Eigenvalue Estimates for 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

We consider the Dirac operator on compact quaternionic Kähler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.

Contents

1 Introduction 1

2 The Dirac Operator of a Quaternionic Kähler Manifold 2 2.1 Preliminaries on Sp(n)–Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Spinor Bundle and Clifford Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Dirac and Twistor Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 The Curvature Tensor on Quaternionic Kähler Manifolds 6 3.1 The Linear Space of Curvature Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 The Bianchi Identity for Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Solutions of the Bianchi Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 The Universal Weitzenböck Formula 12 4.1 The Riemannian Estimate Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 The Weitzenböck Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Associated Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Proof of the Theorem 20

1 Introduction

On a compact Riemannian spin manifold (M^{n}, g) with positive scalar curvature κ the eigenvalues of the Dirac operator D satisfy

λ^{2} ≥ \frac{n}{n − 1} \frac{κ_{0}}{4} ,

∗Supported by the SFB 256 ‘Nichtlineare partielle Differentialgleichungen’ †Supported by the Graduiertenkolleg ‘Algebraische, analytische und geometrische Methoden und ihre Wechselwirkungen in der modernen Mathematik’

‡Supported by the SFB 256 ‘Nichtlineare partielle Differentialgleichungen’

where κ_{0} is the minimum of the scalar curvature. This estimate has been proven by Th. Friedrich (cf. [Fri80]). The lower bound is sharp, since it turns out to be the first eigenvalue on the sphere. A theorem of O. Hijazi and A. Lichnerowicz ([Hij86], [Lic87]) implies that equality cannot be attained on manifolds with non–trivial parallel k−forms 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 manifolds of the first class 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 equal to the first eigenvalue on the complex projective space in odd complex dimensions and on the product of ℂP^{2m+1} with the flat 2–torus in even complex dimensions.

The proof of the corresponding result for the second class of manifolds is object of the present article. A quaternionic Kähler manifold is by definition an oriented 4n–dimensional Riemannian manifold with n ≥ 2 whose holonomy group is contained in the subgroup Sp(n) · Sp(1) ⊂ 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]). In even quaternionic dimensions n any quaternionic Kähler manifold possesses a spin structure, whereas for odd n only the quaternionic projective space is spin (cf. [Sal82]). In this article we prove the following theorem:

Theorem 1.1 Let (M^{4n}, g) be a compact quaternionic Kähler spin manifold of positive scalar curvature κ. Then any eigenvalue λ of the Dirac operator satisfies

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

Note that κ is indeed 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] and Lemma 3.5).

The spectrum of the Dirac operator D on the quaternionic projective space ℍP^{n} has been computed by J.–L. Milhorat in [Mil92]. The lower bound \frac{n+3}{n+2} \frac{κ}{4} turns out to be the first eigenvalue of D^{2} on ℍP^{n}. Hence, the stated estimate is sharp.

This estimate was first conjectured by O. Hijazi and J. –L. Milhorat in [HiM95]. They gave first eigenvalue estimates and proved the conjecture for quaternionic dimensions n = 2 and n = 3 (see also [Hij96]). Their approach was to define quaternionic Kähler twistor operators and to use Weitzenböck formulas to prove inequalities for the eigenvalues. We will follow a similar approach. Nevertheless, by using representation theory of the group Sp(n) · Sp(1) we are lead to define more natural twistor operators.

In fact, we obtain a somewhat more general result. The spinor bundle of a quaternionic Kähler manifold M^{4n} splits into a sum of n subbundles and this decomposition is respected by the square of the Dirac operator. We prove that the spectrum of D^{2} on any subbundle is bounded from below by the first eigenvalue of D^{2} on the corresponding bundle on the quaternionic projective space.

The second author would like to thank D. V. Alekseevskii for many interesting and helpful discussions. All of us would like to thank C. Bär and W. Ballmann for encouragement and support.

2 The Dirac Operator of a Quaternionic Kähler Manifold

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 (some local lift of) P. This bundle exists globally 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 the 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 .

The fundamental role of H and E was first realized by S. Salamon, who introduced the E–H–formalism in [Sal82].

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, e. g.

\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. Using either Gram’s determinant or permanent the symplectic form σ_{E} can be extended to Λ^{s}E or Sym^{r}E. As an immediate consequence we obtain

σ_{E}(e ∧ η_{1}, η_{2}) = σ_{E}(η_{1}, e^{♯} ⌟ η_{2}) η_{1}, η_{2} ∈ Λ^{s}E resp. Sym^{r}E .

The usual formula σ_{E}( . , J.) then defines positive definite Hermitian forms, such that e. g. (e∧)^{∗} = (Je)^{♯}⌟ .

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}. Mutatis mutandis 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 — by Λ^{2}E ≅ Λ^{2}E^{∗} associated — canonical bivector 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 reads:

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

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

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:

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

On H, there are similar equations which relate contraction and symmetric product h· with h ∈ H. However, it is convenient to modify the contraction, setting for α ∈ H^{∗} 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 Spinor Bundle and Clifford Multiplication

The aim of this section is to give an explicit formula for the Clifford multiplication using the H–E–formalism. The spinor module considered as Sp(n) × Sp(1)–representation splits into a sum of n irreducible components. Hence, the spinor bundle of a 4n–dimensional quaternionic Kähler manifold decomposes into a sum of n subbundles which can be expressed using the locally defined bundles E and H. Alternatively, this decomposition can be defined by considering the Kraines form Ω as endomorphism of the spinor bundle acting via Clifford multiplication.

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}) .

The covariant derivative on S(M) induced by the Levi–Civita connection on (M, g) respects the decomposition given above. For further proceeding it is necessary to express the Clifford multiplication on spinors in terms of the H–E–formalism. Recall that the complexified tangent bundle is the tensor product T M^{ℂ} ≅ H ⊗ E. The real structure is induced by the quaternionic structures: \overline{h ⊗ e} := Jh ⊗ Je. Finally, the Riemannian metric on T M^{ℂ} is given by

g_{M} = σ_{H} ⊗ σ_{E} .

Proposition 2.2 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).

Proof. To check whether the above formula indeed defines the Clifford multiplication it suffices to verify the general relation:

μ(X) ∘ μ(Y) + μ(Y) ∘ μ(X) = − 2 g(X, Y) .  (2.1)

We prove this for vectors X = h_{1} ⊗ e_{1} and Y = h_{2} ⊗ e_{2}.

{h_{1}^{♯} ⌟_{∘} ⊗ e_{1} ∧_{∘} , h_{2} · ⊗e_{2}^{♯} ⌟ } = [h_{1}^{♯} ⌟_{∘} , h_{2}·] ⊗ e_{1} ∧_{∘} e_{2}^{♯} ⌟ + h_{2} · h_{1}^{♯} ⌟_{∘} ⊗ {e_{1} ∧_{∘} , e_{2}^{♯} ⌟ }
= − \frac{1}{r+1} h_{1} · h_{2}^{♯} ⌟_{∘} ⊗ e_{1} ∧_{∘} e_{2}^{♯} ⌟ + h_{2} · h_{1}^{♯} ⌟_{∘} ⊗ (σ_{E}(e_{2}, e_{1}) + \frac{1}{n−s+1} e_{2} ∧_{∘} e_{1}^{♯} ⌟)
= − h_{2} · h_{1}^{♯} ⌟_{∘} ⊗ σ_{E}(e_{1}, e_{2}) − \frac{1}{r+1} h_{1} · h_{2}^{♯} ⌟_{∘} ⊗ e_{1} ∧_{∘} e_{2}^{♯} ⌟ + \frac{1}{n−s+1} h_{2} · h_{1}^{♯} ⌟_{∘} e_{2} ∧_{∘} e_{1}^{♯} ⌟

If we symmetrize over (1,2) and substitute s = n − r we obtain

{μ(h_{1} ⊗ e_{1}), μ(h_{2} ⊗ e_{2})} = −2 (h_{2} · h_{1}^{♯} ⌟_{∘} − h_{1} · h_{2}^{♯} ⌟_{∘}) ⊗ σ_{E}(e_{1}, e_{2})
= −2 σ_{H}(h_{1}, h_{2}) σ_{E}(e_{1}, e_{2})
= −2 g(h_{1} ⊗ e_{1}, h_{2} ⊗ e_{2}) .

The second assertion is clear from the definition. □

Thus, Clifford multiplication 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) ,  (2.2)

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

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

and

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

As a first application of Lemma 2.2 we will give a new interpretation of the subbundles S_{r}(M) as eigenspaces of the Kraines form acting via Clifford multiplication on the subbundles S_{r}(M). For this we have to investigate the Clifford multiplication with 4–forms. An easy calculation shows that for an arbitrary 2–form A:

μ(A ∧ A) = μ(A)μ(A) + g(A, A).  (2.3)

The space of 2–forms of a quaternionic Kähler manifold splits into the following components:

Λ^{2}T M ≅ Sym^{2}H ⊕ Sym^{2}E ⊕ (Sym^{2}H ⊗ Λ_{∘}^{2}E) ,

where Sym^{2}H resp. Sym^{2}E is realized in Λ^{2}T M as Sym^{2}H⊗σ_{E} resp. σ_{H} ⊗Sym^{2}E. In consequence, Sym^{2}H acts in two different ways on the spinor bundle, by Clifford multiplication of 2–forms and by the identification Sym^{2}H ≅ 𝔰𝔭(1) ⊗ ℂ. These two actions are related by

Proposition 2.3 Let A be an element in Sym^{2}H ⊂ Λ^{2}T M. Then Clifford multiplication with A on S_{r}(M) = Sym^{r}H ⊗ Λ_{∘}^{n−r}E, (r ≥ 1) is given by

μ(A) = 2 A ⊗ id .

By definition the Kraines form Ω is the trace of the wedge product Sym^{2}H ⊗ Sym^{2}H → Λ^{4}T M:

Ω = I ∧ I + J ∧ J + K ∧ K,

where {I, J, K} is a real orthonormal frame of Sym^{2}H. Considered as elements of Λ^{2}T M real unit vectors of Sym^{2}H satisfy g(I, I) = 2n. Using equation (2.3) and Proposition 2.3 Clifford multiplication with Ω is seen to be equivalent to the action of the Casimir operator C of 𝔰𝔭(1):

μ(Ω) = μ(I)μ(I) + μ(J)μ(J) + μ(K)μ(K) + 6n = 4 C ⊗ id + 6n = (6n − 4r(r + 2))id .

The next step is to introduce a scalar product on the bundles Sym^{p}H ⊗ Λ_{∘}^{q}E which for q = n − p corresponds to the usual Hermitian scalar product on the subbundle S_{p}(M) of the spinor bundle. We have already seen that the symplectic forms σ_{H} and σ_{E} extend to positive definite Hermitean forms on Sym^{r}H and Λ_{∘}^{s}E. Using this we define on the tensor product Sym^{r}H ⊗ Λ_{∘}^{s}E the following twisted Hermitean form

⟨A_{1} ⊗ ω_{1}, A_{2} ⊗ ω_{2}⟩ := \frac{1}{r!} σ_{H}(A_{1}, JA_{2}) σ_{E}(ω_{1}, Jω_{2}) .

The next lemma is an immediate consequence of this definition.

Lemma 2.4

⟨μ_{−}^{+}(X)ψ_{1}, ψ_{2}⟩  =  − ⟨ψ_{1}, μ_{+}^{−}(X̄)ψ_{2}⟩  (2.4)
⟨μ_{+}^{+}(X)ψ_{1}, ψ_{2}⟩  =  ⟨ψ_{1}, μ_{−}^{−}(X̄)ψ_{2}⟩ .  (2.5)

2.3 Dirac and Twistor Operators

The aim of this section is to describe the decompositon of T M ⊗ S_{r}(M) and to introduce the corresponding Dirac and twistor operators. Using the Clebsch–Gordan formulas and similar formulas for the fundamental representations of Sp(n) the tensor product decomposes for r ≥ 1 as follows:

T M ⊗ S_{r}(M) ≅ (H ⊗ E) ⊗ (Sym^{r}H ⊗ Λ_{∘}^{n−r}E)
≅ (Sym^{r+1}H ⊗ Λ_{∘}^{n−r−1}E) ⊕ (Sym^{r−1}H ⊗ Λ_{∘}^{n−r+1}E)
⊕ (Sym^{r+1}H ⊗ Λ_{∘}^{n−r+1}E) ⊕ (Sym^{r−1}H ⊗ Λ_{∘}^{n−r−1}E)  (2.6)
⊕ (Sym^{r+1}H ⊗ K^{n−r}) ⊕ (Sym^{r−1}H ⊗ K^{n−r})
≅ S_{r+1}(M) ⊕ S_{r−1}(M) ⊕ ( S_{r}^{+} ⊕ S_{r}^{−} ⊕ V_{r}^{+} ⊕ V_{r}^{−}) .

Here, K^{n−r} is the summand corresponding to the sum of the highest weights in the decomposition of E⊗Λ_{∘}^{n−r}E. Moreover, we introduced the notation S_{r}^{±} = Sym^{r±1}H⊗Λ_{∘}^{n−r±1}E and V_{r}^{±} = Sym^{r±1}H⊗K^{n−r}. 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}^{+} .  (2.7)

The two components of the Clifford multiplication define natural projections onto the first two summands appearing in the splitting (2.6). 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_{−}^{−} defined in the preceeding paragraph and, finally, the projectors onto V^{±} will be denoted by pr_{V^{±}}. By applying these projectors on 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) ,  (2.8)

where D = D_{−}^{+} + D_{+}^{−} is the full 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}  (2.9)

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_{+}^{−}. The adjoint operators are easily computed if one remembers the scalar product introduced in the previous paragraph.

Lemma 2.5

(D_{−}^{+})^{∗} = D_{+}^{−} ,  (D_{+}^{−})^{∗} = D_{−}^{+} ,  (D_{+}^{+})^{∗} = −D_{−}^{−} ,  (D_{−}^{−})^{∗} = −D_{+}^{+} .

The proof is an easy consequence of Lemma 2.4.

3 The Curvature Tensor on Quaternionic Kähler Manifolds

3.1 The Linear Space of Curvature Tensors

The exterior algebra ΛV^{∗} of a vector space V defines a natural multiplication m : Sym^{2}Λ^{2}V^{∗} → Λ^{4}V^{∗}. Its adjoint is usually called comultiplication and is given by

Δ : Λ^{4}V^{∗} → Sym^{2}Λ^{2}V^{∗}, x ∧ y ∧ z ∧ w ↦ (x ∧ y)(z ∧ w) + (y ∧ z)(x ∧ w) + (z ∧ x)(y ∧ w) .

The curvature tensor of a Riemannian manifold M satisfies Bianchi’s first identity, i.e. it has to vanish on the image of Δ. Thus it is straightforward to call ker m the space of curvature tensors on a vector space V. It is easier, however, to adopt the following equivalent

Definition. The space of curvature tensors Curv V^{∗} on V is the subspace

Curv V^{∗} := span{(α · β) × (γ · δ) := (α ∧ γ)(β ∧ δ) + (α ∧ δ)(β ∧ γ) with α, β, γ, δ ∈ V^{∗}}

of Sym^{2}Λ^{2}V^{∗}. Note that the generators trivially satisfy the ‘dual’ Bianchi identity

(α · β) × (γ · δ) + (α · γ) × (δ · β) + (α · δ) × (β · γ) = 0

and its variants using the apparent symmetries of (α · β) × (γ · δ).

Lemma 3.1 There exists a natural isomorphism of vector spaces

Sym^{2}Λ^{2}V^{∗}  ⟶^{≅}  Curv V^{∗} ⊕ Λ^{4}V^{∗}
(α ∧ β)(γ ∧ δ) ⟼ \frac{1}{3}((α · γ) × (β · δ) − (α · δ) × (β · γ)) ⊕ \frac{1}{3}α ∧ β ∧ γ ∧ δ .

In the same vein

Sym^{2}Sym^{2}V^{∗} ⟶^{≅}  Curv V^{∗} ⊕ Sym^{4} V^{∗}
(α · β)(γ · δ) ⟼ \frac{1}{3}(α · β) × (γ · δ) ⊕ \frac{1}{3}α · β · γ · δ .

This isomorphism reveals an alternative realisation of Curv V^{∗} as the subspace of sectional curvature tensors in Sym^{2}Sym^{2}V^{∗}.

Proof. It is easy to write down an inverse of the first homomorphism using the explicit formula

(α ∧ β)(γ ∧ δ) = \frac{1}{3}((α · γ) × (β · δ) − (α · δ) × (β · γ)) + Δ(\frac{1}{3}α ∧ β ∧ γ ∧ δ) .

The inverse of the second homomorphism needs the analogues of Δ and ×:

Δ :  Sym^{4}V ⟶ Sym^{2}Sym^{2}V
α · β · γ · δ ⟼ (α · β)(γ · δ) + (α · γ)(δ · β) + (α · δ)(β · γ)
Cr^{∗} :  Sym^{2}Λ^{2}V^{∗} ⟶ Sym^{2}Sym^{2}V
(α ∧ β)(γ ∧ δ) ⟼ (α · γ)(β · δ) − (α · δ)(β · γ) .

With these definitions one finds

(α · β)(γ · δ) = Δ(\frac{1}{3}α · β · γ · δ) + Cr^{∗}(\frac{1}{3}(α · β) × (γ · δ)) . □

Remark. Either of the above isomorphisms may be used to deduce

dim Curv V^{∗} = \frac{N^{2}}{12} (N^{2} − 1) with N = dim V .

In the next section the space Curv(H^{∗} ⊗ E^{∗}) will be decomposed into its irreducible SL H × SL Ecomponents. For these calculations we need the additional

Lemma 3.2 There exists a natural isomorphism

Sym^{2}V^{∗} ⊗ Λ^{2}V^{∗} ⟶^{≅}  Λ^{2}Sym^{2}V^{∗} ⊕ Λ^{2}Λ^{2}V^{∗}
α · β ⊗ γ ∧ δ ⟼ \frac{1}{2}((α · γ) ∧ (β · δ) + (β · γ) ∧ (α · δ) ⊕ (α ∧ γ) ∧ (β ∧ δ) + (β ∧ γ) ∧ (α ∧ δ)) .

Proof. The partial inverse for the first summand is given by

(α · γ) ∧ (β · δ) ↦ \frac{1}{2}(α · β ⊗ γ ∧ δ + α · δ ⊗ γ ∧ β + γ · β ⊗ α ∧ δ + γ · δ ⊗ α ∧ β) .

Marginal changes are needed for the partial inverse of the second summand. □

Lemma 3.3 Suppose that V is a symplectic vector space with symplectic form σ. The above isomorphism together with the embedding Sym^{2}V^{∗} ⟶^{⊗σ} Sym^{2}V^{∗} ⊗ Λ^{2}V^{∗} induces mappings

i_{Sym} : Sym^{2}V^{∗} → Λ^{2}Sym^{2}V^{∗} , i_{Λ} : Sym^{2}V^{∗} → Λ^{2}Λ^{2}V^{∗} ,

whereas i_{Sym} is always injective, i_{Λ} is injective if and only if dim V ≠ 2.

Proof. The symplectic form σ = ∑_{i} de_{i} ⊗ e_{i}^{♯} ∈ V^{∗} ⊗ V^{∗} defines multiplication operators σ· and σ∧ on Sym V^{∗} ⊗ Sym V^{∗} and Λ V^{∗} ⊗ Λ V^{∗} respectively, which fit into commutative diagrams

[figure: V^{∗} ⊗ V^{∗}; Sym^{2}V^{∗}; Sym^{2}V^{∗} ⊗ Sym^{2}V^{∗}; Λ^{2}Sym^{2}V^{∗}; σ·; 2i_{Sym}; V^{∗} ⊗ V^{∗}; Sym^{2}V^{∗}; Λ^{2}V^{∗} ⊗ Λ^{2}V^{∗}; Λ^{2}Λ^{2}V^{∗}; σ∧; 2i_{Λ}]

because e. g.

2i_{Sym}(α · β) = \frac{1}{2} ∑_{i}[(α · de_{i}) ∧ (β · e_{i}^{♯}) + (β · de_{i}) ∧ (α · e_{i}^{♯})]

↪ ∑_{i}[de_{i} · α ⊗ e_{i}^{♯} · β + de_{i} · β ⊗ e_{i}^{♯} · α] .

Associated to σ is its canonical bivector L, which in turn defines contraction operators L⌟ = ∑_{i} de_{i}^{♭}⌟ ⊗ e_{i}⌟ in either case. The pairs of operators σ· and L⌟ or σ∧ and L⌟ each generate a 𝔰𝔩_{2}-algebra:

H_{Sym} := [σ·, −L⌟ ] = dim V + l + r on Sym^{l}V^{∗} ⊗ Sym^{r}V^{∗}
H_{Λ} := [L⌟ , σ∧] = dim V − l − r on Λ^{l}V^{∗} ⊗ Λ^{r}V^{∗} .

It remains to appeal to the representation theory of 𝔰𝔩_{2} to conclude that σ· is always injective, whereas σ∧ : Λ^{l}V^{∗} ⊗ Λ^{r}V^{∗} → Λ^{l+1}V^{∗} ⊗ Λ^{r+1}V^{∗} is injective only for l + r < dim V. □

3.2 The Bianchi Identity for Tensor Products

The aim of this section is to give a description of the space Curv(H^{∗} ⊗ E^{∗}) of curvature tensors on a tensor product H^{∗} ⊗ E^{∗} in terms of the first Bianchi identity. To start with, let’s make explicit the usual isomorphisms

Sym^{2}(H^{∗} ⊗ E^{∗}) ⟶^{≅} Sym^{2}H^{∗} ⊗ Sym^{2}E^{∗} ⊕ Λ^{2}H^{∗} ⊗ Λ^{2}E^{∗}
Λ^{2}(H^{∗} ⊗ E^{∗}) ⟶^{≅} Sym^{2}H^{∗} ⊗ Λ^{2}E^{∗} ⊕ Λ^{2}H^{∗} ⊗ Sym^{2}E^{∗}

setting e. g.

(α ⊗ β) ∧ (γ ⊗ δ) ⟼ \frac{1}{2}[α · γ ⊗ β ∧ δ ⊕ α ∧ γ ⊗ β · δ] .

Using all the isomorphisms above, the space Sym^{2}Λ^{2}(H^{∗} ⊗ E^{∗}) can be decomposed along the lines

Sym^{2}Λ^{2}(H^{∗} ⊗ E^{∗})

≅ Sym^{2}(Sym^{2}H^{∗} ⊗ Λ^{2}E^{∗} ⊕ Λ^{2}H^{∗} ⊗ Sym^{2}E^{∗})
≅ Sym^{2}(Sym^{2}H^{∗} ⊗ Λ^{2}E^{∗}) ⊕ (Sym^{2}H^{∗} ⊗ Λ^{2}E^{∗} ⊗ Λ^{2}H^{∗} ⊗ Sym^{2}E^{∗}) ⊕ Sym(Λ^{2}H^{∗} ⊗ Sym^{2}E^{∗})

≅ (CurvH^{∗} ⊗ CurvE^{∗})_{H} CurvH^{∗} ⊗ Λ^{4}E^{∗} Sym^{4}H^{∗} ⊗ CurvE^{∗} Sym^{4}H^{∗} ⊗ Λ^{4}E^{∗} (Λ^{2}Sym^{2}H^{∗} ⊗ Λ^{2}Λ^{2}E^{∗})_{H} ⊕ Λ^{2}Sym^{2}H^{∗} ⊗ Λ^{2}Sym^{2}E^{∗} (Λ^{2}Λ^{2}H^{∗} ⊗ Λ^{2}Sym^{2}E^{∗})_{M} (Λ^{2}Sym^{2}H^{∗} ⊗ Λ^{2}Λ^{2}E^{∗})_{M} Λ^{2}Λ^{2}H^{∗} ⊗ Λ^{2}Λ^{2}E^{∗} ⊕ (CurvH^{∗} ⊗ CurvE^{∗})_{E} CurvH^{∗} ⊗ Sym^{4}E^{∗} Λ^{4}H^{∗} ⊗ CurvE^{∗} Λ^{4}H^{∗} ⊗ Sym^{4}E^{∗} (Λ^{2}Λ^{2}H^{∗} ⊗ Λ^{2}Sym^{2}E^{∗})_{E}

where summands occuring twice are labelled by their respective columns. In exactly the same manner the space Sym^{2}Sym^{2}H^{∗} may be decomposed. With Curv(H^{∗} ⊗ E^{∗}) being contained in either space a look at differences and similarities leads to

Lemma 3.4 In terms of the decomposition of Sym^{2}Λ^{2}(H^{∗}⊗E^{∗}) above the first Bianchi identity is equivalent to the five equations

I  pr_{(CurvH^{∗}⊗CurvE^{∗})_{H}} R = pr_{(CurvH^{∗}⊗CurvE^{∗})_{E}}R
II  pr_{Sym^{4}H^{∗}⊗Λ^{4}E^{∗}}R = 0
II′  pr_{Λ^{4}H^{∗}⊗Sym^{4}E^{∗}}R = 0
III  pr_{(Λ^{2}Sym^{2}H^{∗}⊗Λ^{2}Λ^{2}E^{∗})_{H}} R = pr_{(Λ^{2}Sym^{2}H^{∗}⊗Λ^{2}Λ^{2}E^{∗})_{M}} R
III′  pr_{(Λ^{2}Λ^{2}H^{∗}⊗Λ^{2}Sym^{2}E^{∗})_{E}}R = pr_{(Λ^{2}Λ^{2}H^{∗}⊗Λ^{2}Sym^{2}E^{∗})_{M}} R

(3.10)

Proof. In the first place one has to check whether the space of solutions to I–III has the appropriate dimension. Setting N = dimE^{∗} and M = dimH^{∗} a first step could be to calculate

dim(Sym^{4}E^{∗} ⊕ CurvE^{∗} ⊕ Λ^{4}E^{∗}) = \binom{N + 3}{4} + \frac{N^{2}}{12} (N^{2} − 1) + \binom{N}{4} = \frac{N^{2}}{6} (N^{2} + 5)

to find the dimension of the space of solutions to be

\frac{N^{2}}{6} (N^{2} + 5)dim CurvH^{∗} + \frac{M^{2}}{6} (M^{2} + 5)dim CurvE^{∗} − dim CurvH^{∗} dim CurvE^{∗}
+ \frac{N^{2}}{4} (N^{2} − 1)\frac{M^{2}}{4} (M^{2} − 1)  =  \frac{N^{2}M^{2}}{12} (N^{2}M^{2} − 1) .

Thus it is sufficient to show that the generators (α ⊗ α′) · (γ ⊗ γ′) × (β ⊗ β′) · (δ ⊗ δ′) of Curv(H^{∗} ⊗ E^{∗}) in fact satisfy equations I–III, equations II′ and III′ are then inferred using the symmetry in H and E. To begin with, II is easy, because under the above isomorphisms/projections the element ((α⊗α′)∧(β⊗β′))·((γ⊗γ′)∧(δ⊗δ′)) is mapped to \frac{1}{72}α · β · γ · δ ⊗ α′ ∧ β′ ∧ γ′ ∧ δ′ ∈ Sym^{4}H^{∗} ⊗ Λ^{4}E^{∗}. Symmetrizing (β ⊗ β′) ↔ (δ ⊗ δ′) to get the image of the above generator yields 0. The argument for equation III. is similar, yet slightly more complicated as two images have to be calculated. Symmetrizing the two results show that the generators are mapped to the same element in the two copies of Λ^{2}Sym^{2}H^{∗} ⊗ Λ^{2}Λ^{2}E^{∗}.

The trickiest part of the proof is I, because the ‘dual’ Bianchi identity for the generators of CurvH^{∗} and CurvE^{∗} is used. Focussing on the component (Curv H^{∗} ⊗ Curv E^{∗})_{H}, the chain of isomorphism/projections maps

((α ⊗ α′) ∧ (β ⊗ β′)) · ((γ ⊗ γ′) ∧ (δ ⊗ δ′))
↦ \frac{1}{4}(α · β ⊗ α′ ∧ β′)(γ · δ ⊗ γ′ ∧ δ′)
↦ \frac{1}{8}(α · β)(γ · δ) ⊗ (α′ ∧ β′)(γ′ ∧ δ′)
↦ \frac{1}{72}(α · β) × (γ · δ) ⊗ ((α′ · γ′) × (β′ · δ′) − (α′ · δ′) × (β′ · γ′)) .

The same element is projected to \frac{1}{72}((α · γ) × (β · δ) − (α · δ) × (β · γ)) ⊗ (α′ · β′) × (γ′ · δ′) in the other copy (Curv H^{∗} ⊗ Curv E^{∗})_{E}. Even after symmetrization the results look different, however the ‘dual’ Bianchi identity for the generators implies the following identity, which should fix this:

(α · β) × (γ · δ) ⊗ (α′ · γ′) × (β′ · δ′) + (α · δ) × (γ · β) ⊗ (α′ · γ′) × (δ′ · β′)
= −(α · γ) × (β · δ) ⊗ (α′ · γ′) × (β′ · δ′)
= (α · γ) × (β · δ) ⊗ (α′ · β′) × (δ′ · γ′) + (α · γ) × (β · δ) ⊗ (α′ · δ′) × (γ′ · β′) . □

3.3 Solutions of the Bianchi Identity

With the first Bianchi identity expressed in terms of the H–E–formalism it is now easy to derive the decomposition of the curvature tensor of a quaternionic Kähler manifold (compare [Ale68-1], [Ale68-2] or [Sal82]) into a linear combination of ‘curvature tensors’ of the following type:

Definition. Define 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}}),  (3.11)

where ℜ ∈ Sym^{4}E^{∗} and ℜ_{e_{1},e_{2}} : e ↦ ℜ(e_{1}, e_{2}, e, .)^{♭} is an endomorphism of E.

Lemma 3.5 A quaternionic Kähler manifold M is Einstein, and its curvature tensor is given by

R = −\frac{κ}{8n(n + 2)}(R^{H} + R^{E}) + R^{hyper} .  (3.12)

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.

Proof. By definition the curvature tensor is a 2-form on T M^{ℂ} ≅ H ⊗ E with values in 𝔰𝔭H ⊕ 𝔰𝔭E, i. e. it is an element of R ∈ Sym^{2}(Sym^{2}H^{∗} ⊗ σ_{E} ⊕ σ_{H} ⊗ Sym^{2}E^{∗}). Of course (σ_{E}) considered as an element of Λ(Λ^{2}E) satisfies (σ_{E}) ∧ (σ_{E}) = 0, though σ_{E} ∧ σ_{E} ≠ 0 for dim E ≠ 2. Thus the E-symmetric part of R can be written as

\frac{1}{2}(σ_{H})^{2} ⊗ r^{E} ⊕ \frac{1}{2}(σ_{H})^{2} ⊗ ℜ

for some r^{E} ∈ CurvE^{∗} and ℜ ∈ Sym^{4}E^{∗}. Equation II′ is then trivially satisfied, because σ_{H} ∧ σ_{H} = 0. The H-symmetric part, however, reduces to

r^{H} ⊗ \frac{1}{2}(σ_{E})^{2}

for some r^{H} ∈ CurvH^{∗}, as a nonzero contribution from Sym^{4}H^{∗} is incompatible with equation II with σ_{E} ∧ σ_{E} being nonzero.

Equation III then implies that the mixed part of R, an element of Sym^{2}H^{∗} ⊗ σ_{E} ⊗ σ_{H} ⊗ Sym^{2}E^{∗}, is mapped to 0 under the projection to Λ^{2}Sym^{2}H^{∗} ⊗ Λ^{2}Λ^{2}E^{∗}. On the other hand, Lemma (3.3) ensures this projection to be injective on this particular subspace of Sym^{2}H^{∗} ⊗ Λ^{2}E^{∗} ⊗ Λ^{2}H^{∗} ⊗ Sym^{2}E^{∗}, so that the mixed curvature part has to vanish.

Finally equation I shows that r^{E} is a scalar multiple of pr_{Curv E^{∗}}(\frac{1}{2}(σ_{E})^{2}) and that r^{H} is the same multiple of pr_{Curv H^{∗}}(\frac{1}{2}(σ_{H})^{2}), because this is the only way to satisfy

r^{E} ⊗ pr_{CurvH^{∗}}(\frac{1}{2}(σ_{H})^{2}) = pr_{CurvE^{∗}}(\frac{1}{2}(σ_{E})^{2}) ⊗ r^{H} .

Thus the first Bianchi identity implies that at any point of M the curvature tensor of the quaternionic Kähler manifold is a linear combination of R^{H} + R^{E} and R^{hyper} with:

R^{H} = pr_{CurvH^{∗}}(\frac{1}{2}(σ_{H})^{2}) ⊗ \frac{1}{2}(σ_{E})^{2}
R^{E} = \frac{1}{2}(σ_{H})^{2} ⊗ pr_{CurvE^{∗}}(\frac{1}{2}(σ_{E})^{2})
R^{hyper} = \frac{1}{2}(σ_{H})^{2} ⊗ ℜ .  (3.13)

To determine R completely, it is convenient to calculate the Ricci curvature of M, as its definition

Ric(X, Y) = tr(Z ↦ R_{Z,X}Y), X, Y, Z ∈ T M

is easy to handle. Note that for real vectors X, Y the endomorphism R_{.,X}Y is already defined over ℝ, so that its trace may be calculated over ℝ or ℂ. The contributions from the different components of R are

• Ric^{hyper}(h_{1} ⊗ e_{1}, h_{2} ⊗ e_{2}) is the trace of the factorizable endomorphism

h ⊗ e ↦ σ_{H}(h, h_{1})h_{2} ⊗ ℜ(e, e_{1}, e_{2}, .)^{♭} = −(h_{1}^{♯} ⊗ h_{2})h ⊗ ℜ_{e_{1},e_{2}}e .

Its trace is thus the product of the partial traces, however, ℜ_{e_{1},e_{2}} ∈ Sym^{2}E^{∗} ≅ 𝔰𝔭E is trace-free.

• Ric^{E}(h_{1} ⊗ e_{1}, h_{2} ⊗ e_{2}) is the trace of the endomorphism

h ⊗ e ↦ σ_{H}(h, h_{1})h_{2} ⊗ (σ_{E}(e, e_{2})e_{1} + σ_{E}(e_{1}, e_{2})e) = −(h_{1}^{♯} ⊗ h_{2})h ⊗ ( − e_{2}^{♯} ⊗ e_{1} + σ_{E}(e_{1}, e_{2})id_{E})e .

which factorizes, too. Its trace is thus

Ric^{E}(h_{1} ⊗ e_{1}, h_{2} ⊗ e_{2}) = −(2n + 1)σ_{H}(h_{1}, h_{2})σ_{E}(e_{1}, e_{2}) .

• The same argument goes through for Ric^{H}; the different dimensions of H and E account for the slightly

changed result:

Ric^{H}(h_{1} ⊗ e_{1}, h_{2} ⊗ e_{2}) = −3σ_{H}(h_{1}, h_{2})σ_{E}(e_{1}, e_{2}) .

The Ricci curvature being a multiple of the metric the quaternionic Kähler manifold M is Einstein and the scalar curvature κ is thus constant on M. The coefficient of R^{H} + R^{E} in R is then fixed by

Ric(X, Y) = \frac{κ}{4n}g(X, Y) = −\frac{κ}{8n(n + 2)}(Ric^{H} + Ric^{E})(X, Y) . □  (3.14)

At the end of this section a strange and remarkable feature of the hyperkähler part R^{hyper} of the curvature tensor should be pointed out. This feature is convenient for deriving the Weitzenböck formulas below, nevertheless it implies that essentially new ideas are needed to discuss the limit case in the eigenvalue estimate: R^{hyper} eludes all Weitzenböck formulas!

To be more precise, the symmetric 4-form ℜ induces a natural endomorphism on every vector bundle associated to the principal Sp(1)Sp(n)–bundle in the following way:

Sym^{4}E^{∗} ⟶^{Δ} Sym^{2}E^{∗} ⊗ Sym^{2}E^{∗} ⟶^{m} U(𝔰𝔭E)
ℜ  ⟼  Q_{ℜ} .

where U(𝔰𝔭E) is the universal envelopping algebra of 𝔰𝔭E. This mapping is injective by Poincaré–Birkhoff– Witt, nevertheless the endomorphism induced on the spinor bundle is trivial:

Lemma 3.6 Any symmetric 4-form in Sym^{4}E^{∗} induces the trivial endomorphism on ΛE.

Proof. As the 4th powers span Sym^{4}E^{∗}, it is sufficient to check the lemma only for an element of the form \frac{1}{24}α^{4}. The comultiplication maps it to \frac{1}{2}α^{2} ⊗ \frac{1}{2}α^{2}. Considered as an endomorphism \frac{1}{2}α^{2} denotes A = α⊗α^{♭}, extended as derivation to ΛE. As such it satisfies A^{2} = 0:

A^{2}(e_{1} ∧ … ∧ e_{s}) = ∑_{i=1}^{s} e_{1} ∧ … ∧ A^{2}e_{i} ∧ … ∧ e_{s}

+2 ∑_{i<j} e_{1} ∧ … ∧ Ae_{i} ∧ … ∧ Ae_{j} ∧ … ∧ e_{s} .

In fact, the first sum is zero, because already A^{2} = 0 on E, and the second vanishes identically, because the endomorphism A of E has rank 1. □

Corollary 3.1 The comultiplication of ℜ may be written as

Δℜ = \frac{1}{2} ∑_{i,j} de_{i} · de_{j} ⊗ ℜ(e_{i}, e_{j}, ., .)

such that

∑_{i,j} id ⊗ (de_{j}^{♭} ∧_{∘} de_{i}⌟ + de_{i}^{♭} ∧_{∘} de_{j}⌟ )ℜ_{e_{i},e_{j}} = 0  (3.15)

holds as an operator identity on the spinor bundle.

4 The Universal Weitzenböck Formula

In this section we develop the main tool for the proof of the lower bound. It turns out that all necessary information can be encoded in a single matrix. To motivate the further proceeding, it is useful to remember the situation in the Riemannian case. For deriving the eigenvalue estimate in that case, two things are needed: the Lichnerowicz–Weitzenböck formula and the twistor operator.

4.1 The Riemannian Estimate Revisited

Let (M^{n}, g) be Riemennian spin manifold with tangent resp. spinor bundle T M resp. S(M) associated to the representations V and Σ of Spin(V). The associativity of the tensor product (V ⊗V)⊗Σ ≅ V ⊗(V ⊗Σ) can be thought of as two different ways of decomposing V ⊗ V ⊗ Σ into irreducible Spin(V)–representations; the isomorphism induced is expressed by an invertible matrix W for each isotypical component.

Accordingly, the special section ∇^{2}ψ ∈ Γ(T M ⊗T M ⊗S(M)) splits into sections of the associated bundles inducing two different sets of 2nd order differential operators which are related by the same matrix. We will only be interested in differential operators from the spinor bundle to itself. Since the corresponding matrix contains all relevant Weitzenböck formulas we will call it Weitzenöck matrix.

In the Riemannian case V ⊗ V ⊗ Σ contains two copies of Σ. Hence, the associativity is expressed by a 2 × 2–matrix. Let’s look at (V ⊗ V) ⊗ Σ first. Due to the obvious decomposition

V ⊗ V ≅ Λ^{2}V ⊕ Sym_{∘}^{2}V ⊕ ℂ ,

the projection onto the two summands in (V ⊗ V) ⊗ Σ is given by contracting the two V factors with the metric g and the operation of Λ^{2}V ≅ 𝔰𝔭𝔦𝔫(V) on Σ. The space Sym_{∘}^{2}V ⊗ Σ contains no copy of Σ. To summarize:

pr_{ℂ} : V ⊗ V ⊗ Σ ⟶ Σ
e_{1} ⊗ e_{2} ⊗ ψ ⟼ g(e_{1}, e_{2})ψ ,
pr_{Λ^{2}} : V ⊗ V ⊗ Σ ⟶ Σ
e_{1} ⊗ e_{2} ⊗ ψ ⟼ e_{1}e_{2}ψ + g(e_{1}, e_{2})ψ .

Strictly speaking, there is a factor \frac{1}{2} missing in pr_{Λ^{2}}, but this is of no concern at this point. On the other hand, the projection to the first Σ in V ⊗ (V ⊗ Σ) is merely twice the Clifford multiplication. The complementary projection involves the kernel K ⊂ V ⊗ Σ of the Clifford multiplication and is contraction of the two V –factors in V ⊗ K:

pr_{Σ} : V ⊗ V ⊗ Σ ⟶ Σ
e_{1} ⊗ e_{2} ⊗ ψ ⟼ e_{1}e_{2}ψ ,
pr_{K} : V ⊗ V ⊗ Σ ⟶ V ⊗ K ⊂ V ⊗ V ⊗ Σ  ⟶ Σ
e_{1} ⊗ e_{2} ⊗ ψ ⟼ e_{1} ⊗ (e_{2} ⊗ ψ + \frac{1}{n} ∑_{i} e_{i} ⊗ e_{i} · e_{2} · ψ) ⟼ g(e_{1}, e_{2})ψ + \frac{1}{n}e_{1}e_{2}ψ .

With these calculations it is now easy to provide a first example of a Weitzenböck matrix W. It is the matrix relating the two sets of projections:

( \begin{matrix}pr_{ℂ} \\ pr_{Λ^{2}} \end{matrix} ) = ( \begin{matrix}−\frac{1}{n} & 1 \\ \frac{n−1}{n} & 1 \end{matrix} ) ( \begin{matrix}pr_{Σ} \\ pr_{K} \end{matrix} ) .

It remains to identify the differential operators which are obtained by applying the projections to ∇^{2}ψ. For a section ψ ∈ Γ(S(M)) the section ∇^{2}ψ can be expanded in a sum over an orthonormal basis leading to the following images of the projections onto Σ:

\begin{aligned}pr_{ℂ}(∇^{2}ψ) & = & ∑_{i,j} g(e_{i}, e_{j})∇_{e_{i},e_{j}}^{2}ψ = −∇^{∗}∇ψ \\ pr_{Λ^{2}}(∇^{2}ψ) & = & ∑_{i,j}(e_{i}e_{j} + g(e_{i}, e_{j}))∇_{e_{i},e_{j}}^{2}ψ = \frac{κ}{4}ψ \end{aligned}

and

\begin{aligned}pr_{Σ}(∇^{2}ψ) & = & ∑_{i,j} e_{i}e_{j}∇_{e_{i},e_{j}}^{2}ψ = D^{2}ψ \\ pr_{K}(∇^{2}ψ) & = & ∑_{i,j} (g(e_{i}, e_{j}) + \frac{1}{n}e_{i}e_{j}∇_{e_{i},e_{j}}^{2})ψ = −T^{∗}T ψ . \end{aligned}

The only projection which does not lead to a differential operator is the well–known curvature term appearing in the Lichnerowicz–Weitzenböck formula. The final form of the Riemannian version of the Weitzenböck matrix formula reads:

( \begin{matrix}−∇^{∗}∇ψ \\ \frac{κ}{4}ψ \end{matrix} ) = ( \begin{matrix}−\frac{1}{n} & 1 \\ \frac{n−1}{n} & 1 \end{matrix} ) ( \begin{matrix}D^{2}ψ \\ −T^{∗}T ψ \end{matrix} ) .  (4.16)

This matrix equation is a nice tool to produce some well–known formulas by multiplying it with row vectors from the left, e. g. multiplying with (−1 1) yields the Lichnerowicz–Weitzenböck formula. Taking the L^{2}-product with ψ, we get its integrated version:

( \begin{matrix}−∥∇ψ∥^{2} \\ \frac{1}{4}⟨ψ, κψ⟩ \end{matrix} ) = ( \begin{matrix}−\frac{1}{n} & 1 \\ \frac{n−1}{n} & 1 \end{matrix} ) ( \begin{matrix}∥Dψ∥^{2} \\ −∥T ψ∥^{2} \end{matrix} ) .  (4.17)

Estimating the twistor operator term to zero, the second row immediately gives the well-known estimate of Friedrich [Fri80] for the first eigenvalue of the Dirac operator. This rather simple example of the Riemannian case shows the principle by which the eigenvalue estimate in the quaternionic case will be obtained.

4.2 The Weitzenböck Matrix

The same strategy is pursued in the quaternionic Kähler case, but under holonomy Sp(1)·Sp(n) the splitting of T M ⊗ T M ⊗ S(M) is much more delicate. Like in the Riemannian case, the splitting of ∇^{2}ψ, considered as section in T M ⊗ T M ⊗ S(M) yields two different sets of differential operators related by a matrix W. To calculate it, we have to look at the splitting of

V ⊗ V ⊗ Σ = (H ⊗ E) ⊗ (H ⊗ E) ⊗ ⨁_{r=0}^{n} (Sym^{r}H ⊗ Λ_{∘}^{n−r}E)

into irreducibles, find the Σ-summands and determine how they are embedded in V ⊗ V ⊗ Σ.

There is a simple observation which makes the following calculations feasible. All representations occuring are products of representations of Sp(1) or Sp(n). Hence it suffices to look at the H–part resp. the E–part of the splitting, that means, one can look at the spaces H ⊗ H ⊗ Sym^{r}H and E ⊗ E ⊗ Λ_{∘}^{n−r}E separately to search for their Sym^{r}H– resp. Λ_{∘}^{n−r}E–summands.

First, let’s consider the H–part. The two sides of the isomorphism

(H ⊗ H) ⊗ Sym^{r}H ≅ H ⊗ (H ⊗ Sym^{r}H)

need to be decomposed, where on the left side H ⊗ H is acting as endomorphisms on Sym^{r}H and on the right side H is acting by its part of the Clifford multiplication. On the right–hand side the situation is easy:

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) .

Using Lemma 2.3 the expression for the first projection can be rewritten:

pr_{−+}(h_{1} ⊗ h_{2} ⊗ s) = σ_{H}(h_{1}, h_{2})s + \frac{r}{r + 1}h_{1} · h_{2}^{♯}⌟_{∘}s .

On the left–hand side, we use the isomorphism

H ⊗ H ≅ Sym^{2}H ⊕ Λ^{2}H ≅ Sym^{2}H ⊕ ℂ ,

where ℂ denotes the trivial representation, spanned by σ_{H}. Hence,

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

Elements of Sym^{2}H operate in the obvious manner as endomorphims on H, namely as (h_{1}h_{2})(h) = σ_{H}(h_{1}, h)h_{2} + σ_{H}(h_{2}, h)h_{1}, extended as derivation to the symmetric algebra:

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

This can be written explicitely:

(h_{1}h_{2})(s) = (h_{2} · h_{1}^{♯}⌟ + h_{1} · h_{2}^{♯}⌟ )s = rσ_{H}(h_{1}, h_{2})s + 2h_{1} · h_{2}^{♯}⌟ s = rσ_{H}(h_{1}, h_{2})s + 2rh_{1} · h_{2}^{♯}⌟_{∘}s .

To get an overview of this situation, a diagram will be helpful. The notation [.]_{Sym^{r}H} refers to the irreducible Sym^{r}H-summand of the space in question.

[figure: [ℂ ⊗ Sym^{r}H]_{Sym^{r}H}; σ_{H}(h_{1}, h_{2})s; Sym^{r}H; σ_{H}(h_{1}, h_{2})s + \frac{r}{r+1}h_{1} · h_{2}^{♯}⌟_{∘}s; H ⊗ H ⊗ Sym^{r}H; h_{1} ⊗ h_{2} ⊗ s; Sym^{r}H; h_{1} · h_{2}^{♯}⌟_{∘}s; [Sym^{2}H ⊗ Sym^{r}H]_{Sym^{r}H}; rσ_{H}(h_{1}, h_{2})s + 2rh_{1} · h_{2}^{♯}⌟_{∘}s; pr_{ℂ}; pr_{Sym^{2}H}; pr_{−+}; pr_{+−}]

¿From this diagram the H–part of the Weitzenböck matrix is evident:

W_{H} = ( \begin{matrix}1 & −\frac{r}{r+1} \\ r & \frac{r(r+2)}{r+1} \end{matrix} ) .  (4.18)

Turning to the E–part of the Weitzenböck matrix, we have to consider the two sides of the isomorphism:

(E ⊗ E) ⊗ Λ_{∘}^{n−r}E ≅ E ⊗ (E ⊗ Λ_{∘}^{n−r}E) .  (4.19)

In difference to the H–part there exists a kernel K^{n−r} of multiplication and contraction on the right side:

E ⊗ Λ_{∘}^{n−r}E ≅ Λ_{∘}^{n−r+1}E ⊕ Λ_{∘}^{n−r−1}E ⊕ K^{n−r} .

Therefore, in the decomposition of E ⊗ (E ⊗ Λ_{∘}^{n−r}E) the representation Λ_{∘}^{n−r}E occurs three times, namely twice by the E–part of the Clifford multiplication and, in addition, as an irreducible summand of E ⊗ K^{n−r}. The following lemma gives an explicit expression for the projection onto K^{n−r}:

Lemma 4.1

p̃r_{K} : E ⊗ Λ_{∘}^{n−r}E ⟶ K^{n−r}
e ⊗ ω ⟼ e ⊗ ω − \frac{1}{n−r+1} ∑_{i} e_{i} ⊗ de_{i}⌟ e ∧_{∘} ω
−\frac{r+2}{(n+r+3)(r+1)} ∑_{i} de_{i}^{♭} ⊗ e_{i} ∧_{∘} e^{♯}⌟ ω .

Proof. Simple use of the number operators introduced in Lemma 2.2 immediately shows that the projection of e ⊗ ω onto the kernel of the ∧_{∘}-product is e ⊗ ω − \frac{1}{n−r+1} ∑_{i} e_{i} ⊗ de_{i}⌟ e ∧_{∘} ω. By the same argument e ⊗ ω plus the third summand is the projection onto the kernel of contraction:

∑_{i} de_{i}⌟ e_{i} ∧_{∘} e^{♯}⌟ ω = ∑_{i} (σ_{E}(de_{i}^{♭}, e_{i}) − e_{i} ∧_{∘} de_{i}⌟ + \frac{1}{r+2}de_{i}^{♭} ∧_{∘} e_{i}^{♯}⌟ )e^{♯}⌟ ω
= (2n − (n − r − 1) − \frac{n−r−1}{r+2})e^{♯}⌟ ω
= \frac{(n+r+3)(r+1)}{r+2}e^{♯}⌟ ω .

It remains to show that the second resp. the third summand already lies in the kernel of contraction resp. multiplication. This is clear for the second summand because σ_{E}⌟ e^{♯} ∧_{∘} ω = 0 by definition. Upon ∧_{∘}–multiplication, the third summand reduces to the projection of L_{E} ∧ e^{♯}⌟ ω onto Λ_{∘}E which is zero by definition, too. □

The projection onto the irreducible Λ_{∘}^{n−r}E–summand contained in E ⊗ K^{n−r} ⊂ E ⊗ E ⊗ Λ_{∘}^{n−r}E can be thought of as contracting the two E–factors. All ingrediences for computing the right hand side are now at hand:

pr_{−+} : E ⊗ E ⊗ Λ_{∘}^{n−r}E ⟶ E ⊗ Λ_{∘}^{n−r+1}E  ⟶ Λ_{∘}^{n−r}E
e_{1} ⊗ e_{2} ⊗ ω ⟼ e_{1} ⊗ e_{2} ∧_{∘} ω  ⟼ e_{1}^{♯}⌟ e_{2} ∧_{∘} ω ,
pr_{+−} : E ⊗ E ⊗ Λ_{∘}^{n−r}E ⟶ E ⊗ Λ_{∘}^{n−r−1}E  ⟶ Λ_{∘}^{n−r}E
e_{1} ⊗ e_{2} ⊗ ω ⟼ e_{1} ⊗ e_{2}^{♯}⌟ ω  ⟼ e_{1} ∧_{∘} e_{2}^{♯}⌟ ω ,
pr_{K} : E ⊗ E ⊗ Λ_{∘}^{n−r}E ⟶ E ⊗ K^{n−r}  ⟶ Λ_{∘}^{n−r}E
e_{1} ⊗ e_{2} ⊗ ω ⟼ e_{1} ⊗ p̃r_{K}(e_{2} ⊗ ω) ⟼ σ_{E}(e_{1}, e_{2})ω − \frac{1}{n−r+1}e_{1}^{♯}⌟ e_{2} ∧_{∘} ω
+\frac{r+2}{(n+r+3)(r+1)}e_{1} ∧_{∘} e_{2}^{♯}⌟ ω .

On the left side of (4.19), E ⊗ E is considered as space of endomorphisms, acting as derivations on Λ_{∘}^{n−r}E:

E ⊗ E ≅ Sym^{2}E ⊕ Λ_{∘}^{2}E ⊕ ℂ ,

where Λ_{∘}^{2}E resp. ℂ denotes the trace–free part resp. the trace part of Λ^{2}E. To calculate the trace–free part of an element e_{1} ∧ e_{2} ∈ Λ^{2}E, we note that the 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} ,

its trace being 2σ_{E}(e_{1}, e_{2}). Its trace–free part is thus e_{1} ∧ e_{2} − \frac{1}{n}σ_{E}(e_{1}, e_{2})id . Extended as derivation to Λ_{∘}^{n−r}E, the identity id acts as multiplication with n − r, and a forteriori, the trace–free part of e_{1} ∧ e_{2} becomes the operator e_{2} ∧_{∘} e_{1}^{♯}⌟ − e_{1} ∧_{∘} e_{2}^{♯}⌟ − \frac{n−r}{n} σ_{E}(e_{1}, e_{2}).

With the preceeding calculations, the form of the projections is obvious:

pr_{ℂ} : E ⊗ E ⊗ Λ_{∘}^{n−r}E ⟶ Λ_{∘}^{n−r}E
e_{1} ⊗ e_{2} ⊗ ω ⟼ σ_{E}(e_{1}, e_{2})ω ,
pr_{Sym^{2}E} : E ⊗ E ⊗ Λ_{∘}^{n−r}E ⟶ Λ_{∘}^{n−r}E
e_{1} ⊗ e_{2} ⊗ ω ⟼ e_{2} ∧_{∘} e_{1}^{♯}⌟ ω + e_{1} ∧_{∘} e_{2}^{♯}⌟ ω
= (\frac{r+2}{r+1}e_{1} ∧_{∘} e_{2}^{♯}⌟ − e_{1}^{♯}⌟ e_{2} ∧_{∘} +σ_{E}(e_{1}, e_{2}))ω ,
pr_{Λ_{∘}^{2}E} : E ⊗ E ⊗ Λ_{∘}^{n−r}E ⟶ Λ_{∘}^{n−r}E
e_{1} ⊗ e_{2} ⊗ ω ⟼ e_{2} ∧_{∘} e_{1}^{♯}⌟ ω − e_{1} ∧_{∘} e_{2}^{♯}⌟ ω − \frac{n−r}{n} σ_{E}(e_{1}, e_{2})ω
= ( − \frac{r}{r+1}e_{1} ∧_{∘} e_{2}^{♯}⌟ − e_{1}^{♯}⌟ e_{2} ∧_{∘} +\frac{r}{n}σ_{E}(e_{1}, e_{2}))ω.

To summarize all calculations above let’s consider a diagram:

[figure: [ℂ ⊗ Λ_{∘}^{n−r}E]_{Λ_{∘}^{n−r}E}; σ_{E}(e_{1}, e_{2})ω; Λ_{∘}^{n−r}E; e_{1}^{♯}⌟ e_{2}∧_{∘}; Λ_{∘}^{n−r}E; e_{1} ∧_{∘} e_{2}^{♯}⌟ ω; E ⊗ E ⊗ Λ_{∘}^{n−r}E; e_{1} ⊗ e_{2} ⊗ ω; [E ⊗ K^{n−r}]_{Λ_{∘}^{n−r}E}; σ_{E}(e_{1}, e_{2})ω − \frac{1}{n−r+1}e_{1}^{♯}⌟ e_{2} ∧_{∘} ω; +\frac{r+2}{(n+r+3)(r+1)}e_{1} ∧_{∘} e_{2}^{♯}⌟ ω; [Λ_{∘}^{2}E ⊗ Λ_{∘}^{n−r}E]_{Λ_{∘}^{n−r}E}; e_{2} ∧_{∘} e_{1}^{♯}⌟ ω − e_{1} ∧_{∘} e_{2}^{♯}⌟ ω; −\frac{n−r}{n} σ_{E}(e_{1}, e_{2})ω; [Sym^{2}E ⊗ Λ_{∘}^{n−r}E]_{Λ_{∘}^{n−r}E}; (e_{2} ∧_{∘} e_{1}^{♯}⌟ + e_{1} ∧_{∘} e_{2}^{♯}⌟ )ω; pr_{ℂ}; pr_{Sym^{2}E}; pr_{Λ_{∘}^{2}}; pr_{−+}; pr_{+−}; pr_{K}]

Finally, we are able to give the E–part of the Weitzenböck matrix W_{E}:

W_{E} = ( \begin{matrix}\frac{1}{n−r+1} & −\frac{r+2}{(n+r+3)(r+1)} & 1 \\ −\frac{n−r}{n−r+1} & \frac{(n+r+2)(r+2)}{(n+r+3)(r+1)} & 1 \\ −\frac{(n−r)(n+1)}{n(n−r+1)} & −\frac{r(n+r+2)(n+1)}{n(n+r+3)(r+1)} & \frac{r}{n} \end{matrix} ) .  (4.20)

All projections onto the six S(M)–summands in (T M ⊗ T M) ⊗ S(M) ≅ T M ⊗ (T M ⊗ S(M)) are given as tensor products of the corresponding projections of E resp. H part. Hence, the full Weitzenböck matrix is a 6 × 6–matrix which is obtained as Kronecker product W = W_{E} ⊗ W_{H} of the two partial Weitzenböck matrices, i. e. the nine 2 × 2-blocks of W are obtained by multiplying W_{H} with the corresponding entry in W_{E}.

W = ( \begin{matrix}\frac{1}{n−r+1} & −\frac{r}{(n−r+1)(r+1)} & −\frac{r+2}{(n+r+3)(r+1)} & \frac{r(r+2)}{(n+r+3)(r+1)^{2}} & 1 & −\frac{r}{r+1} \\ \frac{r}{n−r+1} & \frac{r(r+2)}{(n−r+1)(r+1)} & −\frac{r(r+2)}{(n+r+3)(r+1)} & −\frac{r(r+2)^{2}}{(n+r+3)(r+1)^{2}} & r & \frac{r(r+2)}{r+1} \\ −\frac{n−r}{n−r+1} & \frac{r(n−r)}{(n−r+1)(r+1)} & \frac{(n+r+2)(r+2)}{(n+r+3)(r+1)} & −\frac{r(n+r+2)(r+2)}{(n+r+3)(r+1)^{2}} & 1 & −\frac{r}{r+1} \\ −\frac{(n−r)r}{n−r+1} & −\frac{r(r+2)(n−r)}{(n−r+1)(r+1)} & \frac{r(n+r+2)(r+2)}{(n+r+3)(r+1)} & \frac{r(n+r+2)(r+2)^{2}}{(n+r+3)(r+1)^{2}} & r & \frac{r(r+2)}{r+1} \\ −\frac{(n−r)(n+1)}{n(n−r+1)} & \frac{r(n−r)(n+1)}{n(n−r+1)(r+1)} & −\frac{r(n+r+2)(n+1)}{n(n+r+3)(r+1)} & \frac{r^{2}(n+r+2)(n+1)}{n(n+r+3)(r+1)^{2}} & \frac{r}{n} & −\frac{r^{2}}{n(r+1)} \\ −\frac{r(n−r)(n+1)}{n(n−r+1)} & −\frac{r(r+2)(n−r)(n+1)}{n(n−r+1)(r+1)} & −\frac{r^{2}(n+r+2)(n+1)}{n(n+r+3)(r+1)} & −\frac{r^{2}(r+2)(n+r+2)(n+1)}{n(n+r+3)(r+1)^{2}} & \frac{r^{2}}{n} & \frac{r^{2}(r+2)}{n(r+1)} \end{matrix} )

(4.21)

4.3 Associated Differential Operators

In this section the two different sets of projectors onto copies of S(M) related by the associativity isomorphism

(T M ⊗ T M) ⊗ S(M) ≅ T M ⊗ (T M ⊗ S(M))  (4.22)

are applied to the special section ∇^{2}ψ ∈ Γ(T M ⊗T M ⊗S(M)). This defines two sets of differential operators on sections of the spinor bundle related by the matrix W. In the sequel we determine all these differential operators.

On the right hand side of (4.22) four of the projections of ∇^{2}ψ are easily identified with expressions like \frac{1}{2}D_{+}^{−}D_{−}^{+}ψ. Due to the definition of the Clifford multiplication, the explicit expressions have an additional factor \frac{1}{2}.

pr_{−+} ⊗ pr_{−+} : ∇^{2}ψ ⟼ ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2}(dh_{a}⌟_{∘} ⊗ de_{i}⌟ )(dh_{b}^{♭} · ⊗de_{j}^{♭}∧_{∘})∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}ψ
= \frac{1}{2}D_{−}^{−}D_{+}^{+}ψ = −\frac{1}{2}(D_{+}^{+})^{∗}D_{+}^{+}ψ ,
pr_{+−} ⊗ pr_{−+} : ∇^{2}ψ ⟼ ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2}(dh_{a}^{♭} · ⊗de_{i}⌟ )(dh_{b}⌟_{∘} ⊗ de_{j}^{♭}∧_{∘})∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}ψ
= \frac{1}{2}D_{−}^{+}D_{+}^{−}ψ ,
pr_{−+} ⊗ pr_{+−} : ∇^{2}ψ ⟼ ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2}(dh_{a}⌟_{∘} ⊗ de_{i}^{♭}∧_{∘})(dh_{b}^{♭} · ⊗de_{j}⌟ )∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}ψ
= \frac{1}{2}D_{+}^{−}D_{−}^{+}ψ ,
pr_{+−} ⊗ pr_{+−} : ∇^{2}ψ ⟼ ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2}(dh_{a}^{♭} · ⊗de_{i}^{♭}∧_{∘})(dh_{b}⌟_{∘} ⊗ de_{j}⌟ )∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}ψ
= \frac{1}{2}D_{+}^{+}D_{−}^{−}ψ = −\frac{1}{2}(D_{−}^{−})^{∗}D_{−}^{−}ψ .

This is only a particular case of a more general phenomenon:

Lemma 4.2 Let V and Σ be the representations spaces for T M and S(M) and let p : V ⊗ Σ → W be an Sp(n)Sp(1)–equivariant mapping. Consider the equivariant mapping Φ : V ⊗ (V ⊗ Σ) → Σ given by id ⊗ p^{∗}p followed by contraction of the two V –factors. Then the differential operator Φ ∘ ∇^{2} is equal to −P^{∗}P, where P = p ∘ ∇.

This lemma can be applied to the remaining two projectors. To begin with Φ = pr_{−+} ⊗ pr_{K} we consider the multiplication m : H ⊗ Sym^{r}H → Sym^{r+1}H. Its adjoint m^{∗} is given by ∑ h_{i} ⊗ dh_{i}⌟_{∘}. The adjoint of the projection p̃r_{K} : E ⊗ Λ_{∘}^{n−r}E → K^{n−r} is simply the inclusion. By definition, pr_{K} : E ⊗ E ⊗ Λ_{∘}^{n−r}E → Λ_{∘}^{n−r}E is id_{E} ⊗ p̃r_{K} followed by contraction of the two E-factors. Similarly, it is easy to show that pr_{−+} : H ⊗ H ⊗ Sym^{r}H → Sym^{r}H is id_{H} ⊗ m^{∗}m followed by contraction. The assumption of the lemma is thus satisfied and we conclude that (pr_{−+} ⊗ pr_{K}) ∘ ∇^{2} = −(T^{+})^{∗}T^{+}.

The same argument goes through for the projector pr_{+−} ⊗ pr_{K}. The adjoint of the contraction operator c : H ⊗Sym^{r}H → Sym^{r−1}H, however, is given by c^{∗} = − ∑ h_{i} ⊗dh_{i}^{♭}·. Hence, (pr_{+−} ⊗pr_{K})∘∇^{2} = (T^{−})^{∗}T^{−}.

The computed projections of ∇^{2}ψ corresponding to the splitting on the right hand side of (4.22) can be collected as entries of a vector:

( − \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) .  (4.23)

To determine the differential operators occuring on the left hand side of the isomorphism, some technical lemmata are needed.

Lemma 4.3

∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} ((dh_{b}^{♭} · dh_{a}⌟_{∘} + dh_{a}^{♭} · dh_{b}⌟_{∘}) ⊗ σ_{E}(de_{i}^{♭}, de_{j}^{♭}))∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}ψ = \frac{r(r+2)}{n+2} \frac{κ}{4}ψ .

Proof. The left hand side factorizes over the projection of ∇^{2}ψ ∈ Γ(T M ⊗ T M ⊗ S(M)) onto Γ(Sym^{2}H ⊗ ℂ ⊗ S(M) ⊂ Λ^{2}T M ⊗ S(M)). Hence it is a curvature term, and only the H–symmetric part \frac{−κ}{8n(n+2)}R_{H} contributes according to Lemma 3.5.

∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} ((dh_{b}^{♭} · dh_{a}⌟_{∘} + dh_{a}^{♭} · dh_{b}⌟_{∘}) ⊗ σ_{E}(de_{i}^{♭}, de_{j}^{♭}))∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}

= \frac{1}{2} ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} ((dh_{b}^{♭} · dh_{a}⌟_{∘} + dh_{a}^{♭} · dh_{b}⌟_{∘}) ⊗ σ_{E}(de_{i}^{♭}, de_{j}^{♭}))

·\frac{−κ}{8n(n+2)}((h_{b} · h_{a}^{♯}⌟_{∘} + h_{a} · h_{b}^{♯}⌟_{∘}) ⊗ σ_{E}(e_{i}, e_{j}))

= \frac{−κ}{4(n+2)} ∑_{a,b=1}^{2} dh_{a}^{♭} · dh_{b}⌟_{∘}(h_{a} · h_{b}^{♯}⌟_{∘} + h_{b} · h_{a}^{♯}⌟_{∘})

= \frac{−κ}{4(n+2)} ∑_{a,b=1}^{2} (σ_{H}(dh_{b}^{♭}, h_{a})dh_{a}^{♭} · h_{b}^{♯}⌟_{∘} + dh_{a}^{♭} · h_{a} · dh_{b}⌟_{∘}h_{b}^{♯}⌟_{∘}

+dh_{a}^{♭} · h_{b} · dh_{b}⌟_{∘}h_{a}^{♯}⌟_{∘} + σ_{H}(dh_{b}^{♭}, h_{b})dh_{a}^{♭} · h_{a}^{♯}⌟_{∘})
= \frac{−κ}{4(n+2)}( − r − (r − 1)r − 2r)
= \frac{r(r+2)}{n+2} \frac{κ}{4} . □

Lemma 4.4

∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} (σ_{H}(dh_{a}^{♭}, dh_{b}^{♭}) ⊗ (de_{j}^{♭} ∧_{∘} de_{i}⌟ + de_{i}^{♭} ∧_{∘} de_{j}⌟ ))∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2} = \frac{(n+r+2)(n−r)}{n(n+2)} \frac{κ}{4} .

Proof. The situation here is exactly the same as in the preceeding lemma only with the roles of H and E interchanged. The result only depends on the E–symmetric part \frac{−κ}{8n(n+2)}R_{E} because R^{hyper} does not contribute as has been shown in Corollary 3.1.

∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} (σ_{H}(dh_{a}^{♭}, dh_{b}^{♭}) ⊗ (de_{j}^{♭} ∧_{∘} de_{i}⌟ + de_{i}^{♭} ∧_{∘} de_{j}⌟ ))∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}

= \frac{1}{2} ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} (σ_{H}(dh_{a}^{♭}, dh_{b}^{♭}) ⊗ (de_{j}^{♭} ∧_{∘} de_{i}⌟ + de_{i}^{♭} ∧_{∘} de_{j}⌟ ))
·\frac{−κ}{8n(n+2)}0(σ_{H}(h_{a}, h_{b}) ⊗ (e_{j} ∧_{∘} e_{i}^{♯}⌟ + e_{i} ∧_{∘} e_{j}^{♯}⌟ ))
= \frac{−κ}{4n(n+2)} ∑_{i,j=0}^{2n} de_{i}^{♭} ∧_{∘} de_{j}⌟ (e_{i} ∧_{∘} e_{j}^{♯}⌟ + e_{j} ∧_{∘} e_{i}^{♯}⌟ )
= \frac{−κ}{4n(n+2)} ∑_{i,j=0}^{2n} (de_{i}^{♭} ∧_{∘} e_{i} ∧_{∘} de_{j}⌟ e_{j}^{♯}⌟ + σ_{E}(de_{j}^{♭}, e_{i})de_{i}^{♭} ∧_{∘} e_{j}^{♯}⌟ + \frac{1}{r+1}de_{i}^{♭} ∧_{∘} de_{j} ∧_{∘} e_{i}^{♯}⌟ e_{j}^{♯}⌟

−de_{i}^{♭} ∧_{∘} e_{j} ∧_{∘} de_{j}⌟ e_{i}^{♯}⌟ + σ_{E}(de_{j}^{♭}, e_{j})de_{i}^{♭} ∧_{∘} e_{i}^{♯}⌟ + \frac{1}{r+1}de_{i}^{♭} ∧_{∘} de_{j} ∧_{∘} e_{j}^{♯}⌟ e_{i}^{♯}⌟ )
= \frac{−κ}{4n(n+2)} ∑_{i,j=0}^{2n} ( − (n − r) + (n − r − 1)(n − r) − 2n(n − r))
= \frac{(n+r+2)(n−r)}{n(n+2)} \frac{κ}{4} . □

Lemma 4.5

pr_{Sym^{2}H} ⊗ pr_{Λ_{∘}^{2}E}(∇^{2}ψ) = 0 .

Proof. The left hand side is the projection of ∇^{2}ψ onto Γ(Sym^{2}H ⊗ Λ_{∘}^{2}E ⊗ S(M)) ⊂ Γ(Λ^{2}T M ⊗ S(M)). Hence, it is again a part of the curvature tensor of the manifold. But Lemma 3.5 shows that this part has to vanish.

Using the preceeding lemmata we can give the projections of ∇^{2}ψ onto the S(M)–summands of the left–hand side of (T M ⊗ T M) ⊗ S(M) ≅ T M ⊗ (T M ⊗ S(M)) can be given:

pr_{ℂ} ⊗ pr_{ℂ} : ∇^{2}ψ ⟼ ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} σ_{H}(dh_{a}^{♭}, dh_{b}^{♭})σ_{E}(de_{i}^{♭}, de_{j}^{♭})∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}ψ
= −∇^{∗}∇ψ ,
pr_{Sym^{2}H} ⊗ pr_{ℂ} : ∇^{2}ψ ⟼ \frac{r(r+2)}{n+2} \frac{κ}{4}ψ ,
pr_{ℂ} ⊗ pr_{Sym^{2}E} : ∇^{2}ψ ⟼ \frac{(n+r+2)(n−r)}{n(n+2)} \frac{κ}{4}ψ ,
pr_{Sym^{2}H} ⊗ pr_{Sym^{2}E} : ∇^{2}ψ ⟼ ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2}(dh_{b}^{♭} · dh_{a}⌟_{∘} + dh_{a}^{♭} · dh_{b}⌟_{∘})
⊗(de_{j}^{♭} ∧_{∘} de_{i}⌟ + de_{i}^{♭} ∧_{∘} de_{j}⌟ )∇_{h_{a}⊗e_{i},h_{b}⊗e_{j}}^{2}ψ
=: Cψ ,
pr_{ℂ} ⊗ pr_{Λ_{∘}^{2}E} : ∇^{2}ψ ⟼ ∑_{i,j=0}^{2n} ∑_{a,b=1}^{2} σ_{H}(dh_{a}^{♭}, dh_{b}^{♭})
⊗(de_{j}^{♭} ∧_{∘} de_{i}⌟ ω − de_{i}^{♭} ∧_{∘} de_{j}⌟ ω − \frac{n−r}{n} σ_{E}(de_{i}^{♭}, de_{j}^{♭}))ψ
=: Lψ ,
pr_{Sym^{2}H} ⊗ pr_{Λ_{∘}^{2}E} : ∇^{2}ψ ⟼ 0 .

Here, C and L are 2nd order differential operators which are defined by the expressions above. As for the right hand the computed projections can be collected in a vector:

( − ∇^{∗}∇ψ \frac{r(r+2)}{n+2} \frac{κ}{4}ψ \frac{(n+r+2)(n−r)}{n(n+2)} \frac{κ}{4}ψ Cψ Lψ 0) .  (4.24)

Summarizing all results, the full Weitzenböck matrix equation reads as follows:

( \begin{matrix}−∇^{∗}∇ψ \\ \frac{r(r+2)}{n+2} \frac{κ}{4}ψ \\ \frac{(n+r+2)(n−r)}{n(n+2)} \frac{κ}{4}ψ \\ Cψ \\ Lψ \\ 0 \end{matrix} ) = W · ( \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} ) .  (4.25)

5 Proof of the Theorem

The proof of the lower bound for the Dirac spectrum is an easy application of our Weitzenböck matrix W. Nevertheless, before deriving this result we want to give a few other consequences showing how much information is contained in W.

¿From equation (4.25) it is clear that the matrix W yields six lineary independent Weitzenböck formulas involving Dirac and twistor operators. Since we have no further information about the operators C and L, we can use only four of them and look for suitable linear combinations to eliminate certain operators. This is easily done by multiplying equation (4.25) from the left by a row vector (a_{1} … a_{6}), where the a_{j} are arbitrary coefficients. For example, we can eliminate the two twistor operators T^{±} by multiplying the second equation with \frac{r}{n} and subtracting the last. This is obtained by multiplying with (0 \frac{r}{n} 0 0 0 − 1), and thus the corresponding Weitzenböck formula can be written as

\frac{r}{2} D_{−}^{−} D_{+}^{+} + \frac{r(r+2)}{2(r+1)} D_{−}^{+} D_{+}^{−} + \frac{r^{2}}{2(r+1)} D_{+}^{−} D_{−}^{+} + \frac{r^{2}(r+2)}{2(r+1)^{2}} D_{+}^{+} D_{−}^{−} = \frac{r^{2}(r+2)}{n(n+2)} \frac{κ}{4} .  (5.26)

This equation also follows from a theorem of Y. Nagatomo and T. Nitta. In [Nag96] they consider Dirac and twistor operators on arbitrary bundles Sym^{p}H ⊗ Λ^{q}E and prove a general Weitzenböck formula.

Since we have four independent equations for six unknowns (D_{−}^{−}D_{+}^{+}ψ, D_{−}^{+}D_{+}^{−}ψ etc.) it is possible to eliminate all four twistor operators at once. The corresponding row vector for r ≠ 0 is (−1 \frac{1}{n} 1 0 0 − \frac{1}{r}), and we obtain the well–known Lichnerowicz formula:

∇^{∗}∇ + \frac{κ}{4} = D_{−}^{+} D_{+}^{−} + D_{+}^{−} D_{−}^{+} = D^{2} .

Before coming to the most important application of this formalism, the proof of the eigenvalue estimate, let’s have a look on the particular form of eigenspinors of D. Since D^{2} respects the splitting into the subbundles S_{r}(M), some assumptions on the form of an eigenspinor of D can be made. Let ψ_{r} ∈ Γ(S_{r}(M)) be an eigenspinor for D^{2} with eigenvalue λ^{2}. Under D, it is mapped to Dψ_{r} =: ψ_{r−1} + ψ_{r+1} ∈ Γ(S_{r−1}(M) ⊕ S_{r+1}(M)). Because of D^{2}ψ_{r} ∈ Γ(S_{r}(M)) we get D_{+}^{−}ψ_{r−1} = 0 = D_{−}^{+}ψ_{r+1}. But starting with ψ_{r−1}, which itself is an eigenspinor of D^{2} with the eigenvalue λ^{2}, we see that λψ_{r−1} ± Dψ_{r−1} ∈ Γ(S_{r−1}(M) ⊕ S_{r}(M)) is an eigenspinor of D with eigenvalue ±λ. The analogous argument goes through with ψ_{r+1}. Hence, we always can assume that an eigenspinor for D is of the form ψ = ψ_{r} + ψ_{r+1} ∈ Γ(S_{r}(M) ⊕ S_{r+1}(M)).

Theorem 5.1 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{n + r + 3}{n + 2} \frac{κ}{4} .

Proof. We apply formula (4.25) to the spinor ψ_{r} and multiply it from the left with the row vector (0 \frac{n+r+2}{n} r + 2 0 0 − \frac{r+2}{r}). This eliminates the operators T^{−} and D_{−}^{−} and we find after taking the L^{2}-product with ψ_{r}:

\begin{matrix}− \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} \\ = \frac{(r+2)(n+r+2)}{n+2} \frac{κ}{4} ∥ψ_{r}∥^{2} . \end{matrix}

Using D_{+}^{−}ψ_{r} = 0 and estimating ∥D_{+}^{+}ψ∥^{2} and ∥T^{+}ψ∥^{2} by zero leads to the inequality

∥D_{−}^{+}ψ_{r}∥^{2} ≥ \frac{n+r+3}{n+2} \frac{κ}{4} ∥ψ_{r}∥^{2} .

Applying formula (4.25tothespinorψ_{r+1} and using D_{−}^{+}ψ_{r+1} = 0 we obtain the same inequality:

∥D_{−}^{+}ψ_{r+1}∥^{2} ≥ \frac{n+r+3}{n+2} \frac{κ}{4} ∥ψ_{r+1}∥^{2} . □

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