Twisted crystallograpic T-duality via the Baum--Connes isomorphism
aa r X i v : . [ m a t h . K T ] J a n TWISTED CRYSTALLOGRAPIC T-DUALITY VIA THEBAUM–CONNES ISOMORPHISM
KIYONORI GOMI, YOSUKE KUBOTA, AND GUO CHUAN THIANG
Abstract.
We establish the twisted crystallographic T-duality, whichis an isomorphism between Freed-Moore twisted equivariant K-groupsof the position and momentum tori associated to an extension of a crys-tallographic group. The proof is given by identifying the map with theDirac homomorphism in twisted Chabert–Echterhoff KK-theory. Wealso illustrate how to exploit it in K-theory computations.
Contents
1. Introduction 2Acknowledgments 52. Twisted crystallographic group 52.1. Definition 52.2. Magnetic space group 83. Freed–Moore twisted equivariant K-theory 133.1. Two definitions 133.2. Operations in twisted equivariant K-theory 173.3. Classification of gapped topological phases 224. Crystallographic T-duality 255. Proof of Theorem 4.4 275.1. Comparison of twisted equivariant K-theories 275.2. Twisted crystallographic T-duality via Kasparov product 305.3. T-duality and Dirac morphism 325.4. Generalization to irrational twists 34
Department of Mathematics, Tokyo Institute of Technology, 2-12-1Ookayama, Meguro-ku, Tokyo, 152-8551, Japan.Department of Mathematical Sciences, Shinshu University, 3-1-1 Asahi,Matsumoto, Nagano, 390-8621, Japan / RIKEN iTHEMS Program, 2-1 Hiro-sawa, Wako, Saitama, 351-0198, JapanBeijing International Center for Mathematical Research, Peking Univer-sity, 5 Yiheyuan Rd, Beijing, China
E-mail addresses : [email protected], [email protected],[email protected] . Date : February 2, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
T-duality, twisted equivariant K-theory, topological phases ofmatter. pg grey magnetic space group 437. Functoriality of the twisted crystallographic T-duality 477.1. Induction of topological insulators 477.2. Push-forward and functoriality 497.3. Atomic insulator and induction 51Appendix A. Equivariant cohomology 53Appendix B. φ -twisted Chabert-Echterhoff twisted equivariantKK-theory 54B.1. Definitions 54B.2. Partial descent homomorphism 56B.3. Twisted equivariant K-theory and Fredholm operators 57References 601. Introduction
T-duality arose as a certain equivalence between string theories compact-ified on a circle and on a dual circle [Bus87]. That a rich mathematicalstructure lies behind topological T-duality became apparent from [BEM04a],where it was found that the K-theories of a circle bundle twisted by an H -flux, coincided with those of a dually fibered one with a dual twist. Fur-thermore, a C ∗ -algebraic formulation [MR05] revealed the connection ofT-duality to the Baum–Connes assembly map [BCH94].The simplest example comes from the abelian group Z , whose classifyingspace R / Z and the Pontrjagin dual ˆ Z are circles dual to each other. Thebasic T-duality isomorphisms K ∗ ( R / Z ) ∼ = K ∗− (ˆ Z ) can be formulated as aFourier–Mukai transform, or as the Baum–Connes assembly map for Z com-posed with Poincar´e duality. In [GT19], it was shown that every (ordinary)crystallographic space group G acting on d -dimensional Euclidean space V ,induces crystallographic T-duality isomorphisms of twisted K-theories,(1.1) T G : K ∗ + d, − v P ( V / Π) → K ∗ ,σP ( ˆΠ) . Here, Π is the abelian subgroup of translations, P = G/ Π is the pointgroup , while v and σ are equivariant twists canonically associated to the P -spaces V /
Π and ˆΠ respectively. The key point is that
V /
Π and ˆΠ aregenerally inequivalent P -spaces with different untwisted cohomologies, yetthere exists a suitable twist v and σ on each side, such that the twistedequivariant K-theories are in perfect duality.The purpose of this paper is to clarify the mathematical structure under-lying (1.1), and establish a vast generalization, (1.2), which we call twisted WISTED CRYSTALLOGRAPHIC T-DUALITY 3 crystallographic T-duality . These dualities are mediated by twisted crystal-lographic groups , whose associated twisted equivariant K-theories arise inmodern studies of topological phases in physics, as we outline below.From the standpoint of condensed matter physics, the topological T-duality map (1.1) is a topological analogue of the Fourier transform revealinga duality between the position space torus
V /
Π and the momentum spacetorus ˆΠ. This is relevant to the theory of topological insulators, becausethe K-group of the momentum torus (Brillouin torus) classifies topologicalphases protected by a given symmetry [Kit09, SRFL08, FM13].According to Wigner’s theorem [Fre12], a symmetry of quantum me-chanics is a unitary/antiunitary Z -graded projective representation of agroup, which is controlled by a triple ( φ, c, τ ) of group cocycles called atwist. We call such a representation a ( φ, c, τ )-twisted unitary representa-tion. Therefore quantum mechanical symmetries go beyond the standardtheory of unitary group representations. A powerful framework to treattopological phases protected by such quantum mechanical symmetries is theFreed–Moore K-theory [FM13], see also [Thi16, Kub17, Gom17a]. Here weconsider a merger of the crystallographic and quantum mechanical sym-metries, which is precisely described in Section 2. Roughly speaking, atwisted crystallographic group is a quadruple ( G, φ, c, τ ), where G is a dis-crete group acting on the Euclidean space V and ( φ, c, τ ) is a twist of G .Moreover we assume that there is a normal subgroup Π of G acting on V by translations and ( φ, c, τ ) is a twist pulled back from P := G/ Π. Freed–Moore [FM13] proposed that topological phases protected by the twistedcrystallographic group (
G, φ, c, τ ) are classified by the twisted equivariantK-group φ K ∗ , t + σP ( ˆΠ). Here, t is short for ( c, τ ), and σ is another twist aris-ing from viewing G as an extension of P by Π. This is reviewed in Section3.3.The twisted crystallographic T-duality for a twisted crystallographic group( G, φ, c, τ ), the main concern of this paper, is an isomorphism φ T t G : φ K ∗ + d, t − v P ( V / Π) → φ K ∗ , t + σP ( ˆΠ) . (1.2)This map is defined as a generalization of the Fourier-Mukai transform inK-theory and is constructed as the composition of standard operations inFreed–Moore twisted equivariant K-theory. The proof of (1.2) being iso-morphic consists of two steps. First, we construct an isomorphism of thegroups in both sides of (1.2) by using the technology of Kasparov’s KK-theory [Kas88] and the Baum–Connes isomorphism [BCH94]. Second, weshow that the isomorphism constructed in the first step coincides with thetwisted crystallographic T-duality map of our interest. We remark that thesecond step is first considered in this paper even in untwisted cases, whichis studied in the previous work [GT19]. In [GT19], both the Fourier–Mukaiand the Baum–Connes maps are independently used to formulate untwistedcrystallographic T-duality, and only the latter was shown to be an isomor-phism. TWISTED CRYSTALLOGRAPHIC T-DUALITY
This strategy is known to work in the non-equivariant topological T-duality. It is a well-known fact in index theory that the (non-equivariant)topological T-duality map is nothing but the Baum–Connes assembly mapfor the lattice group Z d (which is pointed out in [BCH94, Section 3]). Hencethe Dirac-dual Dirac method [Kas88] provides an alternative proof of theisomorphism of the T-duality map, which is essentially different from theone given in e.g. [MR14]. In order to extend this approach to (twisted)crystallographic T-duality, it is convenient to work in Chabert–Echterhofftwisted equivariant KK-theory [CE01]. For our use, we consider its φ -twistedgeneralization, which is summarized in Appendix B. This is a bivariant ho-mology theory for φ -twisted ( G, Π)-C*-algebras, i.e., a C*-algebra A witha φ -twisted G -action whose restriction to Π is implemented by a unitaryrepresentation (cf. Definition B.1), constructed for the purpose of under-standing the permanence properties of the Baum–Connes conjecture. Atypical example of a φ -twisted ( G, Π)-C*-algebra is the crossed product A ⋊ Π of a φ -twisted G -C*-algebra A . Indeed, the crossed product func-tor A A ⋊ Π extends to the φ -twisted partial descent homomorphism φ j G, Π : φ KK G ( A, B ) → φ KK G, Π ( A ⋊ Π , B ⋊ Π). The isomorphism consideredin the first step explained above is the Kasparov product with the partialdescent of the Dirac homomorphism φ j G, Π ( D ). For comparing the Freed–Moore and Chabert–Echterhoff K-theories, a point is that this comparisondoes not work in the level of categories. Indeed, the categories of φ -twisted( G, Π)-C*-algebras and the twisted P -spaces have very small overlaps.In the rest of the paper, Sections 6 and 7, we introduce applications ofthe twisted crystallographic T-duality isomorphisms. The first is concernedwith the computation of the twisted equivariant K-groups. A standard ap-proach to the calculation of these groups is the Atiyah–Hirzebruch spectralsequence (AHSS), which is well-studied in the physics paper [SSG18]. It isa problem of the AHSS approach that it does not determine the group ingeneral because of the extension problem. In Section 6 we show that thetwisted crystallographic T-duality isomorphism solves the extension problemin some examples.The second is concerned with the induction of topological insulators. Ifone has a subgroup H of G , we can define the “induction of topologicalinsulator” map Ind GH : φ K ∗ , t + σQ ( ˆΣ) → φ K ∗ , t + σP ( ˆΠ) , where Σ := H ∩ R d is the subgroup of translations in H and Q := H/ Σ.When H is a finite subgroup, an element of the image of this map iscalled an atomic insulator and studied in the context of condensed-matterphysics. In Section 7 we show that, through the twisted crystallographic T-duality isomorphism, this induction corresponds to the map K ∗ , t − w Q ( W/ Σ) → K ∗ , t − v P ( V /
Π) induced from the wrong-way functoriality of the twisted equi-variant K-theory, where W ⊂ V is an H -invariant affine subspace on which H acts cocompactly. WISTED CRYSTALLOGRAPHIC T-DUALITY 5
Acknowledgments.
The author KG was supported by JSPS KAKENHIGrant Number JP15K04871 and 20K03606. YK was supported by RIKENiTHEMS and JSPS KAKENHI Grant Numbers 19K14544 and JPMJCR19T2.GCT was supported by Australian Research Council grants DE170100149and DP200100729, and thanks the University of Adelaide for hosting himas a visitor. 2.
Twisted crystallographic group
In this section we introduce the notion of twisted crystallographic group,which is a merger of quantum mechanical and crystallographic symmetriesprotecting topological phases of matter.2.1.
Definition.
Let V denote the d -dimensional Euclidean space, i.e., thereal vector space R d equipped with the standard inner product. The set ofisometries of the space V forms the Euclidean motion group Euc( V ). Thisgroup is isomorphic to the semi-direct product R d ⋊ O ( d ), where R d and O ( d ) acts on V by translations and orthogonal transformations respectively.The subgroup S ⊂ Euc( V ) is called a d -dimensional crystallographic group (or a space group ) if • the subgroup Π := S ∩ R d ⊂ R d is a full-rank lattice of translations,and • the point group P = S/ Π ⊂ O ( d ) is a finite group.That is, the diagram1 / / R d / / ∪ Euc( V ) / / ∪ O ( d ) / / ∪ / / Π / / S / / P / / , commutes.The Euclidean group and its subgroups act naturally on L ( V ), but forphysical reasons, we also need to consider on-site, or internal, quantummechanical symmetries. Let K denote the Hilbert space of internal degreesof freedom (e.g. spinor, gauge, etc.). This is a finite rank complex Hilbertspace which is decomposed as K = K ⊕ K by a Z -grading γ , i.e., γ = γ ∗ and γ = 1. We write Aut qtm ( K ) for the group of unitary/antiunitary andeven/odd operators on K . Let U ( K ) denote the group of even unitaries on K , which is isomorphic to U( K ) × U( K ). Once we fix an antilinear eveninvolution T and a linear odd involution S such that [ T, S ] = 0, there is acanonical isomorphismAut qtm ( K ) ∼ = U ( K ) ⋊ ( Z × Z ) , where Z × Z in the right hand side corresponds the subgroup generatedby T and S . Wigner’s theorem [Fre12] states that the quotient groupAut qtm ( K ) / T ∼ = PU ( K ) ⋊ ( Z × Z ) , TWISTED CRYSTALLOGRAPHIC T-DUALITY where PU ( K ) := U ( K ) / T and T is the scalar subgroup, is canonicallyisomorphic to the automorphism group of the set of quantum states P K :=( K \ { } ) / C × which preserves the transition probabilities and commuteswith γ . Hence it is denoted by Aut qtm ( P K ) hereafter. Definition 2.1.
A subgroup G of the direct product Euc( V ) × Aut qtm ( P K )is said to be a d -dimensional twisted crystallographic group if(1) the subgroup Π := G ∩ ( R d ×
1) is a full-rank lattice of translations,(2) The quotient P = G/ Π, called the twisted point group, is a finitesubgroup of O ( d ) × Aut qtm ( P K ).That is, there is a commutative diagram of exact sequences1 / / R d ∪ / / Euc( V ) × Aut qtm ( P K ) / / ∪ O ( d ) × Aut qtm ( P K ) / / ∪ / / Π / / G / / P / / . We write i : G → Euc( V ) × Aut qtm ( P K ) for the inclusion, as well as j := pr Euc( V ) ◦ i and k := pr Aut qtm ( P K ) ◦ i .The group Aut qtm ( P K ) is naturally equipped with the following threestructures: • The (anti)linearity indicator; the homomorphism ˜ φ : Aut qtm ( P K ) → Z mapping unitaries/antiunitaries to 0/1 respectively. • The grading indicator; the homomorphism ˜ c : Aut qtm ( P K ) → Z mapping even/odd unitaries to 0/1 respectively. • A group extension1 → T → U ( K ) ⋊ ( Z × Z ) → PU ( K ) ⋊ ( Z × Z ) → , which is ˜ φ -twisted, i.e., the adjoint action of Aut qtm ( P K ) on T is g · z := ( z if ˜ φ ( g ) = 0 , ¯ z if ˜ φ ( g ) = 1.(2.2)Each of them corresponds to a group cocycle, • ˜ φ ∈ H (Aut qtm ( P K ); Z ), • ˜ c ∈ H (Aut qtm ( P K ); Z ) and • ˜ τ ∈ H (Aut qtm ( P K ); ˜ φ T )respectively. Here we write ˜ φ T for the abelian group T with the Aut qtm ( P K )-action as (2.2).Let us denote by ˜ φ Z N the cyclic group { z ∈ C | z N = 1 } ⊂ T withthe Aut qtm ( P K )-action given as (2.2). We remark that the element of H (Aut qtm ( P K ); ˜ φ Z N ) corresponding to the extension1 → Z N → SU ( K ) ⋊ ( Z × Z ) → PU ( K ) ⋊ ( Z × Z ) → τ by the homomorphism induced from ˜ φ Z N → ˜ φ T . WISTED CRYSTALLOGRAPHIC T-DUALITY 7
A twisted crystallographic group G is also equipped with the cocycles( φ, c, τ ) obtained by the pull-back of ( ˜ φ, ˜ c, ˜ τ ) with respect to k : G → Aut qtm ( P K ).Note that ( φ, c, τ ) are pulled back through P , in other words the restrictions φ | Π , c | Π , τ | Π are trivial (see Subsection 5.4 for a generalization). Proposition 2.3.
Let ( G, φ, c, τ ) be the following data; • j : G → Euc( V ) is a group homomorphism with finite kernel suchthat S := j ( G ) is a space group, • φ, c ∈ H ( G ; Z ) and τ ∈ Im( H ( G ; φ Z N )) ⊂ H ( G ; φ T ) for some N ∈ N .Then there is a homomorphism k : G → Aut qtm ( P K ) such that (1) the image ( j, k )( G ) is a twisted crystallographic group, (2) k ∗ ( ˜ φ, ˜ c, ˜ τ ) = ( φ, c, τ ) . For this reason, in this paper we often specify a twisted crystallographicgroup by a quadruple (
G, φ, c, τ ). For the proof of this proposition, we usethe following lemma.
Lemma 2.4.
Let → K → H q −→ Π → be an extension of groups suchthat K is finite and Π is free abelian of finite rank. Assume that the adjointaction Π → Out( K ) is trivial. Let N := K , Π N := { t N | t ∈ Π } and H N := q − (Π N ) . Then there is a section s : Π N → H N by homomorphismwhose image commutes with K , i.e., H N ∼ = Π N × K .Proof. The proof is given by induction on the rank of Π. Let Π ′ ≤ Π bea subgroup with Π / Π ′ ∼ = Z . let q : H → Π denote the projection and set H ′ := q − (Π ′ ), Π ′ N := Π N ∩ Π ′ and H ′ N := H N ∩ H ′ . Then by the inductionhypothesis there is a section s ′ : Π ′ N → H ′ N whose image commutes with K .Let us choose an element x ∈ H which is mapped to the generator bythe composition H → Π → Π / Π ′ ∼ = Z . Then there is an isomorphism H = H ′ ⋊ Ad( x ) Z . Since Ad( x ) is an inner action on K by assumption, we mayassume that Ad( x ) = id by replacing x with kx for some k ∈ K if necessary.Moreover, for any t ∈ Π ′ N , there is k t ∈ K such that Ad( x )( s ′ ( t )) = s ′ ( t ) k t holds. Hence we haveAd( x l )( s ′ ( t )) = Ad( x l − )( s ′ ( t ) k t ) = Ad( x l − )( s ′ ( t ) k t ) = · · · = s ′ ( t ) k lt , and in particular, Ad( x N ) acts identically on the subgroup s (Π ′ N ). Finallywe obtain the desired section s : Π N → H N by s | Π ′ = s ′ and s ( q ( x ) N ) = x N . (cid:3) Proof of Proposition 2.3.
We choose a preimage τ ′ ∈ H ( G ; φ Z N ) of τ . Let G τ ′ denote the φ -twisted extension of G corresponding to τ ′ (i.e. the pullbackof (2.1) under k ) and let H τ ′ denote the kernel of the composition G τ ′ → G ( j,φ,c ) −−−−→ Euc( V ) × ( Z × Z ) → O ( d ) × ( Z × Z ) . By assumption, H τ ′ is a finite index normal subgroup of G τ ′ and j ( H τ ′ ) ⊂ Euc( V ) is a free abelian group of rank d . We apply Lemma 2.4 for the TWISTED CRYSTALLOGRAPHIC T-DUALITY extension 1 → K τ ′ → H τ ′ → j ( H τ ′ ) →
1. Then, we obtain a finite indexnormal subgroup s ( j ( H τ ′ ) N ) of H τ ′ ,N := j − ( j ( H τ ′ ) N ). Now we define thenormal subgroup Π of G asΠ := \ gH τ ′ ,N ∈ G τ ′ /H τ ′ ,N gs ( j ( H ) N ) g − . Then Π is a finite index normal subgroup of G τ ′ and is free abelian of rank d . Set P τ ′ := G τ ′ / Π and P := G/ Π. Then φ and c factor through P and τ ∈ H ( G ; φ T ) is the pull-back of the 2-cocycle of P corresponding to the φ -twisted extension 1 → φ Z N → P τ ′ → P →
1. Let us choose a finiterank ( φ, c, τ ′ )-twisted unitary representation K of P (such a representationalways exists, see e.g. [Kub16, Example 2.9]). Then k : P → Aut qtm ( P K ) isthe desired homomorphism. (cid:3) Example . Twisted crystallographic groups include the following classesof symmetries studied in the context of topological phases of matter.(1) Let G := Z × Z be acting on V trivially and let φ, c ∈ Hom( G , Z )be the first and the second projections respectively. Then there are10 choices of a subgroup A ⊂ G and τ ∈ H ( A ; φ T ), each of whichcorresponds to one of 2 complex and 8 real Clifford algebras (upto Morita equivalence). The classification of topological phase withthe symmetry type (Π × A , φ, c, τ ) is studied in [Kit09, SRFL08] andsummarized as the celebrated periodic table.(2) More generally, let S be a d -dimensional crystallographic group andlet ( A , φ, c, τ ) be as above. Set G := S × A . Then ( G, φ, c, τ ) isa d -dimensional twisted crystallographic group. This class includesthe pioneering work of Fu [Fu11] (corresponding to the case that S = Z ⋊ C n , where C n is the group generated by rotation by 2 π/n )and a part of the work of Chiu–Yao–Ryu [CYR13] and Morimoto–Furusaki [MF13] on reflection-invariant topological phases (corre-sponding to the case that S = Z d ⋊ Z , where Z acts on V by areflection).(3) Magnetic space groups, discussed in detail in the next subsection.This class includes the antiferromagnetic topological insulator byMong–Essin–Moore [MEM10]. The classification of topological phaseswith the symmetry of magnetic space groups are studied in [OSS19].2.2. Magnetic space group.
An important and non-trivial class of twistedcrystallographic groups is the magnetic space groups (see [Lif05, Sch84] forexample), which are also called (two-)colour symmetry groups, antisymme-try groups, dichromatic groups, Shubnikov groups, Heesch groups, Opechowski-Guccione groups, etc. They are the groups of symmetries of crystals suchthat two possible values are attached to each site.
WISTED CRYSTALLOGRAPHIC T-DUALITY 9
Definition.
Let us consider the simplest case where the Hilbert space K of internal degrees of freedom is of dimension 1 and equipped with thetrivial Z -grading. In this case, the group Aut qtm ( P K ) is isomorphic to Z ,generated by complex conjugation. Definition 2.6. A magnetic space group is a subgroup of Euc( V ) × Z suchthat:(1) the subgroup Π := G ∩ R d ⊂ R d is a full-rank lattice of translations,(2) The magnetic point group P = G/ Π ⊂ O ( d ) × Z is a finite group.That is, there is a commutative diagram of exact sequences1 / / R d / / ∪ Euc( V ) × Z / / ∪ O ( d ) × Z / / ∪ / / Π / / G / / P / / . In view of the background of our formulation, a symmetry ˜ ϕ = ( ϕ, t ) ∈ Euc( V ) × Z preserves the ‘time’ direction if t is the identity and reversesit if t is the complex conjugation. In other words, the second projection φ = pr Z : Euc( V ) × Z → Z indicates the time-reversal. On the otherhand, the first projection pr Euc( V ) : Euc( V ) × Z → Euc( V ) tells us whatthe induced isometry on V is. From the viewpoint of physics, the two valuesattached to each site can be regarded as the spin directions (up and down),and the magnetic space groups as the symmetries of such spin systems oncrystals. One can also think of the possible two values as the time directions(future and past), and the magnetic space groups as the symmetries ofcrystals which possibly contain the time-reversal symmetry.For a magnetic space group G , the subgroup S := j ( G ) ⊂ Euc( V ) iscalled an associated space group of G . This terminology is justified by thefollowing lemma. Lemma 2.7.
A discrete subgroup G of Euc( V ) × Z is a magnetic spacegroup if and only if S := j ( G ) is a space group.Proof. It suffices to show that Π ⊂ R d is of full-rank if and only if S ∩ R d ⊂ R d is of full-rank. To see this, consider the restriction of j to a surjection G ∩ ( R d × Z ) → S ∩ R d . Since G ∩ ( R d × Z ) includes Π as a finite indexsubgroup, the ranks of Π and S ∩ R d are the same. (cid:3) Type classification and cohomology.
By means of their associated data φ and S , we can classify the magnetic space groups G into three types, whichis the standard formulation in the literature [Lif05, Sch84].To state this, let G := ker φ , let Π := G ∩ Π and let P := G / Π .Moreover, we think of G as a subgroup of S × Z ⊂ Euc( V ) × Z . Thequotient ( S × Z ) /G is either Z or trivial. In the former case, we define thehomomorphism ρ : S × Z → Z as the quotient S × Z → ( S × Z ) /G ∼ = Z .In the latter case, set ρ ≡
1. We write as ρ S := ρ | S and ρ Z := ρ | Z . Notethat G is now characterized as the kernel of ρ = ρ S · ρ Z . Proposition 2.8 ([Sch84]) . Let G be a d -dimensional magnetic group, φ : G → Z its time-reversal indicator, and S the associated space group. Then G isclassified into one of the following three types: (a) (black group) G satisfies one of the following three equivalent condi-tions: (a1) φ is trivial. (a2) ρ S = 1 and ρ Z = id Z . (a3) Π = Π and P = P . (b) (grey group) G satisfies one of the following three equivalent condi-tions: (b1) G = S × Z and φ is the second projection. (b2) ρ S = 1 and ρ Z = 1 . (b3) Π = Π and P = P × Z . (c) (black and white group) G satisfies one of the following three equiv-alent conditions: (c1) G = { ( s, ρ S ( s )) ∈ Euc( V ) × Z | s ∈ S } . (c2) ρ S = 1 and ρ Z = id Z . (c3) Either of the following two conditions holds: (c-i) Π = Π and P is an index normal subgroup of P . (c-ii) Π is an index subgroup of Π and P ∼ = P . Notice that the characterization of types in the above proposition omitsthe case of ρ S = 1 and ρ Z = 1. In this case, we get a magnetic space groupof ‘black type’ (i.e. a space group without time-reversal) whose associatedspace group is an index 2 subgroup of S . Therefore this case is covered inthe case of (a), though the associated subgroup may be different from S .Notice also that, in the characterization (c) of black and white groups,the non-trivial homomorphism ρ S specifies a subgroup ker ρ S ⊂ S of index2. Any subgroup S ′ ⊂ S of index 2 is uniquely characterized as the kernelof a non-trivial homomorphism S → Z . Thus, a black and white group ischaracterized by an index 2 subgroup of S . This characterization is adoptedin [Sch84].Now we provide a classification of magnetic space groups via group coho-mology. Proposition 2.9.
The following hold: (1)
There is only one black and one grey magnetic space group for eachspace group S . (2) For a given space group S whose point group is P , the set of blackand white groups of type (c-i) with j ( G ) = S corresponds one-to-oneto H ( P ; Z ) \ { } . (3) For a given space group G with the point group P , the set of blackand white groups of type (c-ii) with ker ρ S = G corresponds one-to-one to a subset of ˜ H P ( ˆΠ ; Z ) \ { } . WISTED CRYSTALLOGRAPHIC T-DUALITY 11
Proof.
The claim (1) is obvious. To see (2), recall that a black and whitegroup of type (c-i) is obtained from a non-trivial homomorphism ρ P : P → Z as G := ker( ρ S , id Z ), corresponding one-to-one to a non-zero element[ ρ P ] ∈ H ( P ; Z ).We show (3). Let G be a magnetic space group of type (c-ii) such thatker ρ S = ker φ is the given space group S . Then, by taking the Pontrja-gin duals, the P -equivariant extension 1 → Π → Π → Z → P -equivariant Z -Galois covering Z → ˆΠ → ˆΠ (in other words a P -equivariant principal Z -bundle) whose restriction at 0 ∈ ˆΠ is equivariantlytrivial. Hence it corresponds to an element of the reduced equivariant coho-mology group ˜ H P ( ˆΠ ; Z ) ∼ = ker( ι ∗ ), where ι : { } → ˆΠ denotes the inclu-sion. Moreover, if there is another magnetic space group G ′ with ker φ = S corresponding to the same element, then there is a commutative diagram of P -equivariant homomorphisms H ( ˆΠ ; Z ) ∼ = Π u u ❥❥❥❥❥❥❥ * * ❚❚❚❚❚❚❚ H ( ˆΠ; Z ) ∼ = Π ∼ = / / H ( ˆΠ ′ ; Z ) ∼ = Π ′ such that the skew homomorphisms are inclusions Π → Π and Π → Π ′ .Hence we obtain that Π and Π ′ coincide in R d ∼ = Π ⊗ Z R , and hence G = G ′ . (cid:3) We say that two magnetic space groups G and G are equivalent if thereis a (not necessarily isometric) affine transform ϕ : V → V preserving theorientation of V such that Ad( ϕ ) G = G . For a space group S , set N S := N R d ⋊ GL d ( R ) ( S ) ∩ ( R d ⋊ GL + d ( R )) , namely the normalizer subgroup of S taken in R d ⋊ GL + d ( R ). To summarizethe above discussion, we obtain: Proposition 2.10.
For a d -dimensional space group S , we write as Π S := S ∩ R d and P S := S/ Π S . The following hold: (1) The set of equivalence classes of d -dimensional black and white mag-netic space groups of type (c-i) is given by G S ( H ( P S ; Z ) \ { } ) /N S , where S runs over all d -dimensional space groups. (2) The set of equivalence classes of d -dimensional black and white mag-netic space groups of type (c-ii) is given by a subset of G G ( ˜ H P G ( ˆΠ G ; Z ) \ { } ) /N G , where G runs over all d -dimensional space groups. We apply this proposition to the enumeration of magnetic space groups indimensions 1 and 2. The calculation of the equivariant cohomology groups H P (pt; Z ) and ˜ H P ( ˆΠ; Z ) for each space group S in dimension 2 is givenin [Gom17b]. By the method of calculations in this paper, one can alsocalculate the equivariant cohomology groups in question for the space groupsin dimension 1. The results are listed in Table 1 and Table 2 below.Space group S P H P ( ˆΠ; Z ) H P (pt; Z ) ˜ H P ( ˆΠ; Z ) Z Z Z Z ⋊ O (1) O (1) Z ⊕ Z Z Table 1.
The list of H P ( ˆΠ; Z ) in dimension 1Space group S P H P ( ˆΠ; Z ) H P (pt; Z ) ˜ H P ( ˆΠ; Z ) p1 Z ⊕ Z ⊕ p2 Z Z ⊕ Z Z ⊕ p3 Z p4 Z Z ⊕ Z Z p6 Z Z Z pm / pg D Z ⊕ Z Z ⊕ cm D Z ⊕ Z Z pmm / pmg / pgg D Z ⊕ Z ⊕ Z ⊕ cmm D Z ⊕ Z ⊕ Z p3m1 D Z Z p31m D Z Z p4m / p4g D Z ⊕ Z ⊕ Z p6m D Z ⊕ Z ⊕ Table 2.
The list of H P ( ˆΠ; Z ) in dimension 2. We follow[Sch84] for the labels of the 2-dimensional space groups. Inthe column of the point group P , the cyclic group Z n is asubgroup of SO(2) ⊂ O(2), and the dihedral group D n SO(2) of order 2 n contains a reflection.In the case of dimension 1, there are 1 magnetic space group of type (c-i)and 2 magnetic space groups of type (c-ii).In the case of dimension 2, there are 29 non-trivial elements of H P (pt; Z )in total. Among them, there are three degenerations of magnetic spacegroups of type (c-i) by the action of the normalizer in pmm , pgg and cmm space groups. All of these groups are invariant under the adjoint by therotation by π/
2. The induced action on the point group P ∼ = Z × Z is theflip, and hence the induced action onto H P (pt; Z ) ∼ = Z ⊕ Z is also the flip.Also, there are 26 non-trivial elements of the reduced cohomology group˜ H P ( ˆΠ; Z ) in total. Among them, 5 non-trivial elements do not contribute WISTED CRYSTALLOGRAPHIC T-DUALITY 13 to the enumeration of the magnetic space group of type (c-ii); 3 comesfrom the pgg space group and 1 from each of p4m and p4g groups. Indeed,these groups are maximal among 2-dimensional space groups. Moreover,the pmm space group admits the adjoint action by the π/ H P ( ˆΠ; Z ) ∼ = Z ⊕ as the flip, which identifies twonon-trivial elements.These discussions are consistent with the result in [Sch84] that there are 26type (c-i) and 20 type (c-ii) magnetic space groups in dimension 2. Similarly,Proposition 2.9 is consistent with the existence of 2 + 2 + (1 + 2) = 7 friezegroups and 17 + 17 + (26 + 20) = 80 layer groups [KL02], which are so-calledsubperiodic groups in one and two dimensions. Some of their associatedtwisted K -theories were investigated in [GT19].3. Freed–Moore twisted equivariant K-theory
In this section we summarize the foundations of Freed–Moore twistedequivariant K-theory. We recall two definitions, i.e., the Fredholm andKaroubi pictures, and introduce several fundamental operations such as thepull-back, the open embedding, the product, the Thom isomorphism andthe push-forward. Although the twisted equivariant K-group is defined forproper groupoids, here we concentrate on action groupoids X ⋊ P , where P is a compact group and X is a locally compact P -space. We also reviewthe relation between the classification of topological phases protected by thesymmetry of a twisted crystallographic group and the twisted equivariantK -group of the Brillouin torus proposed in [FM13].3.1. Two definitions.
We start with a reminder of the definition of thetwisted equivariant K-groups. For a detailed discussion, we refer to [Gom17a]and [Kub16].We say that a twist of the action P y X is a triplet ( φ, c, σ ), where φ, c are elements of the equivariant cohomology group H P ( X ; Z ) and σ ∈ H P ( X ; φ T ) (note that the cohomology groups are not compactly-supportedones). For details of groupoid cohomology, refer to Appendix A. Remark . In this section, we use the letter σ for a 2-cocycle in H P ( X ; φ T ).As a general convention, a left-superscript φ indicates that complex conju-gation is to be applied whenever φ ( p, x ) = 1. For simplicity of notation, inthis section we represent 1-cocycles φ and c by a function X × P → Z as φ ( p, x ) or c ( p, x ) (although it is possible only for 1-cocycles which are sentto the trivial element in the non-equivariant cohomology by the forgetfulmap).For a fixed choice of φ , the summation on H P ( X ; Z ) × H P ( X ; φ T ) isimposed as ( c, σ ) + ( c ′ , σ ′ ) := ( c + c ′ , σ + σ ′ + ǫ ( c, c ′ )) , (3.2) where ǫ ( c, c ′ ) is a 2-cocycle defined by ǫ ( c, c ′ )( p, q, x ) := ( − c ′ ( p,x ) c ( q,x ) for any x ∈ X and p, q ∈ P . Remark . A 1-cocycle c ∈ H P ( X ; Z ) corresponds to a P -equivariantprincipal Z -bundle on X . Also, a 2-cocycle σ ∈ H P ( X ; φ T ) corresponds toa φ -twisted extension of the action groupoid X ⋊ P (cf. [Gom17a, Definition2.3]), comprising the following data; • a collection { L p } p ∈ P of complex line bundles on X , • a collection of bundle isomorphisms σ ( p, q ) : α ∗ q − L p ⊗ φ ( p ) L q → L pq , i.e., a continuous family σ ( p, q ) x : L p,qx ⊗ φ ( p,qx ) L q,x → L pq,x , withthe compatibility condition α ∗ r − σ ( p, q ) ◦ σ ( pq, r ) = φ ( p ) σ ( q, r ) ◦ σ ( p, qr ) . Here L p,x denotes the fiber of L p at x ∈ X . Remark . For simplicity of notation, hereafter we often use the letter t for a pair ( c, σ ) ∈ H P ( X ; Z ) × H P ( X ; φ T ).For a twist ( φ, t ) of a P -space X , we say that a ( φ, t )-twisted P -equivarianthermitian vector bundle on X is a Z -graded hermitian vector bundle E on X with a collection of C -linear bundle isomorphisms u p : L p ⊗ φ ( p ) E → α ∗ p − E, i.e., a continuous family u p,x : L p,x ⊗ φ ( p,x ) E x → E px , such that • u p,x : L p,x ⊗ φ ( p,x ) E x → E px is even/odd if c ( p ) = 0 /
1, and • u is σ -projective, i.e., the equality u p ◦ u q = u pq ◦ (id E ⊗ σ ( p, q ))holds for p, q ∈ P as bundle maps α ∗ q − L p ⊗ φ ( p ) L q ⊗ φ ( pq ) E → α ∗ ( pq ) − E. In other words, u p is a continuous family of φ -linear c -graded maps u p,x : E x ⊗ L p,x → E px such that u p,qx ◦ u px = u pq,x ◦ (id E ⊗ σ ( p, q ) x ) . Note that, for a fixed φ ∈ H P ( X ; Z ), the graded tensor product E ˆ ⊗ E of ( φ, t i )-twisted P -equivariant hermitian vector bundles E i (for i = 1 , φ, t + t )-twisted P -equivariant hermitian vector bun-dle. Here t + t is the summation (3.2). This is checked as( u p ˆ ⊗ u p )( u q ˆ ⊗ u q )=( − c ( p,x ) c ( q,x ) ( u p u q ) ˆ ⊗ ( u p u q )=( − c ( p,x ) c ( q,x ) ( u pq ˆ ⊗ u pq )(id E ˆ ⊗ E ⊗ σ ( p, q ) ⊗ σ ( p, q )) . WISTED CRYSTALLOGRAPHIC T-DUALITY 15
Remark . A ( φ, t )-twisted P -equivariant Hilbert bundle H is said to beuniversal if any ( φ, t )-twisted vector bundle on X is embedded into H . Auniversal bundle satisfies the following properties;(1) Any P -space X admits a universal Hilbert P -bundle (cf. [FHT11,Corollary A.33]).(2) Any two universal Hilbert bundles are unitarily isomorphic (cf. [FHT11,Lemma A.26]).(3) A universal bundle has the following absorbing property (cf. [FHT11,Lemma A.24]); for any ( φ, t )-twisted Hilbert bundle V , there is aneven P -invariant unitary isomorphism V : H ⊕ V → H . Let B ( H ), K ( H ) and U ( H ) denote the bundle of bounded, compact andunitary operators on H respectively. Also, let Fred( H ) denote the bundle on X whose fiber at x ∈ X is the set of odd self-adjoint operators F x ∈ B ( H x )such that F x − ∈ K ( H x ), on which we put the weakest topology such thata local section F is continuous if F is strongly continuous and F − Definition 3.6.
We write Γ c ( X, Fred( H )) P for the topological space of con-tinuous sections F ∈ Γ( X, Fred( H )) such that(1) F − X and(2) [ u p , F ] = 0 holds (where [ , ] denotes the graded commutator), thatis, u p,x F x u ∗ p,x = ( − c ( p,x ) F px holds for any ( x, p ) ∈ X × P .The twisted equivariant K -group φ K , t P ( X ) is defined to be the set of con-nected components of Γ c ( X, Fred( H )).For the definition of the K-group in other degrees, a symmetry of Cliffordalgebras is taken into account. Remark . Let Cl p,q denote the Clifford algebra (it is customary to use p, q for the integer labels in Cl p,q , and it should not be confused with our use of p, q ∈ P ) associated to R p + q with the bilinear form h· , ·i of signature ( p, q ),that is, the R -algebra generated by elements v for v ∈ R p + q with relations vw + wv = h v, w i
1. Let ∆ p,q denote the real linear space Cl p,q with theinner product h a, b i = tr( b ∗ a ), where tr is the unique tracial state on Cl p,q which is even, i.e., tr( Cl odd p,q ) = 0. Let c : Cl p,q → B (∆ p,q ) denote the graded ∗ -representation given by the multiplication from the left.In this paper we use the same symbols Cl p,q and ∆ p,q for the correspond-ing Real C*-algebra and its Real ∗ -representation (see Remark B.2). Inparticular, G acts on Cl p,q and ∆ p,q by their complex conjugations through φ . Let H p,q := H ˆ ⊗ ∆ p,q and let Fred p,q ( H ) denote the subbundle of Fred( H p,q )consisting of operators such that [ F, c ( v )] = 0. Also, we write Γ c ( X, Fred p,q ( H )) P for the space of continuous sections of Fred p,q ( H ) satisfying (1), (2) of Defi-nition 3.6. Definition 3.8.
We define the twisted equivariant K-group as φ K q − p, t P ( X ) := π (Γ c ( X, Fred p,q ( H )) P ) . This is an abelian group under the direct sum[ F ] + [ F ] := [ V ( F ⊕ F ) V ∗ ] , where V : H p,q ⊕ H p,q → H p,q is a P -equivariant unitary intertwining the Cl p,q -actions.Note that the above summation is well-defined independent of the choiceof V because of the following lemma. Here we write U p,q ( H ) and U p,q ( H )for the fiber bundle of even/odd unitaries on H p,q which graded commuteswith the action of Cl p,q respectively. We consider the space Γ( X, U ip,q ( H )) P of graded P -invariant sections of these bundles. Lemma 3.9.
The spaces Γ( X, U p,q ( H )) P and Γ( X, U p,q ( H )) P are weaklycontractible. For a proof, see [FHT11, Lemma A.36] and [Gom17a, Lemma 3.2]. Weremark that Γ( X, U p,q ( H )) P is a subspace of Γ c ( X, Fred p,q ( H )) P . Remark . We mention a relation of Definition 3.8 with the definitionsgiven in references [Gom17a] and [Kub16]. In this paper, we used the setof self-adjoint Fredholm operators as in [Kub16, Definition 3.10], instead ofskew-adjoint Fredholm operators as in [Gom17a, Definition 3.4]. Indeed, aself-adjoint Fredholm operator is more compatible with the Kasparov theory.In this remark τ denotes a cocycle H P ( X, φ T ) in the terminology of[Gom17a]. It is mentioned in [Gom17a, Remark 4.12] that, the twistedequivariant K-group φ K ( c,τ ) ,q − pP ( X ) given in [Gom17a, Definition 3.4] is iso-morphic to another group φ ´K ( c, ´ τ ) ,p − qP ( X ), where ´ τ := τ + ǫ ( c, c ). The lattergroup is the same thing as our φ K q − p,c, ´ τP ( X ). That is, the convention of τ in this paper is shifted by ǫ ( c, c ) from [Gom17a]. We also remark that therole of p, q in the notation of the Clifford algebra Cl p,q in this paper and[Gom17a] are opposite, i.e., in [Gom17a] the algebra Cl p,q is generated bymutually anticommuting elements e , · · · , e p with e i = − f , · · · , f q with f i = 1.There is another description of the twisted equivariant K-group as atwisted generalization of Karoubi’s K-theory. Here we say that a ( φ, c, σ )-twisted P -equivariant Cl p,q -vector bundle is a Z -graded hermitian vectorbundle E on X with a graded ∗ -representation c : Cl p,q → B ( E ) and a φ -linear σ -twisted P -action u such that u p is even/odd if c ( p, x ) = 0 / c ( v ) , u p ] = 0 for any v ∈ Cl p,q . Definition 3.11 ([Gom17a, Definition 4.13]) . Let X be a locally com-pact P -space. A ( φ, c, σ )-twisted P -equivariant Cl p,q -triple on X is a triple( E, γ , γ ), where • E is a ( φ, σ )-twisted P -vector bundle on X of finite rank, WISTED CRYSTALLOGRAPHIC T-DUALITY 17 • E is equipped with a fiberwise ∗ -representation of Cl p,q , • each γ i is a Z -grading on E which makes E into a ( φ, c, σ )-twisted P -equivariant Cl p,q -vector bundle on X , and • γ − γ is compactly supported.Let φ K q − p,c,σP ( X ) denote the quotient of the monoid of homotopy classes of( φ, c, σ )-twisted P -equivariant triples on X by its submonoid consisting oftriples of the form [ E, γ, γ ].This notation is justified by the fact that the group φ K q − p,c,σP ( X ) dependsonly on the difference q − p as is shown in [Gom17a, Subsection 4.3].It was proved in [Gom17a, Theorem 4.11, Theorem 4.20] (a similar resultis also shown in [Kub16, Theorem 5.14]) that there is an isomorphism ϑ : φ K q − p, t P ( X ) → φ K q − p, t P ( X ) . The map is described in Definition 4.10, Lemma 4.18, Lemma 4.19 of [Gom17a](compare with Remark 3.10).3.2.
Operations in twisted equivariant K-theory.
Next we introduceseveral operations appearing in twisted equivariant K-theory. They are com-pared with the corresponding operations in Kasparov’s KK-theory in Section5.3.2.1.
Pull-back.
Let X and Y be P -spaces and let ( φ, c, σ ) be a twist of Y . Let f : X → Y be a P -equivariant continuous map. The pull-back of( φ, c, τ ) by f is defined to be f ∗ ( φ, c, σ ) := ( f ∗ φ, f ∗ c, f ∗ σ ), where f ∗ φ, f ∗ c ∈ H P ( X ; Z ) and f ∗ σ ∈ H P ( X ; f ∗ φ T ). Let t := ( c, σ ).Let H X and H Y denote the universal f ∗ ( φ, t )-twisted P -equivariant Hilbertbundle on X and the universal ( φ, t )-twisted P -equivariant Hilbert bundle Y respectively. Consider the pull-back f ∗ H Y , which is equipped with acanonical f ∗ ( φ, t )-twisted action of P . Moreover, the pull-back also inducesa continuous map f ∗ : Γ( Y, Fred( H )) P → Γ( X, Fred( f ∗ H )) P . We define the pull-back f ∗ : φ K ∗ , t P ( Y ) → f ∗ φ K ∗ ,f ∗ t P ( X )as f ∗ [ F ] = [ V ( f ∗ F ⊕ G ) V ∗ ] , (3.12)where V : f ∗ H Y ⊕ H X → H X is a P -equivariant even unitary respecting the Cl p,q -action (which exists by Remark 3.5 (3)) and G ∈ Γ( X ; U p,q ( H X )) P .This definition is independent of the choice of V and G by Lemma 3.9. Open embedding.
Let X be a compact P -space and let Y be a P -invariant closed subspace. Let ( φ, c, σ ) be a twist on P y X and let H bethe universal ( φ, c, σ )-twisted Hilbert bundle of P y X . Let t := ( c, σ ).Let F ∈ Γ c ( X \ Y, Fred p,q ( H )) P . Then there is an open subspace U ⊂ X including Y such that F | U \ Y is a unitary-valued section. Since the spaceΓ( U \ Y, U p,q ( H )) P is contractible, there is a continuous path F t ∈ Γ( U \ Y, U p,q ( H )) P such that F = F | U \ Y and F extends to a section on U . Let ρ be a [0 , X supported on U and ρ | Y ≡ F ′ ( x ) := ( F ( x ) x ∈ X \ U,F ρ ( x ) ( x ) x ∈ U, is a section in Γ( X, Fred p,q ( H )) P , which is homotopic to F in Γ c ( X \ Y, Fred p,q ( H )) P .Let ι : X \ Y → X denote the inclusion. Now ι ∗ [ F ] := [ F ′ ] determines thewell-defined open embedding map ι ∗ : φ K p − q, t P ( X \ Y ) → φ K p − q, t P ( X ) . Here we use the same letter t for its restriction to X \ Y for simplicity ofnotation.3.2.3. External and internal products.
Let P be a compact group, let Z bea compact P -space and let φ ∈ H P ( Z ; Z ). Let X and X be P -spaces over Z , that is, X i are equipped with a continuous P -equivariant map f i : X i → Z . Then the fiber product space X × Z X is also a P -space over Z by f ( x , x ) := f ( x ). For i = 1 ,
2, we consider twists of X i of the form( f ∗ i φ, c i , σ i ). Let t i := ( c i , σ i ). Then there is an external product operationas · ⊗ Z · : f ∗ φ K n , t G ( X ) ⊗ f ∗ φ K n , t G ( X ) → f ∗ φ K n + n , t + t G ( X × Z X ) . Here, for simplicity of notation we use the same letter for a twist t i on X i and its pull-back to X × Z X .In the Fredholm picture of twisted equivariant K-theory as in Definition3.6, this homomorphism is defined in the following way. Firstly, consider thegraded tensor product bundle H ˆ ⊗ Z H , i.e., the bundle H ⊗ Z H with the Z -grading γ ⊗ γ . For a pair of (resp. anti) linear bundle maps T : H → H and S : H → H , T ˆ ⊗ S stands for the graded tensor product T ⊗ Z S + T γ ⊗ Z S , where S , S denotes the even and odd parts of S . Now, the collection ofbundle maps u p ˆ ⊗ Z u p : H ˆ ⊗ Z H → H ˆ ⊗ Z H determines the structure of a φ -linear ( t + t )-twisted P -equivariant bundlesince ( u p γ c ( p )1 ⊗ u p )( u q γ c ( q )1 ⊗ u q )=( − c ( p ) c ( q ) σ ( p, q ) σ ( p, q )( u pq γ c ( pq )1 ⊗ u pq ) . WISTED CRYSTALLOGRAPHIC T-DUALITY 19
When the Clifford algebra symmetry is taken into account, we canonicallyidentify Cl p ,q ˆ ⊗ Cl p ,q with Cl p,q where p := p + p and q := q + q . Thisidentification gives rise to isomorphisms ∆ p ,q ˆ ⊗ ∆ p ,q ∼ = ∆ p,q and( H ˆ ⊗ ∆ p ,q ) ˆ ⊗ ( H ) ˆ ⊗ ∆ p ,q → H ˆ ⊗ Z H ˆ ⊗ ∆ p,q . Lemma 3.13.
Let F i ∈ Γ c ( X i , Fred p i ,q i ( H i )) P for i = 1 , . Then the sec-tions (1) F ˆ ⊗ Z (1 − F ) / + 1 ˆ ⊗ Z F , (2) ( F ˆ ⊗ ⊗ F ) · χ ( F ˆ ⊗ ⊗ F ) , where χ ( x ) := max { , | x | − / } ,both lie in Γ c ( X × Z X , Fred p,q ( H ˆ ⊗ H )) P , and are mutually homotopic.Proof. Since F ˆ ⊗ ⊗ F , the functional calculus f ( F ˆ ⊗ , ⊗ F )is defined for any continuous function f : [0 , → C . In general, if we havea pair of continuous functions ( f , f ) such that f i : [0 , → [0 ,
1] such that xf ( x, y ) + yf ( x, y ) ≡ , × { } ∪ { } × [0 , F ˆ ⊗ · f ( F ˆ ⊗ , ⊗ F ) + (1 ˆ ⊗ F ) · f ( F ˆ ⊗ , ⊗ F )lies in Γ c ( X × Z X , Fred( H ˆ ⊗ H )) P . Indeed,1 − (( F ˆ ⊗ · f ( F ˆ ⊗ , ⊗ F ) + (1 ˆ ⊗ F ) · f ( F ˆ ⊗ , ⊗ F )) =1 − ( F ˆ ⊗ f ( F ˆ ⊗ , ⊗ F ) − (1 ˆ ⊗ F ) f ( F ˆ ⊗ , ⊗ F ) =(1 − xf − yf )( F ˆ ⊗ , ⊗ F ) , (3.14)is compact at any fiber since the essential joint spectrum of ( F ˆ ⊗ , ⊗ F )lies in [0 , × { } ∪ { } × [0 , x , x ) ∈ X × Z X either F at x or F at x is aunitary.Now, we obtain (1) if we choose ( f , f ) as f ( x, y ) = (1 − y ) / and f ∼ = 1.Also, if we choose ( f , f ) as f ( x, y ) = f ( x, y ) = χ ( x + y ), then we obtain(2). This finishes the proof since the space of functions ( f , f ) as above isconnected. (cid:3) Let H denote the universal ( φ, c, σ )-twisted P -equivariant Hilbert bundleon X × Z X . Definition 3.15.
We define the external product φ K n , t P ( X ) ⊗ φ K n , t P ( X ) → φ K n + n , t + t P ( X × Z X )by [ F ] ⊗ Z [ F ] := [ V (( F ˆ ⊗ Z (1 − F ) / + 1 ˆ ⊗ F ) ⊕ G ) V ∗ ]= [ V (( F ˆ ⊗ ⊗ F ) · χ ( F ˆ ⊗ ⊗ F ) ⊕ G ) V ∗ ] , where V : ( H ,p ,q ˆ ⊗ Z H ,p ,q ) ⊕ H p,q → H p,q is an even P -equivariant uni-tary commuting with Cl p,q -action and G ∈ Γ( X × Z X , U p,q ( H )) P . Partic-ularly, in the case of X = Z = X = X , the homomorphism φ K n , t P ( X ) ⊗ φ K n , t P ( X ) → φ K n + n , t + t P ( X ) is called the internal (or cup) product.This is well-defined independent of the choice of V . Moreover, this mapis obviously additive in the first and second components. Lemma 3.16.
The map given in Definition 3.15 satisfies the following prop-erties: (1) (associative) ([ F ] ˆ ⊗ [ F ]) ˆ ⊗ [ F ] = [ F ] ˆ ⊗ ([ F ] ˆ ⊗ [ F ]) . (2) (graded commutative) [ F ] ˆ ⊗ [ F ] = ( − n f ∗ [ F ] ˆ ⊗ [ F ] , where f : X × Z X → X × Z X is the flip and n := ( q − p )( q ′ − p ′ ) . By this lemma, the internal product induces a graded commutative ringstructure on L n, t φ K n, t P ( X ). Proof.
For simplicity of notation, in the proof of this lemma, we use thesame letter F for the tensor product F ˆ ⊗ F , F are also used in asimilar fashion). To see (1), note that ( F + F ) χ ( F + F ) = ψ ( F + F ),where ψ ( t ) := t max { , | t | − } . Then the homotopy˜ F t := ψ ( t ( F + F ) + (1 − t ) ψ ( F + F ) + F )connects ψ ( ψ ( F + F ) + F ) with ψ ( F + F + F ) in Γ c ( X, Fred p,q ( H )) P .The same argument also connects ψ ( F + ψ ( F + F )) with ψ ( F + F + F ).The claim (2) follows from the commutativity of the diagram Cl p,q ˆ ⊗ Cl p ′ ,q ′ / / flip (cid:15) (cid:15) Cl p + p ′ ,q + q ′ Ad( x ) (cid:15) (cid:15) / / B (∆ p + p ′ ,q + q ′ ) Ad( c ( x )) (cid:15) (cid:15) Cl p ′ ,q ′ ˆ ⊗ Cl p,q / / Cl p ′ + p,q ′ + q / / B (∆ p + p ′ ,q + q ′ ) , where x = v · · · v k ∈ Cl p,q is a product of vectors in V such that thecomposition R v · · · R v k , where R v denotes the reflection along v , is thelinear automorphism θ : R p + p ′ ,q + q ′ → R p ′ + p,q ′ + q exchanging the basis. Notethat k is odd if and only if θ changes the orientation, i.e., ( p + q )( p ′ + q ′ ) ≡ n is odd. Now f ∗ ˆ ⊗ c ( x ) : ( H ˆ ⊗ Z H ) ˆ ⊗ ∆ p + p ′ ,q + q ′ → ( H ˆ ⊗ H ) ˆ ⊗ ∆ p ′ + p,q ′ + q implements the map between the spaces of Fredholm sections, which maps F to ( − n f ∗ F . (cid:3) Remark . In this paper we only consider the case that Z = pt, in otherwords, φ is the pull-back of an element of H P (pt; Z ) (denoted by the sameletter φ ). In this case the external product φ K n , t P ( X ) ⊗ φ K n , t P ( X ) → φ K n + n , t + t P ( X × X )is defined. Moreover, if φ is chosen in this way, the product given in Defini-tion 3.15 satisfies [ F ] ⊗ Z [ F ] = ∆ ∗ Z ([ F ] ˆ ⊗ [ F ]) , where ∆ Z : X × Z X → X × X is the inclusion. WISTED CRYSTALLOGRAPHIC T-DUALITY 21
Thom isomorphism.
The equivariant Bott periodicity and the Thomisomorphism is discussed in [Gom17a, Subsection 3.6]. Here we describe theThom isomorphism map by means of the product introduced in 3.2.3 for ourconvenience.Let q : E → X be a P -equivariant real vector bundle on a P -space X ofrank r . For simplicity, here we only consider the case that an orientationand a spin structure on E , which is not necessarily P -equivariant, is fixed.Let SO( E ) and Spin( E ) denote the corresponding principal bundles.Set E ′ := E ⊕ R n − r , where R n − r denotes the trivial bundle of rank 8 n − r with the trivial P -action. We associate to E and E ′ the Clifford algebrabundles Cl ( E ) and Cl ( E ′ ) respectively. That is, Cl ( E ) := SO( E ) × SO( r ) Cl r and Cl ( E ′ ) := SO( E ′ ) × SO(8 n ) Cl n . Note that Cl ( E ′ ) is isomorphic to thegraded tensor product Cl ( E ) ˆ ⊗ Cl n − r . Let S denote the spinor bundle of E ′ , i.e., the vector bundle on X whose fiber at x is the unique irreduciblerepresentation of Cl ( E ′ ), i.e., S := Spin( E ) × Spin(8 n ) ∆.The R -linear actions of P onto E and E ′ give rise to algebra actions onto Cl ( E ′ ), and hence a ( ν, µ )-twisted action onto S in the following way. Let ν ∈ H P ( X ; Z ) be the 1-cocycle corresponding to the orientation bundleOr( E ) on which P acts canonically. Also, consider the following data; • the collection of Z -bundles { L p } on X defined as L p := G x ∈ X { T ∈ Hom( S x , S px ) | Ad( T ) = α p } , • the unitary µ ( p, q ) : L p ⊗ L q → L pq given by the composition, and • the unitary u p : S ⊗ L p → α ∗ p S given by ( ξ, T ) T ξ .Then (
L, µ ) corresponds to a 2-cocycle µ ∈ H P ( X ; Z ). Let us use the sameletter µ for its image in H P ( X ; φ T ) through the homomorphism inducedfrom the inclusion of local systems Z ⊂ φ T . Then u determines a ( φ, ν, µ )-twisted P -action on S . We write as v := ( ν, µ ). Note that ν , µ are thesame thing as the P -equivariant Stiefel-Whitney classes w P ( E ) and w P ( E )respectively.Let ξ : E → q ∗ E denote the section defined by ξ ( e ) := e for any e ∈ E x .Let C denote the bounded section C := c ( ξ )(1 + k ξ k ) − / ∈ Γ( E, q ∗ Cl ( E ′ )) , which acts on the bundle q ∗ S on E by the fiberwise multiplication. Moreover[ C, c ( v )] = 0 for any v ∈ Cl n − r, . Let H be a universal ( φ, − v )-twisted P -equivariant bundle on E . We define the Bott element β E := [ V ( C ⊕ G ) V ∗ ] ∈ φ K r, v P ( E ) , (3.18)where V : π ∗ S ⊕ H n − r, → H n − r, is a P -equivariant even unitary respect-ing the Cl n − r, -action and G ∈ Γ( E, U n − r, ( H )) P . (Here we omit 8 n inthe degree of the K-group). Definition 3.19.
The Thom isomorphism map is defined by the cup prod-uct over X asThom E := β E ⊗ X : φ K ∗ , t P ( X ) → φ K ∗ + r, t + v P ( E ) . By Remark 3.17 the Thom isomorphism is written asThom E ( x ) = ∆ ∗ E ( x ⊗ β E ) , (3.20)where ∆ E := (id E , q ) : E → E × X . Proposition 3.21.
For any real P -equivariant vector bundle E on X , thehomomorphism Thom E is an isomorphism.Proof. If E is the trivial bundle X × V , where V is a real representation of P ,then Thom X × V is the external product with β V ∈ φ K r, v P ( V ) over the point.It will be proved in Lemma 5.6 that this is an isomorphism by using theKasparov theory. Now a Mayer–Vietoris argument concludes the proof. (cid:3) Push-forward.
Let M and N be P -manifolds and let f : M → N bea P -equivariant smooth map. We choose a spin representation V of P anda P -equivariant embedding j : M → V . Let E denote the normal bundle ofthe embedding ( j, f ) : M → V × N .Set v = ( µ, ν ) := ( w P ( E ) , w P ( E ))= f ∗ ( w P ( N ) , w P ( N )) − ( w P ( M ) , w P ( M )) . Note that f is said to be spin-oriented if v = 0. Definition 3.22.
We define the push-forward by π as π ! := Thom − N × V ◦ ι ∗ ◦ Thom E : φ K ∗ ,f ∗ t − v P ( M ) → φ K −∗ , t P ( N ) . It is seen in the same way as [AS68, Section 3] that the push-forward isindependent of the choice of j .3.3. Classification of gapped topological phases.
In condensed matterphysics, a (non-interacting) quantum system on a lattice with space groupsymmetry is studied through the Pontrjagin dual of the lattice (Brillouintorus) with an action of the point group. We generalize in this section theaction of the point group on the Pontrjagin dual, to that of the twisted pointgroup.Let (
G, φ, c, τ ) be a twisted crystallographic group in the sense of Defini-tion 2.1. Firstly we define the action of the twisted crystallographic group G on the Pontrjagin dual ˆΠ := Hom(Π , T ) of Π as ρ g ( χ )( t ) := (cid:26) χ ( g − tg ) if φ ( g ) = 1, χ ( g − tg ) if φ ( g ) = 0,(3.23)for χ ∈ ˆΠ and g ∈ G . Since ρ t is trivial for any t ∈ Π, this ρ is reduced tothe action of the twisted point group P . WISTED CRYSTALLOGRAPHIC T-DUALITY 23
We also define the 2-cocycle σ ∈ H P ( ˆΠ , φ T ) in the following way: Let L p be the trivial bundle C over ˆΠ. Let us fix a set-theoretic section s : P → G and define the continuous map σ : P × P → End( C ) as σ ( χ, p, q ) := χ ( s ( p ) s ( q ) s ( pq ) − )(3.24)for any p, q ∈ P . It is straightforward to check that this σ is a 2-cocycle ofthe action ρ : P y ˆΠ with coefficients in φ T .This ( ρ, σ ) is related to unitary representations of G through the Fouriertransform in the following way. Let ( G, φ, c, τ ) be a twisted crystallographicgroup acting on the Euclidean space V and the internal Hilbert space K . Sothere is a φ -twisted projective unitary representation of G on L ( V ) ⊗ K .The full dynamics (Hamiltonian operator which graded-commutes with G )is usually reducible at low energy to an sub-representation space, spannedby an orthonormal set of localized orbitals centered on lattice points thoughtof as atomic positions. Such a basis is called a (localized) Wannier basis, seealso the discussion of atomic insulators in Section 7.3. This is formalized asfollows.Let X ⊂ V be a G -invariant discrete subspace and let us fix a choice X of a fundamental domain of the Π-action onto X , i.e., the image of asection of the quotient map X → X/ Π. Let H denote the Hilbert space ℓ ( X, K ) with the ( φ, c, τ )-twisted unitary representation U := λ ⊗ v of G .Here λ is the regular representation on ℓ ( X ), i.e., ( λ g ξ )( x ) := ξ ( g − x ), and v : G → Aut qtm ( K ) is a lift of k : G → Aut qtm ( P K ). By using this U weidentify H with ℓ (Π , ˜ K ), where ˜ K := ℓ ( X , K ).The canonical basis of ℓ (Π), along with a basis for ˜ K , should be thoughtof as mutually orthogonal wave functions in L ( V ) ⊗ K localized at the pointsof X , and the effective Hilbert space H as a subrepresentation in L ( V ) ⊗ K .When one works at this effective level, the (discrete) Fourier transform givesa unitary isomorphism F : ℓ (Π , ˜ K ) → L ( ˆΠ , ˜ K ) , where the L -space on ˆΠ is defined by the Haar measure normalized asvol( ˆΠ) = 1. Lemma 3.25.
For each g ∈ G , there is a bundle map u g : ˆΠ × ˜ K → ˆΠ × ˜ K covering ρ g such that F U g F ∗ = u g as a φ -linear unitary on L ( ˆΠ , ˜ K ) .Proof. Firstly we remark that, for any t ∈ Π, F U t F ∗ is the multiplicationwith t regarded as a continuous function on ˆΠ by ˆ t ( χ ) := χ ( t ).Let us decompose ˜ K into the direct sum of subspaces K x = ℓ ( { x } , K )where x runs over X . For ξ ∈ L ( ˆΠ , ˜ K ), ξ x denotes its L ( ˆΠ , K x )-componentwith respect to this decomposition. Let λ denote the regular representationof G acting on X/ Π ∼ = X . Then, for any ξ ∈ L ( ˆΠ , ˜ K ) and g ∈ G , we have F U g F ∗ ξ = X x ∈ X U t ( x,g ) · ( λ g ⊗ v g ) ξ x , where t ( x, g ) ∈ Π is the element characterized by gx ∈ t ( x, g ) X . Therefore, u g ( χ, ξ ) := (cid:16) ρ g x, X x ∈ X χ ( t ( x, g )) · ( λ g ⊗ v g ) ξ x (cid:17) is the desired bundle map. (cid:3) Let s : P → G be a set-theoretic section. Then we have u s ( p ) u s ( q ) u − s ( pq ) = σ ( χ, p, q ) · τ ( p, q )for any p, q ∈ P . That is, the collection of bundle maps { u s ( p ) } p ∈ P forms astructure of ( φ, c, τ + σ )-twisted P -equivariant vector bundle on ˆΠ × K .Let H N denote the direct sum of N copies of H , which is isomorphic to ℓ ( X, K N ). For x, y ∈ X and H ∈ B ( H N ), H xy denotes the operator δ y Hδ x ,where δ x denotes the delta function on x , regarded as a bounded operatoron K N . The following lemma is a standard fact in Fourier analysis. Lemma 3.26.
Let H be a bounded operator on H N satisfying the followingtwo properties: (1) There are constants C , C > such that, for any x, y ∈ X , thecoefficient H xy satisfies k H xy k ≤ C e − C d ( x,y ) . (2) The operator H is Π -invariant, i.e., U t HU ∗ t = H for any t ∈ Π .Then F H F ∗ ∈ B ( L ( ˆΠ , ˜ K )) is the multiplication operator with a smoothfunction h : ˆΠ → B ( ˜ K ) . Let H N ( G, φ, t ) denote the set of operators on H N for some N , satisfying(1) of Lemma 3.26 and(2’) H is graded G -invariant, i.e., U g HU ∗ g = ( − c ( g ) H for any g ∈ G ,(3) H is self-adjoint and invertible.We embed H N ( G, φ, t ) into H N +1 ( G, φ, t ) by H (cid:0) H γ (cid:1) and define the setof topological phases with the symmetry ( G, φ, t ) as TP ( G, φ, t ) := (cid:16) [ N H N ( G, φ, t ) (cid:17) / ∼ , where the equivalence relation is given by the homotopy of operators. Theset TP ( G, φ, t ) is obviously related to Karoubi’s picture of the twisted equi-variant K-theory. The following proposition is a slight modification of[FM13, Theorem 10.15] which is essentially proved in [Kub17, Proposition3.4]. Proposition 3.27.
The map [ H ] [ K N , sgn( h ) , γ ] , where h : ˆΠ → B ( K N ) is the continuous function as in Lemma 3.26, is an isomorphism between TP ( G, φ, t ) and φ K , t + σP ( ˆΠ) . In particular, this proposition shows that the twisted equivariant K-group φ K , t + σP ( ˆΠ) is independent of the choice of Π. WISTED CRYSTALLOGRAPHIC T-DUALITY 25
Remark . When φ, c, τ are trivial, a concrete meaning to the class of H being non-trivial is as follows. Namely, the range of the negative-energyprojection − sgn( h )2 cannot be spanned by the Fourier transform of a localizedWannier basis, see [LT20].4. Crystallographic T-duality
In this section we introduce the main research target of the paper, twotwisted equivariant K-groups associated to a twisted crystallographic groupand the crystallographic T-duality homomorphism between them.Let (
G, φ, c, τ ) be a d -dimensional twisted crystallographic group in thesense of Definition 2.1 acting on a d -dimensional Euclidean space V and letΠ := G ∩ R d × { } . Let ρ denote the action of P = G/ Π onto ˆΠ given in(3.23) and let σ ∈ H P ( ˆΠ; φ T ) denote the 2-cocycle given in (3.24). We alsoconsider the P -action onto V /
Π induced from the G -action onto V .Let v = ( µ, ν ) be the P -equivariant Stiefel-Whitney class ( w P ( V ) , w P ( V )) ∈ H P (pt; Z ) ⊕ H P (pt; Z ) of V as a P -bundle over the point. Since the tan-gent bundle T ( V / Π) → V /
Π is P -equivariantly trivial, that is, T ( V / Π) ∼ = V / Π × V , its equivariant Stiefel-Whitney class is the pull-back π ∗ v by thecollapsing map π : V / Π → pt.The crystallographic T-duality map is schematically represented by thediagram V / Π × ˆΠ [ P ] ⊗ (cid:11) (cid:11) ˆ π z z ttttttttt π ❍❍❍❍❍❍❍❍❍❍ V /
Π ˆΠ . Here, π : V / Π → pt and ˆ π : ˆΠ → pt are collapsing maps. Another ingredientis the element [ P ] ∈ φ K ,π ∗ σP ( V / Π × ˆΠ)represented by the Poincar´e line bundle in the following way. Lemma 4.1 ([GT19, Theorem 4.3]) . The φ -twisted G -action on the Poincar´eline bundle P := ( V × ˆΠ × C ) / { ( v, χ, z ) ∼ ( v + t, χ, χ ( t ) z ) } given by g · [ v, χ, z ] = [ gv, χ, φ ( g ) z ] determines a ( φ, , ˆ π ∗ σ ) -twisted P -equivariant vector bundle on V / Π × ˆΠ .Proof. It suffices to show that the Π-action on P is given by the multipli-cation by ˆ π ∗ σ t ∈ C ( V / Π × ˆΠ , T ), where σ t denotes the function ˆ t as before(i.e., the function defined by ˆ t ( χ ) := χ ( t )). This is checked as t · [ v, χ, z ] = [ v + t, χ, z ] = [ v, χ, χ ( t ) z ] = [ v, χ, σ t ( χ ) z ] . (cid:3) The bundle P determines an element of the Freed–Moore K-group (in theKaroubi picture) as [ P , , − ∈ φ K ,π ∗ σP ( V / Π × ˆΠ) , which is simply written as [ P ] hereafter. Definition 4.2.
Let t ∈ H ( P ; Z ) ⊕ H ( P ; φ T ). We call the composition φ T t G := π ! ◦ ([ P ] ⊗ V/ Π × ˆΠ ) ◦ ˆ π ∗ : φ K ∗ + d, t − v P ( V / Π) → φ K ∗ , t + σP ( ˆΠ)(4.3)the crystallographic T-duality map .Here the homomorphisms(1) ˆ π ∗ : φ K ∗ + d, t − v P ( V / Π) → φ K ∗ + d, t − v P ( V / Π × ˆΠ),(2) [ P ] ⊗ V/ Π × ˆΠ : φ K ∗ + d, t − v P ( V / Π × ˆΠ) → φ K ∗ + d, t − v + σP ( V / Π × ˆΠ), and(3) π ! : φ K ∗ + d, t − v + σP ( V / Π × ˆΠ) → φ K ∗ , t + σP ( ˆΠ)are the pull-back (3.12), the internal tensor product (Definition 3.15) with[ P ] and the push-forward (Definition 3.22) respectively, and we have sup-pressed the pullback notation for the twists.Now we state the main theorem of the paper. Theorem 4.4.
The crystallographic T-duality map φ T t G is an isomorphismfor any φ ∈ H ( P ; Z ) and t ∈ H ( P ; Z ) ⊕ H ( P ; φ T ) .Remark . Let φ R ( P ) denote the Grothendieck ring of finite dimensional φ -twisted representations of P . In other words, φ R ( P ) := φ K P (pt) with thering structure given by Definition 3.15. The crystallographic T-duality map(4.3) is actually a homomorphism of φ R ( P )-modules.Here we introduce a simple application of Remark 4.5. Suppose that G acts on V freely, which implies that the induced action of P onto V /
Π is alsofree. Hence the Lemma 4.6 below, which is a generalization of the Atiyah–Segal completion theorem, implies that there is n ∈ Z > such that φ I nP · φ K ∗ , t P ( V /
Π) = 0. Here φ I P denotes the augmentation ideal ker( φ K P (pt) → K (pt)). Now Theorem 4.4 implies that the φ R ( P )-module φ K ∗ , t − v + σP ( ˆΠ)also satisfies φ I nP · φ K ∗ , t − v + σP ( ˆΠ) = 0, although ˆΠ is no longer a free P -space. Lemma 4.6.
Let X be a free finite P -CW-complex. Then the φ R ( P ) -module φ K ∗ , t P ( X ) satisfies φ I nP · φ K ∗ , t P ( X ) = 0 for some n ∈ Z > .Proof. By the assumption, there is an n ∈ Z > and a continuous P -equivariantmap X → E n P , where E n P is the join P ∗ · · · ∗ P of n copies of P . This de-fines the φ K P ( E n P )-module structure on φ K ∗ P ( X ). Hence it suffices to showthat the pull-back by π n : E n P → pt factors through π ∗ n : φ R ( P ) / φ I nP → φ K P ( E n P ). We show this by induction on n . Let x , · · · , x n ∈ φ I P and set x ′ := x · · · · · x n − ∈ φ I n − P , x := x ′ · x n ∈ φ I nP . It suffices to show that WISTED CRYSTALLOGRAPHIC T-DUALITY 27 π ∗ n ( x ) = 0. Let us consider the diagram φ R ( P ) π ∗ n (cid:15) (cid:15) φ R ( P ) π ∗ n − (cid:15) (cid:15) φ K P ( E n P, E n − P ) / / φ K P ( E n P ) i ∗ n / / φ K P ( E n − P ) , where i n : E n − P → E n P denotes the inclusion. Since the right squarecommutes, i ∗ n ◦ π ∗ n ( x ′ ) = π ∗ n − ( x ′ ) = 0 by the induction hypothesis. Theexactness of the second row implies that π ∗ n ( x ′ ) = φ j ∗ n ( ξ ) for some ξ ∈ φ K P ( E n P, E n − P ). On the other hand, since E n P \ E n − P is P -equivariantlyhomeomorphic to the direct product P × C ( π n − ) (where C ( π n − ) is themapping cone space), we have φ I P · φ K P ( E n P, E n − P ) = 0. Hence we ob-tain π ∗ n ( x ) = x n · π ∗ n ( x ′ ) = x n · φ j ∗ n ( ξ ) = φ j ∗ n ( x n · ξ ) = 0 . (cid:3) Proof of Theorem 4.4
In this section we give a proof of Theorem 4.4. To this end, we firstly iden-tify the Freed–Moore twisted equivariant K-groups under consideration withthe corresponding Chabert–Echterhoff KK-group [CE01], and then identifythe twisted crystallographic T-duality map with the Kasparov product withan element, which is known to be a KK-equivalence. After the proof, we alsogive two related discussions; the irrational generalization of the twisted crys-tallographic T-duality and a relation between the twisted crystallographicT-duality and the Baum–Connes assembly map.5.1.
Comparison of twisted equivariant K-theories.
As is summarizedin Appendix B, the φ -twisted Chabert–Echterhoff KK-theory is a bivarianthomology theory for a pair of φ -twisted ( G, Π)-C*-algebras. Here a φ -twisted( G, Π)-C*-algebra is a triple (
A, α, σ ), where A is a C*-algebra, α is a φ -linear action of G onto A , and σ : Π → U ( M ( A )) such that σ t aσ ∗ t = α t ( a )and α t ( σ t ) = σ gtg − for any t ∈ Π, a ∈ A and g ∈ G . In this section wedeal with the following three kinds of φ -twisted ( G, Π)-C*-algebras and theirtensor product:(1) For a locally compact P -space X and a ( φ, t )-twisted P -equivariantvector bundle V on X , the C*-algebra C ( X, K ( V )) of continuoussections of the algebra bundle K ( V ) has a canonical φ -twisted P -action.(2) Let ˆΠ be the Pontrjagin dual of Π, on which G acts by ρ as in(3.23). We put a G -action φ ρ ∗ g ( f ) = φ ( g ) ( ρ ∗ g ( f )) on the C*-algebra C ( ˆΠ) (cf. Remark B.2). Moreover, let us define the implementingunitaries σ t ∈ C ( ˆΠ) as σ t = ˆ t , i.e., σ t ( χ ) = χ ( t ) for χ ∈ ˆΠ. Then( C ( ˆΠ) , φ ρ ∗ , σ ) is a ( G, Π)-C*-algebra, which is denoted by C ( ˆΠ) σ hereafter . (3) The complex number field C equipped with the G -action z φ ( g ) z ,which is simply denoted by C (cf. Remark B.2). More generally,we deal with Clifford algebras regarded as a Real C*-algebra (cf.Remark 3.7).Let A be a φ -twisted Z -graded ( G, Π)-C*-algebra. As is shown in Corol-lary B.16, the twisted equivariant K-group is defined as φ KK G, Π ( Cl p,q , A ) := π Fred p,q ( φ H G, Π A ) G , where φ H G, Π A is the φ -twisted ( G, Π)-equivariant Hilbert A -module as in(B.12). Note that if A is a P -C*-algebra, then φ H G, Π A is isomorphic to φ H PA := φ ℓ ( P ) ⊕∞ ˆ ⊗ A . (Here φ ℓ ( P ) denotes the space of ℓ -functionson P equipped with the usual summation and the complex multiplicationdetermined by ( λ · f )( p ) = φ ( p ) λ · f ( p ), on which P acts by the left regularrepresentation.) Lemma 5.1.
Let X be a locally compact P space. For t = ( c, τ ) ∈ H P ( X ; Z ) ⊕ H P ( X ; φ T ) , let ´ t denote the pair ( c, ´ τ ) where ´ τ := τ + ǫ ( c, c ) . Let ( V , v ) bea ( φ, ´ t ) -twisted P -equivariant vector bundle on X . Then there is an isomor-phism φ K q − p, t P ( X ) ∼ = φ KK P ( Cl p,q , C ( X, K ( V ))) . Note that for any twist ( φ, c, τ ) of a P -space X , there is a twisted P -vectorbundle on X (cf. [Kub16, Example 2.9]). Proof.
Let (´ V , ´ v ) denote the ( φ, t )-twisted P -equivariant vector bundle on X given by ´ V = V and ´ v p := γ c ( p ) v p . Then we have´ V ˆ ⊗ V ∗ ∼ = V ⊗ V ∗ ∼ = K ( V )as φ -twisted Hilbert K ( V )-modules. Set H := X × ( φ ℓ ( P ) ⊕∞ ˆ ⊗ ´ V ) . Then this is a universal ( φ, t )-twisted P -equivariant Hilbert bundle. More-over, there is a canonical isomorphism φ H PC ( X ) ˆ ⊗ K ( V ) ∼ = Γ( X, H ˆ ⊗ V ∗ )as Hilbert C ( X ) ˆ ⊗ K ( V )-modules.We write B ( H ˆ ⊗ V ∗ ) for the bundle whose fiber at x ∈ X is the boundedoperator algebra over the a Hilbert K ( V )-module H x ˆ ⊗ V ∗ . Then, by theabove discussion, there is a canonical identification B ( φ H PC ( X ) ˆ ⊗ K ( V ) ) ∼ = Γ( X, B ( H ˆ ⊗ V ∗ )) . Moreover, the mapΓ( X, B ( H )) → Γ( X, B ( H ˆ ⊗ V ∗ )) , T T ˆ ⊗ WISTED CRYSTALLOGRAPHIC T-DUALITY 29 is a P -equivariant bundle isomorphism. Therefore, we obtain a P -equivariantisomorphism Γ( X, B ( H )) → B ( φ H PC ( X ) ˆ ⊗ K ( V ) ), which restricts to an isomor-phism Γ( X, Fred( H )) P → Fred( φ H PC ( X ) ˆ ⊗ K ( V ) ) P . This finishes the proof forthe p = q = 0 case by taking the π of both sides. The general Clifford-equivariant version is proved in the same way. (cid:3) Remark . In general, for a real P -equivariant vector bundle E , the Clif-ford bundle Cl ( E ) is P -equivariantly Morita equivalent to Cl ( E ⊕ R n ) ∼ = Cl ( E ⊕ R n − d ) ˆ ⊗ Cl d, . The Clifford algebra Cl ( E ⊕ R n − d ) is P -equivariantlyisomorphic to K ( S ), where S is the irreducible representation of Cl ( E ⊕ R n − d ). The group P acts on S as a ( φ, v )-twisted representation. Notethat ´ v = − v holds since 2 ν = 0 and 2 µ = 0 in the Z -coefficient cohomologygroups. Lemma 5.3.
Under the isomorphisms given in Lemma 5.1, the followinghold: (1)
For a continuous map f : X → Y , the pull-back f ∗ : φ K ∗ , t P ( Y ) → φ K ∗ , t P ( X ) corresponds to the Kasparov product with [ f ∗ ] := [ C ( X ) , f ∗ , ∈ φ KK G, Π ( C ( Y ) , C ( X )) . (2) For [ F i ] ∈ φ K ∗ , t i P ( X i ) ( i = 1 , ), the external product [ F ] ⊗ [ F ] ∈ φ K ∗ , t + t P ( X × X ) corresponds to the Kasparov product [ F ] ˆ ⊗ [ F ] . (3) The Bott element β E ∈ φ K r, v P ( E ) corresponds to the KK-element [ C ( E, q ∗ S ) , c , C ] ∈ φ KK P ( Cl n − r, , C ( E, q ∗ Cl ( E ′ ))) . Proof.
The claims (1) and (3) are obvious from the definition. To see (2),notice that the first definition of the external product [ F ] ⊗ [ F ] in Lemma3.13 (1) satisfies the conditions of the Kasparov product given in DefinitionB.7. (cid:3) Let K denote the bundle ˆΠ × ˜ K with the G -action as in Lemma 3.25. Thisbundle is viewed in two ways: • Let us choose a section s : P → G . Then ( K , u s ( p ) ) is a ( φ, σ )-twisted P -equivariant vector bundle on ˆΠ. • The space C ( ˆΠ , K ) of continuous sections is a ( G, Π)-equivariantHilbert C ( ˆΠ) σ -module. Lemma 5.4.
The φ -twisted ( G, Π) -C*-algebras C ( ˆΠ , K ( K )) and C ( ˆΠ) σ areequivariantly Morita equivalent via the imprimitivity bimodule C ( ˆΠ , K ) (cf.Example B.8). In particular, [ K ] := [ C ( ˆΠ , K ) , id , ∈ φ KK G, Π ( C ( ˆΠ , K ( K )) , C ( ˆΠ) σ ) is a φ KK G, Π -equivalence. Twisted crystallographic T-duality via Kasparov product.
Weapply the above discussion to the twisted equivariant K-groups of our inter-est. Before that, we summarize our notations on Dirac operators.
Notation 5.5. let ˆ S denote the spinor representation of the Clifford alge-bra Cl ( V ⊕ − V ) , where − V denotes the linear space V with the negativedefinite inner product. We have ∗ -homomorphisms c : Cl ( V ) → B ( ˆ S ) and h : Cl ( − V ) → B ( ˆ S ) . We write m : C ( V / Π) → B ( L ( V / Π , ˆ S )) for the multi-plication ∗ -representation. Let us define the Dirac operator D := P h ( v i ) ∂ v i on V / Π , where v i runs over an orthonormal basis of T ( V / Π) . Set F := D (1 + D ) − / .Similarly, we write D E and F E for the Dirac operator twisted by a vectorbundle E . Also, we write ˜ D := P v i h ( v i ) ∂ v i for the Dirac operator on V and set ˜ F := ˜ D (1 + ˜ D ) − / . Lemma 5.6.
The Bott map β V ⊗ : φ K ∗ , t − v P (pt) → φ K ∗ , t P ( V ) is an isomor-phism.Proof. This follows from Lemma 5.3 (3). Indeed, it is proved in the sameway as [Kas80b] that the element β V has the φ KK P -inverse α V := [ L ( V, ˆ S ) , m, ˜ F ] ∈ φ KK P ( C ( V ) , C ) . (cid:3) Now we apply Lemma 5.1, Remark 5.2 and Lemma 3.25 for the spaces ofour interest. Let ( φ, t ) be a twist on the group P and let V be a ( φ, t )-twistedrepresentation of P . Proposition 5.7.
There are canonical isomorphisms φ K n + d, t − v P ( V / Π) ∼ = φ KK P ( Cl ,n , C ( V /
Π) ˆ ⊗ K ( V ) ˆ ⊗ Cl ( V )) , φ K n + d, t − v P ( V / Π × ˆΠ) ∼ = φ KK P ( Cl ,n , C ( V / Π × ˆΠ) ˆ ⊗ K ( V ) ˆ ⊗ Cl ( V )) , φ K n + d, t − v + σP ( V / Π × ˆΠ) ∼ = φ KK P ( Cl ,n , C ( V / Π × ˆΠ) σ ˆ ⊗ K ( V ) ˆ ⊗ Cl ( V )) , φ K n, t + σP ( ˆΠ) ∼ = φ KK P ( Cl ,n , C ( ˆΠ) σ ˆ ⊗ K ( V )) . We describe the twisted crystallographic T-duality map as a Kasparovproduct.
Lemma 5.8.
Through the isomorphisms in Proposition 5.7, the pull-back ˆ π ∗ : φ K ∗ , t − v P ( V / Π) → φ K ∗ , t − v P ( V / Π × ˆΠ) corresponds to the Kasparov productwith [ˆ π ∗ ] ˆ ⊗ id C ( V/ Π) , where [ˆ π ∗ ] := [ C ( ˆΠ) , π ∗ , ∈ φ KK P ( C , C ( ˆΠ)) . This is a special case of Lemma 5.3 (1).
Lemma 5.9.
Through the isomorphisms in Proposition 5.7, the internalproduct [ P ] ˆ ⊗ V/ Π × ˆΠ : φ K ∗ , t − v P ( V / Π × ˆΠ) → φ K ∗ , t − v + σP ( V / Π × ˆΠ) correspondsto the Kasparov product with [[ P ]] := [ C ( V / Π × ˆΠ , P ) , m, ∈ φ KK G, Π ( C ( V / Π × ˆΠ) , C ( V / Π × ˆΠ) σ ) . WISTED CRYSTALLOGRAPHIC T-DUALITY 31
Proof.
This follows from Lemma 5.3 (1), (3) and the fact that the cup prod-uct [ P ] ⊗ V/ Π × ˆΠ x coincides with ∆ ∗ ([ P ] ⊗ x ), where ∆ : V / Π × ˆΠ → ( V / Π × ˆΠ) denotes the diagonal embedding. Indeed, ∆ ∗ ([ P ] ⊗ x ) corresponds to x ˆ ⊗ C ( V/ Π) ˆ ⊗ C ( ˆΠ) ([ P ] ˆ ⊗ C ( V/ Π) ˆ ⊗ C ( ˆΠ) [∆ ∗ σ ]) = x ˆ ⊗ C ( V/ Π) ˆ ⊗ C ( ˆΠ) [[ P ]] , where ∆ ∗ σ : C ( V / Π × ˆΠ) ⊗ C ( V / Π × ˆΠ) σ → C ( V / Π × ˆΠ) σ is the pull-backby the diagonal embedding. (cid:3) Lemma 5.10.
Through the isomorphisms in Lemma 5.7, the push-forward π ! : φ K ∗ , t − v + σP ( V / Π × ˆΠ) → φ K ∗ , t + σP ( ˆΠ) corresponds to the Kasparov productwith the element [ π ! ] ⊗ id C (Π) σ , where [ π ! ] := [ L ( V / Π , ˆ S ) , m ⊗ c , F ] ∈ φ KK P ( C ( V / Π) ⊗ Cl ( V ) , C ) . Proof.
Let us choose a P -equivariant embedding V / Π → W . Let E denotethe normal bundle and let ι : E → W denote the embedding. Then thenormal bundle of V / Π × ˆΠ ⊂ W × ˆΠ is E × ˆΠ with the open embedding ι × id ˆΠ .Now, by Definition 3.22, (3.20) and Lemma 5.3 (2), we have • Thom E × ˆΠ ( x ) = x ˆ ⊗ C ( V/ Π × ˆΠ) σ β E × ˆΠ = ( x ˆ ⊗ C ( V/ Π) β E ) ˆ ⊗ id C ( ˆΠ) σ , • ( ι × id ˆΠ ) ∗ ( y ) = y ˆ ⊗ C ( E ) ([ ι ∗ ] ˆ ⊗ id C ( ˆΠ) σ ), • Thom − × W ( z ) = z ˆ ⊗ C ( W ) α W ,for any x ∈ φ K ∗ , t − v P ( V / Π × ˆΠ), y ∈ φ K ∗ , t P ( E × ˆΠ) and z ∈ φ K ∗ , t P ( W × ˆΠ)under the identification as in Proposition 5.7. Here ∆ E denotes the propermap (id E , q ) : E → E × V /
Π.Hence the map π ! corresponds to the Kasparov products over C ( V /
Π)with the KK-element( β E ˆ ⊗ C ( E ) [∆ ∗ E ]) ˆ ⊗ C ( E ) [ ι ∗ ] ˆ ⊗ C ( W ) ˆ ⊗ Cl ( V ) α W of φ KK G, Π ( C ( V /
Π) ˆ ⊗ Cl ( V ) , C ). Note that the Bott element β E is identifiedwith [ C ( E, S W ) , c , C ] ∈ φ KK P ( Cl ( V ) , C ( E )) , where S W denotes the spinor bundle on W , through the isomorphism φ KK P ( Cl n − r, , C ( E, q ∗ Cl ( E ′ ))) ∼ = φ KK P ( Cl ( V ) , C ( E ))given by Cl ( E ′ ) ˆ ⊗ Cl ( V ) ∼ = Cl ( W ) ˆ ⊗ Cl n − r, . Now it is proved that thiselement coincides with [ π ! ] in the same way as the push-forward map in non-equivariant K-theory, given in [CS84, Proposition 2.9] (which is essentiallydue to [CS84, Lemma 2.4]). (cid:3) In summary, the crystallographic T-duality map φ T ∗ , t G is given by theKasparov product with the element[ˆ π ∗ ] ⊗ C ( ˆΠ) [[ P ]] ⊗ C ( V/ Π) [ π ! ] ∈ φ KK G, Π ( C ( V / Π) ⊗ Cl ( V ) , C ( ˆΠ) σ ) . Now we give an explicit representative of this KK-element. Let H P denotethe Hilbert bundle on ˆΠ obtained by the fiberwise L -completion of C ( ˆΠ × V / Π , ˆ S ⊗ P ) and let D P denote the fiberwise Dirac type operator twistedby P , acting on the Hilbert C ( ˆΠ)-module C ( ˆΠ , H P ) as an unbounded oddself-adjoint operator. Set F P := D P (1 + D P ) − / . We write m V/ Π for themultiplication ∗ -representation of C ( V /
Π) onto C ( ˆΠ , H P ). Lemma 5.11.
We have [ˆ π ∗ ] ⊗ C ( ˆΠ) [[ P ]] ⊗ C ( V/ Π) [ π ! ] = [ C ( ˆΠ , H P ) , m ⊗ c , F P ] . (5.12) Proof.
The Kasparov product [ π ∗ ] ⊗ [[ P ]] is represented by the Kasparovbimodule[ C ( V / Π × ˆΠ , P ) , m V/ Π , ∈ φ KK G, Π ( C ( V / Π) , C ( V / Π × ˆΠ) σ ) . Hence the only thing we have to see is that the operator F P is a F -connectionon C ( V / Π × ˆΠ , P ) ⊗ C ( V/ Π) L ( V / Π , S ) ∼ = C ( ˆΠ , H P ) . It holds true because the principal symbol σ ( F P ) coincides with σ ( F ) ⊗ id P . (cid:3) T-duality and Dirac morphism.
The proof of Theorem 4.4 is nowreduced to the invertibility of the twisted equivariant KK-morphism (5.12).In this subsection we show this by relating (4.3), equivalently (5.12), withthe Dirac element (in the sense of [Kas88]), which is known to be invertible.
Definition 5.13.
Let D ∈ φ KK G ( C ( V ) ˆ ⊗ Cl ( V ) , C ) be the Dirac element D := [ L ( V, ˆ S ) , m ⊗ c , ˜ F ] , where ˆ S and ˜ F are as in Notation 5.5.It is proved by Higson–Kasparov [HK01] that this element is a φ KK G -equivalence. By the functoriality of the partial descent homomorphism, theKK-element φ j G, Π ( D ) ∈ φ KK G, Π (( C ( V ) ˆ ⊗ Cl ( V )) ⋊ Π , C ∗ (Π)) , represented by the Kasparov bimodule [ L ( V ) ⋊ Π , ( m ˆ ⊗ c ) ⋊ Π , F Π ] as inDefinition B.9, is also a φ KK G, Π -equivalence.The group C*-algebra C ∗ Π with the twisted ( G, Π)-action as in RemarkB.3 is isomorphic to C ( ˆΠ) σ . Also, the crossed product C*-algebra C ( V ) ⋊ Πis φ KK G, Π -equivalent to C ( V /
Π). Indeed, let X denote the bundle of Hilbertspaces V × Π ℓ (Π) on V /
Π. The G -action on V induces that on X as g · [ v, ξ ] := [ g · v, ξ ] for v ∈ V and ξ ∈ ℓ Π. The continuous section space C ( V / Π , X ) is G -equivariantly isomorphic to the completion of C c ( V ) by the C ( V /
Π)-valued inner product h ξ, η i ( v + Π) := X t ∈ Π ξ ( v + t ) η ( v + t ) . WISTED CRYSTALLOGRAPHIC T-DUALITY 33
This identification extends to the G -equivariant unitary W : L ( V ) → L ( V / Π , X ) . Lemma 5.14.
There is a ( G, Π) -equivariant ∗ -isomorphism ϕ : C ( V ) ⋊ Π ∼ = K ( C ( V / Π , X )) . Proof.
The compact operator algebra K ( C ( V / Π , X )) is isomorphic to thecontinuous section algebra C ( V / Π , V × Π K ( ℓ Π)). By the Takesaki–Takaiduality K ( ℓ Π) ∼ = c (Π) ⋊ Π, we have K ( C ( V / Π , X )) ∼ = C ( V / Π , V × Π K ( ℓ Π)) ∼ = C ( V / Π , V × Π ( c (Π) ⋊ Π)) ∼ = C ( V / Π , V × Π c (Π)) ⋊ Π = C ( V ) ⋊ Π . This gives a ∗ -representation ϕ : C ( V ) ⋊ Π → K ( C ( V / Π , X )), which isexplicitly written as ϕ (( X f t u t ) ξ )( v ) = X f t ( v ) ξ ( v − t )for ξ in the dense subalgebra C c ( V ) ⊂ C ( V / Π , X ). Hence it is G -equivariantand u t ( ξ ) = ϕ ( u t ) ξ , that is, ϕ is ( G, Π)-equivariant. (cid:3)
Lemma 5.14 means that C ( V ) ⋊ Π is φ -twisted ( G, Π)-equivariantlyMorita equivalent with C ( V /
Π) via the imprimitivity bimodule X (cf. Ex-ample B.8). In particular, the element[ X ] := [ C ( V / Π , X ) , ϕ, ∈ φ KK G, Π ( C ( V ) ⋊ Π , C ( V /
Π))is a φ KK G, Π -equivalence with the KK-inverse represented by its adjoint X ∗ . Lemma 5.15.
The φ KK G, Π -equivalence [ X ∗ ] ⊗ C ( V ) ⋊ Π φ j G, Π ( D ) ∈ φ KK G, Π ( C ( V / Π) ⊗ Cl ( V ) , C ( ˆΠ) σ ) is represented by the Kasparov bimodule [ C ( ˆΠ , H P ) , m ⊗ c , F P ] .Proof. We identify the G -equivariant Hilbert C ( ˆΠ) σ -module L ( V ) ⋊ Π with L ( V / Π , X ) ⊗ C ( ˆΠ) σ by the unitary U : L ( V ) ⋊ Π ∋ X t ∈ Π ξ t u t X t ∈ Π W ξ t ⊗ χ t ∈ L ( V / Π , H ) ⊗ C ( ˆΠ) σ , where χ t ∈ C ( ˆΠ) σ is the function χ t ( ξ ) := ξ ( t ). Then we have U ( m ⋊ Π) (cid:16) X h ∈ Π f h u h (cid:17) U ∗ (cid:16) X g ∈ Π ξ g ⊗ χ g (cid:17) = X g,h ∈ Π ϕ ( f h u h )( ξ g ) ⊗ χ hg ,U ( F Π ) U ∗ (cid:16) X g ∈ Π ξ g ⊗ χ g (cid:17) = X g ∈ Π W F W ∗ ξ g ⊗ χ g . In other words, ˜ ϕ := Ad( U ) ◦ ( m ⋊ Π) satisfies˜ ϕ (cid:16) X g ∈ Π f g u g (cid:17) ( α ) = X g ∈ Π α ( g ) ϕ ( f g u g ) for any α ∈ ˆΠ, and U F Π U ∗ = W F W ∗ ⊗
1. The operator
W F W ∗ coincideswith F X = D X (1+ D X ) − / , where D X is the Dirac operator on V /
Π twistedby X with respect to its flat connection.Let Q := X ∗ ⊗ ˜ ϕ X , let D Q be the Dirac operator twisted by Q and let F Q := D Q (1 + D Q ) − / . Then the above discussion shows that[ X ] ∗ ⊗ C ( V ) ⋊ Π φ j G, Π ( D )=[ X ∗ ] ⊗ C ( V ) ⋊ Π [ L ( V / Π , X ⊗ ˆ S V ) ⊗ C ( ˆΠ) σ , ˜ ϕ, F X ⊗ C ( ˆΠ) σ ]=[ C ( ˆΠ , L ( V / Π , Q ⊗ ˆ S )) , m V/ Π , F Q ] . The remaining task is to show that Q is G -equivariantly isomorphic to P .For x = v + Π ∈ V /
Π and α ∈ ˆΠ, the bundle map X ∗ ⊗ ˜ ϕ X ∋ [ x, α, s ∗ ⊗ t ] h v, α, X g ∈ Π α ( g ) s ( v + g ) t ( v + g ) i ∈ ( V × ˆΠ × C ) / Πis well-defined independent from the choice of v . It is everywhere non-zeroand hence is an isomorphism. (cid:3) Proof of Theorem 4.4.
Now the theorem follows from Lemma 5.11 and Lemma5.15. (cid:3)
Generalization to irrational twists.
The description using the Diracmorphism given in the above subsection extends the twisted crystallographicT-duality to the “irrational flux” case, i.e., the case that the twist ( φ, c, τ ) of G is not necessarily obtained as Proposition 2.3, or in other words τ is notnecessarily trivial on Π. The right side of the T-duality will then involvea noncommuative torus rather than ˆΠ. Such a situation may arise whenΠ is projectively represented as magnetic translation operators, as happenswhen an external magnetic field is applied perpendicular to V .Let G be a discrete group acting properly and cocompactly on the affinespace V and let ( φ, c, τ ) be an arbitrary twist of G . Let Π be a free abelianfinite index subgroup of G such that φ | Π and c | Π are trivial (such Π existsby Lemma 2.4). We write G τ and Π τ for the T -extension of G and Πcorresponding to τ respectively. Let ( π, H ) be a ( φ, c, τ )-twisted unitaryrepresentation of G . The Kasparov product with φ j G, Π ( D ⊗ id K ( H ) ) ∈ φ KK G, Π (( C ( V ) ⊗ K ( H )) ⋊ Π , K ( H ) ⋊ Π)induces an isomorphism φ K G, Π (( C ( V ) ⊗ K ( H )) ⋊ Π) ∼ = φ K G, Π ( K ( H ) ⋊ Π) . (5.16)The equivariant K-groups on the left and right hand sides of the aboveisomorphism are simplified. Firstly, let A denote the φ -linear P -equivariantbundle V × Π , Ad( π ) K ( H ) of Z -graded compact operator algebras over V /
Π.By the construction, the equivariant Dixmier–Douady class isDD( A ) = f ∗ t ∈ H P ( V / Π; Z ) ⊕ H P ( V / Π; φ T ) , WISTED CRYSTALLOGRAPHIC T-DUALITY 35 where f : V → pt, through the canonical identification H P ( V / Π; Z ) ⊕ H P ( V / Π; φ T ) ∼ = H G ( V ; Z ) ⊕ H G ( V ; φ T ). Now it is checked in the same wayas Lemma 5.14 that there is a φ -twisted ( G, Π)-equivariant Morita equiva-lence ( C ( V ) ⊗ K ( H )) ⋊ Π ∼ Morita C ( V / Π , A ) . Secondly, we consider the non-commutative torus C ∗ ¯ τ Π (where ¯ τ denotesthe inverse of τ ), i.e., the C*-algebra generated by unitaries { u t } t ∈ Π withthe relation u s u t = ¯ τ ( s, t ) u st . Let α denote the φ -twisted G -action on C ∗ ¯ τ Πdetermined as the φ -linear extension of α g ( u t ) = ¯ τ ( g, t )¯ τ ( gt, g − ) u gtg − (5.17)for any t ∈ Π and g ∈ G τ (note that this α satisfies α g ◦ α h = α gh ). Also, set σ t := u t . Then α t = Ad σ t and σ s σ t = ¯ τ ( s, t ) σ st hold for any t ∈ Π. Hencethe pair ( α, σ ) obviously gives rise to a ( G τ , Π τ )-action onto C ∗ ¯ τ Π such that σ z = ¯ z for any z ∈ T ⊂ Π τ . Lemma 5.18.
The C*-algebras K ( H ) ⋊ Π and C ∗ ¯ τ Π are ( G τ , Π τ ) -equivariantlyMorita equivalent (in the sense of Example B.8).Proof. Before starting the proof we remark that the group C*-algebra C ∗ Π τ of the central extension group Π τ is decomposed as the direct sum L n ∈ Z C ∗ τ n Πsince the subalgebra C ∗ T ∼ = c ( Z ) of C ∗ Π τ lies in the center. This decom-position respects the φ -twisted ( G τ , Π τ )-C*-algebra structure of C ∗ Π τ .As is mentioned in Example B.10, the φ -twisted G τ -equivariant K ( H )- C imprimitivity bimodule H induces a φ -twisted ( G τ , Π τ )-equivariant K ( H ) ⋊ Π τ - C ∗ Π τ imprimitivity bimodule H ⋊ Π τ . Let ϕ : C ∗ Π τ → C ∗ ¯ τ Π denote thequotient onto the direct summand. Then H ⋊ ¯ τ Π := ( H ⋊ Π τ ) ⊗ ϕ C ∗ ¯ τ Πis a G τ -equivariant Hilbert K ( H ) ⋊ Π τ - C ∗ ¯ τ Π bimodule. Moreover, the leftaction of K ( H ) ⋊ Π τ descends to the quotient K ( H ) ⋊ Π since the centralsubgroup T ⊂ Π τ acts on H ⋊ τ Π trivially from the left. That is, H ⋊ τ Π isa K ( H ) ⋊ Π- C ∗ ¯ τ Π imprimitivity bimodule. Since it is ( G τ , Π τ )-equivariantby construction, this finishes the proof. (cid:3) Since an equivariant Morita equivalence induces an isomorphism of theequivariant K-group (cf. Example B.8), the above discussion and the iso-morphism (5.16) conclude the following theorem.
Theorem 5.19.
Let ( φ, c, τ ) be a twist of a discrete group G acting properlyand cocompactly on V . There is an isomorphism φ K ∗ ,f ∗ t − v P ( V / Π) ∼ = −→ φ K G τ , Π τ ∗− d,c ( C ∗ ¯ τ Π) . Even in the irrational case, the K-group lying in the right hand side ofthe above isomorphism is viewed as the set of topological phases with the symmetry type (
G, φ, c, τ ). Let us define a Hilbert space φ ℓ ( G, τ ) as thecompletion of φ C c ( G, τ ) := { ξ ∈ C c ( G τ ) | ξ ( zg ) = z − ξ ( g ) for any z ∈ T } equipped with the L -inner product on G τ with respect to the Haar mea-sure and the C -vector space structure given by the complex multiplica-tion ( λ · ξ )( g ) = φ ( g ) λ · ξ ( g ). By choosing a section s : G → G τ , the leftregular representation ( π l ( h ) ξ )( g ) := ξ ( s ( h ) − g ) is regarded as a ( φ, τ )-twisted representation of G . Similarly, the right regular representation( π r ( h ) ξ )( g ) := ξ ( gs ( h )) is a ( φ, ¯ τ )-twisted representation of G .A choice of sections P → G identifies the underlying complex Hilbertspace φ ℓ ( G, τ ) with the direct sum of subspaces ℓ (Π g, τ ). Then π l (Π)and π r (Π) acts by the regular representations on each component. Theset of operators on φ ℓ ( G, τ ) satisfying assumption (1) of Lemma 3.26 andcommuting with π l (Π) forms a ∗ -algebra. Let A τ denote its closure. It isgenerated by π r (Π) and K ( φ ℓ ( P )), that is, A τ ∼ = C ∗ ¯ τ Π ⊗ K ( φ ℓ ( P ))as C*-algebras. The φ -twisted G τ -action Ad( π l ) on B ( φ ℓ ( G, τ )) descendsto a φ -twisted P -action onto A τ .In the same way as Subsection 3.3, let us define H N ( G, φ, t ) as the set ofbounded operators on ( φ ℓ ( G, τ ) ˆ ⊗ V ) ⊕ N satisfying the conditions (1), (2’)and (3) enumerated below Lemma 3.26, and define the set of topologicalphases as TP ( G, φ, t ) := (cid:0) S N H N ( G, φ, t ) (cid:1) / ∼ . This set is the same thingas the Karoubi picture of the twisted equivariant K-group K P ,c ( A τ ) definedin [Kub16, Theorem 5.14]. Proposition 5.20.
The ( G τ , Π τ ) -C*-algebra C ∗ ¯ τ Π and the P -C*-algebra A τ are ( G τ , Π τ ) -equivariantly Morita equivalent.Proof. Following the idea of [Roe02, Lemma 2.1], we define the φ -twisted( G τ , Π τ )-equivariant Hilbert C ∗ ¯ τ Π-module φ ℓ ( G, τ ) as the closure of φ C c ( G, τ )with respect to the inner product h ξ, η i C ∗ ¯ τ Π := X t ∈ Π h π l ( t − ) ξ, η i ℓ · u t , where h· , ·i ℓ is the inner product of φ ℓ ( G, τ ). It is a routine work to see thatthe above inner product h· , ·i C ∗ ¯ τ Π is compatible with the right C ¯ τ [Π]-modulestructure given by ξ · u t := π l ( t − ) ξ. Moreover, the external tensor product φ ℓ ( G, τ ) ⊗ C ∗ ¯ τ Π ℓ (Π , τ ) (where C ∗ ¯ τ Πacts on ℓ (Π , τ ) by the right regular representation) is canonically isomorphicto φ ℓ ( G, τ ).Now the same argument as [Roe02, Lemma 2.3] shows that · ⊗ C ∗ ¯ τ Π B ( φ ℓ ( G, τ )) → B ( φ ℓ ( G, τ )) WISTED CRYSTALLOGRAPHIC T-DUALITY 37 maps the compact operator algebra K ( φ ℓ ( G, τ )) faithfully onto A τ . Thismeans that φ ℓ ( G, τ ) is an equivariant A τ - C ∗ ¯ τ Π imprimitivity bimodule. (cid:3)
Together with the above discussion, we obtain the desired isomorphism TP ( G, φ, t ) = φ K P ,c ( A τ ) ∼ = K G τ , Π τ ,c ( C ∗ ¯ τ Π) . Relation to the partial Baum–Connes assembly map.
Following[CE01], we define a φ -twisted generalization of the partial Baum–Connesassembly map. Throughout this section we use the following notation φ KK G t ( A, B ) := φ KK G ( A, B ˆ ⊗ K (´ V ))for any Z -graded G -C*-algebras A and B , where ´ V is a ( φ, ´ t )-twisted unitaryrepresentation of G (cf. Lemma 5.1). This notation is consistent with theone given in [Kub16] (because of [Kub16, Proposition 3.10]). Definition 5.21.
The partial assembly map with coefficients in a φ -twisted G -C*-algebra A is defined by the composition φ µ Π G := [ H ] ⊗ C ( V ) ⋊ Π φ j G, Π ( ) : φ KK G ( C ( V ) , A ) → φ KK G, Π ( C , A ⋊ Π) . In particular, when A = K (´ V ) we get a homomorphism from φ KK G t ( C ( V ) , C )to φ KK G, Π t ( C , C ( ˆΠ) σ ). Here we relate this map with the Dirac morphism,and hence the twisted crystallographic T-duality map.The domains of the Baum–Connes and T-duality maps are identified bythe following two isomorphisms.(1) A G -equivariant C ( V )- K (´ V ) bimodule is canonically viewed as a( G, Π)-equivariant C ( V ) ⋊ Π- K (´ V ) bimodule. Hence there is acanonical isomorphism J Π : φ KK G t ( C ( V ) , C ) → φ KK G, Π t ( C ( V ) ⋊ Π , C ) . By definition of the partial descent map, this J Π coincides with thecomposition ι ∗ ◦ φ j G, Π , where ι : pt → ˆΠ denotes the map to theunit (i.e., the trivial character).(2) The Poincar´e duality isomorphismPD V/ Π : φ KK P t ( C , C ( V / Π) ⊗ Cl ( V )) ∼ = φ KK P t ( C ( V / Π) , C )is given by the Kasparov product with[∆ ∗ ] ⊗ C ( V/ Π) [ L ( V / Π , S ) , m ˆ ⊗ c , F ] ∈ φ KK P ( C ( V / Π) ⊗ C ( V / Π) ⊗ Cl ( V ) , C ) , where ∆ : V / Π → V / Π × V /
Π is the diagonal embedding. It is seenin the same way as [Kas80b, Theorem 7] that this Kasparov productis an isomorphism by constructing its inverse as a Kasparov product.Note that [ L ( V / Π , S ) , m, F ] coincides with φ j G, Π ( D ) ⊗ [ ι ∗ ].To summarize, the composition µ Π G ◦ J Π ◦ PD V/ Π is a homomorphism from φ KK P t − v ( C , C ( V /
Π)) to φ KK G, Π t ( C , C ( ˆΠ) σ ). Theorem 5.22.
The map µ Π G ◦J Π ◦ PD V/ Π is the same as the crystallographicT-duality map φ T ∗ , t G . Let p denote the projection V → V /
Π and let ∆ V := (id V , p ) : V → V × V /
Π. This is a proper G -equivariant map (by regarding V /
Π as a G -space through the quotient G → P ) and hence induces∆ ∗ V : C ( V ) ⊗ C ( V / Π) → C ( V ) . Lemma 5.23.
We have [ X ∗ ] ⊗ C ( V ) ⋊ Π φ j G, Π ([∆ ∗ V ]) ⊗ C ( V ) ⋊ Π [ X ] = [∆ ∗ ] . Proof.
By definition we have φ j G, Π ([∆ ∗ V ]) = [∆ ∗ V ⋊ Π], where ∆ ∗ V ⋊ Π isthe induced ∗ -homomorphism ( C ( V ) ⊗ C ( V /
Π)) ⋊ Π → C ( V ) ⋊ Π. This ∗ -homomorphism is explicitly described as(∆ ∗ V ⋊ Π)( x ⊗ f ) = xf, for any x ∈ C ( V ) ⋊ Π and f ∈ C ( V /
Π). Here the product xf is taken in B ( C ( V / Π , X )), where C ( V ) ⋊ Π ∼ = K ( C ( V / Π , X )) acts on C ( V / Π , X ) as inLemma 5.14 and C ( V /
Π) acts by multiplication. Therefore, the Kasparovproduct [∆ ∗ V ⋊ Π] ⊗ C ( V ) ⋊ Π [ X ] is represented by C ( V / Π , X ). This showsthat [∆ ∗ V ⋊ Π] ⊗ C ( V ) ⋊ Π [ X ] = ([ X ] ⊗ id C ( V/ Π) ) ⊗ C ( V/ Π) [∆ ∗ ] , which finishes the proof. (cid:3) Proof of Theorem 5.22.
In the proof, we omit the KK-equivalence [ X ] andidentify C ( V ) ⋊ Π with C ( V /
Π) for simplicity of notation.Let Φ : φ KK G, Π t ( C , C ( V /
Π)) → φ KK G, Π t ( C ( V / Π) , C ( V /
Π)) denote thegroup homomorphism defined byΦ( ξ ) := ( ξ ⊗ id C ( V/ Π) ) ⊗ C ( V/ Π) ⊗ C ( V/ Π) [∆ ∗ ] . Similarly we define Φ V : φ KK G, Π t ( C , C ( V /
Π)) → φ KK G, Π ( C ( V ) , C ( V ))given by Φ V ( ξ ) := ( ξ ⊗ id C ( V ) ) ⊗ C ( V/ Π) ⊗ C ( V ) [∆ ∗ V ]. Here Lemma 5.23is rephrased as Φ = φ j G, Π ◦ Φ V . WISTED CRYSTALLOGRAPHIC T-DUALITY 39
Let us consider the diagram (cid:8) φ KK G t ( C ( V ) , C ( V )) ⊗ C V ) D (cid:15) (cid:15) φ j G, Π / / (cid:8) φ KK G, Π t ( C ( V / Π) , C ( V /
Π)) π ∗ / / ⊗ C ( V/ Π) φ j G, Π ( D ) (cid:15) (cid:15) (cid:8) φ KK G, Π t ( C , C ( V /
Π)) ⊗ C ( V/ Π) φ j G, Π ( D ) (cid:15) (cid:15) Φ V v v Φ s s φ KK G t ( C ( V ) , C ) φ j G, Π / / J Π * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ (cid:8) φ KK G, Π t ( C ( V / Π) , C ( ˆΠ) σ ) π ∗ / / ˆ ⊗ C (ˆΠ) σ [ ι ∗ ] (cid:15) (cid:15) φ KK G, Π t ( C , C ( ˆΠ) σ ) φ KK P t ( C ( V / Π) , C ) . The functoriality of φ j G, Π and the above arguments show that parts of theabove diagram indicated by the circular arrows are commutative. Moreover,by Lemma 5.15 we have φ j G, Π ( D ) ˆ ⊗ C ( ˆΠ) σ [ ι ∗ ] = [ L ( V / Π , S ) , m ˆ ⊗ c , F ], whichshows that PD V/ Π ( ξ ) = Φ( ξ ) ˆ ⊗ C ( V/ Π) ( φ j G, Π ( D ) ˆ ⊗ C ( ˆΠ) σ [ ι ∗ ]) . Therefore a diagram chasing shows that( φ µ Π G ◦ ( J Π ) − ◦ PD V/ Π )( ξ ) = ξ ˆ ⊗ φ j G, Π ( D ) , which finishes the proof by Theorem 4.4. (cid:3) T-duality between Atiyah–Hirzebruch spectral sequences
A promising application of Theorem 4.4 is the calculation of the twistedequivariant K-theory, which is well studied in the literature of condensed-matter physics [SSG17, SSG18]. The crystallographic T-duality implies thatthere are two different spectral sequences converging to the same group; theAtiyah–Hirzebruch spectral sequence (AHSS) of φ K ∗ + d, t − v P ( V /
Π) and that of φ K ∗ , t + σP ( ˆΠ). These two spectral sequences are very different, in the sense thatthe map φ T t G does not respect the Atiyah–Hirzebruch filtrations. Indeed, the(noncommutative) topology of the spaces on either side may be different, asis observed in non-equivariant setting in [BEM04b] and [MT16,HMT16]. Fortrivial φ, t , the (ordinary) crystallographic T-duality isomorphism was ex-ploited in [GT19] Section 8.3, to solve AHSS extension problems on one sideby comparing with the other side where no extension problem occurs. Herewe provide some new examples, involving twisted crystallographic groups,of twisted equivariant K-group calculations via the comparison of Atiyah–Hirzebruch spectral sequences.6.1. 1 -dimensional black and white magnetic space group. Here weconsider the case that V is of dimension 1, G = Z acting on V by translation,and φ : G → Z is the surjective homomorphism. Then G is a twistedcrystallographic group in the sense of Definition 2.1, where the groups Π and P are Π = 2 Z and P ∼ = Z respectively. We write p ∈ P for the uniquenon-trivial element.The group P acts on the torus V / Π ∼ = S as rotation by π . Moreover, thelinear P -action on V is trivial and hence v = ( w P ( V ) , w P ( V )) = 0. We putthe P -equivariant cell decomposition of V /
Π as: ◦ e ◦ e ◦ e e e The action of P = Z maps e to e , e to e , e to e and e to e respec-tively.The P -action on the Pontrjagin dual ˆΠ ∼ = U (1) as in (3.23) is the complexconjugation z ¯ z . We put the P -equivairant cell decomposition as: ◦ e ◦ e ◦ e e e The action of P = Z fixes e and e , and maps e to e and e to e respectively. Moreover, the 2-cocycle σ ∈ H P ( ˆΠ; Z ) corresponding to theextension 1 → Π → G → Z → σ ( z, p, p ) = z for any z ∈ U (1).6.1.1. Type AI: AHSS for φ K ∗ P ( V / Π) . We start with the case that t := ( c, τ )is trivial. Let us consider the AHSS for φ K ∗ P ( V /
Π). Recall that φ K ∗ Z ( Z )is identified with K ∗ (pt). Hence the E -page of this AHSS is as the left ofFigure 1. Moreover, by the following Lemma 6.1, we obtain the E -page ofthis AHSS as the right of Figure 1.7 0 06 Z Z
Z Z
Z Z
Z Z E pq Z Z Z Z Z Z E pq Figure 1. E and E pages of AHSS for φ K ∗ , − v P ( V / Π) Lemma 6.1.
Under the above identification E p,q ∼ = Z for p = 0 , and even q , the differential d ,q : E ,q → E ,q is d ,q = (cid:26) q ≡ mod , q ≡ mod . Proof.
We fix a Real structure, i.e., a complex conjugation on H and extendit to H n := H ⊗ ∆ n, . The complex Clifford algebra C l n := Cl n, ⊗ R C acts WISTED CRYSTALLOGRAPHIC T-DUALITY 41 on H n and K n (pt) is isomorphic to the set of homotopy classes of the spaceof odd self-adjoint operators F with F − ∈ K ( H n ) and [ F, c ( v )] = 0 for any v ∈ C l n . Now the complex conjugation induces an involution T : [ F ] [ ¯ F ]on K n (pt). This involution is T x = ( − n x for any x ∈ K n (pt) . (6.2)We use the trivial bundle H := V / Π × H , on which Z acts by the com-position of the rotation by π and the complex conjugation, as the universal φ -twisted P -equivariant Hilbert bundle. Now an element of φ K n Z ( x ∪ x )is represented by [ F ∪ ¯ F ] for some F ∈ Fred n ( H ). By definition of theboundary homomorphism ∂ : φ K nP ( e ∪ e ) → φ K n +1 P ( e ∪ e ) ∼ = K n +1 ( e ) , it maps the element [ F ∪ ¯ F ] to ([ F ] − [ ¯ F ]) ⊗ β . Now (6.2) finishes theproof. (cid:3) Remark . Here we give a detail on the proof of (6.2). Let M C n (resp. M R n )denote the free abelian group generated by Z -graded representations of thecomplex (resp. real) Clifford algebra C l n (resp. Cl n, ) and let i ∗ : M F n +1 → M F n denote the forgetful map. The Atiyah–Bott–Shapiro isomorphism K n (pt) ∼ = M C n /i ∗ M C n +1 (resp. KO n (pt) ∼ = M R n /i ∗ M R n +1 ) is now given by mapping [ F ]to the Z -graded representation ker( F ) of C l n . This correspondence is com-patible with the complex conjugation, i.e., [ ¯ F ] is mapped to ker ¯ F ∼ = ker F .In n = 4 k , the complex conjugation acts trivially on K n (pt) since it mustbe trivial on the complexification map KO n → K n . In n = 2, the generator of M n /i ∗ M n +1 is S = C on which the Clifford generators e , e acts as (cid:0) − (cid:1) and (cid:0) (cid:1) respectively, and the Z -grading is given by γ = (cid:0) i − i (cid:1) . Now¯ S is the same representation of C l n with the opposite Z -grading. That is,[ ¯ S ] = − [ S ]. Now we obtain (6.2) since the complex conjugation is compatiblewith the multiplication in K-theory.Finally we obtain that φ K ∗ P ( V / Π) ∼ = Z ∗ ≡ , , ∗ ≡ , Z ∗ ≡ . (6.4)6.1.2. Type AI: AHSS for φ K ∗ + σP ( ˆΠ) . We also consider the AHSS for φ K ∗ ,σP ( ˆΠ).Since σ ( e , p, p ) = +1 and σ ( e , p, p ) = −
1, we have φ K ∗ ,σP ( e ) ∼ = KO ∗ and φ K ∗ ,σP ( e ) ∼ = KO ∗ +4 . Similarly we also have φ K ∗ ,σP ( e ∪ e ) = φ K ∗ P ( e ∪ e ) =K ∗ (pt) since σ ∈ H P ( e ∪ e ; φ T ) ∼ = 0. Hence the E -page is as in the left ofFigure 2. Moreover, the differentials d ,q are given by d ,q = ( C C ) : KO q ⊕ KO q +4 → K q , where C : KO ∗ → K ∗ denotes the complexification homomorphism. Hence d ,q are surjective for q = 0 , E -pageof this AHSS is as the right of Figure 2. Z Z Z Z Z Z Z Z Z Z E pq Z Z Z Z Z Z Z Z E pq Figure 2. E and E pages of AHSS for φ K ∗ ,σP ( ˆΠ)Now, thanks to Theorem 4.4, a comparison with (6.4) shows that theextension 0 → E ,q − → φ K q,σP ( ˆΠ) → E ,q → q ≡ φ K ∗ ,σP ( ˆΠ) ∼ = Z ∗ ≡ , , ∗ ≡ , Z ∗ ≡ . (6.5)6.1.3. Other symmetry types.
We consider some variations of the 1-dimensionalblack and white magnetic space group studied above.(1) The case that G , φ are as above, c is trivial and τ ∈ H ( P ; φ T ) isdetermined by τ ( p, p ) = − G , φ are as above and c : G → Z is the surjection.There are two choices of τ determined by τ ( p, p ) = ± G = Z × Z , φ is the composition of the projection pr Z and the surjection Z → Z and c := pr Z . The twisted point groupis P = Z × Z generated by p and c , and there are four choices for τ given by τ ( p, p ) = ± τ ( c, c ) = ±
1, corresponding to symmetrytypes BDI, DIII, CI, and CII.In each of these cases, the E and E pages of the AHSS for φ K ∗ , t − v P ( V /
Π)indicated in Figure 1 is shifted by m ∈ Z / Z depending on the Altland–Zirnbauer (AZ) symmetry type as Table 3. Indeed, as is well-understoodin the study of topological insulators (see e.g. [Kub16, Corollary 5.16]), thefunctor φ K ∗ , t − v P is naturally identified with the type AI (i.e. Real) K-functor φ K ∗ + m, − v Z . Cartan AI BDI D DIII AII CI C CII m Table 3.
The degree shift corresponding to the symmetry type.
WISTED CRYSTALLOGRAPHIC T-DUALITY 43
Therefore, the resulting group φ K ∗ , t P ( V /
Π) is isomorphic to (6.5) shiftedby m , that is, φ K ∗ , t P ( V / Π) ∼ = Z ∗ + m ≡ , , ∗ + m ≡ , Z ∗ + m ≡ . -dimensional pg grey magnetic space group. The second exam-ple is the case that G is the 2-dimensional magnetic space group of type pg .That is, G is the subgroup of Euc( V ) generated by a : ( x, y ) ( x + 1 , y )and b : ( x, y ) ( − x, y + 1). Let Π = h a, b i ⊂ G . Then Π is a free abeliansubgroup of G acting on V by translation. Let φ : G → Z be the quotient G → P := G/ Π ∼ = Z . Then ( G, φ ) is a magnetic space group in the senseof Definition 2.6, and hence (
G, φ, ,
0) is a twisted crystallographic groupin the sense of Definition 2.1. Note that, since b reverses the orientation of V , the twist v = ( w P ( V ) , w P ( V )) (cf. Section 4) is non-trivial.6.2.1. Type AI: AHSS for φ K ∗ P ( V / Π) . Let us consider the AHSS for φ K ∗ , − v P ( V /
Π)with respect to the following P -equivariant cell decomposition of V / Π: ◦ e ◦ e ◦ e ◦ e ◦ e ◦ e e e e e e e e e e The P -action maps e to e and e to e . It also maps e to e and e to e while reversing the x -coordinate.Whatever the twist ( φ, v ) is, the group φ K ∗ , − v Z ( Z ) is identified with thecomplex K-group K ∗ (pt). Hence we have φ K ∗ , − v P ( e ∪ e ) ∼ =K ∗ , φ K ∗ , − v P ( e ∪ e ) ∼ =K ∗ +1 ∼ = φ K ∗ , − v P ( e ∪ e ) , φ K ∗ , − v P ( e ∪ e ) ∼ =K ∗ +2 , and hence the E -page of this AHSS is as the left of Figure 3.Let T := e ∪ e ∪ e ∪ e . Then the P -equivariant maps T inclusion −−−−−→ V / Π projection −−−−−−→ T induces a direct sum decomposition φ K ∗ , − v Z ( V / Π) ∼ = φ K ∗ , − v Z ( T ) ⊕ φ K ∗ , − v Z ( V / Π \ T ) . We apply Lemma 6.1 to obtain that the differential d pq of the AHSS for V /
Πrestricted to the subcomplex T is d q = (cid:26) Z → Z q ≡ , Z → Z q ≡ , and otherwise zero (note that the degree is shifted by 2 due to the twist − v ). We also remark that the Thom isomorphism (Definition 3.19) showsthat φ K ∗ , − v Z ( R , × X ) ∼ = φ K ∗ +3 Z ( X ), where R , is the real line R with the Z -action given by the reflection. Since V / Π \ T ∼ = R , × T , the differential d pq of the AHSS for V /
Π restricted to the subcomplex
V / Π \ T is d q = (cid:26) Z → Z q ≡ , Z → Z q ≡ , and otherwise zero. Therefore, the E -page of this AHSS as the right side ofFigure 3. Now the differential d pq is zero and there is no extension problem.7 0 0 06 Z Z Z Z Z Z Z Z Z Z Z Z E pq Z Z Z Z ⊕ Z Z
Z Z Z Z ⊕ Z Z E pq Figure 3. E and E pages of AHSS for φ K ∗ , − v P ( V /
Π)Hence we obtain that φ K ∗ , − v P ( V / Π) ∼ = Z ∗ ≡ , Z ⊕ Z ∗ ≡ , Z ∗ ≡ , Z ∗ ≡ . (6.6)We remark that essentially the same calculation as above is done by Baraglia[Bar13] in his study of topological T-duality of twisted KR-theory.6.2.2. Type AI: AHSS for φ K ∗ + σP ( ˆΠ) . Next we calculate the AHSS of φ K ∗ , t P ( ˆΠ)with respect to the following cell decomposition. Let ( k x , k y ) denote the co-ordinate of ˆΠ dual to the standard ( x, y )-coordinate of V . We put the cell WISTED CRYSTALLOGRAPHIC T-DUALITY 45 decomposition of ˆΠ as: ◦ e ◦ e ◦ e ◦ e ◦ e ◦ e e e e e e e e e e on which P acts by the reflection along k y -axis. The 2-cocycle σ is deter-mined by σ ( , p, p ) = e πik y . Hence we have φ K ∗ ,σP ( e ) ∼ =KO ∗ ∼ = φ K ∗ +1 ,σP ( e ) , φ K ∗ ,σP ( e ) ∼ =KO ∗ +4 ∼ = φ K ∗ +1 ,σP ( e ) , φ K ∗ +1 ,σP ( e ∪ e ) ∼ =K ∗ ∼ = φ K ∗ +2 P ( e ∪ e ) . Therefore the E -page is as in the left of Figure 2. By comparing thisAHSS with that of e ∪ e ∼ = S , e ∪ e ∼ = S , e ∪ e ∪ e ∪ e ∼ = S and e ∪ e ∪ e ∪ e ∼ = R , × S along the homomorphism induced from theinclusion maps, we obtain that the differential d pq is d q = (cid:16) C C (cid:17) : KO ∗ ⊕ KO ∗ +4 → K ∗ ⊕ KO ∗ ⊕ KO ∗ +4 ,d q = ( C C ) : K ∗ ⊕ KO ∗ ⊕ KO ∗ +4 → K ∗ . Hence the E -page is as in the right of Figure 4.7 Z ⊕ Z Z Z ⊕ Z Z ⊕ Z Z ⊕ Z Z Z ⊕ Z Z Z ⊕ Z Z ⊕ Z Z ⊕ Z Z E pq Z Z Z Z ⊕ Z Z Z Z Z Z Z Z ⊕ Z Z Z Z E pq Figure 4. E and E pages of AHSS for φ K ∗ ,σP ( ˆΠ)It is automatic from Figure 4 that the higher differentials d pqn ( n ≥ p + q ≡ nontrivial, we obtain φ K ∗ ,σP ( ˆΠ) ∼ = Z ∗ ≡ , Z ∗ ≡ , Z ∗ ≡ , Z ⊕ Z ∗ ≡ , by comparing Figure 4 with (6.6) through the twisted crystallographic T-duality isomorphism. Remark . The reasons that φ K n, − v ( V /
Π) and φ K n,σ ( ˆΠ) are periodic withperiod 4 are as follows: • Let τ φ ∈ H ( P ; φ T ) be the cocycle determined by τ ( e, e ) = τ ( e, p ) = τ ( p, e ) = 1 and τ φ ( p, p ) = −
1. Then φ K n,τ φ P ( V /
Π) is identified withthe Quaternionic K-theory. The product line bundle L = V / Π × C gives rise to a Quaternionic line bundle on V /
Π by the Quaternionicstructure ( x, y, z ) ( − x, y + 1 , e πiy ¯ z ). This line bundle definesan element [ L ] ∈ φ K n,τ φ P ( V /
Π), and its internal tensor product in-duces an isomorphism φ K n, − v P ( V / Π) → φ K n, − v + τ φ P ( V /
Π), whose in-verse is the tensor product with the dual line bundle L ∗ . It isknown [Gom17a, Subsection 3.3] that τ φ has the effect of degreeshift by 4. Combining these isomorphisms, we get the periodicity φ K n, − v P ( V / Π) ∼ = φ K n +4 , − v P ( V /
Π). (The same argument can be seenin [Bar13].) • Let ι : ˆΠ → ˆΠ denote the homeomorphism ι ( k x , k y ) = ( k x , k y +1 / P -equivariant, its pull-back induces an isomorphism ι ∗ : φ K n,σP ( ˆΠ) → φ K n,ι ∗ σP ( ˆΠ). Since ι ∗ σ ( p, p )( k x , k y ) = e πi ( k y +1 / = − σ ( p, p )( k x , k y ), we have ι ∗ σ = σ + τ φ , and τ φ has the effect ofthe degree shift by 4. Combining them, we get the periodicity φ K n,σP ( ˆΠ) → φ K n +4 ,σP ( ˆΠ).We remark that the same argument shows that the twisted equivariant K-groups (6.4) and (6.5) have the periodicity with period 4.6.2.3. Other symmetry types.
In the same way as the 1-dimensional case, wemay consider some variations of 2-dimensional pg grey magnetic space group.Each of them corresponds to one of eight real Cartan labels as following.(1) The case that G , φ , c are as above and τ ∈ H ( P ; φ T ) is determinedby τ ( p, p ) = − G , φ are as above and c : G → Z is the surjection.There are two choices of τ determined by τ ( p, p ) = ± G = pg × Z , φ is the composition of pr pg and thesurjection pg → Z and c := pr Z . There are four choices of τ deter-mined by τ ( p, p ) = ± τ ( c, c ) = ±
1, where p, c are generators of P = G/ Π ∼ = Z × Z , corresponding to symmetry types BDI, DIII,CI, CII. WISTED CRYSTALLOGRAPHIC T-DUALITY 47
The same calculation as the case of type AI symmetry shows that in eachof these cases the twisted equivariant K-group of
V /
Π is isomorphic to (6.6)shifted by m , where m ∈ Z / Z is the one indicated in Table 3.7. Functoriality of the twisted crystallographic T-duality
In this section we discuss the functoriality of the twisted crystallographicT-duality map. For the Baum–Connes assembly map, it was proved byValette [Val03] that, for a homomorphism φ : Γ → Γ of groups, there is acommutative diagram K Γ ∗ ( E Γ ) µ Γ1 / / (cid:15) (cid:15) K ∗ ( C ∗ Γ ) (cid:15) (cid:15) K Γ ∗ ( E Γ ) µ Γ2 / / K ∗ ( C ∗ Γ ) . Theorem 5.22 suggests that a similar functoriality holds for twisted crystal-lographic T-duality.Here we treat the following setting: Let G be a twisted crystallographicgroup and let H be a subgroup of G . We write Σ := H ∩ Π and Q := H/ Σ ⊂ P . Then H is also regarded as a twisted crystallographic group bythe following lemma. Lemma 7.1.
There is an affine subspace W ⊂ V of rank d ′ such that α h ( W ) = W for any h ∈ H and Σ ⊂ Euc( W ) ∩ R d ∼ = R d ′ is of full-rank.Proof. Let Π R and Σ R denote the R -linear spans of Π and Σ respectively.Note that Π R is nothing but the group R d of translations. Since h Σ R h − =Σ R , the action of H on V is reduced to the Q -action of the affine space V / Σ R . Since this is an affine action of a finite group, there is a fixed pointin V / Σ R , which is represented by a Σ R -orbit W := v Σ R ∈ V / Σ R . This isthe desired affine subspace of V . (cid:3) Induction of topological insulators.
Now we define the induction oftopological insulators with the symmetry of twisted crystallographic groups.Let X := G · x be a G -orbit in V . We choose the reference point x ∈ V asan element of W . Here we employ the internal degree of freedom K over X as the one of the form K ′ ˆ ⊗ ℓ ( G x ), in the way that ℓ ( X, K ) ∼ = ℓ ( G, K ′ ).Then it is decomposed as the direct sum L gH ∈ G/H ℓ ( H, K ′ ). Definition 7.2.
Let h ∈ H N ( H, φ, t ). We define the induction of topologicalinsulator as Ind GH ( h ) := M gH ∈ G/H U g hU ∗ g ∈ B (cid:16) M gH ∈ G/H ℓ ( H, K ′ ) (cid:17) . We also use the same letter Ind GH for the induced map TP ( H, φ, t ) → TP ( G, φ, t ). Under the identification in Proposition 3.27, this homomorphism is de-scribed as a composition of twisted equivariant K-theory operations. To seethis, we decompose the induction into two steps. For G and H as above, let H ′ denote the intermediate subgroup q − ( Q ) ⊂ G , where q : G → P denotethe projection. Then H ′ is also a twisted crystallographic group acting on V and K , such that H ′ ∩ R d = Π and has point group Q . By definition wehave Ind GH = Ind GH ′ ◦ Ind H ′ H . (7.3)Here we write σ H , σ H ′ and σ G for the 2-cocycles on Q y ˆΣ, Q y ˆΠ and P y ˆΠ associated to the extensions H , H ′ and G as in (3.24) respectively.Firstly we describe the map Ind GH ′ , i.e., the induction in the case thatΣ = Π. Lemma 7.4.
Let P be a finite group and let Q ≤ P be a subgroup. For a Q -space X and a twist ( φ, s ) of Q y X , there is a twist ( ˜ φ, ˜ s ) of P y P × Q X and an isomorphism I PQ : φ K ∗ , s Q ( X ) → ˜ φ K ∗ , ˜ s P ( P × Q X ) . Moreover, if X is a P -space and ( φ, s ) is the restriction of a twist of P y X ,then ( ˜ φ, ˜ s ) is the pull-back of ( φ, s ) by the P -equivariant map ρ : P × Q X → X given by the multiplication [ p, x ] px .Proof. This follows from the fact that the inclusion of action groupoids X ⋊ Q → ( P × Q X ) ⋊ P is a local equivalence. Indeed, the mapΓ c ( P × Q X, Fred( P × Q H )) P → Γ c ( X, Fred( H )) Q given by the restriction of the section to X ⊂ P × Q X is a homeomorphism.The bundle P × Q H over P × Q X is a universal twisted P -equivariant Hilbertbundle, to which a twist ( ˜ φ, ˜ s ) is associated. Finally, consider the case that X is a P -space and H is a twisted P -equivariant Hilbert bundle on X . Then P × Q H is isomorphic to ρ ∗ H , and hence ( ˜ φ, ˜ s ) = ρ ∗ ( φ, s ). (cid:3) Definition 7.5.
Let X be a P -space and let ( φ, s ) be a twist on P y X .We define the induction map as the compositionInd PQ := ρ ! ◦ I PQ : φ K ∗ , s Q ( X ) ∼ = ρ ∗ φ K ∗ ,ρ ∗ s P ( P × Q X ) → φ K ∗ , s P ( X ) . Here ρ ! is the push-forward map with respect to ρ . Lemma 7.6.
Through the isomorphism in Proposition 3.27, the induction
Ind GH ′ in the sense of Definition 7.2 is identified with Ind PQ : φ K ∗ , t + σ H ′ Q ( ˆΠ) → φ K ∗ , t + σ G P ( ˆΠ) .Proof. First of all, the statement makes sense because σ H ′ is the restrictionof σ G to Q y ˆΠ, which follows by definition (3.24). Now the lemma followsfrom the following presentation of the push-forward ρ ! with respect to a finite WISTED CRYSTALLOGRAPHIC T-DUALITY 49 covering map: Let ρ ! H denote the ( φ, t )-twisted P -equivariant Hilbert bun-dle on X defined as ( ρ ! H ) x = L ρ (¯ x )= x H ¯ x and let ρ ! F ∈ Γ c ( X, Fred( ρ ! H )) P given by ρ ! ( F ) x = M ρ (¯ x )= x F ¯ x ∈ Fred( ρ ! H ) x . Then we have ρ ! [ F ] = [ ρ ! ( F )]. This is proved in the same way as the caseof genuine push-forward, given in [CS84, Proposition 2.9]. (cid:3) Next we describe the map Ind H ′ H , i.e., the induction in the case when thetwisted crystallographic groups have the same point group Q . Here, theinclusion Σ → Π induces the surjection of Pontrjagin duals θ : ˆΠ → ˆΣ. Thismap is Q -equivariant and θ ∗ σ H = σ | H ′ holds, which is checked as σ H ′ ( p, q ) χ = χ ( s ( p ) s ( q ) s ( pq ) − ) = σ H ( p, q ) θ ( χ ) , since s ( p ) s ( q ) s ( pq ) − ∈ Σ and θ ( χ ) := χ | Σ . Hence the pull-back by p in-duces a homomorphism between twisted equivariant K-groups under con-sideration. Lemma 7.7.
Through the isomorphism in Proposition 3.27, the induction
Ind H ′ H is identified with θ ∗ : φ K ∗ , t + σ H ′ Q ( ˆΣ) → φ K ∗ , t + σ H ′ P ( ˆΠ) .Proof. Through the Fourier transform, a gapped Hamiltonian h ∈ H N ( H, φ, t )corresponds to a continuous function on ˆΣ taking values in B ( ˜ K ). Now itis a standard fact of Fourier transform that Ind H ′ H h = θ ∗ h as B ( ˜ K )-valuedfunctions on ˆΠ. (cid:3) In summary, we obtain the following description of the induction map.
Proposition 7.8.
Under the isomorphism in Proposition 3.27, the induc-tion
Ind GH in the sense of Definition 7.2 corresponds to the composition Ind PQ ◦ θ ∗ : φ K ∗ , t + σQ ( ˆΣ) → φ K ∗ , t + σQ ( ˆΠ) → φ K ∗ , t + σP ( ˆΠ) . Push-forward and functoriality.
Now we demonstrate the maintheorem of this section. The inclusion W → V extends to a G -equivariantcontinuous map ˜ ι : G × H W → V, [ g, w ] gw, which induces the P -equivariant map ι : P × Q W/ Σ → V / Π . The push-forward by ι gives a group homomorphism ι ! : φ K ∗ , t Q ( W/ Σ) ∼ = φ K ∗ , t P ( P × Q W ) → φ K ∗ , t P ( V / Π) . (7.9) Theorem 7.10.
The diagram φ K ∗− m, t − w Q ( W/ Σ) φ T t H / / ι ! (cid:15) (cid:15) φ K ∗ , t + σQ ( ˆΣ) Ind GH (cid:15) (cid:15) φ K ∗− n, t − v P ( V / Π) φ T t G / / φ K ∗ , t + σP ( ˆΠ)(7.11) commutes. Lemma 7.12.
The diagram (7.11) commutes when
Π = Σ .Proof.
In this case we have ι ! = Ind PQ by definition. Now the lemma followsfrom the commutativity of Ind PQ with the pull-back, the product and thepush-forward with respect to P -equivariant maps. (cid:3) Lemma 7.13.
The diagram (7.11) commutes when P = Q .Proof. In this case Ind GH = θ ∗ holds. Let π ′ and ˆ π ′ denote the projectionsfrom W/ Σ × ˆΣ to ˆΣ and W/ Σ respectively. The lemma follows from thecommutativity of the diagram φ K ∗− m, t − w P ( W/ Σ) ι ! (cid:15) (cid:15) (ˆ π ′ ) ∗ / / ˆ π ∗ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ φ K ∗− m, t − w P ( W/ Σ × ˆΣ) θ ∗ (cid:15) (cid:15) [[ Q ]] / / φ K ∗− m, t − w + σP ( W/ Σ × ˆΣ) π ′ ! / / θ ∗ (cid:15) (cid:15) φ K ∗ , t + σP (ˆΣ) θ ∗ (cid:15) (cid:15) φ K ∗− m, t − w P ( W/ Σ × ˆΠ) [[ P ]] / / ι ! (cid:15) (cid:15) φ K ∗− m, t − w + σP ( W/ Σ × ˆΠ) ι ! (cid:15) (cid:15) π ′ ! ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ (2) φ K ∗− n, t − v P ( V/ Π) ˆ π ∗ / / (1) φ K ∗− n, t − v P ( V/ Π × ˆΠ) [[ P ]] / / φ K ∗− n, t − v + σP ( V/ Π × ˆΠ) π ! / / φ K ∗− m, t − w P ( ˆΠ) . Among them, the square (1) commutes since ι ! and ˆ π ∗ are given by theKasparov product with the elements of the form [ ι ! ] ˆ ⊗ id and id ˆ ⊗ [ˆ π ∗ ] re-spectively. The square (2) also commutes by the same reason. (cid:3) Proof of Theorem 7.10.
The theorem follows from Lemma 7.12 and Lemma7.13 since the diagram φ K ∗ , t − w Q ( W/ Σ) ι ! / / φ T t H (cid:15) (cid:15) φ K ∗− v , t Q ( V / Π) φ T t H ′ (cid:15) (cid:15) ι ! / / φ K ∗ , t − v P ( V / Π) φ T t G (cid:15) (cid:15) φ K −∗ , t + σQ ( ˆΣ) Ind H ′ H / / φ K −∗ , t + σQ ( ˆΠ) Ind GH ′ / / φ K −∗ , t + σP ( ˆΠ)commutes. (cid:3) Combining Theorem 7.10 with Theorem 5.22, we obtain the followingcorollary.
WISTED CRYSTALLOGRAPHIC T-DUALITY 51
Corollary 7.14.
The diagram φ K H ∗ , t ( W ) φ µ Σ H / / ˜ ι ∗ (cid:15) (cid:15) φ K −∗ , t + σQ ( ˆΣ) Ind GH (cid:15) (cid:15) φ K G ∗ , t ( V ) φ µ Π G / / φ K −∗ , t + σP ( ˆΠ) commutes. This is a variation of the result of Valette [Val03] for twisted partialassembly maps.7.3.
Atomic insulator and induction.
We finish the section with an ex-planation of the terminology ”atomic insulators”. An atomic insulator isa (possibly topological) insulator which has nearly flat energy bands withenergies corresponding to the electric spectrum of an isolated atom. In[SSG17], atomic insulators in K -theory are constructed from informationon configurations of atoms, known as Wyckoff positions. This constructionis mathematically formulated by the results in this section.We consider Theorem 7.10 in the case that H is a finite group. In thiscase W as in Lemma 7.1 is a fixed point of the H -action onto V and G × H W corresponds to a G -orbit in V . Definition 7.15. A G -orbit in V is called a Wyckoff position . A topolog-ical insulator induced from the stabilizer subgroup G x is called an atomicinsulator .We remark that, for a space group G , the stabilizer group G x of anypoint x ∈ V is a finite group. In [Hah87], a Wyckoff position is defined tobe a subset of V consisting of all points x ∈ V for which G x are conjugatesubgroups of G . Our definition of Wyckoff positions is slightly modifiedfrom the standard definitions, which is suitable for the purpose of definingatomic insulators. Example . To provide examples, let us take G = Z ⋊ Z to be the2-dimensional space group p4 , where Π = Z ⊂ V = R is the standardsquare lattice and the point group P = Z ⊂ SO (2) acts on V by rotation.To suppress notations, we make use of the identifications V = C , Π = Z ⊕ Z i and P = Z = {± , ± i } . In [SSG17], the K -theory K P ( ˆΠ) is determined asthe following R ( Z )-moduleK P ( ˆΠ) ∼ = R ( Z ) ⊕ R ( Z ) ⊕ (1 − t + t − t ) , where the representation ring R ( Z ) ∼ = Z [ t ] / (1 − t ) of Z is generatedby the irreducible representation Z ⊂ U (1). The first direct summand R ( Z ) is generated by an equivariant line bundle with non-trivial Chernclass, whereas the second direct summand is K P (pt) ∼ = R ( Z ). (a) For the Wyckoff position W a = Π, we have the stabilizer groups H ∼ = Z . For pt ∈ W a , the image of 1 ∈ K H (pt) ∼ = R ( Z ) underInd GH : K H (pt) → K P ( ˆΠ) is 1 ∈ R ( Z ) ∼ = K P (pt).(b) For the Wyckoff position W b = i + Π, we have the stabilizergroups H ∼ = Z . For pt ∈ W b , the image of 1 ∈ R ( Z ) ∼ = K H (pt)under Ind GH : K H (pt) → K P ( ˆΠ) is represented by the product linebundle ˆΠ × C with the Z -action (cid:16) (cid:18) k k (cid:19) , z (cid:17) i (cid:16) (cid:18) − k k (cid:19) , e − πik z (cid:17) . (c) For the Wyckoff position W c = ( + Π) ⊔ ( i + Π), we have thestabilizer groups H ∼ = Z . For pt ∈ W c , the image of 1 ∈ R ( Z ) ∼ =K H (pt) under Ind GH : K H (pt) → K P ( ˆΠ) is represented by the productvector bundle ˆΠ × C with the Z -action (cid:16) (cid:18) k k (cid:19) , (cid:18) zw (cid:19) (cid:17) i → (cid:16) (cid:18) − k k (cid:19) , (cid:18) e − πik (cid:19) (cid:18) zw (cid:19) (cid:17) . (d) For any x
6∈ W a ∪ W b ∪ W c , we have the Wyckoff position W d = F g ∈ Z ( gx + Π), for which the stabilizer groups are H ∼ = 1. Note thatthe Wyckoff positions of this type are homotopic to each other. Forpt = x ∈ W d , the image of 1 ∈ K H (pt) ∼ = Z under Ind GH : K H (pt) → K P ( ˆΠ) is represented by the product bundle ˆΠ × ℓ ( Z ), where ℓ ( Z )is the left regular representation of Z .In [SSG18], the atomic insulators associated to W a and W c are shown togenerate the submodule R ( Z ) ⊕ (1 − t + t − t ) ⊂ K P ( ˆΠ) with trivial Chernclasses.We say that two Wyckoff positions (in our sense) W and W ′ are homotopicif there is a continuous G -equivariant map f : [0 , × G/H → V such that f ( t, · ) is injective, Im( f (0 , · )) = W and Im( f (1 , · )) = W ′ holds. Lemma 7.17.
There is a one-to-one correspondence between homotopyclasses of Wyckoff positions and finite subgroups of G .Proof. Let H be a finite subgroup. Then there is a homeomorphism betweenMap( G/H, V ) G and V H mapping f : G/H → V to f ( eH ) ∈ V . Now V H isan affine subspace of V , and hence is contractible. (cid:3) Now let us rephrase Corollary 7.14 in the case where H is finite. Corollary 7.18. the diagram φ K ∗ , t H (pt) ι ! (cid:15) (cid:15) φ K ∗ , t H (pt) Ind GH (cid:15) (cid:15) φ K ∗− n, t − v P ( V / Π) φ T t G / / φ K ∗ , t + σP ( ˆΠ) . WISTED CRYSTALLOGRAPHIC T-DUALITY 53 commutes.
The right vertical map is a mathematical formulation of the constructionof atomic insulators in [SSG17].
Appendix A. Equivariant cohomology
This appendix is a brief account of equivariant cohomology. We refer to[AP93, Hsi75, Tu06] for more details.Let P be a finite group acting on a topological space X (which we assumeto be “nice” enough, like a smooth manifold or a P -CW complex). Then,for any n ∈ Z , the n th P -equivariant cohomology of X , in the sense of Borel,is defined to be the following (singular) cohomology H nP ( X ; A ) = H n ( EP × P X ; A ) , where A is an abelian group, and EP × P X is the so-called the Borel con-struction or the homotopy quotient of X . This is the quotient of the productof the universal P -bundle Bπ : EP → BP and X under the diagonal actionof P .Being a cohomology of EP × P X , the equivariant cohomology above canbe twisted by an element of H P ( X ; Z ) = H ( EP × P X ; Z ). In partic-ular, a homomorphism φ : P → Z defines an element of H ( BP ; Z ) ∼ =Hom( π ( BP ) , Z ), in view of the fact that the fundamental group of BP is P . Then the natural projection EP × P X → BP induces an element of H P ( X ; Z ) by pull-back. We write H nP ( X ; φ A ) = H n ( EP × P X ; φ A ) for the P -equivariant cohomology of X twisted by φ .Suppose that X = pt comprises a single point. In this case, the Borelconstruction agrees with the classifying space BP of P . A classical fact isthat the (singular) cohomology of BP is isomorphic to the cohomology ofthe group P , so that we have H nP (pt; A ) ∼ = H n ( BP ; A ) ∼ = H n ( P ; A ) . The coefficient A can also be twisted by a homomorphism φ : P → Z : H nP (pt; φ A ) ∼ = H n ( BP ; φ A ) ∼ = H n ( P ; φ A ) . The φ -twisted group cohomology with coefficients in a (right) P -module A isdefined as follows: Let C n ( P ; φ A ) be the group of maps c : n z }| { P × · · · × P → A .We define a homomorphism δ : C n ( P ; φ A ) → C n +1 ( P ; φ A ) by( δc )( p , . . . , p n ) = φ ( a ) c ( p , . . . , p n ) + n − X i =1 ( − i c ( p , . . . , p i − , p i +1 , . . . , p n )+ ( − n c ( p , . . . , p n − ) p n . It turns out that ( C ∗ ( P ; φ A ) , δ ) is a cochain complex, and its n th cohomol-ogy is H n ( P ; φ A ). For instance, in the simple case that φ is trivial and A is a trivial P -module, we immediately see H ( P ; A ) ∼ = Hom( P, A ). In the case that P = Z and φ : P → Z is the identity homomorphism, we have H ( Z ; φ T ) ∼ = Z , where T is trivial as a right Z -module. In the case that P = Z × Z and φ : P → Z is the projection onto the first factor, we have H ( Z × Z ; φ T ) ∼ = Z ⊕ Z . Appendix B. φ -twisted Chabert-Echterhoff twistedequivariant KK-theory Here we summarize definitions and standard facts on the φ -twisted Chabert-Echterhoff KK-theory [CE01] defined for a pair of φ -twisted Z -graded ( G, Π)-C*-algebras.B.1.
Definitions.
Let G be a discrete group and let Π be a normal subgroupof G . Set P := G/ Π and let φ : P → Z be a homomorphism. We say thata continuous action of G or P on a complex Banach space (a Hilbert space,C*-algebra, Hilbert C*-module) is φ -linear if each α g is linear if φ ( g ) = 0and antilinear if φ ( g ) = 1. Definition B.1. A φ -twisted ( G, Π) -C*-algebra is a triple ( A, α, σ ), where A is a C*-algebra, α : G y A is a φ -linear G -action and σ : Π → U ( M ( A )) isa group homomorphism satisfying α g ( σ t ) = σ gtg − and α t = Ad σ t for g ∈ G and t ∈ Π.Moreover, we say that a φ -twisted ( G, Π)-C*-algebra (
A, α, σ ) is Z -graded if A is equipped with a linear involutive ∗ -automorphism which com-mutes with the G -action and preserves each σ t for t ∈ Π. Remark
B.2 . We say that a Real C*-algebra is a C*-algebra equipped withan involutive antilinear automorphism (a reference is e.g., [Goo82]). Typi-cally, if A is a Banach ∗ -algebra over R satisfying the C*-norm condition,then it complexification A ⊗ R C is a complex C*-algebra and the complexconjugation a ⊗ λ a ⊗ ¯ λ is a involutive antilinear automorphism. Themost typical example of Real C*-algebra is the complex field C equippedwith the complex conjugation ¯ · . In this paper we simply write as C for theReal C*-algebra ( C , ¯ · ).If A is a Real G -C*-algebra (i.e., a Real C*-algebra with a G -actioncommuting with the complex conjugation), then it is regarded as a φ ′ -twisted G × Z -C*-algebra, where φ ′ := pr Z : G × Z → Z . Passing through(id , φ ) : G → G × Z , A is also regarded as a φ -twisted G -C*-algebra. Moreexplicitly, G acts on A as φ α g ( a ) := ( α g ( a ) φ ( g ) = 0 ,α g ( a ) φ ( g ) = 1 . WISTED CRYSTALLOGRAPHIC T-DUALITY 55
Remark
B.3 . There is a general construction of a φ -twisted ( G, Π)-C*-algebra associated to a φ -twisted G -C*-algebra, under the assumption that φ | Π is trivial. For a φ -twisted G -C*-algebra A , let A ⋊ Π denote the (max-imal or reduced) crossed product C*-algebra. The group G acts on A ⋊ Πas α g ( X a t u t ) = X α g ( a t ) · u gtg − and let σ t := u t . Then ( A ⋊ Π , α, σ ) is a φ -twisted ( G, Π)-C*-algebra. Thecorrespondence extends to a functor between equivariant KK categories inDefinition B.9.
Definition B.4.
Let A be a Z -graded φ -twisted ( G, Π)-C*-algebra. A φ -twisted G -equivariant Hilbert A -module is a Z -graded Hilbert A -module E with an isometric φ -linear action ρ : G y E preserving the Z -grading suchthat • h ρ g ( ξ ) , ρ g ( η ) i = α g ( h ξ, η i ) and • ρ g ( ξa ) = ρ g ( ξ ) α g ( a ).We remark that if A is a ( G, Π)-C*-algebra then ξ ρ t ( ξ ) · σ t determinesa unitary operator on E for t ∈ Π. We write α E for the G -action on B ( E )given by α Eg ( T ) ξ = ( ρ g ◦ T ◦ ρ − g )( ξ ) for any ξ ∈ E . Definition B.5.
Let (
A, α, σ ) and (
B, β, θ ) be Z -graded φ -twisted ( G, Π)-C*-algebras. A φ -twisted ( G, Π) -equivariant Kasparov A - B bimodule is atriple ( E, ϕ, F ) where • a countably generated φ -twisted G -equivariant Hilbert B -module E , • a Z -graded G -equivariant ∗ -homomorphism ϕ : A → B ( E ) satisfy-ing ρ t ( ξ ) = ϕ ( σ t ) ξ · θ ∗ t , • an odd self-adjoint operator F ∈ B ( E ) such that [ ϕ ( a ) , F ] , ϕ ( a )( F −
1) and ϕ ( a )( α Eg ( F ) − F ) are in K ( E ) for any a ∈ A .We say that two φ -twisted G -equivariant Kasparov A - B bimodules ( E i , ϕ i , F i )are homotopic if there is a ( φ, c, τ )-twisted G -equivariant Kasparov A - B [0 , E, ˜ ϕ, ˜ F ) such that each ev ∗ i ( ˜ E, ˜ ϕ, ˜ F ) is unitarily equivalent to( E i , ϕ i , F i ) for i = 0 , Definition B.6.
We define φ KK G, Π ( A, B ) to be the set of homotopy classesof φ -twisted ( G, Π)-equivariant Kasparov A - B bimodules. This set formsan abelian group under the summation [ E , ϕ , F ] + [ E , ϕ , F ] = [ E ⊕ E , ϕ ⊕ ϕ , F ⊕ F ]. Here the zero element is represented by any degeneratebimodule, i.e., a Kasparov A - B bimodule ( E, ϕ, F ) with [ ϕ ( a ) , F ] = 0, F − α Eg ( F ) − F = 0. Definition B.7.
Let ( E , ϕ , F ) be a φ -twisted ( G, Π)-equivariant Kas-parov A - B bimodule and let ( E , ϕ , F ) be a φ -twisted ( G, Π)-equivariantKasparov B - D bimodule. Let ( E , ϕ , F ) ♯ ( E , ϕ , F ) be the set of opera-tors F on E = E ˆ ⊗ B E such that • ( E, ϕ ⊗ , F ) is a φ -twisted Kasparov bimodule, • F is an F -connection, i.e. T x F − F T x and F T ∗ x − T ∗ x F are compact(where T x ∈ B ( E , E ) is defined by ξ x ⊗ B ξ ), • ϕ ( a )[ F, F ˆ ⊗ ϕ ( a ) is positive modulo compacts.It is proved in the same way as [Ska84] that the set ( E , ϕ , F ) ♯ ( E , ϕ , F )is path-connected and hence[ E , ϕ , F ] ⊗ [ E , ϕ , F ] → [ E, ϕ ⊗ , F ]gives a well-defined twisted Kasparov product φ KK G, Π ( A, B ) ⊗ φ KK G, Π ( B, D ) → φ KK G, Π ( A, D ) . Example
B.8 . Let A , B be φ -twisted ( G, Π)-C*-algebras. An equivari-ant A - B imprimitivity bimodule (cf. [RW98, Definition 3.1]) is a φ -twisted( G, Π)-equivariant Hilbert B -module E equipped with a φ -twisted ( G, Π)-equivariant ∗ -isomorphism ϕ : A → K ( E ). We say that A and B are equiv-ariantly Morita equivalent if there is a ( G, Π)-equivariant A - B imprimitivitybimodule. If E is an equivariant A - B imprimitivity bimodule, the Kasparovbimodule [ E, ϕ, ∈ φ KK G, Π ( A, B ) has a multiplicative inverse. That is, A and B are φ KK G, Π -equivalent. In particular, the equivariant KK-groups φ KK G, Π ( D, A ) and φ KK G, Π ( D, B ) are isomorphic for any D .B.2. Partial descent homomorphism.
We also mention that the partialdescent homomorphism [CE01] is also generalized to the φ -twisted setting.Let ( A, α ) and (
B, β ) be φ -twisted G -C*-algebras and let A ⋊ Π and B ⋊ Π bethe associated φ -twisted ( G, Π)-C*-algebra as in Remark B.3. Let (
E, ϕ, F )be a φ -twisted G -equivariant Kasparov A - B bimodule. We associate to it a( G, Π)-equivariant Kasparov A ⋊ Π- B ⋊ Π bimodule ( E ⋊ Π , ϕ ⋊ Π , F Π ) inthe following way. • E ⋊ Π is the completion of c c (Π , E ) equipped with – the c c (Π , B )-valued inner product h ξ, η i ( t ) := P s ∈ Π h ξ ( s ) , ρ s ( η ( s − t ) i , – the c c (Π , B )-action ( ξb )( t ) := P s ∈ Π ξ ( s ) · β s ( b ( s − t )), and – the φ -twisted G -action ˜ ρ g ( ξ )( t ) := ρ g ( ξ ( g − tg )),with respect to the norm k ξ k := kh ξ, ξ ik / B ⋊ Π , • ϕ ⋊ Π : A ⋊ Π → B ( E ⋊ Π) is given by(( ϕ ⋊ Π)( a ) ξ )( t ) := X s ∈ Π a ( s ) ρ s ( ξ ( s − t )) , • ( F Π ξ )( t ) := F ( ξ ( t )), which is a well-defined bounded operator on E ⋊ Π. Definition B.9 (cf. [CE01, Theorem 4.5]) . The correspondence [
E, ϕ, F ] [ E ⋊ Π , ϕ ⋊ Π , F Π ] gives a group homomorphism φ j G, Π : φ KK G ( A, B ) → φ KK G, Π ( A ⋊ Π , B ⋊ Π) , which we call the φ -twisted partial descent map . WISTED CRYSTALLOGRAPHIC T-DUALITY 57
It is proved in the same way as [Kas88, Theorem 3.11] and [CE01, Theorem4.5] that φ j G, Π is functorial, that is, compatible with the Kasparov product.In particular, if x ∈ KK G ( A, B ) is a KK G -equivalence, then φ j G, Π ( x ) is aKK G, Π -equivalence. Example
B.10 . Let A , B be φ -twisted G -C*-algebras and let E be a φ -twisted G -equivariant imprimitivity A - B bimodule. Then E ⋊ Π is a φ -twisted ( G, Π)-equivariant A ⋊ Π- B ⋊ Π bimodule, and hence A ⋊ Π and B ⋊ Π are equivariantly Morita equivalent in the sense of Example B.8. TheKK-element φ j G, Π [ E ] = [ E ⋊ Π] induces a φ KK G, Π -equivalence of A ⋊ Π and B ⋊ Π.B.3.
Twisted equivariant K-theory and Fredholm operators.
Herewe give a description of the group φ KK G, Π ( C , A ) for a Z -graded φ -twistedC*-algebra A as the set of homotopy classes of Fredholm operators. Herewe assume that the quotient group P := G/ Π is a compact group.We say that a φ -twisted G -equivariant Hilbert A -module E is ( G, Π)-equivariant if ρ t ξ = ξ · σ ∗ t for any t ∈ Π and ξ ∈ E . The following lemma iseasily verified. Lemma B.11.
Let ( E , ρ ) be a φ -twisted Z -graded G -equivariant HilbertA-module and let { θ t } t ∈ Π be the collection of unitaries on E determined by ρ t ξ = θ t ξσ ∗ t . Then the right A -module I G, Π ( E ) := { ξ ∈ C b ( G, E ) | ξ ( t − g ) = θ ∗ t ξ ( g ) for any t ∈ Π } with the A -valued inner product h ξ, η i := X p ∈ P ξ ( s ( p )) η ( s ( p )) ∗ , where s : P → G is a section, and the G -action is given by ˜ ρ h ( ξ )( g ) = ρ h ( ξ ( h − g )) , is a φ -twisted ( G, Π) -equivariant Hilbert A -module. For a φ -twisted G -C*-algebra A , let φ H A denote the direct sum of infinitecopies of A regarded as a φ -twisted G -equivariant Hilbert A -module and let φ H op A := φ H A with the opposite Z -grading. Set φ H G, Π A := I G, Π ( φ H A ⊕ φ H op A ) . (B.12) Lemma B.13 (Kasparov stabilization theorem) . Let A be a φ -twisted ( G, Π) -C*-algebra and let E be a countably generated Z -graded φ -twisted ( G, Π) -equivariant Hilbert A -module. Then there is a ( G, Π) -equivariant unitaryisomorphism E ⊕ φ H G, Π A ∼ = φ H G, Π A . Proof.
The following proof is based on [MP84]. By the (non-equivariant)Kasparov stabilization theorem [Kas80a], there is a possibly non-equivariant Z -graded unitary isomorphism U : E ⊕∞ ⊕ φ H A ⊕ φ H op A → φ H A ⊕ φ H op A . This is by definition Π-equivariant. The induced unitary I G, Π U : I G, Π ( E ⊕∞ ⊕ φ H A ⊕ φ H op A ) → I G, Π ( φ H A ⊕ φ H op A ) , i.e., I G, Π U : I G, Π ( E ) ⊕ φ H G, Π A → φ H G, Π A , determined by ( I G, Π U ( ξ ))( g ) = U · ξ ( g ), is G -equivariant. Since E is a ( G, Π)-equivariant Hilbert A -module, I G, Π ( E ) includes the submodule consisting of constant functions, which isisomorphic to E . Hence I G, Π ( E ⊕∞ ) ⊃ E ⊕∞ absorbs E , i.e., there is a ( G, Π)-equivariant unitary V : E ⊕ I G, Π ( E ⊕∞ ) →I G, Π ( E ⊕∞ ). Finally we obtain the desired unitary I G, Π ( U ) ◦ ( V ⊕ id φ H G, Π A ) ◦ (id E ⊕I G, Π ( U ) ∗ ) . (cid:3) Here we recall the strict (strong ∗ ) topology on the space of boundedadjointable operators on Hilbert C*-modules. A sequence of operators T n ∈ B ( E ) converges to 0 in the strict topology if and only if k T n ξ k → k T ∗ n ξ k → ξ ∈ E . Note that, for a locally compact Hausdorffspace X , an operator T ∈ B ( φ H G, Π A ⊗ C ( X ) ) corresponds to a uniformly boundedstrictly continuous function T : X → B ( φ H G, Π A ). Lemma B.14.
Let φ H G, Π A be as above. (1) There is a strictly continuous family of G -invariant even unitaries V s : φ H G, Π A ⊕ φ H G, Π A → φ H G, Π A for s ∈ [0 , such that the isometry V s | φ H G, Π A ⊕ strictly converges to a G -invariant even unitary V . (2) The space U ( φ H G, Π A ) G of G -equivariant even unitaries is connectedwith respect to the strict topology.Proof. We apply the Kasparov stabilization theorem B.13 for the Hilbert A ⊗ C [0 , φ H G, Π A ⊗ C [0 , to obtain a G -equivariant even unitary V : φ H G, Π A ⊗ C [0 , ⊕ φ H G, Π A ⊗ C [0 , → φ H G, Π A ⊗ C [0 , . This corresponds to a strictly continuous family of G -equivariant even uni-taries V s : φ H G, Π A ⊕ φ H G, Π A → φ H G, Π A , for s ∈ [0 , V s | φ H A,P ⊕ ∈ B ( φ H G, Π A )of even isometries converging strictly to V ∈ U ( φ H G, Π A ). This shows (1).To see (2) we notice that, for any T, S ∈ B ( φ H G, Π A ), the family V s diag( T, S ) V ∗ s converges strictly to V T V ∗ . Moreover, Lemma B.13 also gives a unitary W : ( φ H G, Π A ) ⊕∞ = φ H G, Π A ⊕ ( φ H G, Π A ) ⊕∞ → φ H G, Π A . WISTED CRYSTALLOGRAPHIC T-DUALITY 59
Let U be an arbitrary unitary on φ H G, Π A . Then the above V and W gives ahomotopy U = V ∗ V U V ∗ V ∼ V ∗ V (cid:18) U
00 1 (cid:19) V ∗ V = V ∗ V (cid:18) W (cid:19) (cid:18) U
00 1 ∞ (cid:19) (cid:18) W ∗ (cid:19) V ∗ V . Also, the standard homotopy R θ diag( U, R ∗ θ diag(1 , U ∗ ), using the rotationmatrix R θ = (cid:0) cos θ − sin θ sin θ cos θ (cid:1) , connects 1 with diag( U, U ∗ ). Hence we obtain ahomotopydiag( U, ∞ ) = diag( U, ∞ , ∞ ) ∼ diag( U, U ∞ , U ∗∞ ) = diag( U ∞ , U ∗∞ ) ∼ ∞ of unitaries in ( φ H G, Π A ) ⊕∞ . This gives a homotopy connecting U with 1. (cid:3) We write Fred( φ H G, Π A ) G for the space of G -invariant odd self-adjoint op-erators F ∈ B ( φ H G, Π A ) such that F − ∈ K ( φ H G, Π A ). The above lemmasenable us to put the abelian group structure on π (Fred( φ H G, Π A ) G ) as[ F ] + [ F ′ ] := [ V (diag( F, F ′ )) V ∗ ] , where V is an arbitrary choice of ( G, Π)-equivariant unitary φ H G, Π A → φ H G, Π A ⊕ φ H G, Π A . Theorem B.15.
The φ -twisted equivariant KK-group φ KK G, Π ( C , A ) is iso-morphic to π (Fred( φ H G, Π A ) G ) .Proof. The map π (Fred( φ H G, Π A ) G ) → φ KK G, Π ( C , A ) is given by[ F ] [ φ H G, Π A , , F ] . By definition of the equivalence relation on φ KK G, Π ( C , A ), this is a well-defined group homomorphism.Its surjectivity follows from Lemma B.13.Hence we show its injectivity.Assume that F ∈ Fred( φ H G, Π A ) G is mapped to the trivial element in KK-group. Then there is a degenerate triple [ E , π, F ′ ] such that F ⊕ F ′ is ho-motopic to a degenerate element in B ( E ⊕ φ H G, Π A ). By replacing E with π (1) E ⊕ φ H G, Π A (which is isomorphic to φ H G, Π A by Lemma B.13) and F ′ with F ′ ⊕
1, we may assume that E = φ H G, Π A and π (1) = 1. Then F ′ is degenerateif and only if it is a unitary. In summary, we have a homotopy of operatorsdiag( F, F ′ ) ∼ F ′′ in M ( B ( φ H G, Π A )), where F ′ and F ′′ are unitaries.Now, by Lemma B.14 (1) and (2), we obtain a homotopy F ∼ h diag( F, ∼ h diag( F, F ′ ) ∼ h F ′′ ∼ h . This shows that [ F ] = 0. (cid:3) Let Fred p,q ( φ H G, Π A ) denote the subspace of Fred( φ H G, Π A ˆ ⊗ ∆ p,q ) consistingof Fredholm operators F with [ F, c ( v )] = 0 for any v ∈ Cl p,q . Then[ F ] [ φ H G, Π A ˆ ⊗ ∆ p,q , , F ]gives a homomorphism π Fred p,q ( φ H G, Π A ) → φ KK G, Π ( Cl p,q , A ). Corollary B.16.
The φ -twisted equivariant KK-group φ KK G, Π ( Cl p,q , A ) isisomorphic to π (Fred p,q ( φ H G, Π A ) G ) .Proof. Note that φ H G, Π A ˆ ⊗ Cl q,p is isomorphic to P ( φ H G, Π A ˆ ⊗ ∆ p,q ˆ ⊗ Cl q,p ), where P is a rank 1 even projection of Cl p,q ˆ ⊗ Cl q,p . Now the map ϑ : Fred p,q ( φ H G, Π A ) G → Fred( φ H G, Π A ˆ ⊗ Cl q,p ) G given by F P ( F ˆ ⊗ id Cl q,p ) P is a homeomorphism. This is compatiblewith the isomorphism of equivariant KK-theory θ : φ KK G, Π ( Cl p,q , A ) ˆ ⊗ C id Cl q,p −−−−−−−→ φ KK G, Π ( Cl p,q ˆ ⊗ Cl q,p , A ˆ ⊗ Cl q,p ) [ P ] ˆ ⊗ Cl p,q ˆ ⊗ Cl q,p −−−−−−−−−−−→ φ KK G, Π ( C , A ˆ ⊗ Cl q,p ) . This finishes the proof since the diagram π Fred p,q ( φ H G, Π A ) G / / ϑ (cid:15) (cid:15) φ KK G, Π ( Cl p,q , A ) θ (cid:15) (cid:15) π Fred( φ H G, Π A ˆ ⊗ Cl q,p ) G ∼ = / / φ KK G, Π ( C , A ˆ ⊗ Cl q,p )commutes and the second horizontal map is an isomorphism by TheoremB.15. (cid:3) References [AP93] C. Allday and V. Puppe.
Cohomological methods in transformation groups ,Cambridge Studies in Advanced Mathematics, vol. 32, Cambridge UniversityPress, Cambridge, ISBN 0-521-35022-0, (1993).[AS68] Michael Atiyah and I. M. Singer,
The index of elliptic operators. III , Annalsof Mathematics. Second Series (1968), 546–604.[Bar13] David Baraglia, Conformal Courant algebroids and orientifold T-duality , Inter-national Journal of Geometric Methods in Modern Physics (2013), no. 2,1250084, 35.[BCH94] Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper ac-tions and K -theory of group C ∗ -algebras , C ∗ -algebras: 1943–1993 (San Anto-nio, TX, 1993), 1994, pp. 240–291, Amer. Math. Soc., Providence, RI.[BEM04a] Peter Bouwknegt, Jarah Evslin, and Varghese Mathai, T -duality: Topologychange from H -flux , Communications in Mathematical Physics (2004),no. 2, 383–415.[BEM04b] , Topology and H -flux of T -dual manifolds , Physical Review Letters (2004), no. 18, 181601, 3. WISTED CRYSTALLOGRAPHIC T-DUALITY 61 [Bus87] Thomas Henry Buscher,
A symmetry of the string background field equations ,Physics Letters B (1987), no. 1, 59–62.[CE01] J´erˆome Chabert and Siegfried Echterhoff,
Twisted equivariant KK -theory andthe Baum-Connes conjecture for group extensions , K -Theory. An Interdisci-plinary Journal for the Development, Application, and Influence of K -Theoryin the Mathematical Sciences (2001), no. 2, 157–200.[CS84] Alain Connes and G. Skandalis, The longitudinal index theorem for foliations ,Publication of the Research Institute for Mathematical Sciences (1984),no. 6, 1139–1183.[CYR13] Ching-Kai Chiu, Hong Yao, and Shinsei Ryu, Classification of topological in-sulators and superconductors in the presence of reflection symmetry , PhysicalReview B (August 2013), no. 7, 075142.[FHT11] Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Loop groupsand twisted K -theory III , Annals of Mathematics. Second Series (2011),no. 2, 947–1007.[FM13] Daniel S. Freed and Gregory W. Moore, Twisted equivariant matter , AnnalesHenri Poincar´e. A Journal of Theoretical and Mathematical Physics (2013),no. 8, 1927–2023.[Fre12] Daniel S. Freed, On Wigner’s theorem , Proceedings of the Freedman Fest, 2012,pp. 83–89, Geom. Topol. Publ., Coventry.[Fu11] Liang Fu,
Topological crystalline insulators , Physical Review Letters (March 2011), no. 10, 106802.[Gom17a] Kiyonori Gomi,
Freed–Moore K-theory , preprint (2017). arXiv:1705.09134 [math.KT] .[Gom17b] ,
Twists on the torus equivariant under the 2-dimensional crystallo-graphic point groups , SIGMA. Symmetry, Integrability and Geometry. Methodsand Applications (2017), Paper No. 014, 38.[Goo82] K. R. Goodearl. Notes on real and complex C ∗ -algebras , Shiva MathematicsSeries, vol. 5, Shiva Publishing Ltd., Nantwich, ISBN 0-906812-16-X, (1982).[GT19] Kiyonori Gomi and Guo Chuan Thiang, Crystallographic T-duality , Journal ofGeometry and Physics (2019), 50–77.[Hah87] Theo Hahn (ed.)
International tables for crystallography. Vol. A , Second, Pub-lished for the International Union of Crystallography, Chester; by D. ReidelPublishing Co., Dordrecht, ISBN 90-277-2280-3, (1987).[HK01] Nigel Higson and Gennadi G. Kasparov, E -theory and KK -theory for groupswhich act properly and isometrically on Hilbert space , Inventiones Mathemati-cae (2001), no. 1, 23–74.[HMT16] Keith C. Hannabuss, Varghese Mathai, and Guo Chuan Thiang, T-dualitysimplifies bulk-boundary correspondence: The parametrised case , Advances inTheoretical and Mathematical Physics (2016), no. 5, 1193–1226.[Hsi75] Wu-Yi Hsiang. Cohomology theory of topological transformation groups ,Springer-Verlag, New York-Heidelberg, (1975).[Kas80a] Gennadi G. Kasparov,
Hilbert C ∗ -modules: Theorems of Stinespring andVoiculescu , Journal of Operator Theory (1980), no. 1, 133–150.[Kas80b] , The operator K -functor and extensions of C ∗ -algebras , IzvestiyaAkademii Nauk SSSR. Seriya Matematicheskaya (1980), no. 3, 571–636,719.[Kas88] , Equivariant KK -theory and the Novikov conjecture , Inventiones Math-ematicae (1988), no. 1, 147–201.[Kit09] Alexei Kitaev, Periodic table for topological insulators and supercon-ductors , AIP Conference Proceedings (2009), no. 1, 22–30,https://aip.scitation.org/doi/pdf/10.1063/1.3149495. [KL02] V Kopsk´y and DB Litvin (eds.)
International Tables for Crystallography, vol.E: Subperiodic groups , Springer, (2002).[Kub16] Yosuke Kubota,
Notes on twisted equivariant K-theory for C * -algebras , Inter-national Journal of Mathematics (2016), no. 6, 1650058, 28.[Kub17] , Controlled topological phases and bulk-edge correspondence , Commu-nications in Mathematical Physics (2017), no. 2, 493–525.[Lif05] R. Lifshitz,
Magnetic point groups and space groups , Encyclopedia of condensedmatter physics, 2005, pp. 219–226, Elsevier, Oxford.[LT20] Matthias Ludewig and Guo Chuan Thiang,
Good Wannier bases in Hilbertmodules associated to topological insulators , Journal of Mathematical Physics (2020), no. 6, 061902.[MEM10] Roger S. K. Mong, Andrew M. Essin, and Joel E. Moore, Antiferromagnetictopological insulators , Physical Review B (June 2010), no. 24, 245209.[MF13] Takahiro Morimoto and Akira Furusaki, Topological classification with addi-tional symmetries from Clifford algebras , Physical Review B (September2013), no. 12, 125129.[MP84] J. A. Mingo and W. J. Phillips, Equivariant triviality theorems for Hilbert C ∗ -modules , Proceedings of the American Mathematical Society (1984), no. 2,225–230.[MR05] Varghese Mathai and Jonathan Rosenberg, T -duality for torus bundles with H -fluxes via noncommutative topology , Communications in Mathematical Physics (2005), no. 3, 705–721.[MR14] , T-duality for circle bundles via noncommutative geometry , Advancesin Theoretical and Mathematical Physics (2014), no. 6, 1437–1462.[MT16] Varghese Mathai and Guo Chuan Thiang, T-Duality Simplifies Bulk-BoundaryCorrespondence , Communications in Mathematical Physics (July 2016),no. 2, 675–701.[OSS19] Nobuyuki Okuma, Masatoshi Sato, and Ken Shiozaki,
Topological classificationunder nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology , PhysicalReview B (February 2019), no. 8, 085127.[Roe02] John Roe, Comparing analytic assembly maps , The Quarterly Journal of Math-ematics (2002), no. 2, 241–248.[RW98] Iain Raeburn and Dana P. Williams. Morita equivalence and continuous-trace C ∗ -algebras , Mathematical Surveys and Monographs, vol. 60, American Math-ematical Society, Providence, RI, ISBN 0-8218-0860-5, (1998).[Sch84] R. L. E. Schwarzenberger, Colour symmetry , The Bulletin of the London Math-ematical Society (1984), no. 3, 209–240.[Ska84] Georges Skandalis, Some remarks on Kasparov theory , Journal of FunctionalAnalysis (1984), no. 3, 337–347.[SRFL08] Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, and Andreas W. W. Lud-wig, Classification of topological insulators and superconductors in three spatialdimensions , Physical Review B (November 2008), no. 19, 195125.[SSG17] Ken Shiozaki, Masatoshi Sato, and Kiyonori Gomi, Topological crystalline ma-terials: General formulation, module structure, and wallpaper groups , PhysicalReview B (June 2017), no. 23, 235425.[SSG18] , Atiyah–Hirzebruch Spectral Sequence in Band Topology: General For-malism and Topological Invariants for 230 Space Groups , preprint (2018). arXiv:1802.06694[cond-mat.str-el] .[Thi16] Guo Chuan Thiang,
On the K-theoretic classification of topological phases ofmatter , Annales Henri Poincar´e (2016), no. 4, 757–794.[Tu06] Jean-Louis Tu, Groupoid cohomology and extensions , Transactions of the Amer-ican Mathematical Society (2006), no. 11, 4721–4747 (electronic).
WISTED CRYSTALLOGRAPHIC T-DUALITY 63 [Val03] Alain Valette,