1 Introduction and statement of the results

Throughout this article, let \(M^m\) denote a smooth compact oriented spin manifold. Following [4, sec. 2], we fix a topological spin structure \(\Theta :\widetilde{\mathrm{GL }}^+M \rightarrow \mathrm{GL }^+M\) on \(M\). We denote by \(\mathcal{R }(M)\) the space of all Riemannian metrics on \(M\) endowed with \(\mathcal C ^1\)-topology. For any metric \(g \in \mathcal{R }(M)\), one obtains a metric spin structure \(\Theta ^g:\mathrm{Spin }^g M \rightarrow \mathrm{SO }^g M\) and an associated spinor bundle \(\Sigma ^g M\). The Dirac operator \({{{{\big /}\!\!\!\!{D}}}}^g\) can be viewed as an unbounded operator \(\Gamma _{L^2}(\Sigma ^g M) \rightarrow \Gamma _{L^2}(\Sigma ^gM)\) with domain \(\Gamma _{H^1}(\Sigma ^g M)\), where \(H^1\) denotes the first-order Sobolev space. For a detailed introduction to spin geometry, see [8, 12].

Unfortunately, one cannot directly compare the Dirac operators \({{{{\big /}\!\!\!\!{D}}}}^g\) and \({{{{\big /}\!\!\!\!{D}}}}^h\) for different metrics \(g,h \in \mathcal{R }(M)\), because they are not defined on the same spaces. This problem has been discussed at length and solved in various articles, cf. [4, 5, 14]. The idea is to construct an isometry \(\bar{\beta }_{g,h}:\Gamma _{L^2}(\Sigma ^g M) \rightarrow \Gamma _{L^2}(\Sigma ^h M)\) between the different spinor bundles and then to pull back the operator \({{{{\big /}\!\!\!\!{D}}}}^h\) to an operator \({{{{\big /}\!\!\!\!{D}}}}^h_g\) on the domain of \({{{{\big /}\!\!\!\!{D}}}}^g\). This enables us to view the collection of all these Dirac operators as operators, which depend continuously on the metric \(g\) (the precise results are cited in Theorem 2.1 later). It is, therefore, natural to ask if—and in what sense—the spectrum of the Dirac operator also depends continuously on the metric. In the present article we will investigate this problem and present a solution.

Every Dirac operator \({{{{\big /}\!\!\!\!{D}}}}^g\) is a self-adjoint, elliptic, first-order differential operator. It is well known (see for instance [12, Thm. 5.8]) that the spectrum \({\text{ spec }}{{{{{\big /}\!\!\!\!{D}}}}}^g\) is a subset of the real line, which is closed, discrete and unbounded from both sides. The elements of \({\text{ spec }}{{{{\big /}\!\!\!\!{D}}}}^g\) consist entirely of eigenvalues of finite multiplicity. Intuitively, we would like to enumerate the eigenvalues from \(-\infty \) to \(+ \infty \) as a non-decreasing sequence indexed by \(\mathbb Z \,\). (In the entire article we will always count eigenvalues with their geometric multiplicity.) The problem is that this is not well-defined, because it is unclear which eigenvalue should be the “first” one. Formally, we can avoid this problem via the following definition.

Definition 1.1

For any \(g \in \mathcal{R }(M)\), let \(\mathfrak{s }^g:\mathbb Z \,\rightarrow \mathbb R \,\) be the unique non-decreasing function such that \(\mathfrak{s }^{g}(\mathbb Z \,)={\text{ spec }}{{{{\big /}\!\!\!\!{D}}}}^{g}\),

$$\begin{aligned} \forall \lambda \in \mathbb R \,: \dim \ker ({{{{\big /}\!\!\!\!{D}}}}^g - \lambda ) = \sharp (\mathfrak{s }^g)^{-1}(\lambda ), \end{aligned}$$

and \(\mathfrak{s }^g(0)\) is the first eigenvalue \(\ge 0\) of \({{{{\big /}\!\!\!\!{D}}}}^g\).

Then \(\mathfrak{s }^g\) is well-defined, but as it will turn out, the requirement that \(\mathfrak{s }^g(0)\) be the first eigenvalue \(\ge 0\) has some drawbacks. Namely, the map \(g \mapsto \mathfrak{s }^g(j)\), \(j \in \mathbb Z \,\), will in general not be continuous, see Remark 1.4. To obtain a more natural notion, we introduce the following defintion.

Definition 1.2

(Mon and Conf) Define

$$\begin{aligned} \mathrm{Mon }:= \{ u:\mathbb Z \,\rightarrow \mathbb R \,\mid u \text{ is } \text{ non-decreasing } \text{ and } \text{ proper } \} \subset \mathbb R \,^\mathbb Z \,. \end{aligned}$$

The group \((\mathbb Z \,,+)\) acts canonically on \(\mathrm{Mon }\) via shifts, i.e.

$$\begin{aligned} \tau : \mathrm{Mon }\times \mathbb Z \,&\rightarrow \mathrm{Mon }\nonumber \\ (u,z)&\mapsto (j \mapsto (u.z)(j):=u(j+z)) \end{aligned}$$
(1.1)

and the quotient

$$\begin{aligned} \mathrm{Conf }:= \mathrm{Mon }/ \mathbb Z \,\end{aligned}$$

is called the configuration space. Let \(\pi :\mathrm{Mon }\rightarrow \mathrm{Conf }\), \(u \mapsto \bar{u}\), be the quotient map.

By construction \(\mathfrak{s }^g \in \mathrm{Mon }\) and \(\overline{\mathfrak{s }}^g := \pi (\mathfrak{s }^g) \in \mathrm{Conf }\). This defines maps

$$\begin{aligned} \mathfrak{s }:\mathcal{R }(M) \rightarrow \mathrm{Mon }, \overline{\mathfrak{s }}:\mathcal{R }(M) \rightarrow \mathrm{Conf }. \end{aligned}$$

We would like to say that \(\overline{\mathfrak{s }}\) is continuous. To make formal sense of this, we introduce a topology on \(\mathrm{Mon }\) and \(\mathrm{Conf }\).

Definition 1.3

(arsinh-topology) The topology induced by the metric \(d_a\) defined by

$$\begin{aligned} \forall u,v \in \mathbb R \,^\mathbb{Z \,}: d_a(u,v) := \sup _{j \in \mathbb Z \,}{|\mathrm{arsinh }(u(j)) - \mathrm{arsinh }(v(j))}| \in [0,\infty ] \end{aligned}$$

on \(\mathbb R \,^\mathbb Z \,\) is called the \(\mathrm{arsinh }\)-topology. The group action \(\tau \) acts by isometries with respect to \(d_a\) and the quotient topology on \(\mathrm{Conf }\) is induced by the metric \(\bar{d}_a\) described by

$$\begin{aligned} \forall u \in \bar{u}, v \in \bar{v} \in \mathrm{Conf }: \bar{d}_a(\bar{u}, \bar{v}) = \inf _{j \in \mathbb Z \,}{d_a(u,v.j)}. \end{aligned}$$
(1.2)

The use of this metric on the quotient is common in metric geometry, cf. [6, Lemma 3.3.6].

This allows us to formulate our main result.

Main Theorem 1

The map \(\overline{\mathfrak{s }} = \pi \circ \mathfrak{s }\) admits a lift \(\widehat{\mathfrak{s }}\) against \(\pi \) such that

(1.3)

is a commutative diagram of topological spaces.

From this one can quickly deduce the following result, which is perhaps a little more intuitive.

Main Theorem 2

There exists a family of functions \(\{ \lambda _j \in \mathcal C ^0(\mathcal{R }(M),\mathbb R \,) \}_{j \in \mathbb Z \,}\) such that, for all \(g \in \mathcal{R }(M)\), the sequence \((\lambda _j(g))_{j \in \mathbb Z \,}\) represents all the eigenvalues of \({{{{\big /}\!\!\!\!{D}}}}^g\) (counted with multiplicities). In addition, the sequence \(\{ \mathrm{arsinh }(\lambda _j) \}_{j \in \mathbb Z \,}\) is equicontinuous and non-decreasing, i.e. all \(g \in \mathcal{R }(M)\) satisfy \(\lambda _j(g) \le \lambda _k(g)\), if \(j \le k\).

Proof of Main Theorem 2

Clearly, the evaluation \(\mathrm{ev }_j:(\mathrm{Mon },d_a) \rightarrow \mathbb R \,\), \(u \mapsto u(j)\), is a continuous map for any \(j \in \mathbb Z \,\). Consequently, by Main Theorem 1, the functions \(\{ \lambda _j := \mathrm{ev }_j \circ \widehat{\mathfrak{s }} \}_{j \in \mathbb Z \,}\) have the desired properties.\(\square \)

Remark 1.4

(intuitive explanation) It is a more subtle problem than one might think to choose functions \(\{ \lambda _j \}_{j \in \mathbb Z \,}\), which depend continuously on \(g \in \mathcal{R }(M)\) and represent the entire Dirac spectrum. The functions induced by \(\mathfrak{s }\) (we call these \(\rho _j := \mathrm{ev }_j \circ \mathfrak{s }\) for the moment) are not continuous in general. To see this, imagine a continuous path of metrics \((g_t)_{t \in \mathbb R \,}\) and consider \(\rho _j:\mathbb R \,\rightarrow \mathbb R \,\) as functions of \(t\) (see Fig. 1). Since \(\rho _0(t)\) is the first eigenvalue \(\ge 0\) of \({{{{\big /}\!\!\!\!{D}}}}^{g_t}\), this function will have a jump at point \(t_0\), where \(\rho _0(t_0)>1\) and \(\rho _{-1}(t_0)=0\). This can cause discontinuities in all the other functions \(\rho _j\) as well.

However, for any \(k \in \mathbb Z \,\), the sequence \(\rho _j^{\prime } := \rho _{j+k}\), \(j \in \mathbb Z \,\), gives another enumeration of the spectrum. Intuitively, Main Theorem 1 states that if one uses this freedom in the enumeration of the eigenvalues at each metric in the “right” way, one obtains a globally well-defined family of continuous functions representing all the Dirac eigenvalues.

Fig. 1
figure 1

A zero of \(\rho _{-1}\) at \(t_0\) can cause discontinuities at \(t_0\) in all \(\rho _j\)

Full size image

The rest of this paper is organized as follows: After a short review of some fundamental results in Sect. 2 and a slight generalization of our notation in Sect. 3, the main part of the paper will be Sect. 4, which is devoted to build up some technical results needed in the proof of Main Theorem 1. Finally, in Sect. 5 we will investigate to what extent the functions \(\widehat{\mathfrak{s }}\) and \(\overline{\mathfrak{s }}\) descend to certain quotients of \(\mathcal{R }(M)\) called moduli spaces. Our central result will be that there exists an obstruction, the spectral flow, for \(\widehat{\mathfrak{s }}\) to descend onto \(\mathcal{R }(M) / \mathrm{Diff }^{\mathrm{spin }}(M)\). This will be made precise in Definition 5.1 and Lemma 5.7 and the central result will be stated in Main Theorem 5.

Using these results the actual proof of Main Theorem 1 becomes very short.

Proof of Main Theorem 1

By Theorem 4.11, the map \(\bar{\mathfrak{s }}:(\mathcal{R }(M),\mathcal C ^1) \rightarrow (\mathrm{Conf }, \bar{d}_a)\) is continuous. By Theorem 4.12, the map \(\pi :(\mathrm{Mon },d_a) \rightarrow (\mathrm{Conf },\bar{d}_a)\) is a covering map. Since \(\mathcal{R }(M)\) is path-connected, locally path-connected and simply connected, the result follows from the Lifting Theorem of Algebraic Topology. \(\square \)

Remark 1.5

(uniqueness) From this proof, we see that the lift \(\widehat{\mathfrak{s }}\) is not unique. In fact there are \(\mathbb Z \,\) possibilities of how to lift \(\overline{\mathfrak{s }}\) against \(\pi \). One can use this freedom to arrange that \(\widehat{\mathfrak{s }}^{g_0} = \mathfrak{s }^{g_0}\) for one fixed \(g_0 \in \mathcal{R }(M)\).

2 Fundamental results

Theorem 2.1

(identification of spinor bundles, cf. [14]) Let \(g \in \mathcal{R }(M)\) be a fixed metric. For every \(h \in \mathcal{R }(M)\), there exists an isometry of Hilbert spaces \(\bar{\beta }_g^h: \Gamma _{L^2}(\Sigma ^gM) \rightarrow \Gamma _{L^2}(\Sigma ^hM)\), such that the operator

$$\begin{aligned} {{{{\big /}\!\!\!\!{D}}}}^h_g := \bar{\beta }^g_h \circ {{{{\big /}\!\!\!\!{D}}}}^h \circ \bar{\beta }_g^h: \Gamma _{L^2}(\Sigma ^gM) \rightarrow \Gamma _{L^2}(\Sigma ^gM) \end{aligned}$$

is closed, densely defined on \(\Gamma _{H^1}(\Sigma ^g M)\), isospectral to \({{{{\big /}\!\!\!\!{D}}}}^h\), and such that the map

$$\begin{aligned} {{{{\big /}\!\!\!\!{D}}}}_g:\mathcal{R }(M) \rightarrow B(\Gamma _{H^1}(\Sigma ^gM),\Gamma _{L^2}(\Sigma ^gM)), \quad h \mapsto {{{{\big /}\!\!\!\!{D}}}}_g^h, \end{aligned}$$

is continuous. (Here \(B(\_)\) denotes the space of bounded linear operators endowed with the operator norm.)

Although the following theorem is actually not needed in the proof of Main Theorem 1, it is nevertheless worth mentioning to give an impression of what is already well known about the continuity of Dirac spectra. It implies that a bounded spectral interval of the Dirac operator can be described locally by continuous functions. Consequently, Main Theorem 1 can be thought of as a global analogue of this local result.

Theorem 2.2

[2, Prop. 7.1] Let \((M,g)\) be a closed Riemannian spin manifold with Dirac operator \({{{{\big /}\!\!\!\!{D}}}}^g\) having spectrum \({\text{ spec }}{{{{\big /}\!\!\!\!{D}}}}^g\). Let \(\Lambda > 0\) such that \(-\Lambda ,\Lambda \notin {\text{ spec }}{{{{\big /}\!\!\!\!{D}}}}^g\) and enumerate

$$\begin{aligned} {\mathrm{spec}}\,{{{{\big /}\!\!\!\!{D}}}}^g \cap \mathopen ] -\Lambda ,\Lambda \mathclose [ = \{ \lambda _1 \le \lambda _2 \le \cdots \le \lambda _n \}. \end{aligned}$$

For any \(\varepsilon > 0\), there exists a \(\mathcal C ^1\)-neighbourhood \(U\) of \(g\) such that for any \(g^{\prime } \in U\)

  1. (i)

    \({\text{ spec }}{{{{\big /}\!\!\!\!{D}}}}^{g^{\prime }} \cap \mathopen ] -\Lambda , \Lambda \mathclose [ = \{ \lambda ^{\prime }_1 \le \cdots \le \lambda ^{\prime }_n \}\),

  2. (ii)

    \(\forall 1 \le i \le n: |\lambda _i - \lambda _i^{\prime }| < \varepsilon \).

3 Families of discrete operators

For the proof of Main Theorem 1, we need a version of the function \(\mathfrak{s }^g\) defined for operators slightly more general than Dirac operators. In this section, we introduce the necessary definitions and notational conventions. Let \(X,Y\) be complex Banach spaces. We denote by \(C(X,Y)\) the space of closed unbounded operators \(T:X \supset \mathrm{dom }(T) \rightarrow Y\). Let \(B(X,Y)\) denote the bounded operators \(X \rightarrow Y\). We set \(C(X):=C(X,X)\) and \(B(X):=B(X,X)\). The spectrum of \(T\) is denoted by \({\text{ spec }}T \subset \mathbb{C }\,\).

Definition 3.1

(discrete operator) An operator \(T \in C(X)\) is discrete, if \(T\) has compact resolvent and \({\text{ spec }}T \subset \mathbb R \,\) is unbounded from both sides. (It follows that \({\text{ spec }}T\) is closed and consists solely of eigenvalues of finite multiplicity.)

Definition 3.2

(ordered spectral function) Let \(T \in C(X)\) be discrete. The sequence \(\mathfrak{s }_T \in \mathbb R \,^\mathbb Z \,\), uniquely defined by the properties

  1. (i)

    \(\mathfrak{s }_T(0) = \min \{\lambda \in {\text{ spec }}T \mid \lambda \ge 0 \}\).

  2. (ii)

    \(\forall i,j \in \mathbb Z \,: i \le j \Longrightarrow \mathfrak{s }_T(i) \le \mathfrak{s }_T(j)\).

  3. (iii)

    \(\forall \lambda \in \mathbb R \,: \sharp (\mathfrak{s }_T)^{-1}(\lambda )= \dim \ker (T - \lambda ) \).

is the ordered spectral function of \(T\).

Definition 3.3

(spectral parts) Let \(T \in C(X)\) be discrete. To denote parts of the ordered spectrum, we introduce the following notation: If \(I \subset \mathbb R \,\) is an interval, then \((\mathfrak{s }_T)^{-1}(I) = \{k, k+1, \ldots , l\}\) for some \(k, l \in \mathbb Z \,\), \(k \le l\). The sequence

$$\begin{aligned} \mathfrak{sp }_T(I) := (\mathfrak{s }_T(i))_{k \le i \le l}, \end{aligned}$$

is the spectral part of \(T\) in \(I\).

Definition 3.4

(discrete family) Let \(E\) be any set. A map \(T:E \rightarrow C(X)\) is a discrete family, if for any \(e \in E\), the operator \(T_e\) is discrete in the sense of Definition 3.1. We obtain a function

$$\begin{aligned} \begin{array}{rcl} \mathfrak{s }_T:E &{} \rightarrow &{} \mathrm{Mon } \\ e &{} \mapsto &{} \mathfrak{s }_T^e:=\mathfrak{s }_{T(e)}. \end{array} \end{aligned}$$

Analogously, we set \(\mathfrak{sp }_T^e := \mathfrak{sp }_{T(e)}\).

Remark 3.5

(family of Dirac operators) In view of Theorem 2.1, we can apply the above in particular to Dirac operators. Namely, we fix \(g \in \mathcal{R }(M)\) and set \(X:=L^2(\Sigma ^g M)\) and \(E:=\mathcal{R }(M)\). Then \(h \mapsto {{{{\big /}\!\!\!\!{D}}}}^h_g\) is a discrete family. We will suppress its name in notation and just write \(\mathfrak{s }^h_g\) for the ordered spectral function of \({{{{\big /}\!\!\!\!{D}}}}^h_g\). Since \({{{{\big /}\!\!\!\!{D}}}}^h\) and \({{{{\big /}\!\!\!\!{D}}}}^h_g\) are isospectral, we can ignore the reference metric entirely and simply write \(\mathfrak{s }^h\).

4 Proof of Main Theorem 1

In this section, we carry out the details of the proof of Main Theorem 1. The idea to construct the \(\mathrm{arsinh }\)-topology was inspired by a paper of John Lott, cf. [13, Theorem 2]. The arguments require some basic notions from analytic pertubation theory. A modified version of some results by Kato is needed, cf. [11]. Applying analytic pertubation theory to families of Dirac operators is a technique that is also used in other contexts, cf. [4, 5, 10].

Let \(X\),\(Y\) be complex Banach spaces, and let \(X^{\prime }\) be the topological dual space of \(X\). For any operator \(T\), we denote its adjoint by \(T^*\). Let \(\Omega \subset \mathbb{C }\) be an open and connected subset. Recall that a function \(f:\Omega \rightarrow X\) is holomorphic, if for all \(\zeta _0 \in \Omega \), the limit

$$\begin{aligned} f^{\prime }(\zeta _0) := \lim _{\zeta \rightarrow \zeta _0}{\tfrac{f(\zeta )-f(\zeta _0)}{\zeta -\zeta _0}} \end{aligned}$$

exists in \((X,\Vert \_\Vert _X)\). A family of operators \(T:\Omega \rightarrow B(X,Y)\) is called bounded holomorphic, if \(T\) is a holomorphic map in the above sense. To treat the unbounded case, the following notions are crucial.

Definition 4.1

(holomorphic family of type (A)) A family of operators \(T:\Omega \rightarrow C(X,Y)\), \(\zeta \mapsto T_\zeta \), is holomorphic of type (A), if the domain \(\mathrm{dom }(T_\zeta ) =: \mathrm{dom }(T)\) is independent of \(\zeta \), and for any \(x \in \mathrm{dom }(T)\) the map \(\Omega \rightarrow Y\), \(\zeta \mapsto T_\zeta x\), is holomorphic.

Definition 4.2

(self-adjoint holomorphic family of type(A)) A family \(T:\Omega \rightarrow C(H)\) is self-adjoint holomorphic of type (A), if it is holomorphic of type (A), \(H\) is a Hilbert space, \(\Omega \) is symmetric with respect to complex conjugation, and

$$\begin{aligned} \forall \zeta \in \Omega : T^*_{\zeta } = T_{\bar{\zeta }}. \end{aligned}$$

These families are particularly important for our purposes due to the following useful theorem.

Theorem 4.3

[11, VII.§3.5, Thm. 3.9] Let \(T:\Omega \rightarrow C(H)\) be a self-adjoint holomorphic family of type (A), and let \(I \subset \Omega \cap \mathbb R \,\) be an interval. Assume that \(T_\zeta \) has compact resolvent for all \(\zeta \in \Omega \). Then, there exists a family of functions \(\{\lambda _n \in \mathcal C ^\omega (I,\mathbb R \,)\}_{n \in \mathbb N \,}\) and a family of functions \(\{u_n \in \mathcal C ^\omega (I,H)\}_{n \in \mathbb N \,}\) such that for all \(t \in I\), the \((\lambda _n(t))_{n \in \mathbb N \,}\) represent all the eigenvalues of \(T_t\) (counted with multiplicity), \(T_tu_n(t)=\lambda _n(t)u_n(t)\), and the \((u_n(t))_{n \in \mathbb N \,}\) form a complete orthonormal system of \(H\).

Derivatives of holomorphic families can be estimated using the following theorem.

Theorem 4.4

[11, VII.§2.1, p.375f] Let \(T:\Omega \rightarrow C(X,Y)\) be a holomorphic family of type (A). For any \(\zeta \in \Omega \) define the operator

$$\begin{aligned} T^{\prime }_{\zeta }: \mathrm{dom }(T) \rightarrow Y, \quad \quad u \mapsto T^{\prime }_{\zeta }u := \tfrac{\mathrm{d}}{\mathrm{d}\zeta }(T_{\zeta }u). \end{aligned}$$

Then \(T^{\prime }\) is a map from \(\Omega \) to the unbounded operators \(X \rightarrow Y\) (but \(T^{\prime }_{\zeta }\) is in general not closed). For any compact \(K \subset \Omega \), there exists \(C_K>0\) such that

$$\begin{aligned} \forall \zeta \in K: \; \forall u \in \mathrm{dom }(T): \; \Vert T^{\prime }_{\zeta }u\Vert _Y \le C_K(\Vert u\Vert _X + \Vert T_{\zeta }u\Vert _Y). \end{aligned}$$

If \(\zeta _0 \in K\) is arbitrary, \(Z:=\mathrm{dom }(T)\) and \(\Vert u\Vert _Z := \Vert u\Vert _X + \Vert T_{\zeta _0}u\Vert _Y\), then \(C_K := \alpha _K^{-1} \beta _K\) does the job, where

$$\begin{aligned} \alpha _K := \inf _{\zeta \in K}{ \inf _{\Vert u\Vert _Z=1}{\Vert u\Vert _X + \Vert T_{\zeta }u\Vert _Y}}, \quad \quad \beta _K := \sup _{\zeta \in K}{\Vert T^{\prime }_{\zeta }\Vert _{B(Z,Y)}}. \end{aligned}$$
(4.1)

This can be used to prove the following result about the growth of eigenvalues.

Theorem 4.5

[11, VII.§3.4, Thm. 3.6] Let \(T:\Omega \rightarrow C(H)\) be a self-adjoint holomorphic family of type (A). Let \(I \subset \Omega \cap \mathbb R \,\) be a compact interval, and let \(J \subset I\) be open. Assume that \(\lambda \in \mathcal C ^\omega (J,\mathbb R \,)\) is an eigenvalue function, i.e. for all \( t \in J\) the value \(\lambda (t)\) is an eigenvalue of \(T_t\). Then

$$\begin{aligned} \forall t,t_0 \in J: \; |\lambda (t) - \lambda (t_0)| \le (1 + |\lambda (t_0)|)(\exp (C_I |t-t_0|) - 1), \end{aligned}$$
(4.2)

where \(C_I\) is the constant from Theorem 4.4.

The preceding Theorem 4.5 provides two key insights into the growth of eigenvalue functions. First of all, it is remarkable that the constant \(C_I\) in (4.2) does not depend on the eigenvalue function \(\lambda \). In particular, if we consider a family of eigenvalue functions \(\{\lambda _n\}_{n \in \mathbb N \,}\) as in Theorem 4.4, the constant \(C_I\) is uniform in \(n\). This will be crucial later in the proof of Corollary 4.8. Secondly, we see that, due to the factor \(1 + |\lambda (t_0)|\) in (4.2), an eigenvalue function grows faster the larger it is. This is the reason why we cannot expect the continuity result of Main Theorem 1 to hold for the ordinary supremum norm. However, as we will show in the next corollary, we can get rid of this factor by reformulating (4.2) in terms of the \(\mathrm{arsinh }\)-topology

Corollary 4.6

(Growth of eigenvalues) In the situation of Theorem 4.5, the following holds in addition: For any \(t_0 \in I\) and \(\varepsilon > 0\), there exists \(\delta > 0\) such that for all \(t \in I_{\delta }(t_0) \cap J\) and all eigenvalue functions \(\lambda \in \mathcal C ^\omega (J,\mathbb R \,)\) we have

$$\begin{aligned} |\mathrm{arsinh }(\lambda (t)) - \mathrm{arsinh }(\lambda (t_0))| < \varepsilon . \end{aligned}$$
(4.3)

There exist universal constants (i.e. independent of the family \(T\)) \(C_1,C_2 > 0\) such that

$$\begin{aligned} \delta := C_I^{-1} \ln (\min (C_1, \varepsilon C_2) + 1 ) \end{aligned}$$
(4.4)

does the job.

Proof

The key observation needed is that \(\mathrm{arsinh }(t)\) grows slower the larger \(|t|\) gets. This follows simply from the formula \(\mathrm{arsinh }^{\prime }(t) = (1+t^2)^{-1/2}\). We will show that this neutralizes the \((1+|\lambda (t_0)|)\)-factor in (4.2) when the growth of \(\lambda \) is measured in the \(\mathrm{arsinh }\)-metric. The \(\exp \)-term in (4.2) can be estimated by a standard continuity argument.

Step 1 (\(\exp \)-term): The function \(\alpha :\mathbb R \,\rightarrow \mathbb R \,\), \(t \mapsto \exp (C_I |t-t_0|) - 1\), is continuous and satisfies \(\alpha (t_0)=0\). Notice that for \(b > 0\)

$$\begin{aligned} |\alpha (t)| < b \Longleftrightarrow |t-t_0| < C_I^{-1} \ln (b+1). \end{aligned}$$
(4.5)

In particular there exists \(\delta _1 > 0\) such that

$$\begin{aligned} \forall t \in I_{\delta _1}(t_0): |\alpha (t)| < \tfrac{1}{4}. \end{aligned}$$
(4.6)

So let \(t \in I_{\delta _1}(t_0)\).

Step 2 (preliminary estimate): Setting \(\lambda _0:=\lambda (t_0)\), we can reformulate (4.2) as

$$\begin{aligned} \lambda _0 - (1 + |\lambda _0|)\alpha (t) < \lambda (t) < \lambda _0 +(1+|\lambda _0|) \alpha (t). \end{aligned}$$
(4.7)

Since

$$\begin{aligned} \lim _{|R| \rightarrow \infty }{\tfrac{|R|}{1+|R|}} = 1 , \end{aligned}$$

and the convergence is monotonously increasing, there exists \(R > 0\) such that

$$\begin{aligned} \forall |\eta | \ge R: \tfrac{1}{2} < \tfrac{|\eta |}{1 + |\eta |}. \end{aligned}$$
(4.8)

Now assume \(|\lambda _0| \ge R\). In case \(\lambda _0 \ge R > 0\), we calculate

$$\begin{aligned} \tfrac{\lambda _0}{1 + \lambda _0} > \tfrac{1}{2} \stackrel{\text{(4.6) }}{\ge } 2 \alpha (t) \Longrightarrow \tfrac{1}{2} \lambda _0 \ge \alpha (t)(1+\lambda _0) \Longrightarrow \lambda _0 - \alpha (t)(1 +|\lambda _0|) \ge \tfrac{1}{2} \lambda _0. \end{aligned}$$
(4.9)

Analogously, if \(\lambda _0 \le -R < 0\), we calculate

$$\begin{aligned} \lambda _0 + \alpha (t) (1 + |\lambda _0|) < \tfrac{1}{2} \lambda _0. \end{aligned}$$
(4.10)

Step 3 (\(\mathrm{arsinh }\)-metric): Define the constants

$$\begin{aligned} C_0 := \sup _{t \in \mathbb R \,}{\tfrac{1 + |t|}{\sqrt{1 + t^2}}}, \quad C_1 := \tfrac{1}{4}, \quad C_2 := \min \left( \tfrac{1}{R+1}, \tfrac{1}{2C_0} \right) , \end{aligned}$$

and set

$$\begin{aligned} \delta _2 := C_I^{-1} \ln (\min (C_1, \varepsilon C_2) + 1 ). \end{aligned}$$

By (4.5) this implies

$$\begin{aligned} \forall t \in I_{\delta _2}(t_0): \alpha (t) < \min (C_1,\varepsilon C_2) \le \varepsilon C_2. \end{aligned}$$
(4.11)

So let \(t \in I_{\delta _2}(t_0)\) be arbitrary and set \(c_\pm := \lambda _0 \pm (1 + |\lambda _0|)\alpha (t) \). It follows from the Taylor series expansion of \(\mathrm{arsinh }\) that there exists \(\xi \in [\lambda _0,c_+]\) such that

$$\begin{aligned} \mathrm{arsinh }(c_+) - \mathrm{arsinh }(\lambda _0) = \mathrm{arsinh }^{\prime }(\xi ) (1 + |\lambda _0|)\alpha (t) =\frac{(1 + |\lambda _0|)}{\sqrt{1 + \xi ^2}} \alpha (t). \end{aligned}$$
(4.12)

Now in case \(|\lambda _0| \le R\), we continue this estimate by

$$\begin{aligned} (4.12) \le (1 + |\lambda _0|) \alpha (t) \le (1 + R) \alpha (t) \stackrel{\text{(4.11) }}{<} \varepsilon \end{aligned}$$

In case \(\lambda _0 \ge R\), we continue this estimate by

$$\begin{aligned} (4.12) \le \frac{(1 + |\lambda _0|)}{\sqrt{1 + \lambda _0^2}} \alpha (t) \le C_0 \alpha (t) \stackrel{\text{(4.11) }}{<} \varepsilon . \end{aligned}$$

In case \(\lambda _0 \le -R\), we continue this estimate by

$$\begin{aligned} (4.12) \le \frac{(1 + |\lambda _0|)}{\sqrt{1 + c_+^2}} \alpha (t) \stackrel{\text{(4.10) }}{\le } \frac{(1 + |\lambda _0|)}{\sqrt{1 + \tfrac{1}{4} \lambda _0^2}} \alpha (t) \le 2 C_0 \alpha (t) \stackrel{\text{(4.11) }}{<} \varepsilon . \end{aligned}$$

Consequently, since \(\mathrm{arsinh }\) is strictly increasing, in all cases we obtain

$$\begin{aligned} \mathrm{arsinh }(\lambda (t)) \stackrel{\text{(4.7) }}{<}\mathrm{arsinh }(\lambda _0 + (1 + |\lambda _0|)\alpha (t)) < \mathrm{arsinh }(\lambda _0) + \varepsilon . \end{aligned}$$

By an analogous argument, we obtain

$$\begin{aligned} \mathrm{arsinh }(\lambda (t)) > \mathrm{arsinh }(\lambda _0 - (1+|\lambda _0|)\alpha (t)) \ge \mathrm{arsinh }(\lambda _0) - \varepsilon . \end{aligned}$$

This proves the claim. \(\square \)

In the next step, we will apply the preceding result to discrete families.

Notation 4.7

Since the following proof is somewhat technical, we abbreviate

and set \(d_a(x,y):=|\mathrm{a }(x)-\mathrm{a }(y)|\) for \(x,y \in \mathbb R \,\). For any \(\varepsilon > 0\), the \(\varepsilon \)-neighbourhoods of \(x \in \mathbb R \,\) and \(\varepsilon \)-hulls of a set \(S \subset \mathbb R \,\) will be denoted, respectively, by

$$\begin{aligned} I_\varepsilon (x) := \{ t \in \mathbb R \,\mid |t-x| < \varepsilon \}, \quad I_\varepsilon (S) := \bigcup _{x \in S}{I_\varepsilon (x)}. \end{aligned}$$

Corollary 4.8

(Spectral Growth) Let \(\Omega \subset \mathbb{C }\) be open, let \(I \subset \Omega \cap \mathbb R \,\) be an interval, and let \(T:\Omega \rightarrow C(H)\) be a discrete and self-adjoint holomorphic family of type (A). For any \(t_0 \in I\) and \(\varepsilon > 0\), there exists \(\delta > 0\) such that

$$\begin{aligned} \forall t \in I_\delta (t_0) \cap I: \exists k \in \mathbb Z \,: \forall j \in \mathbb Z \,: d_a(\mathfrak{s }^{t_0}_T(j),\mathfrak{s }^{t}_T(j+k)) < \varepsilon . \end{aligned}$$
(4.13)

Proof

Step 1 (apply Theorem 4.3): Certainly the family of eigenfunctions from Theorem 4.3 can be \(\mathbb Z \,\)-reindexed to a family \(\{ \lambda _j \in \mathcal C ^\omega (I,\mathbb R \,) \}_{j \in \mathbb Z \,}\) satisfying \(\lambda _j(t_0)=\mathfrak{s }_T^{t_0}(j)\), \(j \in \mathbb Z \,\). By Corollary 4.6

$$\begin{aligned} \exists \delta > 0: \forall t \in I_\delta (t_0) \cap I: \forall j \in \mathbb Z \,: |\mathrm{a }(\lambda _j(t_0)) - \mathrm{a }(\lambda _j(t))| < \varepsilon . \end{aligned}$$

Fix any \(t \in I_\delta (t_0) \cap I\) and let \(\sigma :\mathbb Z \,\rightarrow \mathbb Z \,\) be the bijection satisfying \(\mathfrak{s }_T^t(\sigma (j)) = \lambda _j(t)\). This implies

$$\begin{aligned} \forall j \in \mathbb Z \,: d_a(\mathfrak{s }_T^{t_0}(j),\mathfrak{s }_T^t(\sigma (j)) < \varepsilon , \end{aligned}$$
(4.14)

which is almost (4.13), except that \(\sigma \) might not be given by a translation.

Step 2 (general idea): We will show that we may replace the bijection \(\sigma \) by an increasing bijection \(\tau \), which still satisfies (4.14). Since every increasing bijection \(\mathbb Z \,\rightarrow \mathbb Z \,\) is given by a translation \(\tau ^{(k)}:\mathbb Z \,\rightarrow \mathbb Z \,\), \(z \mapsto z + k\), for some \(k \in \mathbb Z \,\), this implies the claim. To that end we will first show how to modify \(\sigma \) on finite subsets and then use the pigeonhole principle to conclude the argument.

Step 3 (on finite subsets): For any \(n \in \mathbb N \,\) set \(I_n := \{-n, \ldots , n\}\) and consider the function \(\sigma _n:=\sigma |_{I_n}:I_n \rightarrow \mathbb Z \,\). This function is injective and satisfies (4.14) for all \(-n \le j \le n\). Furthermore, setting

$$\begin{aligned} \mathfrak{sp }^T_{t_0}([\lambda _{-n}(t_0), \lambda _n(t_0)])&=: (\lambda _{-n}, \ldots , \lambda _{n}) \end{aligned}$$

we obtain numbers \(n^{\prime },m^{\prime } \in \mathbb Z \,\) such that the eigenvalues \(\mu _j := \mathfrak{s }_T^{t}(j)\) satisfy

$$\begin{aligned} I_\varepsilon (\mathrm{a }(\mathfrak{sp }^T_{t}([\lambda _{-n},\lambda _{n}])))&= (\mathrm{a }(\mu _{n^{\prime }}), \ldots , \mathrm{a }(\mu _{m^{\prime }}) ), \end{aligned}$$
(4.15)

and we have the estimate

$$\begin{aligned} \forall -n \le j \le n:&|\mathrm{a }(\lambda _j) - \mathrm{a }(\mu _{\sigma _n(j)}) | < \varepsilon . \end{aligned}$$
(4.16)

We will show that \(\sigma _n\) can be modified to an increasing injection \(\tilde{\sigma }_n\), which satisfies \(\mathrm{im }(\sigma _n) = \mathrm{im }(\tilde{\sigma }_n)\) and (4.16). To that end, choose any \(-n \le i < j \le n\) and assume that \(\sigma _n(j)<\sigma _n(i)\). Notice that by construction

$$\begin{aligned} i < j \Longrightarrow \lambda _i \le \lambda _j , \quad \quad \sigma _n(j) < \sigma _n(i) \Longrightarrow \mu _{\sigma _n(j)} \le \mu _{\sigma _n(i)}. \end{aligned}$$
(4.17)

Define the function \(\tilde{\sigma }_n\) by setting

$$\begin{aligned} \tilde{\sigma }_n|_{\{-n, \ldots , n\} \setminus \{i,j\}} := \sigma _n, \quad \quad \tilde{\sigma }_n(i)=\sigma _n(j), \quad \quad \tilde{\sigma }_n(j)=\sigma _n(i). \end{aligned}$$

It is clear that \(\tilde{\sigma }_n\) is still injective and \(\mathrm{im }(\tilde{\sigma }_n) = \mathrm{im }(\sigma _n)\). To show that it still satisfies (4.16), we distinguish two cases (see Fig. 2). First consider the case that \(\lambda _i = \lambda _j\). Then, it follows automatically that

$$\begin{aligned} |\mathrm{a }(\lambda _i) - \mathrm{a }(\mu _{\tilde{\sigma }_n(i)})| = |\mathrm{a }(\lambda _j) - \mathrm{a }(\mu _{\sigma _n(j)})| < \varepsilon , \end{aligned}$$

and the same for \(j\). In case \(\lambda _i \ne \lambda _j\), it follows that \(\lambda _i < \lambda _j\). This implies

$$\begin{aligned} \mathrm{a }(\lambda _i) - \varepsilon < \mathrm{a }(\lambda _j) - \varepsilon < \mathrm{a }(\mu _{\sigma _n(j)}) \stackrel{\text{(4.17) }}{\le } \mathrm{a }(\mu _{\sigma _n(i)}) < \mathrm{a }(\lambda _i) + \varepsilon < \mathrm{a }(\lambda _j) + \varepsilon , \end{aligned}$$

hence

$$\begin{aligned} \mu _{\sigma _n(i)},\mu _{\sigma _n(j)} \in I_\varepsilon (\mathrm{a }(\lambda _i)) \cap I_\varepsilon (\mathrm{a }(\lambda _j)). \end{aligned}$$

In particular, this intersection is not empty. Consequently, \(\tilde{\sigma }_n\) satisfies (4.16). By repeating this procedure for all index pairs \((i,j)\), \(-n \le i \le n\), \(i<j \le n\), it follows that \(\sigma _n\) can be modified finitely many times in this manner to obtain an increasing injection having the same image, which still satisfies (4.16). For simplicity, denote this function also by \(\tilde{\sigma }_n\) and, define

$$\begin{aligned} \begin{array}{rcl} \tilde{\tau }_n:\mathbb Z \, &{} \rightarrow &{} \mathbb Z \, \\ j &{} \mapsto &{} {\left\{ \begin{array}{ll} \tilde{\sigma }_n(j), &{} -n \le j \le n, \\ \sigma (j), &{} \text{ otherwise }. \end{array}\right. } \end{array} \end{aligned}$$

This function is still bijective, still satisfies (4.14) and is increasing on \(I_n\). Define \(J_n:=\tilde{\tau }_n(I_n)\).

Fig. 2
figure 2

The two possibilities for \(\lambda _j\)

Full size image

Step 4 (pigeonhole principle): Unfortunately, it might happen that \(\tilde{\tau }_{n+1}|_{I_n} \ne \tilde{\tau }_n\). Due to (4.16) however, there exists \(n_1\) such that all \(n \ge n_1\) satisfy \(\tilde{\tau }_n(I_1) \subset J_{n_1}\). Since there are only finitely many functions \(I_{1} \rightarrow J_{n_1}\), there must be at least one such function occurring infinitely often in the sequence \(\{\tilde{\tau }_n|_{I_1}\}_{n \in \mathbb N \,}\). Thus, there exists an infinite subset \(\mathbb N \,_1 \subset \mathbb N \,_0:=\mathbb N \,\) such that \(\tilde{\tau }_n|_{I_1}\) is the same for all \(n \in \mathbb N \,_1\).

Now, the same holds for \(I_2\): There exists \(n_2 \ge n_1\) such that all \(n \ge n_2\) satisfy \(\tau _n(I_2) \subset J_{n_2}\). Again, since there are only finitely many functions \(I_2 \rightarrow J_{n_2}\), one of them must occur infinitely often in the sequence \(\{\tau _n|_{I_2}\}_{n \in \mathbb N \,_1}\). Consequently, there exists an infinite subset \(\mathbb N \,_2 \subset \mathbb N \,_1\) such that \(\tau _n|_{I_2}\) is the same for all \(n \in \mathbb N \,_2\). This process can be continued indefinitely for all the intervals \(I_\nu \), \(\nu \in \mathbb N \,\). Finally, the function

$$\begin{aligned} \begin{array}{rcl} \tau :\mathbb Z \, &{} \rightarrow &{} \mathbb Z \, \\ j &{} \mapsto &{} \tilde{\tau }_n(j), \; j \in I_\nu , n \in \mathbb N \,_{\nu } \end{array} \end{aligned}$$

does the job: It is well-defined, satisfies (4.14), remains injective and is surjective: Since the sets \(\{I_n\}_{n \in \mathbb N \,}\) exhaust all of \(\mathbb Z \,\), and since the \(\tilde{\tau }_n\) are bijective and increasing on \(I_n\), it follows that the \(J_n\) are also sets of subsequent numbers in \(\mathbb Z \,\). Thus, by injectivity of the \(\tilde{\tau }_n\), the \(\{J_n\}_{n \in \mathbb N \,}\) exhaust all of \(\mathbb Z \,\). \(\square \)

The preceding Corollary 4.8 is almost the result we need to conclude the proof of Main Theorem 1, except that it is formulated only for paths of operators. As a last step we provide a framework, which allows us to pass from paths of operators to families of operators parametrized by more general spaces.

Definition 4.9

(discrete family of type (A)) Let \(H\) be a Hilbert space. A discrete family \(T:E \rightarrow C(H)\) is self-adjoint of type (A), if

  1. (i)

    There exists a dense subspace \(Z \subset H\), such that all \(e \in E\) satisfy \(\mathrm{dom }T_e = Z\). We set \(\mathrm{dom }T:=Z\).

  2. (ii)

    For all \(e \in E\), the operator \(T_e\) is self-adjoint.

  3. (iii)

    There exists a norm \(|\_ |\) on \(Z\) such that, for all \(e \in E\), the operator \(T_e:(Z,| \_ |) \rightarrow (H, \Vert \_ \Vert _H)\) is bounded and the graph norm of \(T_e\) is equivalent to \(| \_ |\).

  4. (vi)

    \(E\) is a topological space.

  5. (v)

    The map \(E \rightarrow B(Z,H)\), \(e \mapsto T_e\), is continuous.

Theorem 4.10

Let \(T:E \rightarrow C(H)\) be a discrete self-adjoint family of type (A). For any \(e_0 \in E\) and any \(\varepsilon > 0\) there exists an open neighbourhood \(U \subset E\) of \(e_0\) such that

$$\begin{aligned} \forall e \in U: \exists k \in \mathbb Z \,: \forall j \in \mathbb Z \,: d_a(\mathfrak{s }^{e_0}_T(j),\mathfrak{s }^e_T(j+k)) < \varepsilon . \end{aligned}$$

Proof

Let \(\varepsilon > 0\) and \(e_0 \in E\) be arbitrary. As in Definition 4.9, let \(\Vert \_ \Vert _H\) be the norm in \(H\), \(Z:=\mathrm{dom }T\), and let \(\Vert \_ \Vert _Z\) be the graph norm of \(T_{e_0}\) on \(Z\). Finally, let \(\Vert \_ \Vert \) be the associated operator norm in \(B(Z,H)\) (which is then also equivalent to the operator norm induced by \(| \_ |\)).

Step 1 (setup and strategy): By construction, for any \(e_1 \in E\)

$$\begin{aligned} D_{e_1}(\zeta ) := \zeta T_{e_1} + (1-\zeta ) T_{e_0} = T_{e_0} + \zeta (T_{e_1} - T_{e_0}), \quad \zeta \in \mathbb{C }, \end{aligned}$$

defines a discrete self-adjoint holomorphic family \(D_{e_1}:\mathbb{C }\, \rightarrow C(H)\) of type (A) with domain \(Z\). The idea is to prove the claim using Corollary 4.8. The only problem is that for any two \(e_1, e_2 \in E\), the families \(D_{e_1}\) and \(D_{e_2}\) are different. Hence their constants \(C_{I,e_1}\), \(C_{I,e_2}\) from Theorem 4.4 for the interval \(I\) could differ. Consequently, their associated deltas \(\delta _{e_1}\), \(\delta _{e_2}\) from Corollary 4.8 could also differ. We will show that there exists an open neighbourhood \(U\) around \(e_0\) sufficiently small such that for all \(e_1 \in U\), the \(\delta = \delta _{e_1}\) is \(\ge 1\) , if \(\zeta _0\) is always set to \(\zeta _0:=t_0:=0\). This will prove the claim.

Step 2 (preliminary estimate): Recall that by (4.4) there are \(C_1,C_2 > 0\) such that

$$\begin{aligned} \delta _{e_1} = C_{I,e_1}^{-1} \ln (\min (C_1, \varepsilon C_2) + 1 ). \end{aligned}$$

Since \(\lim _{t \rightarrow 0}{e^t} = 1\), there exists \(\varepsilon _1 > 0\) such that

$$\begin{aligned} \forall |t| \le 2 \varepsilon _1: \exp (t) - 1 \le \min (C_1, \varepsilon C_2). \end{aligned}$$
(4.18)

Step 3 (construction of \(U\)): Since \(T\) is discrete and self-adjoint of type (A), the map \(E \rightarrow B(Z,H)\), \(e \mapsto T_{e}\), is continuous. Consequently, there exists an open neighbourhood \(U\) of \(e_0\) such that

$$\begin{aligned} \forall e_1 \in U: \Vert T_{e_1} - T_{e_0}\Vert < \min \left( \tfrac{1}{2}, \varepsilon _1 \right) . \end{aligned}$$
(4.19)

Now for any \(e_1 \in U\), \(t \in [0,1]\) and \(\varphi \in Z\) we have

$$\begin{aligned} \Vert D_{e_1}(t) \varphi \Vert _H \ge \Vert T_{e_0} \varphi \Vert _H - \Vert T_{e_1} - T_{e_0} \Vert \Vert \varphi \Vert _Z. \end{aligned}$$

Therefore, applying (4.1) to \(D_{e_1}\), we obtain

$$\begin{aligned} \alpha _{I,e_1}&= \inf _{t \in I}{\inf _{\Vert \varphi \Vert _Z=1}}{\Vert \varphi \Vert _H + \Vert D_{e_1}(t) \varphi \Vert _H} \ge 1 - \Vert T_{e_0} - T_{e_0} \Vert \stackrel{\text{(4.19) }}{>} \tfrac{1}{2}, \\ \beta _{I,e_1}&= \sup _{t \in I}{\Vert D^{\prime }_{e_1}(t)\Vert } = \Vert T_{e_1} - T_{e_0}\Vert < \varepsilon _1. \end{aligned}$$

Altogether, we achieved for any \(e_1 \in U\)

$$\begin{aligned} C_{I,e_1} = \alpha _{I,e_1}^{-1} \beta _{I,e_1} < 2 \varepsilon _1. \end{aligned}$$

By (4.18), this implies

$$\begin{aligned} \exp (C_{I,e_1}) - 1 \le \min (C_1, \varepsilon C_2) \Longrightarrow \delta _{e_1} = C_{I,e_1}^{-1}\ln (\min (C_1, \varepsilon C_2)+1) \ge 1, \end{aligned}$$

which proves the claim. \(\square \)

Finally, we apply all our results to Dirac operators.

Theorem 4.11

The map

$$\begin{aligned} \bar{\mathfrak{s }}:(\mathcal{R }(M),\mathcal C ^1) \rightarrow (\mathrm{Conf },\bar{d}_a), \; \; g \mapsto \overline{\mathfrak{s }}^g, \end{aligned}$$

is continuous

Proof

Let \(g_0 \in \mathcal{R }(M)\) and \(\varepsilon > 0\) be arbitrary. By definition of \(\bar{d}_a\), cf. (1.2), it suffices to find an open neighbourhood \(U \subset \mathcal{R }(M)\) such that

$$\begin{aligned} \forall g^{\prime } \in U: \exists k \in \mathbb Z \,: \forall j \in \mathbb Z \,: d_a(\mathfrak{s }^{g}(j), \mathfrak{s }^{g^{\prime }}(j+k)) < \varepsilon . \end{aligned}$$
(4.20)

By Theorem 2.1, the map \(\mathcal{R }(M) \rightarrow B(H^1(\Sigma ^{g_0}M),L^2(\Sigma ^{g_0}M))\), \(h \mapsto {{{{\big /}\!\!\!\!{D}}}}^h_{g_0}\), is a discrete self-adjoint family of type (A). Consequently, by Theorem 4.10, there exists \(U\) such that (4.20) holds. \(\square \)

To apply the Lifting Theorem, we quickly verify that \(\pi \) is a covering map.

Theorem 4.12

The map \(\pi :(\mathrm{Mon },d_a) \rightarrow (\mathrm{Conf }, \bar{d}_a)\) is a covering map with fibre \(\mathbb Z \,\).

Proof

In this proof we also use Notation 4.7. By the definition of \(\tau \) (see (1.1)), \(\mathbb Z \,\) acts on \(\mathrm{Mon }\) by isometries. In particular, \(\tau \) is continuous. We will show that, for each \(u \in \mathrm{Mon }\), there exists an open neighbourhood \(V\) such that

$$\begin{aligned} \pi ^{-1}(\pi (V)) = \dot{\bigcup }_{j \in \mathbb Z \,}{V.j}. \end{aligned}$$
(4.21)

To see this, note that the function \(\mathrm{a }\circ u\) is non-decreasing and proper. The set \(K_0:=(\mathrm{a }\circ u)^{-1}(\mathrm{a }(u(0)))\) is of the form \(K_0 = \{a_0, \ldots , b_0\}\) for some \(a_0 \le b_0\), \(a_0,b_0 \in \mathbb Z \,\). For the same reason, there exist \(b_1\), \(a_{-1} \in \mathbb Z \,\) such that (see also Fig. 3)

$$\begin{aligned} (\mathrm{a }\circ u)^{-1}(\mathrm{a }(u(0)))&= \{a_0, \ldots , b_0\} = K_0, \\ (\mathrm{a }\circ u)^{-1}(\mathrm{a }(u(b_0 + 1)))&= \{b_0+1, \ldots , b_1 \} =: K_1, \\ (\mathrm{a }\circ u)^{-1}(\mathrm{a }(u(a_0 - 1)))&= \{ a_{-1}, \ldots , a_0 - 1\} =:K_{-1}. \end{aligned}$$

Since \(\mathrm{a }(u(\mathbb Z \,))\) is discrete, there exists \(\varepsilon > 0\) such that

$$\begin{aligned} I_{\varepsilon }(\mathrm{a }(u(0))) \cap I_{\varepsilon }(\mathrm{a }(u(b_0+1))) = \emptyset , \quad \quad I_{\varepsilon }(\mathrm{a }(u(0))) \cap I_{\varepsilon }(\mathrm{a }(u(a_0-1))) = \emptyset , \end{aligned}$$

Thus, we obtain open sets

$$\begin{aligned} U_0 := I_{\varepsilon }(\mathrm{a }(u(0))), \quad \quad U_1 := I_{\varepsilon }(\mathrm{a }(u(b_0 + 1))), \quad \quad U_{-1} := I_{\varepsilon }(\mathrm{a }(u(a_0-1))), \end{aligned}$$

which are mutually disjoint. To see that \(V := B_\varepsilon (u)\) satisfies (4.21), suppose to the contrary that there exists \(v \in V\) and \(j \in \mathbb Z \,\) such that \(v.j \in V\). Assume \(j > 0\) (the proof for \(j<0\) is entirely analogous). By hypothesis, this implies that \(\mathrm{a }(v(b_0)) \in U_0\) and \(\mathrm{a }(v(b_0 + j)) = \mathrm{a }((v.j(b_0))) \in U_0 \). But \(\mathrm{a }\circ v\) is non-decreasing, so \(\mathrm{a }(v(b_0 + j)) \ge \mathrm{a }(v(b_0 + 1)) \in U_1\). This implies that \(\mathrm{a }(v(b_0+j)) \notin U_0\), which is a contradiction.

Fig. 3
figure 3

An evenly covered neighbourhood for \(u\)

Full size image

Finally, to see that \(\pi \) is a covering map, let \(u \in [u] \in (\mathrm{Conf },\bar{d}_a)\) be arbitrary. Let \(V\) be an open neighbourhood of \(u\) satisfying (4.21). Then, \(\bar{V} := \pi (V)\) is evenly covered. Thus, \(\pi \) is a covering map. \(\square \)

5 Moduli spaces and spectral flow

In this section, \(M\) is still a compact spin manifold with a fixed topological spin structure \(\Theta \), and \(I:=[0,1]\) denotes the unit interval. Let \(\mathrm{Diff }(M)\) be the diffeomorphism group of \(M\). This group acts canonically on the space of Riemannian metrics via \(\mathcal{R }(M) \times \mathrm{Diff }(M) \rightarrow \mathcal{R }(M)\), \((g,f) \mapsto f^*g\). For any subgroup \(G \subset \mathrm{Diff }(M)\), the quotient space \(\mathcal{R }(M) / G\) is called a moduli space. We investigate when the map \(\bar{\mathfrak{s }}\) (respectively \(\widehat{\mathfrak{s }}\)) from Main Theorem 1 (and hence the family of functions \((\lambda _j)_{j \in \mathbb Z \,}\) from Main Theorem 2) descends to the moduli spaces, where \(G\) is one of the groups

$$\begin{aligned} \mathrm{Diff }^+(M), \quad \mathrm{Diff }^{\mathrm{spin }}(M), \quad \mathrm{Diff }^0(M). \end{aligned}$$

Here, \(\mathrm{Diff }^+(M)\) denotes the subgroup of orientation-preserving diffeomorphisms and \(\mathrm{Diff }^0(M)\) are the diffeomorphisms that are isotopic to the identity. The group \(\mathrm{Diff }^{\mathrm{spin }}(M)\) is defined as follows.

Definition 5.1

(spin diffeomorphism) A diffeomorphism \(f \in \mathrm{Diff }^+(M)\) is a spin diffeomorphism, if there exists a morphism \(F\) of \(\widetilde{\mathrm{GL }}^+_m\)-fibre bundles such that

(5.1)

commutes. We say \(F\) is a spin lift of \(f\), and define

$$\begin{aligned} \mathrm{Diff }^{\mathrm{spin }}(M) := \{ f \in \mathrm{Diff }^+(M) \mid f \text{ is } \text{ a } \text{ spin } \text{ diffeomorphism } \}. \end{aligned}$$

Notice that if \(f\) is a spin diffeomorphism and \(M\) is connected, there are precisely two spin lifts \(F^\pm \) of \(f\), which are related by \(F^+.(-1)=F^-\), where \(-1 \in \widetilde{\mathrm{GL }}^+_m\).

Remark 5.2

(spin isometries) If \(f \in \mathrm{Diff }^{\mathrm{spin }}(M)\), \(h \in \mathcal{R }(M)\), we can set \(g := f^* h\). In this case (5.1) restricts to the analogous diagram

(5.2)

of metric spin structures. We say that \(f\) is a spin isometry in this case. Notice that this implies that \((M,g)\) and \((M,h)\) are Dirac-isospectral, i.e. their sets of Dirac eigenvalues, as well as their multiplicities, are equal. Rephrased in the terminology of the previous chapters, this implies \(\bar{\mathfrak{s }}^g = \bar{\mathfrak{s }}^h\).

These considerations immediately imply the following.

Theorem 5.3

There exists a commutative diagram

(5.3)

With only a little more work, we get an even stronger statement for \(\mathrm{Diff }^0(M)\).

Theorem 5.4

There exists a commutative diagram

(5.4)

Proof

The claim will follow from the universal property of the topological quotient, if we can show that

$$\begin{aligned} \forall g_0 \in \mathcal{R }(M): \forall f \in \mathrm{Diff }^0(M): \widehat{\mathfrak{s }}^{g_0} = \widehat{\mathfrak{s }}^{f^*g_0}. \end{aligned}$$

Let \(H:M \times I \rightarrow M\) be an isotopy from \(H_0 = \mathrm{id }\) to \(H_1 = f\), let \(t \in I\) be arbitrary, and set \(g_t := H_t^*g_0\). Since \(\det ((H_t)_*) \ne 0\) and \(H_0 = \mathrm{id }\), we obtain \(H_t \in \mathrm{Diff }^+(M)\) for all \(t \in I\). Consequently, we obtain a diagram

(5.5)

To be precise, the map \(H_*\) is defined by

$$\begin{aligned} \forall B \in \mathrm{GL }^+M: \forall t \in I: H_*(B,t) := (H_t)_* B. \end{aligned}$$

To show the existence of \(\tilde{H}\), we note that, since \(H\) is an isotopy, it is in particular a homotopy. Consequently, \(H_* \circ \Theta \) is also a homotopy between \((H_0)_* \circ \Theta = \Theta \) and \((H_1)_* \circ \Theta = f_* \circ \Theta \). Clearly \(\widetilde{\mathrm{id }}:\widetilde{\mathrm{GL }}^+M \rightarrow \widetilde{\mathrm{GL }}^+M\) satisfies \(\Theta \circ \widetilde{\mathrm{id }} = \Theta = (H_0)_* \circ \Theta \). Since covering spaces have the homotopy lifting property, there exists \(\tilde{H}\) such that \(\Theta \circ \tilde{H} = H_* \circ \Theta \). We conclude from (5.5) that for any \(t \in I\)

commutes as well. Consequently, for all \(t \in I\), the map \(H_t\) is a spin isometry in the sense of (5.2). Therefore, \((M,g_t)\) and \((M,g_0)\) are Dirac isospectral for all \(t \in I\). This implies \(\widehat{\mathfrak{s }}^{g_0} = \widehat{\mathfrak{s }}^{g_1}\). \(\square \)

Remark 5.5

(a counter-example on the torus) It remains to discuss the group \(\mathrm{Diff }^+(M)\), and one might ask if (5.3) still holds, if \(\mathrm{Diff }^{\mathrm{spin }}(M)\) is replaced by \(\mathrm{Diff }^+(M)\). This is false in general. A counter-example is provided by the standard torus \(\mathbb T \,^3 = \mathbb R \,^3 / \mathbb Z \,^3\) equipped with the induced Euclidean metric \(\bar{g}\). It is well known that the (equivalence classes of) spin structures on \(\mathbb T \,^3\) are in one-to-one correspondence with tuples \(\delta \in \mathbb Z \,_2^3\), see for instance [7]. We denote by \(\mathrm{Spin }^{\bar{g}}_{\delta } \mathbb T \,^3\) the spin structure associated to \(\delta \). The map

$$\begin{aligned} f := \left( \begin{array}{l@{\quad }l@{\quad }l} 1 &{} 1 &{} 0\\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array}\right) :\mathbb R \,^3 \rightarrow \mathbb R \,^3 \end{aligned}$$

preserves \(\mathbb Z \,^3\) and satisfies \(\det (f) = 1\). Hence it induces a diffeomorphism \(\bar{f} \in \mathrm{Diff }^+(\mathbb T \,^3)\). One checks that there is a commutative diagram

The map in the right upper row cannot exist; if it did, the spin structures corresponding to \((1,1,0)\) and \((1,0,0)\) would be equivalent. The left part of the above diagram is a spin isometry analogous to (5.2). Therefore, \({{{{\big /}\!\!\!\!{D}}}}^{f^* \bar{g}}_{(1,0,0)}\) and \({{{{\big /}\!\!\!\!{D}}}}^{\bar{g}}_{(1,1,0)}\) are isospectral, but the spectra of \({{{{\big /}\!\!\!\!{D}}}}^{\bar{g}}_{(1,1,0)}\) and \({{{{\big /}\!\!\!\!{D}}}}^{\bar{g}}_{(1,0,0)}\) are already different as a set. This follows from the explicit computation of the spectra of Euclidean tori, see also [7]. Consequently, \({\text{ spec }}{{{{\big /}\!\!\!\!{D}}}}^{\bar{f}^*g}_{(1,0,0)} \ne {\text{ spec }}{{{{\big /}\!\!\!\!{D}}}}^{\bar{g}}_{(1,0,0)}\), and no diagram analogous to (5.3) can exist for \(\mathrm{Diff }^+(\mathbb T \,^3)\).

Remark 5.6

Notice that in (5.4) the map \(\mathfrak{s }^0\) goes from the moduli space for \(\mathrm{Diff }^0(M)\) to \(\mathrm{Mon }\), whereas in (5.3) the corresponding map \(\mathfrak{s }^{\mathrm{spin }}\) goes to \(\mathrm{Conf }\). Therefore one might ask, if one could improve (5.3) by lifting \(\mathfrak{s }^{\mathrm{spin }}\) to a map \(\widehat{\mathfrak{s }}^{\mathrm{spin }}\) such that

(5.6)

commutes. This question is not so easy to answer, and the rest of this section is devoted to the proof that this is not possible in general. To see where the problem lies, it will be convenient to introduce the following terminology.

Lemma 5.7

(spectral flow)

  1. (i)

    For any \(f \in \mathrm{Diff }^{\mathrm{spin }}(M)\) and \(g \in \mathcal{R }(M)\), there exists a unique \(\mathrm{sf }_g(f) \in \mathbb Z \,\) such that

    $$\begin{aligned} \forall j \in \mathbb Z \,: \widehat{\mathfrak{s }}^{g}(j) = \widehat{\mathfrak{s }}^{f^*g}(j - \mathrm{sf }_g(f)). \end{aligned}$$
    (5.7)

    The induced map \(\mathrm{sf }(f):\mathcal{R }(M) \rightarrow \mathbb Z \,\) is called the spectral flow of \(f\).

  2. (ii)

    Let \(\mathbf g :[0,1] \rightarrow \mathcal{R }(M)\), \(t \mapsto g_t\), be a continuous path of metrics. Let \(\mathfrak{s }:\mathcal{R }(M) \rightarrow \mathrm{Mon }\) be the ordered spectral function for the associated Dirac operators. Take a lift \(\widehat{\mathfrak{s }}:\mathcal{R }(M) \rightarrow \mathrm{Mon }\) of \(\overline{\mathfrak{s }}\) such that \(\widehat{\mathfrak{s }}^{g_0} = \mathfrak{s }^{g_0}\) as in (1.3). There exists a unique integer \(\mathrm{sf }(\mathbf g ) \in \mathbb Z \,\) such that

    $$\begin{aligned} \forall j \in \mathbb Z \,: \widehat{\mathfrak{s }}^{g_1}(j) = \mathfrak{s }^{g_1}(j+\mathrm{sf }(\mathbf g )). \end{aligned}$$

    The integer \(\mathrm{sf }(\mathbf g )\) is called the (Dirac) spectral flow along \(\mathbf g \).

  3. (iii)

    For any \(f \in \mathrm{Diff }^{\mathrm{spin }}(M)\) and any family \(\mathbf g \) joining \(g_0\) and \(f^* g_0\), we have \(\mathrm{sf }_{g_0}(f)=\mathrm{sf }(\mathbf g )\).

Proof

  1. (i)

    By Theorem 5.3, the map \(\overline{\mathfrak{s }}:\mathcal{R }(M) \rightarrow \mathrm{Conf }\) descends to a quotient map \(\mathfrak{s }^{\mathrm{spin }}:\mathcal{R }(M) / \mathrm{Diff }^{\mathrm{spin }}(M) \rightarrow \mathrm{Conf }\). This precisely means that \(\widehat{\mathfrak{s }}^g\) and \(\widehat{\mathfrak{s }}^{f^*g}\) are equal in \(\mathrm{Conf }\). By the definition of \(\mathrm{Conf }\), this implies the existence of \(\mathrm{sf }_g(f)\) as required.

  2. (ii)

    This follows directly from (1.3) and the fact that \(\mathfrak{s }\) and \(\widehat{\mathfrak{s }}\) are equal in \(\mathrm{Conf }\).

  3. (iii)

    Set \(g_1:=f^*g_0\), and let \(\mathbf g \) be a family of metrics joining \(g_0\) and \(g_1\). By Theorem 5.3, we obtain \(\mathfrak{s }^{g_0} = \mathfrak{s }^{g_1}\). Take a lift \(\widehat{\mathfrak{s }}\) satisfying \(\widehat{\mathfrak{s }}^{g_0} = \mathfrak{s }^{g_0}\). This implies for all \(j \in \mathbb Z \,\)

    $$\begin{aligned} \mathfrak{s }^{g_0}(j)&= \widehat{\mathfrak{s }}^{g_0}(j) =\widehat{\mathfrak{s }}^{g_1}(j - \mathrm{sf }_{g_0}(f)) =\mathfrak{s }^{g_1}(j - \mathrm{sf }_{g_0}(f) + \mathrm{sf }(\mathbf g )) \\&= \mathfrak{s }^{g_0}(j - \mathrm{sf }_{g_0}(f) + \mathrm{sf }(\mathbf g )), \end{aligned}$$

    which implies \(\mathrm{sf }(\mathbf g ) - \mathrm{sf }_{g_0}(f) = 0\), since \(\mathfrak{s }^{g_0}\) is non-decreasing and all eigenvalues are of finite multiplicity. \(\square \)

Remark 5.8

(spectral flow) Intuitively, the spectral flow \(\mathrm{sf }(\mathbf g )\) of a path \(\mathbf g :[0,1] \rightarrow \mathcal{R }(M)\) counts the signed number of eigenvalues of the associated path \({{{{\big /}\!\!\!\!{D}}}}^{g_t}\) of Dirac operators that cross \(0\) from below when \(t\) runs from \(0\) to \(1\). The sign is positive, if the crossing is from below, whereas it is negative, if the crossing is from above.

The concept of spectral flow is well known in other contexts. A good introduction can be found in a paper by Phillips, see [15]. Phillips introduces the spectral flow for continuous paths \([0,1] \rightarrow \mathcal F _*^{\mathrm{sa }}\), where \(\mathcal F _*^{\mathrm{sa }}\) is the non-trivial component of the space of self-adjoint Fredholm operators on a complex separable Hilbert space \(H\). In this general setup, the definition of spectral flow is a little tricky, see [15, Prop. 2]. But for paths of Dirac operators, it coincides with the definition given in Lemma 5.7 above (by Theorem 2.1 we can think of all Dirac operators \({{{{\big /}\!\!\!\!{D}}}}^{g_t}\), \(t \in [0,1]\), of a path \(\mathbf g \) as defined on the same Hilbert space). Therefore we have found a convenient alternative for describing the spectral flow in this case using the continuous function \(\widehat{\mathfrak{s }}\).

By [15, Prop. 3], the spectral flow of a path of operators depends only on the homotopy class of the path. Consequently, since \(\mathcal{R }(M)\) is simply-connected, \(\mathrm{sf }(\mathbf g )\) depends only on \(g_0\) and \(g_1\). It follows that \(\mathrm{sf }:\mathrm{Diff }^{\mathrm{spin }}(M) \rightarrow \mathbb Z \,\) is a group homomorphism.

Remark 5.9

The map \(\widehat{\mathfrak{s }}\) certainly descends to

and \(\ker \mathrm{sf }\) is the largest subgroup of \(\mathrm{Diff }^{\mathrm{spin }}(M)\) with this property. Rephrased in these terms, we conclude that the map \(\mathfrak{s }^{\mathrm{spin }}\) from (5.3) lifts to a map \(\widehat{\mathfrak{s }}^{\mathrm{spin }}\) as in (5.6) if and only if \(\mathrm{sf }(f)=0\) for all \(f \in \mathrm{Diff }^{\mathrm{spin }}(M)\). Consequently, we must show the following theorem.

Main Theorem 3

There exists a spin manifold \((M,\Theta )\) and a diffeomorphism \(f \in \mathrm{Diff }^{\mathrm{spin }}(M)\) such that \(\mathrm{sf }(f) \ne 0\).

Proof

The general idea is to use a recent result from differential topology, which implies the existence of a fibre bundle \(P \rightarrow S^1\) with non-vanishing \(\widehat{A}\)-genus and \(2\)-connected fibre type \(M\). Setting \(S^1 = [0,1] / \{0 \sim 1\}\), we can view \(M\) as a fibre \(M=P_{[0]}\). The bundle \(P\) will be isomorphic to a bundle \(P_f\), obtained from the trivial bundle \([0,1] \times M \rightarrow [0,1]\) by identifying \((1,x)\) with \((0,f(x))\), \(x \in M\), for a suitable diffeomorphism \(f \in \mathrm{Diff }^{\mathrm{spin }}(M)\). To show that \(f\) has nontrivial spectral flow, we will cut open the bundle along \([0]\), obtain a trivial fibre bundle \([0,1] \times M \rightarrow [0,1]\), and glue in two infinite half-cylinders at both sides (see Fig. 4). This gives a bundle of the form \(\mathbb R \,\times M \rightarrow \mathbb R \,\), and in each fibre we get a Dirac operator. Using various index theorems, we will show that the \(\widehat{A}\)-genus of \(P\) equals the index of \(\mathbb R \,\times M\), which in turn equals the spectral flow of the associated family of Dirac operators in the fibres, which finally equals the spectral flow of \(f\). The technical details of this proof rely on several other theorems, which are collected in the appendix for convenient reference (one might want to take a look at these first).

Fig. 4
figure 4

\(P\) is cut open at \([0]\), and we obtain a bundle \(P^{\prime } \rightarrow [0,1]\). Then, we glue in two half-cylinders \(Z_0^{\prime }\) and \(Z_1^{\prime }\) to obtain a bundle \(Q \rightarrow \mathbb R \,\)

Full size image

Step 1 (construct a bundle): Apply Theorem 6.1 to \((k,l) = (1,2)\) and obtain a fibre bundle \(P \rightarrow S^1\) with some fibre type \(M\), where \(\dim P = 4n\), \(n\) odd, and

$$\begin{aligned} \widehat{A}(P) \ne 0. \end{aligned}$$
(5.8)

Since \(M\) is \(2\)-connected, \(M\) has a unique spin structure (up to equivalence). It follows that \(m := \dim M = 4n-1 \equiv 3 \mod 4\), and also \(m \equiv 3 \mod 8\), since \(n\) is odd. Therefore, by Theorem 6.2, there exists a metric \(g_0\) on \(M\) such that the associated Dirac Operator \({{{{\big /}\!\!\!\!{D}}}}^M\) is invertible. By Lemma 6.3, \(P\) is isomorphic to \(P_f = [0,1] \times M / f\) for some \(f \in \mathrm{Diff }^{\mathrm{spin }}(M)\). Define \(g_1 := f^* g_0\), and connect \(g_0\) with \(g_1\) in \(\mathcal{R }(M)\) by the linear path \(g_t := t g_1 + (1-t) g_0\), \(t \in [0,1]\). Endow \([0,1] \times M\) with the generalized cylinder metric \(dt^2 + g_t\). Denote by \(\pi :[0,1] \rightarrow S^1\) the canonical projection. We obtain a commutative diagram

By construction, we can push forward the metric \(dt^2 + g_t\) on \([0,1] \times M\) to a metric on \(P_f\), and then further to \(P\) such that the above row consists of local isometries. The right map is actually an isometry along which we can pull back the spin structure on \(P\) to \(P_f\). This map is then a spin isometry and, therefore, we will no longer distinguish between \(P\) and \(P_f\). The left map is an isometry, except that it identifies \(\{0\} \times M\) with \(\{1\} \times M\). Notice that, since \([0,1] \times M\) is simply-connected, the spin structure on \([0,1] \times M\) obtained by pulling back the spin structure on \(P_f\) along \(\pi \) is equivalent to the canonical product spin structure on \([0,1] \times M\).

Step 2 (trivialize): The Riemannian manifold \(P^{\prime }:= ([0,1] \times M,dt^2 + g_t)\) has two isometric boundary components. Geometrically, \(P^{\prime }\) is obtained from \(P\) by cutting \(M=P_{[0]}\) out of \(P\) and adding two boundaries \(P^{\prime }_0\) and \(P^{\prime }_1\), i.e. \(P^{\prime } = (P \setminus P_{[0]}) \coprod P^{\prime }_0 \coprod P^{\prime }_1\), where \(P^{\prime }_0\), \(P^{\prime }_1\) are two isometric copies of \(P_{[0]}\). By Theorem 6.4, we obtain

$$\begin{aligned} \mathrm{index }({{{{\big /}\!\!\!\!{D}}}}_+^P) = \mathrm{index }({{{{\big /}\!\!\!\!{D}}}}_+^{P^{\prime }}). \end{aligned}$$
(5.9)

Step 3 (index of half-cylinders): Now, set

$$\begin{aligned} \begin{array}{ll} Z_0^{\prime } := (\mathclose ] -\infty , 0 \mathclose ] \times M, dt^2 + g_0), &{}Z_1^{\prime } := (\mathopen [ 1,\infty \mathopen [ \times M, dt^2 + g_1), \\ Z_1^{\prime \prime } := (\mathopen [ 0,\infty \mathopen [ \times M, dt^2 + g_0), &{}Z := (\mathbb R \,\times M, dt^2+g_0), \\ Z^{\prime } := Z_0^{\prime } \textstyle \coprod Z_1^{\prime }, &{}Z^{\prime \prime } := Z_0^{\prime } \textstyle \coprod Z_1^{\prime \prime }. \end{array} \end{aligned}$$

Since \(Z\) is a Riemannian product, it follows that

$$\begin{aligned} \forall \psi \in \Gamma _c(\Sigma Z): \Vert {{{{\big /}\!\!\!\!{D}}}}^Z \psi \Vert _{L^2(\Sigma Z)}^2 \ge \lambda _{\min }^2 \Vert \psi \Vert _{L^2(\Sigma Z)}^2, \end{aligned}$$
(5.10)

where \(\lambda _{\min }\) is the eigenvalue of \({{{{\big /}\!\!\!\!{D}}}}^{g_0}\) of minimal absolute magnitude. By construction, \({{{{\big /}\!\!\!\!{D}}}}^{g_0}\) is invertible, thus \(\lambda _{\min } > 0\). Therefore \({{{{\big /}\!\!\!\!{D}}}}^{Z}\) is invertible and coercive at infinity (see Theorem 6.4 for the definition). By Theorem 6.4, this implies

$$\begin{aligned} 0&= \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}^{Z}_+\right) = \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}^{Z^{\prime \prime }}_+\right) = \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}^{Z_0^{\prime }}_+\right) + \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}^{Z_1^{\prime \prime }}_+\right) \nonumber \\&= \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}^{Z_0^{\prime }}_+\right) + \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}^{Z_1^{\prime }}_+\right) = \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}^{Z^{\prime }}_+\right) , \end{aligned}$$
(5.11)

where we used the fact that \(Z_1^{\prime }\) and \(Z_1^{\prime \prime }\) are spin isometric.

Step 4 (glue in the half-cylinders): Now glue \(Z^{\prime }\) to \(P^{\prime }\) (\(Z^{\prime }_0\) at \(\{0\} \times M\) and \(Z^{\prime }_1\) at \(\{1\} \times M\)) and obtain a bundle \(Q =( \mathbb R \,\times M, dt^2 + g_t)\) where \(g_t = g_0\) for \(t \le 0\) and \(g_t = g_1\) for \(t \ge 1\). Since \({{{{\big /}\!\!\!\!{D}}}}^{g_1}\) is invertible as well, it follows that \({{{{\big /}\!\!\!\!{D}}}}^{Z^{\prime }}\) satisfies an estimate analogous to (5.10). We, therefore, see that \({{{{\big /}\!\!\!\!{D}}}}^{Q}\) is also coercive at infinity (take \(K:=P^{\prime }\) as the compact subset). By Theorem 6.4, we obtain

$$\begin{aligned} \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}_+^{Q}\right) = \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}_+^{P^{\prime }}\right) + \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}_+^{Z^{\prime }}\right) \stackrel{\text{(5.11) }}{=} \mathrm{index }\left( {{{{\big /}\!\!\!\!{D}}}}_+^{P^{\prime }}\right) . \end{aligned}$$
(5.12)

Step 5 (apply hypersurface theory): Each \(Q_t = \{t\} \times M\) is a hypersurface in \(Q\), and \(\tilde{\partial }_t \in \mathcal T (Q)\) (horizontal lift of \(\partial _t\)) provides a unit normal field for all \(Q_t\). Therefore, we can apply some standard results about the Dirac operator on hypersurfaces, see [4]: Since \(m\) is odd

$$\begin{aligned} \Sigma Q|_{Q_t} = \Sigma ^+ Q|_{Q_t} \oplus \Sigma ^- Q|_{Q_t} = \Sigma ^+ M_t \oplus \Sigma ^- M_t,&M_t := (M,g_t), \end{aligned}$$

where \(\Sigma ^+ M_t = \Sigma ^- M_t = \Sigma M_t\) as hermitian vector bundles. The Clifford multiplication “\(\cdot \)” in \(\Sigma Q\) is related to the Clifford multiplication “\(\bullet _t^\pm \)” in \(\Sigma ^\pm M_t\) by \(X \bullet _t^\pm \psi = \pm \tilde{\partial }_t \cdot X \cdot \psi \). Setting

$$\begin{aligned} \tilde{{{{\big /}\!\!\!\!{D}}}}^{M_t} := \left( {{{{\big /}\!\!\!\!{D}}}}^{M_t} \oplus \left( -{{{{\big /}\!\!\!\!{D}}}}^{M_t}\right) \right) , \end{aligned}$$
(5.13)

we obtain the Dirac equation on hypersurfaces

$$\begin{aligned} \tilde{\partial }_t \cdot {{{{\big /}\!\!\!\!{D}}}}^{Q} = \tilde{{{{\big /}\!\!\!\!{D}}}}^{M_t} +\tfrac{m}{2}H_t - \nabla ^{\Sigma Q}_{\tilde{\partial }_t}: \Gamma (\Sigma Q|_{Q_t}) \rightarrow \Gamma (\Sigma Q|_{Q_t}) \end{aligned}$$
(5.14)

for all \(t \in \mathbb R \,\).

Step 6 (identification of the Spinor spaces): Let \(\psi \in \Gamma (\Sigma Q)\) be a spinor field. For any \(t \in \mathbb R \,\) this defines a section \(\psi _t \in \Gamma (\Sigma Q|_{Q_t})\). Therefore, we can also think of \(\psi \) as a “section” of

$$\begin{aligned} \bigcup _{t \in \mathbb R \,}{\Gamma (\Sigma Q|_{Q_t})} \rightarrow \mathbb R \,. \end{aligned}$$

and (5.14) tells us how \(\tilde{\partial }_t \cdot {{{{\big /}\!\!\!\!{D}}}}^Q\) acts on these sections under this identifcation. We would like to apply Theorem 6.5 and, therefore, have to solve the problem that for various \(t\) the Hilbert spaces \(\Gamma _{L^2}(\Sigma Q|_{Q_t})\) are different. As discussed in [4], we will use the following identification: For any \(x \in M\), consider the curve \(\gamma ^x:\mathbb R \,\rightarrow \mathbb R \,\times M\), \(t \mapsto (t,x)\). Each spinor \(\psi \in \Gamma (\Sigma Q)\) determines a section \(\psi ^x\) in \(\Sigma Q\) along \(\gamma _x\). Using the connection \(\nabla ^{\Sigma Q}\), we obtain a parallel translation \(\tau ^{t}_{0}:\Sigma _x M_{t} \rightarrow \Sigma _x M_{0}\). Notice that the Clifford multiplication “\(\cdot \)”, the vector field \(\tilde{\partial }_t\) and the volume form \(\omega \) that determines the splitting \(\Sigma Q = \Sigma ^+Q \oplus \Sigma ^-Q\) are all parallel. Identifying \(M\) with \(M_0\), we obtain a map

$$\begin{aligned} \begin{array}{rcl} \tau (\psi ):=\bar{\psi }: \mathbb R \, &{} \rightarrow &{} \Gamma (\Sigma ^+ M) \oplus \Gamma (\Sigma ^- M) \\ t &{} \mapsto &{} (x \mapsto \tau _{t}^0(\psi ^+_{(t,x)})+\tau _{t}^0(\psi ^-_{(t,x)})). \end{array} \end{aligned}$$

This identification defines an isometry \(\tau : \Gamma _{L^2}(\Sigma Q) \rightarrow L^2(\mathbb R \,, \Gamma _{L^2}(\Sigma ^+ M \oplus \Sigma ^- M))\). The operator in (5.14) can be pulled back via a commutative diagram

and the upper row is given by (5.14). We calculate what this equation looks like in the lower row: Since Clifford multiplication and \(\tilde{\partial }_t\) are parallel,

$$\begin{aligned} \tau \circ \left( \tilde{\partial }_t \cdot {{{{\big /}\!\!\!\!{D}}}}^{Q}\right) \circ \tau ^{-1} = \tilde{\partial }_t \cdot \left( \tau \circ {{{{\big /}\!\!\!\!{D}}}}^{Q} \circ \tau ^{-1}\right) =: \tilde{\partial }_t \cdot {{{{\big /}\!\!\!\!{D}}}}^{Q}_{M}. \end{aligned}$$

Since there is a splitting \((\Sigma Q, \nabla ^Q) = (\Sigma ^+Q,\nabla ^{+} \oplus \nabla ^{-})\), it suffices to check the following for a \(\psi \in \Gamma (\Sigma ^+ Q)\): Let \(x \in M\) be arbitrary, and let \(D_x\) be the covariant derivative induced by \(\nabla ^{\Sigma Q}\) along \(\gamma ^x\). For any \(t_0 \in \mathbb R \,\)

$$\begin{aligned} \nabla _{\tilde{\partial }t}^{\Sigma Q}{\psi }|_{{(t_0,x)}} =D_x(\psi ^x)(t_0) =\lim _{t \rightarrow t_0}{\frac{\tau _t^{t_0}(\psi ^x(t)) - \psi ^x(t_0)}{t-t_0}}, \end{aligned}$$

and consequently

$$\begin{aligned} \overline{\nabla _{\tilde{\partial }t}^{\Sigma Q}{\psi }}(t_0)|_x&= \tau _{t_0}^0\left( \nabla _{\tilde{\partial }t}^{\Sigma Q}{\psi }|_{{(t_0,x)}}\right) =\tau _{t_0}^0\left( \lim _{t \rightarrow t_0}{\frac{\tau _t^{t_0}(\psi ^x(t)) - \psi ^x(t_0)}{t-t_0}} \right) \\&= \lim _{t \rightarrow t_0}{\frac{\tau _{t_0}^0(\tau _t^{t_0}(\psi ^x(t))) - \tau _{t_0}^0(\psi ^x(t_0))}{t-t_0}} = \lim _{t \rightarrow t_0}{\frac{\tau _t^{0}(\psi _{(t,x)}) - \tau _{t_0}^0(\psi |_{(t_0,x)})}{t-t_0}} \\&= \lim _{t \rightarrow t_0}{\frac{\bar{\psi }(t)|_x - \bar{\psi }(t_0)|_x}{t-t_0}} =\tfrac{\mathrm{d}\bar{\psi }}{\mathrm{d}t}(t_0)|_x. \end{aligned}$$

This implies for any \(\bar{\psi }\in L^2(\mathbb R \,, \Gamma _{H^1}(\Sigma ^+ M \oplus \Sigma ^- M))\)

$$\begin{aligned} \left( \tau \circ \nabla _{\tilde{\partial }t}^{\Sigma Q} \circ \tau ^{-1}\right) (\bar{\psi }) = \tfrac{\mathrm{d}}{\mathrm{d}t} \bar{\psi }. \end{aligned}$$

All in all (5.14) transforms under \(\tau \circ \_ \circ \tau ^{-1}\) into

$$\begin{aligned} \tilde{\partial }_t \cdot {{{{\big /}\!\!\!\!{D}}}}^{Q}_M = \tilde{{{{\big /}\!\!\!\!{D}}}}^{Q}_M + \tfrac{m}{2}H - \tfrac{\mathrm{d}}{\mathrm{d}t}, \end{aligned}$$
(5.15)

where \(H:\mathbb R \,\rightarrow \mathbb R \,\), \(t \mapsto H_t\), and \(\tilde{{{{\big /}\!\!\!\!{D}}}}^Q_M:\mathbb R \,\rightarrow \Gamma (\Sigma M \oplus \Sigma M)\) is given by \(\tilde{{{{\big /}\!\!\!\!{D}}}}^Q_M(t) = \bar{{{{\big /}\!\!\!\!{D}}}}(t) \oplus (- \bar{{{{\big /}\!\!\!\!{D}}}}(t))\), \(\bar{{{{\big /}\!\!\!\!{D}}}}(t) = \tau \circ {{{{\big /}\!\!\!\!{D}}}}^{M_t} \circ \tau ^{-1}\).

Step 7 (apply Theorem 6.5): Set \(H := L^2(\mathbb R \,, \Gamma _{L^2}(\Sigma M))\), \(W := L^2(\mathbb R \,, \Gamma _{H^1}(\Sigma M))\) and \(A(t) = \bar{{{{\big /}\!\!\!\!{D}}}}(t)\). Using Theorem 6.5, we obtain

$$\begin{aligned} \mathrm{index }({{{{\big /}\!\!\!\!{D}}}}^Q_+)&= \mathrm{index }((\tilde{\partial }_t \cdot {{{{\big /}\!\!\!\!{D}}}}^Q_M - \tfrac{m}{2} H)_+) \stackrel{\text{(5.15) }}{=}\mathrm{index }\left( \left( {{{{\big /}\!\!\!\!{D}}}}^{Q}_M - \tfrac{\mathrm{d}}{\mathrm{d}t}\right) _+\right) \nonumber \\&= \mathrm{index }\left( \left( \bar{{{{\big /}\!\!\!\!{D}}}} - \tfrac{\mathrm{d}}{\mathrm{d}t}\right) _+\right) =\mathrm{index }\left( \tfrac{\mathrm{d}}{\mathrm{d}t} - \bar{{{{\big /}\!\!\!\!{D}}}}\right) =\mathrm{sf }(\bar{{{{\big /}\!\!\!\!{D}}}}). \end{aligned}$$
(5.16)

Now the spectral flow \(\mathrm{sf }(\bar{{{{\big /}\!\!\!\!{D}}}})\) in the sense of Salamon, cf. [16], coincides with the spectral flow in the sense of Lemma 5.7.

Step 8 (final argument): By the classical Atiyah–Singer Index theorem, cf. [12, Thm. III.13.10], we obtain

$$\begin{aligned} \widehat{A}(P) = \mathrm{index }({{{{\big /}\!\!\!\!{D}}}}_+^P). \end{aligned}$$
(5.17)

Consequently, we can put all the steps together to obtain

$$\begin{aligned} 0 \stackrel{\text{(5.8) }}{\ne } \widehat{A}(P)&\stackrel{\text{(5.17) }}{=} \mathrm{index }({{{{\big /}\!\!\!\!{D}}}}_+^P) \stackrel{\text{(5.9) }}{=} \mathrm{index }({{{{\big /}\!\!\!\!{D}}}}_+^{P^{\prime }}) \\&\stackrel{\text{(5.12) }}{=} \mathrm{index }({{{{\big /}\!\!\!\!{D}}}}_+^{Q}) \stackrel{\text{(5.16) }}{=} \mathrm{sf }(\bar{{{{\big /}\!\!\!\!{D}}}}) =\mathrm{sf }_{g_0}(f). \end{aligned}$$

\(\square \)