Continuity of Dirac spectra

Abstract

It is well known that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article, we extend this result to a global version. We view the spectrum of a Dirac operator as a function $\mathbb Z \,\rightarrow \mathbb R \,$ and endow the space of all spectra with an $\mathrm{arsinh }$-uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that, due to spectral flow, these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms in general.

Appendix

Here, we collect some results for convenient reference.

Theorem 6.1

[9, Thm. 1.3] Given $k,l \ge 0$ there is an $N=N(k,l) \in \mathbb N \,_{\ge 0}$ with the following property: For all $n \ge N$, there is a $4n$-dimensional smooth closed spin manifold $P$ with non-vanishing $\widehat{A}$-genus and which fits into a smooth fibre bundle

$$\begin{aligned} X \rightarrow P \rightarrow S^k. \end{aligned}$$

In addition, we can assume that the following conditions are satisfied:

• (i) The fibre $X$ is $l$-connected.

• (ii) The bundle $P \rightarrow S^k$ has a smooth section $s:S^k \rightarrow P$ with trivial normal bundle.

Theorem 6.2

[1, Thm 1.1] Let $M$ be a compact spin manifold of dimension $m$. Assume $m \not \equiv 0 \mod 4$ and $m \not \equiv 1,2 \mod 8$. Then there exists a metric $g \in \mathcal{R }(M)$ such that ${{{{\big /}\!\!\!\!{D}}}}^g$ is invertible.

Lemma 6.3

(Classification of fibre bundles over $S^1$) Let $M$ be any smooth manifold.

• (i) For any $f \in \mathrm{Diff }(M)$ the space obtained by setting

  $$\begin{aligned} P_f := [0,1] \times M / \sim ,&\forall (t,x) \in [0,1] \times M: (1,x) \sim (0,f(x)) \end{aligned}$$

  is a smooth $M$-bundle over $S^1 = [0,1] / (0 \sim 1)$.

• (ii) Let $\mathcal I$ denote the isotopy classes of $\mathrm{Diff }(M)$, and $\mathcal P$ the isomorphism classes of $M$-bundles over $S^1$. The map

  $$\begin{aligned} \begin{array}{rcl} \mathcal I &{} \rightarrow &{} \mathcal P \\ {[f]} &{} \mapsto &{} {[P_f]} \end{array} \end{aligned}$$

  is well-defined, surjective, and if $[P_{f}] = [P_{f^{\prime }}]$, then $f$ is isotopic to a conjugate of $f^{\prime }$.

• (iii) Let $M$ be oriented. Then $P_f$ is orientable if and only if $f \in \mathrm{Diff }^+(M)$.

• (vi) Let $M$ be spin and simply-connected. Then $P_f$ is spin if and only if $f \in \mathrm{Diff }^{\mathrm{spin }}(M)$.

The proof of this is elementary.

Theorem 6.4

[3, Thm. 8.17, Rem. 8.18a)] Let $(M,g)$ be an even-dimensional smooth complete spin manifold with volume element $\mu$ and a fixed spin structure. Let $N$ be a closed, two-sided hypersurface in $M$. Cut $M$ open along $N$ to obtain a manifold $M^{\prime }$ with two isometric boundary components $N_1$ and $N_2$. Consider the pullbacks $\mu ^{\prime }$, $\Sigma M^{\prime }$, ${{{{\big /}\!\!\!\!{D}}}}^{\prime }$ of $\mu$, $\Sigma M$ and ${{{{\big /}\!\!\!\!{D}}}}$. Let ${{{{\big /}\!\!\!\!{D}}}}$ be coercive at infinity, i.e. assume there exists a compact subset $K \subset M$ and a $C>0$ such that

$$\begin{aligned} \Vert \psi \Vert _{L^2(\Sigma M)} \le C \Vert {{{{\big /}\!\!\!\!{D}}}} \psi \Vert _{L^2(\Sigma M)} \end{aligned}$$

for all $\psi \in \Gamma (\Sigma M)$, which are compactly supported in $M \setminus K$. Then ${{{{\big /}\!\!\!\!{D}}}}^{\prime }$ is Fredholm and

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

Here ${{{{\big /}\!\!\!\!{D}}}}^{\prime }$ is to be understood as the Dirac operator with APS-boundary conditions, i.e.

$$\begin{aligned} \mathrm{dom }({{{{\big /}\!\!\!\!{D}}}}^{\prime }) = \{ \psi \in H^1(\Sigma M ^{\prime }) \mid \psi |_{N_1} \in H^{1/2}_{\mathclose ]-\infty ,0\mathclose ]}(\tilde{{{{\big /}\!\!\!\!{D}}}}), \psi |_{N_2} \in H^{1/2}_{\mathopen [0, \infty \mathopen [}(\tilde{{{{\big /}\!\!\!\!{D}}}})\}, \end{aligned}$$

where $\tilde{{{{\big /}\!\!\!\!{D}}}}$ is the Dirac operator on the boundary (resp. its two-fold copy as in (5.13)).

Theorem 6.5

[16, Thm A] Assume we are given the following data.

• (i) A complex separable Hilbert space $(H,\Vert \_\Vert _H)$.

• (ii) A dense subspace $W \subset H$ and a norm $\Vert \_ \Vert _W$ on $W$ such that $W$ is also a Hilbert space and such that the injection $W \hookrightarrow H$ is compact.

• (iii) A family of unbounded self-adjoint operators $\{A(t)_{t\in \mathbb R \,}\}$ on $H$ with time independent domain $W$, such that for each $t \in \mathbb R \,$ the graph norm of $A(t)$ is equivalent to $\Vert \_ \Vert _W$.

• (vi) A map $\mathbb R \,\rightarrow L(W,H)$, $t \mapsto A(t)$, which is continuously differentiable with respect to the weak operator topology.

• (v) Invertible operators $A^\pm \in L(W,H)$ such that $\lim _{t \rightarrow \pm \infty }{A(t)} = A^\pm$ in norm topology.

Then the operator

$$\begin{aligned} D_A := \tfrac{\mathrm{d}}{\mathrm{d}t} - A(t): W^{1,2}(\mathbb R \,,H) \cap L^2(\mathbb R \,,W) \rightarrow L^2(\mathbb R \,,H) \end{aligned}$$

is Fredholm, and its Fredholm index is equal to the spectral flow $\mathrm{sf }(A)$ of the operator family $A=(A(t))_{t \in \mathbb R \,}$.