Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator

We perform a detail study of higher dimensional quantum Hall effects and A-class topological insulators with emphasis on their relations to non-commutative geometry. There are two different formulations of non-commutative geometry for higher dimensional fuzzy spheres; the ordinary commutator formulation and quantum Nambu bracket formulation. Corresponding to these formulations, we introduce two kinds of monopole gauge fields; non-abelian gauge field and antisymmetric tensor gauge field, which respectively realize the non-commutative geometry of fuzzy sphere in the lowest Landau level. We establish connection between the two types of monopole gauge fields through Chern-Simons term, and derive explicit form of tensor monopole gauge fields with higher string-like singularity. The connection between two types of monopole is applied to generalize the concept of flux attachment in quantum Hall effect to A-class topological insulator. We propose tensor type Chern-Simons theory as the effective field theory for membranes in A-class topological insulators. Membranes turn out to be fractionally charged objects and the phase entanglement mediated by tensor gauge field transforms the membrane statistics to be anyonic. The index theorem supports the dimensional hierarchy of A-class topological insulator. Analogies to D-brane physics of string theory are discussed too.

Though in the former articles, the non-abelian monopoles are adopted, there may be another monopole realization for higher dimensional quantum Hall effect. That is to use antisymmetric tensor U (1) monopole. Tensor U (1) monopole is a monopole [40,41] whose gauge group is U (1) but gauge field is not a vector but an antisymmetric tensor 5 . While the non-abelian monopole corresponds to an extension of the Dirac monopole by increasing the internal gauge degrees of freedom, the tensor monopole manifests another extension of the Dirac monopole by increasing the external indices. Therefore, there may be two reasonable generalizations of quantum Hall effect, one is based on the non-abelian monopole and the other is based on the tensor monopole. One may be immediately inclined to ask the following questions. What does quantum Hall effect in tensor monopole background look like and what kind of non-commutative geometry will emerge in the lowest Landau level? If 2D quantum Hall effect has two reasonable generalizations, is there any connection between them? For such questions, researches of non-commutative geometry gives a suggestive hint; There are two (superficially) different formulations for higher dimensional generalizations fuzzy sphere [22,23,24], one of which is the ordinary commutator formulation and the other is the quantum Nambu bracket formulation. Inspired by the observation, we establish connection between the non-abelian and tensor monopole and answer to the questions in this work.
Topological field theory description of the quantum Hall effect [43,44] has brought great progress in understanding non-perturbative aspects of quantum Hall effect. The Chern-Simons effective field theory naturally describes the flux attachment that electron and Chern-Simons fluxes are combined to yield a "new particle" called composite boson [45,46], and the fractional quantum Hall effect is regarded as a superfluid state of the composite bosons [44]. The fundamental object of the A-class topological insulator turns out to be membrane-like objects. Based on the connection between the non-abelian and tensor monopoles, we propose a tensor type Chern-Simons field theory as the effective field theory of the A-class topological insulator. Interestingly, while we start from the non-abelian quantum mechanics in (2k + 1)D space-time, the tensor Chern-Simons field theory is defined in (4k − 1)D space-time. Membranes have a fractional charge and obey anyonic statistics. Ground state of A-class topological insulators is regarded as a superfluid state of composite membrane at magic values of the filling factor. We discuss dimensional condensation of membranes with emphasis on relation to brane-democracy of string theory. We thus integrate so far loosely connected subjects, such as Nambu-bracket, tensor topological field theory and physics of quantum Hall effect, to have an entire picture of A-class topological insulator [ Fig.1].
Though we share several terminologies with string theory such as p-branes and C field, the present analysis is not directly related to the string theory: We do not use either strings or Dbranes. About a realization of topological insulators in string theory, one may consult [47,48]. For C field realization of non-commutative geometry on M-brane, see Refs. [49,50,51].
The paper is organized as follows. In Sec.2, we briefly review the basic mathematics of the fuzzy sphere and its physical realization in the lowest Landau level. Sec.3 describes the two mathematical D-brane [34,35]. 5 Such antisymmetric tensor gauge field is also known as Kalb-Ramond field [42]. formulations for higher dimensional fuzzy spheres. We introduce non-abelian monopole quantum Hall effect with or without spin degrees of freedom in Sec.4. Sec.5 discusses the connection between the tensor and non-abelian monopoles, and gives a tensor monopole realization of the quantum Nambu geometry. In Sec.6, the Chern-Simons tensor field theory is proposed as the effective field theory of A-class topological insulator, where we clarify the fractional charge and anyonic statistics of membranes. We also discuss the hierarchical property of membranes and A-class topological insulator. Sec.7 is devoted to summary and discussions.

Fuzzy Sphere and Dirac Monopole
Here, we briefly review how the fuzzy geometry emerges in the context of the lowest Landau level physics by using the fuzzy two-sphere and Dirac monopole system. The observation is a template for higher dimensional fuzzy sphere in the subsequent sections.
The fuzzy sphere is realized as the lowest Landau level physics. We will show how fuzzy geometry emerges on a two-sphere in Dirac monopole background, respectively from the Lagrange and Hamilton formalisms .

Hopf map and Lagrange formalism
The Lagrangian for the electron on a two-sphere in monopole background is given by where x i (i = 1, 2, 3) are subject to a constraint and the Dirac monopole gauge field is given by with Dirac monopole charge I/2 (I integer) [61]. Relation to the non-commutative geometry will be transparent by introducing the Hopf spinor. The Hopf spinor is the two-component spinor that induces the (1st) Hopf map S 3 S 1 → S 2 : with φ † φ = I.
The x a (7) given by automatically satisfy the condition of two-sphere: The Hopf spinor φ takes the form with e iχ denoting U (1) phase factor, and the monopole gauge field (6) can be derived as In the lowest Landau level, the kinetic energy is quenched and the Lagrangian (4) is reduced to the following form We regard the Hopf spinor as the fundamental variable and derive the canonical momentum of φ as iφ * from (12), and apply the quantization condition to them (not the original coordinates x i ): After the quantization, the Hopf spinor becomes to the Schwinger operator of harmonic oscillator expressed as 6 : and the coordinates on two-sphere (7) turn out to be which satisfy the fuzzy two-sphere algebra (1), and the condition (8) is rewritten as One can readily show that Eq. (15) with (16) indeed satisfies (2). The emergence of fuzzy sphere is based on the Hopf-Schwinger operator and the Pauli matrices in the Lagrange formalism.

Hamilton formalism and angular momentum
The Hamiltonian for a particle in gauge field is given by where D i represent the covariant derivative: and Λ i denote the covariant angular momentum: Hence, the Hamiltonian for a particle on two-sphere (r const.) is given by With the U (1) monopole at the center of the sphere, the total angular momentum L i is given by the sum of the covariant angular momentum and the angular momentum of monopole gauge field: where Since L i are the conserved angular momentum, they satisfy the SU (2) algebra 6 We can derive the same result in the Hamilton formalism. The lowest Landau level eigenstates are given by by the holomorphic function of φ, and the complex conjugate of φ is effectively represented by the derivative of φ.
In the lowest Landau level, the kinetic term is quenched Λ i = 0, and then x i (∝ F i ) can be identified with L i : It is obvious that X i satisfy the fuzzy two-sphere algebra (1). With use of L ij = ǫ ijk L k , (24) is written as Notice the construction of fuzzy sphere coordinates in the Hamilton formalism is based on the angular momentum. Consequently, there are two ways to see the emergence of fuzzy sphere, one of which is the Hopf-Schwinger construction (15) in the Lagrange formalism, and the other is the angular momentum construction (25) in the Hamilton formalism.

Fuzzy sphere algebra
As discussed above the coordinates of fuzzy two-sphere are given by the SO(3) vector operators that satisfy and its minimal representation is given by the 2 × 2 Pauli matrices. Since Pauli matrices are equal to the SO(3) gamma matrices, it may be natural to adopt the SO(2k + 1) gamma matrices as the coordinates of S 2k F with minimum radius. For S 2k F with larger radius, the SO(2k + 1) gamma matrices G a (a = 1, 2, · · · , 2k + 1) of fully symmetric representation 7 , , is adopted as the fuzzy coordinates [55,56]. Indeed X a ≡ αG a satisfy 2k+1 a=1 X a X a = α 2 4 which represents the condition of constant radius of the fuzzy sphere. In the limit I → ∞ with fixed r, (26) is reduced to the condition of the classical 2k-sphere, 2k+1 a=1 x a x a = r 2 . One should notice however, there is a big difference between the fuzzy two-sphere and its higher dimensional counterpart [57,58,59,60]. Though the SO(3) gamma matrices are equivalent to the SO(3) generators and form a closed algebra by themselves, the SO(2k + 1) (k ≥ 2) gamma matrices X a do not satisfy a closed algebra among them but their commutators yield "new" operators, the SO(2k + 1) generators X ab : 7 For several properties of gamma matrix in fuzzy symmetric representation, see Append.A.
The appearance of X ab suggests that the geometry of higher dimensional fuzzy sphere cannot simply be understood only by the original coordinates. To construct a closed algebra for higher dimensional fuzzy sphere, we need to incorporate X ab also to have an enlarged algebra in which X a and X ab amount to the SO(2k + 2) algebra. Around the north pole, (27) reduces to where η µν i denotes the expansion coefficient (for k = 2, η µν i is given by the t'Hooft symbol) and X i stand for SO(2k) generators related to X µν by the relation The extra-degrees of freedom is described by the operators X i , and can be interpreted as the fuzzy fibre over S 2k . Since the corresponding algebra of S 2k F is the SO(2k + 2) algebra, the fuzzy fibre described by the SO(2k) algebra is identified with S 2k−2 F . Due to the existence of the fuzzy bundle, the classical counterpart of S 2k F does not simply realize S 2k ≃ SO(2k + 1)/SO(2k) but SO(2k)/U (k) fibration over S 2k [58] S 2k Here, ∼ denotes the local equivalence. The SO(2k)/U (k)-fibre is the classical counterpart of the extra fuzzy space S 2k−2 F . As we shall see later, such extra degrees of freedom correspond to (fuzzy) membrane excitation.
Though in the commutator formulation, the existence of the fuzzy fibre is explicit, the commutator formulation is rather "awkward" in the sense the algebra does not close within the original fuzzy coordinates. The Nambu bracket gives a more sophisticated formulation. In the d dimension, quantum Nambu bracket (or Nambu-Heisenberg bracket) [21,22,23,24] is defined as where a 1 , a 2 , · · · , a n = 1, 2, · · · , d (with n ≤ d) 8 , and the bracket for the low indices represents the fully anti-symmetric combination about the indices. We have n! terms on the right-hand side of (32). For instance, In the quantum Nambu bracket formulation 9 , the non-commutative algebra for S 2k F is given by [22,23,24] [X a 1 , X a 2 , X a 3 , · · · , X a 2k ] = i k C(k, I)α 2k−1 ǫ a 1 a 2 a 3 ···a 2k+1 X a 2k+1 , 8 For n > d, due to the anti-symmetric property, quantum Nambu bracket always vanishes 9 (33) essentially comes from the following property of the SO(2k + 1) gamma matrices, γ1γ2γ3 · · · γ 2k = i k γ 2k+1 .
For more detail properties of quantum Nambu bracket, see Appendix B. where Thus, the extra operators X ab do not appear in the quantum Nambu bracket formulation of fuzzy sphere, and the closure of algebra is satisfied only by the original fuzzy coordinates. The extra fuzzy-fibre degrees of freedom seems to be completely "hidden" in the quantum Nambu bracket. Around the north-pole X 2k+1 ≃ r, (33) is reduced to the quantum Nambu bracket for non-commutative plane: where For instance, 3.2 Two monopole set-ups for higher dimensional fuzzy sphere As discussed in Sec.2, the fuzzy two-sphere is realized in the Dirac monopole background. The easiest way to find what kind of monopole corresponds to non-commutative geometry is to find the right-hand side of the non-commutative algebra. For instance, the fuzzy two-sphere algebra is given by and one can read off the U (1) monopole field strength from the right-hand side: For higher dimensional fuzzy sphere, in correspondence to the two non-commutative formulations, we will obtain two different types of monopoles.
• Non-abelian monopole Around the north pole, the commutation relation between the fuzzy coordinates (27) becomes where the right-hand side is the SO(2k) generators. This suggests the field strength of the nonabelian monopole where Σ µν denotes the SO(2k) matrix generators. Thus we may identify the one monopole set-up for S 2k F with the SO(2k) non-abelian monopole.
• Tensor monopole set-up Meanwhile, the right-hand side of the quantum Nambu bracket formulation implies antisymmetric tensor monopole field strength: Here two comments are added. Firstly, even though there are two different non-commutative formulations, they describe the same non-commutative object, i.e. the fuzzy sphere. Similarly, the two different types of monopoles are expected to describe same physical system corresponding to fuzzy sphere. In other words, the non-abelian and the tensor monopoles are two different physical set-ups for the same system. Hence, they are expected to be "equal" in some sense. Their connection will be clarified in Sec.5. Secondly, though the quantum Nambu bracket veils the "extra" degrees of freedom of fuzzy-bundle, (2k − 1) rank field (41) on the right-hand side of (35) implies the existence of (2k − 2)-brane whose (2k − 1)-from current naturally coupled to (2k − 1) rank tensor field. This observation will be important in constructing the Chern-Simons tensor field theory in Sec.6.

Non-Abelian Monopole and Higher Dimensional Quantum Hall Effect
Here, we give non-abelian monopole realization for higher dimensional quantum Hall effect [31,32]. The monopole gauge group is chosen SO(2k) so as to be compatible with the holonomy of the basemanifold S 2k10 .

SO(2k) non-Abelian monopole
First let us introduce the generalized Hopf map: 10 The present monopole set-up is quite similar to the Kaluza-Klein monopole in the sense that the geometrical information determines the corresponding monopole gauge group. Kaluza-Klein monopole accompanies with the spontaneous compactification of the Kaluza-Klein theory [62,63], and the isometry of the compactified space is transfered to the gauge symmetry of the uncompactified space. For instance, S 2k−1 compactified space yields the gauge field of SO(2k) non-Abelian monopole [64].
In low dimensions, (59) yields For the SO(2k) fully symmetric representation , the Chern-numbers are calculated as [35] c k=1 = I, which correspond to the monopole charge or the number of magnetic fluxes on spheres.

Non-commutative geometry in the lowest Landau level
Following to the similar step in Sec.2.1, we can find how higher dimensional fuzzy sphere geometry emergies in the lowest Landau level. It should be noted since the monopole gauge field is non-abelian, and then the particle on S 2l carries the SO(2k) color degrees of freedom like a "quark". The Lagrangian is given by where x a x a = r 2 . In the lowest Landau level, the Lagrangian is reduced to with Ψ (48). By imposing the canonical quantization condition on Ψ and Ψ * , x a (42) are realized as the operators which satisfy [X a , where . X a and X ab amount to (2k + 1) + k(2k + 1) = (k + 1)(2k + 1) generators of the SO(2k + 2) algebra, and X ab bring the degrees of freedom of fuzzy fibre S 2k−2 F over S 2k . It should be noted that the coordinates of the external space and those of the internal space are related by (65) and they are same size matrices of the SO(2k + 2) generators. Since they are similarly treated in the fuzzy algebra, there is no reason to distinguish the external and internal spaces in the lowest Landau level. Inversely it may be natural to consider an enlarged space that includes both external and internal spaces. Since the fuzzy-fibre coordinates X ab are the SO(2k + 1) generators, X ab can be represented as Meanwhile we have L ab ∼ r 2 F ab in the lowest Landau level (see Sec.5.4). From these relations, we have which suggests the non-abelian field strength is equivalent to the fuzzy-fibre [see Fig.2]. This identification coincides with the intuitive picture, since the fuzzy-fibre realizes the non-abelian flux of the monopole. In the 2D quantum Hall liquid, the U (1) magnetic flux penetration induces a charged excitation at the point where the flux is pierced. Similarly in higher dimensional quantum Hall liquid, the non-abelian flux penetration will induce an point-like excitation on S 2k . Though the excitation is a "point"-like object on S 2k , the non-abelain flux accommodates the S 2k−2 F geometry as its internal space. Remember that there is no distinction between the external and internal spaces in the lowest Landau level, and so the "internal" space S 2k−2 F can be regared as an extended (2k − 2) dimensional object, (2k − 2)-brane, in the enlarged (4k − 2) dimensional space. In this sense, the non-abelian flux penetration induces (2k − 2)-brane excitation.

The SO(2k + 1) Landau model
In d-dimensional space, one-particle Hamiltonian under the influence of gauge field is given by where D a = ∂ a + iA a and Λ ab = −ix a D b + ix b D a that satisfy where F ab are the components of the field strength, Since the SO(2k) non-abelian monopole (50) is located at the center of d = 2k + 1 dimensional space, its field strength is radially distributed and the system respects the SO(2k + 1) rotational symmetry. Hence, we have the conserved SO(2k + 1) angular momentum: It is straightforward to verify that L ab act as the generators of the SO(2k + 1) rotation: where M ab = L ab , Λ ab , F ab . For a particle on 2k-sphere, (69) is reduced to the SO(2k + 1) Landau Hamiltonian: Due to the existence of the SO(2k + 1) symmetry, one may readily derive the eigenvalues of (73) by a group theoretical method. With the orthogonality Λ ab F ab = F ab Λ ab , (73) is rewritten as where a<b F ab 2 = µ<ν Σ µν 2 was used. We adopt the fully symmetric representation for the SO(2k) Casimir µ<ν Σ + µν 2 , and the irreducible representation for the SO(2k + 1) Casimir a<b L ab 2 (n denotes the Landau level index), and the energy eigenvalues are derived as 13 where C 2k+1 (n, I/2) and C 2k (I/2) respectively represent the SO(2k + 1) and SO(2k) Casimir eigenvalues for (n, I/2) and (I/2): The degeneracy in the nth Landau level is given by (80) In particular for the lowest Landau level (n = 0), the representation is reduced to the SO(2k + 1) fully symmetric spinor repr. (I/2), and the degeneracy becomes to In low dimensions, One may notice that the lowest Landau level degeneracy (82) and the Chern number (61) are related by the following simple formula: This relation is indeed guaranteed by the index theorem for arbitrary k [see Sec.4.4]. 13 In the thermodynamic limit, r, I → ∞ with I/r 2 fixed, the energy eigenvalues (78) are reduced to The lowest Landau level energy, ELLL = I 4M r 2 k, is equal to k times the lowest Landau level energy of the 2D (planar) Landau model, B 2M = I 4M r 2 . This is because that in the thermodynamic limit, the 2kD fuzzy sphere is reduced to k copies of 2D non-commutative plane.

The SO(2k + 1) spinor Landau model and index theorem
Here, we consider a spinor particle on S 2k in the SO(2k) monopole background. The spinor particle carries the SO(2k + 1) spin degrees of freedom coupled to the external SO(2k) magnetic field through Zeeman term. We analyze the SO(2k + 1) spinor Landau problem with use of the formulation explored by Dolan [70]. In the presence the gauge field, the Dirac operator on dD curved manifold is generally given by where α stand for the intrinsic coordinates of the manifold and ω α denotes the spin connection of the manifold, while µ represent the coordinates of the dD flat Euclidean space and γ µ stand for the SO(d) gamma matrices: For symmetric (≡ torsion free) manifold, the square of the Dirac operator is given by the following Lichnerowicz formula [71]: where the Laplacian ∆ and the field strength F αβ are respectively given by and R denotes the scalar curvature. The second term on the right-hand side of (86), σ αβ F αβ = e a α e b β σ ab F αβ , represents the Zeeman term. As readily verified from the Lichnerowicz formula, in the absence of the Zeeman term, the Dirac operator does not have zero-eigenvalue on manifold with positive scalar curvature, since the eigenvalues of Laplacian are semi-positive definite. Meanwhile in the presence of the gauge field strength, the Zeeman term may cancel the contribution from the curvature term to give zero-eigenvalue for (−i D) 2 . This cancellation indeed occurs in the present case, and the zero-modes of the Dirac operator are identified with the lowest Landau level basis states whose spin direction is opposite to the external magnetic field. When the gauge group is identical to the holonomy group of the coset M ≃ G/H, (86) can be represented by the group theoretical quantities [70]: where C(G) represents (quadratic) Caimir for the isometry group G and C(H, R) denotes (quadratic) Casimir for the holonomy group H made by the gauge group representation R. Therefore, now we are able to derive the eigenvalues of (−i D) 2 by using a simple group theoretical method. For S 2k ≃ SO(2k + 1)/SO(2k), we propose the SO(2k + 1) spinor Landau Hamiltonian as where we used the Ricci scalar of S 2k14 For the irreducible representations the Casimir eigenvalues are respectively given by and the eigenvalues of (89) are given by and the nth Landau level degeneracy is derived as For the spinor particle 15 , we take where for + (↑ spin state), I ≥ 0, while for − (↓ spin state), I ≥ 1. This implies that the spin polarization due to the Zeeman effect effectively changes the strength of magnetic flux by ± 1 2 according to the direction of spin. In accordance with ± sector, (93) is block diagonalized as 14 The SO(2k) Casimir for the fundamental representation (79b) (I = 1) is equal to the Ricci scalar of S 2k ; 15 For the scalar particle, we substitute to (89) to derive the energy eigenvalue (78): where whose degeneracies are respectively given by D n (k, I + 1) and D n (k, I − 1) through the formula (94) 16 . In low dimensions, (99) reads as (n + 1)(n + I + 1) 0 0 n(n + I) , n 2 + n(I + 4) + 2(I + 2) 0 0 n 2 + n(I + 2) , The Landau level energy spectrum is bounded by zero for the lowest Landau level basis states (n = 0) with ↓ spin: and the number of the zero-energy states is given by Since the Hamiltonian is the square of the Dirac operator, the zero-energy eigenstates correspond to the zero-modes of the Dirac operator: and the index theorem tells that the number of zero-modes is equal to the topological charge of the non-trivial gauge configuration: where c k denotes the kth Chern number of the SO(2k) monopole (59). Thus, we verified (83) for arbitrary k. 16 It can be confirmed that E+(n)|I=0 (99) and Dn(k, 2J = I + 1)|I=0 = Dn(k, 1) (94) respectively reproduce the eigenvalues and the degeneracy of the free Dirac operator without gauge field [72,73,74,75]: 2M E+(n)|I=0 = n + k,

Laughlin-like wavefunction
For higher dimensional quantum Hall effect, the particles carry the SO(2k) color degrees of freedom with the geometry S 2k−2 F , and the total space will be given by x a x a = r 2 denotes the basemanifold S 2k while y = (y 1 , y 2 , · · · , y 2k−2 ) with 2k−1 i=1 y i y i = r 2 represents the coordinates on (2k−2)-dimensional internal space S 2k−2 (which is regarded as the classical counterpart of fuzzy bundle coordinates X i (30)). The coordinates of the total space S 2k ⊗ S 2k−2 is represented by where ψ denotes 2 k−1 component spinor giving the internal coordinates by the relation: The lowest Landau level basis states can be constructed by taking a fully symmetric products of the components of Ψ(x) : with m 1 + m 2 + · · · + m 2k = I. For m = 1 the particles occupy all the lowest Landau level states on S 2k , and so the total particle number N is where D(k, I) denotes the number of states of the total space S 2k F , and D(k − 1, I) stands for the number of states of the fuzzy-fibre S 2k−2 F . For I/2 → mI/2, the state number on S 2k changes as With use of the Slater determinant, the Laughlin-like groundstate wavefunction is constructed as where A = (m 1 , m 2 , · · · , m 2k ) and m is taken as odd integer to keep the Fermi statistics of the particles. When the power of Ψ A changes from 1 to m, the monopole charge changes from I to mI, and then Ψ Lin is considered to correspond to the 2kD quantum Hall liquid at the filling factor: Notice that since m is an odd inter, ν 2k is also the inverse of an odd integer. From the perspective of the original basemanifold S 2k , Ψ Llin denotes the incompressible liquid made of the particles. However, from the emergent (4k − 1)D space-time point of view, the particle corresponds to (2k − 2)-brane, and Ψ Llin is alternatively interpreted as a many-body state of membranes.

Tensor Monopole Fields from Non-Abelian Monopole Fields
We discussed the non-abelian monopoles whose gauge group is compatible with the holonomy of sphere. In this section, we introduce the other type of monopole, the tensor monopole [40,41] whose gauge group is U (1) whole gauge field is not a vector field but an antisymmetric tensor field 17 .

Tensor monopole fields
To begin with, we review several basic properties of the n-form tensor gauge field [41]: where C a 1 a 2 ···an represent a totally antisymmetric tensor gauge field. Notice that C a 1 a 2 ···an is not a matrix-valued gauge field but a tensor extension of the U (1) gauge field. Like the ordinary U (1) gauge theory, the field strength is defined as where For instance, The U (1) gauge symmetry is incorporated in the following way. The U (1) gauge transformation is given by with It is obvious that the field strength G is invariant under (118). In terms of the tenor components, the gauge transformation is represented as For instance, 17 The antisymmetric tensor gauge field is realized as a solution of the Kalb-Ramond equation and also referred to as the Kalb-Ramond field [42]. Table 1: Relations between the non-abelian monopole and the tensor monopole.

/ Non-abelian monopole Tensor monopole
It is a simple exercise to see that (117) is invariant under (121). The field strength of the U (1) tensor monopole located at the origin of (n + 2)D Euclidean space is given by where g denotes the charge of U (1) tensor monopole. The integral of the gauge field strength over where A(S n+1 ) represents the area of S n+1 .

Relation between field strengths of monopoles
The non-abelian and tensor monopoles are two different extensions of the Dirac monopole in terms of internal and external indices respectively. As discussed in Sec.3, there is no reasonable distinction between the external and internal spaces in the lowest Landau level, and so it is expected that non-abelian and tensor monopoles should be "equivalent" in some sense. Interestingly, for the SU (2) monopole and 3-rank tensor monopole, their connection has already been pointed out, at least for fundamental representation (quaternions) [76] and for the integral form [77]. As a natural generalization of these results, we demonstrate connection between tensor and non-abelian monopoles for fully symmetric representation in arbitrary even dimensions. In the following, we take n as an odd integer, n = 2k − 1 and the monopole at the center of S 2k [see Table 1] and consider the tensor monopole gauge field of the following form We fix the ratio between two monopole charges, c k and g k , by imposing the condition: with and S 2k the relation between two monopole charges is determined as Eq.(125) is rather "trivial", since we are always able to impose (125) by fixing the ratio between the two monopole charges. What we really need to verify is the local non-abelian and tensor monopole relation: To prove (130) we use a brute force method: We substitute the explicit form of F (54) to the right-hand side of (130) to see whether we can derive G (124) on the left-hand side under the identification (129). For the component relation between G a 1 a 2 ···a 2k (a 1 , a 2 , · · · , a 2k = 1, 2, · · · , 2k + 1) and F ab , the local relation (130) can be rewritten as 18 .
(a 1 , a 2 , · · · , a 9 = 1, 2, · · · , 9) Thus we derived the tensor monopole gauge field G from trF k . Furthermore, for a general symmetric representation of the SO(2k) 19 we have a generic expression for the U (1) antisymmetric tensor field strength as where C(k, I) and G (I=1) a 1 as···a 2k+1 are respectively given by (34) and (136). Here, we used the formulae for the symmetric representation (244). It is rather simple to confirm the symmetric representation(139) for I = 1 reproduces (136) by the formula With (140) and the formula about the lowest Landau level degeneracy we finally find that G takes an amazingly simple form 20 : 19 I = 1 corresponds to the spinor representation. 20 In differential form, (144) is represented as and hence the normalized U (1) tensor monopole charge q k (I) ≡ S 2k G 2k , is identical to the Chern number: q k (I) = c k (I).

Relation between gauge fields of monopoles
For non-abelian gauge field, we have [78] tr where L (2k−1) CS represents the Chern-Simons term Meanwhile for the tensor monopole gauge field, we have seen From the non-abelian and tensor monopole relation (130), it is obvious that the tensor monopole gauge field is identical to the Chern-Simons term of the non-abelian gauge field: For instance, Notice that tr(A 3 F 2 ) = tr(AF A 2 F ), since A and F take their values in matrix and are not commutative. For components of (150), we have The SO(2k) gauge transformation for A acts as the U (1) gauge transformation for C 2k−1 . For instance k = 2, the non-abelian (SU (2)) gauge transformation (57) acts to C 3 as The second term on the right-hand side is the total derivative. The third term satisfies 21 and is locally expressed as a total derivative (Poincaré Lemma). Consequently, (152) can be rewritten in the following form In general, the SO(2k) gauge transformation acts as U (1) gauge transformation to tensor gauge field (see Appendix C for more details): For practical applications, it is important to derive the explicit form of the tensor monopole gauge field. From the general formula (151), we derive the tensor monopole gauge field from the non-abelian monopole in low dimensions. We substitute the non-abelian monopole field (50) to the right-hand side of the formula (151). After a long straightforward calculation with use of trace formulae for gamma matrices, we obtain the following expressions for spinor representation: 21 tr(α 2n ) = 0 for any one-form α = dxaαa.
Notice that (2k − 1) rank tensor monopole gauge field exhibits kth power string-like singularity. Similarly for fully symmetric representation, we obtain For I = 1, (157) is reduced to (156). One may also confirm that (157) indeed gives the field strength(145) through the formula:

Quantum Nambu geometry via tensor monopole
In the lowest Landau level, the covariant angular momentum is quenched, and then we have the identification: In 3D, two rank antisymmetric tensor is equivalent to vector, and the angular momentum is directly related to the coordinates of fuzzy two-sphere (24). However in higher dimensions, two rank antisymmetric tensor is no longer equivalent to vector and the angular momentum is not apparently related to the coordinates of fuzzy sphere. As mentioned in Sec.3.2, the quantum Nambu bracket suggests the existence of tensor monopole, and we have shown the non-abelian and tensor monopole relation (130) or The identification (159) suggests that (160) becomes to in the lowest Landau level, and the coordinates of higher dimensional sphere are now regarded as the operators. (161) is a natural generalization of (25). Let us consider the algebra for X a . For this purpose, it is useful to adopt the analogy between the algebras of X a and the covariant derivatives −iD a [30]. For S 2 F case, the algebra of X i is given by while the covariant derivative gives One may notice the analogy: This analogy can hold in higher dimensions [see Sec.3.2], and for evaluation of the Nambu bracket for X a we utilize the following identification: The right-hand side gives and the trace is evaluated as Due to the relation (141), we finally obtain [X a 1 , X a 2 , · · · , X a 2k ] = i k C(k, I)α 2k−1 ǫ a 1 a 2 ···a 2k+1 X a 2k+1 , which is exactly equal to the quantum Nambu algebra for fuzzy sphere (33).

Flux Attachment and Tensor Chern-Simons Field Theory for Membranes
Here we discuss physics of A-class topological insulator based on Chern-Simons tensor field theory. We will see exotic concepts in 2D quantum Hall effect are naturally generalized in higher dimensions: • Flux attachment and composite particles [45,46,44] • Effective topological field theory [43,44] • Fractional statistics of quasi-particle excitations [79] • Haldane-Halperin hierarchy [28,80] . . .

Basic observations
Before going to the details, we summarize basic observations about the relevant physical concepts and associated mathematics in higher dimensions.
• (2k − 1) rank tensor gauge field and (2k − 2)-brane The (2k − 1) rank gauge field is naturally coupled to the (2k − 1) rank current of (2k − 2)-brane. The membrane degrees of freedom is automatically incorporated in the geometry of S 2k F as the fuzzy fibre S 2k−2 Although S 2k−2 F represents the internal non-abelian gauge space of the particle, the internal space is as large as the external space S 2k , and it can be regarded as (2k − 2)-brane in the enlarged space [see Fig.2] that consists of the external S 2k and the "internal" S 2k−2 which membrane occupies. Since membrane is associated with the non-abelian flux of non-abelian monopole, membrane can be considered as a charged excitation induced by a penetration of the non-abelian flux in higher dimensions.
• Emergence of (4k − 1)D space-time and J-homomorphism The left homotopy is related to the SO(2k) monopole at the origin of (2k+1)D space and describes the non-trivial winding from the equator of S 2k to the SO(2k) monopole gauge group, while the right homotopy describes a non-trivial winding from (4k − 1) space(-time) to the base-manifold S 2k on which (2k − 2)-brane lives. In particular for k = 1, (173) gives 22 In general, J-homomorphism represents the homomorphism between the homotopy group of the orthogonal group and that of sphere: π l (SO(M )) → π l+M (S M ).  Table 2: Two (2k − 2)-branes in (4k − 1)D space-time. From the co-dimension 2 space, the membranes are regarded as "point-particles".
The left homotopy guarantees the non-trivial topology of Dirac monopole bundle, while the right homotopy is the 1st Hopf map which represents the underlying mathematics of fractional statistics for 0-brane in 3D space(-time) [82]. The world line of the 0-brane on S 2 corresponds to the S 1 fibre on S 2 , and the non-trivial linking of world lines of two 0-branes indicates the topological number denoted by the 1st Hopf map [84]. Similarly, the non-trivial homotopy π 4k−1 (S 2k ) ≃ Z is related to the fractional statistics in (4k − 1)D space(-time) [85,86,87]. The dimension of the object obeying the fractional statistics can readily be obtained by the following dimensional counting. Since the dimension of the total space(-time) is (4k − 1) and S 2k is the basemanifold, the remaining (4k − 1)− 2k = 2k − 1 dimension should be the dimension of the world volume of the object that obeys the fractional statistics. Indeed the dimension of (2k − 2)-brane world volume is (2k − 1) dimension, and so (2k − 2)-branes are expected to obeys the fractional statistics. Another way to see (2k − 2)-brane can obey fractional statistics is to notice the co-dimension. The necessary condition for the existence of fractional statistics is the co-dimension 2 where the braiding operation has non-trivial meaning. Indeed, the co-dimension of two (non-overlapping) (2k − 2)-branes in (4k − 2) space is 2 [ Table 2]. From the co-dimension, two membranes are regarded as two point-like objects, and the idea of fractional statistics (for point-like object) in 3D can similarly be applied to higher dimensions.

• Physical realization of fractional statistics
The statistical transformation is physically achieved by acquiring Aharonov-Bohm phase [89,90], where the particles acquires a statistical phase during a trip around the magnetic flux. In the fractional quantum Hall effect, the statistical phase accounts for the fractional statistics of fractionally charged quasi-particle excitation [79] and also for the statistical transformation from electron to composite boson at the odd-denominator fillings [45,46,44]. The statistical transformation to composite boson is elegantly described by the Chern-Simons field theory formulation [43,44]. In higher dimensions, there are (2k − 2)-branes coupled to the (2k − 1) rank tensor U (1) gauge field, the statistical transformation is generalized in higher dimensions by adopting tensor version of Chern-Simons field theory for membranes instead of particles. The mathematics of linking and phase interaction mediated by tensor gauge field in higher dimensions have already been formulated in Refs. [85,76,86,88] [see Appendix D]. Based on the results, we discuss the statistical transformation and effective field theory for the A-class topological insulator. We will see that A-class topological insulator cam be regarded as a superfluid state of composite membranes in the same way as the fractional quantum Hall effect is considered as a superfluid state of composite bosons.

Tensor flux attachment
The flux attachment is achieved by applying the singular gauge transformation [89,90,45]. We first generalize this procedure in higher dimensions. Suppose p-brane occupying the dimensions from x 0 to x p in D = 2p + 3 [ Table 3]. (Here, we render p as a non-negative integer not only an even integer.) From the remaining (p + 2) dimension (x p+1 , · · · , x p+2 ) p-brane is regarded as a point-particle. We apply the flux attachment to such a "point-particle" in (p + 2)-dimensional space. Technically, the gauge field associated with the flux readily be obtained by a "dimensional reduction" of the tensor monopole gauge field (157). On the equator of S p+2 (x p+3 = 0), the tensor monopole gauge field (157) is reduced to where µ 1 , µ 2 , · · · , µ p+2 = p + 1, p + 2, · · · , 2p + 2 and r 2 = 2p+2 µ=p+1 x µ x µ . For instance, we have They are regarded as the tensor gauge field on the (p + 2)D plane [ Fig.3]. With use of the Green Figure 3: Flux is attached to membrane and yields the tensor gauge field around the membrane. function in (p + 2)D space 23 , (175) can be represented as which take the form of "pure gauge": where Λ µ 1 µ 2 ···µ p+1 is formally expressed as The corresponding field strength is evaluated as where B represents the flux-like magnetic field: Φ p stands for the strength of the flux. When a p-brane with charge e p moves around the flux, the p-brane acquires the phase: where M p denotes the configuration of p-brane. The phase should be 1: and then Φ p is quantized as 23 G (d) denotes Green function for the d-D Laplace equation: where ∂ 2 = d µ=1 ∂ ∂xµ ∂ ∂xµ . Explicitly, the Green functions are given by with integer n. Hence, the minimum unit of flux is given by 24 Let us consider a (composite) p-brane that carries κ fluxes: where Q p denotes the p-brane charge. In the (D − p − 1)-dimensional space perpendicular to p-brane, (191) can locally be rewritten as where with x µ ⊥ = (x p+1 , x p+2 , · · · , x D−1 ). Furthermore, one may readily derive (192) by integrating over the space parallel to p-brane, x = (x 1 , x 2 , · · · , x p ), with use of Here, J µ 1 µ 2 ···µ p+1 (x) denotes the p-brane current 25 and ρ(x) and B(x) are given by which is consistent with (187). 25 The explicit form of the membrane current is given as follows. We place p-brane in the dimensions, (x 1 , x 2 , · · · , x p ), and we parametrize the coordinates of membrane as x µ = X µ (σ), (µ = 1, 2, · · · , p), where σ = (σ 1 , σ 2 , · · · , σ p ) denotes the intrinsic coordinates of the the p-brane. Non-vanishing component of p-brane current is given by Consequently, one may find the covariant expression for (194): This realizes the tensor flux attachment to p-brane in (2p + 3)D space(-time), and is a natural generalization of the flux attachment in 3D space(-time): 6.3 (2k − 2)-brane as the SO(2k + 1) skyrmion In the realization of the fractional statistics for the SO(3) nonlinear model in (2+1)D [82,83], the statistical gauge field is coupled to the SO(3) skyrmion topological current. The underlying mathematics of the SO(3) skyrmion is given by the 1st Hopf map [82], where the target space S 2 (7) corresponds to the field manifold of skyrmion. Since both of SO(3) non-linear sigma model and the Haldane's two-sphere are based on the 1st Hopf map, the mathematical structure of the SO(3) non-linear sigma model is quite similar to that of the Haldane's two-sphere [28]; The internal field manifold of the SO(3) skyrmion is S 2 and the "hidden" local symmetry is U (1), while in the Haldane's two-sphere the external space is S 2 and the gauge symmetry is U (1). Thus interestingly, we can "interchange" the set-ups for the SO(3) non-linear sigma model and the Haldane's two-sphere by exchanging external and internal spaces. Subsequently, the authors in [76,85,86,77] adopted the 2nd Hopf map (and the 3rd Hopf map also) to construct the SO(5) non-linear sigma model for 2-brane on a four-sphere. We further apply this idea to construct the non-linear sigma model for membrane of higher dimensional quantum Hall effect. Since 2kD quantum Hall effect accommodates the "internal" (2k − 2)-brane on the external space S 2k , the corresponding non-linear sigma model is the SO(2k + 1) non-linear sigma model realizing a skyrmion solution spatially extended over S 2k−2 with S 2k internal space. The internal space coordinates of the SO(2k + 1) skyrmion are given by where n is subject to the condition of S 2k : n a n a = 1.
Following to the Derrick's theorem, there does not exist static soliton solutions in the scalar field theory whose Lagrangian only consists of the second order kinetic term, tr(∂ µ n) † (∂ µ n), and selfinteraction potential in the space-time whose dimension is larger than 2. However, there are at and the total charge Qp is with Vp the volume of the p-brane, Vp ≡ d p σ det( ∂X ∂σ ).
least two ways to evade the Derrick'e theorem. One is to include an extra interaction term to stabilize the soliton configuration, and the other is to adopt a higher derivative kinetic term [91].

Flux cancellation and tensor Chern-Simons theory
Topological features of the fractional quantum Hall effect are nicely captured by the Chern-Simons effective field theory [43,44]. The Chern-Simons field is introduced to cancel the external magnetic field, and the odd number Chern-Simons fluxes attachment transmutes electron to composite boson. In 2D, both of the external magnetic field and the Chern-Simons field are U (1), and then the relation for flux cancellation is rather trivial Meanwhile in higher dimensions, we have to deal with the non-abelian external field and membranes. One may wonder how we can incorporate these two objects to generalize the flux cancellation. The non-abelian and tensor monopole relation (149) gives a crucial hint. We demonstrated that the non-abelian gauge field is "equivalent" to the U (1) tensor gauge field. This suggests that the cancellation of the external non-abelian gauge field by abelian gauge (tensor) field is and so JD−p−1 = 1 (p + 1)!(D − p − 1)! ǫ µ 1 µ 2 ···µ D Jµ 1 µ 2 ···µ p+1 dxµ p+2 dxµ p+3 · · · dxµ D .
The original space-time D.  Table 4: The effective field theory of A-class topological insulators in (2k + 1)D space-time is given by the tensor Chern-Simons theory of (2k − 2)-branes in (4k − 1)D space-time.
possible. We thus consider the U (1) Chern-Simons tensor flux attachment to membrane and the flux cancellation condition is generalized in higher dimensions as In low dimensions, (211) yields Since the membranes are the fundamental object in A-class topological insulator, it is natural to reformulate the theory by using the membrane degrees of freedom. We propose the Chern-Simons tensor field theory as the effective field theory for A-class topological insulators 27 where J p+2 is given by the membrane p + 2 form current (208) and The action (213) is equivalent to the one used in the analysis of linking of membrane currents [86]. For p = 2k − 2, the Chern-Simons coupling is given by whereΦ p denotes the unit-flux (190) and ν 2k stands for the filling factor of (2k − 2)-brane (113). Notice that while the original space-time dimension is (2k + 1) that the tensor Chern-Simons theory is defined in (4k − 1)D space(-time) (for p = 2k − 2) [ Table 4]. Thus, the tensor Chern-Simons theory is formulated in the enlarged space as consistent with the observation in Sec.4.2.
Since there does not exit the kinetic term in the action, C p+1 is not a dynamical field but an auxiliary field determined by the equations of motion 28 In the space-time components, (218) can be written as where ǫ i 1 i 2 ···i 2p+2 ≡ ǫ i 1 i 2 ···i 2p+2 0 and (219a) indeed realizes the generalization the flux attachment for membrane (200) and suggests that the membrane with unit charge e p carries m k fluxes in unit ofΦ p . Meanwhile (219b) gives a generalization of the Hall effect. From the antisymmetric property of the epsilon tensor, we have Since the membranes are always an even dimensional object (p = 2k−2), the Hall effect necessarily holds: Meanwhile if p was odd, the Hall effect would not necessarily hold.

Composite membrane and fractional charge
Integration of the Chern-Simons field in the tensor Chern-Simons action gives a generalized Gauss-Hopf linking between two membrane world volumes, which can alternatively be understood as the winding number from the two higher dimensional "tori" to a higher dimensional sphere [see [86] or Appendix D]: From (223), it is obvious that the non-trivial winding exists for arbitrary k, and so does the linking. Even though the membrane statistics is related to the linking, it does not necessarily mean that 28 In component representation, (213) and (218) are respectively expressed as membranes obey the fractional statistics. For instance in quantum Hall effect, for quasi-excitation to be anyonic, the fractional charge is essential [79]. Similarly, for statistical transmutation from electron to (composite) boson, the odd number flux attachment is crucial, and hence quantum Hall liquid at the magic filling ν = 1/m for odd m is considered as a superfluid state of the composite bosons [45,46,43]. First, we consider the composite boson counterpart in A-class topological insulators. At ν = 1/m k , m k fluxes are attached to the membrane and the membrane becomes to a composite object of membrane and the fluxes. The original statistics of the membrane is fermionic since at ν = 1 membrane corresponds to "quarks" with color degrees of freedom. The statistics of the composite membrane is derived by evaluating the phase interaction between two composite membranes. Under the interchange, the membranes acquires the following statistical phase where we used A = m k Φ 2k−2 (Φ 2k−2 = 2π e 2k−2 ) and m is odd so is m k . Since the membrane acquires the extra minus sign under the interchange of membranes, by the flux attachment the membrane transmutes its statistics from fermion to boson, and obeys the Bose statistics. Notice that such transmutation is only possible when the inverse of the magic filling ν 2k = 1/m k is odd. In the same way as the fractional quantum Hall effect at ν = 1/m is considered as a condensation of composite bosons, A-class topological insulator at ν = 1/m k can be interpreted as a superfluid state of composite membranes. Next let us discuss the statistics of membrane excitation. We first need to specify the membrane charge. When the monopole charge is I/2, the number of states on S 2k is given by and for the filling ν = 1 the (2k − 2)-brane with unit charge e 2k−2 , occupies each state. When the monopole charge change as I ′ = mI, the number of states becomes to In other words, each state occupied by membrane is split to m k states, and so does the membrane charge. Hence at ν 2k = 1 m k , the fractional charge of (2k − 2)-brane is given by 29 : 29 (230) can also be derived from the perspective of 0-branes. When the monopole charge is I/2, the (2k−2)-brane is made of I 1 2 k(k−1) 0-branes, and then (2k − 2)-brane charge is expressed by where κ(k) is a coefficient of dimension of (mass) 2k−2 . At I ′ = mI, the 0-brane charge becomes to and so the (2k − 2)-brane charge reads as Since the (2k − 2)-brane excitation is induced by the flux penetration, (2k − 2)-brane excitation is a "composite" of the fractional charge e ′ 2k−2 and the unit fluxΦ p = 2π e 2k−2 . Therefore, the geometrical phase which a fractionally charged (2k − 2)-brane acuires by the round trip around another (2k − 2)-brane excitation is given by We thus have shown that the statistical phase of membrane excitation is 2πν 2k and hence membrane excitations are anyonic.

Dimensional hierarchy and analogies to string theory
Analogies between the A-class topological insulator and the string theory will be transparent in analyses of membrane properties. According to the Haldane-Halperin picture [28,80], quasiparticles condense on the parent quantum Hall liquid to generate a new incompressible liquid and the filling factor exhibits a hierarchical structure called Haldane-Halperin hierarchy. Similarly in A-class topological insulator, membrane excitations are expected to condense to form a new incompressible liquid and the filling factor will exhibit a generalized Haldane-Halperin like hierarchy: where each of p 1 , p 2 , · · · denotes a natural number. Apart from the Halperin-Haldane hierarchy, the membranes exhibit a unique type of condensation -the dimensional hierarchy [31,32], which reflects the special dimensional pattern of A-class topological insulator. From (81), one may find that there is a relation between 2k and (2k − 2)D lowest Landau level degeneracies: and then D LLL (k, I) ∼ I k · I k−1 · I k−2 · · · I 2 · I = I 1 2 k(k+1) .
Eq.(234) suggests a hierarchical structure in dimensions. This feature can intuitively be understood by the following simple explanations. Each of the SO(2k) monopole fluxes on S 2k occupies an area ℓ 2k B = (αr) k = (2r 2 /I) k , and the number of fluxes on S 2k is given by ∼ r 2k /ℓ 2k B ∼ I k . Since the SO(2k) non-abelian flux is equivalent to (2k − 2)-brane, one may say (2k − 2)-brane occupies the same area ℓ 2k B and ∼ I k is the number of (2k − 2)-branes. Similarly, on S 2k−2 , there are (2k − 4)-branes each of which occupies the area l 2k−2

B
, and the total number of (2k − 4)-branes is ∼ I k−1 . By repeating this iteration from 2kD to the lowest dimension 2D, we obtain the formula (233). The corresponding filling factor (for 0-brane) is given by Similar to the Haldane-Halperin hierarchy, such hierarchical structure may imply a particular condensation of membranes which ranges in dimensions. Inversely, one may see the formula from low dimension and find a physical interpretation of the hierarchy; low dimensional membranes gather to form a higher dimensional incompressible liquid of membranes [ Fig.4]. Most general total filling factor will be give by the combination of (232) and (235), (236) Figure 4: Low dimensional membranes condense to to form a higher dimensional membrane. Since the membrane itself describes fuzzy sphere or A-class topological insulator, one may alternatively state this phenomena as the dimensional hierarchy of A-class topological insulator.
Since ν 2 , ν 4 , · · · , ν 2k are equally treated in (236), one can arbitrarily interchange νs in different dimensions, which suggests a "democratic" property of A-class topological insulator. The filling factor denotes the membrane density and the interchangeability of the filling factors implies a equivalence of membranes in different dimensions. This may immediately remind the brane democracy of string theory; any D-brane can be a starting point to construct another D-brane in different dimensions [94]. Thus, the dimensional hierarchy -the membranes condense to make an incompressible liquid -can be regarded as a physical realization of the brane democracy. The index theorem also suggests the close relations between the A-class topological insular and the string theory. The index theorem tell that the lowest Landau level degeneracy, D LLL (k − 1, I), is equal to the (k − 1)th Chern-number, c k−1 (I + 1). This equality means that the (k − 1)th Chern number is identical to the (2k − 2)-brane charge, since the number of 0-branes is given by the lowest Landau level degeneracy. Analogous phenomena have been reported in the context of Myers effect of string theory [34] where low dimensional D-branes on higher dimensional D-brane are regarded as magnetic fluxes of monopole. In particular, Kimura found that the number of D0-branes that constitute a spherical D(2k − 2)-brane is given by the (k − 1)th Chern-number of non-abelian monopole [35]. The fact that the membrane charge is equal to the lowest Landau level degeneracy -the number of the fundamental elements of the lowest Landau level, implies that membranes themselves should be identified with the fundamental elements of the space(-time). This observation again reminds the idea of the matrix theory [92,93] in which the D0 (D−1)) branes constitute the space(-time) and the spacial coordinates are represented by matrices. It is quite interesting that the ideas of the string theory can be understood in the context of topological insulators.

Summary and Discussions
We discussed physical realization of the quantum Nambu geometry in the context of A-class topological insulator. As the higher dimensional dimensional fuzzy sphere has two different formulations, A-class topological insulator has two physically different realizations, one of which is the non-abelian monopole realization and the other is the tensor monopole realization. We established the connection between these two kinds of monopole through the Chern-Simons term. Based on the non-abelian and tensor connection, we generalized the flux attachment procedure in A-class topological insulator to construct the Chern-Simons tensor effective field theory. We also showed the exotic concepts in 2D quantum Hall effect can be naturally generalized to A-class topological insulators.
For convenience of readers, we summarize the main achievements of the present work. In arbitrary even dimension we established • Non-commutative coordinates of quantum Nambu geometry via angular momentum construction [Sec. 5.4] Subsequently, we discussed their physical consequences in the context of A-class topological insulators: • Tensor flux attachment to membrane and its statistical phase [Sec.6.2] • Higher D generalization of flux cancellation and Chern-Simons tensor field theory [Sec. 6.4] • Fractional charge and anyonic statistics for membrane [Sec. 6.5] Though the original space-time of A-class topological insulators is arbitrary odd dimensional space-time (2k + 1), the effective Chern-Simons tensor field theory lives in (4k − 1) dimensional space-time, not in arbitrary odd dimensions. Since the Chern-Simons theories are defined in arbitrary odd dimensions, one may wonder why we "skipped" the Chern-Simons theories in other odd dimensions, 5, 9, 13, · · · . We will address the issue in the forth coming paper. The edge theory and accompanied Callan Harvey mechanism based on the Chern-Simons tensor field theory may also be interesting. The quantum Nambu bracket has attracted a lot of attentions in recent years since it is expected to provide an appropriate description for M-brane boundstate [95] and plays a vital role in Bagger-Lambert-Gustavsson theory of multiple M-branes [96,97,98]. Non-associative geometry associated with the quantum Nambu bracket has also been vigorously studied [99,100]. Zhang, a pioneer of topological insulator and higher D. quantum Hall effect, noted that the study of condensed matter physics may provide an alternative approach to understand exotic ideas in mathematical and particle physics [101]. We thus enforced his observation by demonstrating quantum Nambu geometry in A-class topological insulators inspired by the recent works [14,15]. We hope the present work will further deepen the understanding of non-commutative geometry and string theory as well as topological insulators. and [G a 1 , G a 2 , · · · , G a 2k ] = i k C ′ (k, I) · ǫ a 1 a 2 ···a 2k+1 G a 2k+1 , where C ′ (k, L) is given by The SO(2k + 1) generators are constructed as G a and G ab satisfy the closed algebra: which is identical to the SO(2k + 2) algebra. X a and X ab operators of S 2n F are constructed as with α = 2r/I (3). For I = 1, G a and G ab are reduced to the fundamental representation, Γ a (44) and Σ ab = −i 1 4 [Γ a , Γ b ]. The SO(2k) group has two Weyl representations, Σ + µν and Σ − µν (µ, ν = 1, 2, · · · , 2k). For the fundamental representation I = 1, the SO(2k) Weyl generators satisfy and for the fully symmetric representation Here, D LLL (k − 1, I) denotes the dimension of the SO(2k) fully symmetric representation equal to the dimension of the SO(2k − 1) fully symmetric representation (233). For the fundamental representation, G µν and Σ ± µν are related by (46), and for the generic fully symmetric representation G µν can be represented as a block diagonal form and Σ ± µν appear in the left-up and right-down blocks:
Λ 2k−2 corresponds to the U (1) transition function of the (2k − 1) form gauge field [see (155)]. The associated U (1) topological charge q k is given by which is exactly equal to the kth Chern number and consistent with (143). We also demonstrate that the pure gauge Chern-Simons action reproduces ρ 2k−1 on the equator S 2k−1 . The SO(2k) non-abelian gauge fields on north and the south hemispheres are related as where g is given by Here, we used On the equator of S 2k , the transition function is reduced to (261): In the pure gauge the Chern-Simons action (147) becomes to where we used On the equator S 2k−1 , (277) is reduced to (268) up to a proportional factor: