The Weitzenböck Machine

Uwe Semmelmann & Gregor Weingart

August 15, 2007

1 Introduction

Weitzenböck formulas are an important tool for linking differential geometry and topology of compact Riemannian manifolds. They feature prominently in the Bochner method, where they are used to prove the vanishing of Betti numbers under suitable curvature assumptions or the non–existence of metrics of positive scalar curvature on spin manifolds with non–vanishing Â–genus. Moreover they are used to proof eigenvalue estimates for Laplace and Dirac type operators. In these applications one tries to find a (positive) linear combination of hermitean squares D^{∗}D of first order differential operators D, which sums to an expression in the curvature only. In this approach one need only consider special first order differential operators D known as generalized gradients or Stein–Weiss operators, which are defined as minimal projections of a covariant derivative ∇. Examples for generalized gradients include the exterior derivative d and its adjoint d^{∗} and the Dirac and twistor operator in spin geometry.

In this article we present two different classifications of all possible linear combinations of hermitean squares D^{∗}D of generalized gradients D, which sum to pure curvature expressions, if the underlying connection is the Levi–Civita connection ∇ of a Riemannian manifold M of reduced holonomy SO(n), U(n), SU(n) or the exceptional holonomies G_{2} and Spin(7). Both classifications are interesting in their own right, the first describes a recursive procedure to calculate a generating set of Weitzenböck formulas, the second classification provides a simple means to decide, whether a given linear combination of hermitean squares of Stein–Weiss operators is actually a pure curvature expression.

In order to describe the setup of the article in more detail we recall that every representation G ⟶ Aut V of the holonomy group G of a Riemannian manifold (M, g) on a complex vector space V defines a complex vector bundle V M on M with a covariant derivative induced from the Levi–Civita connection, in particular the complexified holonomy representation T of G defines the complexified tangent bundle T M. The generalized gradients on V M are the parallel first order differential operators T_{ε} defined as the projection of ∇ : V M ⟶ T M ⊗ V M to the parallel subbundles V_{ε}M ⊂ T M ⊗ V M arising from a decomposition T ⊗ V = ⊕_{ε}V_{ε} into irreducible subspaces. It will be convenient in this article to call every (finite) linear combination ∑_{ε} c_{ε}T_{ε}^{∗}T_{ε} of hermitean squares of generalized gradients a Weitzenböck formula.

It is easy to see that the space 𝔚(V) of all Weitzenböck formulas on a vector bundle V M has a canonical involution, the twist τ : 𝔚(V) ⟶ 𝔚(V), such that a Weitzenböck formula reduces to a pure curvature expression if and only if it is an eigenvector of τ of eigenvalue −1. Of course there are interesting Weitzenböck formulas, which are eigenvectors of τ for the eigenvalue +1, perhaps the most prominent example is the connection Laplacian ∇^{∗}∇. The classical examples of Weitzenböck formulas like

Δ = d d^{∗} + d^{∗} d = ∇^{∗}∇ + q(R)

the original Weitzenböck formula or the Lichnerowicz–Weitzenböck formula

D^{2} = ∇^{∗}∇ + \frac{κ}{4}

reduce in this setting to the statements that Δ − ∇^{∗}∇ and D^{2} − ∇^{∗}∇ respectively are eigenvectors of τ for the eigenvalue −1 and thus pure curvature expressions.

Starting with the connection Laplacian ∇^{∗}∇ we will describe a recursion procedure to construct a basis of the space 𝔚(V_{λ}) of Weitzenböck formulas on an irreducible vector bundle V_{λ}M on M such that the base vectors are eigenvectors of τ with alternating eigenvalues ±1. Interestingly this recursive procedure makes essential use of a second fundamental Weitzenböck formula B ∈ 𝔚(V), the so–called conformal weight operator, which was considered for the first time in the work of Paul Gauduchon on conformal geometry ^{\underline{pg1}} [G91]. The details of this recursion procedure and the first initial elements are discussed for the holonomies SO(n), G_{2} and Spin(7) only, because the discussion of Weitzenböck formulas in the Kähler holonomies U(n) and SU(n) is better done differently, whereas the hyperkähler holonomies Sp(1) Sp(n) and Sp(n) will be discussed in more detail in a forthcoming paper.

We will start to construct the basis of τ–eigenvectors in End_{𝔤}(T ⊗ V) by a recursive procedure with id and B as first elements. It will be necessary to consider a further decomposition of the (±1)–eigenspaces of the involution τ. If we decompose the G– representation T ⊗T = ∑_{α} W_{α} into irreducible summands, then the space of Weitzenböck formulas decomposes as End_{𝔤}(T ⊗ V) ≅ ⊕_{α}Hom_{g}(W_{α}, End V). Our recursion procedure states that for a F ∈ Hom_{g}(W_{a}, End V), which is then automatically a τ–eigenvector, the equivariant homomorphism F_{new} := (B + c_{W_{α}})F is again a τ–eigenvector for the opposite sign. Here c_{W_{α}} is some explicit constant depending only on the representation W_{α}. In particular we find 1_{new} = B.

However in general the homomorphism F_{new} does not belong to a single summand Hom_{𝔤}(W_{α}, End V) and a further argument will be necessary. For holonomy SO(n) it suffices to use the projection onto the orthogonal complement of the summand 1, which by Schur’s Lemma is isomorphic to ℂ, generated by id_{T ⊗V}. In case of holonomy G_{2} and Spin(7) we need additional projections and equivariant homomorphisms in order to proceed with the recursion procedure. Surprisingly this can be obtained by using an interesting link with the universal enveloping algebra of 𝔤. The crucial ingredients are here the surjective algebra homomorphism Hom_{𝔤}(T ⊗ T, U𝔤) ⟶ Hom_{g}(T ⊗ T, End V) and a theorem of Kostant on the decomposition of U 𝔤 as a G–representation.

Eventually we obtain a sequence of B–polynomials p_{i}(B) such that p_{2i}(B) is in the (+1)– and p_{2i−1}(B) is in the (−1)–eigenspace of τ. If b_{ε} are the B-eigenvalues on V_{ε} ⊂ T × V, then the coefficient of T_{ε}^{∗}T_{ε} in the Weitzenböck formula corresponding to p_{i}(B) is given by p_{i}(b_{ε}). An interesting feature appears for holonomy G_{2} and Spin(7). Here we have the decomposition Hom_{𝔤}(Λ^{2}T, End V) ≅ Hom_{g}(𝔤, End V) ⊕ Hom_{𝔤}(𝔤^{⊥}, End V) and because of the holonomy reduction any Weitzenböck formula in the second summand has a zero curvature term.

Finally we will give a short description of the separate sections. In Section ^{\underline{tables}} 2 we consider the decomposition of T ⊗V into irreducible summands. In particular we include tables from which for all relevant holonomies the decomposition can be read off in terms of highest weights.

In Section ^{\underline{wspace}} 3 we introduce the space of Weitzenböck formulas, i. e. the space End_{𝔤}(T ⊗ V) with its two different identifications. We define the twistor operators and the conformal weight operator B. In Corollary ^{\underline{confW}} 3.4 we give an explicit formula for the B–eigenvalues and derive in Proposition ^{\underline{universal}} 3.6 the universal Weitzenböck formula corresponding to B.

2 The Holonomy Representation

tables

For the rest of this article we will essentially restrict to irreducible non–symmetric holonomy algebras 𝔤. Most of the statements easily generalize to holonomy algebras 𝔤 with no symmetric irreducible factor in their local de Rham–decomposition, which could be called properly non–symmetric holonomy algebras. Some of concepts introduced are certainly interesting for symmetric holonomy algebras as well, in particular the central idea used to find the matrix of the twist through the Recursion Formula ^{\underline{rec}} 4.1. Turning to irreducible non–symmetric holonomy algebras 𝔤 leaves us with seven different cases

algebra 𝔤_{ℝ} holonomy representation T_{ℝ}  T
general Riemannian  𝔰𝔬_{n}  defining representation ℝ^{n}  T
Kähler  𝔲_{n} ≅ iℝ ⊕ 𝔰𝔲_{n} defining representation ℂ^{n}  Ē ⊕ E
Calabi–Yau  𝔰𝔲_{n}  defining representation ℂ^{n}  Ē ⊕ E
quaternionic Kähler 𝔰𝔭(1) ⊕ 𝔰𝔭(n) representation ℍ^{1} ⊗_{ℍ} ℍ^{n}  H ⊗ E
hyper–Kähler  𝔰𝔭(n)  defining representation ℍ^{n} ℂ^{2} ⊗ E
exceptional G_{2}  𝔤_{2}  standard representation ℝ^{7}  [7]
exceptional Spin(7)  𝔰𝔭𝔦𝔫_{7}  spinor representation ℝ^{8}  [8]

(2.1) ht

according to a theorem of Berger, where T denotes the complexified holonomy representation T := T_{ℝ}⊗_{ℝ}ℂ endowed with the ℂ–bilinear extension ⟨ , ⟩ of the scalar product. For simplicity we will work with the complexified holonomy representation T and the complexified holonomy algebra 𝔤 := 𝔤_{ℝ} ⊗_{ℝ} ℂ throughout as well as with irreducible complex representations V_{λ} of 𝔤 of highest weight λ. Notations like E or H in the table above fix nomenclature for particularly important representations in special holonomy, say E and Ē refer to the spaces of (1, 0)– and (0, 1)–vectors in T in the Kähler and Calabi–Yau case, while [7] and [8] are the standard 7–dimensional representation of G_{2} and 8–dimensional spinor representation of Spin(7) respectively. In passing we note that the complexified holonomy representation T is not isotypical in the Kähler and the Calabi–Yau case and this is precisely the reason why these two cases differ significantly from the rest.

In order to understand Weitzenböck formulas or parallel second order differential operators it is a good idea to start with parallel first order differential operators usually called generalized gradients or Stein–Weiss operators. Their representation theoretic background is the decomposition of tensor products T ⊗V of the holonomy representation T with an arbitrary complex representation V. The general case immediately reduces to studying irreducible representations V = V_{λ} of highest weight λ. In this section we will see that the isotypical components of T ⊗ V_{λ} are always irreducible for a properly non–symmetric holonomy algebra 𝔤 and isomorphic to irreducible representations V_{λ+ε} of highest weight λ + ε for some weight ε of the holonomy representation T. Thus the decomposition of T ⊗ V_{λ} is completely described by the subset of relevant weights ε:

weights Definition 2.1 (Relevant Weights)

A weight ε of the holonomy representation T is called relevant for an irreducible representation V_{λ} of highest weight λ if the irreducible representation V_{λ+ε} of highest weight λ + ε occurs in the tensor product T ⊗ V_{λ}. We will write ε ⊂ λ for a relevant weight ε for a given irreducible representation V_{λ}.

relw Lemma 2.2 (Characterization of Relevant Weights)

Consider the holonomy representation T of an irreducible non–symmetric holonomy algebra 𝔤 and an irreducible representation V_{λ} of highest weight λ. The decomposition of the tensor product T ⊗ V_{λ} is multiplicity free in the sense that all irreducible subspaces are pairwise non–isomorphic. The complete decomposition of T ⊗ V_{λ} is thus the sum

T ⊗ V_{λ} ≅ ⨁_{ε⊂λ} V_{λ+ε}

over all relevant weights ε. A weight ε ≠ 0 is relevant if and only if λ + ε is dominant. The zero weight ε = 0 only occurs for the holonomy algebras 𝔰𝔬_{n} with n odd and 𝔤_{2}, it is relevant if λ − λ_{Σ} or λ − λ_{T} respectively is still dominant, where λ_{Σ} and λ_{T} are the highest weights of the spinor representation of 𝔰𝔬_{n} and the standard representation of 𝔤_{2}.

Proof: The proof is essentially an exercise in Weyl’s character formula □

A particular consequence of Lemma ^{\underline{relw}} 2.2 is that for sufficiently complicated representations V_{λ} all weights ε of the holonomy representation T are relevant. With this motivation we will call a highest weight λ generic if λ + ε is dominant for all weights ε of the holonomy representation T. A simple consideration shows that λ is generic if and only if λ − ρ is dominant, where ρ is the half sum of positive roots or equivalently the sum of fundamental weights, unless we consider odd–dimensional generic holonomy 𝔤 = 𝔰𝔬_{2r+1} or 𝔤 = 𝔤_{2}. In the latter holonomies the generic weights λ must have λ − ρ − λ_{Σ} or λ − ρ − λ_{T} dominant respectively. In any case the number of relevant weights for the representation V_{λ}

N(G, λ) := ♯ { ε | ε is relevant for λ } ≤ dim T

is bounded above by dim T with equality if and only if λ is generic. In particular there are at most dim T summands in the decomposition of T ⊗ V_{λ} into irreducibles, exactly one copy of V_{λ+ε} for every relevant weight ε.

On the other hand the number N(G, λ) of irreducible summands in the decomposition of T ⊗ V_{λ} agrees with the dimension of the algebra End_{𝔤}( T ⊗ V_{λ}) of 𝔤–invariant endomorphisms of T ⊗V_{λ}, because all isotypical components are irreducible by Lemma ^{\underline{relw}} 2.2. In the next section we will study the identification End_{𝔤}( T ⊗V_{λ}) = Hom_{𝔤}( T ⊗T, End V_{λ}) extensively, which allows us to break up End_{𝔤}( T ⊗ V_{λ}) into interesting subspaces called Weitzenböck classes, whose dimension can be calculated in the following way:

Lemma 2.3 (Dimension of Weitzenböck Classes)

generic

Let us call the space W^{𝔱} := Hom_{𝔱}(ℝ, W) ⊂ W of elements of a G–representation W invariant under a fixed Cartan subalgebra 𝔱 ⊂ 𝔤 the zero weight space of W. The dimension of the zero weight space provides an upper bound

dim Hom_{𝔤}( W, End V_{λ} ) ≤ dim W^{𝔱}

for the dimension of the space Hom_{𝔤}( W, End V_{λ}) for an irreducible representation V_{λ}. For sufficiently dominant highest weight λ in dependence on W this upper bound is sharp.

The lemma follows again from the Weyl character formula, but it is also an elementary consequence of Kostant’s theorem ^{\underline{kostant}} 8.3 formulated below. We will mainly use Lemma ^{\underline{generic}} 2.3 for the subspaces W_{α} occuring in the decomposition T ⊗ T = ⊕ W_{α} into irreducibles. In the case of the holonomy algebras 𝔰𝔬_{n}, 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7} we have the decomposition T ⊗ T = ℂ ⊕ Sym_{0}^{2}T ⊕ 𝔤 ⊕ 𝔤^{⊥} and the following dimensions of the zero weight spaces:

dim T dim [ ℂ]^{𝔱} dim [ Sym_{0}^{2}T ]^{𝔱} dim [ 𝔤]^{𝔱} dim [ 𝔤^{⊥}]^{𝔱}
𝔰𝔬^{n}  n  1  ⌊\frac{n−1}{2}⌋  ⌊\frac{n}{2}⌋  —
𝔤_{2}  7  1  2  3  1
𝔰𝔭𝔦𝔫^{7}  8  1  3  3  1

(2.2) wt

Note in particular that the dimensions of the zero weight spaces sum up to dim T.

Although complete the decision criterion given in Lemma ^{\underline{relw}} 2.2 is not particularly straightforward in general. At the end of this section we want to give a graphic interpretation of this decision criterion for all irreducible holonomy groups in order to simplify the task of finding the relevant weights. For a fixed holonomy algebra 𝔤 the information necessary in this graphic algorithm is encoded in a single diagram featuring the weights of the holonomy representation T and labeled boxes. A weight ε is relevant for an irreducible representation V_{λ} if and only if the highest weight λ = λ_{1}ω_{1} + … + λ_{r}ω_{r} of V_{λ} written as a linear combination of fundamental weights ω satisfies all inequalities labeling the boxes containing ε. The notation introduced for the weights ε of the holonomy representation T and the fundamental weights ω will be used throughout this article.

To begin with let us consider even dimensional Riemannian geometry with generic holonomy 𝔤 = 𝔰𝔬_{2r}, r ≥ 2. In this case the holonomy representation T is the defining representation, whose weights ±ε_{1}, ±ε_{2}, …, ±ε_{r} form an orthonormal basis for a suitable scalar product ⟨ , ⟩ on the dual 𝔱^{∗} of the maximal torus. The ordering of weights can be chosen in such a way that the fundamental weights ω_{1}, …, ω_{r} are given by:

ω_{1} = ε_{1}  ±ε_{1} = ± ω_{1}
ω_{2} = ε_{1} + ε_{2}  ±ε_{2} = ±(ω_{2} − ω_{1})
⋮  ⋮  ⋮  ⋮
ω_{r−2} = ε_{1} + … + ε_{r−2}  ±ε_{r−2} = ±(ω_{r−2} − ω_{r−3})
ω_{r−1} = \frac{1}{2}(ε_{1} + … + ε_{r−1} + ε_{r})  ±ε_{r−1} = ±(ω_{r−1} + ω_{r} − ω_{r−2})
ω_{r} = \frac{1}{2}(ε_{1} + … + ε_{r−1} − ε_{r})  ±ε_{r} = ±(ω_{r−1} − ω_{r})

Every dominant integral weight of 𝔰𝔬_{2r} can be written λ = λ_{1}ω_{1} +…+λ_{r}ω_{r} with natural numbers λ_{1}, …, λ_{r} ≥ 0 and the criterion of Lemma ^{\underline{relw}} 2.2 becomes:

+ε_{1}  λ_{1} ≥ 1 −ε_{1} +ε_{2}  λ_{2} ≥ 1 −ε_{2} +ε_{3}  ·············· λ_{r−2} ≥ 1 −ε_{r−2} +ε_{r−1} λ_{r} ≥ 1 −ε_{r−1} +ε_{r}  λ_{r−1} ≥ 1 −ε_{r}

A weight ε of the holonomy representation T of 𝔰𝔬_{2r} is relevant for the irreducible representation V_{λ} if and only if λ satisfies all the conditions labeling the boxes containing ε. Say the weights −ε_{1} and +ε_{2} are relevant for all irreducible representations V_{λ} with λ_{1} ≥ 1, whereas −ε_{r−1} is relevant for V_{λ} if and only if λ_{r−1} ≥ 1 and λ_{r} ≥ 1.

Odd dimensional Riemannian geometry 𝔤 = 𝔰𝔬_{2r+1}, r ≥ 1, with generic holonomy is of course closely related to 𝔤 = 𝔰𝔬_{2r}. The weights ±ε_{1}, ±ε_{2}, …, ±ε_{r} of the defining holonomy representation T besides the zero weight form an orthonormal basis for a suitable scalar product ⟨ , ⟩ on the dual 𝔱^{∗} of the maximal torus. With a suitable choice of ordering of weights the fundamental weights ω_{1}, …, ω_{r} and the weights of T relate via:

ω_{1} = ε_{1}  ±ε_{1} = ± ω_{1}
ω_{2} = ε_{1} + ε_{2}  ±ε_{2} = ±(ω_{2} − ω_{1})
⋮  ⋮  ⋮  ⋮
ω_{r−1} = ε_{1} + … + ε_{r−1}  ±ε_{r−2} = ±(ω_{r−1} − ω_{r−2})
ω_{r} = \frac{1}{2}(ε_{1} + … + ε_{r−1} + ε_{r})  ±ε_{r} = ±(2ω_{r} − ω_{r−1})

Writing a dominant integral weight λ = λ_{1}ω_{1} + … + λ_{r}ω_{r} as a linear combination of fundamental weights with integers λ_{1}, …, λ_{r} ≥ 0 the criterion of Lemma ^{\underline{relw}} 2.2 becomes:

+ε_{1}  λ_{1} ≥ 1 −ε_{1} +ε_{2}  λ_{2} ≥ 1 −ε_{2} +ε_{3}  ·············· λ_{r−2} ≥ 1 −ε_{r−2} +ε_{r−1} λ_{r−1} ≥ 1 −ε_{r−1} +ε_{r}  λ_{r} ≥ 1 −ε_{r} 0  λ_{r} ≥ 2

Turning from the Riemannian case to the Kähler case 𝔤 = 𝔲_{n} we observe that the weights ±ε_{1}, …, ±ε_{n} of the defining standard representation T = E ⊕ Ē form an orthonormal basis for an invariant scalar product on the dual 𝔱^{∗} of a maximal torus 𝔱 ⊂ 𝔲_{n}, but they become linearly dependent when projected to the dual of a maximal torus of the ideal 𝔰𝔲_{n} ⊂ 𝔲_{n}. In any case the fundamental weights and the weights of T relate as

ω_{1} = ε_{1}  ±ε_{1} = ± ω_{1}
ω_{2} = ε_{1} + ε_{2}  ±ε_{2} = ±(ω_{2} − ω_{1})
⋮  ⋮  ⋮  ⋮
ω_{n} = ε_{1} + … + ε_{n}  ±ε_{n} = ±(ω_{n} − ω_{n−1})

and the criterion of Lemma ^{\underline{relw}} 2.2 becomes:

+ε_{1}  λ_{1} ≥ 1 −ε_{1} +ε_{2}  λ_{2} ≥ 1 −ε_{2} +ε_{3}  ·············· λ_{n−2} ≥ 1 −ε_{n−2} +ε_{n−1} λ_{n−1} ≥ 1 −ε_{n−1} +ε_{n}  −ε_{n}

The quaternionic Kähler and hyperkähler cases are more complicated, because the condition of being relevant has to be checked for both ideals 𝔰𝔭(1) and 𝔰𝔭(n) of 𝔤. For a single ideal however the condition becomes simple again. The weights ±ε_{1}, …, ±ε_{n} of E are again orthonormal for a suitable scalar product ⟨ , ⟩ on the dual 𝔱^{∗} of a maximal torus 𝔱 ⊂ 𝔰𝔭(r) for r = 1 or r = n and relate to the fundamental weights by the formulas:

ω_{1} = ε_{1}  ±ε_{1} = ± ω_{1}
ω_{2} = ε_{1} + ε_{2}  ±ε_{2} = ±(ω_{2} − ω_{1})
⋮  ⋮  ⋮  ⋮
ω_{r} = ε_{1} + … + ε_{r}  ±ε_{r} = ±(ω_{r} − ω_{r−1})

The graphical interpretation of Lemma ^{\underline{relw}} 2.2 is given by the diagram:

+ε_{1}  λ_{1} ≥ 1 −ε_{1} +ε_{2}  λ_{2} ≥ 1 −ε_{2} +ε_{3}  ·············· λ_{r−2} ≥ 1 −ε_{r−2} +ε_{r−1} λ_{r−1} ≥ 1 −ε_{r−1} +ε_{r}  λ_{r} ≥ 1 −ε_{r}

Finally we consider the two exceptional cases 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7}. Recall that the group G_{2} is the group of automorphisms of the octonions 𝕆 as an algebra over ℝ. In this sense the holonomy representation T_{ℝ} is the defining representation Im 𝕆 of 𝔤_{2} with complexification T = [7]. There are too many weights of the holonomy representation to be orthonormal for any scalar product on the dual 𝔱^{∗} of a fixed maximal torus 𝔱 ⊂ 𝔤_{2}, but at least we can choose an ordering of weights for 𝔱^{∗} so that the weights of T become totally ordered +ε_{1} > +ε_{2} > +ε_{3} > 0 > −ε_{3} > −ε_{2} > −ε_{1}. In this notation we have:

ω_{1} = ε_{1}  ±ε_{1} = ± ω_{1}
ω_{2} = ε_{1} + ε_{2}  ±ε_{2} = ±(ω_{2} − ω_{1})
±ε_{3} = ∓(ω_{2} − 2ω_{1})

The scalar product of choice on 𝔱^{∗} is specified by ⟨ ε_{1}, ε_{1} ⟩ = 1 = ⟨ ε_{2}, ε_{2} ⟩ and ⟨ ε_{1}, ε_{2} ⟩ = \frac{1}{2}. Writing a dominant integral weight as λ = aω_{1} + bω_{2}, a, b ≥ 0, we read Lemma ^{\underline{relw}} 2.2 as:

+ε_{1}  a ≥ 1 −ε_{1} +ε_{2} 0 a ≥ 2  −ε_{3}  b ≥ 1 −ε_{2} +ε_{3}

The holonomy representation of the holonomy algebra 𝔤 = 𝔰𝔭𝔦𝔫_{7} is the 8–dimensional spinor representation T = [8]. It is convenient to write the weights ±ε_{1}, …, ±ε_{4} of T and the fundamental weights ω_{1}, ω_{2} and ω_{3} in terms of the weights ±η_{1}, ±η_{2}, ±η_{3}, 0 of the 7–dimensional defining representation of 𝔰𝔭𝔦𝔫_{7}, which form an orthonormal basis for a suitable scalar product on the dual 𝔱^{∗} of the maximal torus. With this proviso the weights ±ε_{1}, …, ±ε_{4} of T and the fundamental weights ω_{1}, ω_{2} and ω_{3} can be written as:

ω_{1} = η_{1}  ±ε_{1} = ±\frac{1}{2}(η_{1} + η_{2} + η_{3}) = ± ω_{3}
ω_{2} = η_{1} + η_{2}  ±ε_{2} = ±\frac{1}{2}(η_{1} + η_{2} − η_{3}) = ±(ω_{2} − ω_{3})
ω_{3} = \frac{1}{2}(η_{1} + η_{2} + η_{3})  ±ε_{3} = ±\frac{1}{2}(η_{1} − η_{2} + η_{3}) = ±(ω_{3} − ω_{2} + ω_{1})
±ε_{4} = ±\frac{1}{2}(η_{1} − η_{2} − η_{3}) = ±(ω_{1} − ω_{3})

and Lemma ^{\underline{relw}} 2.2 for a dominant integral weight λ = aω_{1} + bω_{2} + cω_{3} translates into:

+ε_{1}  c ≥ 1 −ε_{1} +ε_{2} −ε_{3} +ε_{4}  b ≥ 1 −ε_{2} +ε_{3} a ≥ 1  −ε_{4}

3 The Space 𝔚(V) of Weitzenböck Formulas

wspace

In this section we will define twistor operators, Weitzenböck formulas and the space of Weitzenböck formula with its different realizations. Then we will introduce the conformal weight operator, which in many cases generates all possible Weitzenböck formulas. Finally we define the classifying endomorphism and study the corresponding eigenspace decomposition.

3.1 Weitzenböck Formulas

We consider parallel second order differential operators P on sections of a vector bundle V M over a Riemannian manifold M with special holonomy G. By definition these are differential operators, which up to first order differential operators can always be written as the composition

Γ(V M) ⟶^{∇^{2}} Γ(T^{∗}M ⊗ T^{∗}M ⊗ V M) ⟶^{≅} Γ(T M ⊗ T M ⊗ V M) ⟶^{F} Γ(V M)

where F is a parallel section of the vector bundle Hom (T M ⊗ T M ⊗ V M, V M) corresponding to a G–equivariant homomorphism F ∈ Hom_{G}(T ⊗ T ⊗ V, V). A particularly important example is the connection Laplacian ∇^{∗}∇ which arises from the linear map a ⊗ b ⊗ ψ ⟼ −⟨ a, b ⟩ψ. Note that we are only considering reduced holonomy groups G, which are connected by definition, so that G–equivariance is equivalent to 𝔤–equivariance. Taking advantage of this fact we describe other parallel differential operators by means of the following identifications of spaces of invariant homomorphisms:

Hom_{𝔤}( T ⊗ T ⊗ V, V ) = Hom_{𝔤}( T ⊗ T, End V ) = End_{𝔤}( T ⊗ V )

Of course the identification Hom_{𝔤}(T ⊗ T ⊗ V, V) = Hom_{𝔤}(T ⊗ T, End V) is the usual tensor shuffling F (a ⊗ b ⊗ v) = F_{a⊗b}v for all a, b ∈ T and v ∈ V. The second identification Hom_{𝔤}(T ⊗ T ⊗ V, V) = End_{𝔤}(T ⊗ V) depends on the existence of a 𝔤– invariant scalar product on T or the musical isomorphism T ≅ T^{∗} via a summation

F ( b ⊗ v) = ∑_{μ} t_{μ} ⊗ F (t_{μ} ⊗ b ⊗ v) F (a ⊗ b ⊗ v) = (⟨ a, · ⟩ ⌟ ⊗ id) F (b ⊗ v)

over an orthonormal basis {t_{μ}}. Under this identification the identity of T ⊗V becomes the homomorphism a ⊗ b ⊗ ψ ⟼ ⟨ a, b ⟩ψ corresponding to the connection Laplacian −∇^{∗}∇. The composition of endomorphisms turns End_{𝔤}( T ⊗ V ) and thus Hom_{𝔤}(T ⊗ T, End V) into an algebra, for F, F̃ ∈ Hom_{𝔤}(T ⊗ T, End V) the resulting algebra structure reads:

( F ∘ F̃)_{a⊗b} = ∑_{μ} F_{a⊗t_{μ}} ∘ F̃_{t_{μ}⊗b}  (3.3) basi

Last but not least we note that the invariance condition for F ∈ Hom_{𝔤}( T ⊗ T, End V ) is equivalent to the identity [ X, F_{a⊗b}] = F_{Xa⊗b} + F_{a⊗Xb} for all X ∈ 𝔤 and a, b ∈ T.

Assuming that V = V_{λ} is irreducible of highest weight λ we know from Lemma ^{\underline{relw}} 2.2 that the isotypical components of T ⊗ V_{λ} are irreducible for non–symmetric holonomy groups. The algebra End_{𝔤}( T ⊗ V_{λ}) is thus commutative and spanned by the pairwise orthogonal idempotents pr_{ε} projecting onto the irreducible subspaces V_{λ+ε} of T ⊗ V_{λ}. In order to describe the corresponding second order differential operators we introduce first order differential operators T_{ε} known as Stein–Weiss operators or generalized gradients by:

T_{ε} : Γ( V_{λ}M ) ⟶ Γ( V_{λ+ε}M ), ψ ⟼ pr_{ε}(∇ψ) .

A typical example of a Stein–Weiss operator is the twistor operator of spin geometry, which projects the covariant derivative of a spinor onto the kernel of the Clifford multiplication. Straightforward calculations show that the second order differential operator associated to the idempotent pr_{ε} is the composition of T_{ε} with its formal abjoint operator T_{ε}^{∗} : Γ(V_{λ+ε}M) ⟶ Γ(V_{λ}M) in the sense pr_{ε}(∇^{2}) = −T_{ε}^{∗}T_{ε} compare ^{\underline{uwe}} [S06]. In consequence we can write the second order differential operator F (∇^{2}) associated to F ∈ Hom_{𝔤}( T ⊗ T, V_{λ}) as a linear combination of the squares of Stein–Weiss operators:

F ( ∇^{2} ) = − ∑_{ε} f_{ε} T_{ε}^{∗}T_{ε} .  (3.4) nabla

In fact with End_{𝔤}( T ⊗ V_{λ}) = Hom_{𝔤}( T ⊗ T, End V_{λ}) being spanned by the idempotents pr_{ε} every F ∈ End_{𝔤}( T ⊗V_{λ}) expands as F = ∑_{ε} f_{ε} pr_{ε} with coefficients f_{ε} determined by F |_{V_{λ+ε}} = f_{ε} id. A particular instance of equation (^{\underline{nabla}} 3.4) is the identity ∇^{∗}∇ = ∑_{ε} T_{ε}^{∗}T_{ε} associated to the expansion id_{T ⊗V_{λ}} = ∑_{ε} pr_{ε}. Motivated by this and other well–known identities of second order differential operators of the form (^{\underline{nabla}} 3.4) we will in general call all elements F ∈ Hom_{𝔤}( T ⊗ T ⊗ V, V ) = End_{𝔤}(T ⊗ V) Weitzenböck formulas:

stiso Definition 3.1 (Space of Weitzenböck Formulas on V M)

The Weitzenböck formulas on a vector bundle V M correspond bijectively to vectors in:

𝔚( V ) := Hom_{𝔤}( T ⊗ T ⊗ V, V ) = Hom_{𝔤}( T ⊗ T, End V ) = End_{𝔤}( T ⊗ V ) .

Of course we are mainly interested in Weitzenböck formulas inducing differential operators of zeroth order or equivalently “pure curvature terms”. Clearly a Weitzenböck formula F ∈ Hom_{𝔤}( T ⊗ T ⊗ V, V ) skew–symmetric in its two T –arguments will induce a pure curvature term F (∇^{2}), because we can then reshuffle the summation in the calculation:

F ( ∇^{2}v) = \frac{1}{2} ∑_{μν} F ( t_{μ} ⊗ t_{ν} ⊗ (∇_{t_{μ},t_{ν}}^{2} − ∇_{t_{ν},t_{μ}}^{2})v) = \frac{1}{2} ∑_{μν} F_{t_{μ}⊗t_{ν}} R_{t_{μ}, t_{ν}}v . (3.5) cterm

Conversely the principal symbol of the differential operator F (∇^{2}) is easily computed to be σ_{F (∇^{2})}(ξ)v = F_{ξ^{♭}⊗ξ^{♭}}v for every cotangent vector ξ and every v ∈ V M. Hence the principal symbol vanishes identically exactly for the skew–symmetric Weitzenböck formulas. Weitzenböck formulas F leading to a pure curvature term F (∇^{2}) are thus completely characterized by being eigenvectors of eigenvalue −1 for the involution

τ :  𝔚( V ) ⟶ 𝔚( V ),  F ⟼ τ( F )

defined in the interpretation 𝔚(V) = Hom_{𝔤}( T ⊗ T ⊗ V, V ) as precomposition with the twist τ : T ⊗ T ⊗ V ⟶ T ⊗ T ⊗ V, a ⊗ b ⊗ v ⟼ b ⊗ a ⊗ v. In other words a Weitzenböck formula F will reduce to a pure curvature term if and only if τ(F) := F ∘ τ = −F.

Considering the space of Weitzenböck formulas 𝔚(V) as the algebra End_{𝔤}(T ⊗V) we can introduce additional structures on it: the unit 1 := id_{T ⊗V} ∈ 𝔚(V), the scalar product

⟨ F, F̃ ⟩ := \frac{1}{dim V} tr_{T ⊗V} ( F F̃) F, F̃ ∈ 𝔚(V)

satisfying ⟨ F G, F̃ ⟩ = ⟨ F, GF̃ ⟩ and the trace tr F := ⟨ F, 1 ⟩. Clearly the trace of the unit is given by tr 1 = dim T. The definition of the trace can be rewritten in the form

tr F = \frac{1}{dim V} tr_{V} ( v ⟼ ∑_{μ} F_{t_{μ}⊗t_{μ}}v )

so that the trace is invariant under the twist τ. Note that ∑_{μ} F_{t_{μ}⊗t_{μ}} is an equivariant endomorphism of V and hence, by Schur’s Lemma, an multiple of the identity on irreducible representations. A slightly more complicated argument using (^{\underline{basi}} 3.3) shows that the scalar product is invariant under the twist, too. In particular the eigenspaces for τ for the eigenvalues ±1 are orthogonal and all eigenvectors in the (−1)–eigenspace of τ have vanishing trace. Right from the definition of the trace we note that the trace of an element F = ∑ f_{ε}pr_{ε} of 𝔚(V_{λ}) in the irreducible case is given by

tr F = ∑_{ε} f_{ε} \frac{dim V_{λ+ε}}{dim V_{λ}}  (3.6) trace

in particular the idempotents pr_{ε} form an orthogonal basis of 𝔚(V_{λ}):

⟨ pr_{ε}, prε̃ ⟩ = δ_{εε̃} \frac{dim V_{λ+ε}}{dim V_{λ}}

3.2 The conformal weight operator

In Proposition ^{\underline{quotient}} 3.7 we will see that the algebra 𝔚(V) = End_{𝔤}(T ⊗ V) is in many cases a quotient of a polynomial algebra generated by one special endomorphism, the conformal weight operator B, which we will now introduce.

Recall that the scalar product ⟨ , ⟩ on T induces a scalar product on Λ^{2}T. Using this scalar product we can identify 𝔰𝔬 T ≅ Λ^{2}T through ⟨ X, a ∧ b ⟩ = ⟨ Xa, b ⟩ and hence think of the holonomy algebra 𝔤 ⊂ 𝔰𝔬 T as a subspace of the euclidian vector space Λ^{2}T:

defw Definition 3.2 (Conformal Weight Operator)

Consider an ideal 𝔥_{ℝ} ⊂ 𝔤_{ℝ} in the real holonomy algebra. Its complexification 𝔥 := 𝔥_{ℝ}⊗_{ℝ}ℂ is an ideal in 𝔤 and a regular subspace 𝔥 ⊂ 𝔤 ⊂ Λ^{2}T in Λ^{2}T with associated orthogonal projection pr_{𝔥} : Λ^{2}T ⟶ 𝔥. The conformal weight operator B^{𝔥} ∈ 𝔚(V) is defined by

B_{a⊗b}^{𝔥} v := pr_{𝔥}(a ∧ b) v

in the interpretation of Weitzenböck formulas as linear maps T ⊗ T ⟶ End V. Under the identification Hom_{𝔤}(T ⊗ T, End V) = End_{𝔤}(T ⊗ V) discussed above the conformal weight operator B^{𝔥} becomes the following sum over an orthonormal basis {t_{μ}} of T:

B^{𝔥}( b ⊗ v) = ∑_{μ} t_{μ} ⊗ pr_{𝔥}(t_{μ} ∧ b) v

The notation B := B^{𝔤} will be used for the conformal weight operator of the algebra 𝔤.

Most of the irreducible non–symmetric holonomy algebras 𝔤 are simple and hence there is only one weight operator B ( c.f. table (^{\underline{ht}} 2.1)). The exceptions are Kähler geometry 𝔤_{ℝ} = iℝ ⊕ 𝔰𝔲_{n} with a one–dimensional center in dimension 2n and two commuting weight operators B^{iℝ} and B^{𝔰𝔲} and quaternionic Kähler geometry 𝔤_{ℝ} = 𝔰𝔭(1) ⊕ 𝔰𝔭(n) in dimension 4n, n ≥ 2 with two commuting weight operators B^{H} and B^{E}.

fegan Lemma 3.3 (Fegan’s Lemma ^{\underline{fegan}} [F76])

The conformal weight operator B^{𝔥} ∈ 𝔚(V) of an ideal 𝔥 ⊂ 𝔤 ⊂ Λ^{2}T can be written

B^{𝔥} = − ∑_{α} X_{α} ⊗ X_{α} ∈ End_{𝔤}( T ⊗ V )

where {X_{α}} is an orthonormal basis of 𝔥 for the scalar product on Λ^{2}T induced from T.

Proof: Let {t_{μ}} and {X_{α}} be orthonormal bases of T and 𝔥 respectively. Using the characterization ⟨ X, a ∧ b ⟩ = ⟨ Xa, b ⟩ of the identification 𝔰𝔬 T = Λ^{2}T we find:

B^{𝔥}( b ⊗ v) = ∑_{μ} t_{μ} ⊗ pr_{𝔥}( t_{μ} ∧ b) v
= ∑_{μα} t_{μ} ⊗ ⟨ X_{α}, t_{μ} ∧ b ⟩X_{α} v = − ∑_{α} X_{α}b ⊗ X_{α} v .  □

A particularly nice consequence of Fegan’s Lemma is that the conformal weight operators B^{𝔥} and B𝔥̃ associated to two ideals 𝔥, 𝔥̃ ⊂ 𝔤 always commute. In fact two disjoint ideals 𝔥 ∩ 𝔥̃ = {0} of 𝔤 commute by definition, the general case follows easily. Hence the algebra structure on 𝔚(V) allows us to use the evaluation homomorphism

Φ :  ℂ[ { B^{𝔥} | 𝔥 irreducible ideal of 𝔤 } ] ⟶ 𝔚( V )  (3.7) evhom

from the polynomial algebra on abstract symbols {B^{𝔥}} to the algebra End_{𝔤}(T ⊗ V) for studying the fine structure of the space 𝔚(V) of Weitzenböck formulas. Put differently the conformal weight operators generate an algebra of multiplication endomorphisms B^{𝔥} : 𝔚(V) ⟶ 𝔚(V), F ⟼ F B^{𝔥}, on 𝔚(V) such that the image of Φ is the subspace generated by the connection Laplacian 1 ∈ 𝔚(V) under this algebra.

In order to turn Fegan’s Lemma into an effective formula for the eigenvalues of the conformal weight operator B^{𝔥} of an ideal 𝔥 ⊂ 𝔤 we need to calculate the Casimir operator in the normalization given by the scalar product on Λ^{2}T. Recall that the Casimir operator is defined as a sum Cas := ∑_{α} X_{α}^{2} ∈ U 𝔥 over an orthonormal basis {X_{α}} of 𝔥 and is thus determined only up to a constant. Usually it is much more convenient to calculate the Casimir Cas with respect to a scalar product of choice and later normalize it to the Casimir Cas^{Λ^{2}} with respect to the invariant scalar product induced on 𝔥 ⊂ Λ^{2}T.

For a given irreducible ideal 𝔥 ⊂ 𝔤 in an irreducible holonomy algebra 𝔤 the Casimir operator Cas for 𝔥 is now real, symmetric and 𝔤–invariant. Although the holonomy representation T of 𝔤 may not be irreducible itself, it is the complexification of the irreducible real representation T_{ℝ} so that we can still conclude that Cas acts as the scalar multiple Cas_{T} id of the identity on T. The Casimir eigenvalue Cas_{V_{λ}}^{Λ^{2}} of the properly normalized Casimir operator on a general irreducible representation V_{λ} of 𝔤 of highest weight λ can then be calculated from the Casimir Cas using

Cas_{V_{λ}}^{Λ^{2}} = − 2 \frac{dim 𝔥}{dim T} \begin{matrix}Cas & V_{λ} \\ Cas & T \end{matrix} ,  (3.8) cnor

where the ambiguity in the choice of normalization cancels out in the quotient \begin{matrix}Cas & V_{λ} \\ Cas & T \end{matrix}. In fact the normalization (^{\underline{cnor}} 3.8) is readily checked for the holonomy representation V = T

tr_{T} Cas^{Λ^{2}} = dim T · Cas_{T}^{Λ^{2}} = ∑_{α} tr_{T} X_{α}^{2} = − 2 dim 𝔥 ,

because the scalar product induced from T on Λ^{2}T satisfies ⟨ X, Y ⟩ = −\frac{1}{2}tr_{T} XY.

confW Corollary 3.4 (Explicit Formula for Conformal Weights)

Consider the tensor product T ⊗ V_{λ} = ⊕_{ε⊂λ}V_{λ+ε} of the holonomy representation T with the irreducible representation V_{λ} of highest weight λ. For an ideal 𝔥 ⊂ 𝔤 let ε_{max} be the highest weight of T and ρ be the half sum of positive weights of 𝔥 in the dual 𝔱^{∗} of a maximal torus 𝔱. With respect to the basis {pr_{ε}} of idempotents the conformal weight operator B^{𝔥} of the ideal 𝔥 can be expanded B^{𝔥} = ∑_{ε⊂λ} b_{ε}^{𝔥} pr_{ε} with conformal weights

b_{ε}^{𝔥} = 2 \frac{dim 𝔥}{dim T} \frac{⟨ λ + ρ, ε ⟩ − ⟨ ρ, ε_{max} ⟩ + \frac{1}{2} (|ε|^{2} − |ε_{max}|^{2})}{⟨ ε_{max} + 2ρ, ε_{max} ⟩} ,

where ⟨ ·, · ⟩ is an arbitrary scalar product on 𝔱^{∗} invariant under the Weyl group of 𝔥.

Proof: According to Lemma ^{\underline{fegan}} 3.3 the conformal weight operator can be written as a difference B^{𝔥} = −\frac{1}{2} (Cas^{Λ^{2}} − Cas^{Λ^{2}} ⊗ id − id ⊗ Cas^{Λ^{2}}) of properly normalized Casimir operators. In particular its restriction to the irreducible summand V_{λ+ε} ⊂ T ⊗ V_{λ} acts by multiplication with b_{ε}^{𝔥} := −\frac{1}{2}(Cas_{V_{λ+ε}}^{Λ^{2}} − Cas_{T}^{Λ^{2}} − Cas_{V_{λ}}^{Λ^{2}}). The conformal weights b_{ε}^{𝔥} can thus be calculated using Freudenthal’s formula Cas_{V_{λ}} = ⟨ λ + 2ρ, λ ⟩ for the Casimir eigenvalues of irreducible representations V_{λ} and the normalization (^{\underline{cnor}} 3.8). □

It is clear from the definition that the conformal weight operator B^{𝔥} ∈ 𝔚(V) of an ideal 𝔥 ⊂ 𝔤 of the holonomy algebra 𝔤 is in the (−1)–eigenspace of the involution τ and thus induces a pure curvature term B^{𝔥}( ∇^{2} ) on every vector bundle V M associated to the holonomy reduction of M. Explicitly we can describe this curvature term using an orthonormal basis {X_{α}} of the ideal 𝔥 for the scalar product induced on 𝔥 ⊂ Λ^{2}T. Namely the curvature operator R : Λ^{2}T M ⟶ 𝔤M ⊂ Λ^{2}T M, a ∧ b ⟼ R_{a,b}, associated to the curvature tensor R of M allows us to write down a well–defined global section

q^{𝔥}(R) := ∑_{α} X_{α} R(X_{α}) ∈ Γ( U ^{≤2}𝔤M )  (3.9) qendo

of the universal enveloping algebra bundle associated to the holonomy reduction. Fixing a representation G ⟶ Aut(V) of the holonomy group the section q^{𝔥}(R) in turn induces an endomorphism on the vector bundle V M associated to V and the holonomy reduction.

A particularly important example of Weitzenböck formulas is the classical formula of Weitzenböck for the Laplace operator Δ = d^{∗} d + d d^{∗} acting on differential forms. The curvature term in this formula is precisely the curvature endomorphism

Δ = ∇^{∗} ∇ + q(R)  (3.10) classical

for the full holonomy algebra 𝔤, in particular q(R) = Ric on the bundle of 1–forms on M. In passing we recall that the curvature term in the Weitzenböck formula (^{\underline{classical}} 3.10) is known to be the Casimir operator of the holonomy algebra 𝔤 on a symmetric space M = G̃/G, more precisely for every ideal 𝔥 ⊂ 𝔤 the curvature term q^{𝔥}(R) acts as the Casimir operator of the ideal 𝔥 on every homogeneous vector bundle V M over a symmetric space M.

A minor subtlety in the definition of the curvature terms q^{𝔥}(R) should not pass unnoticed, in difference to the conformal weight operators B^{𝔥} the curvature terms q^{𝔥}(R) associated to two ideals 𝔥, 𝔥̃ ⊂ 𝔤 do not in general commute. The problem is that the curvature operator R : Λ^{2}T M ⟶ 𝔤M does not necessarily map the parallel subbundle 𝔥M ⊂ Λ^{2}T M associated to an ideal 𝔥 ⊂ 𝔤 to itself due to the presence of “mixed terms” in the curvature tensor R. In other words the section q^{𝔥}(R) is not in general a section of U ^{≤2}𝔥M ⊂ U ^{≤2}𝔤M, so it is of no use that disjoint ideals centralize each other. For this time the usual suspects, the symmetric spaces, are tame as are the holonomies 𝔰𝔬_{n}, 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7} without non–trivial ideals 𝔥 ⊂ 𝔤 and the quaternionic holonomies 𝔰𝔭(n) and 𝔰𝔭(1) 𝔰𝔭(n), the curvature terms q^{𝔥}(R) for different ideals all commute in all these holonomies. Mixed terms may however spoil commutativity for Kähler manifolds M. According to equation (^{\underline{ccop}} 5.30) the curvature term associated to the center iℝ ⊂ 𝔲_{n} reads

q^{iℝ}( R ) = − \frac{1}{n} I ( IRic ) ∈ Γ( U ^{≤2}𝔲M ) ,

where I is the parallel complex structure and IRic is the composition of I with the symmetric Ricci endomorphism of T M thought of as a section of the holonomy bundle 𝔲M. Evidently the central curvature term q^{iℝ}(R) is a section of U ^{≤2} iℝM if and only if M is Kähler–Einstein, otherwise it will not in general commute with q^{𝔰𝔲}(R). The central curvature term features prominently in the Bochner identity for Kähler manifolds.

Lemma 3.5 B^{𝔥}( ∇^{2} ) = q^{𝔥}( R)

Proof: Expanding the second covariant derivative ∇^{2}ψ = ∑ t_{μ} ⊗ t_{ν} ⊗ ∇_{t_{μ}, t_{ν}}^{2}ψ of the section ψ with an orthonormal basis {t_{μ}} of T and using the same resummation as in the derivation of equation (^{\underline{cterm}} 3.5) we find for an orthonormal basis \{X_{α}\} of the ideal 𝔥:

B^{𝔥}( ∇^{2}ψ ) = \frac{1}{2} ∑_{μν} pr_{𝔥}(t_{μ} ∧ t_{ν}) R_{t_{μ}, t_{ν}}^{V} ψ

= \frac{1}{2} ∑_{αμν} ⟨ t_{μ} ∧ t_{ν}, X_{α} ⟩ X_{α} R_{t_{μ}, t_{ν}}^{V} ψ = q^{𝔥}( R )  □

On the other hand Corollary ^{\underline{confW}} 3.4 tells us how to write the conformal weight operator B^{𝔥} in terms of the basis {pr_{ε}} of projections onto the irreducible summands V_{λ+ε} ⊂ T ⊗ V_{λ}. Using the identification of B^{𝔥}(∇^{2}) with the universal curvature terms q^{𝔥}( R) proved above we obtain some prime examples of Weitzenböck formulas:

Proposition 3.6 (Universal Weitzenböck Formula)

universal

Consider a Riemannian manifold M of dimension n with holonomy group G ⊂ SO(n) and the vector bundle V_{λ}M over M associated to the holonomy reduction of M and the irreducible representation V_{λ} of G of highest weight λ. In terms of the Stein–Weiss operators T_{ε} : Γ(V_{λ}M) ⟶ Γ(V_{λ+ε}M) arising from the decomposition T ⊗V_{λ} = ⊕_{ε⊂λ}V_{λ+ε} the action of the curvature endomorphisms q^{𝔥}(R) can be written

q^{𝔥}(R) = − ∑_{ε⊂λ} b_{ε}^{𝔥} T_{ε}^{∗} T_{ε} ,  (3.11) weizen3

where the b_{ε}^{𝔥} are the eigenvalues of the conformal weight operator B^{𝔥} ∈ End_{𝔤}(T ⊗ V_{λ}).

As a direct consequence of Proposition ^{\underline{universal}} 3.6 and the classical Weitzenböck formula (^{\underline{classical}} 3.10) for the Laplace operator Δ = dd^{∗} + d^{∗}d on the bundle of differential forms we obtain

Δ = ∑_{ε⊂λ} (1 − b_{ε}) T_{ε}^{∗} T_{ε} .

In the case of Riemannian holonomy G = SO(n) the Weitzenböck formula (^{\underline{weizen3}} 3.11) was considered for the first time in ^{\underline{pg1}} [G91]. In this article one can also find the conformal weight operator and its expression in terms of Casimir operators. The operator B appears also in ^{\underline{cgh}} [CGH00]. Similar results can be found in ^{\underline{homma}} [H04].

Considering B as an element of End_{𝔤}(T ⊗ V) we obtain new 𝔤–invariant endomorphisms as powers of B, which as elements of Hom_{𝔤}(T ⊗ T, End V) may be expressed as

B_{a⊗b}^{k} = ∑ pr_{𝔤}(a ∧ e_{i_{1}}) pr_{𝔤}(e_{i_{1}} ∧ e_{i_{2}}) … pr_{𝔤}(e_{i_{k−2}} ∧ e_{i_{k−1}}) pr_{𝔤}(e_{i_{k−1}} ∧ b) .  (3.12) wk

Taking the trace in the two T –variables we obtain an endomorphism of V which corresponds to the action of the element

Cas^{[k]} := ∑ pr_{𝔤}(e_{i_{0}} ∧ e_{i_{1}}) · pr_{𝔤}(e_{i_{1}} ∧ e_{i_{2}}) · . . . · pr_{𝔤}(e_{i_{k−2}} ∧ e_{i_{k−1}}) · pr_{𝔤}(e_{i_{k−1}} ∧ e_{i_{0}}) (3.13) highercasimir

of the universal enveloping algebra U 𝔤. It is well known that all Cas^{[k]}, k ≥ 2, belong to the center of the universal enveloping algebra U 𝔤. Since Cas^{[2]} = −2Cas^{Λ^{2}} they are called higher Casimirs (c.f. ^{\underline{cgh}} [CGH00]). An easy calculation shows that on an irreducible representation V = V_{λ} we have

Cas^{[k]} = tr (B^{k}) id_{V_{λ}} = (∑_{ε} b_{ε}^{k} \frac{dim V_{λ+ε}}{dim V_{λ}} ) id_{V_{λ}} ,  (3.14) kth

where we write B^{k} = ∑ b_{ε}^{k}pr_{ε} and use the second equation from (^{\underline{trace}} 3.6) for computing the trace of B^{k} explicitly.

Furthermore we note the equation Cas^{[3]} = −\frac{1}{2}Cas_{𝔤}^{Λ^{2}}Cas^{Λ^{2}}. Indeed it follows from the recursion formula of Corollary ^{\underline{recursion2}} 4.2 or by direct calculation that B^{2} − \frac{1}{4}Cas_{𝔤}^{Λ^{2}}B is a +1 eigenvector of the involution τ. Thus it is orthogonal to B, which is a −1 eigenvector, i.e.

0 = ⟨ B^{2} − \frac{1}{4}Cas_{𝔤}^{Λ^{2}}B, B ⟩ = tr(B^{3}) − \frac{1}{4}Cas_{𝔤}^{Λ^{2}}tr(B^{2}) = tr(B^{3}) + \frac{1}{2}Cas_{𝔤}^{Λ^{2}}Cas^{Λ^{2}} .

From a slightly more general point of view the evaluation at the conformal weight operator defines an algebra homomorphism Φ : ℂ[B] ⟶ End_{𝔤}(T ⊗ V), whose kernel is generated by the minimal polynomial of B as an endomorphism on T ⊗ V. With B being diagonalizable its minimal polynomial is the product min(B) = ∏_{b ∈ {b_{ε}}}(B − b) over all different conformal weights. In consequence the injective algebra homomorphism

Φ : ℂ[B]/⟨min( B)⟩ ⟶ End_{𝔤}( T ⊗ V) .

is an isomorphism as soon as all conformal weights are pairwise different. Indeed the dimension of End_{𝔤}(T ⊗ V) is the number N(G, λ) of relevant weights or the number of conformal weights counted with multiplicity, while the number of different conformal weights determines the degree of min(B) and so the dimension of ℂ[B]/⟨min(B)⟩.

In section ^{\underline{eigenvalues}} 4.2 we compute the B–eigenvalues in the cases 𝔤 = 𝔰𝔬_{n}, 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7}. It follows that they are pairwise different with the only exceptions of 𝔤 = 𝔰𝔬_{2r} and a representation of highest weight λ = λ_{1}ω_{1}+…+λ_{r}ω_{r}, with λ_{r} = λ_{r−1}, which is equivalent to b_{ε_{r}} = b_{−ε_{r}} and 𝔤 = 𝔰𝔭𝔦𝔫_{7} and a representation with highest weight λ = aω_{1} +bω_{2} +cω_{3} with c = 2a + 1, which is equivalent to b_{−ε_{4}} = b_{ε_{4}}. In these cases the degree of the minimal polynom is reduced by one and hence the image of Φ has codimension one. We thus have proved the following

Proposition 3.7 (Structure of the Algebra of Weitzenböck Formulas)

quotient

Let G be one of the holonomy groups SO_{n}, G_{2} or Spin(7) of non–symmetric manifolds. If V_{λ} is irreducible, then Φ is an isomorphism

Φ : ℂ[B]/⟨min( B)⟩ ⟶^{≅} End_{𝔤}( T ⊗ V_{λ} ) ,

with the only exception of the cases G = SO_{2r} and a highest weight λ with λ_{r−1} = λ_{r}, or G = Spin(7) and a highest weight λ = aω_{1} + bω_{2} + cω_{3} with c = 2a + 1. In both cases the homomorphism Φ is not surjective and its image has codimension one.

3.3 The Classifying Endomorphism

The decomposition of the space 𝔚(V) = Hom_{𝔤}(T ⊗ T, End V) of Weitzenböck formulas into the (±1)–eigenspaces of the involution τ can be written as

Hom_{𝔤}(T ⊗ T, End V) ≅ Hom_{𝔤}(Λ^{2}T, End V) ⊕ Hom_{𝔤}(Sym^{2}T, End V) .

However in general we have a further splitting of T ⊗T leading to a further decomposition of the τ–eigenspaces. Our aim is now to introduce an endomorphism K on 𝔚(V) whose eigenspaces correspond to this finer decomposition.

cls Definition 3.8 (The Classifying Endomorphism)

The classifying endomorphism K^{𝔥} of an ideal 𝔥_{ℝ} ⊂ 𝔤_{ℝ} of the real holonomy algebra 𝔤_{ℝ} is the endomorphism K^{𝔥} : 𝔚(V) ⟶ 𝔚(V) on the space of Weitzenböck formulas defined in the interpretation 𝔚(V) = Hom_{𝔤}(T ⊗ T, End V) by the formula

K^{𝔥}( F )_{a⊗b} v := − ∑_{α} F_{X_{α}a⊗X_{α}b}v

where {X_{α}} is an orthonormal basis for the scalar product induced on the ideal 𝔥 ⊂ Λ^{2}T. As before we denote the classifying endomorphism of the ideal 𝔤 simply by K := K^{𝔤}.

Of course the definition of the classifying endomorphism K^{𝔥} is motivated by Fegan’s Lemma ^{\underline{fegan}} 3.3 for the conformal weight operator B^{𝔥}. Note that for every 𝔤–equivariant map F : T ⊗ T ⟶ End V the map K^{𝔥}(F) : T ⊗ T ⟶ End V is again 𝔤–equivariant, because we sum over an orthonormal basis {X_{α}} of an ideal 𝔥 ⊂ 𝔤 for a 𝔤–invariant scalar product on 𝔥. In the interpretation 𝔚(V) = End_{𝔤}(T ⊗ V) the definition of K^{𝔥} reads

K^{𝔥}( F )(b ⊗ v) = − ∑_{μα} t_{μ} ⊗ F_{X_{α}t_{μ}⊗X_{α}b} v = ∑_{μα} X_{α}t_{μ} ⊗ F_{t_{μ}⊗X_{α}b} v

or more succinctly:

K^{𝔥}( F ) = ∑_{α} ( X_{α} ⊗ id ) F ( X_{α} ⊗ id ) .  (3.15) ksym

In consequence the classifying endomorphisms K^{𝔥} and K𝔥̃ for two ideals 𝔥, 𝔥̃ ⊂ 𝔤 commute on the space 𝔚(V) of Weitzenböck formulas similar to the conformal weights operators. The classifying endomorphisms will be extremely useful in finding the matrix of the twist τ : 𝔚(V) ⟶ 𝔚(V) in the basis of 𝔚(V) given by the orthogonal idempotents pr_{ε}.

Lemma 3.9 (Eigenvalues of the Classifying Endomorphism)

evc

Consider the decomposition of the tensor product T ⊗ T = ⊕_{α} W_{α} into irreducible summands. The classifying endomorphisms K^{𝔥} are diagonalizable on Hom_{𝔤}(T ⊗ T, End V) with eigenspaces Hom_{𝔤}( W_{α}, End V ) ⊂ Hom_{𝔤}(T ⊗ T, End V) for relevant α and eigenvalues:

κ_{W_{α}} = \frac{1}{2} Cas_{W_{α}}^{Λ^{2}} − Cas_{T}^{Λ^{2}} .

In particular K^{𝔥} acts as K^{𝔥}(F) = ∑_{α} κ_{W_{α}} F |_{W_{α}} on the space 𝔚(V) = Hom_{𝔤}(T ⊗ T, End V).

Proof: From the very definition of K^{𝔥} we see that it acts by precomposition with the map − ∑ X_{α} ⊗ X_{α} in the interpretation 𝔚(V) = Hom_{𝔤}(T ⊗ T, End V) of the space of Weitzenböck formulas. The argument used in the proof of Corollary ^{\underline{confW}} 3.4 shows that K^{𝔥} is actually a difference of Casimir operators leading to the stated formula for its eigenspaces and eigenvalues. □

In the case of the holonomies 𝔰𝔬_{n}, 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7} we have T ⊗ T = ℂ ⊕ Sym_{0}^{2}T ⊕ 𝔤 ⊕ 𝔤^{⊥} and we obtain the following K–eigenvalues:

\begin{aligned}& κ_{ℂ} & κ_{Sym_{0}^{2}T} & κ_{𝔤} & κ_{𝔤^{⊥}} \\ 𝔰𝔬^{n} & −(n − 1) & 1 & −1 & — \\ 𝔤_{2} & −4 & \frac{2}{3} & 0 & −2 \\ 𝔰𝔭𝔦𝔫^{7} & −\frac{21}{4} & \frac{3}{4} & −\frac{1}{4} & −\frac{9}{4} \end{aligned}  (3.16) kt

Note that all these K–eigenvalues are different and consequently the twist τ is a polynomial in the classifying endomorphism K. Moreover an invariant homomorphism F ∈ Hom_{𝔤}(T ⊗ T, End V) is an eigenvector of K if and only if F is different from zero one precisely one summand W_{α} ⊂ T ⊗ T. Thus it is clear that 1 and B are special K– eigenvectors.

elem Lemma 3.10 (Properties of the Classifying Endomorphism)

The classifying endomorphism K : 𝔚(V) ⟶ 𝔚(V) is a symmetric endomorphism on the space 𝔚(V) of Weitzenböck formulas equipped with the scalar product ⟨ F, F̃ ⟩ := \frac{1}{dim V} tr_{T ⊗V} F F̃. It commutes with the twist map τ and the special endomorphisms 1 and B for the same ideal 𝔥 ⊂ 𝔤 are K–eigenvectors:

K( 1 ) = Cas_{T}^{Λ^{2}} 1 K( B) = ( Cas_{T}^{Λ^{2}} − \frac{1}{2} Cas_{𝔥}^{Λ^{2}}) B

Proof: The symmetry of K is a trivial consequence of equation (^{\underline{ksym}} 3.15) in the form

⟨ K( F ), F̃ ⟩ = \frac{1}{dim V} ∑_{ν} tr_{T ⊗V} ( (X_{ν} ⊗ id) F (X_{ν} ⊗ id) F̃ )

and the cyclic invariance of the trace, moreover K commutes with τ by definition. Coming to the explicit determination of K(1) and K(B) we observe that the unit of End_{𝔤}(T ⊗V) becomes the equivariant map 1(a ⊗ b) = ⟨ a, b ⟩id_{V} in Hom_{𝔤}(T ⊗ T, End V) and so:

( K1 )(a ⊗ b) = − ∑_{ν} ⟨ X_{ν}a, X_{ν}b ⟩ id_{V} = ∑_{ν} ⟨ a, X_{ν}^{2}b ⟩ id_{V} = Cas_{T}^{Λ^{2}} 1(a ⊗ b) .

The conformal weight operator B considered as an element of Hom_{𝔤}(T ⊗ T, End V) lives by definition in the eigenspace Hom_{𝔤}(𝔤, End V) for the eigenvalue −\frac{1}{2} Cas_{𝔤}^{Λ^{2}} + Cas_{T}^{Λ^{2}} of K, where Cas_{𝔤}^{Λ^{2}} is the Casimir eigenvalue of the adjoint representation. □

On a manifold with holonomy algebra 𝔤 ⊂ 𝔰𝔬_{n} ≅ Λ^{2}T the Riemannian curvature tensor takes values in 𝔤, i.e. it can be considered as an element of Sym^{2}𝔤. This fact has the following important consequence:

Proposition 3.11 (Bochner Identities)

bi

Let F ∈ Hom_{𝔤}(𝔤^{⊥}, End V) be an invariant homomorphism, which factors through the complement 𝔤^{⊥} ⊂ Λ^{2}T. Its corresponding curvature expression F ( ∇^{2} ) = 0 vanishes.

We will call the Weitzenböck formula corresponding to an invariant homomorphism factoring through 𝔤^{⊥} a Bochner identity. If the invariant homomorphism F is written as F = ∑ f_{ε} pr_{ε} then the corresponding Bochner identity is

∑_{ε} f_{ε} T_{ε}^{∗} T_{ε} = 0 .

The Bochner identities of G_{2}– and Spin(7)–holonomy correspond to eigenvectors of the classifying endomorphism K for the eigenvalues −2 and \frac{9}{4} respectively. Since the zero weight space of 𝔤^{⊥} is in both cases one-dimensional it follows from Lemma ^{\underline{generic}} 2.3 that

dim Hom_{𝔤}(𝔤^{⊥}, End V_{λ}) ≤ 1 ,  (3.17) estimate1

i. e. there is at most one Bochner identity. Note that because of Schur’s Lemma we have Hom_{g}(ℂ, End V_{λ}) ≅ ℂ and this K–eigenspace is spanned by 1 ∈ End_{𝔤}(T ⊗ V_{λ}). Moreover, since the zero weight space of 𝔤 coincides with the fixed Cartan subalgebra 𝔱 ⊂ 𝔤, we also obtain from Lemma ^{\underline{generic}} 2.3 the following estimates

dim Hom_{𝔤_{2}}(𝔤_{2}, End V_{λ}) ≤ 2 and dim Hom_{𝔰𝔭𝔦𝔫_{7}}(𝔰𝔭𝔦𝔫_{7}, End V_{λ}) ≤ 3 .

4 The Recursion Procedure for SO(n), G_{2} and Spin(7)

In this section we will derive a relation between the action of K and B on the space of Weitzenböck formulas. As a corollary we will obtain a recursion procedure, which allows us to produce new τ–eigenvectors out of given K–eigenvectors. In the last part of this section we will see that in many cases this recursion procedure is sufficient to produce a complete basis in the space of Weitzenböck formulas.

4.1 The basic recursion procedure

Recall that the twist τ acts on 𝔚(V) = Hom_{𝔤}(T ⊗ T ⊗ V, V) by precomposition with the linear map:

τ : T ⊗ T ⊗ V ⟶ T ⊗ T ⊗ V, a ⊗ b ⊗ v ⟼ b ⊗ a ⊗ v

Generalizing this precomposition we observe that 𝔚(V) is naturally a right module over the algebra End_{𝔤}(T ⊗ T ⊗ V) containing τ. Interestingly both the classifying endomorphism K and the (right) multiplication by the conformal weight operator B are induced by precomposition with elements in End_{𝔤}(T ⊗ T ⊗ V). In fact from the very definition of K we see that it is precomposition with the 𝔤–invariant endomorphism

K : T ⊗ T ⊗ V ⟶ T ⊗ T ⊗ V, a ⊗ b ⊗ v ⟼ − ∑_{ν} X_{ν}a ⊗ X_{ν}b ⊗ v

while (right) multiplication by B is precomposition with the 𝔤–invariant endomorphism

B : T ⊗ T ⊗ V ⟶ T ⊗ T ⊗ V, a ⊗ b ⊗ v ⟼ − ∑_{ν} a ⊗ X_{ν}b ⊗ X_{ν}v

by Fegan’s Lemma ^{\underline{fegan}} 3.3. From this description of the action of the classifying endomorphism K and right multiplication of B on 𝔚(V) we immediately conclude:

K + B + τBτ = −\frac{1}{2}( Cas^{Λ^{2}} − 2Cas_{T}^{Λ^{2}} − Cas_{V_{λ}}^{Λ^{2}} )

Applying once again Schur’s Lemma this relation implies our basic Recursion Formula:

Theorem 4.1 (Recursion Formula)

rec

Let V_{λ} be an irreducible representation of the holonomy algebra 𝔤. Then the action of K, B and τ on 𝔚(V_{λ}) = Hom_{𝔤}(T ⊗ T ⊗ V_{λ}, V_{λ}) by precomposition satisfies:

K + B + τBτ = Cas_{T}^{Λ^{2}} = − 2 \frac{dim 𝔥}{dim T}

We will now explain how this theorem yields a recursion formula for K–eigenvectors. In fact given an eigenvector F ∈ 𝔚(V) for the twist τ and the classifying endomorphism K with eigenvalues t and κ, i.e. τF = tF and KF = κF, the Recursion Formula allows us to produce a new τ–eigenvector F̂ with eigenvalue −t. This simple prescription suffices to obtain a complete eigenbasis for 𝔚(V) of τ– and actually K–eigenvectors in general Riemannian geometry 𝔤 = 𝔰𝔬_{n} and, with some modifications, also in the exceptional cases 𝔤 = 𝔤_{2} and 𝔤 = 𝔰𝔭𝔦𝔫_{7}. In the Kähler and quaternionic Kähler case one has a similar procedure.

Corollary 4.2 (Basic Recursion Procedure)

recursion2

Let F ∈ 𝔚(V) be an eigenvector for the involution τ and the classifying endomorphism K of an ideal 𝔥 ⊂ 𝔤, i. e. K(F) = κF and τ(F) = ±F. The new Weitzenböck formula

F_{new} := ( B − \frac{Cas_{T}^{Λ^{2}} − κ}{2} ) ∘ F

is again a τ–eigenvector in 𝔚(V) with τ(F_{new}) = ∓F_{new}. In particular we find:

1_{new} = B  and  B_{new} = B^{2} − \frac{1}{4}Cas_{𝔤}^{Λ^{2}} B .

Proof: We observe that the Recursion Formula ^{\underline{rec}} 4.1 in the form τBτ = Cas_{T}^{Λ^{2}} − K − B implies under the assumptions K(F) = κF and τ(F) = ±F that

±τ( BF ) = ( Cas_{T}^{Λ^{2}} − κ) F − BF

and consequently:

±τ( BF − \frac{Cas_{T}^{Λ^{2}} − κ}{2} F ) = − ( BF − \frac{Cas_{T}^{Λ^{2}} − κ}{2} F ) .

The formulas for 1_{new} and B_{new} are immediate consequences of Lemma ^{\underline{elem}} 3.10. □

Recall that a K–eigenvector is automatically a τ–eigenvector. In general however the new Weitzenböck formula F_{new} ∈ 𝔚(V) does not need to be an eigenvector for K again and there is no way to iterate the recursion. Nevertheless it is possible to avoid termination of the recursion procedure for most of the irreducible non–symmetric holonomy algebras by using appropriate projections.

We note that any +1–eigenvector of τ orthogonal to 1 is already K–eigenvector in Hom_{𝔤}(Sym_{0}^{2}T, End V_{λ}). This is due to the fact that 1 spans the other summand of the +1–eigenspace of τ. In particular the orthogonal projection of B_{new} onto the orthogonal complement of 1, i. e. the polynomial B^{2} − \frac{1}{4}Cas_{𝔤}^{Λ^{2}}B + \frac{2}{n}Cas_{V_{λ}}^{Λ^{2}}, is a K–eigenvector in Hom_{𝔤}(Sym_{0}^{2}T, End V_{λ}). More generally we have:

Corollary 4.3 (Orthogonal recursion procedure)

orp

Let p_{0}(B), …, p_{k}(B) be a sequence of polynomials obtained by applying the Gram Schmidt orthonormalization procedure to the powers 1, B, B^{2}, …, B^{k} of the conformal weight operator B. If all these polynomials are τ–eigenvectors and p_{k}(B) is moreover a K– eigenvector, then the orthogonal projection p_{k+1}(B) of B^{k+1} onto the orthogonal complement of the span of 1, B, …, B^{k} is a again a τ–eigenvector.

Proof: Since p_{k}(B) is a K–eigenvector the basic recursion procedure shows the existence of a polynomial in B of degree k + 1, which is a τ–eigenvector. It follows that with span{1, B, …, B^{k}} = span{p_{0}(B), …, p_{k}(B)} also span{1, B, …, B^{k+1}} is τ– invariant. Clearly the orthogonal projection p_{k+1}(B) of B^{k+1} onto the orthogonal complement of span{1, B, …, B^{k}} is a polynomial in B of degree k + 1 with:

span{1, B, B^{2}, …, B^{k}} ⊕ ℂ p_{k+1}( B) = span{1, B, B^{2}, …, B^{k}, B^{k+1}} .

Now the involution τ is symmetric with respect to the scalar product on End_{𝔤}(T ⊗ V) and so the orthogonal complement of a τ–invariant space is again τ–invariant. □

4.2 Computation of B–eigenvalues for SO(n), G_{2} and Spin(7)

eigenvalues

In the last part of this section we will apply the recursion procedure of the previous section in case of the holonomies SO(n), G_{2} and Spin(7). We will start with computing the B–eigenvalues for these groups, which is an application of the explicit formula of Corollary ^{\underline{confW}} 3.4. In particular we will see that with only one exception all B–eigenvalues are pairwise different. This information is relevant in the proof of Proposition ^{\underline{quotient}} 3.7.

The SO(n)–case. Recall that in Section ^{\underline{tables}} 2 we fixed the notation for the fundamental weights ω_{1}, …, ω_{r} and the weights ±ε_{1}, …, ±ε_{r} of the defining representation ℝ^{n} of SO(n) with r := ⌊\frac{n}{2}⌋. Moreover the scalar product ⟨ , ⟩ on the dual of a maximal torus was chosen so that the weights ±ε_{1}, …, ±ε_{r} are an orthonormal basis. A highest weight can be written λ = λ_{1}ω_{1} +…+λ_{r}ω_{r} = μ_{1}ε_{1} +…+μ_{r}ε_{r} with integral coefficients λ_{1}, …, λ_{r} ≥ 0 and coefficients μ_{1}, …, μ_{r}, which are either all integral or all half–integral and decreasing. Independent of the parity of n the conformal weights are

b_{+ε_{k}} = μ_{k} − k + 1,  b_{−ε_{k}} = −μ_{k} − n + k + 1,  b_{0} = − r

according to Corollary ^{\underline{confW}} 3.4, where the zero weight only appears for n odd. With only a few exceptions the conformal weights are totally ordered and thus pairwise different. In the case n odd the coefficients μ_{1}, …, μ_{r} are decreasing in the sense μ_{1} ≥ μ_{2} ≥ … ≥ μ_{r} ≥ 0 so that we find strict inequalities

b_{−ε_{1}} < b_{−ε_{2}} < … < b_{−ε_{r}} ≤ b_{0} < b_{+ε_{r}} < … < b_{+ε_{1}}

unless μ_{r} = 0 or equivalently λ_{r} = 0. In the latter case b_{−ε_{r}} ≤ b_{0} happens to be an equality, however Lemma ^{\underline{relw}} 2.2 tells us that the zero weight is irrelevant for highest weights λ with λ_{r} = 0. Without loss of generality we may thus assume all conformal weights to be different for n odd. Similar considerations in the case of even n based on the inequalities μ_{1} ≥ μ_{2} ≥ … ≥ μ_{r−1} ≥ |μ_{r}| satisfied by the coefficients μ_{1}, …, μ_{r} of λ lead to

b_{−ε_{1}} < b_{−ε_{2}} < … < b_{−ε_{r−1}} < { b_{−ε_{r}} , b_{+ε_{r}} } < ♭_{+ε_{r−1}} < … < b_{+ε_{2}} < b_{+ε_{1}}

where nothing specific can be said about the relation between b_{−ε_{r}} and b_{+ε_{r}} due to b_{+ε_{r}} − b_{−ε_{r}} = 2μ_{r}. This should not come too surprising as the outer automorphism of SO(n) with n even acts on 𝔱^{∗} as a reflection along the hyperplane μ_{r} = 0.

The G_{2}–case. We write the highest weight as λ = aω_{1} + bω_{2} with integers a, b ≥ 0 and use the scalar product defined in Section ^{\underline{tables}} 2 by setting ⟨ ε_{1}, ε_{1} ⟩ = 1 = ⟨ ε_{2}, ε_{2} ⟩ and ⟨ ε_{1}, ε_{2} ⟩ = \frac{1}{2}, equivalently ⟨ ω_{1}, ω_{1} ⟩ = 1, ⟨ ω_{2}, ω_{2} ⟩ = 3 and ⟨ ω_{1}, ω_{2} ⟩ = \frac{3}{2}. According to Corollary ^{\underline{confW}} 3.4 the conformal weight of the zero weight is given by b_{0} = − 2, similarly:

b_{±ε_{1}} = − (\frac{5}{3} ∓ \frac{5}{3}) ± (\frac{2}{3}a+b), b_{±ε_{2}} = − (\frac{5}{3} ∓ \frac{4}{3}) ± (\frac{1}{3}a+b), b_{±ε_{3}} = − (\frac{5}{3} ∓ \frac{1}{3}) ± \frac{1}{3}a .

Again all conformal weights or B–eigenvalues are pairwise different and totally ordered:

b_{−ε_{1}} < b_{−ε_{2}} < b_{−ε_{3}} < b_{0} < b_{+ε_{3}} < b_{+ε_{2}} < b_{+ε_{1}} .

The Spin(7)–case. Using the fundamental weights ω_{1}, ω_{2}, ω_{3} and the scalar product ⟨ , ⟩ introduced in Section ^{\underline{tables}} 2 in terms of the weights ±η_{1}, ±η_{2}, ±η_{3} of the represention ℝ^{7} we write the highest weight λ = aω_{1} + bω_{2} + cω_{3} with integers a, b, c ≥ 0 and compute

\begin{aligned}b_{±ε_{1}} & = & − (\frac{9}{4} ∓ \frac{9}{4}) ± (\frac{1}{2}a + b + \frac{3}{4}c) , & b_{±ε_{2}} & = & − (\frac{9}{4} ∓ \frac{7}{4}) ± (\frac{1}{2}a + b + \frac{1}{4}c) , \\ b_{±ε_{3}} & = & − (\frac{9}{4} ∓ \frac{3}{4}) ± (\frac{1}{2}a + \frac{1}{4}c) , & b_{±ε_{4}} & = & − (\frac{9}{4} ∓ \frac{1}{4}) ± (\frac{1}{2}a − \frac{1}{4}c) . \end{aligned}

In this case we obtain the inequalities

b_{−ε_{1}} < b_{−ε_{2}} < b_{−ε_{3}} < { b_{−ε_{4}} , b_{+ε_{4}} } < b_{+ε_{3}} < b_{+ε_{2}} < b_{+ε_{3}} .

however the difference b_{+ε_{4}} − b_{−ε_{4}} = a − \frac{1}{2}c + \frac{1}{2} does not allow to draw conclusions about the relation between b_{−ε_{4}} and b_{+ε_{4}}. In particular for a heighest weight λ with c = 2a + 1 the two conformal weights b_{−ε_{4}} = b_{+ε_{4}} agree.

4.3 Basic Weitzenböck formulas for SO(n), G_{2} and Spin(7)

We will now apply the recursion procedure of Corollary ^{\underline{recursion2}} 4.2 for the holonomy groups SO(n), G_{2} and Spin(7). In the SO(n)–case we get a complete basis.

Let us start with the generic Riemannian holonomy algebra 𝔤 = 𝔰𝔬_{n} with only a single non–trivial ideal 𝔥 = 𝔤. According to table (^{\underline{kt}} 3.16) its classifying endomorphism K has eigenvalues (n−1), 1 and −1 with eigenspaces ℂ1, the orthogonal complement of 1 in the τ–eigenspace for the eigenvalue 1 and the τ–eigenspace for the eigenvalue −1 respectively. The orthogonal projection of any τ–eigenvector to the orthogonal complement of 1 is thus a K–eigenvector and Corollary ^{\underline{orp}} 4.3 implies that a complete eigenbasis of τ–eigenvectors may be obtained by applying the Gram-Schmidt orthonormalization procedure to the powers of B.

Alternatively we may modify the recursion procedure such that it associates to an eigenvector F for τ of eigenvalue −1 the K–eigenvector

F_{new} := ( B + \frac{n − 2}{2} ) F − \frac{1}{n} ⟨ BF, 1 ⟩ 1

for the eigenvalue 1, while a τ–eigenvector F for the eigenvalue +1 orthogonal to 1 is mapped to the K–eigenvector:

F_{new} := ( B + \frac{n}{2} ) F

for the eigenvalue −1. For an irreducible representation V_{λ} of 𝔰𝔬_{n} we thus get a sequence p_{0}(B), p_{1}(B), … of K–eigenvectors in 𝔚(V_{λ}) defined recursively by p_{0}(B) := 1, p_{1}(B) := B and p_{k+1}(B) := (p_{k}(B))_{new} for k ≥ 1. Evidently the different p_{k}(B) are polynomials of degree k in B and so the eigenvectors p_{0}(B), …, p_{d−1}(B) ∈ 𝔚(V_{λ}) with d := deg min B are necessarily linearly independent. According to Proposition ^{\underline{quotient}} 3.7 we always get a complete basis of τ–eigenvectors with the exception of the case 𝔤 = 𝔰𝔬_{2r} and a representation V_{λ} with λ_{r−1} = λ_{r}. Here we still have to add a K–eigenvector F_{spin} spanning the orthogonal complement of the image of ℂ[ B] in 𝔚(V_{λ}).

Note that the polynomials p_{2k+1}(B), k = 0, 1, … are in the −1–eigenspace of τ. Hence the corresponding Weitzenböck formulas give a pure curvature term. Let N the number of irreducible components of T ⊗ V_{λ}, then there are ⌊\frac{N}{2} ⌋ linearly independent equations of this type. This result, which is clear from our construction, was proved for the first time in ^{\underline{branson}} [BH02]. The first eigenvectors in this sequence are p_{0}(B) = 1 and p_{1}(B) = B as well as:

p_{2}(B) = B^{2} + \frac{n−2}{2} B + \frac{2}{n} Cas_{V_{λ}}^{Λ^{2}}  (4.18)
p_{3}(B) = B^{3} + (n − 1) B^{2} + (\frac{2}{n} Cas_{V_{λ}}^{Λ^{2}} + \frac{n(n−2)}{4}) B + Cas_{V_{λ}}^{Λ^{2}}  (4.19) gleichung2

Essentially the same procedure can be used in the case 𝔤 = 𝔤_{2} to compute a complete K–eigenbasis for the space 𝔚(V_{λ}) for an irreducible 𝔤_{2}–representation V_{λ}. Again there is only one non–trivial ideal 𝔥 = 𝔤_{2} and hence only a single classifying endomorphism K. However the τ–eigenspace in 𝔚(V_{λ}) for the eigenvalue −1 decomposes into two K– eigenspaces. The recursion procedure gives the K–eigenvectors

p_{0}(B) = 1, p_{1}(B) = B, p_{2}(B) = B^{2} + 2B + \frac{2}{7} Cas_{V_{λ}}^{Λ^{2}} .  (4.20) g2-1

Using the recursion procedure again gives a polynom of degree 3 in B. Projecting it onto the orthogonal complement of B we obtain

p_{3}(B) = B^{3} + \frac{13}{3} B^{2} + (\frac{1}{2} Cas_{V_{λ}}^{Λ^{2}} + 4) B + \frac{2}{3} Cas_{V_{λ}}^{Λ^{2}} .  (4.21) g2-2

We will see in Theorem ^{\underline{bochig2}} 8.6 that p_{3}(B) is in fact a K–eigenvector for the eigenvalue −2, in other words p_{3}(B) ∈ Hom_{𝔤_{2}}(𝔤_{2}^{⊥}, V_{λ}) is a Bochner identity. Due to the estimate (^{\underline{estimate1}} 3.17) any τ–eigenvector orthogonal to 1 and F_{3} is a K–eigenvector and so we may obtain a complete eigenbasis p_{0}(B), …, p_{6}(B) in the G_{2}–case by applying the Gram–Schmidt orthogonalization to the powers of B and using Corollary ^{\underline{orp}} 4.3.

In order to make the generalized Bochner identity corresponding to the polynomial p_{3}(B) explicit we recall that its coefficients as a linear combination of the basis projections pr_{ε} are the value of the polynomial p_{3} at the corresponding B–eigenvalues b_{ε}. Substituting the explicit formulas for b_{ε} and for Cas_{V_{λ}}^{Λ^{2}} (c.f. Remark ^{\underline{g2cas}} B.5) we obtain:

F_{Bochner} := 27 p_{3}(B) = + a (a + 3b + 3) (2a + 3b + 4) pr_{+ε_{1}}
− (a + 2) (a + 3b + 5) (2a + 3b + 6) pr_{−ε_{1}}
− (a + 2) (a + 3b + 3) (2a + 3b + 4) pr_{+ε_{2}}
+ a (a + 3b + 5) (2a + 3b + 6) pr_{−ε_{2}}
− a (a + 3b + 5) (2a + 3b + 4) pr_{+ε_{3}}
+ (a + 2) (a + 3b + 3) (2a + 3b + 6) pr_{−ε_{3}}
+ 6( a^{2} + 3b^{2} + 3ab + 5a + 9b + 6) pr_{0}

(4.22) g2bx

Eventually let us discuss the example of 𝔤 = 𝔰𝔭𝔦𝔫_{7}. Here the modified recursion procedure gives the three K–eigenvectors

p_{0}(B) = 1, p_{1}(B) = B, p_{2}(B) = B^{2} + \frac{5}{2} B + \frac{1}{4} Cas_{V_{λ}}^{Λ^{2}} .  (4.23) s7-1

and a τ–eigenvector for the eigenvalue −1, which is of third order as a polynomial in B. After projecting it onto the orthogonal complement of B we obtain:

p_{3}(B) = B^{3} + \frac{11}{2} B^{2} + _{\frac{1}{2 Cas_{V_{λ}}^{Λ^{2}}}} (Cas_{V_{λ}}^{[4]} + \frac{55}{2} Cas_{V_{λ}}^{Λ^{2}}) B + \frac{3}{4} Cas_{V_{λ}}^{Λ^{2}} ,  (4.24) s7-2

where Cas_{V_{λ}}^{[4]} is the eigenvalue of the higher casimir Cas^{[4]} on the irreducible representation V_{λ}. Its explicit value is given in the appendix in Remark ^{\underline{spin7cas}} B.6.

However in difference to the 𝔤_{2}–case this is no K–eigenvector. Indeed in Section ^{\underline{kis}} 8 we will see that the space of polynomials in B of degree at most 3 is not invariant under K. Hence there cannot be a further K–eigenvector expressible as a polynomial of order 3 in B. In general the other K–eigenvectors are polynomials of degree 7 in B. They are too complicated to be written down, but surprisingly the K–eigenvector for the eigenvalue −\frac{9}{4}, i. e. the Bochner identity, for a representation of highest weight λ = aω_{1} + bω_{2} + cω_{3} has the following simple explicit expression:

F_{Bochner} = + c (2b + c + 2) (2a + 2b + c + 4) pr_{+ε_{1}}
− (c + 2) (2b + c + 4) (2a + 2b + c + 6) pr_{−ε_{1}}
− (c + 2) (2b + c + 2) (2a + 2b + c + 4) pr_{+ε_{2}}
+ c (2b + c + 4) (2a + 2b + c + 6) pr_{−ε_{2}}
− c (2b + c + 4) (2a + 2b + c + 4) pr_{+ε_{3}}
+ (c + 2) (2b + c + 2) (2a + 2b + c + 6) pr_{−ε_{3}}
+ (c + 2) (2b + c + 4) (2a + 2b + c + 4) pr_{+ε_{4}}
− c (2b + c + 2) (2a + 2b + c + 6) pr_{−ε_{4}}

(4.25) s7b

This formula is proved in Theorem ^{\underline{bochis7}} 8.7. Note that the coefficients of pr_{+ε_{4}} and pr_{−ε_{4}} are different. Hence in the critical case with c = 2a + 1, i. e. where b_{+ε_{4}} = b_{−ε_{4}}, this K–eigenvector F_{Bochner} spans the space orthogonal to ℂ[B] in End_{𝔤}(T ⊗ V_{λ}).

5 The Weitzenböck Machine for Kähler Holonomies

kholon

The Weitzenböck machine for the Kähler holonomy groups U(n) and SU(n) is remarkably different to the machine for the other non–symmetric irreducible holonomies. Perhaps the most distinctive characteristic of Kähler holonomy is that the complexified holonomy representation T = Ē ⊕ E = T^{1,0} ⊕ T^{0,1} is not irreducible. Consequently the set of generalized gradients {T_{ε}} on sections of an irreducible vector bundle V_{λ}M falls apart into two subsets, the sets of first order parallel differential operators factorizing over the complementary projections pr^{0,1} and pr^{1,0} from T ⊗ V_{λ} to T^{0,1} ⊗ V_{λ} and T^{1,0} ⊗ V_{λ} respectively. In turn the space of Weitzenböck formulas on V_{λ} is the direct sum

𝔚( V_{λ}) = 𝔚^{1,0}( V_{λ}) ⊕ 𝔚^{0,1}( V_{λ})  (5.26) hah

of the spaces 𝔚^{1,0}(V_{λ}) := End_{𝔤}( T^{0,1} ⊗ V_{λ}) and 𝔚^{0,1}(V_{λ}) := End_{𝔤}( T^{1,0} ⊗ V_{λ}) of holomorphic and antiholomorphic Weitzenböck formulas. Evidently the space 𝔚^{1,0}(V_{λ}) is spanned by the projections pr_{ε} to the holomorphic relevant weights ε ⊂ λ^{1,0} defined as the relevant weights among the weights −ε_{1}, …, −ε_{n} of T^{0,1}, in particular pr^{0,1} is the sum pr^{0,1} = ∑_{ε⊂λ^{1,0}} pr_{ε} over all relevant holomorphic weights. Similarly the space 𝔚^{0,1}(V_{λ}) is spanned by the projections to the relevant weights ε ⊂ λ^{0,1} among the antiholomorphic weights +ε_{1}, …, +ε_{n} of T^{1,0} with pr^{1,0} = ∑_{ε⊂λ^{0,1}} pr_{ε}. The isomorphisms T^{0,1} = T^{1,0∗} and T^{1,0} = T^{0,1∗} are reponsible for this apparently skewed notation.

In the Weitzenböck machine we are eventually interested in the matrices of the twist τ and the classifying endomorphism K in the basis pr_{ε} of idempotents of 𝔚(V_{λ}), and a simple observation drastically simplifies this task in the case of Kähler holonomies. The subspaces T^{1,0} and T^{0,1} are isotropic subspaces of T so that Hom_{𝔲_{n}}(T^{1,0} ⊗ T^{1,0} ⊗ V_{λ}, V_{λ}) and Hom_{𝔲_{n}}(T^{0,1} ⊗ T^{0,1} ⊗ V_{λ}, V_{λ}) are trivial. Hence the decomposition (^{\underline{hah}} 5.26) becomes

𝔚^{1,0}( V_{λ}) = Hom_{𝔲_{n}}(T^{1,0}⊗T^{0,1}⊗V_{λ}, V_{λ}) 𝔚^{0,1}( V_{λ}) = Hom_{𝔲_{n}}(T^{0,1}⊗T^{1,0}⊗V_{λ}, V_{λ})

in the interpretation 𝔚(V_{λ}) = Hom_{𝔲_{n}}(T ⊗ T ⊗ V_{λ}, V_{λ}) of the space of Weitzenböck formulas. In this interpretation the endomorphisms τ and K are defined by precomposition with a ⊗ b ⊗ ψ ⟼ b ⊗ a ⊗ ψ and a ⊗ b ⊗ ψ ⟼ − ∑_{α} X_{α}a ⊗ X_{α}b ⊗ ψ respectively so that K preserves the decomposition (^{\underline{hah}} 5.26), while τ interchanges 𝔚^{1,0}(V_{λ}) and 𝔚^{0,1}(V_{λ}). In difference to the holonomies SO(n), G_{2} and Spin(7) discussed above the twist τ is thus not a polynomial in the classifying endomorphism K for U(n) and SU(n).

Before calculating the B–eigenvalues let us recall that the weights ±ε_{1}, …, ±ε_{n} of the holonomy representation T form an orthonormal basis for a scalar product ⟨ , ⟩ on the dual 𝔱^{∗} of a maximal torus 𝔱 ⊂ 𝔲_{n}. The decomposition 𝔲_{n} = iℝ ⊕ 𝔰𝔲_{n} of 𝔲_{n} into center iℝ and simple ideal 𝔰𝔲_{n} is accompanied by an analoguous decomposition λ = λ^{iℝ} + λ^{𝔰𝔲} of a weight λ ∈ 𝔱^{∗} along complementary orthogonal projections defined on the basis by:

ε_{k}^{iℝ} := \frac{1}{n}( ε_{1} + … + ε_{n} ) ε_{k}^{𝔰𝔲} := ε_{k} − \frac{1}{n}( ε_{1} + … + ε_{n} ) .

For the calculations below it is useful to keep the following formulas in mind for all k, l:

⟨ ε_{k}^{iℝ}, ε_{l}^{iℝ} ⟩ = \frac{1}{n}  ⟨ ε_{k}^{iℝ}, ε_{l}^{𝔰𝔲} ⟩ = 0  ⟨ ε_{k}^{𝔰𝔲}, ε_{l}^{𝔰𝔲} ⟩ = δ_{kl} − \frac{1}{n} .

Interestingly the half sum ρ = \frac{n−1}{2}ε_{1} + \frac{n−3}{2}ε_{2} + … + \frac{1−n}{2}ε_{n} of positive roots of 𝔲_{n} is not equal to the sum of the fundamental weights ω_{1} := ε_{1}, ω_{2} := ε_{1} + ε_{2}, ω_{3} := ε_{1} + ε_{2} + ε_{3} etc. specified in Section ^{\underline{tables}} 2. Nevertheless ρ = ρ^{𝔰𝔲} is orthogonal to a weight λ^{iℝ} ∈ 𝔱^{∗} due to

ρ = ω_{1}^{𝔰𝔲} + ω_{2}^{𝔰𝔲} + … + ω_{n}^{𝔰𝔲} ,

in particular ⟨ ε_{k}^{𝔰𝔲}, 2ρ ⟩ = ⟨ ε_{k}, 2ρ ⟩ = n − 2k + 1. Having established this much of the notation we need to verify that the chosen scalar product ⟨ , ⟩ is suitable for calculating Casimir eigenvalues. The problem is that there are two linearly independent scalar products on 𝔲_{n} and so we have to check the normalization condition (^{\underline{cnor}} 3.8) for the Casimir eigenvalues for both ideals iℝ and 𝔰𝔲_{n} of 𝔲_{n} separately. Recalling that the holonomy representation T is the complexification of a real irreducible representation T_{ℝ} we observe that the Casimir eigenvalues of the normalized Casimirs Cas^{Λ^{2},iℝ} and Cas^{Λ^{2},𝔰𝔲} of the two ideals iℝ and 𝔰𝔲_{n} are the same on T, T^{1,0} and T^{0,1}. Taking the irreducible subspace T^{1,0} of highest weight ε_{1} for convenience we verify (^{\underline{cnor}} 3.8)

Cas_{T}^{Λ^{2},iℝ} = Cas_{T^{1,0}}^{Λ^{2},iℝ} = − 2 \frac{dim iℝ}{dim T} = − \frac{1}{n} = − ⟨ ε_{1}^{iℝ}, ε_{1}^{iℝ} ⟩
Cas_{T}^{Λ^{2},𝔰𝔲} = Cas_{T^{1,0}}^{Λ^{2},𝔰𝔲} = − 2 \frac{dim 𝔰𝔲}{dim T} = − \frac{n^{2} − 1}{n} = − ⟨ ε_{1}^{𝔰𝔲} + 2ρ, ε_{1}^{𝔰𝔲} ⟩

for the normalized Casimir:

Cas_{V_{λ}}^{Λ^{2}} = − ⟨ λ + 2ρ, λ ⟩ = − ⟨ λ^{iℝ}, λ^{iℝ} ⟩ − ⟨ λ^{𝔰𝔲} + 2ρ, λ^{𝔰𝔲} ⟩ .  (5.27) ckk

Corollary ^{\underline{confW}} 3.4 allows us to calculate the eigenvalues of the conformal weight operator B aka conformal weights. With respect to the orthonormal basis ε_{1}, …, ε_{n} of 𝔱^{∗} the highest weight of an irreducible representation V_{λ} can be written λ = μ_{1}ε_{1} + … + μ_{n}ε_{n} with decreasing integral coefficients μ_{1} ≥ … ≥ μ_{n} so that the conformal weights are

b_{−ε_{k}} = −μ_{k} + k − n  b_{+ε_{k}} = μ_{k} − k + 1  (5.28) kaehler

for the conformal weight operator B of the full holonomy algebra 𝔲_{n}. In particular both the holomorphic and antiholomorphic conformal weights are separately ordered

b_{−ε_{n}} > b_{−ε_{n−1}} > … > b_{−ε_{1}} b_{+ε_{1}} > b_{+ε_{2}} > … > b_{+ε_{n}} ,

but we can not exclude the equality between the conformal weights for a holomorphic and an antiholomorphic weight, moreover there are certainly dominant integral weights λ with b_{−ε_{k}} = b_{+ε_{n−k}} for all k = 1, …, n. For this reason it is prudent to study the central conformal weight operator B^{iℝ} besides the conformal weight operator B of the holonomy algebra 𝔲_{n}, whose eigenvalues or central conformal weights

b_{ε}^{iℝ} = \frac{1}{2} ( ⟨ λ^{iℝ} + ε^{iℝ}, λ^{iℝ} + ε^{iℝ} ⟩ − ⟨ λ^{iℝ}, λ^{iℝ} ⟩ − ⟨ ε_{1}^{iℝ}, ε_{1}^{iℝ} ⟩ ) = ⟨ λ^{iℝ}, ε^{iℝ} ⟩

equal ±\frac{1}{n}(μ_{1}+…+μ_{n}) for holomorphic (−) and antiholomorphic (+) weights respectively. The algebra homomorphism Φ : ℂ[B^{iℝ}, B^{𝔰𝔲}] ⟶ End_{𝔲_{n}}(T ⊗ V_{λ}) is thus surjective for all dominant integral weights λ with λ^{iℝ} ≠ 0.

In order to find the eigenspaces of the classifying endomorphism K and their dimensions we recall that K preserves the orthogonal subspaces 𝔚^{1,0}(V_{λ}) and 𝔚^{0,1}(V_{λ}) of holomorphic and antiholomorphic Weitzenböck formulas. The eigenvalues of K on 𝔚^{1,0}(V_{λ}) agree with the conformal weights for the relevant antiholomorphic weights for T^{0,1} = V_{−ε_{n}} with associated projections in T^{1,0}⊗T^{0,1} ⊂ T ⊗T^{0,1}. The eigenvalues of K on the complementary subspace 𝔚^{0,1}(V_{λ}) correspond similarly to the conformal weights for the relevant holomorphic weights for T^{1,0} = V_{+ε_{1}}. Using the decision criterion of Lemma \underline{relw} 2.2 we find the relevant antiholomorphic weights +ε_{1} and +ε_{n} for λ = −ε_{n} and the relevant holomorphic weights −ε_{1} and −ε_{n} for λ = +ε_{1} with corresponding decompositions

T^{1,0} ⊗ V_{−ε_{n}} = (𝔰𝔲_{n} ⊗_{ℝ} ℂ) ⊕ ℂ T^{0,1} ⊗ V_{+ε_{1}} = ℂ ⊕ (𝔰𝔲_{n} ⊗_{ℝ} ℂ)

where 𝔰𝔲_{n} ⊗_{ℝ} ℂ = V_{+ε_{1}−ε_{n}} is the adjoint representation. Using the explicit values (^{\underline{kaehler}} 5.28) for the conformal weights we conclude that K has eigenvalues −n and 0 with multiplicities 1 and n − 1 respectively on 𝔚^{1,0}(V_{λ}), in other words K vanishes on the orthogonal complement of the one–dimensional eigenspace of eigenvalue −n in 𝔚^{1,0}(V_{λ}). The distinguished projections pr^{0,1} ∈ 𝔚^{0,1}(V_{λ}) and pr^{1,0} ∈ 𝔚^{0,1}(V_{λ}) are natural candidates for eigenvectors of K for the eigenvalue −n and in fact we find in the interpretation 𝔚^{1,0}(V_{λ}) = Hom_{𝔲_{n}}(T^{1,0} ⊗ T^{0,1} ⊗ V_{λ}, V_{λ})

( Kpr^{0,1} )(a ⊗ b ⊗ v) = − ∑_{α}⟨ X_{α}a, pr^{0,1}(X_{α}b) ⟩ v = ∑_{α}⟨ X_{α} X_{α}a, pr^{0,1}b ⟩ v
= Cas_{T^{1,0}}^{Λ^{2}} pr^{0,1}(a ⊗ b ⊗ v) = − n pr^{0,1}(a ⊗ b ⊗ v) ,

where the X_{α} are an orthonormal basis for 𝔲_{n} with respect to the scalar product induced from 𝔲_{n} ⊂ Λ^{2}T. Turning to antiholomorphic Weitzenböck formulas in 𝔚^{0,1}(V_{λ}) a similar argument implies that the classifying endomorphism K vanishes on the orthogonal complement of the eigenvector pr^{1,0} with eigenvalue −n. Put differently this result reads:

Lemma 5.1 (Explicit Form of the Classifying Endomorphism)

cle

On the basis vectors {pr_{ε}} of the space 𝔚(V_{λ}) of Weitzenböck formulas on an irreducible representation V_{λ} the classifying endomorphism K of the holonomy algebra 𝔲_{n} acts by

K pr_{ε} = − \frac{dim V_{λ+ε}}{dim V_{λ}} pr^{0,1}  or  K pr_{ε} = − \frac{dim V_{λ+ε}}{dim V_{λ}} pr^{1,0}

depending on whether ε is a holomorphic or antiholomorphic weight of T respectively.

Proof: Taking the stated formula as a definition of a linear map K : 𝔚(V_{λ}) ⟶ 𝔚(V_{λ}) we apply it to a holomorphic Weitzenböck formula F^{1,0} = ∑_{ε⊂λ^{1,0}} f_{ε}pr_{ε} and find

K( ∑_{ε⊂λ^{1,0}} f_{ε} pr_{ε} ) = − ( ∑_{ε⊂λ^{1,0}} f_{ε} \frac{dim V_{λ+ε}}{dim V_{λ}} ) pr^{0,1} = − ⟨ pr^{0,1}, F^{1,0} ⟩ pr^{0,1}

so that Kpr^{0,1} = −n pr^{0,1} and KF^{1,0} = 0 for F^{1,0} ∈ 𝔚^{1,0}(V_{λ}) orthogonal to pr^{0,1}. Hence the endomorphism K implicitly defined above agrees with the classifying endomorphism on the subspace 𝔚^{1,0}(V_{λ}). Mutatis mutandis we check agreement of K with the classifying endomorphism on the subspace 𝔚^{0,1}(V_{λ}). □

In the case of Kähler holonomies the twist τ is not a polynomial on the classifying endomorphism K and the explicit description of K in Lemma ^{\underline{cle}} 5.1 does not seem to be of any use in understanding τ. It turns out however that the Recursion Formula ^{\underline{rec}} 4.1 considered as a matrix equation for the twist τ already has a unique solution. More precisely the Recursion Formula ^{\underline{rec}} 4.1 can be put into the form

τ K = − ( τ B + B τ + n τ )  (5.29) recz

because τ is an involution and −2\frac{dim 𝔲_{n}}{dim T} = −n. Moreover τ interchanges the subspaces 𝔚^{1,0}(V_{λ}) and 𝔚^{0,1}(V_{λ}) of holomorphic and antiholomorphic Weitzenböck formulas, thus

τ pr_{−ε} = ∑_{+ε̃} τ_{−ε,+ε̃} pr_{+ε̃} τ pr_{+ε} = ∑_{−ε̃} τ_{+ε,−ε̃} pr_{−ε̃}

with as yet unknown rational matrix coefficients τ_{+ε,−ε̃} and τ_{−ε,+ε̃}. Here and in the following the notation −ε refers exclusively to holomorphic, +ε to antiholomorphic weights. Besides interchanging 𝔚^{1,0}(V_{λ}) and 𝔚^{0,1}(V_{λ}) the twist interchanges pr^{0,1} and pr^{1,0}:

( τ pr^{0,1} )(a ⊗ b ⊗ v) = ⟨ b, pr^{0,1}a ⟩ v = ⟨ a, pr^{1,0}b ⟩ v = pr^{1,0}(a ⊗ b ⊗ v) .

In consequence Lemma ^{\underline{cle}} 5.1 tells us the matrix coefficients of the operator τ K

( τ K) pr_{−ε} = − \frac{dim V_{λ−ε}}{dim V_{λ}} τ pr^{0,1} = − ∑_{+ε̃} \frac{dim V_{λ−ε}}{dim V_{λ}} pr_{+ε̃}

while B is diagonal in the basis pr_{ε} with eigenvalues b_{ε}. All in all equation (^{\underline{recz}} 5.29) becomes:

( τ K)_{−ε,+ε̃} = − \frac{dim V_{λ−ε}}{dim V_{λ}} = − ( b_{−ε} + b_{+ε̃} + n ) τ_{−ε,+ε̃}

This equation has a unique solution τ_{−ε,+ε̃}, because b_{−ε} + b_{+ε̃} + n is never zero. In fact

b_{−ε_{k}} + ♭_{+ε_{l}} + n = ( μ_{k} − μ_{l}) + ( l − k) + 1 1 ≤ k, l ≤ n

equals 1 for k = l and is positive for k < l as the coefficients μ_{1} ≥ … ≥ μ_{n} are decreasing. Similarly (μ_{k} − μ_{l}) + (l − k) + 1 is negative for k > l unless k = l + 1 and μ_{l} = μ_{k}, in which case Lemma ^{\underline{relw}} 2.2 ensures that −ε_{l} is irrelevant by λ_{l+1} = 0. Mutatis mutandis the same argument applies to the remaining matrix coefficients τ_{+ε,−ε̃} of the twist operator.

Theorem 5.2 (Matrix Coefficients of the Twist Operator)

tm

The matrix coefficients τ_{−ε,+ε̃} and τ_{+ε,−ε̃} of the twist operator in the basis pr_{ε} of projections in 𝔚(V_{λ}) defined by τpr_{−ε} = ∑_{+ε̃} τ_{−ε,+ε̃} pr_{+ε̃} and τpr_{+ε} = ∑_{−ε̃} τ_{+ε,−ε̃} pr_{−ε̃} are given by:

τ_{−ε,+ε̃} = \frac{1}{b_{−ε} + b_{+ε̃} + n} \frac{dim V_{λ−ε}}{dim V_{λ}} τ_{+ε,−ε̃} = \frac{1}{b_{+ε} + b_{−ε̃} + n} \frac{dim V_{λ+ε}}{dim V_{λ}}

Another interesting consequence of the decomposition of the space 𝔚(V_{λ}) into the subspaces 𝔚^{1,0}(V_{λ}) and 𝔚^{0,1}(V_{λ}) of holomorphic and antiholomorphic Weitzenböck formulas should not pass unnoticed. As the twist τ interchanges 𝔚^{1,0}(V_{λ}) and 𝔚^{0,1}(V_{λ}) every eigenvector for τ is determined by its summand in the subspace 𝔚^{1,0}(V_{λ}). In fact an arbitrarily chosen F^{1,0} ∈ 𝔚^{1,0}(V_{λ}) gives rise to an eigenvector

F := F^{1,0} ± τ( F^{1,0} )

for τ of eigenvalue ±1. Of course we are primarily interested in the eigenvalue −1:

Corollary 5.3 (Pure Curvature Terms in Kähler Holonomy)

purec

Recall that the eigenvectors F = ∑_{ε} f_{ε}pr_{ε} of the twist operator τ for the eigenvalue −1 in the space 𝔚(V_{λ}) of Weitzenböck formulas on an irreducible representation V_{λ} correspond to Weitzenböck formulas reducing to a pure curvature term, compare (^{\underline{nabla}} 3.4) and (^{\underline{cterm}} 3.5):

F ( ∇^{2} ) = − ∑_{ε} T_{ε}^{∗} T_{ε} = \frac{1}{2} ∑_{μ,ν} F_{t_{μ}⊗t_{ν}} R_{t_{μ}, t_{ν}} .

In Kähler holonomies there is a bijection F^{1,0} ⟼ F^{1,0} − τ(F^{1,0}) between 𝔚^{1,0}(V_{λ}) and the eigenspace of τ for eigenvalue −1 or pure curvature term Weitzenböck formulas.

On a Kähler manifold M the holonomy algebra bundle 𝔲M can be identified with the bundle of skew–symmetric endomorphisms of the tangent bundle commuting with the parallel complex structure I, it has a parallel subbundle iℝM ⊂ 𝔲M associated to the center iℝ ⊂ 𝔲_{n} of the holonomy algebra and spanned by I in every point. It is well– known that the curvature tensor R of a Kähler manifold is a (1, 1)–form in the sense R_{IX,IY} Z = R_{X,Y} Z for all X, Y, Z ∈ T M, hence the Ricci endomorphism is not only a symmetric endomorphism of the tangent bundle, but commutes with I, too:

Ric(IZ) = ∑_{μ} R_{IZ,t_{μ}}t_{μ} = − ∑_{μ} R_{Z,It_{μ}}t_{μ} = ∑_{μ} R_{Z,t_{μ}}It_{μ} = I (Ric Z) .

In turn the skew–symmetric endomorphism IRic commuting with I can be thought of as a section of the holonomy algebra bundle 𝔲M with a natural action on every vector bundle V M associated to the holonomy reduction. A different way to understand this action is to note that IRic ∈ Γ(𝔲M) is up to sign the result of applying the curvature operator R : Λ^{2}T M ⟶ 𝔲M to the complex structure, because we find the action

R( I) Z = \frac{1}{2} ∑_{μ} R_{t_{μ},It_{μ}}Z = −\frac{1}{2} ∑_{μ} ( R_{Z,t_{μ}}It_{μ} − R_{Z,It_{μ}}t_{μ} ) = −I Ric Z

on T M. Using ⟨ I, I ⟩ = −\frac{1}{2}I^{2} = n we conclude that the curvature term q^{iℝ}(R) reads:

q^{iℝ}( R) = \frac{1}{n} I R( I) = − \frac{1}{n} I (I Ric)  (5.30) ccop

6 A Matrix Presentation of the Twist Operator τ

matrix

In the last section we have studied the Weitzenböck machine for Kähler holonomies and succeeded in describing the twist τ and the classifying endomorphism K explicitly in the natural basis pr_{ε} of the space of Weitzenböck formulas. The principal idea of this calculation was to read the Recursion Formula ^{\underline{rec}} 4.1, which we used in Section ^{\underline{eigenvalues}} 4.2 for a recursive construction of an eigenbasis for the classifying endomorphism, directly as a matrix equation for the unknown matrix of τ. It turns out that the same idea allows us to derive a closed matrix expression for the twist operator τ and the classifying endomorphism K in the three other holonomies SO(n), G_{2} and Spin(7) considered in this article.

In the case of SO(n)–holonomy the classifying endomorphism K and the twist differ only on the span of the unit 1 ∈ 𝔚(V_{λ}), which becomes the identity 1 = id_{T ⊗V_{λ}} in the interpretation 𝔚(V_{λ}) = End_{𝔰𝔬_{n}}(T ⊗ V_{λ}) and the connection Laplacian 1(∇^{2}) = −∇^{∗}∇ in the interpretation as a second order parallel differential operator. The norm of 1 is simply the dimension ⟨ 1, 1 ⟩ = n of T and so we conclude

K = τ − 1 ⊗ 1  (6.31) kqmod

where 1 ⊗ 1 denotes the endomorphism:

1 ⊗ 1 :  𝔚(V_{λ}) ⟼ 𝔚(V_{λ}),  F ⟼ ⟨ 1, F ⟩ 1 .

In fact in the interpretation 𝔚(V_{λ}) = Hom_{𝔰𝔬_{n}}(T ⊗ T ⊗ V_{λ}, V_{λ}) both K and τ act by precomposition with an endomorphism of T ⊗ T extended to T ⊗ T ⊗ V_{λ} in such a way that the eigenvalues of K and τ on the orthogonal complement Sym_{0}^{2}T ⊕ Λ^{2} of ℂ agree according to Table ^{\underline{kt}} 3.16. Hence we need only check equation (^{\underline{kqmod}} 6.31) on 1, where we can use the trivial statement τ(1) = 1 and K(1) = −(n − 1)1 from Lemma ^{\underline{elem}} 3.10. In light of equation (^{\underline{kqmod}} 6.31) the Recursion Formula ^{\underline{rec}} 4.1 can be rewritten in the form

τ K = id − 1 ⊗ 1 = − ( τ B + B τ + (n − 1) τ )  (6.32) recso

in the case of SO(n)–holonomy, note that τ is an involution with τ ∘ 1 ⊗ 1 = 1 ⊗ 1. In order to turn equation (^{\underline{recso}} 6.32) into a matrix equation for the unknown matrix coefficients

τ pr_{ε} =: ∑_{ε̃} τ_{ε,ε̃} prε̃ τ_{ε,ε̃} ∈ ℚ

of the twist τ we observe that by its very definition (1 ⊗ 1)pr_{ε} = ⟨ 1, pr_{ε} ⟩1 = \frac{dim V_{λ+ε}}{dim V_{λ}} 1. With this additional piece of information equation (^{\underline{recso}} 6.32) becomes

δ_{ε,ε̃} − \frac{dim V_{λ+ε}}{dim V_{λ}} = − ( b_{ε} + bε̃ + n − 1 ) τ_{ε,ε̃}

as again the operator B is diagonal in the basis pr_{ε}. Unlike the Kähler case we can not exclude the possibility that b_{ε} + bε̃ + n − 1 is zero. If we assume ε to be relevant and thus dim V_{λ+ε} ≠ 0 however, then this can only happen on the diagonal ε = ε̃ under the assumption dim V_{λ+ε} = dim V_{λ}. For all relevant, off–diagonal weights ε ≠ ε̃ the matrix coefficients of τ are thus given by

τ_{ε,ε̃} = − \frac{1}{b_{ε} + bε̃ + n − 1} ( δ_{ε,ε̃} − \frac{dim V_{λ+ε}}{dim V_{λ}} ) .  (6.33) som

and this formula is still true on the diagonal ε = ε̃ provided dim V_{λ+ε} ≠ dim V_{λ}. In order to eliminate these rather unpleasant restrictions we note that the equation τ(1) = 1 allows us to recover the diagonal coefficients τ_{ε,ε} of τ with dim V_{λ+ε} = dim V_{λ}. Moreover the matrix coefficients τ_{ε,ε̃} are rational functions in the highest weight λ, because the conformal weights b_{ε} and bε̃ are linear in λ and the dimensions of V_{λ} and V_{λ+ε} are polynomials in λ due to Weyl’s Dimension Formula. Taken together these arguments imply that the troublesome matrix coefficients τ_{ε,ε̃} left undefined by equation (^{\underline{som}} 6.33) agree with the values of the analytic continuation considered as a rational function of λ to values to the discriminant set, where zeroes of the numerator and the denominator cancel out.

Theorem 6.1 (The Twist Operator in SO(n)–Holonomy)

matw

The matrix coefficients τ_{ε,ε̃} of the twist τ : 𝔚(V_{λ}) ⟶ 𝔚(V_{λ}) in the basis of projections pr_{ε} ∈ 𝔚(V_{λ}) are defined by τ pr_{ε} = ∑ε̃ τ_{ε,ε̃} prε̃. In SO(n)–holonomy they are given by

τ_{ε,ε̃} = − \frac{1}{b_{ε} + bε̃ + n − 1} ( δ_{ε,ε̃} − \frac{dim V_{λ+ε}}{dim V_{λ}} )

provided we interprete τ_{ε,ε̃} as a rational function in the highest weight λ and cancel zeroes of the numerator and denominator for values of ε and ε̃, for which b_{ε} + bε̃ + n − 1 = 0.

Turning from the case of SO(n)–holonomy to G_{2} we note that the relation between the twist operator τ and the classifying endomorphism K becomes more complicated than equation (^{\underline{kqmod}} 6.31) due to the presence of the Bochner identity (^{\underline{g2bx}} 4.22). For definiteness we let β := F_{Bochner} be the K–eigenvector for the eigenvalue −2 specified in (^{\underline{g2bx}} 4.22), whose coefficients β = ∑_{ε} β_{ε} pr_{ε} are given explicitly as polynomials in the highest weight λ = aω_{1} + bω_{2}, a, b ≥ 0. The argument in the SO(n)–case extends to the relation

K = \frac{1}{3} τ + \frac{1}{3} − \frac{2}{3} 1 ⊗ 1 − \frac{2}{⟨ β, β ⟩}β ⊗ β

between K and τ in the G_{2}–case so that the Recursion Formula ^{\underline{rec}} 4.1 becomes

τ K = \frac{1}{3} id + \frac{1}{3} τ − \frac{2}{3} 1 ⊗ 1 + \frac{2}{⟨ β, β ⟩}β ⊗ β = − ( τ B + B τ + 4 τ ) ,

because β is an eigenvector for τ of eigenvalue −1. Moreover the calculation

( β ⊗ β) pr_{ε} = ∑_{ε̃} ( β_{ε} \frac{dim V_{λ+ε}}{dim V_{λ}} βε̃ )prε̃

tells us the matrix coefficients of β ⊗ β and so we end up with the formula

τ_{ε,ε̃} = − \frac{1}{3 b_{ε} + 3 bε̃ + 13} ( δ_{εε̃} − 2 ( 1 − 3 \frac{β_{ε} βε̃}{⟨ β, β ⟩} ) \frac{dim V_{λ+ε}}{dim V_{λ}} )  (6.34) g2om

for the matrix coefficients τ_{ε,ε̃} of τ defined by τ pr_{ε} = ∑ε̃ τ_{ε,ε̃}prε̃. Again this formula has to be read as a rational function in the highest weight λ to get the right values for τ_{ε,ε̃} for index tuples ε, ε̃ satisfying 3b_{ε} + 3bε̃ + 13 = 0 by cancelling the zero of the denominator with the apparent zero of the numerator.

Theorem 6.2 (The Twist Operator in G_{2}–Holonomy)

matg2

The matrix coefficients τ_{ε,ε̃} of the twist τ : 𝔚(V_{λ}) ⟶ 𝔚(V_{λ}) in the basis of projections pr_{ε} ∈ 𝔚(V_{λ}) are defined by τ pr_{ε} = ∑ε̃ τ_{ε,ε̃} prε̃. In G_{2}–holonomy they are given by

τ_{ε,ε̃} = − \frac{1}{3 b_{ε} + 3 bε̃ + 13} ( δ_{εε̃} − 2 ( 1 − 3 \frac{β_{ε} βε̃}{⟨ β, β ⟩} ) \frac{dim V_{λ+ε}}{dim V_{λ}} )

where the coefficients β_{ε} of the Bochner identity β = ∑_{ε} β_{ε}pr_{ε} are given by (^{\underline{g2bx}} 4.22).

Eventually we want to sketch briefly the corresponding argument in the case of exceptional holonomy Spin(7). Chosing the Bochner identity β := F_{Bochner} of equation (^{\underline{s7b}} 4.25) with coefficients β = ∑_{ε} β_{ε} pr_{ε} we can cast the relation between K and τ into the form

K = \frac{1}{2} τ + \frac{1}{4} − \frac{3}{4} 1 ⊗ 1 − \frac{2}{⟨ β, β ⟩}β ⊗ β

so that the Recursion Formula ^{\underline{rec}} 4.1 is equivalent to the identity:

τ K = \frac{1}{2} id + \frac{1}{4} τ − \frac{3}{4} 1 ⊗ 1 + \frac{2}{⟨ β, β ⟩}β ⊗ β = − ( τ B + B τ + \frac{21}{4} τ ) ,

All in all we end up with the following formula for the matrix coefficients

τ_{ε,ε̃} = − \frac{1}{2 b_{ε} + 2 bε̃ + 11} ( δ_{εε̃} − ( \frac{3}{2} − 4 \frac{β_{ε} βε̃}{⟨ β, β ⟩} ) \frac{dim V_{λ+ε}}{dim V_{λ}} )  (6.35) s7om

of the twist operator τ defined by τ pr_{ε} = ∑ε̃ τ_{ε,ε̃} prε̃.

7 Examples

examples

In this section we will present a few examples of how to obtain for a given representation V_{λ} the possible Weitzenböck formulas on sections of the associated bundle V_{λ}M. The general procedure is as follows: we first determine the relevant weights ε using the diagrams of Section ^{\underline{tables}} 2. This gives the decomposition of T ⊗ V_{λ} into irreducible summands and defines the twistor operators T_{ε}. Next we compute the B–eigenvalues b_{ε}, e.g. using the general formula of Corollary ^{\underline{confW}} 3.4 and obtain the first Weitzenböck formula as Equation (^{\underline{weizen3}} 3.11). Other Weitzenböck formulas correspond to the B–polynomials constructed in the preceding section. If F = F (B) is such a polynomial then the coefficient of T_{ε}^{∗}T_{ε} is given as −F (b_{ε}).

As a first example we consider the bundle of p–forms on a Riemannian manifold (M^{n}, g) for simplicity we assume n = 2r + 1 and p ≤ r − 1, i.e. 𝔤 = 𝔰𝔬_{2r+1} and λ = ω_{p}, i.e. the associated bundle is the bundle of p–forms. The relevant weights according to the tables of Section ^{\underline{eigenvalues}} 4.2 are ε_{1}, −ε_{p} and ε_{p+1} with the decomposition

T ⊗ V_{λ} = V_{λ+ε_{1}} ⊕ V_{λ−ε_{p}} ⊕ V_{λ+ε_{p+1}} ≅ V_{λ+ε_{1}} ⊕ Λ^{p−1} ⊕ Λ^{p+1}

and twistor operators T_{ε_{1}}, T_{−ε_{p}} and T_{ε_{p}}. To compare our twistor operators with differential and codifferential d, d^{∗} we have to embed Λ^{p−1} resp. Λ^{p+1} into the tensor product T ⊗ Λ^{p}. This leads to the following formula

T_{−ε_{p}}^{∗} T_{−ε_{p}} = \frac{1}{n−p+1} d d^{∗}, T_{+ε_{p}}^{∗}T_{+ε_{p}} = \frac{1}{p+1} d^{∗} d

Next we take the relevant B–eigenvalues from Section ^{\underline{eigenvalues}} 4.2 they are:

b_{+ε_{1}} = 1,  b_{−ε_{p}} = −n + p,  b_{e_{p+1}} = −p .

Since we have only three summands in the decomposition of T ⊗ V_{λ} we obtain only one Weitzenböck formula with a pure curvature term, which is the formula given in Corollary ^{\underline{weizen3}} 3.11:

\begin{matrix}q(R) = & − & T_{+ε_{1}}^{∗} T_{+ε_{1}} & + & (n − p) & T_{−ε_{p}}^{∗} T_{−ε_{p}} & + & p & T_{+ε_{p}}^{∗} T_{+ε_{p}} \\ = & − & T_{+ε_{1}}^{∗} T_{+ε_{1}} & + & \frac{n−p}{n−p+1} & d d^{∗} & + & \frac{p}{p+1} & d^{∗} d \end{matrix}

If we add the Weitzenböck formula (^{\underline{nabla}} 3.4) for ∇^{∗}∇ to this expression for q(R) we obtain the classical Weitzenböck formula for the Laplacian on p–forms:

Δ = ∇^{∗}∇ + q(R) = (n − p + 1) T_{−ε_{p}}^{∗} T_{−ε_{p}} + (p + 1) T_{+ε_{p}}^{∗} T_{+ε_{p}} = d d^{∗} + d^{∗} d

Let (M^{2r}, g) be a Riemannian spin manifold with spinor bundle S = S_{+} ⊕ S_{−}. We consider the two bundles defined by the Cartan summand in S_{±} ⊗ T with highest weights λ_{+} = ω_{1}+ω_{r−1} and λ_{−} = ω_{1}+ω_{r}. Using the tables of Section ^{\underline{tables}} 2 we find the relevant weights +ε_{1}, −ε_{1} and +ε_{2} for both λ_{±} and in addition −ε_{r} or +ε_{r} for λ_{+} or λ_{−} respectively. The corresponding tensor product decomposition is

T ⊗ V_{λ_{±}} = V_{λ_{±}+ε_{1}} ⊕ V_{λ_{±}−ε_{1}} ⊕ V_{λ_{±}+ε_{2}} ⊕ V_{λ_{±}∓ε_{r}} .

Note that λ_{±} −ε_{1} is the defining representation for the bundles S_{±} and that λ_{±} ∓ε_{r} = λ_{∓}. Projecting the covariant derivative of a section of T ⊗ V_{λ_{±}} onto one of these summands defines four twistor operators. The fourth operator T_{∓ε_{r}} : Γ(V_{λ_{±}}) ⟶ Γ(V_{λ_{∓}}) is usually called the Rarita–Schwinger operator. A solution of the Rarita–Schwinger equation is by definition a section of ψ ∈ Γ(V_{λ_{±}}) with both T_{∓ε_{r}}ψ = 0 and T_{−ε_{1}}ψ = 0.

The B–eigenvalues for 𝔰𝔬_{n}–representations were computed in Section ^{\underline{eigenvalues}} 4.2 and in particular:

b_{+ε_{1}} = \frac{3}{2}, b_{−ε_{1}} = −2r + \frac{1}{2}, b_{+ε_{2}} = − \frac{1}{2}, b_{±ε_{r}} = − r + \frac{1}{2}

Since the decomposition of T ⊗ V_{λ_{±}} has four summands we will obtain two Weitzenböck formulas with a pure curvature term. The first one is again Equation ^{\underline{weizen3}} 3.11 corresponding to B, whereas the second corresponds to p_{3}(B), the degree 3 polynomial of the recursion procedure defined in (^{\underline{gleichung2}} 4.19). Its coefficients are the values p_{3}(b_{ε}) for the relevant weights ε. The Casimir operator of an irreducible 𝔰𝔬_{n}–representation V_{λ} with highest weight λ is computed as Cas_{V_{λ}} = −⟨ λ + 2ρ, λ ⟩, where ⟨ ·, · ⟩ is the standard scalar product on ℝ^{r}. In particular we have Cas_{V_{λ_{±}}} = − \frac{1}{4} r (2r + 7). Eventually we obtain the following two Weitzenböck formulas on sections of V_{λ_{±}}:

q(R) = −\frac{3}{2} T_{+ε_{1}}^{∗} T_{+ε_{1}} + (2r − \frac{1}{2}) T_{−ε_{1}}^{∗} T_{−ε_{1}} + \frac{1}{2} T_{+ε_{2}}^{∗} T_{+ε_{2}} + (r − \frac{1}{2}) T_{±ε_{r}}^{∗} T_{±ε_{r}}
p_{3}(B)(∇^{2}) = −(\frac{3}{2} + r)(r − 1) T_{+ε_{1}}^{∗} T_{+ε_{1}} + (2r − 1)(r^{2} − 1) T_{−ε_{1}}^{∗} T_{−ε_{1}}
+ (r − \frac{1}{2})(r + 1) T_{+ε_{2}}^{∗} T_{+ε_{2}} + T_{±ε_{r}}

Note that similar Weitzenböck formulas were obtained in ^{\underline{branson}} [BH02]. More precisely their curvature terms Z_{1} and Z_{2} are related to B and p_{3}(B) by the following equations:

Z_{1} = − \frac{(2r+3)(r−1)}{r(2r+1)} B + \frac{3}{r(2r+1)} p_{3}(B) Z_{2} = − \frac{(2r−1)(r+1)}{r(2r+1)} B + \frac{1}{r(2r+1)} p_{3}(B) .

In the last part of this section we want to describe for G_{2}– and Spin(7)–holonomy all pure curvature Weitzenböck formulas on parallel subbundles of the form bundle. In particular we will present the form Laplacian Δ = d^{∗}d + dd^{∗} = ∇^{∗}∇ + q(R) as a linear combination of the operators T_{ε}^{∗}T_{ε} and discuss the existence of harmonic forms.

We start with the case of G_{2}–holonomy. Let Γ_{a, b} be the irreducible G_{2}–representation with highest weight aω_{1} + bω_{2}, a, b ≥ 0, e.g. Γ_{0, 0} = ℂ is the trivial representation, Γ_{1, 0} = T and Γ_{0, 1} = Λ_{14}^{2} ≅ 𝔤_{2}. Recall that up to dimension 77 irreducible G_{2}–representations are uniquely determined by their dimension. However there are two different irreducible representations in dimension 77, one of them is [77]^{−} := Γ_{3, 0}, the other is Γ_{0,2}, the space of G_{2}-curvature tensors. Moreover

dim Γ_{2, 0} = 27, dim Γ_{1, 1} = 64 .

The spaces of 2– and 3–forms have the following decompositions:

Λ^{2}T ≅ Λ^{5}T ≅ T ⊕ Λ_{14}^{2},  Λ^{3}T ≅ Λ^{4}T ≅ ℂ ⊕ T ⊕ Λ_{27}^{3} ,  (7.36) deco4

where the subscripts denote the dimension of the representation. Next we give the relevant weights for the representations Γ_{1, 0}, Γ_{0, 1} and Γ_{2, 0}. We start with λ = ω_{2}, i.e. the representation V_{λ} = Γ_{0, 1} = Λ_{14}^{2}, here the relevant weights are ε = −ε_{2}, ε_{3}, ε_{1} with λ + ε = ω_{1}, 2ω_{1}, ω_{1} + ω_{2} and the corresponding decomposition reads

T ⊗ Γ_{0, 1} = Γ_{1, 0} ⊕ Γ_{2, 0} ⊕ Γ_{1, 1} = T^{∗} ⊕ Λ_{27}^{3}T^{∗} ⊕ [64] .

In this case equation (^{\underline{weizen3}} 3.11) is the only pure curvature Weitzenböck formula. With the explicit B–eigenvalues given in Section ^{\underline{eigenvalues}} 4.2 we find on sections of Λ_{14}^{2}T^{∗}M:

\begin{aligned}q(R) & = & 4 T_{−ε_{2}}^{∗} T_{−ε_{2}} & + & \frac{4}{3} T_{+ε_{3}}^{∗} T_{+ε_{3}} & − & T_{+ε_{1}}^{∗} T_{+ε_{1}} \\ Δ & = & 5 T_{−ε_{2}}^{∗} T_{−ε_{2}} & + & \frac{7}{3} T_{+ε_{3}}^{∗} T_{+ε_{3}} \end{aligned}

Hence a form ψ in Λ_{14}^{2} ⊂ Λ^{2}T is harmonic if and only if T_{−ε_{2}}ψ = 0 = T_{+ε_{3}}ψ (the manifold is assumed to be compact), i.e. if and and only if ∇ψ = T_{+ε_{1}}ψ or equivalently if and only if ∇ψ is a section of Γ_{11} = [64]. This statement corresponds to the fact that on a compact manifold a form is harmonic if and only if it is closed and coclosed.

For λ = ω_{1}, i.e. the representation V_{λ} = Γ_{1,0} = T the relevant weights are determined as ε = −ε_{1}, 0, +ε_{2}, +ε_{1} with λ + ε = 0, ω_{1}, ω_{2}, 2ω_{1} leading to the decomposition:

T ⊗ Γ_{1,0} = Γ_{0, 0} ⊕ Γ_{1, 0} ⊕ Γ_{0, 1} ⊕ Γ_{2, 0} = ℂ ⊕ T^{∗} ⊕ Λ_{14}^{2}T^{∗} ⊕ Λ_{27}^{3}T^{∗} .

Here we have two pure curvature Weitzenböck formulas. In fact both curvature terms are zero, since q(R) = Ric = 0 on Γ_{1,0} = T. In addition to (^{\underline{weizen3}} 3.11) we have the equation corresponding to the polynomial F_{3} given in (^{\underline{g2bx}} 4.22) with a = 1, b = 0. After substituting the B–eigenvalues we obtain the following Weitzenböck formulas on 1–forms:

0 =  4 T_{−ε_{1}}^{∗} T_{−ε_{1}} + 2 T_{0}^{∗} T_{0}  − \frac{2}{3} T_{+ε_{1}}^{∗} T_{+ε_{1}}
0 = − \frac{16}{3} T_{−ε_{1}}^{∗} T_{−ε_{1}} + \frac{8}{3} T_{0}^{∗} T_{0} − \frac{8}{3} T_{+ε_{2}}^{∗} T_{+ε_{2}} + \frac{8}{9} T_{+ε_{1}}^{∗} T_{+ε_{1}}
Δ =  5 T_{−ε_{1}}^{∗} T_{−ε_{1}} + 3 T_{0}^{∗} T_{0} + T_{+ε_{2}}^{∗} T_{+ε_{2}} + \frac{1}{3} T_{+ε_{1}}^{∗} T_{+ε_{1}}

It follows that Δ ≥ \frac{1}{3} ∇^{∗}∇, i. e. there are no non–parallel harmonic 1–forms, which is of course Bochners theorem in the case of G_{2}–manifolds.

Next we consider the case λ = 2ω_{1}, i. e. the representation V_{λ} = Γ_{2,0} = Λ_{27}^{3}. Here the relevant weights are ε = −ε_{1}, 0, −ε_{3}, ε_{2}, ε_{1} with λ + ε = ω_{1}, 2ω_{1}, ω_{2}, ω_{1} + ω_{2}, 3ω_{1} and with the decomposition

T ⊗ Γ_{2, 0} = Γ_{1, 0} ⊕ Γ_{2, 0} ⊕ Γ_{0, 1} ⊕ Γ_{1, 1} ⊕ Γ_{3, 0}
= T^{∗} ⊕ Λ_{27}^{3}T^{∗} ⊕ Λ_{14}^{2} ⊕ [ 64 ] ⊕ [ 77 ]^{−}

Hence we have two pure curvature Weitzenböck formulas on sections of Λ_{27}^{3}. The first one is the formula for q(R) corresponding to B, while the second corresponds to −\frac{27}{240} p_{3}(B):

q(R) =  \frac{14}{3} T_{−ε_{1}}^{∗} T_{−ε_{1}} + 2 T_{0}^{∗} T_{0} + \frac{8}{3} T_{−ε_{3}}^{∗} T_{−ε_{3}} − \frac{1}{3} T_{+ε_{2}}^{∗} T_{+ε_{2}} − \frac{4}{3} T_{+ε_{1}}^{∗} T_{+ε_{1}}
0 = − \frac{7}{6} T_{−ε_{1}}^{∗} T_{−ε_{1}} + \frac{1}{2} T_{0}^{∗} T_{0} + \frac{5}{6} T_{−ε_{3}}^{∗} T_{−ε_{3}} − \frac{2}{3} T_{+ε_{2}}^{∗} T_{+ε_{2}} + \frac{1}{3} T_{+ε_{1}}^{∗} T_{+ε_{1}}
Δ =  \frac{9}{2} T_{−ε_{1}}^{∗} T_{−ε_{1}} + \frac{7}{2} T_{0}^{∗} T_{0} + \frac{9}{2} T_{−ε_{3}}^{∗} T_{−ε_{3}}

It follows that a form ψ in Λ_{27}^{3} ⊂ Λ^{3}T is harmonic if and only if ∇ψ is a section of Γ_{1,1} ⊕ Γ_{3,0}. Note that the expression for Δ was obtained by adding the Bochner identity, i. e. the second Weitzenböck formula, to the equation for ∇^{∗}∇ + q(R).

Finally we turn to the case of Spin(7)–holonomy. Irreducible Spin(7)–representations are parametrized as Γ_{a, b, c} = aω_{1}+bω_{2}+cω_{2}. Again Γ_{0, 0, 0} = ℂ is the trivial representation and Γ_{0, 0, 1} = T denotes the 8–dimensional holonomy representation. We want to describe the twistor operators for the parallel subbundles of the form bundle. For this we need the following representations, which are also uniquely determined by their dimension:

dim Γ_{1, 0, 0} = 7  dim Γ_{0, 1, 0} = 21  dim Γ_{1, 0, 1} = 48  dim Γ_{1, 1, 0} = 105
dim Γ_{2, 0, 0} = 27  dim Γ_{0, 0, 2} = 35  dim Γ_{2, 0, 1} = 168  dim Γ_{1, 0, 2} = 189 .

In dimension 112 there are two different irreducible representations denoted by [112]^{a} := Γ_{0, 1, 1} and [112]^{b} := Γ_{0, 0, 3}. Using Poincaré duality we can decompose the differential forms

Λ^{2}T^{∗} ≅ Λ_{7}^{2}T^{∗} ⊕ Λ_{21}^{2}T^{∗}  ≅ Λ^{6}T^{∗}
Λ^{3}T^{∗} ≅ T^{∗} ⊕ Λ_{48}^{3}T^{∗}  ≅ Λ^{5}T^{∗}
Λ^{4}T^{∗} ≅ ℂ ⊕ Λ_{7}^{4}T^{∗} ⊕ Λ_{27}^{4}T^{∗} ⊕ Λ_{35}^{4}T^{∗}

(7.37) deco5

into irreducible subspaces, where again the subscripts refer to the dimension.

We start with the representation V_{λ} = Γ_{1, 0, 0} = Λ_{7}^{2} of highest weight λ = ω_{1}. The relevant weights are +ε_{1} and −ε_{4} with B–eigenvalues b_{+ε_{1}} = \frac{1}{2} and b_{−ε_{4}} = −3 leading to the decomposition:

T ⊗ Γ_{1, 0, 0} = Γ_{1, 0, 1} ⊕ Γ_{0, 0, 1} = Λ_{48}^{3} ⊕ T .

Because the bundle defined by Γ_{1, 0, 0} can be considered as the subbundle of the spinor bundle orthogonal to the parallel spinor, the curvature endomorphism q(R) is a multiple of the scalar curvature and hence vanishes. Thus we obtain on sections of Λ_{7}^{2} the only Weitzenböck formula

0 = −\frac{1}{2} T_{+ε_{1}}^{∗} T_{+ε_{1}} + 3 T_{−ε_{4}}^{∗} T_{−ε_{4}}

It follows that Δ ≥ \frac{1}{2} ∇^{∗}∇, i. e. there are no non-parallel harmonic forms in Λ_{7}^{2}.

For the second component of the space of 2–forms Λ_{21}^{2} = Γ_{0, 1, 0} the relevant weights are +ε_{1}, −ε_{2}, e_{3} with B–eigenvalues 1, −5, −\frac{3}{2} in the decomposition:

T ⊗ Γ_{0, 1, 0} = Γ_{0, 1, 1} ⊕ Γ_{0, 0, 1} ⊕ Γ_{1, 0, 1} = [112]^{a} ⊕ T ⊕ Λ_{48}^{3} .

On sections of Λ_{21}^{2} we have the Weitzenböck formula

q(R) = − T_{+ε_{1}}^{∗} T_{+ε_{1}} + 5 T_{−ε_{2}}^{∗} T_{−ε_{2}} + \frac{3}{2} T_{+ε_{3}}^{∗} T_{+ε_{3}} .

Hence a form ψ in Λ_{21}^{2} is harmonic if and only if ∇ψ is a section of Γ_{0, 1, 1}.

The last parallel subbundle of the form bundle with only one pure curvature Weitzenböck formula is V_{λ} = Γ_{2, 0, 0} = Λ_{27}^{4}. The relevant weights are +ε_{1}, −ε_{4} with B–eigenvalues 1, −\frac{7}{2} and the decomposition

T ⊗ Γ_{2, 0, 0} = Γ_{2, 0, 1} ⊕ Γ_{1, 0, 1} = [168] ⊕ Λ_{48}^{3} ,

with [168] := Γ_{2,0,1}. On sections of Λ_{27}^{4} we have the Weitzenböck formula:

q(R) = − T_{+ε_{1}}^{∗} T_{+ε_{1}} + \frac{7}{2} T_{−ε_{2}}^{∗} T_{−ε_{2}} .

Hence a form ψ in Λ_{27}^{4} is harmonic if and only if ∇ψ is a section of Γ_{2, 0, 1}. For the remaining subbundles we have at least two pure curvature Weitzenböck formulas one of which is a Bochner identity, i. e. with a zero curvature term.

We start with the representation T = Γ_{0, 0, 1} describing 1– and 6–forms on M. The relevant weights are +ε_{1}, −ε_{1}, +ε_{2}, +ε_{4} conformal weights or B–eigenvalues \frac{3}{4}, −\frac{21}{4}, −\frac{1}{4} and −\frac{9}{4} respectively. The corresponding decomposition of the representation T ⊗T reads:

T ⊗ Γ_{0, 0, 1} = Γ_{0, 0, 2} ⊕ Γ_{0, 0, 0} ⊕ Γ_{0, 1, 0} ⊕ Γ_{1, 0, 0} = Λ_{35}^{4}T^{∗} ⊕ ℂ ⊕ Λ_{21}^{2}T^{∗} ⊕ Λ_{7}^{2}T^{∗} .

In this case we have the Weitzenböck formula (^{\underline{weizen3}} 3.11) and the Bochner identity (^{\underline{s7b}} 4.25) for (a, b, c) = (0, 0, 1). Since q(R) = Ric on the tangent bundle we obtain two zero curvature Weitzenböck formulas on sections of T:

\begin{aligned}0 & = & − & \frac{3}{4} T_{+ε_{1}}^{∗} T_{+ε_{1}} & + & \frac{21}{4} T_{−ε_{1}}^{∗} T_{−ε_{1}} & + & \frac{1}{4} T_{+ε_{2}}^{∗} T_{+ε_{2}} & + & \frac{9}{4} T_{+ε_{4}}^{∗} T_{+ε_{4}} \\ 0 & = & + & 15 T_{+ε_{1}}^{∗} T_{+ε_{1}} & − & 105 T_{−ε_{1}}^{∗} T_{−ε_{1}} & − & 45 T_{+ε_{2}}^{∗} T_{+ε_{2}} & + & 75 T_{+ε_{4}}^{∗} T_{+ε_{4}} \end{aligned}

Evidently the first equation tells us Δ ≥ \frac{1}{4}∇^{∗}∇ so that every harmonic 1–form is necessarily parallel. Of course this is Bochner’s theorem reproved in the case of Spin(7)– manifolds. Another direct consequence is the well–known fact that any Killing vector field on a compact Spin(7)–manifold has to be parallel. Indeed Killing vector fields are vector fields X ∈ Γ(T M), for which ∇X^{♯} ∈ Γ(T^{∗}M ⊗ T^{∗}M) is skew and thus a 2–form. On Spin(7)–manifolds this implies T_{+ε_{1}}X = 0 = T_{−ε_{1}}X and so all twistor operators vanish on X.

Next we consider the representation Λ_{35}^{4}T^{∗} = Γ_{0, 0, 2} with relevant weights ε_{1}, −ε_{1}, ε_{2}, ε_{4}, conformal weights or B–eigenvalues \frac{3}{2}, −6, 0, −\frac{5}{2} and decomposition

T ⊗ Γ_{0, 0, 2} = Γ_{0, 0, 3} ⊕ Γ_{0, 0, 1} ⊕ Γ_{0, 1, 1} ⊕ Γ_{1, 0, 1} = [112]^{b} ⊕ T^{∗} ⊕ [112]^{a} ⊕ Λ_{48}^{3}T^{∗} .

Thus there are two pure curvature Weitzenböck formulas on sections of Λ_{35}^{4}. For the second we take \frac{1}{96}F_{Bochner} and obtain

q(R) = − \frac{3}{2} T_{+ε_{1}}^{∗} T_{+ε_{1}} + 6 T_{−ε_{1}}^{∗} T_{−ε_{1}}  + \frac{5}{2} T_{+ε_{4}}^{∗} T_{+ε_{4}}
0 =  \frac{1}{2} T_{+ε_{1}}^{∗} T_{+ε_{1}} − 2 T_{−ε_{1}}^{∗} T_{−ε_{1}} − T_{+ε_{2}}^{∗} T_{+ε_{2}} + \frac{3}{2} T_{+ε_{4}}^{∗} T_{+ε_{4}}
Δ =  5 T_{−ε_{1}}^{∗} T_{−ε_{1}}  + 5 T_{+ε_{4}}^{∗} T_{+ε_{4}}

Note that in order to obtain the optimal expression for the operator Δ it was not sufficient to take its definition Δ = ∇^{∗}∇ + q(R), we still had to add a multiple of the Bochner identity. Last but not least we consider the representation Λ_{48}^{3}T^{∗} = Γ_{1, 0, 1}. According to the table at the end of Section ^{\underline{tables}} 2 the relevant weights are +ε_{1}, −ε_{1}, +ε_{2}, −ε_{3}, ε_{4}, −ε_{4} with conformal weights or B–eigenvalues \frac{5}{4}, −\frac{23}{4}, \frac{1}{4}, −\frac{15}{4}, −\frac{7}{4} and −\frac{11}{4} respectively leading to

T ⊗ Γ_{1, 0, 1} = Γ_{1, 0, 2} ⊕ Γ_{1, 0, 0} ⊕ Γ_{1, 1, 0} ⊕ Γ_{0, 1, 0} ⊕ Γ_{2, 0, 0} ⊕ Γ_{0, 0, 2}
= [ 189 ] ⊕ Λ_{7}^{2}T^{∗} ⊕ [ 105 ] ⊕ Λ_{21}^{2}T^{∗} ⊕ Λ_{27}^{4}T^{∗} ⊕ Λ_{35}^{4}T^{∗}

On sections of the associated bundle Λ_{48}^{3}T^{∗}M ⊂ Λ^{3}T^{∗}M one has three curvature Weitzenböck formulas, the formula corresponding to B and the Bochner identity \frac{1}{84}F_{Bochner}:

q(R) = − \frac{5}{4} T_{ε_{1}}^{∗}T_{ε_{1}} + \frac{23}{4} T_{−ε_{1}}^{∗}T_{−ε_{1}} − \frac{1}{4} T_{ε_{2}}^{∗}T_{ε_{2}} + \frac{15}{4} T_{−ε_{3}}^{∗}T_{−ε_{3}} + \frac{7}{4} T_{ε_{4}}^{∗}T_{ε_{4}} + \frac{11}{4} T_{−ε_{4}}^{∗}T_{−ε_{4}}
0 = \frac{1}{4} T_{ε_{1}}^{∗}T_{ε_{1}} − \frac{45}{28} T_{−ε_{1}}^{∗}T_{−ε_{1}} − \frac{3}{4} T_{ε_{2}}^{∗}T_{ε_{2}} + \frac{27}{28} T_{−ε_{3}}^{∗}T_{−ε_{3}} + \frac{5}{4} T_{ε_{4}}^{∗}T_{ε_{4}} − \frac{9}{28} T_{−ε_{4}}^{∗}T_{−ε_{4}}
Δ =  \frac{36}{7} T_{−ε_{1}}^{∗}T_{−ε_{1}}  + \frac{40}{7} T_{−ε_{3}}^{∗}T_{−ε_{3}} + 4 T_{ε_{4}}^{∗}T_{ε_{4}} + \frac{24}{7} T_{−ε_{4}}^{∗}T_{−ε_{4}}

Consequently a 3–form ψ ∈ Γ(Λ_{48}^{3}T^{∗}M) of type is harmonic if and only if its covariant derivative ∇ψ takes values in ([189] ⊕ [105])M ⊂ T^{∗}M ⊗ Λ_{48}^{3}T^{∗}M everywhere.

8 Bochner Identities in G_{2}– and Spin(7)–Holonomy

kis

Aim of this section is to provide a proof of the Bochner identities for the holonomies 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7} and thus to complete the description of the space of Weitzenböck formulas in these cases. Interestingly it seems necessary to introduce a fairly more abstract point of view of Weitzenböck formulas in order to get to this point.

8.1 Universal Weitzenböck Classes and the Kostant Theorem

The essential additional twist we will employ in this section is that we will basis the study of Weitzenböck formulas on the study of the action of central elements of the universal enveloping algebra. As a byproduct we get a explicit formula for a central element of order 4 in the universal enveloping algebra of 𝔰𝔭𝔦𝔫_{7} and a central element of order 4 in the universal enveloping algebra U 𝔤_{2}.

Recall that the universal enveloping algebra U 𝔤 of a Lie algebra 𝔤 is the associative algebra with 1 generated freely by the vector space 𝔤 subject only to the commutator relation X Y −Y X = [X, Y]. Thus U 𝔤 is spanned by monomials of the form X_{1} ·. . .·X_{r} in elements X_{1}, …, X_{r} of 𝔤 and the filtration U ^{≤•}𝔤 by the degree r of these monomials makes U 𝔤 a filtered algebra. Even more important for our purposes is the Hopf algebra structure of U 𝔤 with the cocommutative comultiplication

Δ : U 𝔤 ⟶ U 𝔤 ⊗ U 𝔤,  Q ⟼ ∑ Δ_{L}Q ⊗ Δ_{R}Q  (8.38) comult

defined as the unique algebra homomorphism sending X ∈ 𝔤 to ΔX := X ⊗ 1 + 1 ⊗ X in U 𝔤 ⊗ U 𝔤. Defining Δ in this way clearly implies for all d, r ≥ 0:

Δ( U ^{≤d+r}𝔤 ) ⊂ U ^{<d}𝔤 ⊗ U 𝔤 + U 𝔤 ⊗ U ^{≤r}𝔤 .  (8.39) dft

An integral part of the structure of the universal enveloping algebra U 𝔤 is its induced representation on representations V of 𝔤. For a finite–dimensional representation V it is easy to characterize the image of this representation thought of as a linear map U 𝔤 ⟶ End V:

bicom Lemma 8.1 (Bicommutant Theorem)

Consider a finite dimensional representation V of a semisimple Lie algebra 𝔤 over ℂ and the induced representation U 𝔤 ⟶ End V of U 𝔤. The image of this map is precisely the commutant of the algebra End_{𝔤}V of 𝔤–invariant endomorphisms of V:

im ( U 𝔤 ⟶ End V ) = { A ∈ End V | [ A, F ] = 0 for all F ∈ End_{𝔤}V } .

In particular the map Zent U 𝔤 ⟶ Zent End_{𝔤}V is surjective for V finite dimensional.

The Bicommutant Theorem is actually a special motivating example of von Neumann’s Bicommutant Theorem, observe that every ∗–subalgebra of End V is necessarily von Neumann for a finite dimensional vector space V. The image of U 𝔤 in End V is the subalgebra generated by the ∗–closed subspace 𝔤 of End V and thus von Neumann with commutant End_{𝔤}V. We will give a more geometric proof of this theorem based on the Peter–Weyl Theorem in Appendix ^{\underline{biapp}} A.

Restricting to irreducible representations V_{λ} we conclude that the representation map U 𝔤 ⟶ End V_{λ} is surjective and hence the same is true for the algebra homomorphism

Φ : Hom_{𝔤}( T ⊗ T, U 𝔤) ⟶ Hom_{𝔤}(T ⊗ T, End V_{λ}) ,

where Hom_{𝔤}(T ⊗ T, End V_{λ}) is one of the interpretation of the space 𝔚(V_{λ}) of Weitzenböck formulas on V_{λ}M. With the universal enveloping algebra U 𝔤 being a module over its center the space Hom_{𝔤}(T ⊗ T, U 𝔤) is naturally a module for Zent U 𝔤, too, and the filtration Hom_{𝔤}(T ⊗T, U ^{≤•}𝔤) turns it into a filtered module for the filtration of the center Zent^{≤•}U 𝔤 := Zent U 𝔤 ∩ U ^{≤•}𝔤.

We note that the endomorphisms B^{k} are in the image under Φ of the filtration subspace Hom_{𝔤}( T ⊗T, U ^{≤k}𝔤). Indeed let the invariant map p_{k} : T ⊗T ⟶ U ^{≤k}𝔤 be defined by p_{k}(a⊗b) = ∑ pr(a∧e_{i_{1}})·. . .·pr(e_{i_{k−1}}∧b), where pr is the projection T ⊗T ⟶ Λ^{2} ⟶ 𝔤, then obviously Φ(p_{k}) = B^{k}. With the vector space identification U 𝔤 ≅ Sym 𝔤, we may consider p_{k}(a ⊗ b) as a polynomial on 𝔤 which reads as p_{k}(a ⊗ b)[X] = ⟨ X^{k}a, b ⟩.

Motivated by the surjection Hom_{𝔤}(T ⊗ T, U 𝔤) ⟶ 𝔚(V_{λ}) we will call Hom_{𝔤}(T ⊗ T, U 𝔤) the space of universal Weitzenböck formulas.

uwe Definition 8.2 (Universal Weitzenböck Classes)

The space of universal Weitzenböck formulas 𝔚^{≤•} := Hom_{𝔤}(T ⊗ T, U ^{≤•}𝔤) is a filtered module over the center Zent^{≤•}U 𝔤 of the universal enveloping algebra U 𝔤. It splits into the direct sum of filtered Zent U 𝔤–submodules called universal Weitzenböck classes:

𝔚^{≤•} = ⨁_{α} 𝔚_{W_{α}}^{≤•} := ⨁_{α} Hom_{𝔤}( W_{α}, U ^{≤•}𝔤) .

It is clear from the definition that with F ∈ 𝔚^{≤k} also F |_{W_{α}} ∈ 𝔚_{W_{α}}^{≤k}. Moreover we have seen above that the space of polynomials in B of degree at most k is in the image under the representation map Φ of 𝔚^{≤k}. Perhaps even more important for our considerations below is that the space of universal Weitzenböck formulas is free as a module over Zent U 𝔤:

kostant Theorem 8.3 (Kostant’s Theorem)

For every finite dimensional representation V the space Hom_{𝔤}( V, U 𝔤) is a free Zent U 𝔤– module, whose rank over Zent U 𝔤 agrees with the multiplicity of the zero weight in V:

Hom _{𝔤}^{≤•}( V, U 𝔤) ≅ Zent U 𝔤 ⊗ Hom _{𝔱}^{•}( V, ℂ)

In particular the module Hom _{𝔤}^{≤•}( 𝔤, U 𝔤) ≅ Zent U 𝔤 ⊗ Prim^{•+1} 𝔤 is generated freely as a filtered Zent U 𝔤–module by the primitive elements of Zent U 𝔤 with degrees shifted by −1.

As an example we consider holonomy 𝔤_{2} and the spaces which are mapped under Φ onto the K–eigenspaces. We refer to the appendix for the other holonomies. Then

Hom_{𝔤_{2}}( ℂ , U 𝔤_{2} ) ≅ Zent U 𝔤_{2}
Hom_{𝔤_{2}}( Sym_{0}^{2}T , U 𝔤_{2} ) ≅ Zent U 𝔤_{2} ⟨ F_{2}, F_{4}, F_{6} ⟩
Hom_{𝔤_{2}}( 𝔤_{2} , U 𝔤_{2} ) ≅ Zent U 𝔤_{2} ⟨ F_{1}, F_{5} ⟩
Hom_{𝔤_{2}}( 𝔤_{2}^{⊥} , U 𝔤_{2} ) ≅ Zent U 𝔤_{2} ⟨G_{3}⟩

where F_{1}, F_{2}, G_{3}, F_{4}, F_{5} and F_{6} are free generators of degree 1, 2, …, 6. The numbers of generators, i. e. the dimension of the corresponding zero weight space, can be read off Table (^{\underline{wt}} 2.2). The degree of the generators, also called generalized exponents, can be obtained by decomposing Sym^{k}𝔤 into irreducible components (e. g. using the program LiE) and by determining the multiplicity of W_{α} in this decomposition for sufficiently many k. As mentioned in the theorem: the degrees of the generators F_{1}, F_{5} are the degrees of the generators C_{2}, C_{6} of Zent U 𝔤_{2} shifted by one.

It follows from Kostant’s theorem that a basis in the eigenspace 𝔚_{W_{α}}(V_{λ}) of the classifying endomorphism K may be obtained as the image under the surjective representation map 𝔚_{W_{α}} ⟶ 𝔚_{W_{α}}(V_{λ}) of certain free generators for the universal Weitzenböck classes 𝔚_{α}. Indeed the module multiplication in 𝔚 with central elements Q ∈ Zent U 𝔤 turns into multiplication with the value of the central character

χ_{λ}( Q ) := \frac{1}{dim V_{λ}} tr_{V_{λ}} Q

associated to the highest weight λ in 𝔚(V_{λ}). The value of the central character on Q ∈ Zent U 𝔤 is a polynomial in the highest weight λ invariant under the Weyl group of 𝔤. Note that at least in principle we know the central characters of the higher Casimirs Cas^{[k]} ∈ Zent^{≤k}U 𝔤 defined in equation (^{\underline{highercasimir}} 3.13) as the traces of the k–th powers of the conformal weight operator, since Equation (^{\underline{kth}} 3.14) implies:

χ_{λ}( Cas^{[k]} ) = ∑_{ε} b_{ε}^{k} \frac{dim V_{λ+ε}}{dim V_{λ}}  (8.40) ev1

In order to proceed we use the diagonal Δ of the Hopf algebra U 𝔤 together with the representation of U 𝔤 on the euclidian vector space T to define an algebra homomorphism

Δ : Zent U 𝔤 ⟶^{Δ} ( U 𝔤 ⊗ U 𝔤)^{𝔤} ⟶ Hom_{𝔤}( T ⊗ T, U 𝔤) = 𝔚

by (ΔQ)_{a⊗b} = ∑⟨ a, (Δ_{L}Q)b ⟩Δ_{R}Q for all Q ∈ Zent U 𝔤. A particularly nice property of Δ is that the image of ΔQ ∈ 𝔚 under the representation map Φ : 𝔚 ⟶ 𝔚(V_{λ}) can be written in the following way:

ΔQ = ∑_{ε⊂λ} χ_{λ+ε}( Q) pr_{ε} ∈ 𝔚(V_{λ})  (8.41) ev

In fact working our way through the identification Hom_{𝔤}(T ⊗T, End V_{λ}) = End_{𝔤}(T ⊗V_{λ}) we find the usual tensor product action of Q ∈ Zent U 𝔤 on T ⊗ V_{λ}:

Φ( ΔQ)( b ⊗ v) = ∑_{μ} t_{μ} ⊗ ( ΔQ)_{t_{μ}⊗b} v = ∑_{μ} t_{μ} ⊗ ⟨ t_{μ}, ( Δ_{L}Q) b ⟩ Δ_{R}Q v
= ∑ ( Δ_{L}Q) b ⊗ ( Δ_{R}Q) v = ΔQ( b ⊗ v) = Q( b ⊗ v)

The last Δ is the restriction of the comultiplication to Zent U 𝔤 ⊂ U 𝔤, which is precisely the action of Q ∈ Zent U 𝔤 on T ⊗ V. This calculation also shows that we have the following commutative diagram

Zent U 𝔤 ⟶^{Δ} Hom_{𝔤}( T ⊗ T, U 𝔤) ≅ ⨁_{α} Hom_{𝔤}( W_{α}, U 𝔤)
Δ↓  ↓^{Φ}  ↓
End_{𝔤}(T ⊗ V_{λ}) ⟶^{≅} Hom_{𝔤}( T ⊗ T, End V_{λ}) ≅ ⨁_{α} Hom_{𝔤}( W_{α}, End V_{λ})

where the right vertical arrow is the restriction of the representation map Φ onto 𝔚_{W_{α}}. Recall that the left square consists of algebra and the right square of Zent U 𝔤–module homomorphisms and that moreover all vertical arrows are surjective maps.

Example 8.4 (Conformal Weight Operator)

cwo

A special case of this construction is the relation between the conformal weight operator B and the Casimir. The image of the Casimir Cas^{Λ^{2}} ∈ Zent U 𝔤 becomes

Δ(Cas^{Λ^{2}}) = −2 B + ( Cas_{T}^{Λ^{2}} + Cas_{V_{λ}}^{Λ^{2}} )

because Δ(X^{2}) = X^{2} ⊗ 1 + 2X ⊗ X + 1 ⊗ X^{2} for every X ∈ 𝔤 and Fegan’s Lemma ^{\underline{fegan}} 3.3. Moreover since Δ is an algebra homomorphism we conclude

p(B) = Δ p( − \frac{1}{2} Cas^{Λ^{2}} + \frac{1}{2}(Cas_{T}^{Λ^{2}} + Cas_{V_{λ}}^{Λ^{2}}) )

for every polynomial p(B) in the conformal weight operator B. In particular the space of polynomials in B is in the image under Δ of the subalgebra generated by Cas^{Λ^{2}}.

The crucial additional information we get from introducing the universal Weitzenböck classes is the filtration degree of the generators of the Zent U 𝔤–modules 𝔚_{W_{α}}. In order to prove the Bochner identities for holonomy 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7} we still need the following

dfilt Lemma 8.5 (Filtration Property of Δ)

Consider the Weitzenböck class 𝔚_{W_{α}} associated to an irreducible subspace W_{α} ⊂ T ⊗ T and a degree d ≥ 1 such that there is no non–trivial, 𝔤–equivariant map from W_{α} to U ^{<d}𝔤, i.e. 𝔚_{W_{α}}^{≤d−1} = {0}. Then the composition of Δ with the restriction res_{W_{α}} to 𝔚_{α} is a filtered map:

res_{W_{α}} ∘ Δ : Zent^{≤d+•}U 𝔤 ⟶ 𝔚_{W_{α}}^{≤•}, Q ⟼ ΔQ|_{W_{α}}

In particular the restriction ΔQ|_{W_{α}} = 0 to W_{α} vanishes for every Q ∈ Zent^{≤2d−1}U 𝔤.

Proof: By the filtration property (^{\underline{dft}} 8.39) of the comultiplication we can write the diagonal ΔQ of an element Q ∈ Zent^{≤d+r}U 𝔤 in a not necessarily unique way as a sum of two terms ΔQ = ΔQ^{<d} + ΔQ^{≤r} satisfying ΔQ^{<d} ∈ (U ^{<d}𝔤 ⊗ U 𝔤)^{𝔤} and ΔQ^{≤r} ∈ (U 𝔤 ⊗ U ^{≤r}𝔤)^{𝔤} respectively. Both these summands give rise to 𝔤–equivariant, linear maps T ⊗T ⟶ U 𝔤 through the pairing of T ⊗T with the left U 𝔤–factor, explicitly

( ΔQ^{<d})_{a⊗b} := ∑⟨ a, ΔQ_{L}^{<d} b ⟩ ΔQ_{R}^{<d}

with essentially the same formula for ΔQ^{≤r}. By construction the map T ⊗ T ⟶ U 𝔤 associated to ΔQ^{≤r} maps into U ^{≤r}𝔤, while the map associated to ΔQ^{<d} vanishes upon restriction to W_{α} ⊂ T ⊗ T, because by assumption there is no non–trivial, 𝔤–invariant, pairing of W_{α} with the left image of ΔQ^{<d} defined by:

span{ ΔQ_{L}^{<d} | ΔQ^{<d} = ∑ ΔQ_{L}^{<d} ⊗ ΔQ_{R}^{<d} } ⊂ U ^{<d}𝔤

For the second statement we note ΔQ|_{W_{α}} ∈ 𝔚_{α}^{<d} = {0} for every Q ∈ Zent^{≤2d−1}U 𝔤. □

8.2 Proof of the Bochner Identities in Holonomy 𝔤_{2} and 𝔰𝔭𝔦𝔫_{7}

Let us now discuss the details of the proof of the additional Bochner identity in G_{2}– holonomy. Applying the Gram–Schmidt orthogonalization process ^{\underline{orp}} 4.3 to the powers 1, B, B^{2} and B^{3} of the conformal weight operator we obtained in Equations (^{\underline{g2-1}} 4.20) and (^{\underline{g2-2}} 4.21) a sequence p_{0}(B), p_{1}(B), p_{2}(B), p_{3}(B) of τ–eigenvectors. In order to proceed with the recursion procedure it remains to be shown that p_{3}(B) is a K–eigenvector.

We know that p_{3}(B) is a −1 eigenvector of τ, orthogonal to B and expressible as polynomial in B of degree 3. Thus p_{3}(B) is an element in the image of 𝔚^{≤3} in 𝔚(V_{λ}) and can be written as a sum p_{3}(B) = p_{3}(B)_{𝔤_{2}} + p_{3}(B)_{𝔤_{2}^{⊥}} of two vectors p_{3}(B)_{𝔤_{2}} and p_{3}(B)_{𝔤_{2}^{⊥}} in the image of 𝔚_{𝔤_{2}}^{≤3} and 𝔚_{𝔤_{2}^{⊥}}^{≤3} in 𝔚(V_{λ}) respectively. However the image of 𝔚_{𝔤_{2}}^{≤3} in 𝔚(V_{λ}) is spanned by B, because the filtered Zent U 𝔤_{2}–module 𝔚_{𝔤_{2}} is generated by two elements in degrees 1 and 5 and the representation 𝔚_{𝔤_{2}} ⟶ 𝔚_{𝔤_{2}}(V_{λ}) turns module multiplication into multiplication by the central character χ_{λ}. Consequently the vector p_{3}(B) is orthogonal to the image of 𝔚_{𝔤_{2}}^{≤3} in 𝔚(V_{λ}) and lies in the eigenspace 𝔚_{𝔤_{2}^{⊥}}(V_{λ}) of the classifying endomorphism K:

Theorem 8.6 (Bochner Identity in G_{2}–Holonomy)

bochig2

The following cubic polynomial in the conformal weight operator defines an eigenvector for the classifying endomorphisms K of eigenvalue −2:

p_{3}(B) := B^{3} + \frac{13}{3} B^{2} + ( \frac{1}{2}Cas_{V_{λ}}^{Λ^{2}} + 4 ) B + \frac{2}{3} Cas_{V_{λ}}^{Λ^{2}}

Inserting the eigenvalues or conformal weights b_{ε} of B we arrive at equation ^{\underline{g2bx}} 4.22.

In the last part of this section we will prove the Bochner identity for holonomy 𝔰𝔭𝔦𝔫_{7}. As in the 𝔤_{2}–case we apply the Gram–Schmidt orthogonalization process ^{\underline{orp}} 4.3 to the powers 1, B, B^{2} and B^{3} and obtain in Equations (^{\underline{s7-1}} 4.23) and (^{\underline{s7-2}} 4.24) a sequence p_{0}(B), p_{1}(B), p_{2}(B), p_{3}(B) of τ–eigenvectors. Again p_{3}(B) is a −1 eigenvector of τ, orthogonal to B and expressible as polynomial in B of degree 3. This time p_{3}(B) decomposes as the sum p_{3}(B) = p_{3}(B)_{𝔰𝔭𝔦𝔫_{7}} + p_{3}(B)_{𝔰𝔭𝔦𝔫}⊥_{7} of two vectors p_{3}(B)_{𝔰𝔭𝔦𝔫_{7}} and p_{3}(B)_{𝔰𝔭𝔦𝔫}⊥_{7} in the image of 𝔚_{𝔰𝔭𝔦𝔫_{7}}^{≤3} and 𝔚_{𝔰𝔭𝔦𝔫_{7}^{⊥}}^{≤3} in 𝔚(V_{λ}) respectively. We would like to extract the Bochner identity as p_{3}(B)_{𝔰𝔭𝔦𝔫}⊥_{7}.

However at this point the argument in the Spin(7)–case becomes more complicated, because the Zent U 𝔰𝔭𝔦𝔫_{7}–module 𝔚_{𝔰𝔭𝔦𝔫_{7}} has generators in degree 1, 3 and 5 so that the image of 𝔚_{𝔰𝔭𝔦𝔫_{7}}^{≤3} in 𝔚_{𝔰𝔭𝔦𝔫_{7}}(V_{λ}) has dimension two. Even with p_{3}(B) orthogonal to B we may thus not conclude that the component p_{3}(B)_{𝔰𝔭𝔦𝔫_{7}} = 0 vanishes. The idea to cope with this complication is to construct an element Q_{λ} ∈ Zent U 𝔰𝔭𝔦𝔫_{7} depending polynomially on the highest weight λ such that ΔQ_{λ} ∈ 𝔚_{𝔰𝔭𝔦𝔫_{7}}(V_{λ}) is orthogonal to B. The Bochner identity is then the projection of p_{3}(B) onto the orthogonal complement of ΔQ_{λ}.

In the 4-dimensional space Zent^{≤4}U 𝔰𝔭𝔦𝔫_{7} we look for an element Q_{λ} as a linear combination of the base vectors 1, Cas, Cas^{2} and Cas^{[4]} with unknown coefficients. We know ΔCas and ΔCas^{2} from Example ^{\underline{cwo}} 8.4 and ΔCas^{[4]} from Equations (^{\underline{ev1}} 8.40) and (^{\underline{ev}} 8.41) so that

⟨ ΔQ_{λ}, 1 ⟩ = 0  ⟨ ΔQ_{λ}, B ⟩ = 0  ⟨ ΔQ_{λ}, B^{2} ⟩ = 0

turn into three linear independent equations for the four unknown coefficients. Using a computer algebra system to do the necessary calculations we find the convenient solution

Q_{λ} = 2 Cas_{V_{λ}}^{Λ^{2}} Cas^{[4]} − 160 Cas_{V_{λ}}^{Λ^{2}} (Cas^{Λ^{2}})^{2}
+ ( 320 (Cas_{V_{λ}}^{Λ^{2}})^{2} − 1184 Cas_{V_{λ}}^{Λ^{2}} − 4 Cas_{V_{λ}}^{[4]} ) Cas^{Λ^{2}}

+ ( − 160 (Cas_{V_{λ}}^{Λ^{2}})^{3} + 2 Cas_{V_{λ}}^{Λ^{2}} Cas_{V_{λ}}^{[4]} + 1712 (Cas_{V_{λ}}^{Λ^{2}})^{2} − 9408 Cas_{V_{λ}}^{Λ^{2}} − 21 Cas_{V_{λ}}^{[4]} )

in Zent^{≤4}U 𝔰𝔭𝔦𝔫_{7}, where we denote the eigenvalue of the central element Cas^{[4]} ∈ U 𝔰𝔭𝔦𝔫_{7} on the irreducible representation V_{λ} by Cas_{V_{λ}}^{[4]} in analogy to the eigenvalues of Cas^{Λ^{2}}.

By construction ΔQ_{λ} is orthogonal to 1, B and B^{2} and we conclude that ΔQ_{λ} is indeed an eigenvector for the classifying endomorphism K. In fact the component ΔQ_{λ}|_{𝔰𝔭𝔦𝔫}⊥_{7} = 0 vanishes according to Lemma ^{\underline{dfilt}} 8.5, because Q_{λ} has degree 4 and there is no non–trivial equivariant map 𝔰𝔭𝔦𝔫_{7}^{⊥} ⟶ U ^{<3}𝔰𝔭𝔦𝔫_{7}. Similarly the components ΔQ_{λ}|_{Sym_{∘}^{2}T} = 0 and ΔQ_{λ}|_{ℂ} = 0 are trivial, since the image of 𝔚_{Sym_{∘}^{2}T}^{≤2} in 𝔚(V_{λ}) has dimension 1 spanned by p_{2}(B) while 𝔚_{ℂ}(V_{λ}) is spanned by p_{0}(B). With ΔQ_{λ} = ΔQ_{λ}|_{𝔰𝔭𝔦𝔫_{7}} being an eigenvector of K orthogonal to p_{1}(B) = B the problematic component p_{3}(B)_{𝔰𝔭𝔦𝔫_{7}} of p_{3}(B) must be a multiple of ΔQ_{λ}. In consequence the complementary component p_{3}(B)_{𝔰𝔭𝔦𝔫}⊥_{7} of p_{3}(B) is the projection of p_{3}(B) onto the orthogonal complement of ΔQ_{λ} and may serve as the 𝔰𝔭𝔦𝔫_{7}–Bochner identity:

Theorem 8.7 (Bochner Identity in Spin(7)–Holonomy)

bochis7

The endomorphism F_{Bochner} ∈ 𝔚(V_{λ}) in equation ^{\underline{s7b}} 4.25 is an eigenvector of the classifying endomorphism K for the eigenvalue −\frac{9}{4}.

A Geometric Proof of the Bicommutant Theorem

biapp

Consider a finite–dimensional representation V of a semisimple Lie group G with Lie algebra 𝔤 and the algebra homomorphism U 𝔤 ⟶ End V associated to the infinitesimal representation of 𝔤 on V. Evidently the image of U 𝔤 under this homomorphism is the subalgebra A_{𝔤}V of End V generated by the image of 𝔤. An alternative characterization of this subalgebra can be given using the notion of the centralizer subalgebra or more succinctly the commutant Comm A of a subset A ⊂ End V:

Comm A := { F ∈ End V | [ F, A] for all A ∈ A }

Von Neumann’s famous Bicommutant Theorem states that the commutant of the commutant of a subset A ⊂ End V of the star algebra of bounded operators on a complex Hilbert space V is precisely the von Neumann subalgebra generated by A ∪ A^{∗}. Every subalgebra of End V for a finite–dimensional Hilbert space V is von Neumann and with the image of 𝔤 = 𝔤^{∗} being closed under taking adjoints we can characterize the subalgebra A_{𝔤}V generated by 𝔤 as the commutant of the subalgebra End_{𝔤}V of 𝔤–invariant endomorphisms on V:

double Lemma A.1 (Bicommutant Theorem)

Let V be a finite–dimensional representation of a complex semisimple Lie algebra 𝔤. The image of the representation homomorphism U 𝔤 ⟶ End V is precisely the commutant Comm End_{𝔤}V of the subalgebra End_{𝔤}V ⊂ End V of 𝔤–invariant endomorphisms on V.

Proof: With 𝔤 being a complex semisimple Lie algebra it has a compact real form 𝔤_{ℝ} defined by a real structure X ⟼ \overline{X} on 𝔤. Being a compact real Lie algebra 𝔤_{ℝ} is the Lie algebra of a compact simply connected Lie group G_{ℝ} and hence there exists a 𝔤_{ℝ}–invariant hermitian form (,) on the complex vector space V satisfying X^{∗} = −\overline{X} for all X ∈ 𝔤. Thus the image of 𝔤 in End V is closed under taking adjoints and by von Neumann’s Theorem we conclude that the algebra A_{𝔤}V generated by 𝔤 is the bicommutant Comm^{2}𝔤 = Comm End_{𝔤}V of 𝔤.

Actually alluding to von Neumann’s theorem may seem somewhat strange for this purely representation theoretic lemma, and for this reason we want to sketch a more elementary proof making use of Schur’s Lemma and the Theorem of Peter–Weyl instead. In this way the reader will presumably get a better understanding of the crux of the statement, which interestingly enough was the main observation leading von Neumann to his Bicommutant Theorem in the first place. As V is a representation of a semisimple Lie algebra 𝔤 defined over ℂ we can decompose V completely into irreducible subrepresentations V_{λ} ⊂ V. With respect to the general form of this decomposition

V = ⨁_{λ} Hom_{𝔤}( V_{λ}, V ) ⊗ V_{λ}

the commutant of the image of 𝔤 in End V is precisely the subalgebra

End_{𝔤}V = ⨁_{λ} End Hom_{𝔤}( V_{λ}, V ) ⊗ id_{V_{λ}}

by Schur’s Lemma, which is a direct sum of matrix algebras one for each non–trivial isotypic component. The crucial observation underlying both the theorem above and von Neumann’s theorem is the fact that the commutant of such a subalgebra is given by:

Comm End_{𝔤}V = ⨁_{λ} id_{Hom_{𝔤}(V_{λ},V)} ⊗ End V_{λ}

Skipping the details of this argument using only elementary linear algebra we conclude that without loss of generality all isotypic components of V can be assumed irreducible. In other words we may assume that V = ⨁ V_{λ} is the direct sum of a finite set of irreducible, pairwise non–isomorphic representations V_{λ} and we want to show that the resulting representation homomorphism is surjective:

U 𝔤 ⟶ ⨁ End V_{λ} ⊂ End ( ⨁ V_{λ} ) .

In order to proceed we reformulate this statement thinking of U 𝔤 as the algebra of left–invariant scalar differential operators on the complex–valued functions C^{∞}( G_{ℝ}, ℂ) on G_{ℝ}, in particular there exists a linear map U 𝔤 ⟶ C^{∞}( G_{ℝ}, ℂ)^{∗}, Q ⟼ ev_{e} ∘ Q by composing the action of the differential operator Q ∈ U 𝔤 with the evaluation at the identity. On the other hand the Theorem of Peter–Weyl identifies the direct sum ⨁ End V_{λ} with a subspace of C^{∞}( G_{ℝ}, ℂ) as all V_{λ} are irreducible and non–isomorphic. Working out the details of this identification we see that the resulting linear map reads:

U 𝔤 ⟶ ( ⨁ End V_{λ} )^{∗} , Q ⟼ ( η ⊗ e ⟼ η( Qe) )

Now all End V_{λ} ≅ ( End V_{λ})^{∗} are real representations with invariant symmetric bilinear form given by the trace and η(Qe) = tr( Q ∘ (η ⊗ e) ) implies that the linear map constructed above is just another interpretation of the representation homomorphism. Moreover it is well–known that all matrix coefficients End V_{λ} ⊂ C^{∞}( G_{ℝ}, ℂ) of an irreducible representation are actually analytic functions with respect to the natural analytic structure on G_{ℝ}. Consequently all functions in the finite direct sum ⨁ End V_{λ} are analytic as well and can thus be separated by their partial derivatives at the identity. In other words the second interpretation of the representation homomorphism is surjective simply because the matrix coefficients of an irreducible representation are analytic. □

zentrum Corollary A.2

The representation homomorphism U 𝔤 ⟶ End V for a finite–dimensional representation V of a complex semisimple Lie algebra 𝔤 restricts to a surjective homomorphism:

Zent U 𝔤 ⟶ Zent End_{𝔤}V

Without giving a formal proof of this corollary we remark that the very definition of the commutant reads Zent End_{𝔤}V = End_{𝔤}V ∩ A_{𝔤}V while A_{𝔤}V is the image of the universal enveloping algebra U 𝔤 under the representation homomorphism by Lemma ^{\underline{double}} A.1. Consequently every element in Zent End_{𝔤}V is the image of some element of U 𝔤. With the representation homomorphism being 𝔤–equivariant and 𝔤 semisimple we can actually choose a 𝔤–invariant preimage Q ∈ U 𝔤 for a 𝔤–invariant endomorphism or equivalently a preimage in the center Zent U 𝔤. Thinking a little bit more about the proof of Lemma \underline{double} A.1 we can formulate a slightly different useful result about the structure of Zent End_{𝔤}V:

structure Corollary A.3

The center Zent End_{𝔤}V of the algebra End_{𝔤}V of 𝔤–invariant endomorphisms on a finite– dimensional representation V = ⨁ Hom_{𝔤}( V_{λ}, V ) ⊗ V_{λ} of a complex semisimple Lie algebra 𝔤 is spanned by the 𝔤–invariant projections onto the isotypical components:

Zent End_{𝔤}V = ⨁_{λ} ℂ id_{Hom_{𝔤}(V_{λ},V)} ⊗ id_{V_{λ}}

For applications in the Weitzenböck machine the representation V will usually be a tensor product T ⊗ V_{λ} of an irreducible representation V_{λ} of highest weight λ defined over ℂ with the complexified holonomy representation T of the holonomy group G_{ℝ} with Lie algebra 𝔤_{ℝ}. In this case the algebra End_{g}( T ⊗ V_{λ}) is commutative and thus agrees with its center.

B Structure of the Universal Enveloping Algebra

Remark B.1 (Special Modules for Zent U 𝔰𝔬_{2r+1})

ksoodd

The center of the universal enveloping algebra of 𝔰𝔬_{2r+1}, r ≥ 1, is a free polynomial algebra Zent U 𝔰𝔬_{2r+1} = ℂ[P^{[2]}, P^{[4]}, …, P^{[2r]}] in r generators of degree 2, 4, …, 2r. Moreover:

Hom_{𝔰𝔬_{2r+1}}( ℂ , U 𝔰𝔬_{2r+1} ) ≅ Zent U 𝔰𝔬_{2r+1}
Hom_{𝔰𝔬_{2r+1}}( Sym_{0}^{2}T , U 𝔰𝔬_{2r+1} ) ≅ Zent U 𝔰𝔬_{2r+1} ⟨ F_{2}, F_{4}, …, F_{2r} ⟩
Hom_{𝔰𝔬_{2r+1}}( 𝔰𝔬_{2r+1} , U 𝔰𝔬_{2r+1} ) ≅ Zent U 𝔰𝔬_{2r+1} ⟨ F_{1}, F_{3}, …, F_{2r−1} ⟩

Remark B.2 (Special Modules for Zent U 𝔰𝔬_{2r})

ksoeven

The center of the universal enveloping algebra U 𝔰𝔬_{2r} of 𝔰𝔬_{2r}, r ≥ 2, is a free polynomial algebra Zent U 𝔰𝔬_{2r} = ℂ[P^{[2]}, P^{[4]}, …, P^{[2r−2]}, E^{[r]}] in r − 1 generators of degree 2, 4, …, 2r − 2 respectively and one additional generator in degree r. Moreover:

Hom_{𝔰𝔬_{2r}}( ℂ , U 𝔰𝔬_{2r} ) ≅ Zent U 𝔰𝔬_{2r}
Hom_{𝔰𝔬_{2r}}( Sym_{0}^{2}T , U 𝔰𝔬_{2r} ) ≅ Zent U 𝔰𝔬_{2r} ⟨ F_{2}, F_{4}, …, F_{2r−2} ⟩
Hom_{𝔰𝔬_{2r}}( 𝔰𝔬_{2r} , U 𝔰𝔬_{2r} ) ≅ Zent U 𝔰𝔬_{2r} ⟨ F_{1}, F_{3}, …, F_{2r−3}, G_{r−1} ⟩

Remark B.3 (Special Modules for Zent U 𝔤_{2})

kg2

The center of the universal enveloping algebra U 𝔤_{2} of 𝔤_{2} is a free polynomial algebra Zent U 𝔤_{2} = ℂ[Cas^{[2]}, Cas^{[6]}] in two generators of degree 2 and 6. Moreover:

Hom_{𝔤_{2}}( ℂ , U 𝔤_{2} ) ≅ Zent U 𝔤_{2}
Hom_{𝔤_{2}}( Sym_{0}^{2}T , U 𝔤_{2} ) ≅ Zent U 𝔤_{2} ⟨ F_{2}, F_{4}, F_{6} ⟩
Hom_{𝔤_{2}}( 𝔤_{2} , U 𝔤_{2} ) ≅ Zent U 𝔤_{2} ⟨ F_{1}, F_{5} ⟩
Hom_{𝔤_{2}}( 𝔤_{2}^{⊥} , U 𝔤_{2} ) ≅ Zent U 𝔤_{2} ⟨G_{3}⟩

Remark B.4 (Special Modules for Zent U 𝔰𝔭𝔦𝔫_{7})

kspin_7

The center of the universal enveloping algebra U 𝔰𝔭𝔦𝔫_{7} of 𝔰𝔭𝔦𝔫_{7} is a free polynomial algebra Zent U 𝔰𝔭𝔦𝔫_{7} = ℂ[Cas^{[2]}, Cas^{[4]}, Cas^{[6]}] in three generators of degree 2, 4 and 6. Moreover:

Hom_{𝔰𝔭𝔦𝔫_{7}}( ℂ , U 𝔰𝔭𝔦𝔫_{7} ) ≅ Zent U 𝔰𝔭𝔦𝔫_{7}
Hom_{𝔰𝔭𝔦𝔫_{7}}( Sym_{0}^{2}T , U 𝔰𝔭𝔦𝔫_{7} ) ≅ Zent U 𝔰𝔭𝔦𝔫_{7} ⟨ F_{2}, F_{4}, F_{6} ⟩
Hom_{𝔰𝔭𝔦𝔫_{7}}( 𝔰𝔭𝔦𝔫_{7} , U 𝔰𝔭𝔦𝔫_{7} ) ≅ Zent U 𝔰𝔭𝔦𝔫_{7} ⟨ F_{1}, F_{3}, F_{5} ⟩
Hom_{𝔰𝔭𝔦𝔫_{7}}( 𝔰𝔭𝔦𝔫_{7}^{⊥} , U 𝔰𝔭𝔦𝔫_{7} ) ≅ Zent U 𝔰𝔭𝔦𝔫_{7} ⟨G_{3}⟩

Remark B.5 (Higher Casimirs for G_{2})

g2cas

The eigenvalues of the generators Cas^{[2]}, Cas^{[6]} of Zent U 𝔤_{2} of degrees 2 and 6 respectively on the irreducible representation V_{λ} of highest weight λ = aω_{1} + bω_{2} are given by:

\frac{3}{4} Cas_{V_{λ}}^{[2]} = a^{2} + 3 ab + 3 b^{2} + 5 a + 9 b

\frac{243}{11} Cas_{V_{λ}}^{[6]} = 4 a^{6} + 36 a^{5}b + 117 a^{4}b^{2} + 162 a^{3}b^{3} + 81 a^{2}b^{4}
+ 60 a^{5} + 414 a^{4}b + 954 a^{3}b^{2} + 810 a^{2}b^{3} + 162 ab^{4}
− 408 a^{4} − 2808 a^{3}b − 8829 a^{2}b^{2} − 12636 ab^{3} − 6804 b^{4}
− 6580 a^{3} − 33174 a^{2}b − 61362 ab^{2} − 40824 b^{3}
− 6396 a^{2} − 32508 ab − 27756b^{2}
+ 56520 a + 100440 b

Remark B.6 (Higher Casimirs for Spin(7))

spin7cas

The eigenvalues of the generators Cas^{[2]}, Cas^{[4]} and Cas^{[6]} of Zent U 𝔰𝔭𝔦𝔫_{7} of degrees 2, 4 and 6 on the irreducible representation V_{λ} of highest weight λ = aω_{1} + bω_{2} + cω_{3} are:

2 Cas_{V_{λ}}^{[2]} = 4 a^{2} + 8 b^{2} + 3 c^{2} + 8 ab + 4 ac + 8 bc + 20 a + 32 b + 18 c
32 Cas_{V_{λ}}^{[4]} = 16 a^{4} + 128 b^{4} + 21 c^{4} + 192 a^{2}b^{2} + 72 a^{2}c^{2} + 240 b^{2}c^{2}

+ 32 a^{3}c + 64 a^{3}b + 256 b^{3}c + 256 b^{3}a + 56 c^{3}a + 112 c^{3}b + 192 a^{2}bc + 384 b^{2}ac + 240 c^{2}ab + 160 a^{3} + 1024 b^{3} + 252 c^{3} + 768 a^{2}b + 432 a^{2}c + 1536 b^{2}a + 1632 b^{2}c + 1056 c^{2}b + 552 c^{2}a + 1632 abc + 800 a^{2} + 1152 c^{2} + 3040 b^{2} + 3040 ab + 1760 ac + 3424 bc + 2000 a + 3968 b + 2376 c

512 Cas_{V_{λ}}^{[6]} = 64 a^{6} + 2048 b^{6} + 183 c^{6} + 384 a^{5}b + 192 a^{5}c + 6144 b^{5}c

+ 6144 b^{5}a + 732 c^{5}a + 1464 c^{5}b + 1920 a^{4}b^{2} + 720 a^{4}c^{2} + 7680 b^{4}a^{2} + 9600 b^{4}c^{2} + 1260 c^{4}a^{2} + 4920 c^{4}b^{2} + 1920 a^{4}bc + 15360 b^{4}ac + 4920 c^{4}ab + 5120 a^{3}b^{3} + 1120 a^{3}c^{3} + 8960 b^{3}c^{3} + 7680 a^{3}b^{2}c + 4800 a^{3}c^{2}b + 15360 b^{3}a^{2}c + 19200 b^{3}c^{2}a + 6720 c^{3}a^{2}b + 13440 c^{3}b^{2}a + 14400 a^{2}b^{2}c^{2} + 960 a^{5} + 24576 b^{5} + 3294 c^{5} + 7680 a^{4}b + 4320 a^{4}c + 61440 b^{4}a + 65280 b^{4}c + 11100 c^{4}a + 22080 c^{4}b + 30720 a^{3}b^{2} + 11040 a^{3}c^{2} + 61440 b^{3}a^{2} + 84480 b^{3}c^{2} + 15120 c^{3}a^{2} + 60000 c^{3}b^{2} + 32640 a^{3}bc + 130560 b^{3}ac + 60000 c^{3}ab + 97920 a^{2}b^{2}c + 63360 a^{2}c^{2}b + 126720 b^{2}c^{2}a + 9600 a^{4} + 167424 b^{4} + 32592 c^{4} + 67200 a^{3}b + 38400 a^{3}c + 334848 b^{3}a + 365568 b^{3}c + 88032 c^{3}a + 175584 c^{3}b + 234624 a^{2}b^{2} + 92832 a^{2}c^{2} + 364128 b^{2}c^{2} + 257664 a^{2}bc + 548352 b^{2}ac + 364128 c^{2}ab + 56000 a^{3} + 684032 b^{3} + 193464 c^{3} + 413952 a^{2}b + 251808 a^{2}c + 993024 b^{2}a + 1158912 b^{2}c + 397968 c^{2}a + 790656 c^{2}b + 1125888 abc + 160000 a^{2} + 1321856 b^{2} + 562848 c^{2} + 1189760 ab + 759040 ac + 1607552 bc + 200000 a + 863744 b + 606240 c

Remark B.7 (First, Second and Third Casimir of Spin(7))

spin7coo

The eigenvalues of the generators Cas^{[2]}, Cas^{[4]} and Cas^{[6]} of Zent U 𝔰𝔭𝔦𝔫_{7} of degrees 2, 4 and 6 on the irreducible representation V_{λ} of highest weight λ = aω_{1} + bω_{2} + cω_{3} are:

Cas_{V_{λ}}^{[2]} = \frac{1}{2} ( 4a^{2} + 8ab + 4ac + 8b^{2} + 8bc + 3c^{2} + 20a + 32b + 18c )

Cas_{V_{λ}}^{[4]} = \frac{1}{32} ( \begin{matrix}& 16 & a^{4} & + & 64 & a^{3}b & + & 32 & a^{3}c & + & 192 & a^{2}b^{2} \\ + & 192 & a^{2}bc & + & 72 & a^{2}c^{2} & + & 256 & ab^{3} & + & 384 & ab^{2}c \\ + & 240 & abc^{2} & + & 56 & ac^{3} & + & 128 & b^{4} & + & 256 & b^{3}c \\ + & 240 & b^{2}c^{2} & + & 112 & bc^{3} & + & 21 & c^{4} & + & 160 & a^{3} \\ + & 768 & a^{2}b & + & 432 & a^{2}c & + & 1536 & ab^{2} & + & 1632 & abc \\ + & 552 & ac^{2} & + & 1024 & b^{3} & + & 1632 & b^{2}c & + & 1056 & bc^{2} \\ + & 252 & c^{3} & + & 800 & a^{2} & + & 3040 & ab & + & 1760 & ac \\ + & 3040 & b^{2} & + & 3424 & bc & + & 1152 & c^{2} & + & 2000 & a \\ + & 3968 & b & + & 2376 & c \end{matrix} )

Cas_{V_{λ}}^{[6]} = \frac{1}{512} ( \begin{matrix}& 64 & a^{6} & + & 384 & a^{5}b & + & 192 & a^{5}c \\ + & 1920 & a^{4}b^{2} & + & 1920 & a^{4}bc & + & 720 & a^{4}c^{2} \\ + & 5120 & a^{3}b^{3} & + & 7680 & a^{3}b^{2}c & + & 4800 & a^{3}bc^{2} \\ + & 1120 & a^{3}c^{3} & + & 7680 & a^{2}b^{4} & + & 15360 & a^{2}b^{3}c \\ + & 14400 & a^{2}b^{2}c^{2} & + & 6720 & a^{2}bc^{3} & + & 1260 & a^{2}c^{4} \\ + & 6144 & ab^{5} & + & 15360 & ab^{4}c & + & 19200 & ab^{3}c^{2} \\ + & 13440 & ab^{2}c^{3} & + & 4920 & abc^{4} & + & 732 & ac^{5} \\ + & 2048 & b^{6} & + & 6144 & b^{5}c & + & 9600 & b^{4}c^{2} \\ + & 8960 & b^{3}c^{3} & + & 4920 & b^{2}c^{4} & + & 1464 & bc^{5} \\ + & 183 & c^{6} & + & 960 & a^{5} & + & 7680 & a^{4}b \\ + & 4320 & a^{4}c & + & 30720 & a^{3}b^{2} & + & 32640 & a^{3}bc \\ + & 11040 & a^{3}c^{2} & + & 61440 & a^{2}b^{3} & + & 97920 & a^{2}b^{2}c \\ + & 63360 & a^{2}bc^{2} & + & 15120 & a^{2}c^{3} & + & 61440 & ab^{4} \\ + & 130560 & ab^{3}c & + & 126720 & ab^{2}c^{2} & + & 60000 & abc^{3} \\ + & 11100 & ac^{4} & + & 24576 & b^{5} & + & 65280 & b^{4}c \\ + & 84480 & b^{3}c^{2} & + & 60000 & b^{2}c^{3} & + & 22080 & bc^{4} \\ + & 3294 & c^{5} & + & 9600 & a^{4} & + & 67200 & a^{3}b \\ + & 38400 & a^{3}c & + & 234624 & a^{2}b^{2} & + & 257664 & a^{2}bc \\ + & 92832 & a^{2}c^{2} & + & 334848 & ab^{3} & + & 548352 & ab^{2}c \\ + & 364128 & abc^{2} & + & 88032 & ac^{3} & + & 167424 & b^{4} \\ + & 365568 & b^{3}c & + & 364128 & b^{2}c^{2} & + & 175584 & bc^{3} \\ + & 32592 & c^{4} & + & 56000 & a^{3} & + & 413952 & a^{2}b \\ + & 251808 & a^{2}c & + & 993024 & ab^{2} & + & 1125888 & abc \\ + & 397968 & ac^{2} & + & 684032 & b^{3} & + & 1158912 & b^{2}c \\ + & 790656 & bc^{2} & + & 193464 & c^{3} & + & 160000 & a^{2} \\ + & 1189760 & ab & + & 759040 & ac & + & 1321856 & b^{2} \\ + & 1607552 & bc & + & 562848 & c^{2} & + & 200000 & a \\ + & 863744 & b & + & 606240 & c \end{matrix} )

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