Hopf Maps, Lowest Landau Level, and Fuzzy Spheres

This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an interesting hierarchical structure made of"compounds"of lower dimensional spheres. We give a physical interpretation for such particular structure of fuzzy spheres by utilizing Landau models in generic even dimensions. With Grassmann algebra, we also introduce a graded version of the Hopf map, and discuss its relation to fuzzy supersphere in context of supersymmetric Landau model.


Introduction
Fuzzy two-sphere introduced by Madore [1] was a typical and important fuzzy manifold where particular notions of fuzzy geometry have been cultivated. In subsequent explorations of fuzzy geometry, higher dimensional and supersymmetric generalizations of the fuzzy spheres were launched by Grosse et al. [2,3,4]. Furthermore, as well recognized now, string theory provides a natural set-up for non-commutative geometry and non-anti-commutative geometry [5,6,7]. In particular, classical solutions of matrix models with Chern-Simons term are identified with fuzzy two-spheres [8], fuzzy four-spheres [9] and fuzzy superspheres [10]. Fuzzy manifolds also naturally appear in the context of intersections of D-branes [11,12,13]. In addition to applications to physics, the fuzzy spheres themselves have intriguing mathematical structures. As Ho and Ramgoolam showed in [14] and subsequently in [15] Kimura investigated, the commutative limit of (even-dimensional) fuzzy sphere takes the particular form 1 : S 2k F ≃ SO(2k + 1)/U (k).
From this coset representation, one may find, though S 2k F is called 2k-dimensional fuzzy sphere, its genuine dimension is not 2k but k(k + 1). Furthermore, the fuzzy spheres can be expressed as the lower dimensional fuzzy sphere-fibration over the sphere Thus, S 2k F has "extra-dimensions" coming from the lower dimensional fuzzy sphere S 2k−2 F . One may wonder why fuzzy spheres have such extra-dimensions. A mathematical explanation may go as follows. Fuzzification is performed by replacing Poisson bracket with commutator on a manifold. Then, to fuzzificate a manifold, the manifold has to have a symplectic structure to be capable to define Poisson bracket. However, unfortunately, higher dimensional spheres S 2k (k ≥ 2) do not accommodate symplectic structure, and their fuzzification is not straightforward. A possible resolution is to adopt a minimally extended symplectic manifold with spherical symmetry. The coset SO(2k + 1)/U (k) suffices for the requirement. 1 The odd-dimensional fuzzy spheres can be given by S 2k−1 F ≃ SO(2k)/(U (1) ⊗ U (k − 1)) [16,17]. For instance, Since detail mathematical treatments of fuzzy geometry have already been found in the excellent reviews [18,19,20,21,22,23], in this paper, we mainly focus on a physical approach for understanding of fuzzy spheres, brought by the developments of higher dimensional quantum Hall effect (see as a review [24] and references therein). The lowest Landau level (LLL) provides a nice way for physical understanding of fuzzy geometry (see for instance [25]). As we shall see in the context, generalization of the fuzzy two-spheres is closely related to the generalization of the complex numbers in 19th century. Quaternions discovered by W. Hamilton were the first generalization of complex numbers [26]. Soon after the discovery, the other division algebra known as octonions was also found (see for instance [27]). Interestingly, the division algebras are closely related to topological maps from sphere to sphere in different dimensions, i.e. the Hopf maps [28,29]. While the division algebras consist of complex numbers C, quaternions H and octonions O (except for real numbers R), there is another generalization of complex numbers and quaternions; the celebrated Clifford algebra invented by W. Clifford [30]. The generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-sphere to its higher dimensional cousins. Particular geometry of fuzzy spheres directly reflects features of Clifford algebra. With non-Abelian monopoles in generic even dimensional space, we explain the particular geometry of fuzzy spheres in view of the lowest Landau level physics. There is another important algebra invented by H. Grassmann [31]. Though less well known compared to the original three Hopf maps, there also exists a graded version of the (1st) Hopf map [32]. We also discuss its relation to fuzzy supersphere.
The organization is as follows. In Section 2, we introduce the division algebras and the Hopf maps. In Section 3, with the explicit construction of the 1st Hopf map, we analyze the Landau problem on a two-sphere and discuss its relations to fuzzy two-sphere S 2 F . The graded version of the Hopf map and its relation to fuzzy superspheres are investigated in Section 4. In Section 5, we extend the former discussions to the 2nd Hopf map and fuzzy four-sphere S 4 F . In Section 6, we consider the 3rd Hopf map and the corresponding fuzzy manifolds. In Section 7, we generalize the observations to even higher dimensional fuzzy spheres based on Clifford algebra. Section 8 is devoted to summary and discussions. In Appendix A, for completeness, we introduce the 0th Hopf map and related "Landau problem" on a circle. In Appendix B, the SU (k + 1) Landau model and fuzzy CP k manifold are surveyed.

Hopf maps and division algebras
As the division algebras consist of only three algebras, there exist three corresponding Hopf maps, 1st, 2nd and 3rd. The three Hopf maps are closely related bundle structures of U (1), SU (2), and SO(8) monopoles [33,34,35]. Interestingly, the Hopf maps exhibit a hierarchical structure. Each of the Hopf maps can be understood as a map from a circle in 2D division algebra space to corresponding projective space: For instance, in the 1st Hopf map, the total space S 1 C represents a circle in 2D complex space, i.e. S 3 , and the basespace CP 1 denotes the complex projective space equivalent to S 2 . For the 2nd and 3rd Hopf maps, same interpretations hold by replacing complex numbers C with quaternions H, and octonions O, respectively. For later convenience, we summarize the basic properties of quaternions and octonions. The quaternion basis elements, 1, q 1 , q 2 , q 3 , are defined so as to satisfy the algebra [26] q 2 1 = q 2 2 = q 2 3 = q 1 q 2 q 3 = −1, q i q j = −q j q i (i = j), or equivalently, where ǫ ijk is Levi-Civita antisymmetric tensor with ǫ 123 = 1. Thus, quaternion algebra is noncommutative. As is well known, the quaternion algebra is satisfied by the Pauli matrices with the identification, q i = −iσ i . (We will revisit this point in Section 5.) An arbitrary quaternion is expanded by the quaternion basis elements: where r 0 , r i are real expansion coefficients, and the conjugation of q is given by The norm of quaternion, ||q||, is given by It is noted that q and q * are commutative. The normalized quaternionic space corresponds to S 3 , and the total manifold of the 2nd Hopf map, S 7 , is expressed as the quaternionic circle, S 1 H : Similarly, the octonion basis elements, 1, e 1 , e 2 , . . . , e 7 , are defined so as to satisfy the algebras where I, J, K = 1, 2, . . . , 7. δ IJ denotes Kronecker delta symbol and f IJK does the antisymmetric structure constants of octonions (see Table 1). The octonions do not respect the associativity as well as the commutativity. (The nonassociativity can be read from Table 1, for instance, (e 1 e 2 )e 4 = e 7 = −e 1 (e 2 e 4 ).) Due to their non-associativity, octonions cannot be represented by matrices unlike quaternions. However, the conjugation and magnitude of octonion can be similarly defined as those of quaternion, simply replacing the role of the imaginary quaternion basis elements q i with the imaginary octonion basis elements e I . An arbitrary octonion is given by with real expansion coefficients r 0 , r I , and its conjugation is r I e I . The norm of octonion, ||o||, is Like the case of quaternion, o and o * are commutative. The normalized octonion o * o = 1 represents S 7 , and the total manifold of the 3rd Hopf map, S 15 , is given by the octonionic One may wonder there might exist even higher dimensional generalizations. Indeed, following to the Cayley-Dickson construction [27], it is possible to construct new species of numbers. Next to the octonions, sedenions consisting of 16 basis elements can be constructed. However, the sedenions do not even respect the alternativity, and hence the multiplication law of norms does not hold: ||x|| ||y|| = ||x · y||. Then, the usual concept of "length" does not even exist in sedenions, and a sphere cannot be defined with sedenions and hence the corresponding Hopf maps either. Consequently, there only exit three division algebras and corresponding three Hopf maps.

1st Hopf map and fuzzy two-sphere
In this section, we give a realization of the 1st Hopf map and discuss basic procedure of fuzzification of sphere in LLL.

1st Hopf map and U (1) monopole
The 1st Hopf map can explicitly be constructed as follows. We first introduce a normalized complex two-spinor Thus φ, which we call the 1st Hopf spinor, represents the coordinate on the total space S 3 , and plays a primary role in the fuzzification of sphere as we shall see below.
With the Pauli matrices the first Hopf map is realized as It is straightforward to check that x i satisfy the condition for S 2 : Thus, (3.2) demonstrates the 1st Hopf map. The analytic form of the Hopf spinor except for the south pole is given by 4) and the corresponding fibre-connection is derived as where The curvature is given by which corresponds to the field strength of Dirac monopole with minimum charge (see for instance [36]). The analytic form of the Hopf spinor except for the north pole is given by The corresponding gauge field is (3.6), which suggests the two expressions of the Hopf spinor are related by gauge transformation. Indeed, where g is a U (1) gauge group element Here, the gauge parameter χ is given by tan(χ) = x 2 x 1 . The U (1) phase factor is canceled in the map (3.2), and there always exists such U (1) gauge degree of freedom in expression of the Hopf spinor. With the formula the above gauge fields are represented as and related by the U (1) gauge transformation The non-trivial bundle structure of U (1) monopole on S 2 is guaranteed by the homotopy theorem specified by the 1st Chern number

SO(3) Landau model
Here, we consider a Landau model on a two-sphere in U (1) monopole background. In 3D space, the Landau Hamiltonian is given by where D i = ∂ i + iA i , r = √ x i x i , and i = 1, 2, 3 are summed over. Λ i is the covariant angular The monopole gauge field has the form and the corresponding field strength is The covariant angular momentum Λ i does not satisfy the SU (2) algebra, but satisfies The conserved SU (2) angular momentum is constructed as which satisfies the genuine SU (2) algebra On a two-sphere, the Hamiltonian (3.7) is reduced to the SO(3) Landau model [37] where R is the radius of two-sphere. The SO(3) Landau Hamiltonian can be rewritten as where we used the orthogonality between the covariant angular momentum and the field strength; Λ i F i = F i Λ i = 0. Thus, the Hamiltonian is represented by the SU (2) Casimir operator, and the eigenvalue problem is boiled down to the problem of obtaining the irreducible representation of SU (2). Such irreducible representations are given by monopole harmonics [38]. The SU (2) Casimir operator takes the eigenvalues L 2 i = j(j + 1) with j = I 2 + n (n = 0, 1, 2, . . . ). (The minimum of j is not zero but a finite value I/2 due to the existence of the field angular momentum of monopole.) Then, the eigenenergies are derived as (3.10) In the thermodynamic limit, R, I → ∞, with B = I/2R 2 fixed, (3.10) reproduces the usual Landau levels on a plane: where ω = B/M is the cyclotron frequency. The degeneracy of the nth Landau level is given by d(n) = 2l + 1 = 2n + I + 1. (3.11) In particular, in the LLL (n = 0), the degeneracy is The monopole harmonics in LLL is simply constructed by taking symmetric products of the components of the 1st Hopf spinor where m 1 + m 2 = I (m 1 , m 2 ≥ 0). In the LLL, the kinetic term of the covariant angular momentum is quenched, and the SU (2) total angular momentum is reduced to the monopole field strength Then in LLL, the coordinates x i are regarded as the operator which satisfies the definition algebra of fuzzy two-sphere with α = 2R/I. We reconsider the LLL physics with Lagrange formalism. In Lagrange formalism, importance of the Hopf map becomes more transparent. The present one-particle Lagrangian on a two-sphere is given by with the constraint (3.14) In the LLL, the Lagrangian is represented only by the interaction term From (3.5), the LLL Lagrangian can be simply rewritten as 15) and the constraint (3.14) is It is noted that, in the LLL, the kinetic term drops and only the first order time derivative term survives. The Lagrangian and the constraint can be represented in terms of the Hopf spinor. Usually, in the LLL, the quantization is preformed by regarding the Hopf spinor as fundamental quantity, and the canonical quantization condition is imposed not on the original coordinate on two-sphere, but on the Hopf spinor. We follow such quantization procedure to fuzzificate two-sphere. From (3.15), the conjugate momentum is derived as π = −iIφ * , and the canonical quantization is given by This quantization may remind the quantization procedure of the spinor field theory, but readers should not be confused: The present quantization is for one-particle mechanics, and the spinor is quantized as "boson" (3.17). Then, the complex conjugation is regarded as the derivative (3.18) and the constraint (3.16) is considered as a condition on the LLL basis The previously derived LLL basis (3.12) indeed satisfies the condition. By inserting the expression (3.18) to the Hopf map, we find that x i are regarded as coordinates on fuzzy two-sphere which satisfies the algebra (3.13). Thus, also in the Lagrange formalism, we arrive at the fuzzy two-sphere algebra in LLL. The crucial role of the Hopf spinor is transparent in the Lagrange formalism: The Hopf spinor is first fuzzificated (3.17), and subsequently the coordinates on two-sphere are fuzzificated. This is the basic fuzzification mechanics of sphere in the context of the Hopf map.

Fuzzy two-sphere
The fuzzy two-sphere is a fuzzy manifold whose coordinates satisfy the SU (2) algebraic relation [1] [X i ,X j ] = iαǫ ijkXk .
The magnitude of fuzzy sphere is specified by the dimension of corresponding SU (2) irreducible representation. A convenient way to deal with the irreducible representation is to adopt the Schwinger boson formalism, in which the SU (2) operators are given bŷ Here,φ = (φ 1 ,φ 2 ) t stands for a Schwinger boson operator that satisfy with α, β = 1, 2. Square of the radius of a fuzzy two-sphere reads aŝ The commutative expression (3.3) and the fuzzy expression (3.19) merely differ by the "ground state energy". Thus, the radius of the fuzzy sphere is where I is the integer eigenvalue of the number operatorÎ ≡φ †φ =φ † 1φ 1 +φ † 2φ 2 . In the classical limit I → ∞, the eigenvalue reproduces the radius of commutative sphere The corresponding SU (2) irreducible representation is constructed as where m 1 + m 2 = I (m 1 , m 2 ≥ 0), and the degeneracy is given by d(I) = I + 1. Apparently, there is one-to-one correspondence between the LLL monopole harmonics (3.12) and the states on fuzzy sphere (3.20). Their "difference" is superficial, coming from the corresponding representations: Schwinger boson representation for fuzzy two-sphere, while the SU (2) coherent representation for LLL physics. This is the basic observation of equivalence between fuzzy geometry and LLL physics.

Graded Hopf map and fuzzy supersphere
Before proceeding to the 2nd Hopf map, in this section, we discuss how the relations between the Hopf map and fuzzy sphere are generalized with introducing the Grassmann numbers, mainly based on Hasebe and Kimura [39]. The Grassmann numbers, η a , are anticommuting numbers [31] In particular, η 2 a = 0 (no sum for a). With Grassmann numbers, the graded Hopf map was introduced by Landi et al. [32,40,41], where the number on the left hand side of the slash stands for the number of bosonic (Grassmann even) coordinates and the right hand side of the slash does for the number of fermionic (Grassmann odd) coordinates. For instance, S 2|2 signifies a supersphere with two bosonic and two fermionic coordinates. The bosonic part of the graded Hopf map (4.2) is equivalent to the 1st Hopf map.

Graded Hopf map and supermonopole
As the 1st Hopf map was realized by sandwiching Pauli matrices between two-component normalized spinors, the graded Hopf map is realized by doing U OSp(1|2) matrices between three-component (super)spinors. First, let us begin with the introduction of basic properties of U OSp(1|2) algebra [42,43,44]. The U OSp(1|2) algebra consists of three bosonic generators L i (i = 1, 2, 3) and two fermionic generators L α (α = θ 1 , θ 2 ) that satisfy where ǫ = iσ 2 . As realized in the first algebra of (4.3), the U OSp(1|2) algebra contains the SU (2) as its subalgebra. L i transforms as SU (2) vector and L α does as SU (2) spinor. The Casimir operator for U OSp(1|2) is constructed as and its eigenvalues are given by L(L + 1 2 ) with L = 0, 1/2, 1, 3/2, 2, . . . . The dimension of the corresponding irreducible representation is 4L + 1. The fundamental representation matrix is given by the following 3 × 3 matrices where σ i are the Pauli matrices, τ 1 = (1, 0) t , and τ 2 = (0, 1) t . With l i and l α , the graded Hopf map (4.2) is realized as where ϕ is a normalized three-component superspinor which we call the Hopf superspinor: Here, the first two components ϕ 1 and ϕ 2 are Grassmann even quantities, while the last component η is Grassmann odd quantity. The superadjoint ‡ is defined as 2 and ϕ represents coordinates on S 3|2 , subject to the normalization condition x i and θ α given by (4.4) are coordinates on supersphere S 2|2 , since the definition of supersphere is satisfied: Except for the south-pole of S 2|2 , the Hopf superspinor has an analytic form The corresponding connection is derived as

6)
A i and A α are the super gauge field of supermonopole. The field strength is evaluated by the formula The analytic form of the Hopf superspinor except for the north pole is given by and the corresponding gauge field is obtained as The field strength F ′ = dA ′ is same as (4.7), suggesting the two expressions of the Hopf superspinor are related by the transformation Here, g denotes U (1) gauge group element given by with χ given by tan and A (4.6) and A ′ (4.8) are expressed as Then, obviously the gauge fields are related as and then

U OSp(1|2) Landau model
Next, we consider Landau problem on a supersphere in supermonopole background [39]. The supermonopole gauge field is given by where I 2 (I denotes an integer) is a magnetic charge of the supermonopole. Generalizing the where Λ i and Λ α are the bosonic and fermionic components of covariant angular momentum: with D i = ∂ i + iA i and D α = ∂ α + iA α . Λ i and Λ α obey the following graded commutation relations where F i and F α are Just as in the case of the SO(3) Landau model, the covariant angular momentum is not a conserved quantity, in the sense [Λ i , H] = 0, [Λ α , H] = 0. Conserved angular momentum is constructed as It is straightforward to see L i and L α satisfy the U OSp(1|2) algebra (4.3). Since the covariant angular momentum and the field strength are orthogonal in the supersymmetric sense: Landau Hamiltonian (4.9) can be rewritten as With the Casimir index J = I 2 + n, the energy eigenvalue is derived as and the degeneracy in the nth Landau level is In particular in the LLL (n = 0), the degeneracy becomes The eigenstates of the U OSp(1|2) Hamiltonian are referred to the supermonopole harmonics, and the LLL eigenstates are constructed by taking symmetric products of the components of the Hopf superspinor (4.5): where m 1 + m 2 = n 1 + n 2 + 1 = I (m 1 , m 2 , n 1 , n 2 ≥ 0). The total number of ϕ B (m 1 ,m 2 ) LLL and ϕ F (n 1 ,n 2 ) LLL is (I + 1) + (I) = 2I + 1, which coincides with (4.10). In the LLL, the U OSp(1|2) angular momentum is reduced to and coordinates on supersphere are identified with the U OSp(1|2) operators which satisfy the algebra defining fuzzy supersphere: With Lagrange formalism, we reconsider the LLL physics. The present one-particle Lagrangian on a supersphere is given by In the LLL, the kinetic term is quenched, and the Lagrangian takes the form of with the constraint From the LLL Lagrangian (4.13), the canonical momentum of ϕ is derived as π = iIϕ * , and from the commutation relation between ϕ and π, the complex conjugation is quantized as After quantization, the normalization condition (4.14) is imposed on the LLL basis: One may confirm the LLL basis (4.11) satisfies the condition. By inserting (4.15) to the graded Hopf map (4.4), we obtain Apparently, they satisfy the algebra of fuzzy supersphere (4.12). Thus, in the case of the fuzzy two-sphere, the appearance of fuzzy supersphere in LLL is naturally understood in the context of the graded Hopf map.

Fuzzy supersphere
Fuzzy supersphere is constructed by taking symmetric representation of the U OSp(1|2) group [3,4,46,22]. We first introduce a superspinor extension of the Schwinger operator whereφ 1 andφ 2 are Schwinger boson operators, andη is a fermion operator: With such Schwinger superoperator, coordinates on fuzzy supersphere are constructed aŝ Square of the radius of fuzzy supersphere is given bŷ and then the radius is specified by the integer eigenvalue I of the number operatorÎ ≡φ †φ = ϕ † 1φ 1 +φ † 2φ 2 +η †η as The U OSp(1|2) symmetric irreducible representation is explicitly constructed as with m 1 + m 2 = n 1 + n 2 + 1 = I (m 1 , m 2 , n 1 , n 2 ≥ 0). Thus, the total number of the states constructing fuzzy supersphere is d = (I + 1) + (I) = 2I + 1. There is one-to-one correspondence between the supermonopole harmonics in LLL and the states on fuzzy supersphere. Especially, the Hopf superspinor corresponds to the superspin coherent state of Schwinger superoperator.

2nd Hopf map and fuzzy four-sphere
In this section, we discuss relations between the 2nd Hopf map and fuzzy four-sphere. Though the 2nd and 3rd Hopf maps were first introduced by Hopf [29], we follow the realization given by Zhang and Hu [47] in the following discussions.

2nd Hopf map and SU (2) monopole
Realization of the 2nd Hopf map is easily performed by replacing the imaginary unit with the imaginary quaternions, q 1 , q 2 , q 3 . The Pauli matrices (3.1) are promoted to the following quaternionic Pauli matrices 3) correspond to σ 2 , γ 4 to σ 1 , and γ 5 to σ 3 , respectively. With such quaternionic "Pauli matrices", the 2nd Hopf map is realized as where a = 1, 2, . . . , 5, and ψ is a two-component quaternionic (quaternion-valued) spinor satisfying the normalization condition ψ † ψ = 1. As in the 1st Hopf map, x a given by the map (5.2) automatically satisfies the condition x a x a = (ψ † ψ) 2 = 1. Like the 1st Hopf spinor, the quaternionic Hopf spinor has an analytic form, except for the south pole, as and, except for the north pole, as where i = 1, 2, 3 are summed over. These two expressions are related by the transformation where g is a quaternionic U (1) group element given by Here, χ i and its magnitude χ = χ 2 i are determined by χ i χ tan(χ) = 1 x 4 x i . It is possible to pursue the discussions with use of quaternions, but for later convenience, we utilize the Pauli matrix representation of the imaginary quaternions: The quaternionic U (1) group is usually denoted as U (1, H), and as obvious from the above identification, U (1, H) is isomorphic to SU (2). The quaternionic Pauli matrices (5.1) are now represented by the following SO(5) gamma matrices: Corresponding to the quaternionic Hopf spinor, we introduce a SO (5) four-component spinor (the 2nd Hopf spinor) subject to the constraint With ψ, the 2nd Hopf map (5.2) is rephrased as It is easy to check that x a satisfies the condition of S 4 : x a x a = (ψ † ψ) 2 = 1.
An analytic form of the 2nd Hopf spinor, except for the south pole, is given by where φ is the 1st Hopf spinor representing S 3 -fibre. The connection is derived as They are the SU (2) gauge field of Yang monopole [34]. Here, η + µνi signifies the 'tHooft etasymbol of instanton [48]: The corresponding field strength F = dA + iA ∧ A = 1 2 dx a ∧ dx b F ab : is evaluated as Another analytic form of the 2nd Hopf spinor, except for the north pole, is given by The corresponding connection is calculated as where η − µνi = ǫ µνi4 − δ µi δ ν4 + δ µ4 δ νi , and the field strength is As is well known, the 'tHooft eta-symbol satisfies the self (anti-self) dual relation The two expressions, ψ and ψ ′ , are related by the SU (2) transition function and the gauge fields (5.7) and (5.9) are concisely represented as Then, A and A ′ are related as and their field strengths are also This manifests the non-trivial topology of the SU (2) bundle on a four-sphere. In the Language of the homotopy theorem, the non-trivial topology of the SU (2) bundle is expressed by which is specified by the 2nd Chern number

SO(5) Landau model
We next explore the Landau problem in 5D space [47]. The Landau Hamiltonian is given by where D a = ∂ a + iA a , r = √ x a x a , and Λ ab are the SO(5) covariant angular momentum As in the previous 3D case, Λ ab do not satisfy a closed algebra, but satisfy where F ab are given by (5.8). The SO (5) conserved angular momentum is constructed as On a four-sphere, the Hamiltonian (5.10) is reduced to the SO(5) Landau Hamiltonian which is rewritten as where the orthogonality, Λ ab F ab = F ab Λ ab = 0, was used. Thus, the energy eigenvalue problem of the Hamiltonian is again boiled down to the problem of obtaining irreducible representation of the SO(5) Casimir. Since the SO (5)  which actually coincides with d n=0 of (5.12). Up to now, everything is parallel with the SO(3) Landau model, but emergence of fuzzy four-sphere in LLL is not transparent unlike the fuzzy two-sphere case.
With Lagrange formalism, we revisit LLL physics of the SO(5) Landau model. The present one-particle Lagrangian on a four-sphere is given by with a constraint x a x a = R 2 .
In the LLL, the interaction term survives to yield 16) and the relation (5.6) implies and also the constraint (5.15) is rewritten as Interestingly, in the LLL, the original SO(5) symmetry of the Lagrangian (5.14) is enhanced to the SU (4) symmetry: the rotational symmetry of the 2nd Hopf spinor. We treat the 2nd spinor as the fundamental variable and apply the quantization condition. After quantization, the complex conjugate spinor is regarded as the derivative ψ * = 1 I ∂ ∂ψ , and the normalization condition (5.18) is translated to the LLL condition The LLL states (5.13) indeed satisfy the condition (5.19). Here, we comment on the origin of the SU (4) symmetry and its relation to the 2nd Hopf map. In the LLL, the 2nd Hopf spinor plays a primary role, and the total manifold S 7 naturally appears in LLL. Projecting out the U (1) phase from S 7 , we obtain the structure of This suggests physical equivalence between the LLL of SO(5) Landau model and that of SU (4) Landau model on CP 3 . (Detail discussions on physical equivalence between two Lagrangians (5.16) and (5.17) are found in [51], and see Appendix B.2 also.) The appearance of CP 3 can also be understood as follows. As mentioned in Introduction, S 4 is not a Kähler manifold that accommodates symplectic structure. The "minimally extended" symplectic manifold of S 4 is CP 3 , which is given by the coset SU (4)/U (3), and then the SU (4) structure naturally appears. Such observation is completely consistent with the mathematical expression of the fuzzy foursphere, since where we used SO(6)/SO(5) ≃ S 5 ≃ U (3)/U (2) and SO(6) ≃ SU (4). By inserting the derivative expression of the complex 2nd Hopf spinor to the 2nd Hopf map (5.4), we find that the coordinate on S 4 is expressed by the following operator X a = αψ t γ a ∂ ∂ψ .
As we shall see in the next subsection, the SU (4) structure also appears in the enhanced algebra of X a .

Fuzzy four-sphere
The fuzzy four-sphere is constructed by taking a fully symmetric representation of the SO (5) spinor [2,9,14,15,52,53,54]. As in the fuzzy two-sphere case, the Schwinger boson formalism is useful to construct coordinates on fuzzy four-spherê It is important to notice, unlike the case of fuzzy two-sphere, the fuzzy coordinates do not satisfy a closed algebra by themselves but yield the SO(5) generators. With the SO(5) generatorsX ab , the fuzzy coordinates satisfy the following closed algebra, [X ab ,X cd ] = i α 2 (δ acXbd − δ adXbc + δ bcXad − δ bdXac ).
By identifyingX a6 = 1 2X a andX ab =X ab , we find the above algebra is concisely expressed by the SO(6) algebra, where A, B = 1, 2, . . . , 6. Thus, the algebra defining fuzzy four-sphere is SO(6) ≃ SU (4). We encountered the SU (4) structure again, and the fuzzy manifold naturally defined by SU (4) algebra is fuzzy CP 3 (see Appendix B.1). The enhanced SU (4) algebra with extra X ab coordinates accounts for the existence of extra fuzzy-dimensions [14,15]. Interestingly, CP 3 is locally expressed as the two-sphere fibration over the four-sphere: since CP 3 ≃ S 7 /S 1 ≈ S 4 ×S 3 /S 1 . The "extra dimension" of S 4 F can be understood as two-sphere fibration over the four-sphere.

3rd Hopf map and fuzzy manifolds
Here, we consider realization of the 3rd Hopf map

3rd Hopf map and SO(8) monopole
The 1st and 2nd Hopf maps were realized by sandwiching Pauli and quaternionic Pauli matrices by spinors. One may expect that such realization can be applied to the 3rd Hopf map. However, it is not so straightforward, since octonions cannot be represented by matrices due to their nonassociative property. To begin with, we construct Majorana representation of the SO(9) gamma matrix with the octonion structure constants (Table 1). With e 0 = 1, the octonion algebra (2.1) is expressed as where P, Q, R = 0, 1, . . . , 7. With use of f P QR , the SO(7) gamma matrices −iλ I (I = 1, 2, . . . , 7) are constructed as They are real antisymmetric matrices that satisfy With λ 0 ≡ 1 8 , λ 0 and λ I (I = 1, 2, . . . , 7) are regarded as the SO(8) "Weyl +" gamma matrices.
Utilizing λ 0 and λ I , the SO(9) gamma matrices Γ A are constructed as Again, they are real symmetric matrices that satisfy where A, B, C = 1, 2, . . . , 9. The octonion structure constants appear in the off-diagonal elements of Γ I . The SO(9) generators are constructed as or more explicitly where σ IJ are the SO(7) generators Since Γ A are real matrices, the corresponding SO(9) generators (6.1) are purely imaginary matrices; Σ * AB = −Σ AB . Thus, the present representation is indeed the Majorana representation, in which the charge conjugation matrix is given by unit matrix, and the SO(9) Majorana spinor is simply represented by (16-component) real spinor. The 3rd Hopf spinor is introduced as SO(9) Majorana spinor subject to the normalization condition Ψ t Ψ = 1, (6.2) and the 3rd Hopf spinor is regarded as the coordinate of S 15 . By sandwiching Γ A between the 3rd Hopf spinors, we now realize the 3rd Hopf map as 3) x A in (6.3) are coordinates on S 8 , since A=1,2,...,9 x A x A = (Ψ t Ψ) 2 = 1.
An analytic form of Ψ, except for the south pole, is represented as where Φ is a SO(7) real 8-component spinor subject to the constraint representing the S 7 -fibre. Then, Φ has the same degrees of freedom of the 2nd Hopf spinor ψ. We may assign Φ = (Re ψ, Im ψ) t , and the 3rd Hopf spinor is expressed as Naively anticipated connection A = −iΨ t dΨ vanishes due to the Majorana property of Ψ, however, defining the connection of S 7 -fibre is evaluated as where A A = (A M , A 9 ) (M = 1, 2, · · · , 8) are (σ IJ and σ I8 are pure imaginary antisymmetric matrices.) The field strength which represent the SO(8) monopole gauge field [35]. Similarly, except for the north pole, the 3rd Hopf spinor has an analytic form 6) and the connection is The corresponding field strength is derived as Here, the SO(8) generators σ M N (σ M N ) satisfy a generalized self (anti-self) dual relation: The two expressions (6.4) and (6.6) are related by where g signifies an SO(8) group element which yields Then, the gauge fields, (6.5) and (6.7), are concisely represented as and are related by Similarly, their field strengths are The non-trivial topological structure of the SO(8) monopole bundle is guaranteed by the homotopy theorem which is specified by the Euler number

Fuzzy CP 7 and fuzzy S 8
In the realization of the 3rd Hopf map, we utilized the real (Majorana) spinor. The fuzzification procedure in the previous sections can not be straightforwardly applied to the present case, since we do not have the complex conjugate spinor to be identified with derivative. However, with 16 real components of the 3rd Hopf spinor, we can construct an 8-component normalized complex spinor to be identified with coordinates on CP 7 . Then, in the present case, there exist two different types of fuzzy manifolds, depending on the choice of the irreducible representation of SO (9). The first one is the above mentioned fuzzy CP 7 ≃ S 15 /S 1 specified by the vector representation of SO (9), while the other is the fuzzy eight-sphere S 8 F ≃ SO(9)/U (4) specified by the spinor representation. Both two fuzzy manifolds are reasonable generalizations of the previous low dimensional fuzzy spheres and fuzzy complex projective spaces (see Table 2).

Beyond Hopf maps: even higher dimensional generalization
We have reviewed the construction of fuzzy manifolds based on the Hopf maps. Since the Hopf maps are only three kinds, the corresponding fuzzy manifolds are also limited. However, from the results of the Hopf maps, one may naturally infer two possible generalizations of the fuzzy manifolds, one of which is a series of fuzzy spheres: and the other is that of fuzzy complex projective spaces: In this section, we first introduce mathematics of fuzzy spheres S 2k F in arbitrary even dimensions, and next provide their physical interpretations, mainly based on Hasebe and Kimura [56] (see also Fabinger [57] and Meng [58]). Fuzzy CP k manifolds and their corresponding Landau models are discussed in Appendix B.

Clifford algebra: another generalization of complex numbers
It may be worthwhile to begin with the story of generalization of the complex numbers to Clifford algebra. Generalization from fuzzy two-sphere to its higher dimensional cousins are quite analogous to the generalization of complex numbers. As discussed in Section 2, Cayley-Dickson construction provides one systematic way to construct new numbers by duplicating the original numbers, but this construction method has a fatal problem: If we utilize the method, the resulting algebra loses a nice property of numbers one by one. For instance, in the construction of quaternions, the commutativity of the complex numbers was lost. In the construction of octonions, even the associativity of quaternions was abandoned. Consequently, generalization of complex numbers ends up with the octonions, and there are only three division algebras (except for real numbers). Clifford found another generalization of complex numbers, based on Hamilton's quaternions and Grassmann algebras, known as Clifford algebra. The Clifford algebra, Cliff n , consists of 2 n basis elements {1, e a , e a e b , e a e b e b , . . . , e 1 e 2 e 3 · · · e n } (a = b, a = b = c, . . . ), and its algebraic structure is determined by the relation 4 {e a , e b } = 2δ ab . (7.1) The division algebras except for the octonions are realized as special cases of Clifford algebra, i.e. R = Cliff 0 , C = Cliff 1 , and H = Cliff 2 . The Clifford algebra can also be regarded as the "quantized" Grassmann algebra (compare (7.1) with {η a , η b } = 0 (4.1)). Though the division property is in general lost, the Clifford algebras always maintain the nice associative property, and are represented by gamma matrices that satisfy Importantly, there are analogous geometrical properties between the division algebras and the Clifford algebras: In the division algebras, new numbers are constructed by the Cayley-Dickson construction. Similarly, higher dimensional gamma matrices are constructed by the lower dimensional gamma matrices. Specifically, SO(2k − 1) gamma matrices γ (2k−1) i (i = 1, 2, . . . , 2k − 1) are provided, SO(2k + 1) gamma matrices γ (2k+1) a (a = 1, 2, . . . , 2k + 1) can be constructed as Thus, in any higher dimensional gamma matrices are constructed by repeating the above procedure from the SO(3) gamma matrices, γ As the Hopf maps exhibit the hierarchical structure stemming from the Cayley-Dickson construction, the geometry of higher dimensional fuzzy spheres reflects the iterative construction structure of the gamma matrices as we shall see below.
Similarly, in the symmetric representation, the eigenvalue of the SO(2k + 1) Casimir is 2k+1 a<b X 2 ab = α 2 16 kI(I + 2k), where X ab = −i 1 4 [X a , X b ] are the SO(2k + 1) generators. (Detail calculation techniques for symmetric representation can be found in [21].) Thus, the index I of the symmetric representation determines the magnitude of the radius of fuzzy sphere. As the dimension of the symmetric representation becomes "larger", the corresponding fuzzy sphere becomes "larger".
As in the case of fuzzy-four sphere, the fuzzy coordinates X a do not satisfy a closed algebra by themselves, but X a and X ab satisfy the following enlarged algebra With identification X a,2k+2 = −X 2k+2,a = 1 2 X a and X ab = X ab , the above algebra is found to be equivalent to the SO(2k + 2) algebra where A, B, C, D = 1, 2, . . . , 2k + 2. Thus, the algebra of fuzzy sphere S 2k+1 F is SO(2k + 2) [14], and S 2k F is expressed as where we used the relation SO(2k + 2)/SO(2k + 1) ≃ S 2k+1 ≃ U (k + 1)/U (k). The above coset representation can also be expressed as Fuzzy 2k-sphere is constructed not only by the operators X a but also the "extra" operators X ab , and the very existence of X ab brings the extra fuzzy-space S 2k−2 F over S 2k . From (7.3), one may find the fuzzy sphere is expressed by the hierarchical fibrations of lower dimensional spheres which reflects the iterative construction of gamma matrices from lower dimensions.

Hopf spinor matrix and SO(2k) monopole
To obtain monopole bundles in generic even dimensions, we "extend" the Hopf maps. First, we define "Hopf spinor matrix" of the form where 1 stands for 2 k × 2 k unit matrix, γ i (i = 1, 2, . . . , 2k − 1) are SO(2k − 1) gamma matrices, and x a (a = 1, 2, . . . , 2k + 1) are coordinate on 2k-sphere satisfying 2k+1 a=1 x a x a = 1. The Hopf spinor matrix is a 2 k+1 × 2 k matrix that satisfies where γ a are SO(2k + 1) gamma matrices (7.2). The corresponding monopole gauge field is evaluated as Σ + µν (µ, ν = 1, 2, . . . , 2k) are SO(2k) generators given by These are the SO(2k) non-Abelian monopole gauge field strength [59,60,61]. Another representation of the Hopf spinor matrix is introduced as and the corresponding gauge field, A ′ = −iΨ † dΨ ′ , reads as The gauge field strength F ′ ab is The SO(2k) generators, Σ + µν and Σ − µν , satisfy the generalized self and anti-self dual relations, respectively: The two Hopf spinor matrices (7.5) and (7.7) are related by the transformation where g is an SO(2k) group element With g, the gauge fields (7.6) and (7.8) are concisely represented as Then, the two gauge fields are related as and the field strengths are The homotopy theorem guarantees the non-trivial topology of the SO(2k) bundle fibration over S 2k : which is specified by the Euler number tr F k .

SO(2k + 1) Landau model
In generic d-dimensional space, Landau Hamiltonian is given by [56] where D a = ∂ a + iA a (A a is the SO(2k) monopole gauge field), and The covariant momentum Λ ab does not satisfy the SO(2k + 1) algebra but satisfies The conserved SO(2k + 1) angular momentum is constructed as which satisfies the SO(2k + 1) algebra and generates the SO(2k + 1) transformations, for instance On 2k-sphere, the Landau Hamiltonian (7.9) is reduced to SO(2k + 1) Landau Hamiltonian With the orthogonality Λ ab F ab = F ab Λ ab , the Hamiltonian is rewritten as Table 3. The fuzzy 2k-sphere is physically realized in the LLL of the SO(2k + 1) Landau model. Previously encountered monopoles are understood as the special cases of SO(2k) monopoles, for instance U (1) ≃ SO(2), SU (2)(⊗SU (2)) ≃ SO (4). Note also SU (4) ≃ SO (6). The holonomy groups of spheres are equal to the corresponding monopole gauge groups.
In the thermodynamic limit: R, I → ∞ with I/R 2 fixed, the Landau levels are reduced to The LLL energy, E LLL = I 4M R 2 k, depends on the spacial dimension 2k. In the thermodynamic limit, S 2k is reduced to 2k-dimensional plane, and the zero-point energy B 2M = I 4M R 2 coming from each 2-dimensional plane amounts to E LLL .
As discussed above, coordinates on fuzzy 2k-sphere are given by the SO(2k + 1) gamma matrices in the symmetric spinor representation. Similarly, the LLL basis of the SO(2k + 1) Landau model realizes such a symmetric spinor representation. Thus, the LLL of SO(2k + 1) Landau model provides a physical set-up for 2k-dimensional fuzzy sphere (see Table 3).

Dimensional hierarchy
Here, we give a physical interpretation of the hierarchical geometry of higher dimensional fuzzy spheres (7.4). From the formula of the irreducible representation of SO(2k + 1) [49], the degeneracy in nth LL is given by (I + 2l)!l! (I + l)!(2l)! −→ I · I 2 · I 3 · · · I k−1 · I k = I 1 2 k(k+1) . (7.10) The last expression implies a nice intuitive picture of the hierarchical geometry of fuzzy spheres. Each of the SO(2k) monopole fluxes on S 2k occupies an area ℓ 2k B = (1/B) k = (2R 2 /I) k with the magnetic field B = (2πI)/(4πR 2 ), and the number of fluxes on S 2k is ∼ R 2k /ℓ 2k B ∼ I k . Besides, the monopole flux itself is represented by the generators of the non-Abelian SO(2k) group, and is regarded as a (2k − 2)-dimensional fuzzy sphere. Again, the (2k − 2)-dimensional fuzzy sphere is interpreted as a (2k − 2)-dimensional sphere in SO(2k − 2) monopole background, and then, on S 2k−2 , there are SO(2k − 2) fluxes each of which occupies the area ℓ 2k−2 Similarly, the SO(2k − 2) non-Abelian flux is given by the generators of the SO(2k − 2) group, and regarded as a fuzzy S 2k−4 . Thus, on S 2k , we have I k S 2k−2 , on which I k−1 S 2k−4 , on which I k−2 S 2k−6 , . . . . By this iteration, we obtain the formula (7.10). Inversely, we can view this mechanism from low dimensions: Lower dimensional spheres gather spherically to form a higher dimensional sphere, and such iterative process amounts to construct a higher dimensional fuzzy sphere. The dimensional hierarchy is depicted in Fig. 1.

Summary and discussions
We reviewed the close relations between monopoles, LLL, and fuzzy spheres. The fuzzy 2k-sphere is physically realized in the LLL of the SO(2k + 1) Landau model. In the generalization of fuzzy spheres, three classical algebras; division algebra, Grassmann algebra, and Clifford algebra, played crucial roles. They brought the basic structures of monopole bundles and fuzzy spheres ( Table 4). In particular, the hierarchical geometry of fuzzy spheres is a direct manifestation of their gamma matrix construction: As higher dimensional gamma matrices are constructed by lower dimensional gamma matrices, higher dimensional fuzzy sphere is constructed by lower dimensional spheres. Such dimensional hierarchy can be physically understood in the context of higher dimensional Landau model. It should be mentioned that, in [62], such interpretation was successfully applied to dual description of higher dimensional fuzzy spheres in string theory. We also emphasize the importance of the Hopf map in realizing fuzzy sphere. In the fuzzification of spheres, the total manifolds (Hopf spinor spaces) of the Hopf maps played a fundamental role: The total manifolds were firstly fuzzificated and as "a consequence" the basemanifolds (spheres) are fuzzificated. Interestingly, this fuzzification mechanism coincides with the philosophy of twistor theory (see [63] and references therein).
Finally, we comment on applications to many-body physics. The correspondence between fuzzy geometry and LLL physics argued in the paper was at one-particle level observation. Interestingly, there even exists correspondence at many-body level: Many-body groundstate wavefunction of quantum Hall effect (Laughlin wavefunction) is mathematically analogous to an antiferromagnetic ground state (AKLT state) [88,89]. Accompanied with the higher dimensional and supersymmetric generalizations of the quantum Hall effect 5 , their formalism has begun to be applied to the construction of antiferromagnetic quantum spin states with higher symmetries [89,94].
A 0th Hopf map and SO(2) "Landau model" Real numbers are the "0th" member of the division algebra. For completeness, we introduce the 0th Hopf map and the corresponding Landau model. The 0th Hopf map is realized by identifying "opposite points" on a circle, and is simply visualized as the geometry of Möbius strip whose basemanifold is S 1 and transition function is Z 2 . Unlike the other Hopf maps, the dimension of the basemanifold is odd and the structure group is a discrete group.

A.1 Realization of the 0th Hopf map
With the coordinate on a circle, w = (w 1 , w 2 ) t (a real two-component spinor subject to w t w = w 2 1 + w 2 2 = 1), the 0th Hopf map is realized as Here, σ 1 and σ 3 are the real Pauli matrices. x 1 and x 2 are invariant under the transformation (w 1 , w 2 ) → −(w 1 , w 2 ), and hence (x 1 , x 2 ) is Z 2 projection of (w 1 , w 2 ). From (A.1), one may find x 2 1 + x 2 2 = (w t w) 2 = 1. Inverting the map, w can be expressed as where x 1 and x 2 are parameterized as x 1 = cos θ and y = sin θ. The corresponding connection vanishes: A = −iw t dw = 0. Meanwhile, using a U (1) element ω = w 1 + iw 2 = e i θ 2 , the 0th Hopf map is restated as and the connection, A = −iω * dω = iw t σ 2 dw = dxA x + dyA y , is given by It is straightforward to see the field strength, B = ∂ x A y − ∂ y A x , represents a solenoid-like magnetic field at the origin: A.2 SO(2) "Landau model" Next, we introduce "Landau model" on a circle in the presence of magnetic fluxes [95] B = Iπδ(x, y).
Here, I is the number of magnetic fluxes, and takes an integer value. The corresponding gauge field is given by where r 2 = x 2 + y 2 . Since the magnetic fluxes are at the origin, the classical motion of a charged particle on a circle is not affected by the existence of the magnetic fluxes. In quantum mechanics, however, the result is different. With the covariant derivatives D x = ∂ x +iA x and D y = ∂ y +iA y , Landau Hamiltonian on 2D plane is given by where Λ is the covariant angular momentum On a circle with radius R, the Hamiltonian is reduced to SO(2) "Landau Hamiltonian" This is a one-dimensional quantum mechanical Hamiltonian easily solved. In higher dimensions, the covariant angular momentum is not a conserved quantity, but the present case it is, as simply verified [Λ, H] = 0. Since the magnetic field angular momentum does not exist on the circle, the particle angular momentum itself is conserved. Imposing the periodic boundary condition u(θ = 2π) = u(θ = 0), the eigenvalue problem is classified to two cases: even I and odd I. For even I, the energy eigenvalue are given by where n = 0, 1, 2, 3, . . . . The eigenstates are Except for the lowest energy level n = 0, every excited energy level is two-fold degenerate. The physical origin of the double degeneracy comes from the left and the right movers on the circle. The energy levels are identical to those of free particle on a circle. Thus, for even I, the magnetic flux does not affect the energy spectrum of the system. Meanwhile, for odd I, the energy eigenvalues become and the corresponding eigenstates are All the Landau levels are doubly degenerate even for n = 0. The energy spectra (A.2) are different from those of the free particle on a circle, reflecting the particular role of the gauge field in quantum mechanics. It is also noted that the number of the magnetic fluxes I has nothing to do with the degeneracy in Landau levels unlike the other Landau models (see equation (3.11) for instance).

B.2 Relations to spherical Landau models
(B.7) The SU (4) Landau level energy is different from that of the SO(5) Landau model (5.11), only by the total energy shift 1 4M R 2 I. This discrepancy is understood by a simple geometrical argument [96]: Since CP 3 ≈ S 4 × S 2 , the CP 3 is regarded as S 2 -fibration over S 4 , and the zero-point energy from the extra S 2 -space, B 2M = 1 4M R 2 I, gives rise to the difference. Generally, the degeneracy of Landau levels of the SU (4) Landau model (B.7) is different from that of the SO(5) Landau model (5.12), but in the LLL, both quantities coincide to yield d LLL = 1 6 (I + 1)(I + 2)(I + 3). This manifests the equivalence between the SO(5) and SU (4) Landau models in LLL. By comparing the 2nd Hopf spinor (5.5) with the SU (4) coherent state we find the correspondence and also for the LLL basis elements, (5.13) and (B.5) for k = 3.

Note added
After completion of this work, the author learned the works [101,102]. In the appendices of the papers, the fuzzy spheres (S 2 F , S 3 F , S 4 F and S 8 F ) are constructed in the context of embedding in Moyal spaces, with emphasis on relations to the Hopf maps. Such constructions are completely consistent with the descriptions in the present paper. The author is grateful to Mohammad M. Sheikh-Jabbari for the information.