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Index theory and noncommutative geometry of topological insulator

In this chapter, we report the recent progress in the understanding of the rich mathematical structures of topological insulators in the framework of index theory and noncommutative geometry.


Introduction
Symmetry plays an important role in modern physics and mathematics [58]. In the framework of symmetry protected topological (SPT) orders [9], topological insulators are referred to as U (1) (charge conservation) and time reversal symmetry protected topological order. The goal of this chapter is to understand the geometry and topology of time reversal symmetry in topological insulators.
Due to time reversal symmetry, topological insulators can be characterized by a Z 2 -valued invariant, which is called the topological Z 2 invariant. If this Z 2 invariant originates from K-theory, then one expects an index theorem behind it, since K-theory is the natural receptacle of an index number. As a milestone in mathematics and physics, the Atiyah-Singer index theorem states that the analytical index is equal to the topological index for an elliptic operator on a compact manifold. In physics, index theory can be used to understand quantum anomalies such as the chiral anomaly. As for topological insulators, the topological Z 2 invariant is a parity anomaly, which will be understood as a gauge anomaly via index theory.
This chapter is intended for graduate students interested in the index theory of time reversal invariant topological insulators. The materials in this chapter are not meant to be complete or fully rigorous, rather their goal is to motivate casual readers to learn more about the subjects discussed here by consulting the literature. The geometry of topological insulators is an active research topic in mathematical physics. 2

Topological band theory
Time reversal symmetry is the Z 2 -symmetry that reverses the direction of physical time. In classical physics, a variable is either even or odd with respect to the time reversal symmetry. For example, position and energy are even, and momentum is odd. In quantum mechanics, time reversal symmetry is represented by an anti-unitary operator Θ, called the time reversal operator. For a system with half-integer total spin, time reversal symmetry has the important property Θ 2 = −1, which gives rise to the Kramers degeneracy, that is, there exists a twofold degeneracy in each band. More precisely, if φ is an electronic state, then Θφ is a partner state with the same energy as φ and the pair (φ, Θφ) is called a Kramers pair.
A topological insulator is a gapped fermionic system, the Fermi energy level is assumed to be located in the band gap. Furthermore, the Fermi energy is defined as the zero energy, gapless edge states (between the conduction and valence bands) may pass through the zero energy level and intersect with each other. The fundamental objects to study in topological insulators are Majorana zero modes. Physically, Majorana zero modes are quasi-particle excitations bound to a defect at zero energy. For example, Dirac cones are Majorana zero modes in three-dimensional time reversal invariant fermionic systems. Two Majorana zero modes tend to couple together and behave as an effective composite boson, the presence of an unpaired Majorana zero mode is the characteristic of a non-trivial topological insulator. We have to point out that Majorana zero modes should not be confused with Majorana fermions (or spinors).
One is interested in the band structure of a topological insulator, and the given raw data is a single-particle Hamiltonian. On the one hand, one considers the number of Majorana zero modes, which gives rise to the analytical index. Indeed, the parity of Majorana zero modes defines the topological Z 2 invariant. On the other hand, one models the band structure by a topological vector bundle and obtains an element in topological K-theory. The transition function of such a vector bundle describes a gauge transformation induced by the time reversal symmetry, and it can be used to compute the topological Z 2 invariant. In this section, we will look at the topological band theory in the presence of time reversal symmetry, and give a definition of the topological Z 2 invariant.
In condensed matter physics, one starts with a lattice model and investigates the effective Hamiltonian defined over the lattice in order to understand different phases of matter and possible phase transitions. It is convenient to consider a continuous space rather than the discrete lattice, and the momentum space (or k-space) is used as the base space. Let X be a compact space representing the momentum space, for instance, the torus T d and the sphere S d give two basic examples.
An involutive space (X, τ ) is a compact space X equipped with an involution τ : X → X such that τ 2 = id X . The pair (X, τ ) is also called a Real space with the real structure τ . A typical example is R p,q := R p ⊕ iR q , the real structure τ on R p,q is defined by the complex conjugation, i.e., τ : (x, y) → (x, −y), or equivalently τ | R p = 1 and τ | iR q = −1.
Time reversal symmetry defines an involutive time reversal transformation that flips the sign of a local coordinate. The explicit form of time reversal transformation may vary according to the choice of coordinate system. Definition 1. Time reversal symmetry defines the time reversal transformation on the momentum space X, (1) τ : X → X; x → −x so that (X, τ ) is a Real space.
Example 1. The unit sphere S d in Cartesian coordinates is so the unit sphere with this involution is denoted by S 1,d ⊂ R 1,d . In the same coordinate system, a torus is written as T d = (S 1,1 ) d since a circle is T = S 1,1 .
The band structure of a topological insulator is described by the finite set of band functions {E n : X → R}, and there exists a Kramers pair living in each band. The characteristic property of a topological insulator is completely determined by the edge states passing through the Fermi energy, so it is enough to consider the top band. Applying a partial Fourier transform to the bulk Hamiltonian (or eigenstate), one obtains a boundary Hamiltonian (or eigenstate), from now on we forget about such a partial Fourier transform and only consider the bulk Hamiltonian H. After this identification, the edge states between the conduction and valence bands are adjusted to the corresponding bulk states in the top band, and the Fermi energy level is also shifted to the middle of the top band. In general, a Hilbert bundle is a complex vector bundle equipped with a complete Hermitian metric, which is commonly used to model continuous fields of Hilbert spaces in geometric quantization. The band structure of a topological insulator defines a Hilbert bundle π : H → X, which is of rank 2 since we concentrate on the top band by assumption. If an electronic state is described by a line bundle π : L → X, then the Hilbert bundle is modeled by H = L ⊕ τ * L inspired by the Kramers pair, where τ * L is the pullback of the conjugate line bundle L.
A Real vector bundle (E, ι) over a Real space (X, τ ) is a complex vector bundle E over X with an involution ι such that ι 2 = id E , where ι is an anti-linear bundle isomorphism ι : E → E. This entails that ι induces an anti-linear isomorphism of vector spaces between the fibers E x over x and E τ (x) over τ (x). A Quaternionic vector bundle (E, χ) over (X, τ ) is a complex vector bundle E over X with an anti-linear anti-involution χ such that Example 4. Taking time reversal symmetry into account, the Hilbert bundle π : H → X becomes a Quaternionic vector bundle π : (H, Θ) → (X, τ ), where τ is the time reversal transformation such that τ 2 = 1 and Θ is the time reversal operator satisfying Θ 2 = −1.
The transition function of the Hilbert bundle π : (H, Θ) → (X, τ ) is given by (8) w : (X, τ ) → (U (2), σ) 5 where σ is an involution on the structure group U (2) induced by the time reversal symmetry, The compatibility condition between the involutions τ and σ gives an important property of w, where T denotes the transpose of a matrix. In particular, w becomes a skew-symmetric matrix over the fixed points, , the time reversal transformation is defined by τ (α, β) = (ᾱ, −β), and the fixed points are (α, β) = (±1, 0). In addition, the transition function w is given by In the literature, the transition function w is also called the sewing matrix [20]. In physical terms, w is the gauge transformation induced by the time reversal symmetry. This transition function is the key to constructing a local formula to compute the topological Z 2 invariant. Now we are ready to define the topological Z 2 invariant. Physicists originally proposed two Z 2 invariants from different perspectives. The Kane-Mele invariant was first introduced in the study of the quantum spin Hall effect in graphene [27], and generalized to 3d topological insulators later [19]. On the other hand, the Chern-Simons invariant [49] was defined as (dimensional reductions of) a topological index, which is also called the Wess-Zumino-Witten (WZW) topological term in string theory. These two Z 2 invariants are equivalent, and both will be called the topological Z 2 invariant. The definition of the Kane-Mele invariant is given below, and the topological index will be discussed in detail later.
Recall the relation between the Pfaffian and determinant functions is given by pf 2 (A) = det(A) for a skew-symmetric matrix A. By comparing the Pfaffian with the square root of the determinant of w at x ∈ X τ , the Kane-Mele invariant is the product of the signs of Pfaffians over the fixed points. The Kane-Male invariant reflects the effective quantum field theory of a topological insulator, whose geometric picture is a comparison of orientations between the determinant line bundle and a Pfaffian line bundle. 6 When the Kane-Mele invariant ν = −1, the material is a non-trivial topological insulator, and when ν = 1, the material is a classical insulator. (14) ν = +1, classical insulator −1, topological insulator 3. Topological K-theory As a generalization of the direct sum of two vector spaces, the Whitney sum (as a fiberwise direct sum) defines an additive operation between vector bundles over a compact space X. With the Whitney sum, the set of isomorphism classes of vector bundles becomes a semigroup (or a commutative monoid). Similar to the construction from the monoid of natural numbers N to the abelian group of integers Z, one defines the abelian group K(X), called the K-theory of X, by adding formal differences of two bundles. Such a construction from semigroup to group is called the Grothendieck construction, and the resulting abelian group is called a Grothendieck group. In general, K(X) = KU (X) denotes the complex K-theory in the category of complex vector bundles, and KO(X) denotes the real K-theory constructed from real vector bundles. By definition, topological K-theory classifies stable isomorphism classes of vector bundles over a compact space.
A discrete Z 2 symmetry introduces a real structure, the relevant K-theory is the Real K-theory first introduced by Atiyah [1]. The Real K-theory KR(X, τ ) is defined as the Grothendieck group of Real vector bundles (E, ι) over a compact Real space (X, τ ). Similarly the Quaternionic K-theory KQ(X, τ ) is defined to be the Grothendieck group of finite rank Quaternionic vector bundles (E, χ) over (X, τ ). From the previous section, the topological band theory of a topological insulator is described by a Quaternionic vector bundle over the momentum space π : (H, Θ) → (X, τ ). Hence all possible band structures of a topological insulator can be classified by the KQ-theory KQ(X, τ ).
If the involution τ is understood in the context, then it is always omitted from the notation, e.g., KR(X) = KR(X, τ ). The trivial Quaternionic vector bundle over a point is given by C 2 → pt with the trivial Quaternionic structure χ = 0 −1 1 0 . The reduced Real K-group KR(X) is defined as the kernel of the restriction map i * : KR(X) → KR(pt) = Z, where i is the inclusion of a point i : pt ֒→ X. Higher KR-groups are defined by There exists an isomorphism, the so-called (1, 1)-periodicity of KR-theory, so by convention a KR-group is denoted by KR p−q (X) = KR p,q (X). The Bott periodicity of KR-theory is 8, There exists a canonical isomorphism between KQ-theory and KR-theory, Even though the topological band theory of a topological insulator is classified by KQ-theory, in practice, we tend to use KR-theory due to the developed computational tools in mathematics. When the involution τ on the base manifold X is trivial, notice that τ is obviously trivial on X τ , KR-theory becomes KO-theory, (19) KR n (X, τ = id) = KO n (X) The Bott periodicity of KO-theory is 8, i.e., KO n (X) ∼ = KO n+8 (X). For instance, the KO-theory of a point is given by the following table. i 0 1 2 3 4 5 6 7 The KQ-theory of a sphere or a torus may be computed by iterative decomposition into R 0,k and points, e.g., In the above example, each KQ-theory has at least one Z 2 component, and we know the topological Z 2 invariant belongs to KQ(X). For instance, KQ(T 3 ) has Z 2 components from both KO −1 (pt) and KO −2 (pt). So it is natural to ask where does the topological Z 2 invariant really live in, KO −1 (pt) or KO −2 (pt)? In order to answer this question, one has to look into the local picture of Majorana zero modes. Later index theory will tell us that the topological Z 2 invariant really lives in KO −2 (pt).
If one considers the spectral flow of a family of self-adjoint operators along a path (or a loop), then the relevant K-theory is K(X × S 1 ) = K −1 (X) instead of K(X), the same argument holds for KQ-theory. This is the very trick behind the transition from KQ(X) to KQ −1 (X). It is very important to distinguish the even KQ-group KQ(X) from the odd KQ −1 (X), since the topological index formulas are totally different for the even and odd cases.
For an odd-dimensional space X, it is natural to consider the odd KQ-group KQ −1 (X), which is intimately related to the odd topological index of a gauge transformation [21]. For instance, for a 3d sphere or torus, KQ −1 (X) leads to the desired KO −2 (pt), which is the receptacle of the topological Z 2 invariant.
K-theory is the natural receptacle of index theory, since the index bundle of a family of Fredholm operators gives a prototypical example of a virtual vector bundle. On the other hand, spin geometry provides a beautiful framework for studying the geometry of spinors such as Dirac, Majorana and Weyl fermions. The geometry of Dirac spinors, i.e., Dirac geometry, produces elegant theorems such as the index theory of Dirac operators. It is via Dirac geometry that topological K-theory connects to spin geometry and index theory.
In general, a Fredholm operator F is a bounded linear operator with finite-dimensional kernel kerF and cokernel cokerF , and with closed range ranF . Given a Fredholm operator F , one defines the Fredholm index as the analytical index, (20) ind a (F ) = dim kerF − dim cokerF If F is a family of Fredholm operators parametrized by X, then one considers the virtual vector bundle kerF ⊖ cokerF over X, which is called the index bundle. The index bundle of a Dirac operator (parametrized by X) We are interested in new geometry of Majorana zero modes compared to Dirac geometry. Depending on the given symmetry, an effective Hamiltonian falls into a subclass of Fredholm operators. Analogously, the index bundle of a time reversal invariant Hamiltonian H(x), x ∈ X is expected to give rise to a class in KQ(X). However, things are more complicated when the time reversal symmetry is present, now the input data is the pair (H, Θ). Similar to the canonical decomposition of the Dirac operator D based on a spinor bundle, we define an effective Hamiltonian as follows. The formal difference [kerH] − [kerΘHΘ * ] in K-theory is supposed to describe Majorana zero modes, but as a number dim kerH − dim kerΘHΘ * always vanishes, since a pair of zero modes (φ 0 , Θφ 0 ) has the same energy. Hence a localization procedure is needed to look closer at the local picture of Majorana zero modes, and obtain the topological Z 2 invariant as an index.

Majorana zero modes and analytical index
We introduce the notion of Majorana zero modes and show that one interpretation of the Z 2 invariant is given by the parity of Majorana zero modes. The Majorana zero modes are similar to those found in Bogoliubovde Gennes (BdG) superconductors with particle-hole symmetry. In physics, Majorana zero modes [12] are quasi-particle excitations bound to a defect at zero energy. Two fermionic Majorana zero modes tend to couple together 9 and behave effectively as a boson. For example, Dirac cones are Majorana zero modes in 3d time reversal invariant fermionic systems, and the presence of an unpaired Dirac cone is the characteristic of a 3d non-trivial topological insulator. Majorana zero modes are different from Majorana fermions, which have been understood by the spin representations and spin geometry [38]. Majorana zero modes have a new geometry beyond spin geometry compared to Majorana fermions. Let us look at the geometry of Majorana zero modes, and explain how to get the mod 2 analytical index by localization. Let H be a time reversal invariant single-particle Hamiltonian parametrized by X such that (22) ΘH where Θ is the time reversal operator. The effective Hamiltonian of a (free fermionic) topological insulator is defined by H is constructed to describe a quasi-particle consisting of a real particle (modeled by H) and its mirror image under time reversal symmetry (modeled by ΘHΘ * ). In spin geometry, a Dirac operator is an intertwining operator between spinors, commonly written as Similar to a Dirac operator,H is constructed off-diagonally since a topological insulator is a fermionic chiral system and time reversal symmetry changes chirality. Our definition of the effective Hamiltonian is different from the convention used in physics, where a Hamiltonian is always diagonal such as H ↑ 0 0 H ↓ with H ↑ , H ↓ modeling spinors with opposite chirality. The main reason behind our definition is that we want to construct a skew-adjoint operator, whose analytical index is Z 2 -valued. Notice that the effective HamiltonianH is also time reversal invariant, By assumption, the single-particle Hamiltonian is self-adjoint H * (x) = H(x), and we have which means the effective HamiltonianH is neither self-adjoint nor skewadjoint. In order to construct a skew-adjoint operator based onH, an approximation will be applied to the raw data H. As a remark, we have to point out that the real structures defined by the adjoint * -operation and the time reversal operator Θ are different. There are so many distinct real 10 structures in topological insulators and superconductors, which is the main source of confusions about the God-given real structure. The effective HamiltonianH acts on a Kramers pair (φ, Θφ) ∈ Γ(X, H), which is viewed as a quasi-particle state Φ = (φ, Θφ). A real structure in operator algebra is an anti-linear isometry J such that J 2 = ±1. The time reversal operator Θ gives a real structure such that Θ 2 = −1. Define a new real structure induced by Θ as (26) J satisfying J * = J and J 2 = 1. With respect to J , the Kramers pair Φ = (φ, Θφ) is a Majorana state, since Φ satisfies the real condition J Φ = Φ, i.e., Our definition of Majorana states is a generalized version of the definition used in physics. For example from [12], Majorana states (or Majorana zero modes) are defined as self-adjoint fermionic operators commuting with the Hamiltonian (in the CAR algebra). In our definition, we follow the conventions used in the Dirac geometry, see Example 13. So the dagger †-operation (commonly used in physics) is replaced by a real structure J, and in order for a state φ to be Majorana (or real), the condition φ † = φ is generalized to the real condition Jφ = φ.
Our notation of Majorana states can be translated to the familiar Majorana operators commonly used in physics, and vice versa. Suppose γ 1 and γ 2 are Majorana operators, f + and f − are fermionic (annihilation and creation) operators related to γ i by (28) f Or equivalently, the Majorana operators can be expressed in terms of fermionic operators, There exists a state-operator correspondence involved in the discussions here. By a Bogoliubov transformation, a Majorana state Φ = (φ, Θφ) can be written as The matrix B = 1 i −i 1 induces the Bogoliubov transformation, in this context, the Majorana states γ i are also called Bogoliubov quasi-particles. The fermionic states can be recovered as Notice that the real structure used here is the time reversal operator Θ instead of i in the original case. The Bogoliubov quasi-particles are equivalently obtained by fermionic states, Notice that the imaginary unit i in γ 1 = φ + iΘφ is a formal symbol to connect two particles and form a quasi-particle. In order to avoid unnecessary confusions introduced by i (as another real structure), we tend to use the vector form Φ = (φ, Θφ) to denote a Majorana state. If a Kramers pair (φ, Θφ) is written as a Bogoliubov quasi-particle γ 1 = φ + iΘφ, then the effective HamiltonianH = 0 ΘHΘ * H 0 is accordingly written as H 1 = H + iΘHΘ * . Similarly, for the other Bogoliubov quasiparticle γ 2 = Θφ − iφ, the corresponding Hamiltonian is The multiplication by Θ maps (φ, Θφ) to (Θφ, −φ), so γ 2 is the mirror image of γ 1 under the time reversal symmetry, i.e., γ 2 = Θγ 1 . Define a new Hamiltonian acting on Bogoliubov quasi-particles (γ 1 , At the first glance, the analytical index ofĤ is which is always zero since ΘHΘ * − iH = −i(H + iΘHΘ * ) and H 1 , H 2 have the same kernel. Therefore, the non-trivial invariant is expected to be defined as a mod 2 version of the chiral part, which is the parity of zero modes of Bogoliubov quasi-particles. This heuristic reasoning leads to the desired definition of the topological Z 2 invariant, but we still need a skew-adjoint operator to define a mod 2 analytical index. A Bogoliubov quasi-particle φ + iΘφ is the same as a Majorana state (or Kramers pair) Φ = (φ, Θφ). Zero modes of a Majorana state can only be found around the set of fixed points, since chiral states φ and Θφ could pass through the zero energy level only at a fixed point.
Let us look at a pair of localized Majorana zero modes (γ 0 1 , γ 0 2 = Θγ 0 1 ) around a fixed point, At this fixed point, a winding matrix (or a transition function of the Hilbert bundle) takes one of the following form A direct matrix-vector multiplication shows that In practice, we mainly focus on the real part Φ + 0 = (φ 0 , Θφ 0 ) = γ 0 1 , and the imaginary part (with respect to Θ) Φ − 0 = (Θφ 0 , −φ 0 ) = γ 0 2 is skipped implicitly but can be easily restored. Inspired by spin geometry, we are actually interested in a Majorana state that changes sign, i.e. γ 0 1 → −γ 0 2 , so we define it as a Majorana zero mode. We stress that a Majorana state that remains the same sign, i.e., γ 0 1 → γ 0 2 , will not be counted as a Majorana zero mode.

Definition 4. A Majorana zero mode is defined as a localized Majorana
and changes sign at that fixed point.
The local geometry of a Majorana zero mode (after linearization) is a conical singularity V (x 2 + y 2 − z 2 ) in 3d, that is, φ 0 and Θφ 0 intersect with each other at that fixed point and form a Dirac cone. The intersection point (or center) of a Dirac cone is called a Dirac point by physicists.
Definition 5. The topological Z 2 invariant of a topological insulator is defined as the parity of Majorana zero modes.
Atiyah and Singer introduced a mod 2 analytical index of a real skewadjoint elliptic operator P in [3], (36) ind a (P ) := dim R ker P mod 2 Near a fixed point x ∈ X τ , the single-particle Hamiltonian H(x) ∼ D(x) can be approximated by a Dirac operator D. Accordingly, the effective HamiltonianH ∼D is approximated by a skew-adjoint operatorD, The big difference is that D is self-adjoint D * = D, butD is skew-adjoint D * = −D, so it makes sense to take the mod 2 index ofD.
Theorem 1. By localization,H can be approximated by the skew-adjoint operatorD, the topological Z 2 invariant ν can be interpreted as the mod 2 analytical index ofH, which can be computed by the spectral flow of H modulo 2.
Notice that we actually take the quaternionic dimension in the above analytical index since a zero mode of H is a complex state, (39) ind a (H) = dim H kerH mod 2 Because of the Kramers degeneracy, a Majorana zero mode consists of two complex zero modes of H, that is, dim H kerH = dim C ker H. Furthermore, zero modes of H can only be found around a fixed point, where H is effectively approximated by a Dirac operator D, so dim C ker H = dim C ker D.
Putting this together, we have The classifying spaces of KR-groups are constructed by different subspaces of Fredholm operators in [2], which are also connected to different discrete Z 2 symmetries of general topological insulators in [23]. In the following, we identify the mod 2 analytical index ofH as an element in KO −2 (pt). Let F(H , J) denote the space of Fredholm operators on a Real Hilbert space (H , J), where H is a complex Hilbert space and J is a real structure such that J 2 = 1. In addition, letF(H , J) denote the subspace of skew-adjoint Fredholm operators, which is a classifying space of KR −1 , i.e., for a compact Real space (X, τ ). The effective HamiltonianH is approximated by a skew-adjoint Fredholm operator acting on the Real Hilbert space (L 2 (X, H), J ), so the analytical index belongs to KO −2 (pt), Note that if H is a complex Hilbert space, then the analytical index lives in KO −1 (pt). However, in our case L 2 (X, H) is a quaternionic Hilbert space since for a typical section (φ, Θφ) and any point x ∈ X, (φ(x), Θφ(x)) ∈ H ∼ = C ⊕ ΘC. Hence for the analytical index we first take the quaternionic dimension of the kernel space, and then mod 2 to get ind a (H) ∈ KO −2 (pt). The geometry of Majorana zero modes studies a quasi-particle consisting of a Dirac fermion and its time reversal partner, which is beyond the Dirac geometry but still in the framework of spin geometry. Two Dirac fermions are coupled together in a way that the relevant operator is essentially given by a skew-adjoint Fredholm operator by localization. The geometry of Majorana zero modes is described by a coupled product of two Dirac geometries, which will be clear when we discuss about KR-cycles. The same strategy is widely used in index theory, where a general elliptic operator is always reduced to a Dirac operator.

Topological index formula
The Atiyah-Singer index theorem teaches us that the analytical index can be computed by a topological index. In this section, we will discuss about the mod 2 topological index, which is originally called the Chern-Simons invariant by physicists [49]. The key observation is that the parity anomaly of the topological Z 2 invariant can be translated into a gauge anomaly, and the local formula is the odd topological index of a specific gauge transformation induced by time reversal symmetry. In fact, the topological index gives the bulk theory of a topological insulator, which is an integral over the momentum space as the bulk. 14 As a prelude, let us first briefly review the integer quantum Hall effect, whose topological invariant is the first Chern number. In the quantum Hall effect, the Hall conductance can be quantized into intergers, which is called the TKNN invariant by physicists. This Z-valued invariant can be understood by topological band theory and topological K-theory. The band structure (or Bloch states) of the quantum Hall system defines a complex vector bundle over the momentum space X, which is called the Bloch bundle. As a two-dimensional system, the base manifold is usually taken as a 2sphere X = S 2 or a 2-torus X = T 2 . Without loss of generality, we assume that the Bloch bundle is of rank one. As a consequence, the topological band theory is classified by complex K-theory, and the topological invariant belongs to K(X), e.g., K(T 2 ) = K(S 2 ) = Z. In addition, the topological index used to compute this Z invariant is the integral of the first Chern character, also called the first Chern number c 1 , where p is a projection (s.t. p 2 = p = p * ) representing the Bloch bundle.
Example 8. The Hopf fibration S 1 ֒→ S 3 h −→ S 2 defines the Hopf bundle as a principal U (1)-bundle over a 2-sphere π : P → S 2 . In complex coordinates, the unit 3-sphere S 3 = (z 1 , z 2 ) ∈ C 2 : |z 1 | 2 + |z 2 | 2 = 1 , the Hopf map is defined by The complex coordinates can be changed to real coordinates by the identification, In real coordinates, The Hopf bundle can be represented by a projection, (46) p : The first Chern character is the standard volume form on S 2 , so the first Chern number c 1 = 1.
where 1 S 2 is the trivial line bundle over S 2 . In the K-theory ring K * (S 2 ), the above relation can be rewritten as The relationship between K-theory and reduced K-theory is K(X) = K(X)⊕ Z. If one defines the Bott element as then β is the generator of the reduced K-theory K( Consider a magnetic monopole with a unit magnetic charge placed at the origin in R 3 , the space R 3 \ {0} has the same homopoty type as S 2 , and the corresponding electromagnetic field is modeled by the canonical line bundle L H → S 2 . The first Chern number c 1 = 1 confirms the fact that the magnetic monopole has a unit charge. Indeed, an electromagnetic field can be expressed by the vector potential A as 1-forms in de-Rham cohomology, and the field strength F = dA is basically given by the first Chern character, then the integral of F gives us a charge Q. Laughlin argument told us that the quantization of Hall conductance is a result of the gauge invariance. In the present geometry, a charge pump can be applied to change the magnetic charge, then indirectly change the Hall conductance via the Faraday's law. If a magnetic flux is pumped adiabatically from infinity to the origin, which induces a gauge transformation (51) g : then the total charge now is the integral of F ′ , and the net change of charges ∆Q is the integral of ∆F (or roughly the gauge transformation g) [5].
From integer quantum Hall effect (with a 2d momentum space), one learned that the relevant topological invariant is the first Chern number. For a quantum system with time reversal symmetry, it is natural to consider a 4d momentum space and expect that the second Chern number would give us the right topological invariant. So for a real material with time reversal symmetry, the relevant topological invariant could be obtained by dimensional reduction from 4d to lower dimensions. It is well-known that the first Chern number equals zero for a time reversal invariant system.
In four dimensions, the topological index is the integral of the second Chern character ch 2 (F ), where F is a curvature 2-form. If c 1 = 0, then the second Chern number c 2 is the same as the 4d topological index. For instance, if the momentum space X = S 4 , one decomposes S 4 into two hemispheres, then the topological index can be computed as the winding number over the equator S 3 , (2) is a gauge transformation defined on the overlap of two hemispheres (i.e., S 3 ), for details see [44]. The right hand side is the winding number of g around the 3-sphere, i.e., the degree of g.
In condensed matter physics, the non-abelian Berry connection and Berry curvature are defined by As characteristic classes, the second Chern character ch 2 (f ) and the Chern- In fact, cs 3 (a, f ) is the boundary term of ch 2 (f ), The magneto-electric polarization P 3 over T 3 is defined as the Chern-Simons action of the Berry connection, In general, the Chern-Simons action is gauge invariant up to a winding number. More precisely, under a gauge transformation, The Chern-Simons invariant of a time reversal invariant system is originally defined by the change of magneto-electric polarization under the gauge transformation induced by the time reversal symmetry modulo two [49]. If w is the transition function of the Hilbert bundle, then the Chern-Simons invariant is defined by (59) υ : One has to take the mod 2 version since the topological invariant of a time reversal invariant system is Z 2 -valued from physical observations. Indeed, we can prove that the above winding number is naturally Z 2 -valued, so "mod 2" is redundant. From the above discussion, the second Chern number c 2 over a 4d momentum space is a universal topological invariant for a time reversal invariant system. The 4d topological index is the integral of the second Chern character, which can be reduced to a 3d winding number of a gauge transformation. In topological insulators, there is a unique choice for such a gauge transformation, that is, the specific gauge transformation representing time reversal symmetry. Now let us look at the general theory of the winding number as a topological index. 17 For an odd-dimensional closed manifold X, the odd Chern character of a differentiable map g : X → U (n) (the unitary group) is defined by which is a closed differential form of odd degree. The topological index in odd dimensions is given by the odd index theorem, when the Dirac or A-roof genusÂ(X) = 1 (e.g., Hence the winding number (or degree) of g is the same as an odd topological index. The above discussion can be rephrased rigorously as the Chern-Weil theory of the relative Chern-Simons form, whose quantization gives the odd index theorem.
Proposition 1. For a 3d topological insulator, the topological Z 2 invariant can be computed by the odd topological index of the transition function representing time reversal symmetry (8) Based on the odd topological index, one can understand why the topological invariant is Z 2 -valued as follows. Time reversal symmetry introduces the Kramers degeneracy for an electronic system, e.g., the quantum spin Hall effect. More precisely, each band is doubly degenerate, or there exists a Kramers pair in each band due to the time reversal symmetry. However, there is no a priori ordering between the Kramers pair. If one switches the ordering between the Kramers pair, the transition function w changes sign, and further ind t (w) changes sign. The winding number ind t (w) as a global invariant is independent of the choice of orderings, so ind t (w) must be Z 2valued to eliminate the ambiguity introduced by the undetermined ordering. As a gauge anomaly, Witten already pointed out that the WZW topological term is Z 2 -valued in [59].
The odd topological index formula is also valid in one dimensions, e.g., the Kitaev Majorana chain. However, the topological index formula in two dimensions is more mysterious, the first Chern character modulo 2 cannot give a local formula for the topological Z 2 invariant since the first Chern number is zero, i.e. c 1 = 0. Heuristically, if one considers the net change of charges ∆Q passing through the zero energy level, that is always zero. Indeed, a Majorana zero mode has two opposite-moving chiral states, one chiral state contributes +1 to ∆Q, at the same time the other chiral state contributes −1 to ∆Q. 18 Inspired by the relation between weak and strong topological Z 2 invariants, we propose a topological index formula below by reducing the dimension from 2d to 1d. Let X = T 2 be the momentum space (or the Brillouin torus), so EBZ = S 1,1 × I and ∂EBZ = S 1,1 N ⊔ S 1,1 S , where S 1,1 N , S 1,1 S are the north and south circles in the cobordism S 1,1 × I. Based on the Z 2 -cellular decomposition of (T 2 , τ ), T 2 is decomposed into two copies of S 1,1 × I and the Hilbert bundle π : H → T 2 can be reconstructed from the restricted bundles H S 1,1 ×I by the clutching map w : . The topological index formula on T 2 is defined as the product of the 1d indices from the boundaries S 1,1 We use multiplication instead of addition on the right hand side, since the product of two Z 2 is still Z 2 but the addition of two Z 2 is not Z 2 .
Proposition 2. For a 2d topological insulator over T 2 , the topological Z 2 invariant can be computed by a variation of the odd topological index formula is the transition function restricted to the boundary of the effective Brillouin torus, and it is naturally Z 2 -valued.
In complex K-theory, the odd K-group K −1 (X) has a different interpretation compared to K(X). Roughly, K −1 (X) can be defined based on K(S 1 × X), which is closely related to the spectral flow of a family of self-adjoint operators along a circle. For a class in an odd K-group, one expects an odd topological index. In index theory, the odd topological index is a generalization of the winding number of the determinant function det : U (n) → C × , which was first understood as the index of a Toeplitz operator in operator algebra. In the topological band theory of a topological insulator, the information about the band structure is essentially encoded in the time reversal gauge w, so the 2d topological index is also reduced to a variation of the odd topological index. In particular, when X = T 2 , S is the boundary of the effective Brillouin torus T × I. In sum, the local formula of the topological Z 2 invariant is an odd topological index (or its variation) of the transition function.

Mod 2 index theorem
The topological Z 2 invariant counts the parity of Majorana zero modes, which is interpreted as an analytical index in §4. Such a parity anomaly is translated into a gauge anomaly, which has an interpretation as an odd topological index in §5. By the Atiyah-Singer index theorem, the analytical index is equal to the topological index through spectral flow. In addition, for a topological insulator, the analytical index and topological index both are naturally Z 2 -valued (indeed KO −2 (pt) = Z 2 ). Theorem 2. The Z 2 invariant of topological insulators can be understood as a mod 2 index theorem, whereH is the effective Hamiltonian and w is the time reversal gauge.
There is a famous slogan by Quillen: "Dirac operators are a quantization of connections, and index theory is a quantization of the Chern character." If one spells out this slogan for the the odd index theorem, then one obtains the proof of the above theorem. Let us briefly review the ideas involved in the proof in the following paragraph.
Given two connections A 0 and A 1 on a vector bundle, the relative Chern-Simons form is defined by There exists a transgression formula connecting the relative Chern-Simons form and the Chern character in Chern-Weil theory, where the Chern character of a connection A is defined as ch(A) := tr(e A 2 ). A connection of a vector bundle has a local form A = d + ω, d is the differential (as the trivial connection) and ω ∈ Ω 1 (u(n)) is a 1-form. For a differentiable map g : X → U (n), choose two connections of the trivial bundle X × C n as the relative Chern-Simons form cs(d, g −1 dg) gives the odd Chern character by Taylor expansion, After quantization, the right hand side gives the odd topological index, and the left hand side is a spectral flow sf (D, g −1 Dg) with the connection d replaced by a Dirac operator D. In our case, the analytical index ind a (H) can be computed by the mod 2 spectral flow of the self-adjoint Hamiltonian H, and the time reversal gauge w plays the role of the map g. Putting it together, the mod 2 index theorem of a topological insulator can be proved by Quillen's slogan. Index theory can be best understood in the framework of K-theory. First, from the discussion about the analytical index in §4, we know the topological 20 Z 2 invariant (as an index number) belongs to KO −2 (pt). Second, from the topological index in §5, the essential information is encoded in the time reversal gauge w, which determines a class in KQ −1 (X). If the starting point is an elliptic operator, then its symbol class gives rise to a class in the K-theory of the cotangent bundle over X, i.e., π : T * X → X. In general, the topological index map (constructed by Atiyah) has T * X as the source and an abstract point (or a Euclidean space R N ) as the target space. For a topological insulator, the effective HamiltonianH can be approximated by a skew-adjoint operator, so the relevant topological index map is For a d-dimensional Real space (X, τ ), the Thom isomorphism in KR-theory is given by Combining these two maps gives a map from KR-theory (or KQ-theory) to KO −2 (pt), and we still call it the topological index map.
Example 9. When X = T 3 , the topological index map is a map from When X = T 2 , the topological index map is a map from KQ(T 2 ) to KO −2 (pt), Or equivalently, By the reduction from T 2 to ∂(T × I) = S 1 ⊔ S 1 , we lift the index map to where ∆ is a boundary map in KQ-theory. Since KQ(T 2 ) = KO −2 (pt) and in the reduced KQ-theory, one has which basically is the map where % is the modulo operation in arithmetic. We stress that the bulk theory in 2d is given by KQ(T 2 ), so the topological index map is from KQ(T 2 ) to KO −2 (pt). However, in order to find a local topological index formula, one is forced to reduce from T 2 to ∂(T×I) or lift from KQ(T 2 ) to KQ −1 (S 1 ⊔S 1 ) and make use of the class [w] ∈ KQ −1 (S 1 ⊔ S 1 ).

KR-homology and Index pairing
As discussed in §5, the topological index is the integral of the odd Chern character Ch(g) over the momentum space X, which can be viewed as a pairing between a K-theoretic class [g] ∈ K −1 (X) with the fundamental class [X].
In a modern language, an index can be obtained by the index pairing between K-homology and K-theory (as a generalized cohomology theory).
The index pairing between K-homology and K-theory can be generalized to a paring between cyclic homology and cohomology using Connes-Chern characters. We reformulate the mod 2 index theorem of a topological insulator into an index pairing between KR-homology and KR-theory, which is a preparation for the index pairing in noncommutative geometry.
In spin geometry, one models Majorana spinors by KR-cycles in KRhomology. As an analogy, we will define a KQ-cycle (as a generalized KRcycle) to model Majorana zero modes. Let us first briefly review the definition of a KR-cycle in operator algebra, which gives a canonical representative for the fundamental class in KR-homology.
A K-cycle for a * -algebra of operators A is a triple (A, H , D), also called a spectral triple in noncommutative geometry, where H is a complex Hilbert space, and A has a faithful * -representation on H as bounded operators, i.e., π : A → B(H ). D is a self-adjoint (typically unbounded) operator satisfying the following conditions.
A bounded linear operator T : H → H is called compact if T maps any bounded sequence {ξ k } ∈ H to a sequence {T ξ k } ∈ H with a convergent subsequence. If the representation π is understood in the context, it is always skipped from the notation, and one treats A as a subalgebra of B(H ) (the algebra of bounded linear operators with the norm topology). If the operator D is a Dirac-type operator, then (A, H , D) is called a Dirac-type spectral triple. 22 Example 10. Conisder the circle S 1 parametrized by the angular parameter θ, the canonical K-cycle on S 1 is given by (C ∞ (S 1 ), L 2 (S 1 ), D). Here the function algebra is the smooth algebra C ∞ (S 1 ), the Hilbert space is the square integrable functions L 2 (S 1 ), and the Dirac operator is D = d idθ . A K-cycle (A, H , D) is even (or graded) if there exists a grading operator γ with the properties γ * = γ and γ 2 = 1 such that Dγ = −γD and γπ(a) = π(a)γ for all a ∈ A. Otherwise, a K-cycle is odd (or ungraded). If there exists a grading γ, then the Hilbert space H is also assumed to be Z 2 -graded, which is thought of as the space of (square integrable) sections of a spinor bundle. A typical example of grading operators is given by γ = 1 0 0 −1 .
A (general) real structure is defined by an anti-linear isometry J on H . For example, the time reversal symmetry introduces a real structure. Here we give the representation theory of spinors in spin geometry [38], also called spin representation. By the representation theory of the real Clifford algebra Cl 2k,0 , when 2k = 0 mod 8, it has a unique real pinor representation (i.e., Majorana pinor) and there are two inequivalent real spinor representations (i.e., Majorana-Weyl spinors). When 2k = 2 mod 8, it has a unique quaternionic pinor representation (i.e., symplectic Majorana pinor) and there are two inequivalent complex spinor representations (i.e., Majorana-Weyl spinors). When 2k = 4 mod 8, it has a unique quaternionic pinor representation (i.e., symplectic Majorana pinor) and there are two inequivalent quaternionic spinor representations (i.e., symplectic Majorana-Weyl spinors). When 2k = 6 mod 8, it has a unique real pinor representation (i.e., Majorana pinor) and there are two inequivalent complex spinor representations (i.e., Majorana-Weyl spinors).  The representation theory of the real Clifford algebra in odd dimensions is easier, when 2k − 1 = 1, 7 mod 8, there is a unique real spinor representation; when 2k − 1 = 3, 5 mod 8, there is a unique quaternionic spinor representation.  [25]. The following example is the classical Dirac geometry modeling Majorana spinors in spin geometry [38,56]. As a summary, a real spectral triple or KR-cycle satisfies the commutation relations given in the following table.
j mod 8 0 1 2 3 4 5 6 7 In addition, a KR j -cycle is said to have the KO-dimension j mod 8. KOdimension of a real spectral triple, first introduced by Connes in [11], is the shift in the grading on KR-homology. The idea comes from the KOorientation represented by a fundamental class in KO-homology, which induces the Poincaré duality for KO-theory. Let us construct a generalized KR-cycle to model the geometry of Majorana zero modes. Due to the Kramers degeneracy, each band has a Kramers pair with opposite spins. The Hilbert bundle is an analogy of a spinor bundle, by assumption, Definition 8. The KQ-cycle of a topological insulator is defined as the quintuple The effective HamiltonianH is not self-adjoint, so (C ∞ (X), L 2 (X, H),H) is not a K-cycle in the classical sense. However, by localization, a Majorana zero mode consists of an electronic chiral state and its time reversal partner, and each chiral state has a well-defined KR-cycle. If we switch off the time reversal symmetry, then a K-cycle (C ∞ (X), L 2 (X, H), H) can be constructed based on the single-particle Hamiltonian H. Furthermore, if H is approximated by a Dirac operator D, then we get the Dirac-type spectral triple (C ∞ (X), L 2 (X, H), D). Now let us switch on the time reversal symmetry, the quadruple In sum, the localization of the KQ-cycle can be reduced to two KR 5 -cycles. As a result, the geometry of a Majorana zero mode can be modeled by a coupled product of two KR 5 -cycles, which is viewed as a KR 10 -cycle with KO-dimension 2. Similar to the spin representation when 2k ≡ 2 mod 8, the KQ-cycle models two inequivalent complex spinors (φ, Θφ) ∈ 25 L 2 (X, C) ⊕ L 2 (X, ΘC), and it can be viewed as a quaternionic pinor Φ ∈ L 2 (X, H). Hence the localization of the KQ-cycle determines a class in the KR 2 -homology group, in order to get an index number, it is expected to be paired with a K-theoretic class in the KR −2 -group. From the discussions in §6, the topological index map has the source as the KR −2 -group of the cotangent bundle π : T * X → X and identifies KO −2 (pt) as the right place where the topological Z 2 invariant really lives in. This fact matches perfectly with the results on the geometry of a Majorana zero mode, which is viewed as a KR 10 -cycle with KO-dimension 2. In other words, the Z 2 index is obtained from the pairing between an element in the KR −2 -group of the cotangent bundle and a reduced KR 2 -cycle representing a Majorana zero mode.
Here f is a group homomorphism. There are two cases: (1) f is the zero map, which maps {0, 1} to 0; (2) f is the identity map, which maps 0 to 0 and 1 to 1. The map f really represents the pairing between Z 2 from homology and cohomology. Since K-theory is dual to K-homology, the above index pairing is essentially the Poincaré duality theorem. By the reduction from the effective HamiltonianH to the Dirac operator D, the geometry of a Majorana zero mode can be reduced to the KR 5cycle of a chiral zero mode (C ∞ (X), L ⊕ ΘL, D, Θ), which determines a fundamental class [D, Θ] ∈ KR 5 (X). On the other hand, the time reversal gauge w determines a K-theoretic class [w] ∈ KQ −1 (X) = KR −5 (X). So the above index pairing can be reduced to a more familiar form as in the mod 2 index theorem. Proposition 3. The index pairing for the topological Z 2 invariant is given by where the right hand side can be spelled out as the mod 2 index theorem

Bulk-boundary correspondence
In general, the bulk-boundary (or bulk-edge) correspondence is a realization of the holographic principle in condensed matter physics. The philosophy of the holographic principle, learned from the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, is that the boundary field theory determines the geometry in the bulk, and conversely the bulk geometry 26 determines the boundary theory up to some equivalence. Even though the general holographic principle still remains a conjecture, the Chern-Simons/Wess-Zumino-Witten (CS/WZW) correspondence is well understood as a theorem in mathematical physics. The bulk-boundary correspondence can be understood on different levels. First of all, on the level of effective Hamiltonian, the boundary Hamiltonian can be obtained from the bulk Hamiltonian by a partial Fourier transform along a direction with translational invariance. Such correspondence realized by the partial Fourier transform holds for all dimensions, so there exists a one-to-one correspondence between the bulk and boundary Hamiltonians. In the topological band theory, we identify the bulk state with the corresponding boundary state via the partial Fourier transform. As a result, the geometry of this correspondence is trivial, since the partial Fourier transform always induces a map from the d-dimensional bulk to the (d−1)-dimensional boundary as momentum spaces. In terms of the topological Z 2 invariant, if the bulk theory gives us KO −2 (pt), then via this correspondence KO −1 (pt) is obtained for the boundary of codimension 1. This is not the desired bulkboundary correspondence, which is supposed to be an equivalence between effective (quantum) field theories from the bulk and boundary.
On the level of gauge theory, the Chern-Simons/Wess-Zumino-Witten (CS/WZW) duality establishes an equivalence between the bulk and boundary field theories. For 3d topological insulators, the Z 2 invariant is the WZW topological term in string theory (or the odd topological index in index theory), which is an important component of the Chern-Simons gauge theory. In other words, the bulk-boundary correspondence on the level of action functional is given by the duality between the bulk Chern-Simons action and the boundary WZW term (also related to currrent algebra in vertex operator algebras). If we further consider quantum field theories, then the bulk is described by a topological quantum field theory (TQFT), and the boundary is characterized by a rational conformal field theory (RCFT). So the bulk-boundary correspondence on the level of QFT/CFT is given by CS3/WZW2 as a correspondence between the 3d CS TQFT and 2d WZW RCFT. On the other hand, for a given gauge group G such as SU (2), there exists a one-to-one correspondence between the WZW model and Chern-Simons theory, both characterized by their level k ∈ Z. If the relevant gauge theory is described by group cohomology, then the bulk-boundary correspondence is given by the Dijkgraaf-Witten theory. If the Dijkgraaf-Witten transgression map is τ : H 4 (BG, Z) → H 3 (G, Z), then τ is the correspondence between the Chern-Simons theory and WZW model associated with G.
As different aspects of the topological Z 2 invariant, the topological Z 2 index and the Kane-Mele invariant are equivalent. The topological index is an integral of the Chern character over the bulk, and the Kane-Mele invariant has a Pfaffian formalism derived from an effective fermionic field theory. In other words, the bulk theory is an action functional given by 27 the WZW term, and the Kane-Mele invariant is an effective quantum field theory defined over the set of fixed points. We propose that the effective boundary is defined as the fixed points, which is different from the geometric boundary of the bulk. Hence the equivalence between the topological index and the Kane-Mele invariant is viewed as the bulk-boundary correspondence on the level of topological invariant.
Theorem 3. The topological index ind t (w) and the Kane-Mele invariant ν are equivalent, which is a bulk-boundary correspondence if the set of fixed pionts X τ is identified as the effective boundary.
From the proof of the mod 2 index theorem, spectral flow is the key idea behind the odd topological index. For a Majorana zero mode, the chiral zero modes can pass through the zero energy level only at a fixed point. In the spirit of localization, the spectral flow of chiral states also indicates that the effective boundary is given by the fixed pionts. In a sense, the above equivalence is a fixed point theorem of the time reversal symmetry.
Inspired by the Kane-Mele invariant defined over the fixed points, we propose X τ as the effective boundary. The Kane-Mele invariant is a collective effect over X τ , so we further project it to an abstract point, i.e., p : X τ → pt, as in the topological index map. Otherwise, we will obtain a family index theorem with values in KO −2 (X τ ) instead of a topological index in KO −2 (pt). To stress the topological Z 2 invariant ν as an index number, we realize the abstract point {pt} in the topological index map as a specific fixed point x 0 ∈ X τ so that ν ∈ KO −2 (x 0 ).

Example 14.
The KR-theory of T d can be computed based on the decomposition of the fixed points where d k is the binomial coefficient, so that The (strong) topological Z 2 invariant only depends on the fixed point x 0 with "top codimension", i.e., In fact, one has the relation d − j = −2, for instance j = 4, d = 2, In the decomposition of the fixed points (T d ) τ , this specific fixed point x 0 is taken as the fixed point corresponding to the last (i.e., k = d) summand, that is the fixed point of "top codimension". This fixed point x 0 ∈ X τ with "top codimension" is naturally identified with S d (d = dim X) by the Poincaré duality in K-theory.
On the level of K-theory, the bulk-boundary correspondence is a KKcycle connecting the bulk and boundary K-theories. The bulk is the ddimensional momentum space X, and the effective boundary is assumed to be the fixed points X τ (or the fixed point with top codimension x 0 ). By the topological band theory, the bulk theory is modeled by the Quaternionic K-theory KQ 2−d (X), which is canonically isomorphic to the Real K-theory KR −2−d (X). Along the same logic, the boundary theory is modeled by the real K-theory KO −2 (X τ ) (or KO −2 (x 0 ) as a number). Since the fixed points X τ are real points with respect to the time reversal transformation τ , one can identify KR −2 (X τ ) = KO −2 (X τ ). Thus the bulk-boundary correspondence is a KKR-cycle connecting the KR-theory of the bulk and the KO-theory of the effective boundary, which is a generalization of the topological index map Example 15. When X = T d is a torus, it is identified as T d = R d /Z d by the translational symmetry group Γ = Z d . The real assembly map µ R in the real Baum-Connes conjecture for Γ = Z d is an isomorphism, called the Baum-Connes isomorphism or dual-Dirac isomorphism, On the right hand side, the KO-theory of the group C * -algebra C * (Z d , R) is since the KR-theory of (T d , τ ) can be identified with the KO-theory of the real function algebra C(T d , τ ), where In fact, the dual-Dirac isomorphism can be realized by an invertible real KKO-class β ∈ KKO(C R (T d ), C(T d , τ )), Let α be the inverse of β, that is, α ∈ KKO(C(T d , τ ), C R (T d )) such that α • β = id, so α realizes the Dirac isomorphism In addition, if i 0 : x 0 ֒→ T d is the inclusion map, then it induces the restriction map in KO-theory In this setting, the bulk-boundary correspondence for the torus T d is a surjective map realized as the composition map where the middle isomorphism is the Poincaré duality in KO-theory. In this case, the Dirac isomorphism is applied to convert the KR-theory of the bulk to a KO-homology, then the Poincaré duality is used to get back the KO-theory of the bulk. So the KKR-cycle is reduced to a KKO-cycle (via isomorphisms) to realize the bulk-boundary correspondence between the KOtheories of the bulk and boundary.
From the above, the bulk-boundary correspondence on the level of Ktheory is essentially a KKO-cycle or KKR-cycle connecting the bulk and boundary K-theories, which is not necessary to be an isomorphism.
On the other hand, we have seen that it is interesting to understand the bulk theory by the geometry of Majorana zero modes using KR-cycles or KR-homology, a similar idea is carried out in [7]. So the bulk-boundary correspondence in KR-homology has the following form, where the isomorphism is again the Poincaré duality. This bulk-boundary correspondence on the level of K-homology is viewed as a generalized version of the analytical index of the KR 2 -cycle of the effective Hamiltonian.
Therefore, if the bulk-boundary correspondence is realized as a correspondence connecting K-homologies, then it is a generalization of the analytical index map; if the bulk-boundary correspondence is realized as a correspondence connecting K-theories, then it is a generalization of the topological index map. Finally, we have the following diagram based on Kasparov products if K-homology and K-theory are written in KKR-theory or KKOtheory, Example 16. In particular, the bulk-boundary correspondence offers a new perspective to view the mod 2 index theorem behind the topological Z 2 invariant.

Family index theorem
Let us briefly review the relation between the determinant and Pfaffian line bundles and a family index theorem, and apply it to the index problem of Majorana zero modes. The classical results on the determinant line bundle in the context of index theory can be found in [16]. This section gives a preparation for the proof of the equivalence between the noncommutative topological index and the noncommutative Kane-Mele invariant.
Let Y be a compact spin manifold, and D + : H + → H − be a Dirac operator mapping H + to H − , where H + and H − are finite dimensional Hilbert spaces of spinors. Taking the top exterior products of the Hilbert spaces, denoted by det H ± = ∧ max H ± , D + induces a natural map Or equivalently, det D + can be viewed as an endomorphism The above construction can be promoted to vector bundles. Let π : Z → X be a vector bundle with a typical fiber Z x ∼ = Y , which is assumed to be spin. Now let D + be a family of Fredholm operators parametrized by the base manifold X, In other words, D + is a bundle map between two Hilbert bundles, On the other hand, for each x ∈ X, the Fredholm operator D +,x has finite dimensional kernel and cokernel, and the Fredholm index is the analytical index, where D * +,x is the adjoint of D +,x . Therefore, one defines the index bundle as the formal difference bundle, where kerD + and kerD * + are vector bundles over X. Obviously, the index bundle of D + gives rise to an element in the K-theory of the base manifold, If the typical fiber Y is spin and (8k + 2)-dimensional, then there exists a Pfaffian line bundle, There exists a push-forward map, called the Gysin map, between the real K-groups of the total and base spaces in π : Z → X, If one considers the index bundle π : Ind(D + ) → X in this real case, then one has the identity, where [1] is a certain generator in KO(Z) [16]. Moreover, by the Chern-Weil theory, there exists a map induced by the Pfaffian form, where H 2 (X, Z) is the 2nd cohomology group characterizing complex line bundles over X. By a theorem in [16], one has the relation, In order to get the topological index (as a number) from the above family index, one has to project the set of fixed points to an abstract point p : X τ → pt, which induces the push-forward map in real K-theory p * : KO −2 (X τ ) → KO −2 (pt). As mentioned above, we realized the abstract point {pt} as a specific fixed point x 0 ∈ X τ with top codimension, and the projection p * can be viewed as a summation : KO −2 (X τ ) → KO −2 (x 0 ) to count the collective effect over the fixed points.

Noncommutative topology index
As a modern geometric framework, noncommutative geometry (NCG) has been providing computational tools for studying topological invariants in disordered fermionic systems, e.g. the noncommutative model of quantum Hall effect [6] and disordered topological insulators [47,53]. We generalize the index theory and K-theory of the topological Z 2 invariant onto the noncommutative 2-torus T 2 θ and 3-sphere S 3 θ [31]. When the base manifold is generalized to be a noncommutative manifold defined by a C * -algebra, the topological index can be defined with the help of the Connes-Chern characters. As a generalization of the Chern character from K-theory to de-Rham cohomology, the Connes-Chern characters are 33 maps from K-theory (or K-homology) to cyclic homology (or cyclic cohomology). By the machinery of noncommutative topology, the topological index can be computed by pairing periodic cyclic cohomology and homology after applying the Connes-Chern characters [10,36]. In this section, we define the noncommutative Z 2 invariant for disordered topological insulators as a topological index in noncommutative topology. In order to fix the notations, let us briefly review the Connes-Chern characters acting on K-theory and K-homology, and we only care about the odd case motivated by the mod 2 index theorem behind the topological Z 2 invariant.
Let A be a unital C * -algebra representing a noncommutative manifold, the odd Connes-Chern character map from K-theory to cyclic homology groups is defined by where T r is the operator trace, and there are (n + 1)-pairs of unitary operators u and u −1 in the trace. It is convenient to equivalently define the above Connes-Chern character by a pairing between cyclic cohomology (dual to cyclic homology) and K-theory, Furthermore, it induces a pairing between periodic cyclic cohomology with K-theory, More precisely, for an odd (b, B)-cocycle φ = (φ 1 , φ 3 , · · · , φ 2k+1 , · · · ) and a unitary element u ∈ A, the pairing between HP 1 (A) and K 1 (A) is defined by Let (H, F ) be an odd p-summable Fredholm module over a C * -algebra A and let n be an integer such that 2n ≥ p. The odd Connes-Chern character of (H, F ), denoted by Ch 2m−1 (H, F ), is defined by (144) For any integer m ≥ n, the class of the cyclic cocycle Ch 2m−1 (H, F ) is stabilized in the odd periodic cyclic cohomology group, call it the stable odd Connes-Chern character of (H, F ), i.e., provided n is in the stable range (i.e., 2n ≥ p).
The above discussion can be summarized in the commutative diagram [36], Example 18. Let θ ∈ R \ Q be an irrational real parameter, the noncommutative 2-torus T 2 θ = C * (u, v) is the universal C * -algebra generated by two unitary operators u, v satisfying uv = e 2πiθ vu. One concrete realization of T 2 θ is given by the irrational rotation algebra or C * -crossed product C(S 1 ) ⋊ Z. Let L 2 (S 1 ) be the Hilbert space, the irrational rotation algebra is generated by C(S 1 ) and the rotation action by θ. More precisely, for any g ∈ L 2 (S 1 ), two unitary generators are defined by U = M f for f ∈ C(S 1 ) and V = V θ , Example 19. The noncommutative 3-sphere S 3 θ is defined as the universal C * -algebra generated by two normal operators α and β (s.t. αα * = α * α, ββ * = β * β) satisfying the relations, (156) αβ = λβα, α * β =λβα * , αα * + ββ * = 1 for the complex parameter λ = e 2πiθ and irrational θ ∈ R \ Q. There exists a natural parametrization of the generators in S 3 θ by Hopf coordinates, where u, v are the generators of the noncommutative 2-torus T 2 θ . Definition 9. The time reversal symmetry is defined as an anti-linear automorphism Θ on the given C * -algebra A such that Θ 2 = −1, so Θ is a real structure and (A, Θ) is a Real C * -algebra.
Example 20. If the C * -algebra A is generated by two operators, then the time reversal symmetry is defined by Θ := iσ 2 * = 0 * − * 0 , where * (a) = a * is the adjoint operation.
LetD be a localized skew-adjoint Fredholm operator approximating the effective Hamiltonian of a topological insulator, (158)D := 0 −D D 0 : where D is a self-adjoint Dirac operator acting on H C .D is an unbounded operator if and only if D is an unbounded operator.D is a skew-adjoint (i.e.,D * = −D) Fredholm operator, so its analytical index is As before, we define the real structure J = 0 Θ * Θ 0 which statisfies J = J * and J 2 = 1, and the grading operator γ = 1 0 0 −1 as usual.
where the phase of D is F = D|D| −1 and the projection P = (1 − F )/2.
For a topological insulator, we are interested in the occupied bands below the Fermi energy, which is assumed to be located at the zero energy level, we define the (Fermi) projection P as the negative projection. By assumption [w] ∈ KQ 1 (A) is a Z 2 generator in the odd KQ-group, which generalizes the transition function of the Hilbert bundle. If A = C ∞ (T 3 ) is the smooth function algebra on the 3-torus, ind(P wP ) gives us the strong Z 2 invariant as in the classical case.
Hence the Fredholm index of P wP is computed by the index pairing between [(H, D)] (KR-homology class of the KR 5 -cycle (A, H, D, Θ)) and [w] ∈ KR 5 (A) = KQ 1 (A), In the classical case, the even Chern character modulo 2 cannot give the topological Z 2 invariant for 2d spaces such as T 2 . By the free Z 2 -action, T 2 can be decomposed into two cylinders, and the Hilbert bundle is determined by the clutching function w : S 1 ⊔ S 1 → U (2), where S 1 ⊔ S 1 = S 1 × Z 2 is the boundary of a cylinder S 1 × I. The topological index is a resolution of the gauge anomaly as a boundary term, see Prop. 2, The topological Z 2 index on the noncommutative 2-torus T 2 θ can be defined similarly. Consider the irrational rotation algebra C(S 1 )⋊Z = C * (U, V ) = C * (M f , V θ ) for any non-zero continuous function f ∈ C(S 1 ), the action of the time reversal operator Θ is spelled out as, Due to the effect Θ : θ → −θ, C(S 1 ) ⋊ Z can be decomposed into two copies of C(S 1 ) ⋊ N 0 (N 0 = N ∪ {0}) analogous to the decomposition of T 2 . Of course the fixed points of Θ on the real parameter are θ = {0, ∞}, that is, the boundary of C(S 1 ) ⋊ N 0 is C(S 1 ) × {0, ∞} = C(S 1 ) × Z 2 . Let w be a unitary operator on C(S 1 ) × Z 2 representing the time reversal operator Θ, i.e., w = ΘK (K is the complex conjugation), then the topological Z 2 index can be defined by dimensional reduction from T 2 θ to C(S 1 ) × Z 2 , which gives a generalization of the classical case.
generates a Z 2 element, then the 2d noncommutative topological Z 2 invariant is defined by pairing boundary classes, where the phase of D ∂ is F ∂ = D ∂ |D ∂ | −1 and the projection P ∂ = (1−F ∂ )/2.

Fixed point algebra
We have seen the noncommutative 2-torus T 2 θ and 3-sphere S 3 θ as concrete examples, the time reversal operator has been introduced for such noncommutative manifolds with two generators. We will define the fixed point C * -algebra of the time reversal symmetry, which is the noncommutative analog of the set of fixed points. We mainly focus on T 2 θ , and the fixed point algebra in S 3 θ can be treated similarly. In the classical case, if the 2-torus T 2 is parametrized by the angles (165) then the time reversal transformation on T 2 is defined by So (T 2 , τ ) is a Real space with the real structure τ (s.t. τ 2 = 1), and the set of fixed points is T τ = {(0, 0), (0, π), (π, 0), (π, π)}. Let C(T 2 ) denote the continuous functions on T 2 , if the complex conjugation is denoted by K, then (C(T 2 ), K) is a Real function algebra with the real structure K (s.t. K 2 = 1). The invariant subspace of C(T 2 ) with respect to τ and K is defined as which is a commutative real C * -algebra [51]. The time reversal operator Θ is an anti-unitary operator acting on a C *algebra such that Θ 2 = −1. Θ acting on T 2 θ can be defined as where * (a) = a * is the adjoint operation. Therefore, the action of Θ on the generators of T 2 θ is given by More explicitly, the action of Θ on the generators is The action of Θ respects the order of a product since we are dealing with a fermionic system, for example, As a consequence, this action is compatible with the relation uv = e 2πiθ vu. If x = ie πiθ uv * , y = ie −πiθ vu * , then x, y are two unitary operators invariant under Θ, i.e., x = ie πiθ uv * , y = ie −πiθ vu * By the relation x * = −y, as a C * -algebra, the fixed point algebra (T 2 θ ) Θ is actually generated by the unitary operator x. In addition, (T 2 θ ) Θ is a real C * -algebra with respect to the real structure Θ.
The real K-theory of the fixed point algebra (T 2 θ ) Θ can be computed as follows, which is the receptacle of the topological Z 2 invariant from the boundary K-theory if the fixed point algebra is viewed as the boundary.
Let us look at the geometry of the fixed point algebra (T 2 θ ) Θ . From the concrete form of the generator x = ie πiθ uv * , it creates a virtual "channel" 39 between u and v * . If one thinks of the relation uv = e 2πiθ vu as a braiding rule, say to rotate u about v, then one gets a phase e 2πiθ after a twist. Along the same logic, e πiθ in x is viewed as a half-twist. The same analysis applies to the other generator y = ie −πiθ vu * , which creates another channel from v to u * in the opposite direction. Putting x and y (or the overlapping channels) together, they creates a virtual double cone with a crossing as in a Majorana zero mode (or a Dirac cone). However, one channel is enough to characterize the geometry since the opposite channel is easily recovered by x * = −y.
If we define a matrix by putting x and y off-diagonally, i.e., 0 y x 0 = 0 −x * x 0 , then it is analogous to the skew-adjoint operatorD = 0 −D D 0 .
Hence the reduction from the pair (x, y) to x reminds us of the reduction fromD to D. If the skew-adjoint operatorD is indexed by the fixed point X τ , then the index ofD lives in KO −2 (X τ ). Corresponding to D, x induces the generator of KO 1 [(T 2 θ ) Θ ], where (T 2 θ ) Θ is the noncommutative generalization of X τ .

Noncommutative Kane-Mele invariant
The Kane-Mele invariant can be understood geometrically as the comparison between orientations of the Pfaffian and determinant line bundles over the fixed points. If the determinant line bundle is normalized to be a trivial bundle, then the Kane-Mele invariant carries information about the orientation of the Pfaffian line bundle. We will generalize the Kane-Mele invariant onto the noncommutative 2-torus.
Quillen's construction of the determinant line bundle on Riemann surfaces can be generalized to the noncommutative 2-torus [15]. Let F (H) be the space of Fredholm operators over a separable infinite dimensional complex Hilbert space H, the determinant line bundle π : Det → F (H) can be defined so that for any Fredholm operator T ∈ F (H), Det T = ∧ max (ker T ) * ⊗ ∧ max (ker T ) Assume that the single particle Hamiltonian H is parametrized by the noncommutative 2-torus, that is, H(a) is a family of self-adjoint Fredholm operators for a ∈ T 2 θ . Then the effective Hamiltonian Consider Fredholm operators acting on a real Hilbert space F R := F (H R ), and let F sa R ⊂ F R be the subspace of self-adjoint Fredholm operators, which is a classifying space of KO 1 , i.e., KO 1 (A) ∼ = [A, F sa R ] for a real C * -algebra A. Similar to the determinant line bundle, one defines the Pfaffian line bundle as P f → F sa R such that for any T ∈ F sa R , (185) P f T = ∧ max ker T If one complexifies the underlying real Hilbert space, then one has the relation between the Pfaffian and determinant line bundles, Or equivalently, for a real self-adjoint Fredholm operator T ∈ F sa R , the fibers at T can be identified as a vector space, The range of f ∈ C R (S 1 ) × is C × , after the normalization C × → S 1 , f is viewed as a continuous function f : S 1 → S 1 such that f (ϑ) = f (−ϑ). For example, one has f n : ϑ → e inϑ (or f n (z) = z n , z = e iϑ ) as an element in C R (S 1 ) × . In fact, {z n ,z n } forms an orthonormal basis for C R (S 1 ) × , by the relation (T 2 θ ) Θ,× ∼ = C R (S 1 ) × , the generator x ∈ (T 2 θ ) Θ,× (resp. y) is identified with z ∈ C R (S 1 ) × (resp.z). On the other hand, the Gelfand transform of the fixed point algebra (T 2 θ ) Θ ∼ = C * R (Z) can be identified with the Fourier transform on the circle.
The fixed point algebra (T 2 θ ) Θ is generated by two unitaries x and y (with x * = −y). We only care about the orientation of the real Pfaffian line bundle π : L 1/2 (D) → (T 2 θ ) Θ ∼ = C R (S 1 ), so its structure group can be reduced from GL(1, R) = R × to O(1) = Z 2 . Indeed, the orientation of L 1/2 (D) is essentially determined by the value of ̺ on the generators, The reduction in KO-theory from KO −2 to KO −1 agrees with the reduction fromD to a Dirac operator D.
Let us get back to the bulk theory, in the classical case, the topological Z 2 index lives in KR −4 (T 2 ) = KQ(T 2 ), (199) ind(w) ∈ KR −4 (T 2 ) = KO −2 (pt) ⊕ KO −4 (pt) By the same reduction fromD to D, we conclude that the noncommutative topological Z 2 index belongs to KR 2 (T 2 θ ), (200) ind(P wP ) ∈ KR 2 (T 2 θ ) Proposition 6. The KR-theory of the noncommutative 2-torus T 2 θ is computable by the Pimsner-Voiculescu method. In particular, we have (201) KR 2 (T 2 θ ) = KO −1 (pt) ⊕ 2KO −2 (pt) which is the receptacle of the topological Z 2 index from the bulk K-theory. The above theorem tells us that the bulk-boundary correspondence is basically an index theorem identifying the topological index from the bulk and the analytical index from the boundary. On the level of K-theory, the bulk-boundary correspondence involves a KKR-cycle connecting the Ktheories of the bulk and boundary, KKR((T 2 θ ) Θ , T 2 θ ) × KR 2 (T 2 θ ) → KO 1 [(T 2 θ ) Θ ] that identifies the KO −1 (pt) component from both sides.