We describe explicit generators for the “real” K-theory of “real” spheres in van Daele’s picture. Pulling these generators back along suitable maps from tori to spheres produces a family of Hamiltonians used in the physics literature on topological insulators. We compute their K-theory classes geometrically based on the wrong-way functoriality of K-theory and the geometric version of bivariant K-theory, which we extend to the “real” case.

Topological insulators are insulators that, nevertheless, conduct electricity on their boundaries. Even more, conducting states on the boundary are forced to exist by topological obstructions. This suggests that the boundary conducting states are quite robust under disorder. In the one-particle approximation, topological insulators may be classified by the topological K-theory of the observable C-algebra. In translation-invariant tight-binding models, the observable algebra is isomorphic to a matrix algebra over the algebra of continuous functions on the d-torus $Td$, where d is the dimension of the material. Many interesting phenomena arise when the Hamiltonian enjoys extra symmetries that are anti-unitary or anticommute with it. Altogether, there are ten symmetry types, which correspond to the two complex and the eight real K-theory groups. More precisely, the torus appears through the Fourier transform, and the relevant observable algebra is the group C-algebra of $Zd$ with real coefficients. Under the Fourier transform, this becomes the real C-algebra,
$f:Td→C:f(z̄)=f(z)̄ for all z∈Td.$
The conjugation map used here is an involution and makes the torus a “real” space. This is the same as a space with an action of the group $Z/2$. In the following, we denote “real” structures as $r$. For the d-torus, we get
$Td=(x1,…,xd,y1,…,yd)∈R2d:xj2+yj2=1 for j=1,…,d,rTd(x1,…,xd,y1,…,yd):=(x1,…,xd,−y1,…,−yd).$
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As a result, the relevant K-theory for the study of topological phases is the “real” K-theory $KR*(Td)$ of $(Td,rTd)$ as defined by Atiyah.1 This appearance in physics has renewed the interest in “real” K-theory.

This article generalizes geometric bivariant K-theory as a tool for K-theory computations to the “real” case and uses this to compute the K-theory classes of certain Hamiltonians studied in the complex case already in Ref. 2. In addition, we describe explicit generators for the “real” K-theory of spheres, extending a formula by Karoubi in Ref. 3 for the complex K-theory of even-dimensional spheres.

There have always been several different ways to describe the K-theory of a space or a C-algebra. As noted by Kellendonk,4 the K-theory picture that is closest to the classification of topological insulators is van Daele’s picture. His definition applies to a real or complex C-algebra A with a $Z/2$-grading. It is based on odd, self-adjoint unitaries in matrix algebras over A. A $Z/2$-grading may be interpreted physically as a chiral symmetry, and an odd self-adjoint unitary is just the spectral flattening of a Hamiltonian with a spectral gap at zero that respects the given chiral symmetry. Systems without chiral symmetry are treated by doubling the number of degrees of freedom to introduce an auxiliary chiral symmetry.

The starting point of this article was the discussion by Prodan and Schulz-Baldes2 of certain examples of Hamiltonians Hm in any dimension d, namely,
$Hm:=12i∑j=1d(Sj−Sj*)⊗γj+m+12∑j=1d(Sj+Sj*)⊗γ0∈C*(Zd)⊗Cl1,d$
with Clifford generators γ0, …, γd and translations Sj in coordinate directions for j = 1, …, d and a mass parameter m (see Secs. 2.2.4 and 2.3.3 in Ref. 2). The self-adjoint element Hm has a spectral gap at zero if and only if m ∉ {−d, −d + 2, …, d − 2, d}. Then, it defines an insulator. The top-degree Chern character of its K-theory class and its jumps at the values in {−d, −d + 2, …, d − 2, d} are computed in the physics literature [see Eq. (2.26) in Ref. 2 and also Refs. 5 and 6]. Here, we explain a possible mathematical origin of Hm: it is the pullback of a generator of the reduced KR-theory of a sphere along a map $φm:Td→S1,d$. Then, we proceed to compute the class of Hm in “real” K-theory for all m and all dimensions d.
The first step for this is to describe explicit generators for the KR-theory of “real” spheres in van Daele’s picture. Let $Ra,b$ denote $Ra×Rb$ with the involution $r(x,y):=(x,−y)$ for $x∈Ra$, $y∈Rb$. Let $Sa,b⊆Ra,b$ be the unit sphere with the restricted real involution. Let Cla,b denote the Clifford algebra with a + b anticommuting, odd, self-adjoint, unitary generators γ1, …, γa+b and such that γ1, …, γa and iγa+1, …, iγa+b are real. Then,
$βa,b:=∑j=1a+bxjγjSa,b$
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is an odd, self-adjoint, unitary, and real element of the C-algebra $C(Sa,b)⊗Cla,b$. Therefore, it defines a class in its van Daele K-theory, which is isomorphic to the “real” K-theory group $KRa−b−1(Sa,b)$. We check that its image in the reduced “real” K-theory is a generator in the sense that the exterior product map with it defines an isomorphism from the “real” K-theory of a point to the reduced “real” K-theory of $Sa,b$. Our proof that βa,b generates the “real” K-theory is based on the proof of Bott periodicity by the proof of Kasparov7 and Roe of the isomorphism between $KK0(R,A)$ and van Daele’s K-theory of A for any $Z/2$-graded real C-algebra A.
The spectral flattening of the Hamiltonian Hm for $m∈R\{−d,−d+2,…,d−2,d}$ is an odd, self-adjoint, real unitary on the “real” d-torus $Td$ in (1). This may be written as the pull-back of β1,d along the real map,
$φm:Td→S1,d,(x1,…,xd,y1,…,yd)↦(x1+⋯+xd+m,y1,y2,…,yd)(x1+⋯+xd+m,y1,y2,…,yd).$
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Our task is to compute this pull back in the KR-theory of the torus.
There is another way to describe the generator of $KRa−b−1(Sa,b)$ for a > 0. Let
$N:=(1,0,…,0),S:=(−1,0,…,0).$
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These two “poles” are fixed by the “real” involution on $Sa,b$. The stereographic projection identifies $Sa,b\{N}$ with $Ra−1,b$ as a “real” manifold. Thus, Bott periodicity identifies the reduced KR-theory of $Sa,b$ with $KRa−b−1(Ra−1,b)≅KR0(pt)≅Z$. The resulting map $KR0(pt)→KRa−b−1(Sa,b)$ is an example of the wrong-way functoriality of K-theory, namely, it is iS! for the inclusion map $iS:{S}↪Sa,b$.

Shriek maps and pull-back maps, such as iS! and $φm*$, are among the building blocks of geometric bivariant K-theory. This theory also allows us to compute the composites of such maps geometrically. In our case, this says that $φm*◦iS!$ is the sum of the shriek maps for all points in $φm−1(S)$, equipped with appropriate orientations. Our main result, Theorem 40, computes the image of this in the usual direct sum decomposition of $KR*(Td)$.

To make this computation valid in the “real” case, we show that the geometric bivariant K-theory as developed in Refs. 8 and 9 still works for KR-theory. These articles define geometric bivariant K-theory in a slightly different way than suggested originally by Connes and Skandalis,10 in order to extend it more easily to the equivariant case. A “real” involution on a space is the same as a $Z/2$-action. The “real” K-theory is not the same as $Z/2$-equivariant K-theory. The difference is that $Z/2$-equivariant K-theory looks at $Z/2$-actions on vector bundles that are fiberwise linear, whereas “real” K-theory looks at $Z/2$-actions on vector bundles that are fiberwise conjugate-linear. This change in the setup does not affect the properties of equivariant K-theory that are needed to develop bivariant equivariant K-theory. We only comment on this rather briefly in this article. More details may be found in the master’s thesis, Ref. 11.

So far, Hm is a real, odd, self-adjoint element in the C-algebra $C(Td,Cl1,d)$. In Sec. VIII, we explain how to turn this into a translation-invariant self-adjoint operator on a Hilbert space of the form $ℓ2(Zd,C2k)$ that has certain extra symmetries. Depending on d mod 8, these will be a combination of time-reversal and/or particle–hole symmetries. The point here is that the real odd self-adjoint elements of Cl1,d should be exactly those matrices that commute or anticommute with certain antiunitary operators on $C2k$. Since the symmetry type of Hm is dictated by d, we only get one specific symmetry type in each dimension. In order to get generators for other direct summands in $KR*(Td)$, we need to compute exterior products with the generators of KOj(pt) for j = 1, 2, 4. We compute these exterior products quite explicitly in Sec. IX. As a result, we may get generators for all the direct summands in $KR*(Td)$ from the Hamiltonians Hm. They are all defined by the same formula, combined with a specific representation of the relevant Clifford algebra.

Throughout this article, we let $Ra,b$ for $a,b∈N$ be $Ra×Rb$ with the involution $r(x,y):=(x,−y)$ for $x∈Ra$, $y∈Rb$, and we let $Sa,b$ be the unit sphere in $Ra,b$, so the dimension of $Sa,b$ is a + b − 1. Our $Ra,b$ and $Sa,b$ are denoted as $Rb,a$ and $Sb,a$ by Atiyah;1 our notation is that of Kasparov.7

A “real” structure on a C-algebra A is a conjugate-linear, involutive *-homomorphism $r:A→A$. Then,
$AR:=a∈A:r(a)=a$
is a real C-algebra such that $A≅AR⊗C$ with the involution $r(a⊗λ):=a⊗λ̄$. Thus, “real” C-algebras are equivalent to real C-algebras. Any commutative “real” C-algebra is isomorphic to C0(X) with the real involution $r(f)(x):=f(r(x))̄$ for all xX for a “real” locally compact space $(X,r)$.

Let Cla,b denote the complex Clifford algebra with a + b anticommuting, odd, self-adjoint, unitary generators γ1, …, γa+b such that γ1, …, γa and iγa+1, …, iγa+b are real. This is a $Z/2$-graded “real” C-algebra. Let $⊗̂$ denote the graded-commutative tensor product for $Z/2$-graded (“real”) C-algebras.

For $R1,d$ and $S1,d⊆R1,d$ and for Cl1,d, it is convenient to start numbering at 0. That is, the coordinates in $R1,d$ are x0, …, xd and Cl1,d has the anticommuting, odd, self-adjoint generators γ0, …, γd such that γ0, iγ1, …, iγd are real.

We are going to write down explicit generators for the reduced K-theory groups of spheres. This depends, of course, on the definition of K-theory that we are using. In terms of vector bundles given by gluing, Atiyah, Bott, and Shapiro12 described how to go from irreducible Clifford modules to these generators. They do treat KO-theory, but not the “real” case. For the description of K0 through projections, Karoubi3 has written down explicit generators for the complex K-theory of even-dimensional spheres. Kasparov7 has written down explicit generators for his bivariant KK-theory $KKR0(C,C0(Ra,b)⊗Cla,b)$; here, we write KKR to highlight that this means the “real” version of the theory.

We want explicit generators in van Daele’s K-theory for $Z/2$-graded “real” C-algebras; this version of K-theory is ideal for the applications to topological insulators that we have in mind (see Ref. 4), and it is also helpful to treat “real” spheres of all dimensions simultaneously. Actually, our generators are just those of Kasparov, translated along a canonical isomorphism between Kasparov’s KK-theory and van Daele’s K-theory. This isomorphism is an auxiliary result due to Roe.13 Roe’s isomorphism is discussed in greater detail in Ref. 14, which also relates several variants of van Daele’s K-theory.

The following definitions make sense for both real and complex C-algebras. We write them down in the “real” case because this is what we are going to use later. The effect of the condition $r(a)=a$ below is to replace a “real” C-algebra A by its real subalgebra $AR:=a∈A:r(a)=a$.

Definition 1.
Let A be a unital, $Z/2$-graded “real” C-algebra and $n∈N≥1$. Let
$FUn(A):=a∈MnA:a=a*,r(a)=a,a2=1,a odd,$
equipped with the norm topology of $MnA$. Two elements in $FUn(A)$ in the same path component are called homotopic. We define the maps
$⊕:FUn(A)×FUm(A)→FUn+m(A),(a,b)↦a00b.$
Let $FUn(A):=π0(FUn(A))$. We abbreviate $FU(A):=FU1(A)$ and FU(A) ≔ FU1(A). We call A balanced if $FU(A)≠∅$.

The condition a = a* may be dropped (see Proposition 2.5 in Ref. 15).

The operation ⊕ is associative and commutative up to homotopy (see Proposition 2.7 in Ref. 15). Hence, $⨆n=1∞FUn(A)$ becomes an Abelian semigroup. Let GFU(A) be its Grothendieck group.

Definition 2.

Let A be a balanced, unital, $Z/2$-graded “real” C-algebra. The van Daele K-theory DK(A) of A is defined as the kernel of the homomorphism $d:GFU(A)→Z$ defined by $d|FUn(A)=n$ for all $n∈N≥1$.

Van Daele’s original definition uses an element $e∈FU(A)$ with e ∼ (−e) to define stabilization maps,
$FUn(A)→FUn+1(A),[x]↦[x⊕e].$
The colimit of the resulting inductive system becomes an Abelian group under ⊕. This group is isomorphic to DK(A) as defined above (see Refs. 13 and 14).
Definition 2 is generalized beyond the balanced, unital case as follows. First, let A be a unital, $Z/2$-graded “real” C-algebra. Then, $A⊗̂Cl1,1$ is balanced because $1⊗̂γ1∈FU(A⊗̂Cl1,1)$. If A is already balanced, then there is a natural stabilization isomorphism $DK(A)≅DK(A⊗̂Cl1,1)$. This justifies defining $DK(A):=DK(A⊗̂Cl1,1)$, in general. For a C-algebra A without unit, let A+ be its unitization, equipped with the canonical augmentation homomorphism $A+→C$. Then,
$DK(A):=KerDK(A+)→DK(C).$
This reproduces the previous definition if A is unital because then $A+≅A⊕C$ and DK is additive. Since there is an isomorphism DK(A) ≅ KKR0(Cl1,0, A), the functor DK is stable with respect to matrix algebras, Morita invariant, homotopy invariant, and exact for all extensions of $Z/2$-graded C-algebras. See also Ref. 14 for direct proofs of these properties of van Daele’s K-theory.

The following proposition is due to Roe.13 The isomorphism DK(A) ≅ KKR0(Cl1,0, A) is equivalent and proven in Ref. 14.

Proposition 3

(see Ref. 13). Let A be a unital, $Z/2$-graded “real” C-algebra. Then, $DK(A⊗̂Clr+1,s)≅KKR0(C,A⊗̂Clr,s)$.

This justifies defining the “real” K-theory KR(A) and its graded version for a $Z/2$-graded “real” C-algebra A as
$KR(A):=DK(A⊗̂Cl1,0),KRn(A):=DK(A⊗̂Cl1,n)$
for $n∈N$. Actually, this only depends on n mod 8 so that we may also take $n∈Z/8$. If A is trivially graded, then KR(A) is naturally isomorphic to the ordinary K0 of the real Banach algebra $AR$. This is sometimes denoted as $KO0(AR)$ to highlight that $AR$ is only a real C-algebra.
For a “real” locally compact space X, we define its “real” K-theory KR(X) as KR(C0(X)) with the induced “real” structure. This is equivalent to the definition by Atiyah.1 For $n∈Z/8$, we also let
$KRn(X):=KR−n(C0(X)).$
To construct explicit generators for the KR-theory of spheres, we shall need an elementary auxiliary result used in the proof in Ref. 13. Let $HA$ be the standard graded “real” Hilbert A-module $ℓ2(N,C)⊗A$; the “real” involution is the tensor product of complex conjugation on $ℓ2(N,C)$ and the given “real” involution on A, and the $Z/2$-grading is the tensor product of the $Z/2$-grading on $ℓ2(N,C)$ induced by the parity operator and the given one on A. A cycle for $KKR0(C,A⊗̂Clr,s)$ is homotopic to one of the form $(HA⊗̂Clr,s,1,F)$, where 1 denotes the left action of $C$ by scalar multiplication, and $F∈B(HA)⊗̂Clr,s$ is real, odd, and self-adjoint and satisfies $F2−1∈K(HA)⊗̂Clr,s$. Let $K=K(ℓ2(N,C))$, identify $K(HA)$ with $K⊗A≅K⊗̂A$, and let $M(K⊗A)$ denote the stable multiplier algebra of A. Then, $B(HA⊗̂Clr,s)$ is isomorphic to $M(K⊗A)⊗̂Clr,s$. Let $Q(K⊗A):=M(K⊗A)/K⊗A$. Then, F is mapped to a “real,” odd, self-adjoint unitary in $Q(K⊗A)⊗̂Clr,s$. Conversely, any “real,” odd, self-adjoint unitary in $Q(K⊗A)⊗̂Clr,s$ lifts to an operator $F∈B(HA⊗̂Clr,s)$ such that $(HA⊗̂Clr,s,1,F)$ is a cycle for $KKR0(C,A⊗̂Clr,s)$. Thus, we get an obvious surjective map from $FU(Q(K⊗A)⊗̂Clr,s)$ onto $KKR0(C,A⊗̂Clr,s)$. Since $HA$ has infinite multiplicity, $B(HA⊗̂Clr,s)≅MnB(HA⊗̂Clr,s)$, and the same holds for the quotient by the compact operators. Thus, the construction above also defines a map
$⨆n=1∞FUn(Q(K⊗A)⊗̂Clr,s)→KKR0(C,A⊗̂Clr,s).$

Lemma 4.
The map above induces a group isomorphism,
$DK(Q(K⊗A)⊗̂Clr,s)→∼KKR0(C,A⊗̂Clr,s).$

Proof.

Our map is a semigroup homomorphism because the sum in DK and KKR is defined as the direct sum. As such, it induces a group homomorphism on the Grothendieck group, which we may restrict to the subgroup DK(A). There is a real, odd, self-adjoint unitary operator $F0∈B(HA⊗̂Clr,s)$, and any such operator defines a degenerate Kasparov cycle. Such degenerate cycles represent zero in KK-theory. If $F∈FU(Q(K⊗A)⊗̂Clr,s)$, then [F] − [F0] ∈ DK(A) is mapped to the same KK-class as F. Thus, the induced map on DK(A) remains surjective. Roe13 showed that it is also injective; the main reason for this is that “homotopic” KK-cycles become “operator homotopic” after adding degenerate cycles.□

Roe further combined the isomorphism in Lemma 4 with the boundary map in the long exact sequence for DK for the extension of C-algebras,
$K⊗A⊗̂Clr,s↣M(K⊗A)⊗̂Clr,s↠Q(K⊗A)⊗̂Clr,s,$
to prove Proposition 3. The latter boundary map is an isomorphism because van Daele’s K-theory, like ordinary K-theory, vanishes for stable multiplier algebras.
Now, we combine Lemma 4 with Kasparov’s proof of Bott periodicity (see p. 545f and Theorem 7 in Ref. 7) in bivariant KK-theory. Fix $a,b∈N$; at some point, we will assume that a ≥ 1, but for now, we do not need this. Let
$D:=C0(Ra,b)⊗Cla,b=C0(Ra,b)⊗̂Cla,b.$
Kasparov constructed an element in $KKR0(C,D)$ and proved that it is invertible by defining an inverse and computing the Kasparov products. The definition of Kasparov’s KK-class is based on the unbounded continuous function,
$β̃a,b:Ra,b→Cla,b,(x1,…,xa+b)↦∑i=1a+bxiγi.$
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Lemma 5.

$β̃a,b$ is self-adjoint, odd, and real, and $β̃a,b(x)2=x2$.

Proof.

First, $β̃a,b$ is self-adjoint because $xi*=xi$ and $γi*=γi$ for all i. It is odd because all γi are odd and the grading on $C0(Ra,b)$ is trivial. Finally, $β̃a,b$ is real because γi is real whenever xi is real and γi is imaginary whenever xi is imaginary. The formula $β̃a,b(x)2=x2$ holds because $γi2=1$ for all 1 ≤ ia + b and γi anticommute.□

The bounded transform of $β̃a,b$ is
$β̃a,b′:Ra,b→Cla,b,x↦β̃a,b(x)/(1+β̃a,b(x)2)1/2.$
It satisfies $1−(β̃a,b′)2=(1+x2)−1/2∈D$ and is odd, self-adjoint, and real because $β̃a,b$ is. Hence, $[β̃a,b′]∈KKR0(C,D)$. Kasparov proved that $[β̃a,b′]$ generates $KKR0(C,D)≅Z$. To plug $[β̃a,b′]$ into the isomorphism in Lemma 4, we choose a degenerate cycle F0. Adding F0 replaces the underlying Hilbert module D of $[β̃a,b′]$ by $HD$. Thus, $β̃a,b′⊕F0$ and F0 are elements of $FUQ(C0(Ra,b,K⊗Cla,b))$ and $[β̃a,b′⊕F0]−[F0]$ represents a class in $DKQ(C0(Ra,b,K⊗Cla,b))$. Since degenerate KK-cycles represent zero, $[β̃a,b′⊕F0]−[F0]$ is a generator for $KKR0(C,D)≅Z$; here, the choice of F0 does not matter.
We may also interpret the passage from $β̃a,b$ to $β̃a,b′$ as replacing $Ra,b$ by
$Ba,b:=x∈Ra,b:x<1,$
which is homeomorphic to $Ra,b$ through the map $x↦x/(1+x2)1/2$. Let $B̄a,b$ be the closure of Ba,b in $Ra,b$. The boundary $B̄a,b\Ba,b$ is the a + b − 1-dimensional unit sphere $Sa,b⊆Ra,b$ with the “real” involution of $Ra,b$. There is a canonical embedding $C(B̄a,b)⊗Cla,b↪Cb(Ba,b)⊗Cla,b=M(C0(Ba,b)⊗Cla,b)$. It induces a canonical embedding $C(Sa,b)⊗Cla,b↪M(C0(Ra,b)⊗Cla,b)/(C0(Ra,b)⊗Cla,b)$. Its image contains the image of $β̃a,b′$ in this quotient. Namely, $[β̃a,b′]$ comes from $βa,b:=β̃a,b|Sa,b∈FU(C(Sa,b)⊗Cla,b)$; this is the same βa,b as in (2).

We assume from now on that a ≥ 1. Then, the first Clifford generator γ1 ∈ Cla,b is an odd self-adjoint real unitary. The constant function with value γ1 belongs to $FU(C(B̄a,b)⊗Cla,b)$ and is mapped to a degenerate KK-cycle for $KKR0(C,D)$. Hence, we may pick F0 to be a direct sum of countably many copies of γ1 above.

Lemma 6.

Let a ≥ 1. The class of $[βa,b]−[γ1]∈DK(C(Sa,b)⊗Cla,b)$ is mapped to a generator of $KKR0(C,D)$.

Proof.

The inclusion of $C(Sa,b)⊗Cla,b$ into $Q(C0(Ra,b)⊗K⊗Cla,b)$ is not unital. The map on van Daele K-theory that it induces sends [e] − [f] for $e,f∈FU(C(Sa,b)⊗Cla,b)$ to [eF0] − [fF0]. For our choice of F0 above, it therefore sends [βa,b] − [γ1] to $[βa,b⊕F0]−[F0]∈DKQ(C0(Ra,b)⊗K⊗Cla,b)$. We already know that Roe’s isomorphism maps the latter to a generator of $KKR0(C,D)$.□

Since a ≥ 1, the two poles S and N in (4) are fixed by the “real” involution on $Sa,b$. We choose $N∈Sa,b$ as a point at infinity and identify $Sa,b\{N}≅Ra−1,b$ by stereographic projection. The C-algebra extension
splits by taking constant functions. Since van Daele’s K-theory is split exact,
$DK(C(Sa,b)⊗Cla,b)≅DK(C0(Ra−1,b)⊗Cla,b)⊕DK(Cla,b).$

Proposition 7.
There is a Bott periodicity isomorphism,
$DK(C0(Ra−1,b)⊗Cla,b)≅KKR0(C,C0(Ra−1,b)⊗Cla−1,b)≅Z.$
Under the two isomorphisms above, $[βa,b]−[γ1]∈DK(C(Sa,b)⊗Cla,b)$ is mapped to $(1,0)∈Z⊕DK(Cla,b)$.

Proof.
The value of βa,b at N is γ1. Hence, evaluation at N maps [βa,b] − [γ1] to zero. We compute $DK(C0(Ra−1,b)⊗Cla,b)$ through the long exact sequences for DK applied to the morphism of C-algebra extensions,
The vertical arrow $C(B̄a,b\{N})↪M(C0(Ra,b)⊗K)$ combines the map from bounded continuous functions on $Ra,b$ to multipliers of $C0(Ra,b)$ and the corner embedding into $K$. It induces an isomorphism on DK because the latter is homotopy invariant, stable under tensoring with $K$, and vanishes on stable multiplier algebras. The left vertical map induces an isomorphism on DK as well. Therefore, the vertical map on the quotients also induces an isomorphism,
$DK(C0(Sa,b\{N}))≅DK(Q(C0(Ra,b)⊗K))≅KKR0(C,D)≅Z.$
Lemma 6 shows that the image of [βa,b] − [γ1] is mapped to a generator of $Z$ under these isomorphisms.□

We have described explicit elements in the van Daele K-theory of the spheres $Sa,b$ for $a,b∈N$. Topological phases are, however, described by the K-theory of the “real” torus $Td$, where d is the dimension of the physical system. One way to transfer K-theory classes from one space to another is the pullback functoriality. In Sec. VII, we are going to use this to pull our K-theory generators on $S1,d$ back to K-theory classes on $Td$ along certain maps $Td→S1,d$.

We now describe this pull back functoriality in detail. Let $(X,rX)$ and $(Y,rY)$ be “real” locally compact spaces, and let b: XY be a proper continuous map that is “real” in the sense that $rY◦b=b◦rX$. Such a map induces a “real” *-homomorphism,
$b*:C0(Y)→C0(X),ψ↦ψ◦b,$
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which for $c,d∈N$ induces maps in van Daele’s K-theory,
$b*:DK(C0(Y)⊗Clc,d)→DK(C0(X)⊗Clc,d).$
We denote it by b* as well because no confusion should be possible.
There are other ways to transfer K-theory classes between spaces. A famous example is the Atiyah–Singer index map for a family of elliptic differential operators. There are two ways to compute this index map, one analytic and one topological. We are going to use the topological approach. One of its ingredients is functoriality for open inclusions: let UX be an open subset with $rX(U)=U$ so that the “real” structure $rX$ restricts to one on U. Then, extension by zero defines a “real” *-homomorphism,
$ιU:C0(U)→C0(X),$
which for $c,d∈N$ induces maps in van Daele’s K-theory,
$ιU,*:DK(C0(U)⊗Clc,d)→DK(C0(X)⊗Clc,d).$
The other ingredient of the topological index map is the Thom isomorphism. Let EX be a $Z/2$-equivariant vector bundle. Let ClE be the Clifford algebra bundle of E; this is a locally trivial bundle of finite-dimensional $Z/2$-graded, real C-algebras over X. Each fiber C0(Ex) ⊗ Cl(Ex) carries a canonical Kasparov generator $β̃Ex′$. Letting x vary, these combine to a class
$[β̃E′]∈KKR0(C0(X),C0(E)⊗ClE).$
(7)
This class is also invertible and produces a raw form of the Thom isomorphism.

Definition 8.

A KR-orientation on a $Z/2$-equivariant vector bundle EX is a $Z/2$-graded, “real,” C0(X)-linear Morita equivalence between C0(X, ClE) and C0(X) ⊗ Cla,b for some $a,b∈N$. That is, it is a full Hilbert bimodule M over C0(X, ClE) and C0(X) ⊗ Cla,b with a compatible “real” structure and $Z/2$-action and with the extra property that the two actions of C0(X) by multiplication on the left and right are the same. A vector bundle is called KR-orientable if it has such a KR-orientation. We call $a−bmod8∈Z/8$ the KR-dimension dimKRE of E.

Morita equivalence for “real” $Z/2$-graded C-algebras with an action of a locally compact groupoid is explored by Moutuou.16 Our definition is the special case where the groupoid is the space X with only identity arrows. The idea to describe K-theory orientations through Morita equivalence goes back to Plymen.17

Two Clifford algebras Cla,b and Cla′,b are Morita equivalent as “real” $Z/2$-graded C-algebras if and only if aba′ − b′ mod 8; this is part of the computation of the Brauer group of the point in Appendix A in Ref. 16. This shows that the KR-dimension is well defined and that it suffices to consider, say, the cases b = 0, a ∈ {0, 1, …, 7} in Definition 8. A trivial vector bundle $X×Ra,b↠X$ is, of course, KR-oriented because $ClX×Ra,b=X×Cla,b$. Its KR-dimension is ab mod 8.

If a = b = 0, then a KR-orientation is a $Z/2$-graded, “real” Morita equivalence between C0(X, ClE) and C0(X). This is equivalent to a $Z/2$-graded “real” Hilbert C0(X)-module $S$ with a C0(X)-linear isomorphism $K(S)≅C0(X,ClE)$. This forces $S$ to be the space of C0-sections of a complex vector bundle over X, equipped with a $Z/2$-grading and a “real” involution. Then, the isomorphism $K(S)≅C0(X,ClE)$ means that the fiber $Sx$ of this vector bundle carries an irreducible representation of $ClEx$. Thus, $S$ is the space of C0-sections of a “real” spinor bundle for E. Allowing a, b ≠ 0 gives an appropriate analog of the spinor bundle for “real” vector bundles of all dimensions.

The KR-orientation induces a KKR-equivalence because KKR is Morita invariant. Together with the KKR-equivalence in (7), this gives a KKR-equivalence,
$τE∈KKR0C0(X),C0(E)⊗Cla,b≅KKR−dimKREC0(X),C0(E),$
called the Thom isomorphism class. It induces Thom isomorphisms in K-theory,
$KRn(X)≅KRn+dimKRE(E)for n∈Z/8.$

Example 9.

Let EX be a complex vector bundles with a “real” structure. A Thom isomorphism for E is defined in Theorem 2.4 in Ref. 1. It is analogous to the Thom isomorphism for complex vector bundles in ordinary complex K-theory. Atiyah’s Thom isomorphism may also be defined using a KR-orientation of dimension 0 as defined above. Namely, the sum of the complex exterior powers of E provides a spinor bundle for E, which also carries a canonical “real” structure to become a KR-orientation of KR-dimension 0 as in Definition 8.

Lemma 10.
Let E1, E2 be two $Z/2$-equivariant vector bundles over X. If two of the vector bundles E1, E2, E1E2 are KR-oriented, then this induces a canonical KR-orientation on the third one such that
$dimKRE1+dimKRE2=dimKR(E1⊕E2).$

Proof.

A C0(X)-linear Morita equivalence is the same as a continuous bundle over X whose fibers are Morita equivalences. Such bundles may be tensored together over X. With the graded tensor product over X, we get $ClE1⊕E2≅ClE1⊗̂ClE2$. Therefore, KR-orientations for E1 and E2 induce one for E1E2. Assume, conversely, that E1E2 and E2 are KR-oriented. Then, the bundle $ClE2$ is Morita equivalent to the trivial bundle with fiber Cla,b for some $a,b∈N$. Hence, $ClE2⊗̂Clb,a$ is Morita equivalent to a trivial bundle of matrix algebras, making it Morita equivalent to the trivial rank-1 bundle $X×C$. Therefore, $ClE1$ is Morita equivalent to $ClE1⊗̂ClE2⊗̂Clb,a≅ClE1⊕E2⊗̂Clb,a$. Since E1E2 is KR-oriented, this is also Morita equivalent to some trivial Clifford algebra bundle. This gives a KR-orientation on E1.□

The following proposition clarifies how many KR-orientations a KR-orientable vector bundle admits.

Proposition 11.

Two KR-orientations for the same $Z/2$-equivariant vector bundle become isomorphic after tensoring one of them with a $Z/2$-graded “real” complex line bundle LX, which is determined uniquely.

Proof.

Let $S1$ and $S2$ be two C0(X)-linear “real” graded Morita equivalences between C0(X, ClE) and C0(X) ⊗ Cla,b. Then, the composite of $S1$ and the inverse of $S2$ are C0(X)-linear “real” graded Morita self-equivalences of C0(X) ⊗ Cla,b. Taking the exterior product with Cla,b is clearly bijective on isomorphism classes of Morita self-equivalences if a = b because, then, Cla,b is a matrix algebra. If ab, then it is also bijective on isomorphism classes because when we tensor first with Cla,b and then with Clb,a, we tensor with Cla+b,a+b, which is already known to be bijective on isomorphism classes. Therefore, we may replace the self-equivalence of C0(X) ⊗ Cla,b by one of C0(X). Such a Morita self-equivalence of C0(X) is well known to be just a complex line bundle L. In our case, the line bundle must also carry a “real” involution and a $Z/2$-grading. Unraveling the bijections on isomorphism classes, we see that $S2$ is isomorphic to $L⊗̂S1$.□

Remark 12.

Assume that X is connected. Let LX be a “real” complex line bundle. Then, there are two ways to define a $Z/2$-grading on L: we may declare all of L to have parity 0 or 1. When we tensor a KR-orientation with the trivial line bundle in negative parity, we merely flip its $Z/2$-grading. Flipping the $Z/2$-grading on a KR-orientation is also called orientation-reversal.

We are going to construct a geometric bivariant KR-theory for spaces with a “real” involution. We follow Refs. 8 and 9, where equivariant geometric bivariant K-theory is developed for spaces with a groupoid action. While it is mentioned there that the theory also works for KO-theory, the more general KR-theory is not mentioned there explicitly. Nevertheless, the theory developed there applies because KR-theory comes from a $Z/2$-equivariant cohomology theory. This is checked in some detail in Ref. 11. The proof is similar to the proof of the same result for $Z/2$-equivariant KO-theory, which only differs in that the group $Z/2$ acts linearly instead of conjugate-linearly. Therefore, we do not repeat the proof here.

We clarify, however, how to define the relevant cohomology theory because we will use this later anyway. KR-theory, like ordinary K-theory, is not a cohomology theory because it is only functorial for proper continuous maps. The cohomology theory from which K-theory comes is called representable K-theory. Adapting the approach in Ref. 18 for groupoid-equivariant K-theory to the “real” case, we define the representable analog of KR-theory as
$KRXn(X):=KKR−nXC0(X),C0(X).$
(8)
The right-hand side means the C0(X)-linear “real” version of Kasparov theory, which Kasparov19 denoted by $RKK$.

The following result is shown in Ref. 11 and is what is needed to apply the machinery developed in Refs. 8 and 9 to KR-theory.

Theorem 13

(see Ref. 11). Representable KR-theory is a multiplicative $Z/2$-equivariant cohomology theory, and KR-oriented vector bundles are oriented for it.

Our notation for representable KR-theory alludes to a further generalization, namely, the KR-theory $KRZ*(X)$ of a space X with Z-compact support, given a map b: XZ. As in Ref. 18, we define this by
$KRZn(X):=KKR−nZC0(Z),C0(X).$
(9)
If Z = pt, then KRZ = KR is the KR-theory as defined above. Representable KR-theory is the special case where Z = X and b is the identity map. More generally, if b is proper, then $KRZ*(X)=KRX*(X)$ is the representable KR-theory of X. In particular, if X is compact, then $KRZ*(X)=KR*(X)$ for all b: XZ.

Let X be a $Z/2$-manifold. Then, any “real” complex vector bundle E over X defines a class in $KRX0(X)$. With our Kasparov theory definition, this is the Hilbert C0(X)-module of sections of E, with C0(X) acting by pointwise multiplication also on the left. This only defines a class in the usual KR0(X) if X is compact. If X is compact, we also know that $KR0(X)=KRX0(X)$ is the Grothendieck group of the monoid of such “real” vector bundles.

For a finite-dimensional CW-complex X, it is well known that its representable K-theory and KO-theory are the Grothendieck groups of the monoids of complex and real vector bundles over X. The analogous result for $Z/2$-equivariant K-theory is false, however, as shown by the counterexample in Example 3.11 in Ref. 20. This counterexample for $KZ/2$, however, does not work like this for KR-theory. Therefore, it is possible that the representable KR-theory of a finite-dimensional $Z/2$-CW-complex is always isomorphic to the Grothendieck group of the monoid of “real” complex vector bundles over X. We have not investigated this question.

More generally, let us add a $Z/2$-map b: XZ. Choose two “real” complex vector bundles E± over X together with an isomorphism $φ:E+|X\A→∼E−|X\A$ for an open subset AX whose closure is Z-compact in the sense that $b|Ā:Ā→Z$ is proper. We may equip E± with inner products. We may arrange these so that φ is unitary. Using the Tietze extension theorem, we may extend φ to a continuous section $φ̃$ of norm 1 of the vector bundle Hom(E+, E) on all of X. Then, we define a cycle for $KKR0ZC0(Z),C0(X)$ as follows. The Hilbert module consists of the space of sections of E+E with the $Z/2$-grading induced by this decomposition and with the “real” structure induced by the “real” structures on E±. A function h ∈ C0(Z) acts on this by pointwise multiplication with hb. The Fredholm operator is pointwise multiplication with $φ̃$. This is indeed a cycle for $KKR0ZC0(Z),C0(X)$ because $φ̃$ is unitary outside a Z-compact subset.

With the definition in (9), a class $ξ∈KRZn(X)$ clearly yields $[ξ]∈KKR−nC0(Z),C0(X)$, which induces maps ξ: KRa(Z) → KRa+n(X) for $a∈Z/8$. This generalizes both the pull back functoriality for continuous proper maps b: XZ and the map on KR*(X) that multiplies with a vector bundle on X.

In the following, we shall specialize some of the theory in Refs. 8 and 9 and simplify it a bit for our more limited purposes. First, we take the groupoid denoted as $G$ in Refs. 8 and 9 to be the group $Z/2$. This is because a “real” structure is the same as a $Z/2$-action. Hence, the object space of $G$, which is denoted as Z in Refs. 8 and 9, is just the one-point space pt. Thus, fiber products over Z become ordinary products.

Second, we work in the smooth setting, that is, with smooth manifolds without boundary and with smooth maps only. We also assume smooth manifolds to be finite-dimensional, which is automatic if they are connected. We briefly call a finite-dimensional smooth manifold with a smooth $Z/2$-action a $Z/2$-manifold.

Remark 14.

Swan’s theorem says that any vector bundle over a paracompact space of finite covering dimension is a direct summand in a trivial vector bundle. This basic result may fail for groupoid-equivariant K-theory even when the spaces are compact and the groupoid is a bundle of Lie groups (see Example 2.7 in Ref. 8). This creates the need to speak of “subtrivial” equivariant vector bundles in the general setting considered in Refs. 8 and 9. However, for the finite group $Z/2$, any $Z/2$-equivariant vector bundle over a $Z/2$-manifold is subtrivial by Theorem 3.11 in Ref. 8. We use this occasion to point out that the hypotheses in that theorem are wrong: it should be assumed that the space Y and not X is finite-dimensional. Any smooth finite-dimensional manifold X with a smooth $Z/2$-action has a structure of finite-dimensional $Z/2$-CW-complex. In the following, we may therefore drop the adjective “subtrivial” as long as we restrict attention to $Z/2$-manifolds.

Remark 15.

Since $Z/2$ is a finite group, its regular representation is a “full vector bundle” over pt. Such a vector bundle is needed for several results in Refs. 8 and 9, and it comes for free in our case. Any linear representation of $Z/2$ is isomorphic to a direct sum of copies of the two characters of $Z/2$. That is, it is isomorphic to $Ra,b$ where the generator of $Z/2$ acts by the “real” involution on that space.

After these preliminary remarks, we construct wrong-way functoriality or shriek maps for KR-oriented smooth maps. This is based on factorizing smooth maps in a certain way. The factorization is called a normally nonsingular map in Refs. 8 and 9. Under the extra assumptions that we impose, any smooth map has such a factorization and it is unique up to “smooth equivalence;” this implies that the shriek map is independent of the factorization. Hence, the factorization becomes irrelevant in the special case that we consider here. This fails already for smooth maps between smooth manifolds with boundary (see Example 4.7 in Ref. 8). Hence, the theory of normally nonsingular maps is needed to treat this more general class of spaces. Our applications, however, concern only manifolds without boundary. Therefore, we will define KR-oriented correspondences only in this special case to simplify the theory. Nevertheless, we include the basic definition of a normally nonsingular map to clarify what would be needed to extend the theory to more general spaces than $Z/2$-manifolds.

Definition 16.

Let X and Y be $Z/2$-manifolds. A (smooth) normally nonsingular $Z/2$-map from X to Y consists of the following data:

• V, a $Z/2$-equivariant $R$-vector bundle over X.

• $E=Ra,b$ for some $a,b∈N$, an $R$-linear representation of the group $Z/2$, which we treat as a $Z/2$-equivariant vector bundle over pt.

• $f̂:V↪Y×Ra,b$, a $Z/2$-equivariant diffeomorphism between the total space of V and an open subset of $Y×Ra,b=Y×ptE$.

Here, $V$ is the total space of the vector bundle V. Let $ζV:X→V$ be the zero section of V, and let $πY:Y×Ra,b→Y$ be the coordinate projection. The trace of $(V,E,f̂)$ is the dotted composite map,
(10)

The following definition describes two ways to change a normally nonsingular map. The important feature is that they do not change the resulting shriek map.

Definition 17.

Let $(V,E,f̂)$ be a normally nonsingular map, and let E0 be another linear representation of $Z/2$. Then, $(V⊕(X×E0),E⊕E0,f̂×idE0)$ is another normally nonsingular map, called a lifting of $(V,E,f̂)$. A normally nonsingular map from X × [0, 1] to Y × [0, 1] where the map $f̂$ is a map over [0,1] is called an isotopy between the two normally nonsingular maps that arise by restricting to the end points of [0,1]. Two smooth normally nonsingular maps are called smoothly equivalent if they have liftings that are isotopic.

Proposition 18

(Theorems 3.25 and 4.36 in Ref. 8). Any smooth $Z/2$-map is the trace of a normally nonsingular map, which is unique up to smooth equivalence. Two smooth $Z/2$-maps are smoothly homotopic if and only if their lifts to normally nonsingular maps are smoothly equivalent.

We recall how to lift a smooth $Z/2$-map f: XY to a normally nonsingular map. There is a $Z/2$-equivariant, proper embedding $i:X→Ra,b$ for some $a,b∈N$. Let $E:=Ra,b$. The map $(f,i):X→Y×Ra,b$ is still a $Z/2$-equivariant, proper embedding. Let V be its normal bundle. By the tubular neighborhood theorem, the total space of V is $Z/2$-equivariantly diffeomorphic to a neighborhood of the image of (f, i). This gives the open embedding $f̂$ for a normally nonsingular map.

Definition 19.

Let f: XY be a smooth $Z/2$-map. A stable normal bundle for f is a vector bundle VX such that VTX is isomorphic to $f*(TY)⊕(X×Ra,b)$ for some $a,b∈N$. A KR-orientation for f is a stable normal bundle V for f together with a KR-orientation of V. Its KR-dimension is the difference of the KR-dimension of V and ab mod 8, the KR-dimension of the trivial bundle $X×Ra,b↠X$.

Lemma 20.

Let $(V,E,f̂)$ be a normally nonsingular $Z/2$-map, and let f: XY be its trace. Then, a KR-orientation for f is equivalent to a KR-orientation of the vector bundle V.

Proof.

The bundle V is a stable normal bundle for f. Therefore, a KR-orientation of V induces one for f. Any two stable normal bundles of f are stably isomorphic. Thus, Lemma 10 implies that a KR-orientation for one stable normal bundle of f induces KR-orientations on all other stable normal bundles of f.□

Definition 21.

A KR-orientation on a $Z/2$-manifold X is a KR-orientation on its tangent vector bundle TX. The KR-dimension dimKRX is defined as dimKRTX.

If X and Y are two KR-oriented $Z/2$-manifolds, then any smooth map f: XY inherits a KR-orientation by Lemma 10, and its KR-dimension is
$dimKRf=dimKRY−dimKRX.$
(11)

Example 22.

We are going to describe KR-orientations on the $Z/2$-manifolds $Ra,b$, $Sa,b$, and $Td$. The tangent bundle of $Ra,b$ is the trivial bundle with fiber $Ra,b$. Hence, its Clifford algebra bundle is already isomorphic to the trivial Clifford algebra bundle with fiber Cla,b.

The covering $R→T$, t ↦ exp(it), becomes a “real” covering $R0,1→T$. This induces an isomorphism between the tangent bundle of $T$ and the trivial bundle with fiber $R0,1$. Taking a d-fold product, we get an isomorphism from the tangent bundle of $Td$ to the trivial bundle with fiber $R0,d$. This induces an isomorphism between the Clifford algebra bundle of $TTd$ and the trivial bundle $Td×Cl0,d$.

The outward pointing radial vector field /∂r on the unit sphere spans the normal bundle of the inclusion map $Sa,b→Ra,b$. Since the involution on $Ra,b$ is linear, it preserves this vector field, that is, the normal bundle is $Z/2$-equivariantly isomorphic to the trivial bundle with fiber $R1,0$. Hence, $TSa,b⊕(Sa,b×R1,0)≅Sa,b×Ra,b$. Then, Lemma 10 gives a KR-orientation on $TSa,b$. We also find
$dimKR(Ra,b)=a−b,dimKR(Td)=−d,dimKR(Sa,b)=a−b−1,$
where the three numbers are understood as elements of $Z/8$. Note that the above argument for $Sa,b$ also works for a = 0.

Let $(V,E,f̂)$ be a KR-oriented, normally nonsingular $Z/2$-map, and let f: XY be its trace. Then, there are Thom isomorphisms for both the vector bundle VX and the trivial vector bundle $Y×Ra,b↠Y$. The open inclusion $f̂$ identifies $C0(V)$ with an ideal in $C0(Y×Ra,b)$. Thus, each of the solid maps in (10) induces a map in KR-theory. Taking the composite gives induced maps,
$f!:KRn(X)→KRn+dimKRf(Y)for n∈Z/8.$
Since the Thom isomorphisms come from KKR-classes, the map f! is the Kasparov product with an element in $KKR−dimKRf(C0(X),C0(Y))$, which we will also denote by f!. It turns out that f! is preserved under lifting and isotopy and is functorial in the sense that
$f!◦g!=(f◦g)!$
for two composable $Z/2$-maps f, g (see Ref. 8). In particular, by Proposition 18, f! only depends on the smooth map f and its KR-orientation, justifying the notation f!. The map f! is called the shriek map of f, and the map that sends f to f! is the wrong-way functoriality of KR-theory.

Example 23.

Let EX be a KR-oriented vector bundle. Then, the zero section ζE: XE and the bundle projection πE: EX are KR-oriented in a canonical way such that their shriek maps are the Thom isomorphism and its inverse $KRn(X)↔KRn+dimKRE(E)$.

Example 24.

Let X be a $Z/2$-manifold. Let TX be its tangent bundle. As a complex manifold, it carries a canonical KR-orientation of KR-dimension 0 (see Example 9). This induces a KR-orientation on the constant map f: TX → pt to the one-point space. The shriek map $f!:KR0(TX)→KR0(pt)=Z$ is the Atiyah–Singer topological index map on X.

Remark 25.

If f is a smooth submersion, then f! has an analytic variant f!,an, given by the class in Kasparov theory of the family of Dirac operators along the fibers of f. This analytic version is equal to the topological one, that is, f!,an = f! holds in $KKR−dimKRf(C0(X),C0(Y))$. The proof is the same as for Theorem 6.1 in Ref. 8, which deals with the analogous statement in KK, that is, when we forget the “real” structures. This is equivalent to the family version of the Atiyah–Singer index theorem for the family of Dirac operators along the fibers of the submersion f.

We are particularly interested in the shriek maps for an inclusion f: pt → X, where pt denotes the one-point space with the trivial “real” structure. Hence, f(pt) ∈ X must be a fixed point of the “real” involution on X. The map f is KR-orientable because any vector bundle over a point is trivial and thus KR-orientable. A KR-orientation for X chooses a canonical KR-orientation for f. Let us specialize to the case where $X=Sa,b$ with a > 1 so that S and N in (4) are fixed points. Let $iS:pt→Sa,b$ be the inclusion of the south pole. Give $Sa,b$ the KR-orientation from Example 22. Then, we get a canonical map
$S!:Z≅KR0(pt)→KRa−b−1(Sa,b)≅DK(C(Sa,b)⊗Cla,b).$

Lemma 26.

There is χ ∈ {±1} such that the map S! sends the generator $1∈Z$ to χ · ([βa,b] − [γ1]).

Proof.

The stereographic projection at the north pole induces a $Z/2$-equivariant diffeomorphism $Sa,b\{N}≅Ra−1,b$. We could use this as a tubular neighborhood for the embedding iS. Therefore, S! factors through an isomorphism onto the direct summand $KRa−b−1(Sa,b\{N})≅Z$. Since [βa,b] − [γ1] generates this summand, S! must send the generator $1∈Z$ to ±([βa,b] − [γ1]).□

The sign χ depends on the choices of KR-orientations and signs in boundary maps, and we do not compute it. It will appear in several formulas below.

The power of geometric bivariant K-theory is that there often is a simple way to compute composite maps $b∗$f! for a KR-oriented map f: XY and a proper continuous map b: ZY. We will use this to compute the pullback of [β1,d] to the van Daele K-theory of $Td$ using only geometric considerations. This circumvents rather messy Chern character computations in the physics literature (see Refs. 5 and 6), and it also works in the real case. The following proposition makes precise when and how we may compute $b∗$f!:

Proposition 27.
Let X and Y be $Z/2$-manifolds. Let f: XY be a KR-oriented smooth $Z/2$-map, and let b: ZY be a proper smooth $Z/2$-map. Assume that b and f are transverse. Then, the fiber product X × YZ is a smooth $Z/2$-manifold. The projection πX: X × YZX is proper, and πZ: X × YZZ inherits from f a KR-orientation of the same KR-dimension. The following diagram commutes:
(12)

Proof.
This is a special case of Theorem 2.32 in Ref. 9 about composing KR-oriented correspondences. We will discuss a more general result later. Here, we only describe the KR-orientation that the projection πZ inherits. The transversality assumption implies that there is an exact sequence of vector bundles over X × YZ as follows:
$T(X×YZ)↣πX*(TX)⊕πZ*(TZ)↠π*(TY).$
Since any such extension splits $Z/2$-equivariantly, it follows that
$π*(TY)⊕T(X×YZ)≅πX*(TX)⊕πZ*(TZ).$
Since f is KR-oriented, there are $a,b∈N$ and a KR-oriented vector bundle VX such that $f*(TY)⊕(X×Ra,b)≅TX⊕V$. Then, $π*(TY)⊕(X×YZ×Ra,b)≅πX*(TX)⊕πX*(V)$ and, hence,
$πX*(TX)⊕πZ*(TZ)⊕(X×YZ×Ra,b)≅πX*(TX)⊕πX*(V)⊕T(X×YZ).$
Since the $Z/2$-equivariant vector bundle $πX*(TX)$ is subtrivial, we may add another $Z/2$-equivariant vector bundle to arrive at an isomorphism,
$πZ*(TZ)⊕(X×YZ×Ra+a′,b+b′)≅(X×YZ×Ra′,b′)⊕πX*(V)⊕T(X×YZ).$
Thus, $(X×YZ×Ra′,b′)⊕πX*(V)$ is a stable normal bundle for πZ. It inherits a KR-orientation from the obvious KR-orientation on the trivial bundle and the given KR-orientation of V by Lemma 10.□

We will later need the special case when f is the inclusion of a point y0Y so that X = pt. A KR-orientation of f is the same as a KR-orientation on the tangent space $Ty0Y$. In this case, the coordinate projection πZ is a bijection between X × YZ and the preimage b−1(y0) of y0 in Z. Transversality means in this case that the differential of b is surjective in all points of b−1(y0). Then, b−1(y0) is a smooth submanifold. It is also compact because b is proper. The normal bundle of the inclusion of b−1(y0) into Z is canonically isomorphic to the trivial bundle with fiber $Ty0Y$. This gives the induced KR-orientation of πZ.

The commuting diagram (12) is best understood in the setting of geometric bivariant KR-theory. This theory describes KKR(C0(X), C0(Y)) for two $Z/2$-manifolds X and Y in a geometric way. Its main ingredients are
$b*∈KKR0(C0(Y),C0(Z)),f!∈KKR−dimKR(f)(C0(X),C0(Y))$
for proper smooth maps b: ZY and KR-oriented smooth maps f: XY. The details are explained in Sec. VI.

We now define KR-oriented correspondences between two $Z/2$-manifolds X and Y as in Ref. 9. These produce a geometric version of bivariant “real” Kasparov theory. The definition of a correspondence in Ref. 9 differs slightly from the original definition of correspondences by Connes and Skandalis in Sec. III in Ref. 10 by allowing maps b that fail to be proper. This greatly simplifies the proof that the geometric and analytic bivariant KR-theories agree. We shall use smooth maps, whereas the definition in Ref. 9 uses normally nonsingular maps. This makes no difference for $Z/2$-manifolds because of Proposition 18.

Definition 28.

Let X and Y be $Z/2$-manifolds. A (smooth) KR-oriented correspondence from X to Y is a quadruple (M, b, f, ξ), where

• M is a $Z/2$-manifold,

• b: MX is a smooth $Z/2$-map,

• f: MY is a KR-oriented smooth $Z/2$-map, and

• $ξ∈KRXa(M)$ for some $a∈Z/8$ (here, X-compact support in M refers to the map b: MX).

The KR-dimension of the correspondence is defined as a + dimKR(f). We often depict a correspondence as

The letters f and b in the definition stand for “forwards” and “backward.”

Example 29.
A proper, smooth $Z/2$-map b: YX yields the following correspondence from X to Y:
Here, idY is the identity map with its canonical KR-orientation, which is indeed a unit arrow in the category of KR-oriented smooth maps, and $1∈KRX0(Y)$ comes from the trivial “real” vector bundle of rank 1.

Example 30.
A KR-oriented smooth $Z/2$-map f: XY yields a KR-oriented correspondence from X to Y,

Example 31.
Any class ξ in the representable KR-theory $KRX*(X)$ of X yields a correspondence from X to itself,

We shall mainly consider correspondences where the map b is proper and ξ is the unit element in $KRX0(X)$ as in Example 29.

A KR-oriented correspondence (M, b, f, ξ) induces an element
$f!◦[ξ]∈KKR−a−dimKR(f)C0(X),C0(Y),$
which induces maps
$f!◦ξ*:KRp(X)→KRp+a+dimKR(f)(Y)for all p∈Z/8.$
For instance, the correspondence in Example 29 gives the class in $KKR0C0(X),C0(Y)$ of the *-homomorphism b* as in (6), the correspondence in Example 30 gives f!, and the correspondence in Example 31 gives the image of ξ in $KKR0C0(X),C0(X)$ under the forgetful map.

The geometric bivariant KR-theory $KR̂(X,Y)*$ is defined as the set of equivalence classes of KR-oriented correspondences, where “equivalence” means the equivalence relation generated by bordism (see Definition 2.7 in Ref. 9) and Thom modification (see Definition 2.8 in Ref. 9), which replaces M in a correspondence by the total space of a KR-oriented vector bundle over it. We shall not use the precise form of these relations below and therefore do not repeat them here. It is shown in Ref. 9 that the disjoint union of correspondences makes $KR̂(X,Y)*$ an Abelian group.

Theorem 32.
Let $n∈Z/8$, and let X and Y be $Z/2$-manifolds. The map above from KR-oriented correspondences to Kasparov theory defines an isomorphism
$KR̂(X,Y)n≅KKR−nC0(X),C0(Y).$

Proof.

This is shown by following the proof in Ref. 9 for equivariant KK-theory. The same arguments as in the Proof of Theorem 4.2 in Ref. 9 show that the map is well defined and a functor for the composition of KR-oriented correspondences defined in Ref. 9. Next, Theorem 2.25 in Ref. 9 for the $Z/2$-equivariant cohomology theory KR shows that the map in the theorem is bijective if X = pt. The bivariant case is reduced to this easy case using Poincaré duality. Any $Z/2$-manifold X admits a “symmetric dual” in KR-theory because of Theorem 3.17 in Ref. 9. This is another $Z/2$-manifold P with a smooth $Z/2$-map PX such that there are duality isomorphisms $KR̂(X,Y)n≅KRXn+a(Y×P)$ in geometric bivariant KR-theory. As in Theorem 4.2 in Ref. 9, the geometric bivariant KR-theory classes that give this duality also give an isomorphism $KKR−nC0(X),C0(Y)≅KRXn+a(Y×P)$.□

Thus, KR-oriented correspondences provide a purely geometric way to describe Kasparov cycles between the C-algebras of functions on $Z/2$-manifolds. An important feature is that the Kasparov product may be computed geometrically under an extra transversality assumption.

Theorem 33.
Let two composable smooth KR-oriented correspondences be given, as described by the solid arrows in the following diagram:
(13)
Assume that the smooth maps f1 and b2 are transverse, that is, if m1M1, m2M2 satisfy yf1(m1) = b2(m2), then $Df1(Tm1M1)+Db2(Tm2M2)=TyY$. Then, the following holds. First, M1 × YM2 is a smooth $Z/2$-manifold. Second, the exterior tensor product $ξ:=π1*(ξ1)⊗Yπ2*(ξ2)$ has X-compact support with respect to bb1π1, that is, it belongs to $KRX*(M1×YM2)$. Third, the composite map ff2π2 inherits a KR-orientation from f1 and f2, whose KR-dimension is dimKRf = dimKRf1 + dimKRf2. Finally, the resulting KR-oriented correspondence (M1 × YM2, b, f, ξ) is the composite of the two given KR-oriented correspondences, that is, its image in Kasparov theory is the Kasparov product of the Kasparov theory images of (M1, b1, f1, ξ1) and (M2, b2, f2, ξ2).

Proof.

In Ref. 9, the composition of correspondences is first defined in a special case. Any KR-oriented correspondence is equivalent to one where the forward map is the restriction of the coordinate projection $Y×Ra,b↠Y$ to an open subset $M⊆Y×Ra,b$ (see Theorem 2.24 in Ref. 9). Then, f is a submersion and hence transverse to any smooth map. The composition of these “special” correspondences gives geometric bivariant KR-theory a category structure, and the canonical map to Kasparov theory is a functor. With this preparation, the claim in our theorem mostly follows from Theorem 2.32 and Example 2.31 in Ref. 9. That theorem says that the “intersection product” in (13) is equivalent to the composite in geometric bivariant KR-theory. The example in Ref. 9 says that the usual transversality notion from differential geometry implies the transversality assumption that is assumed in the theorem in Ref. 9 (which makes sense for correspondences with a normally nonsingular forward map).□

Proposition 27 is the special case of Theorem 33 where b1 and f2 are identity maps, b2 is proper, and ξ1 and ξ2 are the units in representable KR-theory.

Remark 34.

The exterior product of two KR-oriented correspondences is defined by simply taking the product of all spaces and maps and the exterior product of the KR-theory classes involved. This defines a symmetric monoidal structure on geometric bivariant KR-theory by Theorem 2.27 in Ref. 9. It is easy to check that it lifts the exterior product on Kasparov theory (compare Theorem 4.2 in Ref. 9).

In this section, we compute the topological phase of certain translation-invariant Hamiltonians that have been considered in the mathematical physics literature. They are given by the formula
$Hm:=12i∑j=1d(Sj−Sj*)⊗γj+m+12∑j=1d(Sj+Sj*)⊗γ0∈C*(Zd)⊗Cl1,d$
for $m∈R$ (see Secs. 2.2.4 and 2.3.3 in Ref. 2). Here, $Si∈C*(Zd)$ is the unitary for the element $ei∈Zd$. Throughout this section, we start numbering Clifford generators for Cl1,d at 0 and not at 1 as before. We give $C*(Zd)$ the trivial $Z/2$-grading and the “real” involution where the generators Si are real. Then, the element Hm is self-adjoint, odd, and real because γ0, iγ1, …, iγd are real by our conventions. The Fourier transform identifies the commutative C-algebra $C*(Zd)$ with the C-algebra of continuous functions on the torus $Td$. Our “real” structure on $C*(Zd)$ transforms under this isomorphism to the “real” structure on $Td$ in (1).
Define $β̃1,d:R1,d→Cl1,d$ as in (5), and define
$φ̃m:Td→R1,d,(x,y)↦(x1+⋯+xd+m,y1,…,yd).$
This map is “real.” Since Si and $Si*$ have the Fourier transforms xi ± iyi, the Fourier transform of Hm in $C(Td,Cl1,d)$ is equal to $β̃1,d◦φ̃m$.

Lemma 35.

If m ∉ {−d, −d + 2, …, d − 2, d}, then $φ̃m(Td)⊆R1,d\0$.

Proof.

Assume $φ̃m(x,y)=0$. Then, x1 +⋯+ xd + m = 0 and y1 = y2 =⋯= yd = 0. The latter forces xi = ±1 for i = 1, …, d, and then, $m=−∑i=1dxi$ must belong to {−d, −d + 2, …, d − 2, d}.□

From now on, we assume m ∉ {−d, −d + 2, …, d + 2, d}. By Lemma 35, this is equivalent to Hm being invertible, which is needed to define its topological phase.

Under our assumption on m, there is a well-defined “real” function,
$φm:Td→S1,d,φm(z):=1φ̃m(z)⋅φ̃m(z),$
as in (3). Since $β̃1,d(x)2=x2$, the “spectral flattening” of Hm with spectrum {±1} is the operator whose Fourier transform in $C(Td,Cl1,d)$ is $β̃1,d◦φm$. Since φm takes values in $S1,d$ by construction, this is just β1,dφm with the real, odd, self-adjoint unitary $β1,d∈C(S1,d)⊗Cl1,d$ in (2). The composite $β1,d◦φm∈FU(C(Td)⊗Cl1,d)$ is its pullback along φm.

To get a class in van Daele’s K-theory, we must consider a formal difference [β1,dφm] − [f] for some $f∈FU(C(Td)⊗Cl1,d)$. Physically, f describes the topological phase that we choose to call “trivial.” An obvious choice in our case is f = γ0, the constant function on $Td$ with value γ0. Another obvious choice would be −γ0. In the complex case, these two are homotopic. In the “real” case, however, these two choices turn out to have different classes in KR-theory for d ≤ 2 (see Lemma 39). Hence, the sign choice here actually matters.

Lemma 36.
Up to the sign χ from Lemma 26, the class [β1,dφm] − [γ0] in $DK(C(Td)⊗Cl1,d))≅KR−d(Td)$ is the composite of the KR-oriented correspondences,

Proof.

The correspondence S! represents the class χ · ([β1,d] − [γ0]) by Lemma 26. Here, γ0 denotes the constant function on $S1,d$ with value γ0, and we changed the numbering of Clifford generators to start at 0. Composing with the correspondence denoted as $φm*$ pulls this back along the map φm. This gives χ · ([β1,dφm] − [γ0]), where now γ0 denotes the constant function on $Td$ with value γ0.□

We need to introduce certain signs. Let $z=(x1,…,xd,0,…,0)∈Td$; then, xj ∈ {−1, +1} for j = 1, …, d. We define
$sign(x1,…,xd,0,…,0):=x1⋅x2⋯xd∈{−1,+1}.$

Lemma 37.
The two correspondences in Lemma 36 are transverse. Therefore, their composite is their intersection product. The fiber product $Td×S1,dpt$ in the intersection product is diffeomorphic to the finite subset $φm−1(S)⊆Td$, and
$[β1,d◦φm]−[γ0]=χ∑z∈φm−1(S)sign(z)z!$
holds in $DK(C(Td)⊗Cl1,d)≅KR−d(Td)$.

Proof.

We must show that the differential of φm is a surjective map onto $TSS1,d$ at all points $(x,y)∈Td$ with φm(x, y) = S. The tangent space $TSS1,d$ is the subspace ${0}×Rd$, spanned by the basis vectors e1, …, ed. If φm(x, y) = S, then y1 = y2 =⋯= yd = 0 follows as in the proof of Lemma 35. At these points, the tangent space of $Td$ is spanned by the vectors in the directions y1, …, yd. The differential of $φ̃m$ maps these to the vectors e1, …, ed. On the preimage of S, the differential of the radial projection map $R1,d\{0}→S1,d$, $z↦z/z$, just multiplies ej for j = 1, …, d with a positive constant, so that the images still span $TSS1,d$. Thus, $φm:Td→S1,d$ is transverse to $iS:pt→S1,d$.

The canonical map $πTd:Td×S1,dpt→Td$ is a diffeomorphism onto the closed submanifold $φm−1(S)⊆Td$ by the definition of the fiber product. Since the differential of φm is bijective at all points in the preimage of S, this preimage is discrete. Since $Td$ is compact, it must be finite. In fact, we may compute it easily: it consists of all points $(x,0)=(x1,…,xd,0,…,0)∈Td$ with $∑j=1dxj+m<0$.

Let $z=(x,y)∈Td$ satisfy $r(z)=z$. Then, y = 0 and hence xi ∈ {±1} for i = 1, …, d. These points satisfy φm(z) ∈ {N, S} for the north and south pole in (4) simply because φm is “real” and N, S are the only points fixed by the involution on $S1,d$. Give $Td$ the KR-orientation described in Example 22. This induces a KR-orientation on the fiber $TzTd$. Since the vector field generated by the exponential function points upward at $(1,0)∈T1$ and downward at $(−1,0)∈T1$, the projection to the y-coordinate $T(1,0)T1→R0,1$ preserves the orientation at +1 and reverses it at −1. Therefore, the projection to the y-coordinate $TzTd→R0,d$ multiplies the orientation with sign(z). The sum in geometric bivariant KR-theory is the disjoint union of correspondences. Therefore, the discrete set $φm−1(S)$ in the composite correspondence contributes the sum of sign(z)z! over all $z∈φm−1(S)$. Lemma 36 identifies this sum with $χ⋅[β1,d◦φm]−[γ0]$.□

If d − 2 < m < d, then $φm−1(S)$ has only one element, namely, $(−1,−1,…,−1,0,…,0)∈Td$ with sign (−1)d. Thus, we get
$[β1,d◦φm]−[γ0]=χ⋅(−1)d⋅(−1,…,−1,0,…,0)!.$
(14)
Up to a sign, this is the generator of $KR−d(Td)$ that is mapped to a generator of the KR-theory of the Roe C-algebra of $Zd$ (see Ref. 21). It is argued in Ref. 21 why this generator describes strong topological phases. For other values of m, we would like to simplify the formula in Lemma 37 further by comparing the point inclusions $z!:pt→Td$ for different $z∈Td$ with $r(z)=z$. If we work in complex K-theory, then all point inclusions are homotopic and therefore give equivalent correspondences. This fails, however, in the “real” case. We need some preparation to explain this.

Lemma 38.

If m < −d, then Hm is homotopic toγ0 in $FU(C(Td)⊗Cl1,d)$.

Proof.
The Hamiltonians Hs for s ∈ (−∞, m] give a homotopy of real, odd, self-adjoint unitaries $Hs−1Hs$ between $Hm−1Hm$ and
$lims→−∞Hs−1Hs=−γ0.$

The same argument shows that Hm is homotopic to γ0 for m > d. This is consistent with Lemma 37, which says that [Hm] − [γ0] = 0 in $KR−d(Td)$ for m > d.

Combining Lemmas 37 and 38 gives
$[−γ0]−[γ0]=χ∑z∈Td,r(z)=zsign(z)z!$
because if m < −d, then φm maps all points in $Td$ with $r(z)=z$ to S. In particular, for d = 1, this says that
$(1,0)!=(−1,0)!+χ⋅[−γ0]−[γ0].$
(15)
The difference [−γ0] − [γ0] is represented by constant functions and thus is in the image of $DK(Cl1,1)≅KO−1(pt)≅Z/2$ in $KR−1(T1)$. Since this group is two-torsion, we may drop the sign χ in (15).

Lemma 39.

The isomorphism above maps [−γ0] − [γ0] to the nontrivial element $μ∈KO−1(pt)≅Z/2$.

Proof.
Identify $Mn(Cl1,1)≅M̂2n$. The odd self-adjoint real unitaries in $Mn(Cl1,1)$ are identified with the 2n × 2n-matrices,
$0UU*0,$
for an orthogonal n × n-matrix U. Two such matrices are homotopic among odd self-adjoint unitaries if and only if the orthogonal matrices are in the same connected component of the orthogonal group or, equivalently, have the same determinant. The elements ±γ0 correspond, however, to ±1 ∈ O(1) = {±1}, which lie in the two different connected components.□

We recall how to describe the KR-theory of tori. There is a split extension of “real” C-algebras,
It induces a KKR-equivalence $C(T1)≅C0(R0,1)⊕C$. Since $C(Td)$ is the tensor product of d copies of $C(T1)$ and the tensor product bifunctor is additive on KKR, it follows that $C(Td)$ is KKR-equivalent to a direct sum of tensor products A1 ⊗⋯⊗ Ad, where each Aj is either $C0(R0,1)$ or $C$. We label such a summand by the set I ⊆ {1, …, d} of those factors that are $C$.

Theorem 40.

Let m ∈ (−d + 2n, −d + 2n + 2) for some n ∈ {0, …, d − 1}, and let I ⊆ {1, …, d}. Let χ be the sign from Lemma 26. The image of $[Hm]−[γ0]∈KR−d(Td)$ in the summand $KR−d(R0,d−I)$ where the ith factor is $C$ for iI and $C0(R0,1)$ for iI is computed as follows:

• If I = ∅, the image in $KR−d(R0,d)≅Z$ is $(−1)dχ⋅d−1n$.

• If $I=1$ and n ≥ 1, the image in $KR−d(R0,d−1)≅Z/2$ is $d−2n−1mod2$.

• If $I=2$ and n ≥ 2, the image in $KR−d(R0,d−2)≅Z/2$ is $d−3n−2mod2$.

• It is zero in the other cases, that is, for $I≥3$ or $n.

Proof.

We first consider the image of (x, 0)! in the summand labeled by I. The shriek map $(x1,…,xd,0,…,0)!$ is the exterior product of $(xi,0)!$ for i = 1, …, d (see Remark 34). In the decomposition of $KR−1(T1)≅KR−1(R0,1)⊕KO−1(pt)≅Z⊕Z/2$, (−1, 0)! becomes (pt!, 0) with the standard generator pt! of $KR−1(R0,1)≅Z$. Equation (15) shows that (1, 0)! = (pt!μ) with the nontrivial element $μ∈KO−1(pt)≅Z/2$. Hence, (x, 0)! is the exterior product of d factors that are (pt!, 0) if xi = −1 and (pt!μ) if xi = 1. The component in the summand labeled by I is zero unless xi = +1 for all iI. If xi = +1 for all iI, then we get the exterior product of $pt!∈KR−1(R0,1)$ for all iI and μ ∈ KO−1(pt) for all iI. The exterior product $μ⊗μ∈KO−2(pt×pt)≅Z/2$ is known to be the nontrivial element (this also follows from the discussion in Sec. IX), whereas μμμ and hence also all higher exterior products of μ vanish because KO−3(pt) = 0. Thus, the image of (x, 0)! is zero for all summands with $I≥3$ or xi = −1 for some iI and the standard generator of $KR−d(R0,d−I)$ if $I≤2$ and xi = +1 for all iI.

Now, we sum up sign(x, 0)(x, 0)! over all $(x,0)∈Td$ with φm(x, 0) = S or, equivalently, $∑i=1dxi+m<0$. The latter means that the number of +1 among the coordinates xi is at most n. There are $dj$ points (x, 0) with xi = +1 for exactly j indices i. For all of them, the image of (x, 0)! in the direct summand $KR−d(R0,d)≅Z$ for I = ∅ is the same standard generator. Hence, the image of [Hm] − [γ0] in this direct summand is
$(−1)dχ⋅∑j=0n(−1)n−jdj=(−1)dχ⋅d−1n$
because of (14). Now, let $I=1$ so that I = {i0} for some i0 ∈ {1, …, d}. The corresponding direct summand $KR−d(R0,d−1)≅Z/2$ only sees $(x,0)∈φm−1(S)$ with $xi0=1$. There are $d−1j−1$ points with $xi0=1$ and xi = +1 for exactly j indices i = 1, …, d. Thus, the overall contribution in this summand isomorphic to $Z/2$ vanishes if n = 0 and otherwise is equal to the class mod 2 of
$∑j=1n(−1)n−jd−1j−1=−∑j=0n−1(−1)n−jd−1j=d−2n−1.$
We could leave out all signs because +1 = −1 in $Z/2$. Similarly, for I ⊆ {1, …, d} with $I=2$, we get $d−3n−2$ if n ≥ 2 and 0 if n ≤ 1.□

The formula in Theorem 40 is compatible with the Chern character computation in Eq. (2.26) in Ref. 2. The latter, however, only gives partial information about the K-theory class, even in the complex case, because it only concerns the top-dimensional part of the Chern character.

As a result, we find that the Hamiltonian Hm for d − 2 < m < d represents a generator of the direct summand $KR−d(R0,d)≅Z$ in $KR−d(Td)$. We get different KR-classes by stacking insulators in lower dimension along some direction. In our framework, this means the following: choose 0 ≤ kd and 1 ≤ i1 < i2 <⋯< ikn, consider the projection $ρ:Td↠Tk$ that only keeps the coordinates $xij,yij$ for j = 1, …, k, and pull back the k-variable Hamiltonian $Hm∈C*(Zk)⊗Cl1,k$ to an odd, real, self-adjoint, invertible element
$Hm′:=12i∑j=1k(Sij−Sij*)⊗γj+m+12∑j=1k(Sij+Sij*)⊗γ0$
(16)
in $C*(Zd)⊗Cl1,k$. If m ∈ (k − 2, k), then $[Hm′]−[γ0]$ generates the direct summand $KR−k(R0,k)≅Z$ in $KR−k(Td)$ that corresponds to the subset
$I={1,…,n}\{i1,…,ik}.$
Here, we use the decomposition of $KR*(Td)$ explained above Theorem 40. We will explain in Sec. IX how to also get generators for the summands $KR−k−j(R0,k)$ for j = 1, 2, 4, which gives all cases where this group is not zero.

In this section, we are going to identify the set of odd, self-adjoint invertible elements in the C-algebra $C(Td)⊗Cla,b$ with $a,b,d∈N$ with the set of all invertible, self-adjoint operators on a suitable Hilbert space that have suitable symmetries, in addition to being translation-invariant and controlled. The symmetries that we need depend only on jba + 1 mod 8; we use this parameter because $DK(C(Td)⊗Cla,b)=KR−j(Td)$. For instance, the case j ≡ 4 mod 8 corresponds to a time-reversal symmetry of square −1.

The correspondence between j above and different symmetry types is well known and can be found, for instance, in Table 1 in Ref. 22. We believe that our discussion is more succinct than in other sources, such as Refs. 4 and 23. The specific Hamiltonians Hm occur for a = 1 and b = d, but there is no need to impose these restrictions, and we will see other combinations of indices in Sec. IX.

We represent $C(Td)$ on the Hilbert space $ℓ2(Zd)$ in the usual way. A bounded operator on $ℓ2(Zd)$ belongs to $C(Td)$ if and only if it (a) commutes with the translation operators Sx for $x∈Zd$ defined by Sxf(y) ≔ f(yx) for $x,y∈Zd$, $f∈ℓ2(Zd)$ and (b) is “controlled” in the sense that it belongs to the closure of the finite-propagation operators (see Ref. 21). We will not discuss this any further here and mention it only for completeness and to point out that mere translation-invariance is not enough, as it gives the group von Neumann algebra of $Zd$, which is identified with $L∞(Td)$ by the Fourier transform.

We are mainly interested in the Clifford algebra part, which is responsible for the different symmetries. We are going to represent Cla,b on $C2k$ for suitable $k∈N$ such that the representation maps odd self-adjoint unitaries in Cla,b bijectively to elements of $M2kC$ with suitable symmetries. The kinds of symmetries that are needed here depend on j. We distinguish a number of cases.

Case: j even

We first assume that j is even or, equivalently, b and a have different parity. Then, there is an isomorphism
$Cla,b≅M2kC⊕M2kC$
(17)
for k = (a + b − 1)/2. In addition, we may arrange that the $Z/2$-grading automorphism on Cla,b becomes the flip isomorphism α(x, y) ≔ (y, x) on $M2kC⊕M2kC$. Thus, self-adjoint, odd elements of Cla,b become elements of the form $(x,−x)∈M2kC⊕M2kC$ for some $x∈M2kC$ with x = x*. As our representation $Cla,b→M2kC$, we simply choose the projection to the first summand.

Since the “real” involution $r$ on Cla,b is a ring automorphism, it permutes the two direct summands $M2kC$ in (17). It induces either the trivial or the nontrivial permutation, leading to two subcases.

1. Assume $r$ induces the trivial permutation. Then, $r$ restricts to the same “real” involution on both summands $M2kC$ in (17) because it commutes with α. This involution is implemented as conjugation by Θ for an antiunitary operator $Θ:C2k→C2k$. The pair (x, −x) for $x∈M2kC$ is fixed by $r$ if and only if x commutes with Θ. Thus, we identify “real” self-adjoint odd unitaries in Cla,b with self-adjoint unitaries in $M2kC$ that commute with Θ. In other words, we are dealing with systems with a time-reversal symmetry Θ. Since Θ induces an antilinear involution, Θ2 = ±1.

2. Assume that $r$ flips the two summands $M2kC$. Then, $r◦α$ maps each direct summand into itself and induces the same real involution on both summands $M2kC$. Thus, we may implement $r◦α$ by an antiunitary operator $Θ:C2k→C2k$ as in case 1. The difference is that a pair (x, −x) is real if and only if Θ anticommutes with x. Thus, Θ is now a particle–hole symmetry. Once again, Θ2 = ±1.

There are four possible even values of j, and we found four possible symmetry types, namely, a time-reversal or a particle–hole symmetry with square ±1. We claim that these possibilities correspond to each other bijectively. Instead of working out the isomorphism (17) and then finding Θ, it is more convenient to look at the subalgebras of real and of real, even elements in each symmetry type. We will see that these subalgebras are different in the four symmetry types so that we may distinguish them by looking only at these subalgebras.

In subcase 1 with Θ2 = +1, the real subalgebra $(M2kC)R$ fixed by $r$ is $M2kR$, and so $(Cla,b)R≅M2kR⊕M2kR$. In subcase 1 with Θ2 = −1, we must have k ≥ 1 and $(M2kC)R≅M2k−1H$ and $(Cla,b)R≅M2k−1H⊕M2k−1H$ for the quaternions $H$. In subcase 2, the real subalgebra $(Cla,b)R$ is always isomorphic to $M2kC$, identified with the subalgebra of $(x,Θ−1xΘ)∈M2kC⊕M2kC$ for $x∈M2kC$. To distinguish the two signs for Θ2 = ±1, we also look at the even subalgebra of $(Cla,b)R$. If Θ2 = +1, then the even subalgebra of $(Cla,b)R$ is $M2kR$. If Θ2 = −1, then k ≥ 1 and the even subalgebra of $(Cla,b)R$ is $M2k−1H$. Summing up, we see how to distinguish the four symmetry types that are possible by looking at the subalgebras of real elements and of real, even elements.

For Cl1,0, the real subalgebra is $R⊕R$, so we must be in the case of a time-reversal symmetry with square +1. The symmetry type only depends on $j∈Z/8$ because Clifford algebras with the same j are Morita equivalent as graded “real” algebras. Therefore, the same symmetry type occurs whenever j ≡ 0 mod 8. For Cl0,3, the real subalgebra is well known to be $H⊕H$ so that we must be in the case of a time-reversal symmetry with square −1. This statement for Cl0,3 then extends to all Clifford algebras with j ≡ 4 mod 8.

For Cl0,1, the real subalgebra is $C$, and the even real subalgebra is $R⊆C$. This only happens for a particle–hole symmetry of square +1. The same symmetry type occurs whenever j ≡ 2 mod 8. Finally, for Cl3,0, the real subalgebra is $M2C$ and the even real subalgebra is isomorphic to $H$. This only happens for a particle–hole symmetry of square −1. This case must occur whenever j ≡ 6 mod 8.

Case: j odd

Now, we treat the case where j is odd or, equivalently, a and b have the same parity. Then, $Cla,b≅M2kC$ for k = (a + b)/2. We exclude the case k = 0 because, then, we are dealing with $Cl0,0=C$, which is not balanced. The $Z/2$-grading and the “real” involution are implemented by a unitary operator $Ξ:Ck→Ck$ and an antiunitary operator $Θ:Ck→Ck$, respectively. A real, self-adjoint odd unitary in Cla,b then becomes a self-adjoint unitary in $M2kC$ that anticommutes with Ξ and commutes with Θ. Thus, it has Θ as a time-reversal and Ξ as a chiral symmetry. Then, ΘΞ is a particle–hole symmetry. Multiplying Ξ with a scalar, we may arrange that Ξ2 = 1. Since Θ induces an antiunitary involution, Θ2 = ±1. Since $r$ commutes with the grading, Ξ−1ΘΞ = ±Θ. This is equivalent to (ΞΘ)2 = ±Θ2. Hence, there are four possibilities for the signs.

As in the case of even j, we want to distinguish the possible sign combinations by looking at the subalgebras of real and real, even elements. The subalgebra of real elements detects the sign of Θ because $(Cla,b)R≅M2kR$ if Θ2 = +1 and $(Cla,b)R≅M2k−1H$ (and k ≥ 1) if Θ2 = −1. The even subalgebra of Cla,b is always $M2k−1C⊕M2k−1C$. If Ξ−1ΘΞ = Θ, then the real involution restricted to the even part preserves the two direct summands $M2k−1C$ so that the even real subalgebra of Cla,b is a direct sum of two simple algebras. If Ξ−1ΘΞ = −Θ, however, then the real involution restricted to the even part flips the two direct summands so that the even real subalgebra of Cla,b is simple. Thus, we may distinguish the symmetry types by inspecting the subalgebras of real and real, even elements that they determine. Looking at the representative cases Cla,b, with (a, b) being (1, 1), (2, 0), (0, 2), and (0, 4), we find the following:

• Θ2 = +1, Ξ−1ΘΞ = +Θ, and (ΞΘ)2 = +1 for Cl1,1 with j ≡ 1 mod 8;

• Θ2 = +1, Ξ−1ΘΞ = −Θ, and (ΞΘ)2 = −1 for Cl2,0 with j ≡ 7 mod 8;

• Θ2 = −1, Ξ−1ΘΞ = −Θ, and (ΞΘ)2 = +1 for Cl0,2 with j ≡ 3 mod 8; and

• Θ2 = −1, Ξ−1ΘΞ = +Θ, and (ΞΘ)2 = −1 for Cl0,4 with j ≡ 5 mod 8.

The correspondence between j and the symmetry types is the same as in Table 1 in Ref. 22.

Our model Hamiltonian is an odd self-adjoint unitary in $C(Td)⊗Cl1,d$. When we realize it on a Hilbert space as in Sec. VIII, then we get a specific symmetry type in each dimension d. For instance, the most important case of time-reversal symmetry with square −1 occurs for d ≡ 4 mod 8; note that j in Sec. VIII is d for Cl1,d. In this section, we explain, in particular, how to find model Hamiltonians with odd time-reversal symmetry in dimensions 2 and 3 instead.

The K-theory of any “real” space X is a module over the graded ring KO*(pt). This is a special case of the exterior product maps,
$KRi(X)⊗KRj(Y)→KRi+j(X×Y).$
We have found above that [Hm] − [−γ0] for d − 2 < m < d generates the direct summand $KR−d(R0,d)⊆KR−d(Td)$. The exterior product with the generator of $KO−j(pt)≅Z/2$ for j = 1, 2 or $KO−4(pt)≅Z$ gives a generator for the direct summand $KR−d−j(R0,d)$. In this section, we describe such exterior products by real, odd, self-adjoint unitaries in $C(Td,Cl1,d+j)$ for j = 1, 2 and in $p+C(Td,Cl1,d+4)p+$ for a certain real even projection p+ ∈ Cl0,4 for j = 4.

The outcome for j = 1, 2 is particularly simple to state. We may embed Cl1,d ↪ Cl1,d+1 ↪ Cl1,d+2 by adding extra generators. Thus, an odd self-adjoint invertible element in $C(Td)⊗Cl1,d$ gives one in $C(Td)⊗Cl1,d+1$ and $C(Td)⊗Cl1,d+2$ as well. We will see that these elements represent the exterior products with the generators of KO−1(pt) and KO−2(pt), respectively. In particular, the same elements Hm and γ0, just viewed in a different Clifford algebra, give a generator for the summands $KR−d−j(R0,d)≅Z/2$ for j = 1, 2. In particular, for d = 3 and j = 1, we get Hm viewed as an element of $C(T3)⊗Cl1,4$. When we realize this on a Hilbert space as in Sec. VIII, then we get a Hamiltonian with odd time-reversal symmetry in dimension 3 with a nontrivial strong topological phase. Here, $Cl1,4≅M2(Cl0,3)≅M4(C)⊕M4(C)$ so that our Hamiltonian will act on $ℓ2(Z3)⊗C4$. The original Hamiltonians take values in $Cl1,3≅M4(C)$ and so act on the same Hilbert space. In Sec. VIII, we have identified odd real elements of Cl1,3 with elements of $M4(C)$ that have both an odd time-reversal and an even particle–hole symmetry; replacing Cl1,3 by Cl1,4 to shift to a different KR-group amounts to forgetting the particle–hole symmetry.

To get a generator for the summand $KR−4(R0,2)≅Z/2$ in $KR−4(T2)$, we view $Hm,γ0∈C(T2,Cl1,2)$ as functions taking values in Cl1,4. In Sec. VIII, we have identified odd, real, self-adjoint elements of Cl1,2 with elements of $M2(C)$ that have an even particle-hold symmetry (this is the case j = 2). The embedding Cl1,2 ↪ Cl1,4 induces a nontrivial map from such matrices in $M2(C)$ to matrices in $M4(C)$ with an odd time-reversal symmetry.

The result about exterior products claimed above may already be known. However, since we do not know a reference, we include a proof. The first step in this proof is to simplify the cycles for $KKR0(C,Clc,d)$. This will simplify the exterior products with them in Kasparov theory.

Lemma 41.

Any cycle for $KKR0(C,Clc,d)$ is homotopic to one with $C$ acting by scalar multiplication and Fredholm operator equal to 0. Isomorphism classes of such cycles are in bijection with finitely generated modules over the real subalgebra $(Clc+1,d)R$ in Clc+1,d.

Proof.

Let $(H,φ,F)$ be a cycle for $KKR0(C,Clc,d)$. First, we may replace it by a homotopic one where $C$ acts just by scalar multiplication. Then, we may use the functional calculus for F to arrange that the spectrum of F consists only of {0, 1, −1}. The direct summand where F has spectrum ±1 is unitary and thus gives a degenerate cycle. Removing that piece gives a cycle with F = 0, still with $C$ acting by scalar multiplication. Thus, the only remaining data are the underlying $Z/2$-graded “real” Hilbert Clc,d-module, which we still denote by $H$. To give a KK-cycle with F = 0, the identity operator on $H$ must be compact.

Next, we claim that $H≅p⋅Clc,d2n$ for some real, even projection $p∈M̂2n(Clc,d)$; here, half of the summands in $Clc,d2n$ have the flipped grading. First, the Kasparov stabilization theorem implies that there is a real, even unitary $H⊕(Clc,d)∞≅(Clc,d)∞$, where $(Clc,d)∞$ denotes the standard “real” graded Hilbert Clc,d-module, with the $Z/2$-grading where half of the summands carry the flipped grading. This gives a real, even projection $p0∈B((Clc,d)∞)$ with $H≅p0(Clc,d)∞$. Since the identity on $H$ is compact, $p0∈K((Clc,d)∞)$. Then, p0 is Murray–von Neumann equivalent—with even real partial isometries—to a nearby projection $p∈M̂2n(Clc,d)$. This implies an isomorphism $H≅p⋅Clc,d2n$ of “real” $Z/2$-graded Hilbert modules.

It is well known that any idempotent in a C-algebra is Murray–von Neumann equivalent to a projection. The proof is explicit using the functional calculus. Therefore, if the idempotent we start with is even and real, then so are the equivalent projection and the Murray–von Neumann equivalence between the two. Therefore, the isomorphism class of $H$ as a “real” graded Hilbert module is still captured by the Murray–von Neumann equivalence class of p as a real, even idempotent element. Let R be the ring of even, real elements in $M̂2(Clc,d)$. What we end up with is that the possible isomorphism classes of $H$ are in bijection with Murray–von Neumann equivalence classes of idempotents in matrix algebras over R. These are, in turn, in bijection to isomorphism classes of finitely generated projective modules over R. The subalgebra of even elements of $M̂2(Clc,d)$ is isomorphic to the crossed product $Clc,d⋊Z/2$—this is an easy special case of the Green–Julg theorem (see Ref. 24), identifying a crossed product for an action of a compact group G on a C-algebra A with the fixed-point algebra of the diagonal G-action on $A⊗K(L2G)$. The nontrivial element of $Z/2$ gives an extra Clifford generator that commutes with even and anticommutes with odd elements of Clc,d. Thus, the even part of $M̂2(Clc,d)$ is isomorphic to Clc+1,d. This implies $R≅(Clc+1,d)R$. This $R$-algebra is semisimple because its complexification is a sum of matrix algebras. Hence, all finitely generated modules over it are projective.□

Remark 42.

The lemma does not yet compute the group $KKR0(C,Clc,d)$ because it does not say when two modules give the same element. We do not need this computation, but sketch it anyway. Consider a degenerate cycle for $KKR0(C,Clc,d)$ that is also finitely generated. The operator F on it is real and odd with F2 = 1 and commutes with Clc,d. Multiplying F with the grading gives a real, odd operator that anticommutes with the generators of Clc,d and has square −1. This shows that a $(Clc+1,d)R$-module admits an operator F that makes it a degenerate cycle for $KKR0(C,Clc,d)$ if and only if the module structure extends to $(Clc+1,d+1)R$. Call two $(Clc+1,d)R$-modules stably isomorphic if they become isomorphic after adding restrictions of $(Clc+1,d+1)R$-modules to them. Since the direct sum of $(Clc+1,d)R$-modules corresponds to the direct sum of Kasparov cycles, stably isomorphic $(Clc+1,d)R$-modules give the same class in $KKR0(C,Clc,d)$. Now, Atiyah, Bott, and Shapiro12computed the stable isomorphism classes of $(Clc+1,d)R$-modules and found that they give the KO-theory of the point. As a result, two $(Clc+1,d)R$-modules give the same class in $KKR0(C,Clc,d)$ if and only if they are stably isomorphic.

We are going to describe the generators of $KKR0(C,Cl0,j)≅Z/2$ for j ≡ 1, 2 mod 8. It follows from Lemma 41 that the generator of $KKR0(C,Clc,d)≅Z/2$ corresponds to a $(Clc+1,d)R$-module. Since the direct sum of modules becomes the sum in KKR-theory, we may pick a simple module.

For c = 0 and d = 1, we get $(Cl1,1)R=M2(R)$. Up to isomorphism, there is a unique two-dimensional simple $(Cl1,1)R$-module. Turn Cl0,1 into a $Z/2$-graded “real” Hilbert module over itself and give it the operator F = 0. The constructions in the proof of Lemma 41 turn this into a two-dimensional $(Cl1,1)R$-module. Therefore, Cl0,1 with F = 0 represents the generator of $KKR0(C,Cl0,1)≅Z/2$.

Next, let c = 0 and d = 2. Then, $(Cl1,2)R=M2(C)$. It has $C2$ as its unique simple module. Therefore, any $(Cl1,2)R$-module of dimension 4 over $R$ is simple. Since Cl0,2 as a $Z/2$-graded “real” Hilbert module over itself also yields a four-dimensional module over $(Cl1,2)R$, the latter is simple. Thus, Cl0,2 with F = 0 represents the generator of $KKR0(C,Cl0,2)≅Z/2$.

The computations above show that the generator of $KKR0(C,Cl0,j)≅Z/2$ for j = 1, 2 is represented by the grading-preserving “real” *-homomorphism $C→Cl0,j$ induced by the unit element. Computing the exterior product in KK with such a *-homomorphism is very easy: for a class in KKR0(A, B), we simply apply the functoriality of KKR for the *-homomorphism $B=B⊗̂C→B⊗̂Cl0,j$. In the case $A=C$, we may identify KKR0(A, B) with van Daele’s K-theory (see Proposition 3). This is how the exterior product in Kasparov theory induces one in van Daele’s K-theory as well. It follows that the exterior product with the generator of $KKR0(C,Cl0,j)≅Z/2$ for j = 1, 2 in van Daele’s K-theory maps a class in DK(A) by applying functoriality for the canonical inclusion map $A=A⊗̂C→A⊗̂Cl0,j$.

Proposition 43.

Let j = 1, 2; define $Hm′$ as in (16). The exterior product of the generator of KOj(pt) and $[Hm′]−[γ0]∈KR−k(Td)$ belongs to $DK(C(Td,Cl1,k+j))$. It is $[Hm′]−[γ0]$, where now $Hm′$ and γ0 are viewed as self-adjoint odd unitaries in $C(Td,Cl1,k+j)$ with j extra Clifford generators.

For j = 3, the unital inclusion $C→Cl0,j$ represents the zero element in $KKR0(C,Cl0,j)$ because KO−3(pt) = 0. This remains so for j ≥ 3 because this inclusion is the exterior product of several copies of the unital inclusion $C→Cl0,1$. Therefore, the generator of $KKR0(C,Cl0,4)≅Z$ cannot have this simple form. It is known that $(Cl1,4)R≅M2(Cl0,3)R≅M2(H)⊕M2(H)$. Hence, there are two simple $(Cl1,4)R$-modules, and both have $R$-dimension 8. When we turn Cl0,4 into a $(Cl1,4)R$-module as above, we get the direct sum of these two simple modules, and this represents the zero element. Therefore, the KKR-classes of the two simple $(Cl1,4)R$-modules are additive inverses of each other, and any $(Cl1,4)R$-module of $R$-dimension 8 represents a generator of the group $KKR0(C,Cl0,4)≅Z$.

To describe the two simple modules, we use the “orientation element”
$ω:=γ1γ2γ3γ4∈Cl0,4$
(compare Lemma in Sec. 2.14 on p. 525 in Ref. 7). This element is real, even, self-adjoint, and unitary, and it anticommutes with the Clifford generators γ1, …, γ4. As a result, $p±:=12(1±ω)$ are real, even, self-adjoint projections in Cl0,4. There is no preferred sign here because −ω = γ2γ1γ3γ4 is built in the same way as ω, just for a differently oriented orthonormal basis. We may decompose
$Cl0,4=p+⋅Cl0,4⊕p−⋅Cl0,4$
as “real” $Z/2$-graded right Hilbert Cl0,4-modules. The constructions in the proof of Lemma 41 turn the two summands into $(Cl1,4)R$-modules of $R$-dimension 8. These are the two simple $(Cl1,4)R$-modules. Hence, the “real” $Z/2$-graded right Hilbert Cl0,4-modules p±Cl0,4 with F = 0 give a generator of $KKR0(C,Cl0,4)$ and its additive inverse. These KK-cycles are equal to those associated with the *-homomorphisms $C→Cl0,4$, λλp±.

Therefore, the exterior product of a class in van Daele’s K-theory DK(A) with the generator of $KO−4(pt)≅Z$ applies the functoriality for one of these two nonunital *-homomorphisms. This first produces an element in $DK(A⊗̂p±Cl0,4p±)$ because the homomorphism is not unital, which is then mapped to one in $DK(A⊗̂Cl0,4)$ using the Morita invariance of DK. For our purposes, however, we eventually want to make a Hamiltonian on a Hilbert space with extra symmetries as in Sec. VIII, and then, it is simpler to leave out this Morita invariance step. First, realize the odd self-adjoint elements in Cl1,d+4 as matrices in $M2kC$ with specific symmetries. The underlying vector space $C2k$ carries a representation of Cl1,d+4, and so p± act on it by complementary projections. The time-reversal, particle–hole, or chiral symmetries produced in Sec. VIII commute with p± because it is real and even, and so these symmetries restrict to the subspace $p±C2k≅C2k−1$. Then, the real, odd, self-adjoint elements in p±Cl1,d+4p± become identified with self-adjoint operators on $p±C2k≅C2k−1$ that are compatible with the restrictions of the symmetries from Sec. VIII. In this way, we also get explicit physical systems with a nontrivial strong topological phase in $KR−d−4(Td)$.

Summing up, we have seen how to produce explicit Hamiltonians that realize all the generators of $KR*(Td)$ for both the strong and weak topological phases. Our method is, however, not helpful to compute the topological phase of a given topological insulator coming from the experiment. Our starting point was to pull the Bott generator of $KRa+b−1(Sa,b)$ for some $a,b∈N$ back along a continuous map $Td→Sa,b$. This can only produce Hamiltonians of the special form $∑i=1dfiγi$ with functions $fi∈C(Td)$ and the standard generators γi of Cla,b. Generic odd self-adjoint elements in a Clifford algebra may, however, involve products of the γi as well. Therefore, we cannot expect Hamiltonians that come from the experiment to be of this simple form.

The authors have no conflicts to disclose.

Collin Mark Joseph: Investigation (equal); Writing – original draft (lead); Writing – review & editing (supporting). Ralf Meyer: Conceptualization (lead); Investigation (equal); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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