DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 319–334

THE HIGHER SPIN DIRAC OPERATORS

JAROLÍM BUREŠ

Abstract. There is a certain family of conformally invariant first order elliptic systems which include the Dirac operator as its first and simplest member. Their general definition is given and some of their basic properties are described. A special attention is paid to the Rarita-Schwinger operator, the second simplest operator in the row. Its basic properties are described in more details. In the last part indices of discussed operators are computed.

1. Introduction

Let M be a smooth oriented compact n-dimensional manifold, endowed with a Riemannian metric and a spin structure. A lot of information has been collected concerning basic invariant differential operators on M as the Laplace (of the second order) and the Dirac operator (of the first order). These invariant operators and their properties are strongly related with geometry and topology of M.

Their behavior under conformal change of metric allowed us to study their conformal invariance and put it into general scheme of invariant operators on AHS manifolds. There is a list of conformal invariant operators of different order, all of them are defined on any Riemannian spin manifold. For this operators similar notions can be defined and properties and relations as for basic operators (Dirac and Laplace) can be studied.

Recently, a growing interest is paid to properties of more complicated invariant first order differential operators on M. A prototype of them is the Rarita-Schwinger operator (see [Fra1, Fra2, FraS, MP, Pe1, Pe2, Pe3, Pe4, Pe5, RaS, US, Wa]). It acts on sections of the bundle associated to a more complicated representation of the group Spin(n).

It it also included in a series of operators on M which are called higher spin Dirac operators and also in a series of operators called higher Rarita-Schwinger operators which are defined on spin valued symmetric tensor fields and has been intensively used also in Clifford analysis in connection with monogenic differential forms (see [DSS, So1, So2, SoS, So3]) or symmetric functions (see [Som1]). In general, there is a question which new geometrical and topological characterizations of Riemannian spin manifold can be obtained from properties of this additional invariant operators on it. To try to see an answer, it is first necessary to learn more about properties of these operators. The aim of the paper is to collect and review facts which exist mostly only in a preprint form and to add some new facts (index computations). In the contribution, the operators are defined including their normalisation and some of their properties are described. Main attention is concentrated to the Rarita-Schwinger operator. Basic questions discussed in the paper are:

1991 Mathematics Subject Classification. 58G25, 53C25.

Key words and phrases. Conformally invariant operators, Rarita-Schwinger operator.

The research was supported by grant GAUK 713.

This paper is in final form and no version of it will be submitted for publication elsewhere.

1. The description of conformally invariant first order differential operators (including the Rarita-Schwinger operator).

2. The spectrum of Rarita Schwinger operator on the flat model, i.e. on spheres.

3. A complete description of polynomial solutions of the Rarita-Schwinger equation.

4. The index of Rarita Schwinger operator and higher spin Dirac operators.

The first three parts have a review character, most results there are taken from papers which are at present only in a preprint form, the index properties are new results.

2. First order conformally invariant operators

There is a scheme for a construction of conformal invariant operators let us recall it shortly from ([BuSo]).

Let M be a compact oriented spin manifold with a conformal structure. Fix a Riemannian metric g in the given conformal class then we have on M principal fibre bundles P̃ ≡ P̃_{Spin} → P_{SO} → M .

Finite-dimensional irreducible representations V_{λ} of the group Spin(n) are determined by their highest weights λ ∈ Λ^{+}, where for n = 2k even, we have

Λ^{+} = \{λ = (λ_{1}, ..., λ_{k}); λ_{1} ≥ λ_{2} ≥ ... ≥ λ_{k−1} ≥ |λ_{k}|\}, λ_{i} ∈ Z ∪ \frac{1}{2}Z

and for n = 2k + 1 odd, we have

Λ^{+} = \{λ = (λ_{1}, ..., λ_{k}); λ_{1} ≥ λ_{2} ≥ ... ≥ λ_{k−1} ≥ λ_{k} ≥ 0\} , λ_{i} ∈ Z ∪ \frac{1}{2}Z .

Invariant operators are acting among spaces of sections of the corresponding associated bundles V_{λ} = P̃ ×_{Spin(n)} V_{λ} over M. Let us consider the Levi-Civita connection ω of the chosen Riemannian metric on P and let ω̃ be its (unique) lift to P̃. For any choice of λ ∈ Λ^{+}, we have the associated covariant derivative ∇_{λ} : Γ(V_{λ})→Γ(V_{λ} ⊗ T^{∗}(M)) . Tensor product V_{λ} ⊗ C_{n} can be decomposed into irreducible components

V_{λ} ⊗ C_{n} = ⊕_{λ′∈A}V_{λ′} ,

where A is the set of highest weights of all irreducible components (multiplicities included). There are simple rules how to describe A = A(λ) explicitely for any λ (see [F]). Let π_{λ′} be the projection from V_{λ} ⊗ C_{n} to V_{λ′}. Then operators

D_{λ,λ′} : Γ(V_{λ}) → Γ(V_{λ′}) , D_{λ,λ′} := π_{λ′} ∘ ∇^{λ}

are first order conformally invariant differential operators and all such operators can be constructed in this way.

Any conformally invariant first order differential operator is uniquely determined (up to a constant multiple) by a choice of allowed λ and λ′ but there is no natural normalization in general. To study spectral properties, it is necessary fix a scale of the operator, to choose appropriate normalization. For the Dirac operator, the choice of normalization is given by the Clifford action. By using twisted Dirac operators, we shall extend this normalization to a wide class of first order operators (which includes our higher spin Dirac perators as well as higher Rarita-Schwinger operators).

Definition 1. [BuSo] Let S_{\frac{1}{2}} (for n = 2k + 1), resp. S_{\frac{1}{2}} = S+_{\frac{1}{2}} ⊕ S−_{\frac{1}{2}} (for n = 2k),

denote the basic spinor representations with highest weights σ = (\frac{1}{2}, …, \frac{1}{2}, \frac{1}{2}), resp. σ^{±} = (\frac{1}{2}, …, \frac{1}{2}, ±\frac{1}{2}).

Let λ ∈ Λ^{+}, (for n = 2k + 1), resp. λ^{±} ∈ Λ^{+} (for n = 2k) be dominant weights with λ = (λ_{1}, ..., λ_{k−1}, \frac{1}{2}), resp. λ^{±} = (λ_{1}, ..., λ_{k−1}, ±\frac{1}{2}). Denote further λ′ = λ − σ ∈ Λ^{+}, resp. λ′ = λ^{+} − σ^{+} ∈ Λ^{+}. In even dimensions, we shall use the notation V_{λ} = V_{λ^{+}} ⊕ V_{λ^{−}} .

The representation V_{λ} appears with multiplicity one in the decomposition of the tensor product S_{\frac{1}{2}} ⊗ V_{λ′} (it is the Cartan product of both representations). Hence we can write the product as S_{\frac{1}{2}} ⊗ V_{λ′} = V_{λ} ⊕ W where W is the sum of all other irreducible components in the decomposition.

Let D_{λ′}^{T} be the twisted Dirac operator on S_{\frac{1}{2}} ⊗ V_{λ′}. If we write the operator D_{λ′}^{T} in the block form as

[figure: Γ(S_{\frac{1}{2}} ⊗ V_{λ′}); Γ(S_{\frac{1}{2}} ⊗ V_{λ′}); Γ(V_{λ}); Γ(V_{λ}); Γ(W); Γ(W); ∥; ∥; ⊕; ⊕; D_{λ′}^{T}; D_{λ}]

we have defined four invariant operators, one of them being the operator

D_{λ} : Γ(V_{λ}) → Γ(V_{λ}) .

Operators D_{λ} defined in such a way will be called generalized (higher spin) Dirac operators.

The case of Rarita-Schwinger operator is included at the beginning of the scheme, as follows.

A certain subclass of invariant operators discussed above is related with the following higher dimensional generalizations of holomorphic differential forms (see [DSS, So2]). Spinor valued differential forms are elements of the twisted de Rham sequence ([BuSo]):

Γ(S±_{\frac{1}{2}}) →^{∇^{S}} … Γ(Ω_{c}^{k} ⊗ S±_{\frac{1}{2}}) →^{∇^{S}} … →^{∇^{S}} Γ(Ω_{c}^{n} ⊗ S±_{\frac{1}{2}})

where ∇^{S} denotes the associated covariant derivative on spinor bundles extended to S_{\frac{1}{2}} -valued forms (see [So2, VSe]).

Every representation Λ^{k}(C_{n}) ⊗ S_{\frac{1}{2}} can be split into irreducible pieces. There are no multiplicities in the decomposition, so the irreducible pieces are well defined. For k forms (k ≤ [n/2]), there are k pieces in the decomposition and the decomposition is symmetric with respect to the action of the Hodge star operator. The space of spinor valued k-forms Γ(Ω_{c}^{k} ⊗ S^{±}) (k ≤ [n/2]) can be written as the sum ⊕_{j=1}^{k}E^{k,j} and it can be checked (see [DSS, VSe, So2]) that E^{k,j} is the bundle associated with the representation with the highest weight λ_{j} = (\frac{3}{2}, …, \frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}, ±\frac{1}{2}), where the number j indicates that the component \frac{3}{2} appears with multiplicity equal to j. Signs ± at the last components are relevant only in even dimensions (more details can be found in [VSe]). The whole splitting can be described by the following triangle shaped diagram (in odd dimensions, there are two columns of the same length in the middle).

E^{0,0} ⟶^{D_{0}} E^{1,0} ⟶^{D_{0}} … ⟶^{D_{0}} E^{k,0} ⟶^{D_{0}} … ⟶^{D_{0}} E^{2k−1,0} ⟶^{D_{0}} E^{2k,0}
⊕ ⊕ ⊕ ⊕ ⊕
E^{1,1} ⟶^{D_{1}} … ⟶^{D_{1}} E^{k,1} ⟶^{D_{1}} … ⟶^{D_{1}} E^{2k−1,1}
⊕  ⊕  ⊕
… ⟶^{D_{j}}  ⋮ ⟶^{D_{j}} …
⊕
E^{k,k}

The general construction of invariant operators described above can be used in the special case of spinor valued forms. The covariant derivative ∇^{S} restricted to E^{k,j} and projected to E^{k+1,j′} is an example of this general construction. It can be shown that if |j − j′| > 1, then the corresponding invariant operator is trivial. We shall be mainly interested in ’horizontal arrows’, i.e. in operators D_{j} given by restriction to E^{k,j} and projection to E^{k+1,j}. They are indicated in the above scheme.

The simplest cases among them are well known. The operator D_{0} is (a multiple of) the Dirac operator.

The operator D_{1} is (an elliptic version of) the operator called Rarita-Schwinger operator by physicists (see [RaS, Wa]).

All of them are elliptic operators (see [So1, Bra]).

Note that all operators D_{j} on the same row in the scheme above cannot be identified without further comments. To compare them, it is necessary first to choose an equivariant isomorphism among corresponding bundles. Then they coincide up to a constant multiple.

To compare the operators D_{j} in the above scheme with the higher spin Dirac operators (see Def.1), we shall choose a certain identification of the corresponding source and target bundles. We shall do it for the first operator D_{j} in the row.

Let us characterize an algebraic operator Y : Γ(Ω_{c}^{k+1} ⊗ S_{\frac{1}{2}} ) → Γ(Ω_{c}^{k} ⊗ S_{\frac{1}{2}} ) by a local formula

Y (ω ⊗ s) = − ∑_{i} ι(e_{i})ω ⊗ e_{i} · s ,

where {e_{i}} is a (local) orthonormal basis of T M and ι denotes the contraction of a differential form by a vector. As shown in [VSe], the map Y : E^{k+1,j} → E^{k,j}, j < k < [n/2] is an isomorphism.

The twisted Dirac operator D^{T} maps the space Γ(Ω_{c}^{k} ⊗ S_{\frac{1}{2}} ) to itself. In [VSe], it was proved that we have a relation ∇ ∘ Y + Y ∘ ∇ = −D^{T} . Let us denote the projection from Ω_{c}^{k} ⊗ S onto E^{k,j} by π_{k,j}. Symbols D̃_{j}, 0 ≤ j < [n/2] will denote operators

D̃_{j} = Y ∘ D_{j} = π_{j,j} ∘ Y ∘ ∇^{S}|_{E^{j,j}} ,

mapping the space of sections of E^{j,j} to itself. Then Y |_{E^{j,j}} = 0 implies that

D̃_{j} = π_{j,j} ∘ Y ∘ ∇^{S}|_{E^{j,j}} = −π_{j,j} ∘ D^{T} |_{E^{j,j}} = −D_{λ_{j}},

where D_{λ_{j}} is the generalized higher spin Dirac operator corresponding to the bundle V_{λ_{j}}, λ_{j} = (\frac{3}{2}, …, \frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}) (component \frac{3}{2} appearing j times). More precisely, there are no signs in odd dimensions, while in even dimension, V_{λ_{j}} = V_{λ_{j}^{+}} ⊕ V_{λ_{j}^{−}}.

3. Definition and general properties of Rarita-Schwinger operator

In this section, the attention is concentrated to the simplest special case of the general definition given in above, i.e. to the Rarita-Schwinger operator. This operator is mentioned in [RaS] and was studied in more details in the paper by M.Y.Wang ([Wa]) and the dissertation of U. Semmelman ([US]) and also in the physical context e.g in [Fra1, E, Pe1, Pe2, Pe3]. It is also related to the Stein-Weiss operators studied by Branson ([Bra]).

In the paper of Wang, it is studied mainly on compact Einstein spin manifolds admitting nonzero Killing spinors and in the context of deformation of Einstein metrics on manifolds.

In the Dissertation of U. Semmelmann, the results of Wang are extended and also some eigenvalue problems as the relations between eigenvalues of the Dirac operator D, the twisted Dirac operator D_{T} and Rarita-Schwinger operator D_{\frac{3}{2}} are added.

Let us first describe (a slight modification of) the Wang definition and his decription of the basic properties of operators involved ([Wa]).

We shall use in the following section the notation

S_{\frac{k}{2}} = V_{λ} for λ = (\frac{k}{2} , \frac{1}{2}, ..., \frac{1}{2}) .

Let e = {e_{1}, ..., e_{n}} be an orthogonal frame on some open set on M and {ϵ^{1}, ..., ϵ^{n}} the dual coframe.

Remark 1. The classical Dirac operator on M is the operator

D : Γ(S_{\frac{1}{2}} ) → Γ(S_{\frac{1}{2}} )

and has with respect to frame e the form

D(σ) = ∑_{i} e_{i} ∇_{e_{i}}^{S}σ .

There is a fundamental diagram, which presents the relation among operators in discussion.

From the theory of representation of the group Spin(n) we have a decomposition of the tensor product

S_{\frac{1}{2}} ⊗ C^{n∗} = S′_{\frac{1}{2}} ⊕ S_{\frac{3}{2}}

into irreducible Spin(n)-modules, where S′_{\frac{1}{2}} is isomorphic to S_{\frac{1}{2}} .

We shall use the following canonical identifications

ι : S_{\frac{1}{2}} → S_{\frac{1}{2}} ⊗ C^{n∗}

ι(σ) := −\frac{1}{n} ∑_{i} e_{i} σ ⊗ ϵ^{i}

which is the isomorphisms onto S′_{\frac{1}{2}} and S_{\frac{3}{2}} ≡ Ker(μ) where

μ : S_{\frac{3}{2}} ⊗ C^{n∗} → S_{\frac{1}{2}}

is the Clifford multiplication

μ(∑_{i} ψ_{i} ⊗ ϵ^{i}) = ∑_{i} e_{i} ψ_{i} .

Moreover we have the projections π_{\frac{1}{2}} , π_{\frac{3}{2}} from S_{\frac{3}{2}} ⊗ C^{n∗} onto the individual irreducible components given by

π_{\frac{1}{2}} (∑_{i} ψ_{i} ⊗ ϵ^{i}) = −\frac{1}{n} ∑_{j} e_{j} (∑_{i} e_{i} ψ_{i}) ⊗ ϵ^{j}

and

π_{\frac{3}{2}} (∑_{i} ψ_{i} ⊗ ϵ^{i}) = ∑_{j}(ψ_{j} + \frac{1}{n}e_{j} ∑_{i} e_{i} ψ_{i}) ⊗ ϵ^{j}

For the corresponding operators we get

[figure: Γ(S_{\frac{1}{2}} ); Γ(S_{\frac{1}{2}} ⊗ T_{C}^{∗}); Γ(S_{\frac{1}{2}} ⊗ T_{C}^{∗}); Γ(S_{\frac{1}{2}} ); Γ(S_{\frac{1}{2}} ); Γ(S_{\frac{3}{2}} ); Γ(S_{\frac{3}{2}} ); ∥; ∥; ⊕; ⊕; ∇; D_{T}; D′; R; D; T; T′; T^{∗}]

with

∇(σ) = ∑_{i} ∇_{e_{i}}^{S}σ ⊗ ϵ^{i}

and the (transported) Dirac operator

D(σ) := π_{\frac{1}{2}} (∇^{S}σ) = −\frac{1}{n} ∑_{i} e_{i} Dσ ⊗ ϵ^{i} = ι(D(σ))

the Twistor operator

T (σ) := π_{\frac{3}{2}} (∇^{S}σ) = ∑_{j}(∇_{e_{j}}^{S}σ + \frac{1}{n}e_{j} Dσ) ⊗ ϵ^{j}

and another operators which with respect to the above identifications are

D′ = −\frac{n − 2}{n}ι ∘ D ∘ ι^{−1}

T′ = \frac{2}{n}T ∘ ι^{−1}

and if we denote the operator

δ : Γ(S_{\frac{3}{2}} ) → Γ(S_{\frac{1}{2}} )

by

δ(∑_{i} ψ_{i} ⊗ ϵ^{i}) = − ∑_{i} ∇_{e_{i}}ψ_{i}

then we get

T^{∗}(ψ) = 2.ι(δ(ψ)) .

Theorem 1. ([Wa]) The operator D_{T} with respect to the decomposition

Γ(S_{\frac{1}{2}} ⊗ T_{C}^{∗}) = Γ(S_{\frac{1}{2}} ) ⊕ Γ(S_{\frac{3}{2}} )

has the following form

(\begin{matrix}\frac{2−n}{n}ι ∘ D ∘ ι^{−1} & 2.ι ∘ δ \\ \frac{2}{n}(T ∘ ι^{−1}) & R \end{matrix})

4. Eigenvalues on spheres

The spectrum of the Dirac operator on the sphere is well known for some time already (see [Baer]). The spectra of operators D̃_{j} on sphere were computed in ([BuSo]), the case of a general elliptic first order operators can be found in [Br1].

Lemma 1. The eigenvalues of the Dirac operator on the sphere S_{n} with standard metric are

μ_{l} = ± (\frac{n}{2} + l); l = 0, 1, 2, ….

with multiplicity

2^{[ \frac{n}{2} ]}\binom{l + n − 1}{l} .

The main result of the paper [BuSo] is given in the following theorem.

Theorem 2. Let D_{λ_{j}} = −D̃_{j}, 0 < j < n/2, be the higher spin Dirac operators defined above, considered on the sphere S_{n} with the standard metric. Then their eigenvalues are:

μ_{l}^{1} = ± (\frac{n}{2} + l); l = 1, 2, ….

with multiplicity

2^{[ \frac{n}{2} ]}\binom{n + 1}{j + 1}\binom{l + n}{l − 1}\frac{(n − 2j)(j + 1)}{(l + j)(l + n − j)}

and

μ_{l}^{2} = ± [\frac{n − 2j}{n − 2j + 2} (\frac{n}{2} + l)]; l = 1, 2, ….

with multiplicity

2^{[ \frac{n}{2} ]}\binom{n + 1}{j}\binom{l + n}{l − 1}\frac{(n − 2j + 2)j}{(l + j − 1)(l + n − j + 1)} .

It is proved using modification of results of Branson et others in [BOO].

The Rarita-Schwinger operator is the operator D̃_{1} and its eigenvalues are:

μ_{l}^{1} = ± (\frac{n}{2} + l); l = 1, 2, ….

with multiplicity

2^{[ \frac{n}{2} ]}\binom{n + 1}{2}\binom{l + n}{l − 1}\frac{2(n − 2)}{(l + 1)(l + n − 1)}

and

μ_{l}^{2} = ± [\frac{n − 2}{n} (\frac{n}{2} + l)]; l = 1, 2, ….

with multiplicity

2^{[ \frac{n}{2} ]}(n + 1)\binom{l + n}{l − 1}\frac{(n)}{(l)(l + n)} .

The second series of eigenspaces on sphere comes out as the image by Twistor operator of the eigenspaces of the Dirac operator on sphere.

5. Homogeneous solutions of Rarita-Schwinger equation on R_{0}^{n+1}

The solutions of the Dirac equation on the flat space are studied traditionally in Clifford analysis. Special attention is paid to the solutions on R_{0}^{n+1} := R^{n+1}−{0} which are polynomial (homogeneous) of some degree k, namely to the functions

ϕ : R_{0}^{n+1} → S_{\frac{1}{2}}

which satisfy

(1) Dϕ = 0

(2) ϕ(λx) = λ^{k}ϕ(x) for all x ∈ R_{0}^{n+1}.

Such solutions are strongly related with the eigenfunctions of the Dirac operator on the unit sphere S^{n} ⊂ R_{0}^{n+1}, the relation follows simply from the restriction of solution to the sphere. Such functions are also called spherical monogenics (of degree k).

We make a simple generalization of the problem, instead of the Dirac operator we shall study the Rarita-Schwinger operator, which is define on some special spinor-valued one forms on R_{0}^{n+1}and is the next operator in the serie of first order elliptic invariant operators on Riemannian spin manifolds.

We would like again to study homogenic solutions of the Rarita-Schwinger operator on R_{0}^{n+1}and their relations with the eigenfunctions of the corresponding operators (Dirac and the Rarita-Schwinger ) on the sphere S^{n}. Of course the situation is more complicated here.

There is a possibility to study the same problem for other higher spin operators on spinor valued forms or higher spin Rarita-Schwinger operators.

Let me mention only the main results, the full description of results and another related topics will be presented in fortcoming paper [Bu], see also [BuLN].

We shall use on R_{0}^{n+1}fixed cartesian coordinates x = (x_{1}, ..., x_{n+1}) and also:

e_{i} = \frac{∂}{∂x_{i}} , x = ∑_{i} x_{i}e_{i} ∈ R^{n} ,

and we have

ϵ^{i} = dx^{i} , ∇_{e_{i}} = \frac{∂}{∂x_{i}} .

Let us take

ψ = ∑_{1}^{n+1} ψ_{i} ⊗ dx^{i} ∈ S_{\frac{1}{2}} ⊗ R^{n∗}

and suppose that

ψ ∈ Γ(S_{\frac{3}{2}} )( ⟺ μ(ψ) = ∑_{i} e_{i} ψ_{i} = 0) .

Then the Rarita-Schwinger operator R on ψ has the form

Rψ = ∑_{i}(Dψ_{i} + \frac{1}{n + 1}e_{i} (∑_{k} e_{k} Dψ_{k})) ⊗ dx^{i} .

Remark 2. The spinor valued 1-form ψ satisfying the condition μ(ψ) = 0 on M is solution of Rarita-Schwinger equation iff there is a spinor field ϕ on M such that

D_{T} ψ = ι(ϕ) .

Let us denote by P_{k}(1) the space of all polynomial k-homogeneous solutions of Rarita-Schwinger equation on and by P_{k}(0) the space of polynomial k-homogeneous solutions of the Dirac equation on R_{0}^{n+1}.

We would like to find a good description of the space P_{k}(1), namely its decomposition into some well defined and natural pieces.

Let us define the map

L : Γ(S_{\frac{3}{2}} ) → Γ(S_{\frac{1}{2}} )

by

L(ψ) := ∑_{i} x_{i}ψ_{i}

Then the map is homomorphism of the corresponding bundles, namely

L(ψ + ϕ) = L(ϕ) + L(ϕ), L(f.ψ) = f.L(ψ) forf ∈ C^{∞} .

Lemma 2. Suppose ψ ∈ P_{k}(1), then

D^{3}(L(ψ)) = 0 .

Let ψ_{0} be a (k-1) homogeneous solution of the Dirac equation on R_{0}^{n+1}, then k-homogeneous solution ψ of the Rarita-Schwinger equation satisfying

D_{T} ψ = ∑_{i} e_{i} ψ_{0} ⊗ dx^{i} ( eq)

can be constructed in the following way. Let

Ξ : Γ(S_{\frac{1}{2}} ) → Γ(S_{\frac{1}{2}} ⊗ Λ^{1})

be the map defined by:

Ξ(ψ_{0}) = \frac{1}{2.(n + k)}(∥x∥^{2}T (ψ_{0}) + ∑_{j} x_{j}.ψ_{0} ⊗ dx^{j} + ∑_{j} e_{j} (x.ψ_{0}) ⊗ dx^{j})

Lemma 3. Ξ(ψ_{0}) is a k-homogeneous solution of the equation (eq).

Theorem 3. Let ψ be a k-homogeneous solution of the Rarita-Schwinger equation. Then there is a unique decomposition of ψ into (k-homogeneous) pieces

ψ = ψ_{1} + ψ_{2} + ψ_{3}

with ψ_{j} ∈ M^{j} where

M^{1} = \{ψ ∈ Γ(R_{0}^{n+1}, S_{\frac{3}{2}} ) | L(ψ) = 0\}
M^{2} = T′(KerD) ⊂ \{ψ ∈ Γ(R_{0}^{n+1}, S_{\frac{3}{2}} ) | DL(ψ) = 0\}
M^{3} = Ξ(P_{k−1}(0)) ⊂ \{ψ ∈ Γ(R_{0}^{n+1}, S_{\frac{3}{2}} ) | DL(ψ) ≠ 0 for ψ ≠ 0\} .

We can also describe the space P_{k}(1) of k-homogeneous solutions of Rarita-Schwinger equation from representation point of view. Any space M^{j} is an irreducible representation of the group Spin(n) and its type is determined by its highest weight λ. We shall speak about the space with highest weight λ as about the space of representation type λ. Recall that the Rarita-Schwinger equation for ψ ∈ Γ(R_{0}^{n+1}, S_{\frac{3}{2}} ) has a form:

D_{T} ψ = ι(ϕ) ,

with

Dϕ = 0 .

The space P_{k}(1) is a direct sum of three spaces (representation types) of solutions, namely M^{1} :

The space is characterized by the condition L(ψ) = 0. It corresponds (by restrictions of fields) to the eigenspace of induced Rarita-Schwinger operator on unit sphere of type (\frac{2k+1}{2}, \frac{3}{2}, \frac{1}{2}, ..., \frac{1}{2}).

M^{2}:

The space is characterized by the condition L(ψ) ≠ 0 for ψ ≠ 0, D_{0}L(ψ) = 0, The corresponding space P_{k}^{B1}(1) is constructed from the space {σ ∈ P_{k+1}(0)} using the Twistor operator and is of the type (\frac{2k+3}{2}, \frac{1}{2}, ..., \frac{1}{2}).

Both preceding types are characterized by condition ϕ = 0 and consists of solutions non only of RS-equation but of the whole twisted Dirac equation.

M^{3}:

The space is characterized by the condition D_{0}(ψ) ≠ 0 for ψ ≠ 0, and can be constructed from P_{k−1}(0) the space of (k-1)-homogenic solutions of Dirac equation, and is of the type (\frac{2k−1}{2}, \frac{1}{2}, ..., \frac{1}{2}).

Remark 3. All types of solutions can be uniquely determined by its restriction to unit sphere and can be described by pure spherical data (see [BuLN]).

Remark 4. The classification of polynomial solutions of Rarita-Schwinger equation can be done also for polynomial solutions of the general higher spin Dirac equation as well as for higher Rarita-Schwinger equations. There are several papers in preparation for publication by the authors P. Van Lancker (Gent), F. Sommen (Gent), V. Souček (Prague) and myself.

6. Index of elliptic differential operator

Let E → M and F → M be complex vector bundles over a compact mdimensional manifold M. Denote Γ(E) the space of smooth sections of E.

Let

D : Γ(E) → Γ(F)

be an elliptic differential operator between E and F, and

D^{∗} : Γ(F) → Γ(E)

its adjoint. Both operators D, D^{∗} have finite dimensional kernels.

The index of the operator D is defined as

IndD = dim KerD − dim KerD^{∗} .

There are formulas, characterizing the index of an operator in terms of topological invariants of the manifold M and the corresponding bundles.

In the formulas appear some characteristic classes of complex vector bundle E on M, which can be expressed through closed differential forms representing Chern classes c_{i}(E) namely : Todd genus

td(E) = 1 + \frac{1}{2}c_{1}(E) + \frac{1}{12}(c_{2}(E) + c_{1}(E)^{2}) + ...

and Chern character

Ch(E) =

dim(E) + c_{1}(E) + \frac{1}{2}(c_{1}(E)^{2} − 2c_{2}(E) + \frac{1}{6}(c_{1}(E)^{3} − 3c_{1}(E)c_{2}(E) + 12c_{3}(E)) + ...

Remark 5. Chern classes c_{i}(E) are represented by closed differential forms of degree 2i, computable from curvature form of any connection on E.

From the index theorem it follows, that index of the operator does not depend on operator itself but only on bundles E and F.

In the following section, suppose that the dimension of M is m = 2n even, even if will be not especially mentioned, because odd dimensional cases are trivial.

For the index of Dirac operator and its twisted version there are well-known index formula, containing the invariant Â-genus.

Theorem 4. (Atiyah-Singer, [BGV]). Let M be an oriented spin manifold of dimension m = 2n. Let S be the spin bundle and let W be an arbitrary vector bundle over M, If D_{W} is the twisted Dirac operator on Γ(M, W ⊗ S) then

indD_{W} = (2πi)^{−n} ∫_{M} Â(M)ch(W) .

We shall use normalization of forms representing characteristic classes as is given in [BGV].

The invariant Â(M) is the Hirzebruch’s Â genus, given by the formula

Â(M) = 1 − \frac{1}{24}p_{1}(M) + (\frac{7}{5760}p_{1}(M)^{2} − \frac{1}{1440}p_{2}(M)) + ...

where

p_{i}(M) = (−1)^{i}c_{2i}(T_{C}(M))

are the Pontrjagin forms.

Remark 6. Let us remark, that index of the classical Dirac (called also Atiyah-Singer [LM] ) operator

D+_{\frac{1}{2}} : Γ(S^{+}) → Γ(S^{−})

is

IndD+_{\frac{1}{2}} := Â_{\frac{1}{2}} [M] = (2πi)^{−n} ∫_{M} Â(M) .

The Chern character Ch(E) of the bundle E satisfies the conditions

Ch(E ⊕ F) = Ch(E) + Ch(F)
Ch(E ⊗ F) = Ch(E).Ch(F)

Now we shall try to use general theory for computation of the index of our operators D_{j} in discussion, namely

indD_{j}^{+} = dim kerD_{j}^{+} − dim kerD_{j}^{−}

332 J. BUREŠ

First of all we have from the Th.1 : Corollary 1. The index of the twisted Dirac operator on M, dim M = 2n,

D_{T}^{+} : Γ(T_{C}^{∗}(M) ⊗ S^{+}) → Γ(T_{C}^{∗}(M) ⊗ S^{−})

is given by the formula

IndD_{T}^{+} = ∫_{M} Ch(T_{C}^{∗}(M))Â(M) =
(2πi)^{−n} ∫_{M}(2n + p_{1}(M) + \frac{1}{12}(p_{1}(M)^{2} − 2p_{2}(M)) + .)
(1 − \frac{1}{24}p_{1}(M) + \frac{1}{5760}(7p_{1}(M)^{2} − 4p_{2}(M)) + .) .

6.1. Index of the Rarita-Schwinger operator. Because of equality

S±_{\frac{1}{2}} ⊗ T^{∗}(M) = S∓_{\frac{1}{2}} ⊕ S±_{\frac{3}{2}}

we have

Ch(S+_{\frac{3}{2}}) − Ch(S−_{\frac{3}{2}}) = (Ch(T_{C}^{∗}(M) + 1)(Ch(S+_{\frac{1}{2}}) − Ch(S−_{\frac{1}{2}})) .

We can simply compute

(Ch(T_{C}^{∗}(M)) + 1 = 2n + 1 + p_{1}(M) + \frac{1}{12}(p_{1}(M)^{2} − 2p_{2}(M)) + ....

and finally we get

IndD_{\frac{3}{2}} := Â_{\frac{3}{2}} [M] = (2πi)^{−n} ∫_{M}(Ch(T_{c}^{∗}(M)) + 1)Â(M).

Especially for dimension 2n = 4 we have

IndD_{\frac{3}{2}} =

(2πi)^{−2} ∫_{M}(5 + p_{1}(M) + ...)(1 − \frac{1}{24}p_{1}(M) + ...)

= (2πi)^{−n} ∫_{M}(5 + \frac{19}{24}p_{1}(M)) = −19(2πi)^{−2} ∫_{M} Â(M).

So we have the relation between the index of Dirac operator and Rarita-Schwinger operator by

IndD_{\frac{3}{2}} = −19.IndD_{\frac{1}{2}} .

Next nontrivial case is dimension m = 2n = 8, we have:

IndD_{\frac{3}{2}} =

= (2πi)^{−4} ∫_{M}(9 + p_{1}(M) + \frac{1}{12}(p_{1}(M)^{2} − 2p_{2}(M)) + .) ·

·(1 − \frac{1}{24}p_{1}(M) + \frac{1}{5760}(7p_{1}(M)^{2} − 4p_{2}(M)) + .) =

= (2πi)^{−4} ∫_{M}(9 + \frac{19}{24}p_{1}(M) + \frac{1}{5760}(543p_{1}(M)^{2} − 996p_{2}(M)) + .) .

So we do not have the simple relation between the index of Dirac operator and Rarita-Schwinger operator as above, only we have

IndD_{\frac{3}{2}} = 249Ind D_{\frac{1}{2}} − \frac{21}{144}(2πi)^{−4} ∫_{M} p_{1}(M)^{2}.

There is a possibility to find a manifold of dimension 8 without harmonic spinors, but with nontrivial kernel of Rarita-Schwinger operator.

6.2. Index of the Dirac higher spin operators. For the computation of the index for operator D_{j}^{+} for 2 ≤ j < n there is an induction procedure.

We have

S^{±} ⊗ Λ^{j}T_{C}^{∗}(M) ≃ V_{0}^{∓} ⊕ V_{1}^{±} ⊕ ... ⊕ V_{j}^{∓} j even
≃ V_{0}^{±} ⊕ V_{1}^{∓} ⊕ ... ⊕ V_{j}^{±} j odd.

and

Ch(S^{±}).Ch(Λ^{j}T_{C}^{∗}(M)) = ∑_{k=0}^{j} Ch(V_{k}^{∓sgn(k)})

for j even,

Ch(S^{±}).Ch(Λ^{j}T_{C}^{∗}(M)) = ∑_{k=0}^{j} Ch(V_{k}^{±sgn(k)})

for j odd, with sgn(k) = ± if k is even or odd.

Together we get

Ch(V_{j}^{+}) − Ch(V_{j}^{−}) = (−1)^{j+1}(Ch(Λ^{j}T_{C}^{∗}) + Ch(Λ^{j−1}T_{C}^{∗})).(Ch(S^{+}) − Ch(S^{−})).

Theorem 5. The index of the operator D_{j} is :

indD_{j} := Â_{\frac{j}{2}} [M] = (2πi)^{−n}. ∫_{M}(Ch(Λ^{j−1}) − Ch(Λ^{j}))Â(M).

Using the theorem we can study problem of existence of solutions of equation D̃_{j}ψ = 0.

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