Manifolds with many Rarita-Schwinger fields

The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected K\"ahler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.


Introduction
The goal of this note is to establish the existence of a sequence of compact Riemannian spin manifolds in any fixed dimension n ≥ 4 for which the number of Rarita-Schwinger fields tends to infinity. When n is divisible by 4, these manifolds are Kähler-Einstein with negative Einstein constant, and arise as complete intersections in some higher dimensional complex projective space. For other values of n we take products of such Kähler-Einstein spaces with flat tori. The Rarita-Schwinger operator Q is the twisted Dirac operator acting on 3 /2-spinors, and perhaps somewhat confusingly, Rarita-Schwinger fields are solutions of this twisted Dirac operator which are also divergence free. This is an overdetermined right-elliptic operator and so the existence of nontrivial solutions is nongeneric. This operator Q appeared initially in the 1941 paper of Rarita and Schwinger [8] to describe the wave functions of particles of spin 3 /2, and has been used extensively in the physics literature since then. In mathematics it is relatively unstudied, and has appeared mostly as one example amongst many in the family of all twisted Dirac operators. Notably, Semmelmann computed its spectrum on spheres and complex projective spaces [9]. Bureš [3] introduced it from the point of representation theory, studied some basic examples of solutions and computed its index in general. Branson and Hijazi [2] proved its conformal covariance and determined the Weitzenböck formula for Q 2 . Unlike the simpler Dirac operator, this Weitzenböck formula has lower order terms which do not have a sign under standard geometric hypotheses, so the existence of solutions (divergence-free or not) does not have any immediate connection to geometry. M. Wang [11] studied the role of Rarita-Schwinger fields in the deformation theory of Einstein metrics with parallel spinors, see also his earlier paper [10]. The problem of counting Rarita-Schwinger fields was considered only quite recently by Homma and Semmelmann [5], with related work more recently still in Homma-Tomihisa [6]. The goal of [5] was to find manifolds admitting any nontrivial Rarita-Schwinger fields; as part of this they obtain a complete classification of positive quaternion-Kähler manifolds and spin symmetric spaces admitting such fields, but their mention of the negative Einstein case is rather brief. We refer to all of these papers for a more extended discussion and some description of the use of Rarita-Schwinger fields in physics. Our main observation is that the negative Kähler-Einstein case provides a particularly rich set of examples. To set the stage, suppose that (M, g) is a Riemannian manifold carrying a spin structure. Denote by ΣM its spin bundle, with its chirality decomposition Σ + M ⊕ Σ − M when dim M is even. The standard Dirac operator acts on sections of ΣM ; coupling to the Levi-Civita connection on T M we may then define the twisted Dirac operator D acting on sections of ΣM ⊗ T M . As we explain in the next section, this bundle splits as the direct sum of an isometric copy of ΣM and another bundle ΣM , which is the bundle of 3 /2-spinors. The Rarita-Schwinger operator Q is the restriction of D to ΣM , i.e., the projection of Dψ to ΣM , where ψ ∈ Γ( ΣM ). The complementary projection of Dψ onto the copy of ΣM in ΣM ⊗ T M is essentially the divergence of ψ. The space of Rarita-Schwinger fields is the set of such sections ψ for which both components of Dψ vanish, and we define RS(M ) = dim{ψ ∈ Γ( ΣM ) : Dψ = 0}.
For even dimensional manifolds we can also define RS ± (M ), the dimension of the space of even or odd chirality Rarita-Schwinger fields. Our main result is Theorem 1.1. Let n be any positive integer greater than or equal to 4 and C any positive constant. Then there exists a compact Riemannian spin manifold M n such that RS(M ) > C. If n is divisible by 4, then we can take M to be a simply connected compact Kähler-Einstein manifold of complex dimension m = n/2 which is spin and has negative Einstein constant.
This is a corollary of a more precise theorem which, in the case where M is Kähler-Einstein and spin with negative Einstein constant, estimates from below RS(M ) (and RS ± (M )) as the difference of a certain characteristic number of M and a number which is essentially the dimension of the space of parallel spinors, see Theorem 2.9. Moreover, for each even m ≥ 2 we find a complex m-dimensional simply connected compact Calabi-Yau manifold with many Rarita-Schwinger fields. This means that the dimension of the space of these fields is much larger than that of flat tori of the same dimension, see Corollary 5.2 for a precise formulation.
The key tool in the proof is a certain elliptic complex involving the Rarita-Schwinger operator which is only well-defined on Einstein manifolds. The index of this complex leads to the aforementioned estimate of RS(M ). The existence of arbitrarily many linearly independent Rarita-Schwinger fields in a given dimension provides a stark contrast with the behavior of other betterknown overdetermined elliptic operators which appear in geometry. For example, the Killing operator is said to be overdetermined of finite type, which corresponds to the fact that there is a sharp bound for the dimension of the space of solutions of the Killing operator which depends only on the dimension of the manifold. The dimension of the space of twistor spinors can be bounded similarly. Our results show that the Rarita-Schwinger operator is not of this type. This finite type phenomenon was studied exhaustively in the work of Kodaira and Spencer.
He was also supported by SPP 2026 funded by Deutsche Forschungsgemeinschaft. The second author was supported by the NSF grant DMS-1608223.

Rarita-Schwinger fields
We start by describing the geometric setup for the study of Rarita-Schwinger fields.
For an introduction to general aspects of spin geometry see e.g. [7].
2.1. The algebra of vector-spinors. We equip R n with its standard Euclidean metric ·, · and standard orientation. Let Σ n be the complex spinor module of the spin group Spin(n). Then there is a linear map γ : R n ⊗ Σ n → Σ n satisfying the Clifford relations where we have used the notation X · φ = γ(X ⊗ φ). We denote the kernel of γ by Σ n ⊂ R n ⊗ Σ n . The Euclidean metric on R n and the Hermitean metric on Σ n induce a Hermitean metric on R n ⊗ Σ n and hence on Σ n . Let e 1 , . . . , e n be the standard basis of R n . We define The factor − 1 n is chosen such that γ • ι = id, hence ι • γ is a projection. The complementary projectionπ = id −ι • γ has image Σ n . The maps γ, ι, andπ are Spin(n)-equivariant. One easily checks that in particular, ι is injective, and that γ * = nι.
The metric on R n yields a contraction R n ⊗ (ι(Σ n ) ⊕ Σ n ) = R n ⊗ R n ⊗ Σ n → Σ n which we denote by X ⊗ Φ → Φ(X), i.e. ( n j=1 e j ⊗ φ j )(X) = n j=1 e j , X φ j . If n is even, Σ n decomposes as a Spin(n)-module into spinors of positive and negative chirality, Σ n = Σ + n ⊕ Σ − n . Clifford multiplication interchanges chirality, γ(R n ⊗ Σ ± n ) = Σ ∓ n . We get a corresponding splitting Now let M be an n-dimensional Riemannian spin manifold. Associating the Spin(n)modules R n , Σ n , and Σ n to the spin structure of M , we obtain the tangent bundle T M , the spinor bundle ΣM and the bundle of 3 /2-spinors ΣM , respectively. If n is even, we furthermore have the bundles Σ ± M and Σ ± M of spinors and 3 /2-spinors of positive and negative chirality. Since γ, ι, andπ are Spin(n)-equivariant, they induce vector bundle morphisms which we again denote by γ, ι, andπ.
2.2. Dirac, twistor, and Rarita-Schwinger operator. The Levi-Civita connection on T M induces one on ΣM and hence on ΣM . We will denote all these connections by ∇. Note that the connection on the spinor bundle is a differential operator ∇ : Γ(ΣM ) → Γ(T M ⊗ ΣM ). The Dirac operator is defined as and the twistor or Penrose operator as The Dirac operator is formally self-adjoint, D = D * , while the adjoint of P is given on 3 /2-spinors Φ by be the twisted Dirac operator acting on vector-spinor fields. We write it as a 2×2-matrix with respect to the decomposition This follows from straight-forward computation, see also [11,Prop. 2 It is a formally self-adjoint first order elliptic differential operator. If n is even, the Dirac operators D and D interchange chirality, hence we get operators (The reader should beware here: by our conventions, the formula for D ± contains D ∓ in its upper left entry, which is perhaps awkward, but is coherent with other notation.) Thus the Rarita-Schwinger operator also interchanges chirality, Q ± : 3. An exact sequence on Einstein manifolds. As it turns out, the Rarita-Schwinger operator has some especially nice features on Einstein manifolds.
is an elliptic complex. If n is even, then (1) restricts to elliptic complexes If, in addition, M is compact without boundary, then the Euler number of (2) is given by HereÂ(M ) is theÂ-class of M , ch(T C M ) is the Chern character of the complexified tangent bundle, and [M ] is the fundamental cycle in homology.
Remark 2.2. It seems reasonable to expect that (1) and (2) might be the initial parts of a BGG complex that exists in certain situations. To our knowledge, this does not seem to be the case.
Proof. Using the Weitzenböck formula for D 2 one computes where Ric 0 is the traceless part of Ricci curvature, see [11,Prop. 2.9(b)]. Since M being Einstein means Ric 0 = 0, we see that (1) is a complex. Then, for any covector ξ ∈ T * x M , the corresponding sequence of principal symbols is a complex too. Let ξ = 0. To show ellipticity of (1) it suffices, for dimensional reasons, to show that the two principal symbols have maximal rank. But this is clear: since D is elliptic the first principal symbol is injective and since Q is elliptic the second is surjective. It remains to compute the Euler numbers of (2). Since Euler numbers are invariant under continuous deformations of elliptic complexes, it suffices to do this for α = 0.
so that the Euler number is given by For α ∈ R consider the operator Again using (4), we see that the off-diagonal terms of D 2 α vanish for Einstein manifolds and we get that is elliptic for every choice of α. Thus all D α are elliptic. Therefore Inserting this into (5) we get, using the Atiyah-Singer index theorem for twisted Dirac operators, Remark 2.3. The proof above shows that the index of the Rarita-Schwinger operator Q ± is given by See [5,Sec. 3] for a different derivation of this fact. Remark 2.4. We have presented this computation phrased using the elliptic complexes (1) and (2). It would be straightforward to wrap each of these complexes into a single elliptic operator (depending on the parameter α) and compute its index. Neither approach seems shorter than the other and the end results are the same. The complex has the advantage that its index can be refined by its cohomology which might turn out to be useful in some other context.
Note that this is equivalent to Qψ = 0 and P * ψ = 0. Since Q is elliptic, the Rarita-Schwinger equation is overdetermined. By elliptic regularity theory, (distributional) Rarita-Schwinger fields are smooth. We use the following notation for the dimension of the space of Rarita-  Table 1. Maximal number of linearly independent parallel spinors on manifolds without flat factor Lemma 2.7. Let M be a complete simply connected Riemannian spin manifold of dimension n. If the dimension of the space of parallel spinors is larger than N (n) then M has a flat factor, i.e. M is isometric to a Riemannian product N × R k with k ≥ 1.
Proof. If M is irreducible, i.e. does not split as an isometric product, and carries parallel spinors then M must have one of the holonomy groups from the following table (see [10]): Spin (7) 1   (7) will not occur in such a maximal product because products of K3-surfaces (holonomy SU (2)) yield a higher-dimensional space of parallel spinors. More than three G 2 -factors will not occur either because four such factors can be replaced by seven SU(2)-factors, again yielding a higher-dimensional space of parallel spinors. ThusÑ (n) is realized by products of K3-surfaces and up to three G 2 -manifolds. This shows thatÑ (n) = N (n) as given in Definition 2.6. The additional factor of 2 in the case of two or three G 2 -factors is due to the space of parallel spinors on the product of two (odd-dimensional!) G 2 -manifolds being 2-dimensional.
Remark 2.8. The proof shows that the bound in Lemma 2.7 is sharp. There exist complete simply connected Riemannian spin manifolds X and Y with holonomy SU(2) and G 2 , respectively. Then Proof. The elliptic complex (2) with α = 1 reads as By Hodge theory, see e.g. [4, p. 159, eq. (1)]. In particular, hence the first Chern class is Since the second Stiefel-Whitney class is the mod-2 reduction of c 1 , the manifold M is spin if m + r + 1 − (a 1 + . . . + a r ) is even.
2 ) −1 and thus the Pontryagin class of M is given by Let H be the complex line bundle over M whose first Chern class is given by h.
The Chern character of a vector bundle is determined by its Chern class, except for the 0-degree term which is given by the rank of the bundle. Thus For a ∈ N let H a := H a ⊕H −a , considered as a real bundle of rank 4. Its Pontryagin class is given by p(H a ) = 1 + a 2 h 2 . By (8), Since h m , [M ] = a 1 · · · a r (see e.g. [4, p. 160]) we find that cosh(a j h) .
Here coeff(h m , f (h)) denotes the coefficient of h m in the power series f (h). (This notation may seem extraneous, but will be useful below in §5.) We analyze this coefficient in the case of hypersurfaces (r = 1).
Lemma 3.1. Let m ∈ N be even. Then is a polynomial in a of degree m + 1.
Proof. The power series under consideration is of the form where f 1 (x), g 1 (x), f 2 (x), g 2 (x) ∈ Q x are even power series. Indeed, this holds with the choice Clearly, the coefficient of h m in the power series f j (h)g j (ah) is a polynomial in a of degree at most m. To see that the degree equals m, we determine the coefficient of a m h m in f j (h)g j (ah) when considered as a power series in the two variables h and a. It is given by f j (0) times the coefficient of x m in g j (x). For j = 1 we get For j = 2 we find, using g 2 (x) = sinh(3x/2)−sinh(x/2) Thus the coefficient of a m+1 in the coefficient of h m in (10) is given by and hence does not vanish.  Proof. Expand all the terms in (9) in Taylor series to obtain a quotient of two formal series in h. Inverting in C[a 1 , . . . , a r ][[h]] (the field of formal power series in h with coefficients polynomials in the vector a) shows that the coefficient of h m is a polynomial in the entries of a. This polynomial is obviously symmetric. An examination of the degrees of each of the series involved shows that the series contains only even powers of h. In other words, we only need consider the case where m is even. It is also not hard to see that the highest power of a in the coefficient of h m is m + 1. Finally, writing the expression in (9) by F r (a 1 , . . . , a r ), then we also see that F r (a 1 , . . . , a s , 1, . . . , 1) = F s (a 1 , . . . , a s ). In particular, evaluating F r on (a 1 , 1, . . . , 1), we reduce to the case considered in Lemma 3.1, and we have in fact calculated the coefficient of the highest term a m+1 1 explicitly. This proves the result.

Products
Using suitable complex submanifolds in complex projective spaces, we found compact Riemannian spin manifolds of any real dimension divisible by 4 with arbitrarily large space of Rarita-Schwinger fields. To treat the remaining dimensions we consider products of these hypersurfaces with flat tori. We start with some general considerations about spinor fields on product manifolds, see [1,Sec. 1] for details. Let X and Y be Riemannian spin manifolds of dimension n and m, respectively. We will only need the case when n is even which we assume from now on. If M := X × Y , then T M is naturally identified with π * 1 T X ⊕ π * 2 T Y and the spinor bundle ΣM with π * 1 ΣX ⊗π * 2 ΣY . Here π 1 : M → X and π 2 : M → Y are the obvious projections. If m is also even, then M . This proves that π * 1 ΣX ⊗ π * 2 ΣY ⊂ ΣM. Now let Φ be a Rarita-Schwinger field on X and ψ a parallel spinor on Y . Using a local orthonormal frame e 1 , . . . , e n , e n+1 , . . . , e n+m of M where the e i are tangent to X (i ≤ n) and the e n+i to Y , we compute Proposition 4.1. Let X and Y be Riemannian spin manifolds, with X evendimensional. Then As an immediate consequence, we can now produce manifolds of any dimension n ≥ 4 with an arbitrarily large dimensional space of Rarita-Schwinger fields: Proof. Write n = 2m + k where m is even and k ∈ {0, 1, 2, 3}. Let X be a Khler-Einstein manifold of complex dimension m as in Corollary 3.2 such that RS(X) > C. If k = 0 then we simply take M := X, while in the other cases, we take M := X ×T k where T k is a flat torus. Endow T k with the (unique) spin structure for which it has nontrivial parallel spinors. Then by Proposition 4.1, RS(M ) ≥ RS(X) > C.

Calabi-Yau manifolds
The simplest examples of Ricci flat manifolds with Rarita-Schwinger fields are provided by flat tori, again equipped with the spin structure which admits parallel spinors. For flat metrics, a section of ΣM is a Rarita-Schwinger field if and only if it is parallel. Thus for M = T n a flat torus, RS(T n ) equals the rank of ΣT n , i.e. RS(T n ) = (n − 1) · 2 [ n /2] . Complete intersections yield more interesting examples. For the sake of simplicity, we focus on hypersurfaces. Thus, using the previous notation, set r = 1 and a = m + 2; a hypersurface M m ⊂ CP m+1 of degree a has vanishing first Chern class by (7). Thus M m is spin and by the Calabi-Yau theorem carries a Ricci flat Kähler metric. According to (9) we have Since the power series is even, we restrict ourselves to the case of even m, i.e., the real dimension of M m is divisible by 4.  (1) to Proposition 4.6 in [5]. We do not know whether the bound is also sharp in higher dimensions.
Thus our simply connected compact Calabi-Yau manifolds have more linearly independent Rarita-Schwinger fields than flat tori of the same dimension. This is illustrated by the following  Table 3. Linearly independent Rarita-Schwinger fields on Calabi-Yau manifolds