ω_{N}. Obviously, the cone is again spin and the spin structure reduces to the principal Spin(n)–bundle P̂ := ℝ^{+} × P. Forms on P give rise to forms on P̂ by extending them constantly along the ℝ^{+}–direction. It easy to see that the soldering resp. the connection form of N̂ on P̂ are given by

θN̂ = dt + e^{λ}θ_{N} resp. ωN̂ = ω_{N} − e^{λ} \frac{∂λ}{∂t} θ_{N} ∧ Ξ,

where Ξ = \frac{∂}{∂t} denotes the vertical unit vector.

A spinor ψ on N can be interpreted as a Spin(n)–equivariant function on P with values in the spinor module Σ_{n} with a fixed Clifford module structure. With the canonical isomorphism ℂl_{n} ≅ ℂl_{n+1}^{0}, e_{i} ↦ e_{i} · Ξ in mind, we can consider Σ_{n} as an ℂl_{n+1}^{0}– and therefore as an Spin(n + 1)–representation. The values of ψ have now to be interpreted as lying in the Spin(n + 1)–representation. Let ψ be a Killing spinor on N. Interpreted as function on P, it satisfies:

dψ + (ω_{N} − μθ_{N}) ψ = 0 ,

where μ is the Killing constant. Extending ψ constantly in ℝ^{+}–direction and using the isomorphism of the Clifford algebras above defines a spinor on N̂. As a function on P̂, it satisfies

dψ + (ω_{N} − μπ^{∗}θ_{N}Ξ) ψ = 0 .

If the warping function is chosen such that μ = \frac{1}{2}e^{λ} \frac{∂λ}{∂t}, the expression in brackets is equal to ωN̂, and therefore ψ has to be parallel on N̂. The other way round, it is also clear that, if ψ is a parallel spinor on N̂, then its associated function on P̂ is constant in ℝ^{+}–direction and it projects onto a Killing spinor on N.

B The Curvature Tensor of M̂

In this appendix we will prove of Proposition 4.16 relating the curvature tensor of M̂ to the hyperkähler part R^{hyper} of the curvature of M. We recall that the connection form of M̂ restricted to PM̂ is pulled back from P and so is its curvature ΩM̂. We have seen in section 6.1 that the most convenient way to read the connection form ωM̂ given in Lemma 4.15 is

ωM̂ = ω𝔰𝔭(1) ; M̂ + ω𝔰𝔭(n) ; M̂ + ωℍ^{n} ; M̂ ,

where ω𝔰𝔭(1) ; M̂ and ω𝔰𝔭(n) ; M̂ are the pull–backs of ω_{M}^{𝔰𝔭(1)} and ω_{M}^{𝔰𝔭(n)} respectively, and

ωℍ^{n} ; M̂ := \sqrt{\frac{κ}{16n(n+2)}} (θ_{M} ∧ 1 + iθ_{M} ∧ I + jθ_{M} ∧ J + kθ_{M} ∧ K) .

Defining [α ∧ β]( X, Y ) := [α(X), β(Y)] − [α(Y), β(X)] = [β ∧ α]( X, Y ) for Lie algebra valued 1–forms α, β, the curvature 2–form Ω_{M} of M on P can be written

\begin{aligned}Ω_{M} & = & dω_{M} + \frac{1}{2}[ω_{M} ∧ ω_{M}] \\ & = & (dω_{M}^{𝔰𝔭(1)} + \frac{1}{2}[ω_{M}^{𝔰𝔭(1)} ∧ ω_{M}^{𝔰𝔭(1)}]) + (dω_{M}^{𝔰𝔭(n)} + \frac{1}{2}[ω_{M}^{𝔰𝔭(n)} ∧ ω_{M}^{𝔰𝔭(n)}]) . \end{aligned}

Of course, [ω_{M}^{𝔰𝔭(1)} ∧ ω_{M}^{𝔰𝔭(n)}] = 0 since 𝔰𝔭(1) and 𝔰𝔭(n) centralize each other in 𝔰𝔭(1) ⊕ 𝔰𝔭(n). Using the naturality of the exterior differential and ω_{M}^{𝔰𝔭(1)} = π_{S}^{∗}ω we conclude that the first summand is equal to π_{S}^{∗}Ω.

Thus, it corresponds to −\frac{κ}{8n(n+2)} R^{H} in the sense of decomposition (3.8) as shown in the proof of Lemma 3.10. We conclude that the second summand corresponds to −\frac{κ}{8n(n+2)} R^{E} + R^{hyper}. The curvature ΩM̂

of M̂ can be calculated similarly. With 𝔰𝔭(1) and 𝔰𝔭(n) centralizing each other in 𝔰𝔭(n + 1) we still have [ω𝔰𝔭(1) ; M̂ ∧ ω𝔰𝔭(n) ; M̂] = 0, and ΩM̂ is the sum of the three terms:

\begin{matrix}dω𝔰𝔭(1) ; M̂ & + & \frac{1}{2} [ω𝔰𝔭(1) ; M̂ ∧ ω𝔰𝔭(1) ; M̂] & + & \frac{1}{2} [ωℍ^{n} ; M̂ ∧ ωℍ^{n} ; M̂]^{𝔰𝔭(1)} \\ dωℍ^{n} ; M̂ & + & [ω𝔰𝔭(1) ; M̂ ∧ ωℍ^{n} ; M̂] & + & [ω𝔰𝔭(n) ; M̂ ∧ ωℍ^{n} ; M̂] \\ dω𝔰𝔭(n) ; M̂ & + & \frac{1}{2} [ω𝔰𝔭(n) ; M̂ ∧ ω𝔰𝔭(n) ; M̂] & + & \frac{1}{2} [ωℍ^{n} ; M̂ ∧ ωℍ^{n} ; M̂]^{𝔰𝔭(n)} \end{matrix}  (B.27)

where [ωℍ^{n} ; M̂ ∧ ωℍ^{n} ; M̂] is projected onto its two components in 𝔰𝔭(1) and 𝔰𝔭(n) according to the Cartan decomposition 𝔰𝔭(n + 1) = (𝔰𝔭(1) ⊕ 𝔰𝔭(n)) ⊕ ℍ^{n}. Using the formula

[ a_{1} ∧ b_{1}, a_{2} ∧ b_{2} ] = ⟨a_{1}, a_{2}⟩b_{1} ∧ b_{2} − ⟨b_{1}, a_{2}⟩a_{1} ∧ b_{2} − ⟨a_{1}, b_{2}⟩b_{1} ∧ a_{2} + ⟨b_{1}, b_{2}⟩a_{1} ∧ a_{2}

we find

\frac{1}{2}[ωℍ^{n} ; M̂ ∧ ωℍ^{n} ; M̂]

= \frac{κ}{32n(n+2)}(θ_{M} ∧ θ_{M} + iθ_{M} ∧ iθ_{M} + jθ_{M} ∧ jθ_{M} + kθ_{M} ∧ kθ_{M}
+ 2⟨θ_{M} ∧ iθ_{M}⟩1 ∧ I + 2⟨θ_{M} ∧ jθ_{M}⟩1 ∧ J + 2⟨θ_{M} ∧ kθ_{M}⟩1 ∧ K
+ 2⟨iθ_{M} ∧ jθ_{M}⟩I ∧ J + 2⟨iθ_{M} ∧ kθ_{M}⟩I ∧ K + 2⟨jθ_{M} ∧ kθ_{M}⟩J ∧ K)
= \frac{κ}{32n(n+2)}(θ_{M} ∧ θ_{M} + iθ_{M} ∧ iθ_{M} + jθ_{M} ∧ jθ_{M} + kθ_{M} ∧ kθ_{M})  (B.28)
+\frac{κ}{16n(n+2)}(⟨θ_{M} ∧iθ_{M}⟩(1∧I−J∧K) + ⟨θ_{M} ∧jθ_{M}⟩(1∧J−K∧I) + ⟨θ_{M} ∧kθ_{M}⟩(1∧K−I∧J)) (B.29)

We remark that by construction of the base 1, I, J and K the infinitesimal action of i, j and k ∈ 𝔰𝔭(1) on H can be written

i : 1 ↦ −I  J ↦ K  j : 1 ↦ −J  K ↦ I  k : 1 ↦ −K  I ↦ J I ↦ 1  K ↦ −J  J ↦ 1  I ↦ −K  K ↦ 1  J ↦ −I ,

i. e. i corresponds to −(1 ∧ I − J ∧ K). Thus, the summand (B.29) is equal to −π_{S}^{∗}Ω according to Lemma 3.10, i. e. equal to \frac{κ}{8n(n+2)}R^{H}. Without proof we state that the summand (B.28) is equal to \frac{κ}{8n(n+2)}R^{E} as this is certainly true on the quaternionic projective space. Consequently, in the decomposition (B.27) of ΩM̂ the first term vanishes as does the second, because straightforward calculations show that it depends linearly on the torsion of the Levi–Civitá connection ω_{M} of M. Hence, the curvature ΩM̂ reduces to

ΩM̂ = dω𝔰𝔭(n) ; M̂ + \frac{1}{2} [ω𝔰𝔭(n) ; M̂ ∧ ω𝔰𝔭(n) ; M̂] + \frac{κ}{8n(n+2)}R^{E} = R^{hyper} .

In this way ΩM̂ operates only on the subbundle PM̂ ×_{Sp(n)} E of the tangent bundle of M̂. We remark that this subbundle is canonically isomorphic to PM̂ ×_{Sp(n)} T, i. e. to π^{∗}T M. □

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