Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

Abstract

In order to study continuous models of disordered topological phases, we construct an unbounded Kasparov module and a semifinite spectral triple for the crossed product of a separable C^∗-algebra by a twisted ${\mathbb {R}}^{d}$-action. The spectral triple allows us to employ the non-unital local index formula to obtain the higher Chern numbers in the continuous setting with complex observable algebra. In the case of the crossed product of a compact disorder space, the pairing can be extended to a larger algebra closely related to dynamical localisation, as in the tight-binding approximation. The Kasparov module allows us to exploit the Wiener–Hopf extension and the Kasparov product to obtain a bulk-boundary correspondence for continuous models of disordered topological phases.

Appendix: Summary of Non-unital Index Theory

In what follows we will assume that the algebras we deal with are separable. A useful exposition of KK-theory and its applications can be found in [13, 75] and [20] for the unbounded setting.

Given a real or complex C^∗-module E_B over a ${\mathbb {Z}}_{2}$-graded C^∗-algebra B, we denote by End_B(E) the algebra of adjointable endomorphisms of E subject to the B-valued inner-product (⋅∣⋅)_B. The algebra of finite rank endomorphisms ${\text {End}}_{B}^{00}(E)$ is the algebraic span of the operators ${\Theta }_{e_{1},e_{2}}$ for e₁,e₂ ∈ E such that

$${\Theta}_{e_{1},e_{2}}(e_{3}) = e_{1}\cdot (e_{2}\mid e_{3})_{B} $$

with e ⋅ b the (possibly graded) right-action of B on E_B. The algebra of compact endomorphisms ${{\text {End}}_{B}^{0}}(E)$ is the C^∗-closure of ${\text {End}}^{00}_{B}(E)$.

Definition A.1

Let A and B be real ${\mathbb {Z}}_{2}$-graded C^∗-algebras. A real unbounded Kasparov module $(\mathcal {A},{~}_{\pi }{E}_{B}, D)$ is a ${\mathbb {Z}}_{2}$-graded real C^∗-module E_B, a graded representation of A on E_B, π : A →End_B(E), and an unbounded self-adjoint, regular and odd operator D such that for all $a\in \mathcal {A}\subset A$, a dense ∗-subalgebra,

$$\begin{array}{@{}rcl@{}} [D,\pi(a)]_{\pm} \in {\text{End}}_{B}({E}), \qquad \pi(a)(1+D^{2})^{-1/2}\in {{\text{End}}_{B}^{0}}({E}). \end{array} $$

For a complex Kasparov module, one simply replaces all spaces and algebras with complex ones. Where unambiguous, we will omit the representation π and write unbounded Kasparov modules as $(\mathcal {A}, {E}_{B}, D)$. The results of Baaj and Julg [7] continue to hold for real Kasparov modules, so given an unbounded module $(\mathcal {A},{E}_{B},D)$ we apply the bounded transformation to obtain the real Kasparov module (A,E_B,D(1 + D²)^− 1/2).

1.1 A.1 Semifinite Theory

An unbounded A-${\mathbb {C}}$ or A-${\mathbb {R}}$ Kasparov module is precisely a complex or real spectral triple as defined by Connes. Complex spectral triples satisfying additional regularity properties have the advantage that the local index formula by Connes and Moscovici [30] gives computable expressions for the index pairing with K-theory, a special case of the Kasparov product

$$K_{j}(A) \times KK^{j}(A,{\mathbb{C}}) \to K_{0}({\mathbb{C}})\cong {\mathbb{Z}}. $$

We can extend this general framework by working with semifinite spectral triples.

Let τ be a fixed faithful, normal, semifinite trace on a von Neumann algebra $\mathcal {N}$. We let $\mathcal {K}_{\mathcal {N}}$ be the τ-compact operators in $\mathcal {N}$ (that is, the norm closed ideal generated by the projections $P\in \mathcal {N}$ with $\tau (P)<\infty$).

Definition A.2

A semifinite spectral triple $(\mathcal {A},\mathcal {H},D)$ relative to $(\mathcal {N},\tau )$ is given by a ${\mathbb {Z}}_{2}$-graded Hilbert space $\mathcal {H}$, a graded ∗-algebra $\mathcal {A}\subset \mathcal {N}$ with (graded) representation on $\mathcal {H}$ and a densely defined odd unbounded self-adjoint operator D affiliated to $\mathcal {N}$ such that

• (1) [D,a]_± is well-defined on Dom(D) and extends to a bounded operator on $\mathcal {H}$ for all $a\in \mathcal {A}$,

• (2) $a(1+D^{2})^{-1/2}\in \mathcal {K}_{\mathcal {N}}$ for all $a\in \mathcal {A}$.

For complex algebras and spaces, we can also remove the gradings, in which case the semifinite spectral triple is called odd (otherwise even).

If we take $\mathcal {N}=\mathcal {B}(\mathcal {H})$ and τ = Tr, then we recover the usual definition of a spectral triple.

Theorem A.3 ([23, 44])

Let $(\mathcal {A},\mathcal {H},D)$ be a complex semifinite spectral triple associated to $(\mathcal {N},\tau )$ with A the C^∗-completion of $\mathcal {A}$. Then $(\mathcal {A},\mathcal {H},D)$ determines a class in KK(A,C) with C a subalgebra of $\mathcal {K}_{\mathcal {N}}$. If A is separable, we can take C to be separable. If$(\mathcal {A},\mathcal {H},D)$ is odd, then the triple determines a class in KK¹(A,C).

1.1.1 A.1.1 The Semifinite Index Pairing

Semifinite spectral triples $(\mathcal {A},\mathcal {H},D)$ with $\mathcal {A}$ separable and ungraded can be paired with K-theory elements by the following composition

$$ K_{j}(A) \times KK^{j}(A, C) \to K_{0}(C) \xrightarrow{\tau_{\ast}} {\mathbb{R}}, $$

(1)

with the class in KK^j(A,C) coming from Theorem A.3. The image of the semifinite index pairing is a countably generated subset of ${\mathbb {R}}$ and, as such, can potentially detect finer invariants than the usual pairing of K-theory with K-homology. We call the map from (1) the semifinite index pairing of a K-theory class with a semifinite spectral triple and use the notation $\langle [e],[(\mathcal {A},\mathcal {H},D)]\rangle$ for [e] ∈ K_j(A) to represent this pairing.

We can also describe the semifinite index pairing analytically using the semifinite Fredholm index. Given a semifinite von Neumann algebra $(\mathcal {N},\tau )$, an operator $T\in \mathcal {N}$ that is invertible modulo $\mathcal {K}_{\mathcal {N}}$ has semifinite Fredholm index

$$\text{Index}_{\tau}(T) = \tau(P_{\text{Ker}(T)}) - \tau(P_{\text{Ker}(T^{*})}). $$

We can use the semifinite index to write down an analytic formula for the semifinite pairing. Let $\mathcal {A}^{\sim }=\mathcal {A}\oplus {\mathbb {C}}$ be the minimal unitisation of $\mathcal {A}$. Given $b\in M_{n}(\mathcal {A}^{\sim })$ we let

$$\hat{b} = \begin{pmatrix} b & 0 \\ 0 & 1_{b} \end{pmatrix}, $$

where 1_b = πⁿ(b) and $\pi ^{n} : M_{n}(\mathcal {A}^{\sim }) \to M_{n}({\mathbb {C}})$ is the quotient map coming from the unitisation.

Proposition A.4 ([23], Proposition 2.13)

Let $(\mathcal {A},\mathcal {H},D)$ be a complex semifinite spectral triple relative to$(\mathcal {N},\tau )$ with $\mathcal {A}$ separableand D invertible. Let e be a projector in $M_{n}(\mathcal {A}^{\sim })$, which represents $[e]\in K_{0}(\mathcal {A})$ and u a unitary in $M_{n}(\mathcal {A}^{\sim })$ representing $[u]\in K_{1}(\mathcal {A})$. In the even case, define $T_{\pm } = \frac {1}{2}(1\mp \gamma )T\frac {1}{2}(1\pm \gamma )$ withγ the grading on $\mathcal {H}$. Then with F = D|D|^− 1 andP = (1 + F)/2, the semifinite index pairing is represented by

$$\begin{array}{@{}rcl@{}} \langle [e] - [1_{e}], (\mathcal{A},\mathcal{H},D)\rangle &=& \text{Index}_{\tau\otimes \text{Tr}_{{\mathbb{C}}^{2n}}}(\hat{e}(F \otimes 1_{2n})_{+} \hat{e}), \qquad \textrm{even case}, \\ \langle [u], (\mathcal{A},\mathcal{H},D) \rangle &=& \text{Index}_{\tau\otimes \text{Tr}_{{\mathbb{C}}^{2n}}}\left( (P \otimes 1_{2n})\hat{u}(P \otimes 1_{2n}) \right), \qquad \textrm{odd case}. \end{array} $$

If D is not invertible, we take m > 0 and define the double spectral triple $(\mathcal {A},\mathcal {H}\oplus \mathcal {H},D_{m})$ relative to $(M_{2}(\mathcal {N}),\tau \otimes \text {Tr}_{{\mathbb {C}}^{2}})$, where the operator D_m and the action of $\mathcal {A}$ is given by

$$\begin{array}{@{}rcl@{}} &D_{m} = \begin{pmatrix} D & m \\ m & -D \end{pmatrix}, \qquad a\mapsto \begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix} \end{array} $$

for all $a\in \mathcal {A}$. If $(\mathcal {A},\mathcal {H},D)$ is graded by γ, then the double is graded by $\hat {\gamma } = \gamma \oplus (-\gamma )$. Doubling the spectral triple does not change the K-homology class and ensures that the unbounded operator D_m is invertible [29].

Remark A.5 (Semifinite spectral triples and torsion invariants)

The pairing of (1) is valid in both the complex and real setting. The pairing is, however, unhelpful in the case of torsion invariants as if [x] ∈ K₀(C) has finite order, then τ_∗([x]) = 0. In particular, torsion invariants are common in real K-theory and play an important role in, for example, characterising the topological phase of a free-fermionic system [54]. In order to access torsion invariants we do not take the induced trace and consider the more general product

$$ K_{j}(A) \times KK^{d}(A, C) \to K_{j-d}(C) $$

(2)

for real or complex algebras. Hence we work with Kasparov modules directly. Unbounded Kasparov theory is a useful method for computing internal Kasparov products, as one must when the product represents a torsion class.

The image of the K-theoretic pairing from (2) can be interpreted as a Clifford module, a finitely generated subspace of a countably generated C^∗-module E_C with graded Clifford action. We can associate an analytic index to elements in KO_j−d(C) or K_j−d(C) via an analogue of Atiyah–Bott–Shapiro theory of Clifford modules, see [4, 18].

1.1.2 A.1.2 Kasparov Modules to Semifinite Spectral Triples

We can associate a Kasparov module to a semifinite spectral triple by Theorem A.3. One may ask if the converse is true. Given an unbounded Kasparov A-B module with B containing a faithful semifinite norm lower-semicontinuous trace (or tracial weight), we can often construct semifinite spectral triples using the dual trace construction (see Section ?? or [68] for a simple example).

The dual-trace method of constructing semifinite spectral triples has the advantage that the algebra B is often more closely related to the problem under consideration than the algebra C from Theorem A.3. In particular, the semifinite index pairing from (1) can be rewritten with B in the place of C.

Given a sufficiently regular (complex) semifinite spectral triple from an unbounded Kasparov module, we may use the semifinite local index formula to compute the K-theoretic semifinite index pairing. As the local index formula is a cyclic expression involving traces and derivations, semifinite spectral triples and index theory can be employed in order to more easily compute pairings of K-theory classes with unbounded Kasparov modules as in (1).

1.2 A.2 Summability of Non-unital Spectral Triples

Spectral triples often contain more than just K-homological data. Hence we introduce extra structure on spectral triples that have the interpretation of a differential structure and measure theory. If the algebra is non-unital and non-local in the sense of [78], then we require the noncommutative measure theory developed in [22, 23]. Our brief exposition follows [32, Section 2]. In order to discuss smoothness and summability for non-unital spectral triples, we need to introduce an analogue of L^p-spaces for operators and weights over a semifinite von Neumann algebra $(\mathcal {N}, \tau )$.

Definition A.6

Let D be a densely defined self-adjoint operator affiliated to $\mathcal {N}$. Then for each $p\geqslant 1$ and s > p we define a weight φ_s on $\mathcal {N}$ by

$$\varphi_{s}(T) = \tau\left( (1+D^{2})^{-s/4} T (1+D^{2})^{-s/4}\right) $$

for T a positive element in $\mathcal {N}$. We define the subspace $\mathcal {B}_{2}(D,p)$ of $\mathcal {N}$ by

$$\begin{array}{@{}rcl@{}} \mathcal{B}_{2}(D,p) &=& \bigcap_{s>p}\left( \text{Dom}(\varphi_{s})^{1/2}\cap (\text{Dom}(\varphi_{s})^{1/2})^{*}\right),\\ \text{Dom}(\varphi_{s})^{1/2} &=& \big\{ T\in \mathcal{N}\,:\, \varphi_{s}(T^{*}T) < \infty \big\}. \end{array} $$

Take $T\in \mathcal {B}_{2}(D,p)$. The norms

$$\mathcal{Q}_{n}(T) = \left( \|T\|^{2} + \varphi_{p + 1/n}(|T|^{2}) + \varphi_{p + 1/n}(|T^{*}|^{2})\right)^{1/2} $$

for n = 1, 2,… take finite values on $\mathcal {B}_{2}(D,p)$ and provide a topology on $\mathcal {B}_{2}(D,p)$ stronger than the norm topology. The space $\mathcal {B}_{2}(D,p)$ is a Fréchet algebra [23, Proposition 1.6] and can be interpreted as the bounded square integrable operators.

To introduce the bounded integrable operators, first take the span of products, $\mathcal {B}_{2}(D,p)^{2}$, and define the norms

$$\mathcal{P}_{n}(T) = \inf\left\{\sum\limits_{i = 1}^{k} \mathcal{Q}_{n}(T_{1,i})\mathcal{Q}_{n}(T_{2,i})\,:\, T= \sum\limits_{i = 1}^{k} T_{1,i}T_{2,i},\ T_{1,i},\,T_{2,i}\in\mathcal{B}_{2}(D,p) \right\}, $$

where the sums are finite and the infimum is over all possible such representations of T. It is shown in [23, p12–13] that $\mathcal {P}_{n}$ are norms on $\mathcal {B}_{2}(D,p)^{2}$.

Definition A.7

Let D be a densely defined and self-adjoint operator and $p\geqslant 1$. We define $\mathcal {B}_{1}(D,p)$ to be the completion of $\mathcal {B}_{2}(D,p)^{2}$ with respect to the family of norms $\{\mathcal {P}_{n}\,:\, n = 1,2,\ldots \}$.

Definition A.8

A semifinite spectral triple $(\mathcal {A},\mathcal {H},D)$ relative to $(\mathcal {N}, \tau )$ is said to be finitely summable if there exists s > 0 such that for all $a\in \mathcal {A}$, $a(1+D^{2})^{-s/2}\in \mathcal {L}^{1}(\mathcal {N},\tau )$. In such a case we let

$$p = \inf\{s>0\,:\, \forall a\in\mathcal{A}, \,\tau(|a|(1+D^{2})^{-s/2}) < \infty \} $$

and call p the spectral dimension of $(\mathcal {A},\mathcal {H},D)$.

Note that $|a|(1+D^{2})^{-s/2}\in \mathcal {L}^{1}(\mathcal {N},\tau )$ by the polar decomposition a = v|a|, which does not require |a| to be in $\mathcal {A}$. For the definition of spectral dimension to have meaning, we require that $\tau (a(1+D^{2})^{-s/2})\geqslant 0$ for $a\geqslant 0$, a fact that follows from [12, Theorem 3]. For a semifinite spectral triple $(\mathcal {A},\mathcal {H},D)$ to be finitely summable with spectral dimension p, it is a necessary condition that $\mathcal {A}\subset \mathcal {B}_{1}(D,p)$ [23, Proposition 2.17].

Definition A.9

Given a densely-defined self-adjoint operator D, set $\mathcal {H}_{\infty } = \bigcap _{k\geqslant 0}\text {Dom}(D^{k})$. For an operator $T:\mathcal {H}_{\infty }\to \mathcal {H}_{\infty }$, we define

$$\begin{array}{@{}rcl@{}} \delta(T) = [|D|,T], \quad L(T) =\! (1+D^{2})^{-1/2}[D^{2},T],\!\! \quad R(T) \,=\, [D^{2},T](1+D^{2})^{-1/2}. \end{array} $$

One has that (cf. [26, 30])

$$\bigcap_{n\geqslant 0} \text{Dom}(L^{n}) = \bigcap_{n\geqslant 0} \text{Dom}(R^{n}) = \bigcap_{k,l\geqslant 0}\text{Dom}(L^{k}\circ R^{l}) = \bigcap_{n\geqslant 0} \text{Dom}(\delta^{n}). $$

We see that to define δ^k(T), we require that $T: \mathcal {H}_{k} \to \mathcal {H}_{k}$ for $\mathcal {H}_{k} = \bigcap _{l = 0}^{k} \text {Dom}(D^{l})$.

Definition A.10

Let D be a densely defined self-adjoint operator affiliated to $\mathcal {N}$ and $p\geqslant 1$. Then define for k = 0, 1,…

$${\mathcal{B}_{1}^{k}}(D,p) = \left\{ T\in\mathcal{N} \,\left\vert \, T:\mathcal{H}_{l}\to\mathcal{H}_{l} \text{ and }\delta^{l}(T)\in\mathcal{B}_{1}(D,p)\,\, \forall l = 0,\ldots,k \right. \right\}, $$

$${\mathcal{B}_{2}^{k}}(D,p) = \left\{ T\in\mathcal{N} \,\left\vert \, T:\mathcal{H}_{l}\to\mathcal{H}_{l} \text{ and }\delta^{l}(T)\in\mathcal{B}_{2}(D,p)\,\, \forall l = 0,\ldots,k \right. \right\} $$

as well as

$$\mathcal{B}_{1}^{\infty}(D,p) = \bigcap_{k = 0}^{\infty} {\mathcal{B}_{1}^{k}}(D,p), \qquad \mathcal{B}_{2}^{\infty}(D,p) = \bigcap_{k = 0}^{\infty} {\mathcal{B}_{2}^{k}}(D,p). $$

For any k (including $\infty$), we equip ${\mathcal {B}_{1}^{k}}(D,p)$ with the topology induced by the seminorms

$$\mathcal{P}_{n,l}(T) = \sum\limits_{j = 0}^{l} \mathcal{P}_{n} (\delta^{j}(T)) $$

for $T\in \mathcal {N}$, l = 0,…,k and $n\in {\mathbb {N}}$.

If we are interested in index theory in the non-compact setting, we need to control the integrability of both functions and their derivatives. The noncommutative analogue of this regularity turns out to be a finitely summable spectral triple but with additional smoothness properties.

Definition A.11

Let $(\mathcal {A},\mathcal {H},D)$ be a semifinite spectral triple relative to $(\mathcal {N},\tau )$. We say that $(\mathcal {A},\mathcal {H},D)$ is QC^k-summable if it is finitely summable with spectral dimension p and

$$\mathcal{A} \cup [D,\mathcal{A}] \subset {\mathcal{B}_{1}^{k}}(D,p). $$

We say that $(\mathcal {A},\mathcal {H},D)$ is smoothly summable if it is QC^k-summable for all $k\in {\mathbb {N}}$, that is

$$\mathcal{A}\cup [D,\mathcal{A}] \subset \mathcal{B}_{1}^{\infty}(D,p). $$

For a smoothly summable spectral triple $(\mathcal {A},\mathcal {H},D)$, we can introduce the δ-φ topology on $\mathcal {A}$ by the seminorms

$$ \mathcal{A} \ni a \mapsto \mathcal{P}_{n,k}(a) + \mathcal{P}_{n,k}([D,a]) $$

(3)

for $n,k\in {\mathbb {N}}$. The completion of $\mathcal {A}$ in the δ-φ topology is Fréchet and closed under the holomorphic functional calculus [23, Proposition 2.20]. We finish this section with a sufficient and checkable condition of finite summability of spectral triples.

Proposition A.12 ([23], Proposition 2.16)

Let $(\mathcal {A},\mathcal {H},D)$ be a semifinite spectral triple. If $\mathcal {A} \subset \mathcal {B}_{1}^{\infty }(D, p)$ for some $p\geqslant 1$, then $(\mathcal {A},\mathcal {H},D)$ is finitely summable with spectral dimension given by the infimum of such p’s. More generally, if $\mathcal {A}\subset \mathcal {B}_{2}(D,p)\mathcal {B}_{2}^{\lfloor p\rfloor + 1}(D,p)$ for $p\geqslant 1$, then $(\mathcal {A},\mathcal {H},D)$ is finitely summable with spectral dimension given by the infimum of such p’s.

Lemma 5.2 offers a slight variation of this result in order to get sharp results on localisation. Lemma 5.2 has essentially the same conclusion, though different statement and proof, as [22, Proposition 6.6].

1.3 A.3 The Local Index Formula

We now briefly recall the semifinite local index formula from [23], which is an extension of previous formulas, [26, 27, 30, 79], to non-unital and non-local semifinite (complex) spectral triples. We note that the local index formula requires a smoothly summable semifinite spectral triple of finite spectral dimension. This may seem restrictive, but turns out to be satisfied in our examples.

To define the resolvent cocycle, we first establish the notation R_s(λ) = (λ − (1 + s² + D²))^− 1.

Definition A.13 ([23, 26, 27])

Let $(\mathcal {A},\mathcal {H},D)$ be a smoothly summable complex semifinite spectral triple relative to $(\mathcal {N},\tau )$ with spectral dimension p. For a ∈ (0, 1/2), let ℓ be the vertical line $\ell =\{a+iv\,:\,v\in \mathbb {R}\}$. We define the resolvent cocycle $({\phi _{m}^{r}})_{m = 0}^{M}$ for R(r) > (1 − m)/2 as

$$\begin{array}{@{}rcl@{}} {\phi_{m}^{r}}(a_{0},\ldots,a_{m}) = \frac{\eta_{m}}{2\pi i} {\int}_{0}^{\infty} s^{m}\, \!\tau\!\left( \!\gamma\! {\int}_{\ell} \lambda^{-p/2-r} a_{0} R_{s}(\lambda) [D,a_{1}] R_{s}(\lambda) {\cdots} [D,a_{m}]R_{s}(\lambda) \,\mathrm{d}\lambda\right)\mathrm{d}s, \end{array} $$

where

$$\eta_{m} = \left( -\sqrt{2i}\right)^{\bullet} 2^{m + 1} \frac{{\Gamma}(m/2 + 1)}{{\Gamma}(m + 1)} $$

with ∙ = 0, 1 depending on whether the spectral triple is even or odd.

The integral over ℓ is well-defined by [23, Lemma 3.3]. The index formula is a pairing of a cocycle with an algebraic chain. If $e\in \mathcal {A}^{\sim }$ is a projection, we define Ch⁰(e) = e and for $k\geqslant 1$,

$$\text{Ch}^{2k}(e) = (-1)^{k}\frac{(2k)!}{k!}\,(e-1/2)\otimes e\otimes\cdots\otimes e\in(\mathcal{A}^{\sim})^{\otimes (2k + 1)}. $$

If $u\in \mathcal {A}^{\sim }$ is a unitary, then we define for $k\geqslant 0$

$$\text{Ch}^{2k + 1}(u) = (-1)^{k} k!\, u^{*}\otimes u\otimes \cdots\otimes u^{*}\otimes u \in (\mathcal{A}^{\sim})^{\otimes (2k + 2)}. $$

We split up the theorem into odd and even cases.

Theorem A.14 ([23, 30, 79])

Let $(\mathcal {A}, \mathcal {H}, D)$ be an odd smoothly summable complex semifinite spectral triple relative to $(\mathcal {N},\tau )$ and with spectral dimension p. Let $N = \lfloor \frac {p}{2}\rfloor + 1$, where ⌊⋅⌋ is the floor function, and let u be a unitary in the unitisation of $\mathcal {A}$. The semifinite index pairing can be computed with the resolvent cocycle

$$\langle [u],[(\mathcal{A},\mathcal{H},D)]\rangle = \frac{-1}{\sqrt{2\pi i}} \text{res}_{r=(1-p)/2} \sum\limits_{m = 1,\text{odd}}^{2N-1} {\phi_{m}^{r}}(\text{Ch}^{m}(u)) $$

and the function $r\mapsto \sum \limits _{m = 1,\text {odd}}^{2N-1} {\phi _{m}^{r}}(\text {Ch}^{m}(u))$ analytically continues to a deleted neighbourhood of r = (1 − p)/2.

Theorem A.15 ([23, 30, 79])

Let $(\mathcal {A}, \mathcal {H}, D)$ be an even smoothly summable complex semifinite spectral triple relative to $(\mathcal {N},\tau )$ and with spectral dimension p. Let $N = \lfloor \frac {p + 1}{2}\rfloor$ and$e\in \mathcal {A}^{\sim }$ be a self-adjoint projection. The semifinite index pairing can be computed by the resolvent cocycle

$$\langle [e]-[1_{e}], [(\mathcal{A},\mathcal{H},D)]\rangle = \text{res}_{r=(1-p)/2} \sum\limits_{m = 0,\text{even}}^{2N} {\phi_{m}^{r}}(\text{Ch}^{m}(e) - \text{Ch}^{m}(1_{e})) $$

and the function $r\mapsto \sum \limits _{m = 0,\text {even}}^{2N} \phi _{m}(\text {Ch}^{m}(e))$ analytically continues to a deleted neighbourhood of r = (1 − p)/2.