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.

## I. INTRODUCTION

^{∗}-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,

*d*-torus, we get

^{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.

^{2}of certain examples of Hamiltonians

*H*

_{m}in any dimension

*d*, namely,

*γ*

_{0}, …,

*γ*

_{d}and translations

*S*

_{j}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

*H*

_{m}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

*H*

_{m}: it is the pullback of a generator of the reduced KR-theory of a sphere along a map $\phi m:Td\u2192S1,d$. Then, we proceed to compute the class of

*H*

_{m}in “real” K-theory for all

*m*and all dimensions

*d*.

_{a,b}denote the Clifford algebra with

*a*+

*b*anticommuting, odd, self-adjoint, unitary generators

*γ*

_{1}, …,

*γ*

_{a+b}and such that

*γ*

_{1}, …,

*γ*

_{a}and i

*γ*

_{a+1}, …, i

*γ*

_{a+b}are real. Then,

^{∗}-algebra $C(Sa,b)\u2297Cla,b$. Therefore, it defines a class in its van Daele K-theory, which is isomorphic to the “real” K-theory group $KRa\u2212b\u22121(Sa,b)$. 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

*β*

_{a,b}generates the “real” K-theory is based on the proof of Bott periodicity by the proof of Kasparov

^{7}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*.

*H*

_{m}for $m\u2208R\{\u2212d,\u2212d+2,\u2026,d\u22122,d}$ is an odd, self-adjoint, real unitary on the “real”

*d*-torus $Td$ in (1). This may be written as the pull-back of

*β*

_{1,d}along the real map,

*a*> 0. Let

*i*

_{S}! for the inclusion map $iS:{S}\u21aaSa,b$.

Shriek maps and pull-back maps, such as *i*_{S}! and $\phi m*$, 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 $\phi m*\u25e6iS!$ is the sum of the shriek maps for all points in $\phi m\u22121(S)$, 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, *H*_{m} 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 $\u21132(Zd,C2k)$ 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 Cl_{1,d} should be exactly those matrices that commute or anticommute with certain antiunitary operators on $C2k$. Since the symmetry type of *H*_{m} 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 KO^{−j}(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 *H*_{m}. They are all defined by the same formula, combined with a specific representation of the relevant Clifford algebra.

### A. Some basic conventions

Throughout this article, we let $Ra,b$ for $a,b\u2208N$ be $Ra\xd7Rb$ with the involution $r(x,y):=(x,\u2212y)$ for $x\u2208Ra$, $y\u2208Rb$, 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}

^{∗}-algebra

*A*is a conjugate-linear, involutive *-homomorphism $r:A\u2192A$. Then,

^{∗}-algebra such that $A\u2245AR\u2297C$ with the involution $r(a\u2297\lambda ):=a\u2297\lambda \u0304$. Thus, “real” C

^{∗}-algebras are equivalent to real C

^{∗}-algebras. Any commutative “real” C

^{∗}-algebra is isomorphic to C

_{0}(

*X*) with the real involution $r(f)(x):=f(r(x))\u0304$ for all

*x*∈

*X*for a “real” locally compact space $(X,r)$.

Let Cl_{a,b} denote the complex Clifford algebra with *a* + *b* anticommuting, odd, self-adjoint, unitary generators *γ*_{1}, …, *γ*_{a+b} such that *γ*_{1}, …, *γ*_{a} and i*γ*_{a+1}, …, i*γ*_{a+b} are real. This is a $Z/2$-graded “real” C^{∗}-algebra. Let $\u2297\u0302$ denote the graded-commutative tensor product for $Z/2$-graded (“real”) C^{∗}-algebras.

For $R1,d$ and $S1,d\u2286R1,d$ and for Cl_{1,d}, it is convenient to start numbering at 0. That is, the coordinates in $R1,d$ are *x*_{0}, …, *x*_{d} and Cl_{1,d} has the anticommuting, odd, self-adjoint generators *γ*_{0}, …, *γ*_{d} such that *γ*_{0}, i*γ*_{1}, …, i*γ*_{d} are real.

## II. EXPLICIT K-THEORY OF SPHERES

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 Shapiro^{12} 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 K_{0} through projections, Karoubi^{3} has written down explicit generators for the complex K-theory of even-dimensional spheres. Kasparov^{7} has written down explicit generators for his bivariant KK-theory $KKR0(C,C0(Ra,b)\u2297Cla,b)$; 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.

### A. van Daele K-theory for graded C*-algebras and Roe’s isomorphism

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 $AR:=a\u2208A:r(a)=a$.

*A*be a unital, $Z/2$-graded “real” C

^{∗}-algebra and $n\u2208N\u22651$. Let

*homotopic*. We define the maps

*A*) ≔ FU

_{1}(

*A*). We call

*A*

*balanced*if $FU(A)\u2260\u2205$.

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, $\u2a06n=1\u221eFUn(A)$ becomes an Abelian semigroup. Let GFU(*A*) be its Grothendieck group.

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 $d:GFU(A)\u2192Z$ defined by $d|FUn(A)=n$ for all $n\u2208N\u22651$.

*e*∼ (−

*e*) to define stabilization maps,

*A*) as defined above (see Refs. 13 and 14).

*A*be a unital, $Z/2$-graded “real” C

^{∗}-algebra. Then, $A\u2297\u0302Cl1,1$ is balanced because $1\u2297\u0302\gamma 1\u2208FU(A\u2297\u0302Cl1,1)$. If

*A*is already balanced, then there is a natural stabilization isomorphism $DK(A)\u2245DK(A\u2297\u0302Cl1,1)$. This justifies defining $DK(A):=DK(A\u2297\u0302Cl1,1)$, in general. For a C

^{∗}-algebra

*A*without unit, let

*A*

^{+}be its unitization, equipped with the canonical augmentation homomorphism $A+\u2192C$. Then,

*A*is unital because then $A+\u2245A\u2295C$ and DK is additive. Since there is an isomorphism DK(

*A*) ≅ KKR

_{0}(Cl

_{1,0},

*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.

### B. Roe’s isomorphism

The following proposition is due to Roe.^{13} The isomorphism DK(*A*) ≅ KKR_{0}(Cl_{1,0}, *A*) is equivalent and proven in Ref. 14.

*(see Ref. 13).* *Let* *A* *be a unital,* $Z/2$*-graded “real”* C^{∗}*-algebra. Then,* $DK(A\u2297\u0302Clr+1,s)\u2245KKR0(C,A\u2297\u0302Clr,s)$*.*

*A*) and its graded version for a $Z/2$-graded “real” C

^{∗}-algebra

*A*as

*n*mod 8 so that we may also take $n\u2208Z/8$. If

*A*is trivially graded, then KR(

*A*) is naturally isomorphic to the ordinary K

_{0}of the real Banach algebra $AR$. This is sometimes denoted as $KO0(AR)$ to highlight that $AR$ is only a real C

^{∗}-algebra.

*X*, we define its “real” K-theory KR(

*X*) as KR(C

_{0}(

*X*)) with the induced “real” structure. This is equivalent to the definition by Atiyah.

^{1}For $n\u2208Z/8$, we also let

*A*-module $\u21132(N,C)\u2297A$; the “real” involution is the tensor product of complex conjugation on $\u21132(N,C)$ and the given “real” involution on

*A*, and the $Z/2$-grading is the tensor product of the $Z/2$-grading on $\u21132(N,C)$ induced by the parity operator and the given one on

*A*. A cycle for $KKR0(C,A\u2297\u0302Clr,s)$ is homotopic to one of the form $(HA\u2297\u0302Clr,s,1,F)$, where 1 denotes the left action of $C$ by scalar multiplication, and $F\u2208B(HA)\u2297\u0302Clr,s$ is real, odd, and self-adjoint and satisfies $F2\u22121\u2208K(HA)\u2297\u0302Clr,s$. Let $K=K(\u21132(N,C))$, identify $K(HA)$ with $K\u2297A\u2245K\u2297\u0302A$, and let $M(K\u2297A)$ denote the stable multiplier algebra of

*A*. Then, $B(HA\u2297\u0302Clr,s)$ is isomorphic to $M(K\u2297A)\u2297\u0302Clr,s$. Let $Q(K\u2297A):=M(K\u2297A)/K\u2297A$. Then,

*F*is mapped to a “real,” odd, self-adjoint unitary in $Q(K\u2297A)\u2297\u0302Clr,s$. Conversely, any “real,” odd, self-adjoint unitary in $Q(K\u2297A)\u2297\u0302Clr,s$ lifts to an operator $F\u2208B(HA\u2297\u0302Clr,s)$ such that $(HA\u2297\u0302Clr,s,1,F)$ is a cycle for $KKR0(C,A\u2297\u0302Clr,s)$. Thus, we get an obvious surjective map from $FU(Q(K\u2297A)\u2297\u0302Clr,s)$ onto $KKR0(C,A\u2297\u0302Clr,s)$. Since $HA$ has infinite multiplicity, $B(HA\u2297\u0302Clr,s)\u2245MnB(HA\u2297\u0302Clr,s)$, and the same holds for the quotient by the compact operators. Thus, the construction above also defines a map

*The map above induces a group isomorphism,*

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 $F0\u2208B(HA\u2297\u0302Clr,s)$, and any such operator defines a degenerate Kasparov cycle. Such degenerate cycles represent zero in KK-theory. If $F\u2208FU(Q(K\u2297A)\u2297\u0302Clr,s)$, then [*F*] − [*F*_{0}] ∈ DK(*A*) is mapped to the same KK-class as *F*. Thus, the induced map on DK(*A*) remains surjective. Roe^{13} showed that it is also injective; the main reason for this is that “homotopic” KK-cycles become “operator homotopic” after adding degenerate cycles.□

^{∗}-algebras,

*a*≥ 1, but for now, we do not need this. Let

$\beta \u0303a,b$ *is self-adjoint, odd, and real, and* $\beta \u0303a,b(x)2=x2$*.*

First, $\beta \u0303a,b$ is self-adjoint because $xi*=xi$ and $\gamma i*=\gamma i$ for all *i*. It is odd because all *γ*_{i} are odd and the grading on $C0(Ra,b)$ is trivial. Finally, $\beta \u0303a,b$ is real because *γ*_{i} is real whenever *x*_{i} is real and *γ*_{i} is imaginary whenever *x*_{i} is imaginary. The formula $\beta \u0303a,b(x)2=x2$ holds because $\gamma i2=1$ for all 1 ≤ *i* ≤ *a* + *b* and *γ*_{i} anticommute.□

*F*

_{0}. Adding

*F*

_{0}replaces the underlying Hilbert module

*D*of $[\beta \u0303a,b\u2032]$ by $HD$. Thus, $\beta \u0303a,b\u2032\u2295F0$ and

*F*

_{0}are elements of $FUQ(C0(Ra,b,K\u2297Cla,b))$ and $[\beta \u0303a,b\u2032\u2295F0]\u2212[F0]$ represents a class in $DKQ(C0(Ra,b,K\u2297Cla,b))$. Since degenerate KK-cycles represent zero, $[\beta \u0303a,b\u2032\u2295F0]\u2212[F0]$ is a generator for $KKR0(C,D)\u2245Z$; here, the choice of

*F*

_{0}does not matter.

*B*

^{a,b}in $Ra,b$. The boundary $B\u0304a,b\Ba,b$ is the

*a*+

*b*− 1-dimensional unit sphere $Sa,b\u2286Ra,b$ with the “real” involution of $Ra,b$. There is a canonical embedding $C(B\u0304a,b)\u2297Cla,b\u21aaCb(Ba,b)\u2297Cla,b=M(C0(Ba,b)\u2297Cla,b)$. It induces a canonical embedding $C(Sa,b)\u2297Cla,b\u21aaM(C0(Ra,b)\u2297Cla,b)/(C0(Ra,b)\u2297Cla,b)$. Its image contains the image of $\beta \u0303a,b\u2032$ in this quotient. Namely, $[\beta \u0303a,b\u2032]$ comes from $\beta a,b:=\beta \u0303a,b|Sa,b\u2208FU(C(Sa,b)\u2297Cla,b)$; this is the same

*β*

_{a,b}as in (2).

We assume from now on that *a* ≥ 1. Then, the first Clifford generator *γ*_{1} ∈ Cl_{a,b} is an odd self-adjoint real unitary. The constant function with value *γ*_{1} belongs to $FU(C(B\u0304a,b)\u2297Cla,b)$ and is mapped to a degenerate KK-cycle for $KKR0(C,D)$. Hence, we may pick *F*_{0} to be a direct sum of countably many copies of *γ*_{1} above.

*Let* *a* ≥ 1*. The class of* $[\beta a,b]\u2212[\gamma 1]\u2208DK(C(Sa,b)\u2297Cla,b)$ *is mapped to a generator of* $KKR0(C,D)$*.*

The inclusion of $C(Sa,b)\u2297Cla,b$ into $Q(C0(Ra,b)\u2297K\u2297Cla,b)$ is not unital. The map on van Daele K-theory that it induces sends [*e*] − [*f*] for $e,f\u2208FU(C(Sa,b)\u2297Cla,b)$ to [*e* ⊕ *F*_{0}] − [*f* ⊕ *F*_{0}]. For our choice of *F*_{0} above, it therefore sends [*β*_{a,b}] − [*γ*_{1}] to $[\beta a,b\u2295F0]\u2212[F0]\u2208DKQ(C0(Ra,b)\u2297K\u2297Cla,b)$. We already know that Roe’s isomorphism maps the latter to a generator of $KKR0(C,D)$.□

*a*≥ 1, the two poles

*S*and

*N*in (4) are fixed by the “real” involution on $Sa,b$. We choose $N\u2208Sa,b$ as a point at infinity and identify $Sa,b\{N}\u2245Ra\u22121,b$ by stereographic projection. The C

^{∗}-algebra extensionsplits by taking constant functions. Since van Daele’s K-theory is split exact,

*There is a Bott periodicity isomorphism,*

*Under the two isomorphisms above,*$[\beta a,b]\u2212[\gamma 1]\u2208DK(C(Sa,b)\u2297Cla,b)$

*is mapped to*$(1,0)\u2208Z\u2295DK(Cla,b)$

*.*

*β*

_{a,b}at

*N*is

*γ*

_{1}. Hence, evaluation at

*N*maps [

*β*

_{a,b}] − [

*γ*

_{1}] to zero. We compute $DK(C0(Ra\u22121,b)\u2297Cla,b)$ through the long exact sequences for DK applied to the morphism of C

^{∗}-algebra extensions,The vertical arrow $C(B\u0304a,b\{N})\u21aaM(C0(Ra,b)\u2297K)$ 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,

*β*

_{a,b}] − [

*γ*

_{1}] is mapped to a generator of $Z$ under these isomorphisms.□

## III. SOME CANONICAL MAPS IN KR-THEORY

We have described explicit elements in the van Daele K-theory of the spheres $Sa,b$ for $a,b\u2208N$. 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 $Td\u2192S1,d$.

*b*:

*X*→

*Y*be a proper continuous map that is “real” in the sense that $rY\u25e6b=b\u25e6rX$. Such a map induces a “real” *-homomorphism,

*b** as well because no confusion should be possible.

*U*⊆

*X*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,

*E*↠

*X*be a $Z/2$-equivariant vector bundle. Let Cl

_{E}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 C

_{0}(

*E*

_{x}) ⊗ Cl(

*E*

_{x}) carries a canonical Kasparov generator $\beta \u0303Ex\u2032$. Letting

*x*vary, these combine to a class

A KR*-orientation* on a $Z/2$-equivariant vector bundle *E* ↠ *X* is a $Z/2$-graded, “real,” C_{0}(*X*)-linear Morita equivalence between C_{0}(*X*, Cl_{E}) and C_{0}(*X*) ⊗ Cl_{a,b} for some $a,b\u2208N$. That is, it is a full Hilbert bimodule *M* over C_{0}(*X*, Cl_{E}) and C_{0}(*X*) ⊗ Cl_{a,b} with a compatible “real” structure and $Z/2$-action and with the extra property that the two actions of C_{0}(*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 $a\u2212bmod8\u2208Z/8$ the KR*-dimension* dim_{KR} *E* 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 Cl_{a,b} and Cl_{a′,b′} are Morita equivalent as “real” $Z/2$-graded C^{∗}-algebras if and only if *a* − *b* ≡ *a*′ − *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 $X\xd7Ra,b\u21a0X$ is, of course, KR-oriented because $ClX\xd7Ra,b=X\xd7Cla,b$. Its KR-dimension is *a* − *b* mod 8.

If *a* = *b* = 0, then a KR-orientation is a $Z/2$-graded, “real” Morita equivalence between C_{0}(*X*, Cl_{E}) and C_{0}(*X*). This is equivalent to a $Z/2$-graded “real” Hilbert C_{0}(*X*)-module $S$ with a C_{0}(*X*)-linear isomorphism $K(S)\u2245C0(X,ClE)$. This forces $S$ to be the space of C_{0}-sections of a complex vector bundle over *X*, equipped with a $Z/2$-grading and a “real” involution. Then, the isomorphism $K(S)\u2245C0(X,ClE)$ means that the fiber $Sx$ of this vector bundle carries an irreducible representation of $ClEx$. Thus, $S$ is the space of C_{0}-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.

*Thom isomorphism class*. It induces Thom isomorphisms in K-theory,

Let *E* ↠ *X* 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.

*Let*

*E*

_{1},

*E*

_{2}

*be two*$Z/2$

*-equivariant vector bundles over*

*X*

*. If two of the vector bundles*

*E*

_{1},

*E*

_{2},

*E*

_{1}⊕

*E*

_{2}

*are*KR

*-oriented, then this induces a canonical*KR

*-orientation on the third one such that*

A C_{0}(*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 $ClE1\u2295E2\u2245ClE1\u2297\u0302ClE2$. Therefore, KR-orientations for *E*_{1} and *E*_{2} induce one for *E*_{1} ⊕ *E*_{2}. Assume, conversely, that *E*_{1} ⊕ *E*_{2} and *E*_{2} are KR-oriented. Then, the bundle $ClE2$ is Morita equivalent to the trivial bundle with fiber Cl_{a,b} for some $a,b\u2208N$. Hence, $ClE2\u2297\u0302Clb,a$ is Morita equivalent to a trivial bundle of matrix algebras, making it Morita equivalent to the trivial rank-1 bundle $X\xd7C$. Therefore, $ClE1$ is Morita equivalent to $ClE1\u2297\u0302ClE2\u2297\u0302Clb,a\u2245ClE1\u2295E2\u2297\u0302Clb,a$. Since *E*_{1} ⊕ *E*_{2} is KR-oriented, this is also Morita equivalent to some trivial Clifford algebra bundle. This gives a KR-orientation on *E*_{1}.□

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

*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* *L* ↠ *X**, which is determined uniquely.*

Let $S1$ and $S2$ be two C_{0}(*X*)-linear “real” graded Morita equivalences between C_{0}(*X*, Cl_{E}) and C_{0}(*X*) ⊗ Cl_{a,b}. Then, the composite of $S1$ and the inverse of $S2$ are C_{0}(*X*)-linear “real” graded Morita self-equivalences of C_{0}(*X*) ⊗ Cl_{a,b}. Taking the exterior product with Cl_{a,b} is clearly bijective on isomorphism classes of Morita self-equivalences if *a* = *b* because, then, Cl_{a,b} is a matrix algebra. If *a* ≠ *b*, then it is also bijective on isomorphism classes because when we tensor first with Cl_{a,b} and then with Cl_{b,a}, we tensor with Cl_{a+b,a+b}, which is already known to be bijective on isomorphism classes. Therefore, we may replace the self-equivalence of C_{0}(*X*) ⊗ Cl_{a,b} by one of C_{0}(*X*). Such a Morita self-equivalence of C_{0}(*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 $L\u2297\u0302S1$.□

Assume that *X* is connected. Let *L* ↠ *X* 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*.

## IV. REPRESENTABLE KR-THEORY AND KR-THEORY WITH SUPPORTS

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.

*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

_{0}(

*X*)-linear “real” version of Kasparov theory, which Kasparov

^{19}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.

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

*X*with

*Z*-compact support, given a map

*b*:

*X*→

*Z*. As in Ref. 18, we define this by

*Z*= pt, then KR

_{Z}= 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*:

*X*→

*Z*.

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 C_{0}(*X*)-module of sections of *E*, with C_{0}(*X*) acting by pointwise multiplication also on the left. This only defines a class in the usual KR^{0}(*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*: *X* → *Z*. Choose two “real” complex vector bundles *E*_{±} over *X* together with an isomorphism $\phi :E+|X\A\u2192\u223cE\u2212|X\A$ for an open subset *A* ⊆ *X* whose closure is *Z*-compact in the sense that $b|A\u0304:A\u0304\u2192Z$ 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 $\phi \u0303$ 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* ∈ C_{0}(*Z*) acts on this by pointwise multiplication with *h*◦*b*. The Fredholm operator is pointwise multiplication with $\phi \u0303$. This is indeed a cycle for $KKR0ZC0(Z),C0(X)$ because $\phi \u0303$ is unitary outside a *Z*-compact subset.

With the definition in (9), a class $\xi \u2208KRZn(X)$ clearly yields $[\xi ]\u2208KKR\u2212nC0(Z),C0(X)$, which induces maps *ξ*_{∗}: KR^{a}(*Z*) → KR^{a+n}(*X*) for $a\u2208Z/8$. This generalizes both the pull back functoriality for continuous proper maps *b*: *X* → *Z* and the map on KR*(*X*) that multiplies with a vector bundle on *X*.

## V. WRONG-WAY FUNCTORIALITY OF KR-THEORY

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*.

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.

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.

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 $a,b\u2208N$, an $R$-linear representation of the group $Z/2$, which we treat as a $Z/2$-equivariant vector bundle over pt.

$f\u0302:V\u21aaY\xd7Ra,b$, a $Z/2$-equivariant diffeomorphism between the total space of

*V*and an open subset of $Y\xd7Ra,b=Y\xd7ptE$.

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.

Let $(V,E,f\u0302)$ be a normally nonsingular map, and let *E*_{0} be another linear representation of $Z/2$. Then, $(V\u2295(X\xd7E0),E\u2295E0,f\u0302\xd7idE0)$ is another normally nonsingular map, called a *lifting* of $(V,E,f\u0302)$. A normally nonsingular map from *X* × [0, 1] to *Y* × [0, 1] where the map $f\u0302$ 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.

*(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*: *X* → *Y* to a normally nonsingular map. There is a $Z/2$-equivariant, proper embedding $i:X\u2192Ra,b$ for some $a,b\u2208N$. Let $E:=Ra,b$. The map $(f,i):X\u2192Y\xd7Ra,b$ 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 $f\u0302$ for a normally nonsingular map.

Let *f*: *X* → *Y* be a smooth $Z/2$-map. A *stable normal bundle* for *f* is a vector bundle *V* ↠ *X* such that *V* ⊕ *TX* is isomorphic to $f*(TY)\u2295(X\xd7Ra,b)$ for some $a,b\u2208N$. 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 *a* − *b* mod 8, the KR-dimension of the trivial bundle $X\xd7Ra,b\u21a0X$.

*Let* $(V,E,f\u0302)$ *be a normally nonsingular* $Z/2$*-map, and let* *f*: *X* → *Y* *be its trace. Then, a* KR*-orientation for* *f* *is equivalent to a* KR*-orientation of the vector bundle* *V**.*

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*.□

A KR*-orientation* on a $Z/2$-manifold *X* is a KR-orientation on its tangent vector bundle *TX*. The KR*-dimension* dim_{KR} *X* is defined as dim_{KR} *TX*.

*X*and

*Y*are two KR-oriented $Z/2$-manifolds, then any smooth map

*f*:

*X*→

*Y*inherits a KR-orientation by Lemma 10, and its KR-dimension is

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 Cl_{a,b}.

The covering $R\u2192T$, *t* ↦ exp(i*t*), becomes a “real” covering $R0,1\u2192T$. 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 $Td\xd7Cl0,d$.

*∂*/

*∂r*on the unit sphere spans the normal bundle of the inclusion map $Sa,b\u2192Ra,b$. 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, $TSa,b\u2295(Sa,b\xd7R1,0)\u2245Sa,b\xd7Ra,b$. Then, Lemma 10 gives a KR-orientation on $TSa,b$. We also find

*a*= 0.

*f*:

*X*→

*Y*be its trace. Then, there are Thom isomorphisms for both the vector bundle

*V*↠

*X*and the trivial vector bundle $Y\xd7Ra,b\u21a0Y$. The open inclusion $f\u0302$ identifies $C0(V)$ with an ideal in $C0(Y\xd7Ra,b)$. Thus, each of the solid maps in (10) induces a map in KR-theory. Taking the composite gives induced maps,

*f*

_{!}is the Kasparov product with an element in $KKR\u2212dimKRf(C0(X),C0(Y))$, 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*(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.

Let *E* ↠ *X* be a KR-oriented vector bundle. Then, the zero section *ζ*_{E}: *X* ↪ *E* and the bundle projection *π*_{E}: *E* ↠ *X* are KR-oriented in a canonical way such that their shriek maps are the Thom isomorphism and its inverse $KRn(X)\u2194KRn+dimKRE(E)$.

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 $f!:KR0(TX)\u2192KR0(pt)=Z$ is the Atiyah–Singer topological index map on *X*.

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 $KKR\u2212dimKRf(C0(X),C0(Y))$. 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*.

*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 $iS:pt\u2192Sa,b$ be the inclusion of the south pole. Give $Sa,b$ the KR-orientation from Example 22. Then, we get a canonical map

*There is* *χ* ∈ {±1} *such that the map* *S*_{!} *sends the generator* $1\u2208Z$ *to* *χ* · ([*β*_{a,b}] − [*γ*_{1}])*.*

The stereographic projection at the north pole induces a $Z/2$-equivariant diffeomorphism $Sa,b\{N}\u2245Ra\u22121,b$. We could use this as a tubular neighborhood for the embedding *i*_{S}. Therefore, *S*_{!} factors through an isomorphism onto the direct summand $KRa\u2212b\u22121(Sa,b\{N})\u2245Z$. Since [*β*_{a,b}] − [*γ*_{1}] generates this summand, *S*_{!} must send the generator $1\u2208Z$ to ±([*β*_{a,b}] − [*γ*_{1}]).□

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 $b\u2217$ ◦*f*_{!} for a KR-oriented map *f*: *X* → *Y* and a proper continuous map *b*: *Z* → *Y*. We will use this to compute the pullback of [*β*_{1,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 $b\u2217$ ◦*f*_{!}:

*Let*

*X*

*and*

*Y*

*be*$Z/2$

*-manifolds. Let*

*f*:

*X*→

*Y*

*be a*KR

*-oriented smooth*$Z/2$

*-map, and let*

*b*:

*Z*→

*Y*

*be a proper smooth*$Z/2$

*-map. Assume that*

*b*

*and*

*f*

*are transverse. Then, the fiber product*

*X*×

_{Y}

*Z*

*is a smooth*$Z/2$

*-manifold. The projection*

*π*

_{X}:

*X*×

_{Y}

*Z*→

*X*

*is proper, and*

*π*

_{Z}:

*X*×

_{Y}

*Z*→

*Z*

*inherits from*

*f*

*a*KR

*-orientation of the same*KR

*-dimension. The following diagram commutes:*

*π*

_{Z}inherits. The transversality assumption implies that there is an exact sequence of vector bundles over

*X*×

_{Y}

*Z*as follows:

*f*is KR-oriented, there are $a,b\u2208N$ and a KR-oriented vector bundle

*V*↠

*X*such that $f*(TY)\u2295(X\xd7Ra,b)\u2245TX\u2295V$. Then, $\pi *(TY)\u2295(X\xd7YZ\xd7Ra,b)\u2245\pi X*(TX)\u2295\pi X*(V)$ and, hence,

*π*

_{Z}. 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 *y*_{0} ∈ *Y* 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 *π*_{Z} is a bijection between *X* × _{Y}*Z* and the preimage *b*^{−1}(*y*_{0}) of *y*_{0} in *Z*. Transversality means in this case that the differential of *b* is surjective in all points of *b*^{−1}(*y*_{0}). Then, *b*^{−1}(*y*_{0}) is a smooth submanifold. It is also compact because *b* is proper. The normal bundle of the inclusion of *b*^{−1}(*y*_{0}) into *Z* is canonically isomorphic to the trivial bundle with fiber $Ty0Y$. This gives the induced KR-orientation of *π*_{Z}.

_{∗}(C

_{0}(

*X*), C

_{0}(

*Y*)) for two $Z/2$-manifolds

*X*and

*Y*in a geometric way. Its main ingredients are

*b*:

*Z*→

*Y*and KR-oriented smooth maps

*f*:

*X*→

*Y*. The details are explained in Sec. VI.

## VI. GEOMETRIC BIVARIANT KR-THEORY

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.

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*:*M*→*X*is a smooth $Z/2$-map,*f*:*M*→*Y*is a KR-oriented smooth $Z/2$-map, and$\xi \u2208KRXa(M)$ for some $a\u2208Z/8$ (here,

*X*-compact support in*M*refers to the map*b*:*M*→*X*).

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

*b*:

*Y*→

*X*yields the following correspondence from

*X*to

*Y*:Here, id

_{Y}is the identity map with its canonical KR-orientation, which is indeed a unit arrow in the category of KR-oriented smooth maps, and $1\u2208KRX0(Y)$ comes from the trivial “real” vector bundle of rank 1.

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

*M*,

*b*,

*f*,

*ξ*) induces an element

*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* $KR\u0302(X,Y)*$ 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 $KR\u0302(X,Y)*$ an Abelian group.

*Let*$n\u2208Z/8$,

*and let*

*X*

*and*

*Y*

*be*$Z/2$

*-manifolds. The map above from*KR

*-oriented correspondences to Kasparov theory defines an isomorphism*

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 *P* → *X* such that there are duality isomorphisms $KR\u0302(X,Y)n\u2245KRXn+a(Y\xd7P)$ 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 $KKR\u2212nC0(X),C0(Y)\u2245KRXn+a(Y\xd7P)$.□

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.

*Let two composable smooth*KR

*-oriented correspondences be given, as described by the solid arrows in the following diagram:*

*Assume that the smooth maps*

*f*

_{1}

*and*

*b*

_{2}

*are transverse, that is, if*

*m*

_{1}∈

*M*

_{1}

*,*

*m*

_{2}∈

*M*

_{2}

*satisfy*

*y*≔

*f*

_{1}(

*m*

_{1}) =

*b*

_{2}(

*m*

_{2})

*, then*$Df1(Tm1M1)+Db2(Tm2M2)=TyY$

*. Then, the following holds. First,*

*M*

_{1}×

_{Y}

*M*

_{2}

*is a smooth*$Z/2$

*-manifold. Second, the exterior tensor product*$\xi :=\pi 1*(\xi 1)\u2297Y\pi 2*(\xi 2)$

*has*

*X*

*-compact support with respect to*

*b*≔

*b*

_{1}◦

*π*

_{1}

*, that is, it belongs to*$KRX*(M1\xd7YM2)$

*. Third, the composite map*

*f*≔

*f*

_{2}◦

*π*

_{2}

*inherits a*KR

*-orientation from*

*f*

_{1}

*and*

*f*

_{2}

*, whose*KR

*-dimension is*dim

_{KR}

*f*= dim

_{KR}

*f*

_{1}+ dim

_{KR}

*f*

_{2}

*. Finally, the resulting*KR

*-oriented correspondence*(

*M*

_{1}×

_{Y}

*M*

_{2},

*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*(

*M*

_{1},

*b*

_{1},

*f*

_{1},

*ξ*

_{1})

*and*(

*M*

_{2},

*b*

_{2},

*f*

_{2},

*ξ*

_{2})

*.*

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 $Y\xd7Ra,b\u21a0Y$ to an open subset $M\u2286Y\xd7Ra,b$ (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 *b*_{1} and *f*_{2} are identity maps, *b*_{2} is proper, and *ξ*_{1} and *ξ*_{2} are the units in representable KR-theory.

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).

## VII. THE MODEL HAMILTONIANS

_{1,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

*S*

_{i}are real. Then, the element

*H*

_{m}is self-adjoint, odd, and real because

*γ*

_{0}, i

*γ*

_{1}, …, i

*γ*

_{d}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).

*S*

_{i}and $Si*$ have the Fourier transforms

*x*

_{i}± i

*y*

_{i}, the Fourier transform of

*H*

_{m}in $C(Td,Cl1,d)$ is equal to $\beta \u03031,d\u25e6\phi \u0303m$.

*If* *m* ∉ {−*d*, −*d* + 2, …, *d* − 2, *d*}*, then* $\phi \u0303m(Td)\u2286R1,d\0$*.*

Assume $\phi \u0303m(x,y)=0$. Then, *x*_{1} +⋯+ *x*_{d} + *m* = 0 and *y*_{1} = *y*_{2} =⋯= *y*_{d} = 0. The latter forces *x*_{i} = ±1 for *i* = 1, …, *d*, and then, $m=\u2212\u2211i=1dxi$ 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 *H*_{m} being invertible, which is needed to define its topological phase.

*m*, there is a well-defined “real” function,

*H*

_{m}with spectrum {±1} is the operator whose Fourier transform in $C(Td,Cl1,d)$ is $\beta \u03031,d\u25e6\phi m$. Since

*φ*

_{m}takes values in $S1,d$ by construction, this is just

*β*

_{1,d}◦

*φ*

_{m}with the real, odd, self-adjoint unitary $\beta 1,d\u2208C(S1,d)\u2297Cl1,d$ in (2). The composite $\beta 1,d\u25e6\phi m\u2208FU(C(Td)\u2297Cl1,d)$ is its pullback along

*φ*

_{m}.

To get a class in van Daele’s K-theory, we must consider a formal difference [*β*_{1,d}◦*φ*_{m}] − [*f*] for some $f\u2208FU(C(Td)\u2297Cl1,d)$. Physically, *f* describes the topological phase that we choose to call “trivial.” An obvious choice in our case is *f* = *γ*_{0}, the constant function on $Td$ with value *γ*_{0}. Another obvious choice would be −*γ*_{0}. 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.

The correspondence *S*_{!} represents the class *χ* · ([*β*_{1,d}] − [*γ*_{0}]) by Lemma 26. Here, *γ*_{0} denotes the constant function on $S1,d$ with value *γ*_{0}, and we changed the numbering of Clifford generators to start at 0. Composing with the correspondence denoted as $\phi m*$ pulls this back along the map *φ*_{m}. This gives *χ* · ([*β*_{1,d}◦*φ*_{m}] − [*γ*_{0}]), where now *γ*_{0} denotes the constant function on $Td$ with value *γ*_{0}.□

*x*

_{j}∈ {−1, +1} for

*j*= 1, …,

*d*. We define

*The two correspondences in Lemma*36

*are transverse. Therefore, their composite is their intersection product. The fiber product*$Td\xd7S1,dpt$

*in the intersection product is diffeomorphic to the finite subset*$\phi m\u22121(S)\u2286Td$

*, and*

*holds in*$DK(C(Td)\u2297Cl1,d)\u2245KR\u2212d(Td)$

*.*

We must show that the differential of *φ*_{m} is a surjective map onto $TSS1,d$ at all points $(x,y)\u2208Td$ with *φ*_{m}(*x*, *y*) = *S*. The tangent space $TSS1,d$ is the subspace ${0}\xd7Rd$, spanned by the basis vectors *e*_{1}, …, *e*_{d}. If *φ*_{m}(*x*, *y*) = *S*, then *y*_{1} = *y*_{2} =⋯= *y*_{d} = 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 *y*_{1}, …, *y*_{d}. The differential of $\phi \u0303m$ maps these to the vectors *e*_{1}, …, *e*_{d}. On the preimage of *S*, the differential of the radial projection map $R1,d\{0}\u2192S1,d$, $z\u21a6z/z$, just multiplies *e*_{j} for *j* = 1, …, *d* with a positive constant, so that the images still span $TSS1,d$. Thus, $\phi m:Td\u2192S1,d$ is transverse to $iS:pt\u2192S1,d$.

The canonical map $\pi Td:Td\xd7S1,dpt\u2192Td$ is a diffeomorphism onto the closed submanifold $\phi m\u22121(S)\u2286Td$ by the definition of the fiber product. Since the differential of *φ*_{m} 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 $(x,0)=(x1,\u2026,xd,0,\u2026,0)\u2208Td$ with $\u2211j=1dxj+m<0$.

Let $z=(x,y)\u2208Td$ satisfy $r(z)=z$. Then, *y* = 0 and hence *x*_{i} ∈ {±1} for *i* = 1, …, *d*. These points satisfy *φ*_{m}(*z*) ∈ {*N*, *S*} for the north and south pole in (4) simply because *φ*_{m} 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 $(1,0)\u2208T1$ and downward at $(\u22121,0)\u2208T1$, the projection to the *y*-coordinate $T(1,0)T1\u2192R0,1$ preserves the orientation at +1 and reverses it at −1. Therefore, the projection to the *y*-coordinate $TzTd\u2192R0,d$ multiplies the orientation with sign(*z*). The sum in geometric bivariant KR-theory is the disjoint union of correspondences. Therefore, the discrete set $\phi m\u22121(S)$ in the composite correspondence contributes the sum of sign(*z*)*z*_{!} over all $z\u2208\phi m\u22121(S)$. Lemma 36 identifies this sum with $\chi \u22c5[\beta 1,d\u25e6\phi m]\u2212[\gamma 0]$.□

*d*− 2 <

*m*<

*d*, then $\phi m\u22121(S)$ has only one element, namely, $(\u22121,\u22121,\u2026,\u22121,0,\u2026,0)\u2208Td$ with sign (−1)

^{d}. Thus, we get

^{∗}-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 $z!:pt\u2192Td$ for different $z\u2208Td$ 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.

*If* *m* < −*d**, then* *H*_{m} *is homotopic to* −*γ*_{0} *in* $FU(C(Td)\u2297Cl1,d)$*.*

*H*

_{s}for

*s*∈ (−∞,

*m*] give a homotopy of real, odd, self-adjoint unitaries $Hs\u22121Hs$ between $Hm\u22121Hm$ and

The same argument shows that *H*_{m} is homotopic to *γ*_{0} for *m* > *d*. This is consistent with Lemma 37, which says that [*H*_{m}] − [*γ*_{0}] = 0 in $KR\u2212d(Td)$ for *m* > *d*.

*m*< −

*d*, then

*φ*

_{m}maps all points in $Td$ with $r(z)=z$ to

*S*. In particular, for

*d*= 1, this says that

*γ*

_{0}] − [

*γ*

_{0}] is represented by constant functions and thus is in the image of $DK(Cl1,1)\u2245KO\u22121(pt)\u2245Z/2$ in $KR\u22121(T1)$. Since this group is two-torsion, we may drop the sign

*χ*in (15).

*The isomorphism above maps* [−*γ*_{0}] − [*γ*_{0}] *to the nontrivial element* $\mu \u2208KO\u22121(pt)\u2245Z/2$*.*

*n*× 2

*n*-matrices,

*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 ±

*γ*

_{0}correspond, however, to ±1 ∈ O(1) = {±1}, which lie in the two different connected components.□

^{∗}-algebras,It induces a KKR-equivalence $C(T1)\u2245C0(R0,1)\u2295C$. 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

*A*

_{1}⊗⋯⊗

*A*

_{d}, where each

*A*

_{j}is either $C0(R0,1)$ or $C$. We label such a summand by the set

*I*⊆ {1, …,

*d*} of those factors that are $C$.

*Let* *m* ∈ (−*d* + 2*n*, −*d* + 2*n* + 2) *for some* *n* ∈ {0, …, *d* − 1}, *and let* *I* ⊆ {1, …, *d*}*. Let* *χ* *be the sign from Lemma* 26*. The image of* $[Hm]\u2212[\gamma 0]\u2208KR\u2212d(Td)$ *in the summand* $KR\u2212d(R0,d\u2212I)$ *where the* *i**th factor is* $C$ *for* *i* ∈ *I* *and* $C0(R0,1)$ *for* *i* ∉ *I* *is computed as follows:*

*If**I*= ∅*, the image in*$KR\u2212d(R0,d)\u2245Z$*is*$(\u22121)d\chi \u22c5d\u22121n$*.**If*$I=1$*and**n*≥ 1*, the image in*$KR\u2212d(R0,d\u22121)\u2245Z/2$*is*$d\u22122n\u22121mod2$*.**If*$I=2$*and**n*≥ 2*, the image in*$KR\u2212d(R0,d\u22122)\u2245Z/2$*is*$d\u22123n\u22122mod2$*.**It is zero in the other cases, that is, for*$I\u22653$*or*$n<I$*.*

We first consider the image of (*x*, 0)_{!} in the summand labeled by *I*. The shriek map $(x1,\u2026,xd,0,\u2026,0)!$ is the exterior product of $(xi,0)!$ for *i* = 1, …, *d* (see Remark 34). In the decomposition of $KR\u22121(T1)\u2245KR\u22121(R0,1)\u2295KO\u22121(pt)\u2245Z\u2295Z/2$, (−1, 0)_{!} becomes (pt_{!}, 0) with the standard generator pt_{!} of $KR\u22121(R0,1)\u2245Z$. Equation (15) shows that (1, 0)_{!} = (pt_{!}, *μ*) with the nontrivial element $\mu \u2208KO\u22121(pt)\u2245Z/2$. Hence, (*x*, 0)_{!} is the exterior product of *d* factors that are (pt_{!}, 0) if *x*_{i} = −1 and (pt_{!}, *μ*) if *x*_{i} = 1. The component in the summand labeled by *I* is zero unless *x*_{i} = +1 for all *i* ∈ *I*. If *x*_{i} = +1 for all *i* ∈ *I*, then we get the exterior product of $pt!\u2208KR\u22121(R0,1)$ for all *i* ∉ *I* and *μ* ∈ KO^{−1}(pt) for all *i* ∈ *I*. The exterior product $\mu \u2297\mu \u2208KO\u22122(pt\xd7pt)\u2245Z/2$ is known to be the nontrivial element (this also follows from the discussion in Sec. IX), whereas *μ* ⊗ *μ* ⊗ *μ* 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 $I\u22653$ or *x*_{i} = −1 for some *i* ∈ *I* and the standard generator of $KR\u2212d(R0,d\u2212I)$ if $I\u22642$ and *x*_{i} = +1 for all *i* ∈ *I*.

*x*, 0)(

*x*, 0)

_{!}over all $(x,0)\u2208Td$ with

*φ*

_{m}(

*x*, 0) =

*S*or, equivalently, $\u2211i=1dxi+m<0$. The latter means that the number of +1 among the coordinates

*x*

_{i}is at most

*n*. There are $dj$ points (

*x*, 0) with

*x*

_{i}= +1 for exactly

*j*indices

*i*. For all of them, the image of (

*x*, 0)

_{!}in the direct summand $KR\u2212d(R0,d)\u2245Z$ for

*I*= ∅ is the same standard generator. Hence, the image of [

*H*

_{m}] − [

*γ*

_{0}] in this direct summand is

*I*= {

*i*

_{0}} for some

*i*

_{0}∈ {1, …,

*d*}. The corresponding direct summand $KR\u2212d(R0,d\u22121)\u2245Z/2$ only sees $(x,0)\u2208\phi m\u22121(S)$ with $xi0=1$. There are $d\u22121j\u22121$ points with $xi0=1$ and

*x*

_{i}= +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

*I*⊆ {1, …,

*d*} with $I=2$, we get $d\u22123n\u22122$ 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.

*H*

_{m}for

*d*− 2 <

*m*<

*d*represents a generator of the direct summand $KR\u2212d(R0,d)\u2245Z$ in $KR\u2212d(Td)$. We get different KR-classes by stacking insulators in lower dimension along some direction. In our framework, this means the following: choose 0 ≤

*k*≤

*d*and 1 ≤

*i*

_{1}<

*i*

_{2}<⋯<

*i*

_{k}≤

*n*, consider the projection $\rho :Td\u21a0Tk$ that only keeps the coordinates $xij,yij$ for

*j*= 1, …,

*k*, and pull back the

*k*-variable Hamiltonian $Hm\u2208C*(Zk)\u2297Cl1,k$ to an odd, real, self-adjoint, invertible element

*m*∈ (

*k*− 2,

*k*), then $[Hm\u2032]\u2212[\gamma 0]$ generates the direct summand $KR\u2212k(R0,k)\u2245Z$ in $KR\u2212k(Td)$ that corresponds to the subset

*j*= 1, 2, 4, which gives all cases where this group is not zero.

## VIII. TRANSFER TO HILBERT SPACE

In this section, we are going to identify the set of odd, self-adjoint invertible elements in the C^{∗}-algebra $C(Td)\u2297Cla,b$ with $a,b,d\u2208N$ 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 *j* ≔ *b* − *a* + 1 mod 8; we use this parameter because $DK(C(Td)\u2297Cla,b)=KR\u2212j(Td)$. 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 *H*_{m} 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 $\u21132(Zd)$ in the usual way. A bounded operator on $\u21132(Zd)$ belongs to $C(Td)$ if and only if it (a) commutes with the translation operators *S*_{x} for $x\u2208Zd$ defined by *S*_{x}*f*(*y*) ≔ *f*(*y* − *x*) for $x,y\u2208Zd$, $f\u2208\u21132(Zd)$ 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 $L\u221e(Td)$ 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 Cl_{a,b} on $C2k$ for suitable $k\u2208N$ such that the representation maps odd self-adjoint unitaries in Cl_{a,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**

*j*is even or, equivalently,

*b*and

*a*have different parity. Then, there is an isomorphism

*k*= (

*a*+

*b*− 1)/2. In addition, we may arrange that the $Z/2$-grading automorphism on Cl

_{a,b}becomes the flip isomorphism

*α*(

*x*,

*y*) ≔ (

*y*,

*x*) on $M2kC\u2295M2kC$. Thus, self-adjoint, odd elements of Cl

_{a,b}become elements of the form $(x,\u2212x)\u2208M2kC\u2295M2kC$ for some $x\u2208M2kC$ with

*x*=

*x**. As our representation $Cla,b\u2192M2kC$, we simply choose the projection to the first summand.

Since the “real” involution $r$ on Cl_{a,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.

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 $\Theta :C2k\u2192C2k$. The pair (*x*, −*x*) for $x\u2208M2kC$ is fixed by $r$ if and only if*x*commutes with Θ. Thus, we identify “real” self-adjoint odd unitaries in Cl_{a,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, Θ^{2}= ±1.Assume that $r$ flips the two summands $M2kC$. Then, $r\u25e6\alpha $ maps each direct summand into itself and induces the same real involution on both summands $M2kC$. Thus, we may implement $r\u25e6\alpha $ by an antiunitary operator $\Theta :C2k\u2192C2k$ 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, Θ^{2}= ±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 Θ^{2} = +1, the real subalgebra $(M2kC)R$ fixed by $r$ is $M2kR$, and so $(Cla,b)R\u2245M2kR\u2295M2kR$. In subcase 1 with Θ^{2} = −1, we must have *k* ≥ 1 and $(M2kC)R\u2245M2k\u22121H$ and $(Cla,b)R\u2245M2k\u22121H\u2295M2k\u22121H$ for the quaternions $H$. In subcase 2, the real subalgebra $(Cla,b)R$ is always isomorphic to $M2kC$, identified with the subalgebra of $(x,\Theta \u22121x\Theta )\u2208M2kC\u2295M2kC$ for $x\u2208M2kC$. To distinguish the two signs for Θ^{2} = ±1, we also look at the even subalgebra of $(Cla,b)R$. If Θ^{2} = +1, then the even subalgebra of $(Cla,b)R$ is $M2kR$. If Θ^{2} = −1, then *k* ≥ 1 and the even subalgebra of $(Cla,b)R$ is $M2k\u22121H$. 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 Cl_{1,0}, the real subalgebra is $R\u2295R$, so we must be in the case of a time-reversal symmetry with square +1. The symmetry type only depends on $j\u2208Z/8$ 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 Cl_{0,3}, the real subalgebra is well known to be $H\u2295H$ so that we must be in the case of a time-reversal symmetry with square −1. This statement for Cl_{0,3} then extends to all Clifford algebras with *j* ≡ 4 mod 8.

For Cl_{0,1}, the real subalgebra is $C$, and the even real subalgebra is $R\u2286C$. This only happens for a particle–hole symmetry of square +1. The same symmetry type occurs whenever *j* ≡ 2 mod 8. Finally, for Cl_{3,0}, 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, $Cla,b\u2245M2kC$ 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 $\Xi :Ck\u2192Ck$ and an antiunitary operator $\Theta :Ck\u2192Ck$, respectively. A real, self-adjoint odd unitary in Cl_{a,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 Ξ^{2} = 1. Since Θ induces an antiunitary involution, Θ^{2} = ±1. Since $r$ commutes with the grading, Ξ^{−1}ΘΞ = ±Θ. This is equivalent to (ΞΘ)^{2} = ±Θ^{2}. 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 $(Cla,b)R\u2245M2kR$ if Θ^{2} = +1 and $(Cla,b)R\u2245M2k\u22121H$ (and *k* ≥ 1) if Θ^{2} = −1. The even subalgebra of Cl_{a,b} is always $M2k\u22121C\u2295M2k\u22121C$. If Ξ^{−1}ΘΞ = Θ, then the real involution restricted to the even part preserves the two direct summands $M2k\u22121C$ so that the even real subalgebra of Cl_{a,b} is a direct sum of two simple algebras. If Ξ^{−1}ΘΞ = −Θ, however, then the real involution restricted to the even part flips the two direct summands so that the even real subalgebra of Cl_{a,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 Cl_{a,b}, with (*a*, *b*) being (1, 1), (2, 0), (0, 2), and (0, 4), we find the following:

Θ

^{2}= +1, Ξ^{−1}ΘΞ = +Θ, and (ΞΘ)^{2}= +1 for Cl_{1,1}with*j*≡ 1 mod 8;Θ

^{2}= +1, Ξ^{−1}ΘΞ = −Θ, and (ΞΘ)^{2}= −1 for Cl_{2,0}with*j*≡ 7 mod 8;Θ

^{2}= −1, Ξ^{−1}ΘΞ = −Θ, and (ΞΘ)^{2}= +1 for Cl_{0,2}with*j*≡ 3 mod 8; andΘ

^{2}= −1, Ξ^{−1}ΘΞ = +Θ, and (ΞΘ)^{2}= −1 for Cl_{0,4}with*j*≡ 5 mod 8.

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

## IX. MODEL HAMILTONIANS FOR OTHER SYMMETRY TYPES

Our model Hamiltonian is an odd self-adjoint unitary in $C(Td)\u2297Cl1,d$. 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 Cl_{1,d}. In this section, we explain, in particular, how to find model Hamiltonians with odd time-reversal symmetry in dimensions 2 and 3 instead.

*X*is a module over the graded ring KO*(pt). This is a special case of the exterior product maps,

*H*

_{m}] − [−

*γ*

_{0}] for

*d*− 2 <

*m*<

*d*generates the direct summand $KR\u2212d(R0,d)\u2286KR\u2212d(Td)$. The exterior product with the generator of $KO\u2212j(pt)\u2245Z/2$ for

*j*= 1, 2 or $KO\u22124(pt)\u2245Z$ gives a generator for the direct summand $KR\u2212d\u2212j(R0,d)$. 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*

_{+}∈ Cl

_{0,4}for

*j*= 4.

The outcome for *j* = 1, 2 is particularly simple to state. We may embed Cl_{1,d} ↪ Cl_{1,d+1} ↪ Cl_{1,d+2} by adding extra generators. Thus, an odd self-adjoint invertible element in $C(Td)\u2297Cl1,d$ gives one in $C(Td)\u2297Cl1,d+1$ and $C(Td)\u2297Cl1,d+2$ 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 *H*_{m} and *γ*_{0}, just viewed in a different Clifford algebra, give a generator for the summands $KR\u2212d\u2212j(R0,d)\u2245Z/2$ for *j* = 1, 2. In particular, for *d* = 3 and *j* = 1, we get *H*_{m} viewed as an element of $C(T3)\u2297Cl1,4$. 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, $Cl1,4\u2245M2(Cl0,3)\u2245M4(C)\u2295M4(C)$ so that our Hamiltonian will act on $\u21132(Z3)\u2297C4$. The original Hamiltonians take values in $Cl1,3\u2245M4(C)$ and so act on the same Hilbert space. In Sec. VIII, we have identified odd real elements of Cl_{1,3} with elements of $M4(C)$ that have both an odd time-reversal and an even particle–hole symmetry; replacing Cl_{1,3} by Cl_{1,4} to shift to a different KR-group amounts to forgetting the particle–hole symmetry.

To get a generator for the summand $KR\u22124(R0,2)\u2245Z/2$ in $KR\u22124(T2)$, we view $Hm,\gamma 0\u2208C(T2,Cl1,2)$ as functions taking values in Cl_{1,4}. In Sec. VIII, we have identified odd, real, self-adjoint elements of Cl_{1,2} with elements of $M2(C)$ that have an even particle-hold symmetry (this is the case *j* = 2). The embedding Cl_{1,2} ↪ Cl_{1,4} 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.

*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* Cl_{c+1,d}*.*

Let $(H,\phi ,F)$ 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 Cl_{c,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 $H\u2245p\u22c5Clc,d2n$ for some real, even projection $p\u2208M\u03022n(Clc,d)$; 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 $H\u2295(Clc,d)\u221e\u2245(Clc,d)\u221e$, where $(Clc,d)\u221e$ denotes the standard “real” graded Hilbert Cl_{c,d}-module, with the $Z/2$-grading where half of the summands carry the flipped grading. This gives a real, even projection $p0\u2208B((Clc,d)\u221e)$ with $H\u2245p0(Clc,d)\u221e$. Since the identity on $H$ is compact, $p0\u2208K((Clc,d)\u221e)$. Then, *p*_{0} is Murray–von Neumann equivalent—with even real partial isometries—to a nearby projection $p\u2208M\u03022n(Clc,d)$. This implies an isomorphism $H\u2245p\u22c5Clc,d2n$ 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 $M\u03022(Clc,d)$. 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 $M\u03022(Clc,d)$ is isomorphic to the crossed product $Clc,d\u22caZ/2$—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 $A\u2297K(L2G)$. The nontrivial element of $Z/2$ gives an extra Clifford generator that commutes with even and anticommutes with odd elements of Cl_{c,d}. Thus, the even part of $M\u03022(Clc,d)$ is isomorphic to Cl_{c+1,d}. This implies $R\u2245(Clc+1,d)R$. This $R$-algebra is semisimple because its complexification is a sum of matrix algebras. Hence, all finitely generated modules over it are projective.□

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 *F*^{2} = 1 and commutes with Cl_{c,d}. Multiplying *F* with the grading gives a real, odd operator that anticommutes with the generators of Cl_{c,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 Shapiro^{12}computed 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 $KKR0(C,Cl0,j)\u2245Z/2$ for *j* ≡ 1, 2 mod 8. It follows from Lemma 41 that the generator of $KKR0(C,Clc,d)\u2245Z/2$ 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 Cl_{0,1} 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, Cl_{0,1} with *F* = 0 represents the generator of $KKR0(C,Cl0,1)\u2245Z/2$.

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 Cl_{0,2} 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, Cl_{0,2} with *F* = 0 represents the generator of $KKR0(C,Cl0,2)\u2245Z/2$.

The computations above show that the generator of $KKR0(C,Cl0,j)\u2245Z/2$ for *j* = 1, 2 is represented by the grading-preserving “real” *-homomorphism $C\u2192Cl0,j$ induced by the unit element. Computing the exterior product in KK with such a *-homomorphism is very easy: for a class in KKR_{0}(*A*, *B*), we simply apply the functoriality of KKR for the *-homomorphism $B=B\u2297\u0302C\u2192B\u2297\u0302Cl0,j$. In the case $A=C$, we may identify KKR_{0}(*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 $KKR0(C,Cl0,j)\u2245Z/2$ for *j* = 1, 2 in van Daele’s K-theory maps a class in DK(*A*) by applying functoriality for the canonical inclusion map $A=A\u2297\u0302C\u2192A\u2297\u0302Cl0,j$.

*Let* *j* = 1, 2*; define* $Hm\u2032$ *as in (16). The exterior product of the generator of* KO^{−j}(pt) *and* $[Hm\u2032]\u2212[\gamma 0]\u2208KR\u2212k(Td)$ *belongs to* $DK(C(Td,Cl1,k+j))$*. It is* $[Hm\u2032]\u2212[\gamma 0]$*, where now* $Hm\u2032$ *and* *γ*_{0} *are viewed as self-adjoint odd unitaries in* $C(Td,Cl1,k+j)$ *with* *j* *extra Clifford generators.*

For *j* = 3, the unital inclusion $C\u2192Cl0,j$ 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 $C\u2192Cl0,1$. Therefore, the generator of $KKR0(C,Cl0,4)\u2245Z$ cannot have this simple form. It is known that $(Cl1,4)R\u2245M2(Cl0,3)R\u2245M2(H)\u2295M2(H)$. Hence, there are two simple $(Cl1,4)R$-modules, and both have $R$-dimension 8. When we turn Cl_{0,4} 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 $KKR0(C,Cl0,4)\u2245Z$.

*γ*

_{1}, …,

*γ*

_{4}. As a result, $p\xb1:=12(1\xb1\omega )$ are real, even, self-adjoint projections in Cl

_{0,4}. There is no preferred sign here because −

*ω*=

*γ*

_{2}

*γ*

_{1}

*γ*

_{3}

*γ*

_{4}is built in the same way as

*ω*, just for a differently oriented orthonormal basis. We may decompose

_{0,4}-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 Cl

_{0,4}-modules

*p*

_{±}Cl

_{0,4}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 $C\u2192Cl0,4$,

*λ*↦

*λp*

_{±}.

Therefore, the exterior product of a class in van Daele’s K-theory DK(*A*) with the generator of $KO\u22124(pt)\u2245Z$ applies the functoriality for one of these two nonunital *-homomorphisms. This first produces an element in $DK(A\u2297\u0302p\xb1Cl0,4p\xb1)$ because the homomorphism is not unital, which is then mapped to one in $DK(A\u2297\u0302Cl0,4)$ 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 Cl_{1,d+4} as matrices in $M2kC$ with specific symmetries. The underlying vector space $C2k$ carries a representation of Cl_{1,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 $p\xb1C2k\u2245C2k\u22121$. Then, the real, odd, self-adjoint elements in *p*_{±}Cl_{1,d+4}*p*_{±} become identified with self-adjoint operators on $p\xb1C2k\u2245C2k\u22121$ 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 $KR\u2212d\u22124(Td)$.

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 $KRa+b\u22121(Sa,b)$ for some $a,b\u2208N$ back along a continuous map $Td\u2192Sa,b$. This can only produce Hamiltonians of the special form $\u2211i=1dfi\gamma i$ with functions $fi\u2208C(Td)$ and the standard generators *γ*_{i} of Cl_{a,b}. Generic odd self-adjoint elements in a Clifford algebra may, however, involve products of the *γ*_{i} as well. Therefore, we cannot expect Hamiltonians that come from the experiment to be of this simple form.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

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

## DATA AVAILABILITY

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

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