Spectral flows of Toeplitz operators and bulk-edge correspondence

Abstract

We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even, this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of a family of Toeplitz operators on a strongly pseudoconvex domain in ${\mathbb {C}}^n$. This result is similar to the Boutet de Monvel’s computation of the index of a single Toeplitz operator on a strongly pseudoconvex domain. Finally, we show that the bulk-boundary correspondence in the tight-binding model of topological insulators is a special case of our result. In “Appendix,” Koen van den Dungen reviewed the main result in the context of (unbounded) KK-theory.

Appendix: A perspective from (unbounded) KK-theory (by Koen van den Dungen)

Mathematisches Institut der Universität Bonn, Endenicher Allee 60 D-53115 Bonn, E-mail address: kdungen@uni-bonn.de

We consider the assumptions and notation of Sect. 2. The aim of this short “Appendix” is to review Theorem 2.1 from the perspective of (unbounded) $K\!K$-theory [3, 23]. For simplicity, we will assume that , viewed as an $M_k({\mathbb {C}})$-valued function on $S^1\times M$, is chosen such that vanishes at infinity. This assumption ensures that the operator (multiplication by ), acting on the Hilbert $C_0(S^1\times M)$-module $\varGamma _0(S^1\times M,E\otimes {\mathbb {C}}^k)$, has compact resolvents, so that is an unbounded Kasparov ${\mathbb {C}}$-$C_0(S^1\times M)$-module. It also means we do not need the (sufficiently large) constant $r>0$, and we simply set $r=1$.

Theorem 2.1 states that we have the equality

(A.1)

In the context of KK-theory the right-hand side of this equality should be viewed as an element in $K\!K^0({\mathbb {C}},{\mathbb {C}})$. The left-hand side naturally defines an element in $K\!K^1({\mathbb {C}},C(S^1))$ (cf. [29, §2.3]), given as the (odd!) class of the regular self-adjoint Fredholm operator

on the Hilbert $C(S^1)$-module $C(S^1,({\mathcal {H}}^+\oplus {\mathcal {H}}^-)\otimes {\mathbb {C}}^k)$, where is given by , and ${\mathcal {H}}= {\mathcal {H}}^+\oplus {\mathcal {H}}^-$ denotes the kernel of D. Of course, these $K\!K$-groups are both isomorphic to ${\mathbb {Z}}$, and we have a natural isomorphism $\cdot \otimes _{C(S^1)} [-i\partial _t] :K\!K^1({\mathbb {C}},C(S^1)) \rightarrow K\!K^0({\mathbb {C}},{\mathbb {C}})$ (which sends the spectral flow of a family A(t) to the index of $\partial _t-A$, as described in Sect. 5.2). Thus, we rewrite Lemma 5.3 as [cf. Lemma 5.3]

Now let us consider the right-hand side of this equality. It is well understood that the index class of the Callias-type operator is given by the Kasparov product , cf. [12]. The class of ${\mathscr {D}}$ is simply given as the exterior Kasparov product $[{\mathscr {D}}] = [ D] \otimes [-i\partial _t]$ of the Dirac operator D on M with $-i\partial _t$ on $S^1$. Using the properties of the Kasparov product, we then obtain

Since the Kasparov product with $[-i\partial _t]$ gives an isomorphism, Eq. (A.1) can be rewritten as

The Kasparov product on the right-hand side can be computed [9, Example 2.38] and is represented by the regular self-adjoint operator (with compact resolvents)

on the Hilbert $C(S^1)$-module $C(S^1,L^2(M,E^+\oplus E^-))$. Theorem 2.1 can then be reproven by showing the equality in $K\!K^1({\mathbb {C}},C(S^1))$. The proof of this spectral flow equality is in fact very similar to the proof of the index equality in [20, §4] and is analogous to the proof of Lemma 5.3.

Proposition A.1

We have the equality

Proof

Let $P=P^+\oplus P^-$ denote the projection onto the kernel of D, and write $Q=1-P$. Since $P D P=0$, we have the equality (where we used the definition of the Toeplitz operators $T_{f_t} := P M_{f_t} P$). Hence, we need to show that and define the same class in $K\!K^1({\mathbb {C}},C(S^1))$. By Corollary 5.1 we know that

is compact, and similarly for . This implies that and are both Fredholm, and that . Rescaling the function by $r>0$, we see that the operator is Fredholm for any $r>0$. Furthermore, since D is invertible on ${\text {Ran}}Q$, we find for $r=0$ that is invertible, and therefore, its class in $K\!K^1({\mathbb {C}},C(S^1))$ is trivial. Since we have a continuous path of Fredholm operators for $0\le r\le 1$, we conclude that the class of is also trivial. Thus, we obtain

$\square$

The statement and proof of Proposition A.1 do not rely on the notion of spectral flow, but merely consider the Fredholm operator and its odd $K\!K$-class. Hence, Proposition A.1 can straightforwardly be generalized to the case where we replace $S^1$ by an arbitrary compact space. We thus obtain the following:

Theorem A.2

Let $E = E^+\oplus E^-$ be a graded Dirac bundle over a complete Riemannian manifold M, and let D be the associated Dirac operator. Assume that zero is an isolated point of the spectrum of D, and let P denote the projection onto the kernel of D. Let S be a compact topological space, and let be given by a continuous family of smooth $M_k({\mathbb {C}})$-valued functions $f_t$ on M such that vanishes at infinity. We consider the Toeplitz operator on the Hilbert C(S)-module $C(S,{\mathcal {H}}\otimes {\mathbb {C}}^k)$. Then we have the equality