1 Introduction

1.1 Parallel spinors

Let \((\overline{N},\bar{h})\) be a connected \((n+1)\)-dimensional oriented and time-oriented Lorentzian manifold with a fixed spin structure. The bundle of complex spinors will be denoted by \(\Sigma \overline{N}\). The spinor bundle carries a natural Hermitian form \(\langle \!\langle \,\cdot \,,\,\cdot \,\rangle \!\rangle \) of split signature, a compatible connection, and Clifford multiplication. In the present article we search for manifolds with a parallel spinor, i.e. a (non-trivial) parallel section of \(\Sigma \overline{N}\). Understanding Lorentzian manifolds with parallel spinors is interesting for several reasons.

The first reason is that Riemannian manifolds with parallel spinors provide interesting structures. We want to briefly sketch some of them: Parallel spinors provide a powerful technique to obtain Ricci-flat metrics on compact manifolds: All known closed Ricci-flat manifolds carry a non-vanishing parallel spinor on a finite covering. Parallel spinors are also linked to “stability”, defined in the sense that the given compact Ricci-flat metric cannot be deformed to a positive scalar curvature metric. This condition in turn is linked to dynamical stability of a Ricci-flat metric under Ricci flow: A compact Ricci-flat metric is dynamically stable under the Ricci flow if and only if it cannot be deformed to a positive scalar curvature metric ([20] and [23, Theorem 1.1]).

Infinitesimal stability was proven for metrics with a parallel spinor in [39] and local stability (for manifolds with irreducible holonomy) in [15]. Manifolds with parallel spinors provide interesting moduli spaces, see [3]. Furthermore parallel spinors help to understand the space of metrics with non-negative scalar curvature. The stability property explained above implies that every homotopy class in the space of positive scalar curvature metrics which is known to be non-trivial also remains non-trivial in the space of metrics with non-negative scalar curvature, see [35]. It is interesting and challenging to see to which extent it is possible to find Lorentzian analogues to these results.

A second reason to be interested in parallel spinors on arbitrary semi-Riemannian manifolds is that their existence implies that the holonomy is special [8, 11, 13, 22, 27, 28]. Thus the construction of Lorentzian manifolds with parallel spinors provides examples of manifolds with special holonomy.

A third reason is that parallel spinors are relevant in many fields of theoretical physics. For example parallel spinors on Lorentzian manifolds are often viewed as generators for the odd symmetries in a supersymmetric theory (see e.g. [16]). Parallel spinors are also important in several physical theories of Kaluza-Klein type, i.e. involving additional compactified dimensions. Mathematically we model them by a semi-Riemannian submersion \(\pi :T\rightarrow B\) from a high-dimensional Lorentzian manifold T to a macroscopically visible 3+1-dimensional space-time B. In this context one has to make sure that spectral and other analytic properties of Dirac operators on T are comparable to the corresponding properties on B. This requires the existence of harmonic spinors on the fibers \(M:=\pi ^{-1}(b)\), \(b\in B\). If the scalar curvature assumed to be non-negative, these harmonic spinors are parallel which in turn implies that the fibers are Ricci-flat. Varying b yields a family of metrics on M with parallel spinors. One thus obtains a map \(B\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\), where \({{\mathcal {M}\textit{od}}}_\parallel (M)\) is the (pre-)moduli space of metrics with a parallel spinor on some covering of M, which is also a central ingredient in this article.

1.2 The Cauchy problem for parallel light-like spinors

Important progress about Lorentzian manifolds with parallel spinors was recently achieved by H. Baum, T. Leistner and A. Lischewski [10, 29, 30], see also [9] for associated lecture notes. In particular, these authors showed the well-posedness of an associated Cauchy problem which we will now describe in more detail and which will be the main topic of the present article.

Let again \((\overline{N},\bar{h})\) be a time- and space-oriented Lorentzian spin manifold, and let \(\langle \!\langle \,\cdot \,,\,\cdot \,\rangle \!\rangle \) be the Hermitian product on \(\Sigma \overline{N}\) with split signature. The Clifford multiplication on \((\overline{N},\bar{h})\) will be denoted by \(\bullet \).

Note that for any spinor \(\varphi \) on a Lorentzian manifold one defines the Lorentzian Dirac current \(V_\varphi \in \Gamma (T\overline{N})\) of \(\varphi \) by requiring that

$$\begin{aligned} \bar{h}(X,V_\varphi )=-\langle \!\langle X\bullet \varphi ,\varphi \rangle \!\rangle \end{aligned}$$

holds for all \(X\in T\overline{N}\). Recall that on Lorentzian manifolds the Clifford action of vector fields on spinors is symmetric, in contrast to the Riemannian case, where it is anti-symmetric.

If \(\varphi \) is parallel, then \(V_\varphi \) is parallel as well. One can show that \(V_\varphi \) is a future oriented causal vector [9, Sec. 1.4.2, Prop. 2]. Thus \(V_\varphi \) is either time-like or light-like everywhere. In the time-like case, the Lorentzian manifold locally splits as a product \((N,h)\times (\mathbb {R},-dt^2)\), where (Nh) is a Riemannian manifold with a parallel spinor. So with respect to a suitable Cauchy hypersurface the understanding of such metrics directly relies on the corresponding results on Riemannian manifolds (Fig. 1).

In this article we are concerned with the case, that \(\varphi \) is a parallel spinor with a light-like Dirac-current \(V_\varphi \). This problem was studied in [10, 30], and [29].

Fig. 1
figure 1

Notation for manifolds, Clifford multiplications and Weingarten maps

Let N be a spacelike hypersurface in this Lorentzian manifold with induced metric h. For a future-oriented (time-like) normal vector field T with \(\bar{h}(T,T)=-1\) we define the Weingarten map \(\overline{W}:=-\bar{\nabla } T\). We use the symbol \(\overline{W}\) instead of W, as the latter one will be used for the Weingarten map of hypersurfaces M in N, which will play a central role in our article. The Hermitian product \(\langle \!\langle \,\cdot \,,\,\cdot \,\rangle \!\rangle \) and the Clifford multiplication \(\bullet \) on \(\Sigma \overline{N}\) induce a positive definite Hermitian product and a Clifford multiplication \(\star \) on \(\Sigma \overline{N}|_N\) which can be characterized by the formulas

$$\begin{aligned} \langle \varphi ,\psi \rangle =\langle \!\langle T\bullet \varphi ,\psi \rangle \!\rangle ,\qquad X\star \varphi = iT\bullet X \bullet \varphi \end{aligned}$$

for all \(x\in N\), \(\varphi ,\psi \in \Sigma _x \overline{N}\), \(X\in T_xN\). The spin structure on \(\overline{N}\) induces a spin structure on the spacelike hypersurface N. Let \(\Sigma N\) be the associated spinor bundle of N. With standard tools about Clifford modules, one sees that there is a bundle monomorphism \(\iota :\Sigma N\rightarrow \Sigma \overline{N}|_N\) over the identity of N such that the Clifford multiplication and the Hermitian product on \(\Sigma N\) are mapped to \(\star \) and \(\langle \,\cdot \,,\,\cdot \,\rangle \) on \(\Sigma \overline{N}|_N\). The bundle monomorphism \(\iota \) is a bundle isomorphism if and only if n is even. However note that \(\iota \) is not parallel, i.e. the connection is not preserved under \(\iota \). More precisely \(\nabla \iota \) is a linear pointwise expression in \(\overline{W}\). From now on we identify \(\Sigma N\) with its image in \(\Sigma \overline{N}|_N\) under \(\iota \), taking the connection from \(\Sigma N\). As already mentioned above we will also consider hypersurfaces M of N and the spinor bundle \(\Sigma M\) of M. However the relation between spinors on M and on N is a bit easier than between spinors on \(\overline{N}\) and on N. One can work with an embedding \(\Sigma M\) into \(\Sigma N|_M\) which preserves the scalar product \(\langle \,\cdot \,,\,\cdot \,\rangle \). We can even choose the embedding such that the Clifford multiplications coincide, however in the literature another embedding is often used which does no longer preserve the Clifford multiplication. In our application in Sect. 5 it depends on the parity of n which embedding is more convenient for us, see also Sect. 2.3 and Appendix B for further information about hypersurfaces and spinors. Thus we want to use \(\cdot \) for the Clifford multiplication on M in contrast to \(\star \) which is used for the Clifford multiplication on N.

For any spinor \(\varphi \in \Gamma (\Sigma N)\) we associate—see e.g. [29, (1.7)]—its Riemannian Dirac current \(U_\varphi \in \Gamma (TN)\) by requiring

$$\begin{aligned} h(U_\varphi ,X)=-i \langle X\star \varphi ,\varphi \rangle ,\qquad \forall X\in TN. \end{aligned}$$

By Lemma 20 in Appendix A we see, that the spinor \(\varphi \) satisfies \(h(U_\varphi ,U_\varphi ) =\Vert \varphi \Vert ^4\) if and only if we have for any \(p\in N\)

$$\begin{aligned} i\varphi (p)\in \{V\star \varphi \mid V\in T_pN\}. \end{aligned}$$

Now, if we assume that \((\overline{N},\bar{h})\) carries a non-vanishing parallel spinor, then this spinor induces a spinor \(\varphi \) on (Nh) such that

$$\begin{aligned} \nabla ^N_X\varphi&= \frac{i}{2} \overline{W}(X)\star \varphi ,\qquad \forall X\in TN, \end{aligned}$$
$$\begin{aligned} U_\varphi \star \varphi&= i u_\varphi \varphi , \end{aligned}$$

where \(U_\varphi \in \Gamma (TN)\) is defined as above and where

$$\begin{aligned} u_\varphi :=\sqrt{h(U_\varphi ,U_\varphi )}, \end{aligned}$$

see [10, (4) and following] or [29, (1.6) and (1.8)].

Note that the constraint Eqs. (1)–(5) are not independent from each others. We comment on this in Appendix A.

Conversely, if (Nh) is given as an abstract Riemannian spin manifold, and if \(\overline{W}\in \Gamma ({\mathrm {End}}(TN))\), and \(\varphi \in \Gamma (\Sigma N)\) satisfy (2), (3) and (4) with \(U_\varphi \) and \(u_\varphi \) defined by (1) and (5), then there is a Lorentzian manifold \((\overline{N},\bar{h})\) with N as a Cauchy hypersurface and a parallel spinor, such that h is the induced Riemannian metric, \(\overline{W}\) the Weingarten map and \(\varphi \) is induced by a parallel spinor on \((\overline{N},\bar{h})\), see [29, Consequence of Theorems 2 and 3]. This was proven by solving the associated wave equations by using the technique of symmetric hyperbolic systems. A simpler approach, going back to a remark by P. Chrusciel was later given in [36, Chap. 4].

The question arises on how to solve these constraint equations. In the present article we will describe a new method to obtain solutions of these constraint equations. We will see how any smooth curve in the (pre-)moduli space of closed \(m=(n-1)\)-dimensional Riemannian manifolds with a parallel spinor together with a scaling function yields a solution to the constraint equations, see our Main Construction 15 in Sect. 6. And thus the well-definedness of the Cauchy problem implies the existence of an associated \((n+1)\)-dimensional Lorentzian with a parallel spinor. Such a relation between families of metrics with special holonomy and solutions of the constraint equations was already conjectured by Leistner and Lischewski, see [29]. We essentially show that the conditions in [29, Table 1] is satisfied if and only if the divergence condition (31) in our Appendix D is satisfied.

We also derive versions of this construction to obtain initial data on a compact Cauchy hypersurface (without boundary). A first idea is to use a smooth closed curve in the (pre-)moduli space of closed m-dimensional Riemannian manifolds with a parallel spinor together with a scaling function. However this will in general not lead to a solution of the original constraint equation, but to a twisted version thereof, see Main Construction 17. This will lead to a Lorentzian manifold with a parallel twisted spinor. Here the twist bundle is always a complex line bundle with a flat connection.

In special cases—more precisely assuming the “fitting condition” introduced in Sect. 6 we however obtain solutions of the constraint equations in the original (i.e. untwisted) sense, see Main Construction 16.

We consider it as remarkable that compared to other classical diffeomorphism invariant Cauchy problems in Lorentzian geometry, e.g. the Cauchy problem for Ricci-flat metrics, we get a large quantity of solutions to the constraint equations. Furthermore it is amazing that the solutions in our situation correspond to curves in the moduli space, while the set of solutions of the constraints in classical problems have no similar description. An important input for our article was the smooth manifold structure of the premoduli space \({{\mathcal {M}\textit{od}}}_\parallel (M)\) and the fact that the BBGM connection preserves parallel spinors along divergence free Ricci-flat families of metrics. Ricci-flat deformations of a metric thus preserve the dimension of the space of parallel spinors. The infinitesimal version of this should be seen as some kind of Hodge theory: infinitesimal Ricci-flat deformations can be viewed as elements of \(\ker (D^{T^*M})^2\) while infinitesimal deformations with parallel spinors can be viewed as elements of \(\ker D^{T^*M}\)—and on compact manifolds Hodge theory tells us that \(\ker (D^{T^*M})^2=\ker D^{T^*M}\). Let us also compare the results of this paper to the tightly related recent preprint [2]. While the current paper constructs initial data for Lorentzian manifolds with a parallel spinor, one of the major themes in [2] is to get topological obstructions for a closed spin manifold to be a spacelike hypersurface in a Lorentzian manifold with a parallel spinor. We also mentioned that the special case of \((3+1)\)-dimensional Lorentzian manifolds (i.e. initial data on 3-dimensional manifolds) was studied in the recent preprint [32].

The structure of the article is as follows. In Sect. 2 we fix some conventions, recall and extend known facts about Clifford modules and about spinors on hypersurfaces. We also explain how to differentiate a spinorial expression such as the Levi-Civita derivative \(\nabla ^g \varphi \) of a spinor \(\varphi \) with respect to the Riemannian metric g. The known results for defining this differential along a path of metrics is recalled in Sect. 2.4, this includes the BBGM parallel transport and the associated connection which arises from the universal spinor bundle construction; the concrete calculations are a central tool of the article and are carried out in Sect. 3. In Sect. 4 we show that for divergence free Ricci-flat deformations, the BBGMconnection preserves parallel spinors, a result which we consider an interesting outcome of the article independently of the application to the constraint equations, mainly discussed in the article. In Sect. 5 we use this construction to obtain solutions of the constraint Eqs. (1)–(5). In Sect. 6 we establish the connection to curves in (pre-)moduli spaces and we also discuss on how to obtain solutions on compact Cauchy hypersurfaces. The article ends with several appendices where we provide some details about facts which are already well-known, but where adequate literature was not available. We hope that these appendices help to make the article sufficiently self-contained.

2 Preliminaries

2.1 Conventions

All Hermitian scalar products in this article are complex linear in the first entry and complex anti-linear in the second one. Let E be a Riemannian vector bundle over a compact Riemannian manifold (Mg) equipped with a metric connection \(\nabla \) and \(k\in \mathbb {N}_0\). Then the Sobolev norm \(H^k\) of a section \(s\in \Gamma (E)\) is

$$\begin{aligned} \left\| s\right\| _{H^k}=\left( \sum _{l=0}^k\int _M |\nabla ^ls|^2dV\right) ^{1/2}. \end{aligned}$$

where dV is the volume element of g. As usual, we denote \(L^2=H^0\). We will write \(H^k(g)\) if we wish to emphasise the dependence of the norm on g.

We use the symbol \(\bigodot ^2\) for the symmetrized tensor product, i.e. for a finite-dimensional real vector space V the space of symmetric bilinear forms \(V\times V\rightarrow \mathbb {R}\) is denoted by \(\bigodot ^2V^*\). Let \(S V^*\subset \bigodot ^2V^*\) be the subset of positive definite symmetric forms on V. Applying this fiberwise to the tangent bundle TM we obtain the vector bundle \(\bigodot ^2 T^*M\) and the bundle \(ST^*M=: S M\).

Definition 1

On a Riemannian manifold M the Einstein operator is the elliptic differential operator \(\Delta _E:\Gamma (\bigodot ^2 T^*M) \rightarrow \Gamma (\bigodot ^2 T^*M)\), given by

$$\begin{aligned} \Delta _Eh=\nabla ^*\nabla h-2\mathring{R}h,\qquad \mathring{R}h(X,Y):=h(R_{e_i,X}Y,e_i) \end{aligned}$$

where \(\left\{ e_1,\ldots e_n\right\} \) is a local orthonormal frame. Here, the curvature tensor is defined with the sign convention such that \(R_{X,Y}Z=\nabla ^2_{X,Y}Z-\nabla ^2_{Y,X}Z\).

The Einstein operator is linked to the deformation theory of Ricci-flat metrics as follows: Let g be a Ricci-flat metric and \(g_s\) a smooth family of Ricci-flat metrics through \(g_0=g\), then

$$\begin{aligned} \frac{d}{ds}\Big |_{s=0}g_s=h+\mathcal {L}_Xg \end{aligned}$$

where X is a vector field and \(h\in \ker ({{\mathrm {div}}_g})\cap \ker (\Delta _E)\), i.e. the essential part of a Ricci-flat deformation is an element in \(\ker (\Delta _E)\). In particular, if \(g_s\) is a family of metrics with a parallel spinor, the essential part of its s-derivative is contained in \(\ker (\Delta _E)\). For more details on the deformation theory of Ricci-flat metrics, see [12, Chapter 12 D].

2.2 Some facts about Clifford modules

In this subsection we briefly summarize some facts about representations of Clifford algebras. The interested reader might consult the first Chapter of the book by Lawson and Michelsohn [26] for further details.

Let \((E_1,\ldots ,E_n)\) be the canonical basis of \(\mathbb {R}^n\). The complexified Clifford algebra for \(\mathbb {R}^n\) with the Euclidean scalar product will be denoted by \({\mathbb {C}\mathrm {l}}_n\). In this section all Clifford multiplications are written as \(\cdot \), independently on n; we use the convention that \(\cdot \) does not depend on n. In the following section we will then explain that the associated bundle construction then turns this Clifford multiplication both into the Clifford multiplication \(\cdot \) on M and the Clifford multiplication \(\star \) on N.

We define the complex volume element \(\omega ^\mathbb {C}_n\) as

$$\begin{aligned} \omega ^\mathbb {C}_n:= \epsilon _n E_1\cdot E_2 \cdot \ldots \cdot E_n\in {\mathbb {C}\mathrm {l}}_n, \end{aligned}$$

where \(\epsilon _{4k}=\epsilon _{4k+3}=1\), \(\epsilon _{4k+1}=-i\) and \(\epsilon _{4k+2}=i\). The choices imply \((\omega ^\mathbb {C}_n)^2=1\). The choice of sign for \(\epsilon _n\) in the literature varies between different sources, our particular and unconventional choice yields an easy formulation of Lemma 2.

For n even, there is a unique irreducible complex represention of \({\mathbb {C}\mathrm {l}}_n\). Here and in the following “unique” will always “unique up to isomorphism of representation”. This unique representation will be denoted by \(\Sigma _n\). It comes with a grading \(\Sigma _n=\Sigma _n^+ \oplus \Sigma _n^-\) given by the eigenvalues \(\pm 1\) of \(\omega ^\mathbb {C}_n\).

For n odd there are two irreducible representations, and as \(\omega ^\mathbb {C}_n\) is central, Schur’s Lemma implies that \(\omega ^\mathbb {C}_n\) acts either as the identity or minus the identity which allows us to distinguish the two representations. For n odd, we will assume that \(\Sigma _n\) is the irreducible representation on which \(\omega ^\mathbb {C}_n\) acts as the identity. The other representation is called \(\Sigma ^{\#}_n\). In the following \(\Sigma _{n}^{(\#)}\) denotes either \(\Sigma _n\) or \(\Sigma ^{\#}_n\) for odd n and \(\Sigma _n\) for even n.

The standard inclusion \(\mathbb {R}^{n-1}\rightarrow \mathbb {R}^n\), \(x\mapsto (x,0)\) induces an inclusion \({\mathbb {C}\mathrm {l}}_{n-1}\rightarrow {\mathbb {C}\mathrm {l}}_n\). This turns \(\Sigma _n\) into a \({\mathbb {C}\mathrm {l}}_{n-1}\)-module. In particular, the Clifford action of \(X\in \mathbb {R}^{n-1}\) is the same if X is viewed as an element of \({\mathbb {C}\mathrm {l}}_{n-1}\) or \({\mathbb {C}\mathrm {l}}_n\).Footnote 1

Thus, if n is odd, we can choose (and fix from now on) isometric isomorphisms of \({\mathbb {C}\mathrm {l}}_{n-1}\)-modules \(J_n:\Sigma _{n-1}\rightarrow \Sigma _n\) and \(J_n^\#:\Sigma _{n-1}\rightarrow \Sigma ^{\#}_n\). If n is even, then we can choose (and fix from now on) an isometric isomorphism of \({\mathbb {C}\mathrm {l}}_{n-1}\)-modules \(I_n:\Sigma _{n-1}\oplus \Sigma ^{\#}_{n-1}\rightarrow \Sigma _n\).

Lemma 2

For n odd, \(J_n:\Sigma _{n-1}\rightarrow \Sigma _n\) and \(J_n^{\#}:\Sigma _{n-1}\rightarrow \Sigma ^{\#}_n\) are \({\mathbb {C}\mathrm {l}}_{n-1}\)-linear isomorphisms of complex \({\mathrm {Spin}}(n-1)\)-representations and they satisfy

$$\begin{aligned} e_n\cdot J_n(\varphi )= & {} i J_n(\omega _{n-1}^{\mathbb {C}}\varphi ), \end{aligned}$$
$$\begin{aligned} e_n\cdot J_n^{\#}(\varphi )= & {} - i J_n^{\#}(\omega _{n-1}^{\mathbb {C}}\varphi ). \end{aligned}$$

for all \(\varphi \in \Sigma _{n-1}\). For n even there are isometric monomorphisms of complex \({\mathrm {Spin}}(n-1)\)-representations \(J_n^{\pm }:\Sigma _{n-1}\rightarrow \Sigma _n\) and \(J_n^{\#,\pm }:\Sigma ^{\#}_{n-1}\rightarrow \Sigma _n\), satisying

$$\begin{aligned} e_n\cdot J_n^{\pm }(\varphi )= & {} \pm i J_n^{\pm }(\varphi ), \end{aligned}$$
$$\begin{aligned} e_n\cdot J_n^{\#,\pm }(\varphi )= & {} \pm i J_n^{\#,\pm }(\varphi ). \end{aligned}$$

for all \(\varphi \in \Sigma _{n-1}^{(\#)}\) and for any \(\pm \in \{+,-\}\).


If n is odd, then we see that \(\epsilon _n=-i\epsilon _{n-1}\), thus \(\omega _n^\mathbb {C}=-i\omega _{n-1}^\mathbb {C}\cdot e_n=-ie_n\cdot \omega _{n-1}^\mathbb {C}\). This implies \(e_n\cdot \omega _n^\mathbb {C}=i\omega _{n-1}^\mathbb {C}\). Hence, we obtain (6):

$$\begin{aligned} e_n\cdot J_n(\varphi )= e_n\cdot \omega _n^\mathbb {C}\cdot J_n(\varphi )=i\omega _{n-1}^\mathbb {C}J_n(\varphi )=J_n(i\omega _{n-1}^\mathbb {C}\varphi ). \end{aligned}$$

The verification of (7) is analogous. As these morphisms are \({\mathbb {C}\mathrm {l}}_{n-1}\)-linear, and as \({\mathrm {Spin}}(n-1)\) is contained in \({\mathbb {C}\mathrm {l}}_{n-1}\), the morphisms are also \({\mathrm {Spin}}(n-1)\)-equivariant.

If n is even, then we define

$$\begin{aligned} J_n^{(\#),\pm }(\varphi ):= \frac{1}{\sqrt{2}} \left( I_n(\varphi )\mp i e_n\cdot I_n(\varphi )\right) . \end{aligned}$$

Equations (8) and (9) are obvious. The morphisms are no longer \({\mathbb {C}\mathrm {l}}_{n-1}\)-linear. However \(e_n\) commutes with \(({\mathbb {C}\mathrm {l}}_{n-1})_0\) which is defined as the even part of \({\mathbb {C}\mathrm {l}}_{n-1}\), i.e. the sub-algebra generated by elements of the form \(X\cdot Y\) with \(X,Y\in \mathbb {R}^{n-1}\). As a consequence the morphisms are \(({\mathbb {C}\mathrm {l}}_{n-1})_0\)-linear. As \({\mathrm {Spin}}(n-1)\) is contained in \(({\mathbb {C}\mathrm {l}}_{n-1})_0\), the morphisms are \({\mathrm {Spin}}(n-1)\)-equivariant.

To show that the morphisms \(J_n^{(\#),\pm }\) are isometric (and thus also injective) it suffices to show \(I_n(\varphi ) \perp i e_n\cdot I_n(\varphi )\), where \(\varphi \in \Sigma _{n-1}\) or \(\varphi \in \Sigma ^{\#}_{n-1}\). We argue only for \(\varphi \in \Sigma _{n-1}\), as the other case is completely analogous. As \(\omega _{n-1}^\mathbb {C}\) and \(e_n\) anticommute, \(i e_n\cdot I_n(\varphi )\) is an eigenvector of \(\omega _{n-1}^\mathbb {C}\) to the eigenvalue \(-1\), while \(I_n(\varphi )\) is an eigenvector to the eigenvalue 1. As \(\omega _{n-1}^\mathbb {C}\) is self-adjoint, orthogonality follows. \(\square \)

Remark 3

Let n be even. Clifford multiplication with \(e_n\) anticommutes with all odd elements in \({\mathbb {C}\mathrm {l}}_{n-1}\), in particular with \(\omega _{n-1}^{\mathbb {C}}\) and vectors \(\mathbb {R}^{n-1}\subset {\mathbb {C}\mathrm {l}}_{n-1}\). Thus, the map \(\varphi \mapsto I_n^{-1}(e_n\cdot I_n(\varphi ))\) restricts to a vector space isomorphism \(A:\Sigma _{n-1}\rightarrow \Sigma ^{\#}_{n-1}\) which anticommutes with Clifford multiplication by vectors in \(\mathbb {R}^{n-1}\).

2.3 Spinors on hypersurfaces

In this section we want to describe how one can restrict a spinor on an n-dimensional Riemannian spin manifold (Nh) to a spinor on an oriented hypersurface M carrying the induced metric g. As, this restriction is local, we can assume—by restricting to a tubular neighborhood of M and using Fermi coordinates, i.e. normal coordinates in normal directions—that \(N= M\times (a,b)\) and \(h=g_s+ds^2\) where \(s\in (a,b)\).

Conversely, given a family of metrics \(g_s\) and spinors \(\varphi _s\in \Gamma (\Sigma ^{g_s}M)\), \(s\in (a,b)\), we want to obtain a spinor on \(N:= M\times (a,b)\), \(h=g_s+ds^2\).

Note that this description does not include the passage from a Lorentzian manifold to a spacelike hypersurface and vice versa, described in the introduction, which plays an important role in the work of Baum, Leistner and Lischewski. For this Lorentzian version analogous techniques are well presented in the lecture notes [9, Sec. 1.5].

For the Riemannian case, which is the goal of this section, a standard reference is [6]. However for the purpose of our article it seems to be more efficient to choose some other convention, following e.g. [1, Sec. 5.3]. To have a fluently readable summary, we do not include detailed proofs in this section. For self-containedness we include them in Appendix B.

In the following let \(T_MN\) be the vertical bundle of the projection \(N\rightarrow (a,b)\), i.e. \(T_MN=\coprod _{s\in (a,b)}T(M\times \{s\})\), viewed as a bundle over N.

For each \(s\in I:=(a,b)\) let \(P_{{\mathrm {SO}}}(M,g_s)\) be the \({\mathrm {SO}}(n-1)\)-principal bundle of positively oriented orthornormal frames on \((M,g_s)\), and let \(P_{{\mathrm {SO}}}(N)\) be defined as the corresponding \({\mathrm {SO}}(n)\)-bundle over (Nh). We note \(\nu :=\partial /\partial s\). The union of the bundles \(P_{{\mathrm {SO}}}(M,g_s)\) is an \({\mathrm {SO}}(n-1)\)-principal bundle over \(N=M\times I\), which we will denoted by \(P_{{\mathrm {SO}},M}(N)\rightarrow N\) (whose topology and bundle structure is induced by the \({\mathrm {SO}}(n-1)\)-reduction of the \({\mathrm {GL}}(n-1)\)-principal bundle of all frames over N containing \(\nu \)). We get a map \(P_{{\mathrm {SO}},M}(N)\rightarrow P_{{\mathrm {SO}}}(N)\), mapping \((e_1,\ldots ,e_{n-1})\) to \((e_1,\ldots ,e_{n-1},\nu )\) which yields an isomorphism \(P_{{\mathrm {SO}}}(N)\cong P_{{\mathrm {SO}},M}(N)\times _{{\mathrm {SO}}(n-1)}{\mathrm {SO}}(n)\). Any topological spin structure on M yields a \({\mathrm {Spin}}(n-1)\)-principal bundle \(P_{{\mathrm {Spin}},M}(N)\) with a \({\mathrm {Spin}}(n-1)\rightarrow {\mathrm {SO}}(n-1)\)-equivariant map \(P_{{\mathrm {Spin}},M}(N)\rightarrow P_{{\mathrm {SO}},M}(N)\) in the usual way. This map induces

$$\begin{aligned} P_{{\mathrm {Spin}}}(N)\!:=\! P_{{\mathrm {Spin}},M}(N)\times _{{\mathrm {Spin}}(n-1)}{\mathrm {Spin}}(n)\rightarrow P_{{\mathrm {SO}}}(N)\cong P_{{\mathrm {SO}},M}(N)\times _{{\mathrm {SO}}(n-1)}{\mathrm {SO}}(n) \end{aligned}$$

which is a spin structure on \(N=M\times I\). In particular this induces a bijection from the set of spin structures on M (up to isomorphism) to spin structures on N (up to isomorphism).

The complex volume elements \(\omega _{n-1}^\mathbb {C}\in {\mathbb {C}\mathrm {l}}_{n-1}\) and \(\omega _n^\mathbb {C}\in {\mathbb {C}\mathrm {l}}_n\) then provide associated complex volume elements

$$\begin{aligned} \omega _M^\mathbb {C}=\epsilon _{n-1} e_1\cdot e_2 \cdot \ldots \cdot e_{n-1}\in {\mathbb {C}\mathrm {l}}(TM)\text { and }\omega _N^\mathbb {C}=\epsilon _n e_1\star e_2 \star \ldots \star e_n\in {\mathbb {C}\mathrm {l}}(TN), \end{aligned}$$

where \((e_1,e_2,\ldots )\) is any positively oriented basis in TM resp. TN.

Using the associated bundle construction, we obtain for n odd:

$$\begin{aligned} \Sigma _M N&:= P_{{\mathrm {Spin}},M}(N)\times _{{\mathrm {Spin}}(n-1)}\Sigma _{n-1}=\coprod _{s\in (a,b)} \Sigma ^{g_s} M,\\ \Sigma N&:= P_{{\mathrm {Spin}}}(N)\times _{{\mathrm {Spin}}(n)}\Sigma _{n},\\ \Sigma ^{\#}N&:= P_{{\mathrm {Spin}}}(N)\times _{{\mathrm {Spin}}(n)}\Sigma ^{\#}_{n}. \end{aligned}$$

The grading \(\Sigma _{n-1}=\Sigma _{n-1}^+ \oplus \Sigma _{n-1}^-\) yields a grading \(\Sigma _M N=(\Sigma _M N)^+\oplus (\Sigma _M N)^-\) which is the grading by eigenvalues \(\pm 1\) of \(\omega _M^\mathbb {C}\).

For n even we have

$$\begin{aligned} \Sigma _M N&:= P_{{\mathrm {Spin}},M}(N)\times _{{\mathrm {Spin}}(n-1)}\Sigma _{n-1}=\coprod _{s\in (a,b)} \Sigma ^{g_s} M,\\ \Sigma ^{\#}_M N&:= P_{{\mathrm {Spin}},M}(N)\times _{{\mathrm {Spin}}(n-1)}\Sigma ^{\#}_{n-1}=\coprod _{s\in (a,b)} \Sigma ^{\#,g_s} M,\\ \Sigma N&:= P_{{\mathrm {Spin}}}(N)\times _{{\mathrm {Spin}}(n)}\Sigma _{n}, \end{aligned}$$

and the grading \(\Sigma _n=\Sigma _n^+ \oplus \Sigma _n^-\) yields a grading \(\Sigma N=(\Sigma N)^+ \oplus (\Sigma N)^-\), given by eigenvalues \(\pm 1\) of \(\omega _N^\mathbb {C}\).

The Clifford multiplications \(\mathbb {R}^{n-1}\otimes \Sigma _{n-1}^{(\#)}\rightarrow \Sigma _{n-1}^{(\#)}\), \(\mathbb {R}^n\otimes \Sigma _n^{(\#)}\rightarrow \Sigma _n^{(\#)}\), induce Clifford multiplications \(\cdot :T_MN\otimes \Sigma _M^{(\#)}N\rightarrow \Sigma _M^{(\#)}N\), \(\star :TN\otimes \Sigma ^{(\#)} N\rightarrow \Sigma ^{(\#)} N\). As indicated before we use the symbols “\(\cdot \)” and “\(\star \)” to distinguish properly between the Clifford multiplication of M and the one of N.

Furthermore the maps \(J_n\), \(J_n^\#\), \(J_n^{\pm }\), and \(J_n^{\#,\pm }\) from Lemma 2 induce vector bundle maps over \({\mathrm {id}}:N\rightarrow N\) which are isometric injective \(\mathbb {C}\)-linear maps in each fiber. For n odd, we obtain fiberwise isomorphisms

$$\begin{aligned}&J:\Sigma _MN\rightarrow \Sigma N\text { and }J^{\#}:\Sigma _MN\rightarrow \Sigma ^{\#}N. \end{aligned}$$

These maps commute with Clifford multiplication by vectors tangent to M. They satisfy

$$\begin{aligned} \nu \star J(\varphi )= & {} i J_n(\omega _M^{\mathbb {C}}\cdot \varphi ), \end{aligned}$$
$$\begin{aligned} \nu \star J^{\#}(\varphi )= & {} - i J^{\#}(\omega _M^{\mathbb {C}}\cdot \varphi ). \end{aligned}$$

for all \(\varphi \in \Sigma _MN\).

On the other hand we get for n even

$$\begin{aligned}&J^\pm :\Sigma _M N\rightarrow \Sigma N\text { and }J^{\#,\pm }:\Sigma ^{\#}_MN\rightarrow \Sigma N. \end{aligned}$$

These maps do not commute with Clifford multiplication by vectors tangent to M, and they are not surjective in any fiber. However, they satisfy

$$\begin{aligned} \nu \star J^{\pm }(\varphi )= & {} \pm i J^{\pm }(\varphi ), \end{aligned}$$
$$\begin{aligned} \nu \star J^{\#,\pm }(\varphi )= & {} \pm i J^{\#,\pm }(\varphi ). \end{aligned}$$

for all \(\varphi \in \Sigma _MN\) and for any \(\pm \in \{+,-\}\).

The bundle maps defined above do not preserve the (partially defined) Levi-Civita connections on the bundles. They modify the connection by terms depending on the second fundamental form of \(M\times \{s\}\) in N. This is made precise in the following lemma.

Lemma 4

Let \(J^{(\#),(\pm )}\) be one of the bundle maps defined above. For \(X\in \Gamma (T_MN)\), and \(\varphi \in \Gamma (\Sigma _MN)\) resp. \(\Gamma (\Sigma ^{\#}_MN)\) we have

$$\begin{aligned} \nabla ^N_XJ^{(\#),(\pm )}(\varphi ) = J^{(\#),(\pm )}(\nabla ^{M,g_s}_X\varphi ) + \frac{1}{2} W(X)\star \nu \star J^{(\#),(\pm )}(\varphi ), \end{aligned}$$

where the Weingarten map W is defined through \(g(W(X),Y)\nu =\nabla _X^NY-\nabla _X^M Y\).

A proof will be given in Appendix B.

In combination with Eqs. (10), (11), (12), and (13) we will see later in this article that families of \((M,g_s)\)-parallel spinors will lead to solutions of the generalized imaginary Killing spinor Eq. (3) which is one of the constraint equations.

2.4 BBGM connection

Let M be a compact spin manifold. We denote the space of all Riemannian metrics on M by \(\mathcal {M}\). For every metric \(g\in \mathcal {M}\) we define

$$\begin{aligned} \mathcal {F}_g = \Gamma (\Sigma ^gM), \end{aligned}$$

and the disjoint union

$$\begin{aligned} \mathcal {F}:=\coprod _{g\in \mathcal {M}} \mathcal {F}_g. \end{aligned}$$

One can equip \(\mathcal {F}\) and \(\mathcal {M}\) naturally with the structure of a Fréchet bundle \(\pi ^\mathcal {F}:\mathcal {F}\rightarrow \mathcal {M}\). This bundle structure is needed to define a connection on this bundle. The oldest references that we used are by Bourguignon and Gauduchon [14] resp. by Bär, Gauduchon and Moroianu [6], this is why we choose the abbreviation BBGM for the four names. However we were told that there was also work by Bismut. The concepts were later properly formalized under the name ’universal spinor bundle’ in [4] and [31], where a finite-dimensional fiber bundle with a partial connection is constructed whose sections correspond to the elements of \(\mathcal {F}\) such that the parallel transport corresponds to the BBGM parallel transport. The connection is given in terms of horizontal spaces \(\mathcal {H}_{(g,\varphi )}\), i.e. vector spaces satisfying

$$\begin{aligned} T_{(g,\varphi )}\mathcal {F}= \mathcal {H}_{(g,\varphi )}\oplus \Gamma (\Sigma ^g M) \end{aligned}$$

in the sense of topological vector spaces such that

$$\begin{aligned} d\pi ^\mathcal {F}|_{(g,\varphi )}:T_{(g,\varphi )}\mathcal {F}\rightarrow T_g\mathcal {M}=\Gamma (\bigodot \nolimits ^2 T^*M) \end{aligned}$$

restricts to an isomorphism \(I_{(g,\varphi )}:\mathcal {H}_{(g,\varphi )}\rightarrow \Gamma (\bigodot \nolimits ^2 T^*M)\). In other words we obtain an injective map of Fréchet spaces \(L_{(g,\varphi )}:\Gamma (\bigodot \nolimits ^2 T^*M)\rightarrow T_{(g,\varphi )}\mathcal {F}\) by postcomposing \((I_{(g,\varphi )})^{-1}\) with the inclusion of \(\mathcal {H}_{(g,\varphi )}\subset T_{(g,\varphi )}\mathcal {F}\).

The space \(\mathcal {H}_{(g,\varphi )}\) will smoothly depend on \((g,\varphi )\) and will be compatible with the vector bundle structure of \(\mathcal {F}\). We do not require more knowledge about Fréchet manifolds in our article, so we do not introduce this Fréchet structure in more detail.

To describe the horizontal space precisely we give the maps \(L_{(g,\varphi )}\): We assume that \(h\in T_{g}\mathcal {M}=\Gamma (\bigodot \nolimits ^2 T^*M)\). Let \(g_t\) be a smooth path of metrics such that \(g_0=g\) and \(\frac{d}{dt}|_{t=0} g_t=h\). We consider the cylinder \(Z=M\times [0,1]\) with the metric \(g_t+dt^2\). Then, similar to [6] we can extend the spinor \(\varphi _0:=\varphi \), defined on \(M\times \{0\}\), uniquely to a spinor \(\varphi _\bullet \) on \(M\times [0,1]\) satisfying \(\nabla _{\partial _t}\varphi _\bullet =0\) where \(\nabla \) denotes the Levi-Civita connection on the cylinder. Note that \(\varphi _\bullet \) may be interpreted as a family, parametrized over [0, 1], of spinor fields \(\varphi _t\) with respect to the family of metrics \(g_t\). The restriction of \(\varphi _\bullet \) to \(M\times \{t\}\), denoted by \(\varphi _t\) is a spinor for the metric \(g_t\), and \(t\mapsto \varphi _t\) is a smooth path in \(\mathcal {F}\). We now define

$$\begin{aligned} L_{(g,\varphi )}(h)=\frac{d}{dt}\Big |_{t=0}\varphi _t. \end{aligned}$$

Lemma 5

The map \(L_{(g,\varphi )}\) is well-defined, linear, and smooth in \((g,\varphi )\).

The lemma follows from the construction of the universal spinor bundle given in [31]. The strategy in that paper is as follows: Let \(\pi ^S:SM\rightarrow M\) be the bundle of positive definite symmetric bilinear forms on TM. A complex vector bundle \(\pi ^\mathrm {univ}:\Sigma M \rightarrow SM\) is constructed, called the universal spinor bundle, which carries a scalar product, a Clifford multiplication with vectors in TM and partial connection on \(\Sigma M\) with respect to \(\pi ^S:SM\rightarrow M\). The Clifford multiplication is given by a bilinear map \({\mathrm {cl}}_g:T_{\pi ^S(g)}M\times \Sigma _g M\rightarrow \Sigma _g M\) for every \(g\in SM\). Note that g is a scalar product on one single tangent space, namely on \(T_{\pi ^S(g)}M\), and thus \(\Sigma _g\) is thus a fiber of the vector bundle \(\pi ^\mathrm {univ}:\Sigma M\rightarrow SM\). This Clifford multiplication shall satisfy the Clifford relations and shall depend smoothly on g. By “partial connection” we mean that for any section \(\varphi \) of the bundle \(\pi ^\mathrm {univ}:\Sigma M \rightarrow SM\), the covariant derivative \(\nabla _X\varphi \) is defined for some \(X\in T(SM)\) if and only \(d\pi ^S(X)=0\) (i.e., X is vertical for \(\pi ^S\)). This partial connection comes from the vertical connection defined in [31, Definition 2.10], and allows to define a map \(L_{(g,\varphi )}\) as above. In particular this shows that \(\frac{d}{dt}|_{t=0}\varphi _t\) does not depend on how we choose \(g_t\), but only its derivative a \(t=0\).

3 Variation of the Parallel Spinor Equation

Let us first recall a result from [4]: For \((g,\varphi ) \in \mathcal {F}\) let \(\hat{\kappa }_{g,\varphi }: \mathcal {H}_{(g,\varphi )} \rightarrow \Gamma ( \Sigma ^gM \otimes T^*M)\) be the map defined by

$$\begin{aligned} \hat{\kappa }_{g,\varphi }(h):= & {} \tfrac{1}{4} \sum _{i \ne j} (\nabla ^g_{e_i} h)(\,\cdot \,\,,e_j) e_i \cdot e_j \cdot \varphi . \end{aligned}$$

In [4, Sec. 4.3] a similar map was defined, namely \(\kappa _{g,\varphi }: T_{(g,\varphi )} \mathcal {F}=\mathcal {H}_{(g,\varphi )}\oplus \Gamma (\Sigma ^g M)\rightarrow \Gamma ( \Sigma ^gM \otimes T^*M)\) which is related to \(\hat{\kappa }_{g,\varphi }\) by the formula \(\kappa _{g,\varphi }(h,\psi )=\hat{\kappa }_{g,\varphi }(h)+\nabla _\cdot ^g\psi \). It was shown in [4, Lemma 4.12] that this is related to the differential of the map \(K_X:\mathcal {F}\rightarrow \mathcal {F}\), \((g,\varphi )\mapsto (g,\nabla ^g_X\varphi )\) as follows:

$$\begin{aligned} (d_{(g,\varphi )} K_X)(h, \psi ) = (h, \kappa _{g,\varphi }(h,\psi )(X)) = (h, \hat{\kappa }_{g,\varphi }(h)(X)+ \nabla ^g_X \psi ). \end{aligned}$$

We define the Wang map \(\mathcal {W}_{g,\varphi }\) as the composition

$$\begin{aligned} \bigodot \nolimits ^2 T^*M\hookrightarrow \bigotimes \nolimits ^2 T^*M\xrightarrow {{\mathrm {cl}}^g(\,\cdot \,,\varphi )\otimes {\mathrm {id}}} \Sigma ^gM\otimes T^*M \end{aligned}$$

so that

$$\begin{aligned} \mathcal {W}_{g,\varphi }(h)=(X\mapsto \sum _{j=1}^n h(X,e_j)e_j\cdot \varphi ). \end{aligned}$$

It is easy to see (c. f. [15, Lemma 2.3, 1.]) that

$$\begin{aligned} |\mathcal {W}_{g,\varphi }(h)|=|\varphi |\cdot |h| \end{aligned}$$


$$\begin{aligned} \nabla _X \mathcal {W}_{g,\varphi }(h) = \mathcal {W}_{g,\varphi }(\nabla _X h) + \mathcal {W}_{g,\nabla _X\varphi }(h). \end{aligned}$$

Lemma 6

If \(\varphi \) is a parallel spinor on (Mg), then the diagram

figure a

commutes. In other words we have for any \(h\in \bigodot \nolimits ^2 T^*M\)

$$\begin{aligned} D^{T^*M}\circ \mathcal {W}_{g,\varphi } (h)= 4\hat{\kappa }_{g,\varphi }(h) + \varphi \otimes {\mathrm {div}}^g(h), \end{aligned}$$

where, for a vector bundle E over M with connection we define the Dirac operator \(D^E\) on \(\Sigma ^gM \otimes E\) by \({\mathrm {cl}}^g \circ \nabla ^{\Sigma ^gM \otimes E}\) where \(\nabla ^{\Sigma ^gM \otimes E} \) is the product connection.


Let \(h \in \Gamma (\bigodot ^2 T^*M)\). Then \(\mathcal {W}_{g,\varphi }(h)=\sum _{j=1}^n h(\,\cdot \,\,,e_j)e_j\cdot \varphi \). We now assume that \((e_1,\ldots ,e_n)\) is a local orthonormal frame satisfying \(\nabla e_j|_p=0\) at \(p\in M\) for \(1\le j\le n\), and we calculate in p

$$\begin{aligned} D^{T^*M}\left( \mathcal {W}_{g,\varphi }(h)\right)&= \sum _{i=1}^n e_i\cdot \nabla _{e_i}\left( \mathcal {W}_{g,\varphi }(h)\right) \\&= \sum _{i,j=1}^n(\nabla _{e_i}h)(\,\cdot \,\,,e_j) e_i\cdot e_j\cdot \varphi \\&= 4 \hat{\kappa }_{g,\varphi }(h) + \sum _{i=1}^n(\nabla _{e_i}h)(\,\cdot \,\,,e_i) e_i\cdot e_i\cdot \varphi \\&= 4 \hat{\kappa }_{g,\varphi }(h) + \varphi \otimes \underbrace{\sum _{i=1}^n(-\nabla _{e_i}h)(\,\cdot \,\,,e_i)}_{={\mathrm {div}}^g h}. \end{aligned}$$

\(\square \)

Very similarly we prove for arbitrary sections \(\varphi \) and h:

$$\begin{aligned} \begin{aligned} D^{T^*M}\left( \mathcal {W}_{g,\varphi }(h)\right) (X)&= 4 \hat{\kappa }_{g,\varphi }(h)(X) + ({\mathrm {div}}^g h)(X)\varphi \\ {}&\quad - 2 \nabla _{h^\#(X)}\varphi - (h^\#(X))\cdot D^g\varphi , \end{aligned} \end{aligned}$$

where \(h^\#\in {\mathrm {End}}(TM)\) is defined as \(h^\#(X):=h(\,\cdot \,,X)^\#\).

The following lemma is a straightforward generalization of a Lemma by McKenzie Wang [39, Lemma 3.3 (d) for \(k=1\)], see also [15, Prop. 2.4].

Lemma 7

For \(\varphi \in \Gamma (\Sigma ^gM)\) and \(h\in \Gamma (\bigodot \nolimits ^2 T^*M)\) we have

$$\begin{aligned} (D^{T^*M})^2\mathcal {W}_{g,\varphi }(h)&= \mathcal {W}_{g,\varphi }(\Delta _E h) - 2 \sum _\ell \mathcal {W}_{g,\nabla _{e_\ell }\varphi }(\nabla _{e_\ell }h)+\mathcal {W}_{g,D^2 \varphi }(h)\\&\quad -2(X\mapsto \sum _{\ell }h^{\sharp }(e_{\ell })\cdot R_{X,e_{\ell }}\varphi ). \end{aligned}$$

In particular, if \(\varphi \) is a parallel spinor, then the diagram

figure b



We can locally write

$$\begin{aligned} \mathcal {W}_{g,\varphi }(h)=\sum _{i,j}h_{ij}e_j\cdot \varphi \otimes e_i^* \end{aligned}$$

where \(\left\{ e_1,\ldots e_n\right\} \) is a local orthonormal frame with respect to g with \(\nabla e_i=0\) at p. We define \(\nabla _kh_{ij}{:}{=}(\nabla _{e_k} h)(e_i,e_j)\), \(\nabla ^2_{k,l}h_{ij}{:}{=}(\nabla ^2_{e_k,e_l} h)(e_i,e_j)\), and let \(R_{e_k,e_l}h_{ij}=\nabla ^2_{k,l}h_{ij}-\nabla ^2_{l,k}h_{ij}\) be the associated curvature. By using the Clifford relations, we get on the domain of the frame:

$$\begin{aligned} D^{T^*M}(\mathcal {W}_{g,\varphi }(h))&=\sum _{i,j,k}(\nabla _kh_{ij}e_k\cdot e_j\cdot \varphi +h_{ij}e_k\cdot e_j\cdot \nabla _{e_k}\varphi )\otimes e_i^*\\&=\sum _{i,j,k}(\nabla _kh_{ij}e_k\cdot e_j\cdot \varphi -h_{ij}e_j\cdot e_k\cdot \nabla _{e_k}\varphi -2h_{ij}\delta _{kj}\nabla _{e_k}\varphi )\otimes e_i^*\\&=\underbrace{\sum _{i,j,k}\nabla _kh_{ij}e_k\cdot e_j\cdot \varphi \otimes e_i^*}_{=:(\mathrm {A})} + \underbrace{(-\mathcal {W}_{g,D\varphi }(h))}_{=: (\mathrm {B})} + \underbrace{(-2\sum _{i,j}h_{ij}\nabla _{e_j}\varphi \otimes e_i^*)}_{=: (\mathrm {C})} \end{aligned}$$

Now we apply \(D^{T^*M}\) to this equation and we use the notation \(\psi =D\varphi \). Then we get from the previous equation at the point p

$$\begin{aligned} D^{T^*M}((\mathrm {B}))&=-D^{T^*M}(\mathcal {W}_{g,\psi }(h))\\ {}&=-\sum _{i,j,k}\nabla _kh_{ij}e_k\cdot e_j\cdot \psi \otimes e_i^*+\mathcal {W}_{g,D\psi }(h)+2\sum _{i,j}h_{ij}\nabla _{e_j}\psi \otimes e_i^*. \end{aligned}$$


$$\begin{aligned} D^{T^*M}((\mathrm {C}))&=-2\sum _{i,j,k}(h_{ij}e_k\cdot \nabla ^2_{e_k,e_j}\varphi +\nabla _kh_{ij}e_k\cdot \nabla _{e_j}\varphi )\otimes e_i^*\\&=-2\sum _{i,j,k}(h_{ij}e_k\cdot \nabla ^2_{e_j,e_k}\varphi +h_{ij}e_k\cdot R_{e_k,e_j}\varphi +\nabla _kh_{ij}e_k\cdot \nabla _{e_j}\varphi ) \otimes e_i^*\\&=-2\sum _{i,j}(h_{ij}\nabla _{e_j}\psi +\sum _kh_{ij}e_k\cdot R_{e_k,e_j}\varphi +\sum _k\nabla _kh_{ij}e_k\cdot \nabla _{e_j}\varphi ) \otimes e_i^* \end{aligned}$$


$$\begin{aligned} D^{T^*M}((\mathrm {A}))&=\underbrace{\sum _{i,j,k,l}\nabla ^2_{lk}h_{ij}e_l\cdot e_k\cdot e_j\cdot \varphi \otimes e_i^*}_{=: (\mathrm {E})}+\underbrace{\sum _{i,j,k,l}\nabla _kh_{ij}e_l\cdot e_k\cdot e_j\cdot \nabla _{e_l}\varphi \otimes e_i^*}_{=: (\mathrm {F})}. \end{aligned}$$

By using the relation \(e_l\cdot e_k\cdot e_j= e_k\cdot e_j\cdot e_l+2e_k\delta _{lj}-2\delta _{kl}e_j\),

$$\begin{aligned} (\mathrm {F})=\sum _{i,j,k}(\nabla _kh_{ij}e_k\cdot e_j\cdot \psi +2\nabla _kh_{ij}e_k\cdot \nabla _{e_j}\varphi -2\nabla _kh_{ij}e_j\cdot \nabla _{e_k}\varphi )\otimes e_i^*. \end{aligned}$$

By adding up, a lot of terms cancel and we are left with

$$\begin{aligned} (D^{T^*M})^2(\mathcal {W}_{g,\varphi }(h))&=(\mathrm {E})+\mathcal {W}_{g,D^2\varphi }(h) -2\sum _{i,j,k}\nabla _kh_{ij}e_j\cdot \nabla _{e_k}\varphi \otimes e_i^*\\&\quad +2\sum _{i,j,k}h_{ij}e_k\cdot R_{e_j,e_k}\varphi \otimes e_i^* \end{aligned}$$

so it remains to consider the term \((\mathrm {E})\). We have

$$\begin{aligned} (\mathrm {E})=&-\sum _{i,j,k}\nabla ^2_{kk}h_{ij}e_j\cdot \varphi \otimes e_i^*+\sum _{\begin{array}{c} i,j,k,l\\ k\ne l \end{array}}\nabla ^2_{lk}h_{ij}e_l\cdot e_k\cdot e_j\cdot \varphi \otimes e_i^*\\ =&\sum _{i,j}(\nabla ^*\nabla h)_{ij}e_j\cdot \varphi \otimes e_i^*+\sum _{i,j,k,l}\frac{1}{2}R_{e_k,e_l}h_{ij}e_k\cdot e_l\cdot e_j\cdot \varphi \otimes e_i^*. \end{aligned}$$

By the formula which expresses the curvature of \(T^*M\otimes T^*M\) in terms of the curvature of TM, we obtain \(R_{e_k,e_l}h_{ij}=-\sum _{r}(R_{klir}h_{rj}+R_{kljr}h_{ir})\). Using the relation \(e_k\cdot e_l\cdot e_j=e_j\cdot e_k\cdot e_l-2e_k \delta _{jl}+2e_l\delta _{jk}\), we get

$$\begin{aligned}&-\frac{1}{2}\sum _{i,j,k,l,r}R_{klir}h_{rj}e_k\cdot e_l\cdot e_j\varphi \otimes e_i^*\\&\quad = -\frac{1}{2}\sum _{i,j,k,l,r}R_{klir}h_{rj}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^*\\&\quad +\sum _{i,k,l,r}(R_{klir}h_{rl}e_k-R_{klir}h_{rk}e_l)\cdot \varphi \otimes e_i^*\\&\quad =-\frac{1}{2}\sum _{i,j,k,l,r}R_{klir}h_{rj}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^*-2 \sum _{i,j}(\mathring{R}h)_{ij}e_j\cdot \varphi \otimes e_i^* \end{aligned}$$


$$\begin{aligned}&-\frac{1}{2}\sum _{i,j,k,l,r}R_{kljr}h_{ir}e_k\cdot e_l\cdot e_j\cdot \varphi \otimes e_i^*\\&\quad = -\frac{1}{2}\sum _{i,j,k,l,r}R_{kljr}h_{ir}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^*\\&\quad +\sum _{i,k,l,r}(R_{kllr}h_{ir}e_k-R_{klkr}h_{ir}e_l)\cdot \varphi \otimes e_i^*\\&\quad =-\frac{1}{2}\sum _{i,j,k,l,r}R_{kljr}h_{ir}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^* +2\sum _{i,j,r}{\mathrm {Ric}}_{jr}h_{ir}e_j\cdot \varphi \otimes e_i^*. \end{aligned}$$

We get

$$\begin{aligned} (D^{T^*M})^2(\mathcal {W}_{g,\varphi }(h))&=\mathcal {W}_{g,\varphi }\bigl (\underbrace{\nabla ^*\nabla h-2\mathring{R}h }_{=\Delta _Eh}\bigr )-\frac{1}{2}\sum _{i,j,k,l,r}R_{klir}h_{rj}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^*\\&\quad -\frac{1}{2}\sum _{i,j,k,l,r}R_{kljr}h_{ir}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^*\\&\quad +2\sum _{i,j,r}{\mathrm {Ric}}_{jr}h_{ir}e_j\cdot \varphi \otimes e_i^*+\mathcal {W}_{g,D^2\varphi }(h)\\&\quad -2\sum _{i,j,k}\nabla _kh_{ij}e_j\cdot \nabla _{e_k}\varphi \otimes e_i^*+2\sum _{i,j,k}h_{ij}e_k\cdot R_{e_j,e_k}\varphi \otimes e_i^*. \end{aligned}$$

From the standard identities

$$\begin{aligned} R_{e_i,e_j}\varphi =\frac{1}{4}\sum _{k,l}R_{ijkl}e_k\cdot e_l\cdot \varphi ,\qquad \sum _ie_i\cdot R_{e_i,e_j}\varphi =\frac{1}{2}{\mathrm {Ric}}(e_j)\cdot \varphi , \end{aligned}$$

we deduce

$$\begin{aligned}&-\frac{1}{2}\sum _{i,j,k,l,r}R_{klir}h_{rj}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^* =-2\sum _{i,j,r}h_{rj}e_j\cdot R_{e_i,e_r}\varphi \otimes e_i^*,\\&-\frac{1}{2}\sum _{i,j,k,l,r}R_{kljr}h_{ir}e_j\cdot e_k\cdot e_l\cdot \varphi \otimes e_i^* =-2\sum _{i,j,r}h_{ri}e_j\cdot R_{e_j,e_r}\varphi \otimes e_i^*\\&=-\sum _{i,j,r}h_{ri}{\mathrm {Ric}}_{jr}e_j\cdot \varphi \otimes e_i^*, \mathrm{and}\\&2\sum _{i,j,k}h_{ij}e_k\cdot R_{e_j,e_k}\varphi \otimes e_i^*=-\sum _{i,j,k}h_{ij}{\mathrm {Ric}}_{jk}e_k\cdot \varphi \otimes e_i^*, \end{aligned}$$

which yields the final result. \(\square \)

4 The BBGM Parallel Transport Preserves Parallel Spinors

Proposition 8

Let \(I\subset \mathbb {R}\) be an interval, \((g_s)_{s\in I}\) be a path of Ricci-flat metrics with a divergence free derivative, \(0\in I\), and let \(\varphi _0\) be a parallel spinor on \((M,g_0)\). Let, for all \(s \in I\), \(\varphi _s\) be a spinor on \((M,g_s)\) such that \(\frac{d}{ds}\varphi _s=0\). Then \(\varphi _s\) is a parallel spinor on \((M,g_s)\) for all \(s\in I\).

Here \(\frac{d}{ds}\varphi _s=0\) should be understood as a derivation with respect to the BBGM connection.

In the proof of Proposition 8, we need a statement about continuous dependence of eigenvalues of the Dirac operator on the metric.

Theorem 9

There exists a family of functions \(\lambda _j:\mathcal {M}\rightarrow \mathbb {R}\), \(j\in \mathbb {Z}\) such that

  • \(\left\{ \lambda _j(g)\right\} _{j\in \mathbb {Z}}=\mathrm {spec}(D_g)\) for all \(g\in \mathcal {M}\)

  • The family is nondecreasing, i.e. \(\lambda _i\le \lambda _j\) whenever \(i\le j\).

  • \(\sup _{j\in \mathbb {Z}}|\mathrm {arsinh}(\lambda _j(g))-\mathrm {arsinh}(\lambda _j(\tilde{g})|\le C\left\| g-\tilde{g}\right\| _{C^1}\)

The theorem in the above version is proven in detail in [33, Main Theorem 2], but the statement we need was well-known long before.

To prove the proposition, we will at first prove the same statement under more restrictive assumptions.

Lemma 10

Let \((g_s)_{s\in I}\) be a path of Ricci-flat metrics with a divergence free derivative, \(0\in I\), and let \(\varphi _0\) be a parallel spinor on \((M,g_0)\). Assume that \(\dim \ker D^{T^*M}\) is constant along I. Let \(\varphi _s\) be a spinor on \((M,g_s)\) such that \(\frac{d}{ds}\varphi _s=0\). Then \(\varphi _s\) is a parallel spinor on \((M,g_s)\) for all \(s\in I\).

Proof of Lemma 10

Without loss of generality we can assume that I is compact. Using Rayleigh-Ritz type arguments, explained e.g. in detail in [33], one can show the existence of continuous functions \(I\ni s\mapsto \lambda _i(s)\in \mathbb {R}\), \(\lambda _i(s)\le \lambda _{i+1}(s)\) such that \((\lambda _i(s))_{i \in \mathbb {Z}}\) are the eigenvalues (including the correct multiplicity) of \(D^{T^*M}\). If the kernel of \(D^{T^*M}\) has constant dimension k, then we can assume \(\lambda _0(s)< \lambda _1(s)=\cdots =\lambda _k(s)=0<\lambda _{k+1}(s)\). Then for

$$\begin{aligned} \gamma :=\min _{s\in I}\min \{-\lambda _0(s),\lambda _{k+1}(s)\}>0 \end{aligned}$$

the operator \(D^{T^*M}_{g_s}\) has no eigenvalue in \((-\gamma ,0)\cup (0,\gamma )\). Because of this, there are bounded operators

$$\begin{aligned} \bigl (D^{T^*M}_{g_s}\bigr )^{-1}:L^2_{g_s}\cap \ker (D^{T^*M}_{g_s})^{\perp }\rightarrow L^2_{g_s}\cap \ker (D^{T^*M}_{g_s})^{\perp } \end{aligned}$$

that invert \(D^{T^*M}_{g_s}\) on the orthogonal complement of the kernel and that are uniformly bounded by \(\gamma ^{-1}\). By elliptic theory, see e.g. [26, Chapter III §5] we have isomorphisms

$$\begin{aligned} D^{T^*M}_{g_s}:H^1_{g_s}\cap \ker (D^{T^*M}_{g_s})^{\perp }\rightarrow L^2_{g_s}\cap \ker (D^{T^*M}_{g_s})^{\perp }. \end{aligned}$$

Because of \(\mathrm {im}(D^{T^*M}_{g_s})\subset \ker (D^{T^*M}_{g_s})^{\perp }\), there is a constant \(C_0>0\) such that for any \(s\in I\) and for any \(\sigma \in \Gamma (\Sigma ^{g_s}M\otimes T^*M)\) we have

$$\begin{aligned} \Vert D^{T^*M}_{g_s}\sigma \Vert _{H^1_{g_s}}\le C_0\Vert \left( D^{T^*M}_{g_s}\right) ^2\sigma \Vert _{L^2_{g_s}}. \end{aligned}$$

Moreover, \(g'_s\in \ker (\Delta _{E,g_s})\) by the facts collected in Sect. 2.1. We calculate using Lemma 7

$$\begin{aligned} (D^{T^*M}_{g_s})^2\mathcal {W}_{g_s,\varphi _s}(g'_s)=&- 2 \sum _\ell \mathcal {W}_{g_s,\nabla _{e_\ell }\varphi _s}(\nabla _{e_\ell } g'_s)+\mathcal {W}_{g_s, D^2 \varphi _s}(g'_s)\\&-2(X\mapsto \sum _{\ell }({g}'_s)^{\sharp }(e_{\ell })\cdot R_{X,e_{\ell }}\varphi _s), \end{aligned}$$

thus pointwise \((D^{T^*M}_{g_s})^2\mathcal {W}_{g_s,\varphi _s}( g'_s)\) is a linear expression in \(\nabla \varphi _s\) and its first derivative. Thus

$$\begin{aligned} \left\| \big (D^{T^*M}_{g_s}\bigr )^2 \mathcal {W}_{(g_s,\varphi _s)}(g'_s) \right\| _{L^2} \le C \Vert \nabla \varphi _s\Vert _{H^1}, \end{aligned}$$

where C depends on \(\sup _{s\in I}\left\| {g}'_s\right\| _{C^1(g_s)}\). This implies, using Eq. (20):

$$\begin{aligned} \left\| D^{T^*M}_{g_s} \mathcal {W}_{(g_s,\varphi _s)}(g'_s) \right\| _{H^1} \le C C_\gamma \Vert \nabla \varphi _s\Vert _{H^1}. \end{aligned}$$

We now differentiate \(\nabla ^{g_s}\varphi _s\) using formulae (16) and (17) and Lemma 6.

$$\begin{aligned}&\frac{d}{ds}\nabla ^{g_s}\varphi _s = \hat{\kappa }_{g_s,\varphi _s}( g_s') +\nabla ^{g_s} \underbrace{\varphi '_s}_{=0}\\&\quad = \frac{1}{4}\Biggl (D^{T^*M}_{g_s}\Bigl (\mathcal {W}_{(g_s,\varphi _s)}(g'_s)\Bigr ) - \varphi _s\otimes \underbrace{{\mathrm {div}}^{g_s} g_s'}_{=0} + 2 \nabla _{(g_s')^\#(\,\cdot \,)}\varphi _s + (g_s')^\#(\,\cdot \,)\cdot D^{g_s}\varphi _s\Biggr )\\&\quad = \frac{1}{4} D^{T^*M}_{g_s} \mathcal {W}_{(g_s,\varphi _s)}( g_s')+\frac{1}{2} \nabla _{( g_s')^\#(\,\cdot \,)}\varphi _s + \frac{1}{4} (g_s')^\#(\,\cdot \,)\cdot D^{g_s}\varphi _s. \end{aligned}$$

This implies, with Eq. 21:

$$\begin{aligned} \left\| \frac{d}{ds} \nabla ^{g_s}\varphi _s\right\| _{H^1(g_s)}&\le \tilde{C}\Biggl ( \Vert \nabla ^{g_s}\varphi _s\Vert _{H^1(g_s)} + \underbrace{\Vert D^{g_s}\varphi _s\Vert _{H^1(g_s)}}_{\le \sqrt{n}\Vert \nabla ^{g_s}\varphi _s\Vert _{H^1(g_s)}}\Biggr ) \le C_1 \Vert \nabla ^{g_s}\varphi _s\Vert _{H^1(g_s)} \end{aligned}$$

where \(\tilde{C}\) and \(C_1\) depend on \(\sup _{s\in I}\left\| {g}'_s\right\| _{C^1(g_s)}\).

We set \(\ell (s):=\left\| \nabla ^{g_s}\varphi _s\right\| _{H^1(g_s)}^2\). In order to derive \(\ell (s)\) one has to be aware that also the evaluating metric depends on s. The dependence on the metrics effects \(\ell (s)\) in the metric contractions, the volume element and in the covariant derivatives used to define by \(\Vert \sigma \Vert _{H^1(g)}^2= \Vert \sigma \Vert _{L^2(g)}^2+ \Vert \nabla ^g\sigma \Vert _{L^2(g)}^2\). Due to compactness of I, these effects lead to a number \(C_2>0\), constant in s, such that

$$\begin{aligned} |\ell '(s)- 2 \langle \frac{d}{ds} \nabla ^{g_s}\varphi _s, \nabla ^{g_s}\varphi _s\rangle _{H^1(g_s)}|\le C_2 \ell (s) \end{aligned}$$

and we get

$$\begin{aligned} \ell '(s)\le 2 |\langle \frac{d}{ds} \nabla ^{g_s}\varphi _s, \nabla ^{g_s}\varphi _s\rangle _{H^1(g_s)} |+C_2\ell (s) \le (2C_1 +C_2)\,\ell (s). \end{aligned}$$

As \(\ell (0)=0\) this implies with Grönwall’s inequality that \(\ell (s)\) vanishes for all s. \(\square \)

Remark 11

In the proof above the derivative \(\frac{d}{ds}\nabla ^{g_s}\varphi _s\) should be taken with some care. Here we derive an s-dependent family \(\nabla ^{g_s}\varphi _s\in \Gamma (T^*M\otimes \Sigma ^{g_s}M)\) with respect to the metric variation given by s. This is the BBGM-derivative in the spinorial part. On the cotangential part, one could also use a BBGM-kind of derivative, but this is not what was used. On the cotangential part, we simply used the derivative in the usual sense, i.e. the derivative of a curve in the vector space \(\Gamma (T^*M)\).

Proof of Proposition 8

In a first step we prove the Proposition for analytic families \(g_s\), \(s\in I\) of Ricci-flat metrics, in other words we assume that the map \(I\rightarrow \bigodot ^2T^*M\) defined by \(g_s\) is analytic.

Then \(D^{T^*M}_{g_s}\) is an analytic family of operators. This implies that the eigenvalues \(\lambda _i(s)\) of \(D^{T^*M}_{g_s}\) can be numbered such that \(s\mapsto \lambda _i(s)\) is analytic in s [21, Appendix A]. Due to Theorem 9, only finitely many \(\lambda _i(s)\) have zero sets on a compact interval. Let \(\mu := \min \dim \ker D^{T^*M}_{g_s}\), and \(J:=\{s\in I\mid \dim \ker D^{T^*M}_{g_s}>\mu \}\) which is a closed discrete subset due to analyticity. (We conjecture that \(J=\emptyset \), but we were unable to prove it.) Let \(I_0\) be a connected component of \(I\setminus J\). The previous lemma states that the Bourguignon-Gauduchon parallel transport along \(s\mapsto g_s\) maps parallel spinor \(\varphi \in \Gamma (\Sigma ^{g_s}M)\) to a parallel spinor \(P_{s,r}(\varphi )\in \Gamma (\Sigma ^{g_r}M)\) for any \(s,r\in I_0\). Let \(\bar{r}\in \partial I_0\). Then by continuity,

$$\begin{aligned} P_{s,\bar{r}}(\varphi )= \lim _{r\rightarrow \bar{r}}P_{s, r}(\varphi ) \end{aligned}$$

is a parallel section of \(\Gamma (\Sigma ^{g_{\bar{r}}}M)\). We use the fact that the dimension of the space of parallel spinors is locally constant (see [3]), thus for any \(s\in I_0\) and and \(r\in \bar{I}_0\) (including \(r\in \partial I_0\)) the monomorphism \(P_{s,r}\) from parallel spinors in \(\Gamma (\Sigma ^{g_s}M)\) to parallel spinors in \(\Gamma (\Sigma ^{g_r}M)\) is an isomorphism. Thus BBGM parallel transport perserves parallel spinors along \(\bar{I}_0\), and by an induction argument (using that J is finite in compact intervals) this also holds along I. The proposition is thus proven for analytic families \(s\mapsto g_s\) and thus also for piecewise analytic families \(s\mapsto g_s\).

Now, as the Riemannian metrics form an open cone in the vector space of bilinear forms, an arbitrary smooth family \(s\mapsto g_s\) can be approximated by piecewise analytic paths in the \(C^1\)-norm. We claim that the BBGM parallel transport is continuous in this limit. This can be seen most easily in the universal spinor bundle formulation: There, for SM being the bundle of symmetric positive definite bilinear forms, a Clifford bundle \(\pi ^\mathrm {univ}: \Sigma M \rightarrow SM\) is constructed such that, for each \(g \in \Gamma (SM) \), \( \Sigma ^gM \) is isomorphic as a Clifford bundle to \(g^* \Sigma M \). Furthermore, the vector bundle \(\pi ^\mathrm {univ}:\Sigma M \rightarrow SM\) carries a vertical (w.r.t. \(S M \rightarrow M\)) covariant derivative \(\nabla \) whose parallel transport P is linked to the BBGM parallel transport \(\hat{P}\) as follows: Let \(g_\bullet :[0,\ell ]\rightarrow \Gamma (S M)\), \(s\mapsto g_s\) be a \(C^1\)-curve of Riemannian metrics on M. Let \(p \in M\) and let \(g|_p: s \mapsto g_s (p)\) and \(\sigma \in (\pi ^\mathrm {univ})^{-1 } (g_0(p))=\Sigma ^{g_0}_p M \), then for any \(\tilde{\sigma } \in \Gamma (\Sigma ^{g_0}M)\) with \(\tilde{\sigma }|_p = \sigma \) we have

$$\begin{aligned} \hat{P}_{g_\bullet } (\tilde{\sigma })\big |_p = P_{g|_p} (\sigma ). \end{aligned}$$

The parallel transport along a curve with respect to a connection is given by a first order ordinary differential equation (ODE), satisfying the conditions of the theorem of Picard-Lindelöf. Using the universal spinor bundle formalism we argued that the BBGM-parallel transport is given by such an ODE and that its coefficient functions converge uniformly (i.e. in the \(C^0\)-norm) when a path of metrics converges in the \(C^1\)-norm to a limit path of metrics.Footnote 2 Thus the theorem of Picard-Lindelöf, taking into account that both M and \([0,\ell ]\) are compact, implies that the BBGM parallel transport converges uniformly when we approximate a smooth path of metrics \(s\mapsto g_s\) by piecewise analytic paths of metrics in the \(C^1\)-norm. Thus the BBGM parallel transport also preserves parallel spinors along the smooth family \(s\mapsto g_s\). \(\square \)

5 Construction of Solutions to the Constraint Equations

In this section we want to use the BBGM connection to construct solutions of the constraint equation on suitable manifolds of the form \((M\times I, g_s+ds^2 )\) where I is an interval and where M is an \(m=(n-1)\)-dimensional manifold.

Assume that \(g_s\) is a family of Ricci-flat metrics with divergence-free derivative and that for some \(s_0\) there is a non-trivial parallel spinor \(\varphi _{s_0}\) on \((M,g_{s_0})\). By rescaling we can achieve that its norm is 1 in every point. We shift it in the s-direction parallely with the BBGM parallel transport. By Prop. 8 we obtain a family \(\varphi _s\) of \(g_s\)-parallel spinors of constant norm 1 on M, for every \(s\in I\). This yields a fiberwise parallel section of \(\Sigma _M^{(\#)}N\). Recall \(\nu =\frac{\partial }{\partial s}\).

In the following we assume that \(f:I\rightarrow \mathbb {R}\) is a given smooth function and choose \(s_0\in I\). We define \(F(s):=\exp \left( -\frac{1}{2}\int _{s_0}^s f(\sigma )\,d\sigma \right) \), i.e. \(-\frac{1}{2} f(s)=F'(s)/F(s)\).

Case n is odd, i.e. \(m=\dim M\) even. By possibly changing the orientation and using Proposition 25 in Appendix C we can assume that all \(\varphi _s\in \Gamma (\Sigma ^{g_s} M)\) have the positive parity for the splitting given by the volume element of M, i.e. we can assume \(\omega _\mathbb {C}^M\cdot \varphi _s=\varphi _s\). For every \((x,s)\in N=M\times I\) we use the map J from Sect. 2.3 to get identifications \(\Sigma _xM\cong {\Sigma _M}_{(x,s)} N\cong \Sigma _{(x,s)} N\). Again the Clifford multiplication for \((M,g_s)\) will be denoted by \(\cdot \), and for the one for (Nh), \(h=g_s+ds^2\) we will use \(\star \). From Eq. (10) we get \(\nu \star \varphi =i\varphi \). Using Lemma 4 we obtain

$$\begin{aligned} \nabla ^N_{\nu }\varphi= & {} 0,\\ \nabla ^N_X\varphi= & {} \frac{1}{2} W(X)\star \nu \star \varphi = \frac{i}{2} W(X)\star \varphi . \end{aligned}$$

We set

$$\begin{aligned} \Psi (x,s):= F(s) \varphi _s(x). \end{aligned}$$

Assuming that X is tangent to M, we obtain

$$\begin{aligned} \nu \star \Psi= & {} i\Psi ,\\ \nabla ^N_{\nu }\Psi= & {} \frac{F'(s)}{F(s)}\Psi =-\frac{1}{2} f(s)\Psi = \frac{i}{2} f(s) \nu \star \Psi ,\\ \nabla ^N_X\Psi= & {} \frac{1}{2} W(X)\star \nu \star \Psi = \frac{i}{2} W(X)\star \Psi . \end{aligned}$$

Thus \(\Psi \) is an imaginary \(\overline{W}\)-Killing spinor with \(\overline{W}= W+f\nu ^b\otimes \nu \), i.e. it satisfies (3).

The first relation and the defining Eq. (22) also imply \(-i\langle \nu \star \Psi , \Psi \rangle = F^2\). For X tangent to M, the real part of \(\langle X\star \Psi , \Psi \rangle \) vanishes and

$$\begin{aligned} {\mathrm {Im}}\langle X\star \Psi , \Psi \rangle = - {\mathrm {Re}}\langle X\star i\Psi , \Psi \rangle =- {\mathrm {Re}}\langle X\star \nu \star \Psi , \Psi \rangle {\mathop {=}\limits ^{(*)}}{\mathrm {Re}}\langle \Psi , X\star \nu \star \Psi \rangle , \end{aligned}$$

where in \((*)\) we used the skew-symmetry of Clifford multiplication with vectors twice and \(X\star \nu =-\nu \star X\). On the other hand one \({\mathrm {Re}}\langle X\star \nu \star \Psi , \Phi \rangle ={\mathrm {Re}}\langle \Phi , X\star \nu \star \Psi \rangle \), which in particular holds for \(\Phi :=\Psi \). We thus get \({\mathrm {Im}}\langle X\star \Psi , \Psi \rangle =0\).

Thus \(- i \langle X \star \Psi , \Psi \rangle = F^2 h(\nu ,X)\) for every vector X and consequently, the Dirac current \(U_\Psi \) of \(\Psi \), defined by \(h(U_\Psi , X) = - i \langle X \star \Psi , \Psi \rangle \) for every vector X (see (1)) is \(U_\Psi =F^2\nu \). We obtain

$$\begin{aligned} U_\Psi \star \Psi = i F^2 \Psi , \end{aligned}$$

which is the constraint Eq. (4) for \(u_\Psi :=\sqrt{h(U_\Psi ,U_\Psi )} \equiv F^2\). Note that here we used (5) as definition for \(u_\psi \). We thus have obtained solutions of the constraint equations.

Case n even, i.e. \(m=\dim M\) odd. We use the map \(J^{(\#),+}\) defined in Sect. 2.3 to view \(\Sigma _x M\cong \Sigma _{M (x,s)}N\) as a subbundle of \(\Sigma _{(x,s)} N\). In particular, \((x,s)\mapsto \varphi _s(x)\) then yields a section of constant length 1 of the bundle \(\Sigma N\rightarrow N\).

Because of Eqs. (12) and (13) we have \(\nu \star \varphi = i\varphi \). Using Lemma 4 we obtain

$$\begin{aligned} \nabla ^N_{\nu }\varphi= & {} 0\\ \nabla ^N_X\varphi= & {} \frac{1}{2} W(X)\star \nu \star \varphi = \frac{i}{2} W(X)\star \varphi \end{aligned}$$

We set

$$\begin{aligned} \Psi (x,s):=F(s) \varphi _s(x). \end{aligned}$$

Then for X tangent to M,

$$\begin{aligned} \nu \star \Psi= & {} i\Psi ,\\ \nabla ^N_{\nu }\Psi= & {} \frac{F'(s)}{F(s)}\Psi =-\frac{1}{2} f(s)\Psi = \frac{i}{2} f(s) \nu \star \Psi ,\\ \nabla ^N_X\Psi= & {} \frac{1}{2} W(X)\star \nu \star \Psi = \frac{i}{2} W(X)\star \Psi . \end{aligned}$$

Thus, \(\Psi \) is an imaginary \(\overline{W}\)-Killing spinor with \(\overline{W}= W+f\nu ^b\otimes \nu \), i.e. it satisfies (3). With the same arguments as in the other case, we can prove that (1), (2), (4), and (5) are satisfied, i.e. we have found a solution to the constraint equations.

Example 12

Let \(M=T^{n-1}\), g be a flat metric on M and \(\varphi \) a parallel spinor on it.

  • Let \(g_s\equiv g\) with \(s\in I\). The BBGM parallel transport leaves \(\varphi \) invariant and we obtain initial data to the Cauchy problem on the metric \(g_s+ds^2\) on either \(T^{n-1}\times \mathbb {R}\) or \(T^{n-1}\times S^1 =T^{n}\). The Minkowski metric together with a parallel spinor (or in the \(S^1\) case a \(\mathbb {Z}\)-quotient of it) is then a solution of the associated Cauchy problem.

  • Let \(g_s=e^{2s}g\) with \(s\in \mathbb {R}\). For the submanifolds \(M\times \{s\}\subset N\) we have \(W=-{\mathrm {id}}|_{TM}\) for alle \(s\in \mathbb {R}\). We take the function \(f(s)=-1\), i.e. \(F(s):=\exp (s/2)\) and \(\overline{W}=-{\mathrm {id}}|_{TN}\). The metric we obtain is now the hyperbolic metric \(h=e^{2s}g+ds^2\) on \(T^{n-1}\times \mathbb {R}\) together with an imaginary Killing spinor with Killing constant \(-i/2\). The Lorentzian cone \((T^{n-1}\times \mathbb {R}\times \mathbb {R}_{>0},r^2 h-dr^2)\), where \(r\in \mathbb {R}_{>0}\), together with a parallel spinor solves solves the associated Cauchy problem. Note that this cone is the quotient by a \(\mathbb {Z}^{n-1}\)-action of \(I_+(0)\subset \mathbb {R}^{n,1}\), defined as the set of all future-oriented time-like vectors in the \((n+1)\)-dimensional Minkowski space \(\mathbb {R}^{n,1}\).

Remark 13

In this example we have seen two different ways of reconstructing Lorentzian Ricci-flat metrics on quotients of subsets of Minkowski space together with a parallel spinor. However, our construction allows many more interesting examples.

Example 14

  • Any imaginary Killing spinor on a complete, connected Riemannian spin manifold arises this way. This was proven by Baum, Friedrich, Grunewald and Kath in [7, Chap. 7], more precisely in Theorem 1 on page 160 and Cor. 1 on page 167 [7, Chap. 7].

  • This was generalized by Rademacher [34], see also [18, Theorem A.4.5]. Rademacher proved that any generalized imaginary Killing spinor with \(W=\alpha {\mathrm {id}}\), \(\alpha \in C^\infty (N,\mathbb {R})\) arises by our construction.

  • Our construction generalizes previous constructions of imaginary Killing spinors, as e.g. [19, Prop. 4.6 and Cor. 4.7].

6 From Curves in the Moduli Space to Initial Data Sets

Let \({\mathrm {Diff}}_0(M)\) be the identity component of the diffeomorphism group of M, acting on the space \(\mathcal {M}_\parallel (M)\) of structured Ricci-flat metrics by pullback (as usual, a semi-Riemannian manifold (Mg) is called structured iff its semi-Riemannian universal covering \((\tilde{M}, \tilde{g}) \) admits a parallel spinor). Furthermore, let \({{\mathcal {M}\textit{od}}}_\parallel (M):=\mathcal {M}_\parallel (M)/{\mathrm {Diff}}_0(M)\) be the associated premoduli space. It was shown in [3] (using previous work about the special case of simply connected manifolds with irreducible holonomy) that \({{\mathcal {M}\textit{od}}}_\parallel (M)\) “naturally” carries the structure of a finite-dimensional smooth manifold. The smooth structure on \({{\mathcal {M}\textit{od}}}_\parallel (M)\) can be described by the following properties

  • \({{\mathcal {M}\textit{od}}}_\parallel (M)\) carries the quotient topology, if we equip \(\mathcal {M}_\parallel (M)\) and \({\mathrm {Diff}}_0(M)\) with the standard Fréchet topology,

  • any smooth family \(N\rightarrow \mathcal {M}_\parallel (M)\), \(y\mapsto g_y\) yields a smooth map \(N\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\), \(y\mapsto [g_y]\),

  • in the case \(N=(a,b)\) one has \(\frac{d}{ds}|_{s=s_0}[g_s]=0\) if and only if \(\frac{d}{ds}|_{s=s_0}g_s\) is tangent to the \({\mathrm {Diff}}_0(M)\)-orbit through \(g_{s_0}\).

If I is an interval, then any smooth curve \(I\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\) can be written as \(s\mapsto [g_s]\) with a family of metrics \({\mathrm {div}}^{g_{s_0}}\left( \frac{d}{ds}|_{s=s_0}g_s\right) =0\), see Appendix D.

The main results of this article is a procedure to construct initial data for the constraint equations for Lorentzian metrics with parallel spinors.

Main Construction 15

(Initial data on an open manifold). Let I be an interval. For any smooth curve \(I\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\) \(s\mapsto [g_s]\) and every positive smooth function \(F:I\rightarrow \mathbb {R}_{>0}\) we obtain a solution of the initial data Eqs. (3) and (4) on \(M\times I\) with norm F(t) at any \((x,t)\in M\times I\).

Here the metric on \(M\times I\) is given by \(g_s+ds^2\) provided that \(g_s\) is chosen such that \({\mathrm {div}}^{g_{s_0}}\frac{d}{ds}|_{s=s_0}g_s=0\) and the spinor is given by (22) resp. (23). In particular the spinor can be normalized such that it has norm F(t) at any \((x,t)\in M\times I\). As derived in the preceding section, (3) are then satisfied, as well as (1), (2), (4), and (5).

The situation is slightly more complicated if we want to obtain solutions of the constraint equations on a closed manifold. We start with a closed curve \(S^1\cong \mathbb {R}/L\mathbb {Z}\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\). If we identify \(\mathbb {R}/L\mathbb {Z}\) with \([0,L]/0\sim L\), then every curve \(\mathbb {R}/L\mathbb {Z}\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\) can be written as \([0,L]\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\), \(s\mapsto [g_s]\) with \({\mathrm {div}}\left( \frac{d}{ds}g_s\right) =0\), but in general we will have \(g_0\ne g_L\) although \((M,g_0)\) and \((M,g_L)\) are isometric with respect to an isometry \(\zeta \in {\mathrm {Diff}}_0(M)\). We then glue \((M\times \{0\},g_0)\) with \((M\times \{L\},g_L)\) isometrically using the diffeomorphism \(\zeta \in {\mathrm {Diff}}_0(M)\). This yields a closed Riemannian manifold (Nh) diffeomorphic to \(M\times S^1\). In order to equip it with a spin structure we have to lift \(d\zeta ^{\otimes n}:P_{{\mathrm {SO}}}(M,g_0)\rightarrow P_{{\mathrm {SO}}}(M,g_L)\) to a map between the corresponding spin structures \(\zeta _\#:P_{{\mathrm {Spin}}}(M,g_0)\rightarrow P_{{\mathrm {Spin}}}(M,g_L)\). This yields a spin structure and a spinor bundle on N. Let \(F:S^1\rightarrow \mathbb {R}_{>0}\) be given. For any parallel spinor \(\varphi _0\) on \((M,g_0)\) Eq. (22) resp. Equation (23) yields a generalized imaginary Killing spinor \(\Psi \) on \(M\times [0,L]\) as in Main Construction 15. The gluing described above allows to view \(\varphi _L:=\Psi _{M\times \{L\}}\) as a parallel spinor on \((M,g_0)\). However, in general \(\varphi _L\) will differ from \(\varphi _0\). Let \(\Gamma _\parallel (\Sigma _{g_0}M)\) denote the space of parallel spinors on \((M,g_0)\). Then \(\varphi _0\mapsto \varphi _L\) yields a unitary map \(P:\Gamma _\parallel (\Sigma _{g_0}M)\rightarrow \Gamma _\parallel (\Sigma _{g_0}M)\). The map P does neither depend on F nor on the parametrization of the curve \(s\mapsto [g_s]\). We say that \((M,g_s,\varphi _0)\) satisfies the fitting condition if \(P(\varphi _0)=\varphi _0\) for a suitable choice of spin structure on N. The fitting condition is always satisfied in the following cases:

  1. (1)

    \((M,g_0)\) is a 7-dimensional manifold with holonomy \(G_2\)

  2. (2)

    \((M,g_0)\) is an 8-dimensional manifold with holonomy \({\mathrm {Spin}}(7)\)

  3. (3)

    \((M,g_0)\) is a Riemannian product of manifolds of that kind and of at most one factor diffeomorphic to \(S^1\).

  4. (4)

    finite quotients of such manifolds

As this statement is not within the core of this article, we only sketch the proof. In the first two cases the spinor bundle \(\Sigma _g M\) is the complexification of the real spinor bundle \(\Sigma _g^\mathbb {R}M\), and thus \(\Gamma _\parallel (\Sigma _g M)=\Gamma _\parallel (\Sigma _g^\mathbb {R}M)\otimes \mathbb {C}\). The real spinor representations of \(G_2\) and \({\mathrm {Spin}}(7)\) on \(\Sigma _n^\mathbb {R}\) have a 1-dimensional invariant subrepresentation, thus \(\dim _\mathbb {R}\Gamma _\parallel (\Sigma _g^\mathbb {R}M)=1\). Thus P is either \(+{\mathrm {id}}\) or \(-{\mathrm {id}}\), and the \(+\)-sign can be achieved by a suitable choice of the lift \(\zeta _\#\). On \(S^1\) it follows from a direct calculation.

The map P behaves “well” under taking products and finite quotients, thus the other two statements follow as well.

For manifolds with a least one factor of holonomy \({\mathrm {SU}}(k)\) or \({\mathrm {Sp}}(k)\), or also for tori of dimension \(>1\), however, we expect that generically the fitting condition does not hold. In this case, we expect that the space of closed paths \(s\rightarrow [g_s]\) for wich P has finite order (in the sense \(\exists \ell \in \mathbb {N}:\, P^\ell ={\mathrm {id}}\)) is dense in the space of all closed paths \(s\rightarrow [g_s]\) with respect to the \(C^\infty \)-topology. This is in fact a consequence of work in progress by Bernd Ammann, Klaus Kröncke and Hartmut Weiß.

Then passing to an \(\ell \)-fold cover of N obtained from running along the path \(s\rightarrow [g_s]\) not just once, but \(\ell \) times, we obtain a solution of the constraint equation on \(M\times S^1\).

Main Construction 16

(Initial data on a closed manifold). Let \(L>0\). Let \(\mathbb {R}/L\mathbb {Z}\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\), \([s]\mapsto [g_s]\) be a smooth path and let \(\varphi _0\in \Gamma _\parallel (\Sigma ^{g_0} M)\) be given such that \((M,g_s,\varphi _0)\) satisfies the fitting condition. Then for any function \(F:S^1\rightarrow \mathbb {R}_{>0}\) we obtain a solution of the initial data Eqs. (3) and (4) on \(M\times S^1\).

It also seems interesting to us to allow a slight generalization of our initial problem, by considering spin\(^c\) spinors with a flat associated line bundle instead of spinors in the usual sense. Assume that \(\vartheta \in \mathbb {C}\) has norm 1. Identifying \((v,t)\in \mathbb {C}\times \mathbb {R}\) with \((\vartheta v,t+L)\) yields a complex line bundle \(L_\vartheta \) over \(S^1 =\mathbb {R}/L\mathbb {Z}\). On \(L_\vartheta \) we choose the connection such that local sections with constant v are parallel. By pull back we obtain complex line bundles with flat, metric connections on \(N=M\times S^1\) and on \(\overline{N}=N\times (-\epsilon ,\epsilon )\). These line bundles will also be denoted by \(L_\vartheta \). The bundle \(\Sigma _h N\otimes L_\vartheta \) resp. \(\Sigma _{\bar{h}} {\overline{N}}\otimes L_\vartheta \) is then a spin\(^c\)-spinor bundle with flat associated bundle \(L_\vartheta \). The objects tensored by \(L_\vartheta \) will be called \(L_\vartheta \)-twisted. All the results of this article immediately generalize to \(L_\vartheta \)-twisted spinors. We ask for \(L_\vartheta \)-twisted parallel spinors, i.e. parallel sections of \(\Sigma _{\bar{h}} {\overline{N}}\otimes L_\vartheta \) instead of parallel spinors in the usual sense. This leads to \(L_\vartheta \)-twisted constraint equations, and the \(L_\vartheta \)-twisted Cauchy problem can be solved the same way as the untwisted.

Let \(P\in {\mathrm {U}}\bigl (\Gamma _\parallel (\Sigma _{g_0}M)\bigr )\) be as above. As P is unitary, there is a basis of \(\Gamma _\parallel (\Sigma _{g_0}M)\) consisting of eigenvectors of P for complex eigenvalues of norm 1.

Main Construction 17

(Spin\(^c\)-version). Let \(L>0\). Let \(\mathbb {R}/L\mathbb {Z}\rightarrow {{\mathcal {M}\textit{od}}}_\parallel (M)\), \([s]\mapsto [g_s]\) be a smooth path. Let \(\varphi _0\in \Gamma _\parallel (\Sigma _{g_0}M)\) be an eigenvalue of P to the eigenvalue \(\vartheta \). Then for any function \(F:S^1\rightarrow \mathbb {R}_{>0}\) we obtain a solution of the \(L_\vartheta \)-twisted version of the constraint Eqs. (3) and (4) on \(M\times S^1\).