$SO(4)$ Landau Models and Matrix Geometry

We develop an in-depth analysis of the $SO(4)$ Landau models on $S^3$ in the $SU(2)$ monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the $SU(2)$ monopole are introduced to provide a concrete coordinate representation of $SO(4)$ operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the $SO(4)$ covariance of the eigenfunctions. With the spin connection of $S^3$, we construct an $SO(4)$ invariant Weyl-Landau operator and analyze its eigenvalue problem with explicit form of the eigenstates. The obtained results include the known formulae of the free Weyl operator eigenstates in the free field limit. Other eigenvalue problems of variant relativistic Landau models, such as massive Dirac-Landau and supersymmetric Landau models, are investigated too. With the developed $SO(4)$ technologies, we derive the three-dimensional matrix geometry in the Landau models. By applying the level projection method to the Landau models, we identify the matrix elements of the $S^3$ coordinates as the fuzzy three-sphere. For the non-relativistic model, it is shown that the fuzzy three-sphere geometry emerges in each of the Landau levels and only in the degenerate lowest energy sub-bands. We also point out that Dirac-Landau operator accommodates two fuzzy three-spheres in each Landau level and the mass term induces interaction between them.

: Landau models on low dimensional spheres and associated monopoles

Introduction
The Landau models are physical models that manifest the non-commutative geometry in a most obvious way. It is well known [1,2] that the fuzzy two-sphere geometry [3,4,5] is realized in the SO(3) Landau model [6,7] that provides a set-up of the 2D quantum Hall effect [8]. Similarly the set-up of the SO(5) Landau model [9,10] is used for the construction of the 4D quantum Hall effect [11] whose underlying geometry is the fuzzy four-sphere [12,13,14]. The correspondence was further explored on S 2k [15,16] and the SO(2k + 1) Landau model was shown to realize the geometry of fuzzy 2k-sphere [17,18]. Besides spheres, there are many manifolds that incorporate non-commutative geometry, and Landau models have been constructed on various manifolds, i.e. CP n , supermanifolds, hyperboloids, etc. [19,20,21,22,23,24,25,26,27,28,29]. The works have brought deeper understanding of the Landau physics and the associated fuzzy geometry as well. The magnetic field is the vital for the realization of the non-commutative geometry in the Landau model, and for spheres, the magnetic field is brought by the monopole at the center of the spheres. Since the monopole charge mathematically corresponds to the Chern number that is defined on even dimensional manifold, all of the manifolds used in the above works are even dimensional. Also in the viewpoint of the non-commutative geometry, adoption of the even dimensional manifolds is quite reasonable, because the geometric quantization is performed by replacing the Poisson bracket with the commutator, and even dimensional symplectic manifold generally accommodates non-commutative structure by such a quantization procedure.
From above point of view, the Landau model on S 3 which Nair and Randjbar-Daemi first proposed [30] 1 was rather exotic, though the model nicely fits in between the SO(3) and SO(5) Landau models (see Table 1). In the model, the quantization of the SU (2) monopole charge was assumed, but there is no reason to justify the assumption: The Chern number is not defined in odd dimensions, and so the monopole charge quantization is not guaranteed. Also for odd dimensional manifolds, the symplectic structure cannot be embedded and then the geometric quantization procedure mentioned above is useless. Even if we adopt the quantum Nambu three-bracket instead of the usual commutator [31], we encounter other problems, such as the violation of the Jacobi identity [32]. It thus seemed to exist fundamental difficulties for Landau models and non-commutative geometry in odd dimensional space. In Refs. [33,34,35] however, it was pointed out that the usage of the odd dimensional bracket can be circumvented by treating the odd dimensional bracket as a sub-bracket of the one-dimension higher even bracket, which indicates that the odd dimensional non-commutative space is not apparently consistent by itself but consistent as a subspace of one-dimension higher even dimensional space. Inspired by this observation, we proposed a resolution for the difficulty of odd dimensional Landau model. We showed that the SO(4) Landau model is naturally embedded in the SO(5) Landau model, and the monopole charge quantization is accounted for by that on one-dimension higher space S 4 [36]. We also demonstrated that similar relation holds for arbitrary odd and even dimensional Landau models [37] and the dimensional relation has its origin in differential topology; the dimensional ladder of anomaly or the spectral flow of Atiyah-Patodi-Singer. Though the foundation of the odd dimensional Landau models was thus established, there are merely a handful of works about them up to the present [30,36,37,38]. (See also [39,40,41] for odd dimensional topological insulator Landau models based on the Dirac oscillator.) In this paper, we revisit the SO(4) Landau model -the minimal model of the odd dimensional Landau models. Through a full investigation of the SO(4) Landau model, we learn properties specific to the odd dimensional Landau model and associated non-commutative geometry whose analyses are technically difficult in higher dimensions. The main achievements are as follows: (i) We introduce the Schwinger gauge and the Dirac gauge for S 3 and solve the Landau problem with explicit form of the wavefunction and the operators. The gauge fixing enables us to demonstrate important algebraic relations, such as the SO(4) invariance of the Dirac-Landau operator and covariance of the SO(4) Landau level eigenstates. (ii) We analyze relativistic Landau operators on S 3 with spin connection. 2 Besides the eigenvalues, we derive a concrete coordinate representation of the eigenstates. It is shown that the obtained results indeed include the known formulae of the (free) relativistic operator [45,46,47] in the free background limit. (iii) The matrix elements of the arbitrary Landau levels of the SO(4) Landau models are derived explicitly. We demonstrate that the obtained matrix geometry is identical to that of the fuzzy three-sphere. This is the first derivation of the odd dimensional matrix geometry in the context of the Landau model. Especially, we point out that the mass parameter of the Dirac-Landau model induces interaction between two fuzzy three-spheres realized in each of the relativistic Landau levels.
This paper is organized as follows. In Sec.2, we introduce the Schwinger and Dirac gauges for geometric quantities of three-sphere. Sec.3 discusses the non-relativistic Landau model in the Dirac gauge. We analyze the eigenvalue problem of the spinor Landau model with synthesized connection in Sec.4. Subsequently, the eigenvalue problem of relativistic Landau models is solved for Weyl-type, Dirac-type and supersymmetrictype in Sec. 5. The matrix geometries of the Landau models are identified as fuzzy three-sphere in Sec. 6. Sec.7 is devoted to summary and discussions.
The matrix form of the spin connection is then constructed as where σ ab (a, b = 1, 2, 3) are the SO(3) generators in the spinor representation: γ a (a = 1, 2, 3) denote the SO(3) gamma matrices, which throughout the paper we will take 3 Since the present dreibein (6) is not the Maurer-Cartan one-form, they do not satisfy ω ab = ǫ abc ec (Appendix B of [1]). Non-zero components of the Riemann curvature 2 form, R a b = dω a b + ω a c ω c b , are derived as For spheres, a special relation, R ab = ea ∧ e b , holds in arbitrary dimensions (p.378 in [48]). Reading off the Riemann curvature R a bcd from R a b = 1 2 R a bcd ec ∧ e d , we obtain R 1212 = R 2323 = R 3131 = 1 (other non-zero components are determined by the symmetry of R abcd ) to construct the scalar curvature R = R a bab = 6.
(14) is now represented as ω S = 1 2 − cos θ dφ i cos χ dθ + cos χ sin θ dφ −i cos χ dθ + cos χ sin θ dφ cos θ dφ . (17) Notice that the holonomy of (2), and the spin connection (17) is formally equivalent to the SU (2) monopole gauge field with minimal charge (see Sec.3). In the Dirac gauge (7), the spin connection is given by (17) and (18) are related by the SU (2) gauge transformation: where g(θ, φ) is the SU (2) group element corresponding to the SO(3) element (9), 4 For the local polar coordinates on S 3 , the gauge field takes a simple form in the Schwinger gauge (17), while in the Dirac gauge the representation is rather clumsy (18). On the other hand, in the target space Cartesian coordinates, the Schwinger gauge representation (17) becomes lengthy while the Dirac gauge representation (18) is much concise Thus, the Schwinger gauge is an appropriate gauge in the usage of the local polar coordinates on S 3 , and the Dirac gauge in the target space Cartesian coordinates. The spin connection (24) has the singularity both at the north pole x 4 = 1 and the south pole x 4 = −1, and hence the name, the Schwinger gauge [50,1]. Meanwhile, the singularity of (25) is only at the south pole x 4 = −1, and the name the Dirac gauge. 4 In other words, or

Non-relativistic Landau Model
In this section, we perform a through investigation of the eigenvalue problem of the SO(4) Landau Hamiltonian. The obtained results are utilized throughout the paper, and we provide a detail explanation for readers not to stumble against any logical gap or technical difficulty. We first present an expanded discussions of [30,36] about the mathematical background of the SO(4) Landau model (Sec.3.1), the SO(4) operators and their algebraic relations (Sec.3.2) and the SO(4) Landau problem (Sec.3.3). Next in Sec.3.4, we fix the gauge and provide new results about the basic properties of the SO(4) monopole harmonics (Sec.3.4.2) and the SO(4) covariance (Sec. 3.4.3) in which we give a verification of the SO(4) monopole harmonics to be the eigenstates of the SO(4) Landau Hamiltonian. In Sec.3.5, we check that the derived SO(4) monopole harmonics indeed reduce to the known SO(4) spherical harmonics in the free background limit.
For notational brevity, we adopt the following abbreviation of the angular coordinates in (1): and The sign flip of χ represents the parity transformation on S 3 : which interchanges the left-handed and right-handed coordinate systems, and so we call it the LR transformation.

The chiral Hopf map and the SU(2) monopole gauge field
The underlying geometry of the SO(4) Landau model is the chiral Hopf map [36], Here, the projected space S 3 denotes the base-manifold, and S 3 D ≃ SU (2) corresponds to the SU (2) fiber, and the map gives a set-up of the SO(4) Landau model on S 3 with SU (2) monopole at the center. We represent the coordinates on S L and S R as two two-component chiral Hopf spinors, ψ L = (ψ L1 ψ L2 ) and (29) can be expressed as where q µ andq µ are the quaternions and conjugate-quaternions, From (31), we have x µ are invariant under the simultaneous SU (2) transformation of ψ L and ψ R , ψ L/R → e αiqi ψ L/R , and we denote such diagonal SU (2) rotation as SU (2) D . The chiral Hopf spinors, ψ L and ψ R , are represented as where φ = (φ 1 φ 2 ) is a normalized two-component spinor φφ † = 1/2 representing the S 3 D -fibre. Ψ (1/2) L and Ψ (1/2) R are given by each of which is an SU (2) group element and their squares yield Ψ , we can derive the SU (2) monopole connection as Note that the gauge connection (36) is formally identical to the spin connection (25). With the angular coordinates, Ψ L and Ψ R can be expressed as 5 wherex signifies a position on the (S 2 -)equator of S 3 : x = (sin θ cos φ, sin θ sin φ, cos θ).
Promoting the Pauli matrices in (38) to SU (2) arbitrary matrices with spin magnitude I/2, we introduce and Ψ Ψ L and Ψ R are interchanged by the LR transformation (28). They satisfy In a same manner to (36), we obtain the SU (2) monopole gauge field in the Dirac gauge or A The winding number associated with the SU (2) gauge field (43) is 5 Indeed, where and (45) is evaluated as It should be mentioned that the chiral Hopf map and the present SU (2) monopole are naturally understood by embedding S 3 in one-dimension higher S 4 [36,37].

SO(4) operators
The SU (2) magnetic field is perpendicular to S 3 surface, and so the present system respects the SO(4) rotational symmetry. We construct total angular momentum operators that generate the simultaneous SO(4) rotations of the base-manifold and the gauge space. We clarify analogies and differences to the U (1) monopole system on S 2 [1].

SO(4) angular momentum operators
For A µ (µ = 1, 2, 3, 4) (44), let us introduce the covariant derivative and the field strength is given by with With the dreibein (7), (50) is concisely represented as The SO(4) covariant angular momentum is introduced as and the conserved SO(4) angular momentum operator consists of particle angular momentum and the field angular momentum of the monopole: In the Dirac gauge, L µν (53) are expressed as where l µν denote the free SO(4) angular momentum operators With the explicit coordinate representations, it is straightforward to check that T µν = L µν , Λ µν and F µν transform as two-rank tensors under the SO(4) transformations generated by L µν : From the orthogonality between Λ µν and F µν the SO(4) Casimir is given by which can be rewritten as with l the free SO(3) angular momentum operator: The first term is the SO(4) free angular momentum Casimir, and the second term is formally equivalent to the spin-orbit coupling in three-dimension. Meanwhile, the SO(3) Casimir of the SO(3) Landau model on S 2 is represented as A i = −I 1 2(1+x3) ǫ ij3 x j is the U (1) monopole gauge field andF i = ǫ ijk ∂ jÂk = I 2 x i . Comparison between (59) and (61) shows the the last term of (59), − 1 (1+x4) 2 (x · S (I/2) ) 2 , is specific to the SO(4) Landau model with the non-Abelian gauge field. 6 3.2.2 SU (2) L ⊗ SU (2) R group generators Since SO(4) ≃ SU (2) L ⊗SU (2) R , we can construct su(2) L ⊕su(2) R generators from the so(4) generators: which satisfy Here, η i µν andη i µν are the 't Hooft symbols: The two independent su(2) algebras (64) can be verified by the so(4) algebra and properties of the 't Hooft symbols (65). 7 The SO(4) Casimir is also given by a simple sum of the two SU (2) Casimirs: 6 When I = 1, this term is reduced to − 1 The following properties will be useful: whose eigenvalues are readily obtained as µ<ν L µν 2 = 2l L (l L + 1) + 2l R (l R + 1).
Here, l L and l R denote the SU (2) L and SU (2) R Casimir indices, and their sum is bounded below by the monopole charge (n = 0, 1, 2, · · · ) (70) n comes from the particle angular momentum, while I/2 comes from the field angular momentum of the monopole (recall (53)). In the case I = 0, the SO(4) irreducible representation is reduced to the SO(4) spherical harmonics with the SO(4) indices (l L , l R ) = (n/2, n/2) (Appendix B.1), and so each of l L and l R is bounded below by n/2: Consequently, for given n, (l L , l R ) can take the following (I + 1) distinct values Introducing the chirality parameter s [36] s ≡ l L − l R = I 2 , we specify (l L , l R ) as Essentially, the Landau level n corresponds to the sum of the two SU (2) indices, while the chirality parameter s their difference. The SO(4) Casimir eigenvalue (68) is now represented as 4 µ<ν=1 L µν 2 = (n + I 2 )(n +

SU (2) diag generators and "boost" generators
In the Dirac gauge, (63) are explicitly represented as L i andL i are interchanged by the LR transformation (28). For the comparison to the SO(3) Landau model [1], it is useful to introduce the SU (2) diagonal group generators and "boost" generators. From the two sets of SU (2) operators, we define the SU (2) diagonal group generators, where L D i do not depend on x 4 and are formally identical to the angular momentum operators of a free particle with higher spin I/2 in three dimension 8 that obviously satisfy the su(2) algebra. In the SO(3) Landau model on S 2 [1], the SO(2) operatorL z (62) consists of the free angular momentum and the U (1) gauge generator,L z = −i ∂ ∂φ − I 2 , and then (77) may be regarded as its SU (2) generalization. From analogy to the Lorentz group, we refer to the remaining SO(4) operators as the "boost" operators: It is easy to see that L diag and K satisfy [30] [L diag K i thus transform as the SU (2) diag vector. Though K 2 (and L diag 2 ) is not invariant under general SO(4) While L diag 2 represents SU (2) Casimir, the square of the boost operators yields [30] In the SO(3) Landau model [1], the boost generators K i correspond toL x andL y (62), and the sum of their squares givesL The SO(3) Landau Hamiltonian can be obtained from (84) by replacing the eigenvalues ofL z with the U (1) gauge generators I/2L We heuristically apply this procedure to the present case and replace L D in (83) with the gauge group generator S (I/2) to obtain which may give SO(4) Landau Hamiltonian. We shall confirm it in Sec.3.3.

SO(4) Landau levels and subbands
From the Landau Hamiltonian on R 4 8 Recall that in the beginning S (I/2) i were the SU (2) gauge group generators and the particle was a spinless particle, but here we reinterpret S (I/2) i as the intrinsic spin of particle. This interpretation is similar to that of Wilczek [51].
the SO(4) Landau Hamiltonian on S 3 is obtained as which is invariant under the SO(4) global rotations generated by L µν . Using the relation (58), we can express (88) as where the second term on the right-hand side is equal to the SO(3) Casimir and then Notice that the energy eigenvalues depend both on n and s. While n denotes the Landau level index, the chirality parameter s corresponds to subband of each Landau level. 9 Notice that the energy eigenvalue (92) is invariant under the sign flip of the chirality parameter: which is a direct consequence of the LR symmetry, since the SU (2) L and SU (2) R quantum numbers (74) are interchanged by the LR transformation. The dimension of the SO(4) irreducible representation specified by n and s is d n (s) = (2l L + 1)(2l R + 1) = (n + Since E n (s) depends on s 2 (92), the SO(4) eigenstates with (n, s) and (n, −s) are degenerate. The degeneracy of the subband of the Landau level E n (|s|) for s = 0 is given by When I is odd, s can take s = 0 and the degeneracy is A schematic picture is given by Fig.1.

Landau level eigenstates
In [36], we constructed the lowest Landau level (n = 0) basis states by taking the symmetric product of the chiral Hopf spinors. Here, we provide a precise meaning of the construction. For n = 0, the SO(4) ≃ SU (2) L ⊗ SU (2) R indices are given by For each of SU (2), we take the fully symmetric product of the chiral Hopf spinors (34): with m L = l L , l L − 1, · · · , −l L and m R = l R , l R − 1, · · · , −l R , and the lowest Landau level basis states are constructed as Recall that the S 3 D -fibre φ is common to ψ L and ψ R (34). As the expansion basis of Φ (0,s,I/2) mL,mR , we adopt the SU (2) D higher spin representation made of φ : Using the basis (100), we expand Φ (0,s,I/2) mL,mR as to define the expansion coefficients carrying the internal SU (2) gauge index A. In particular for s = I/2 and s = −I/2, the coefficients are obtained as where Ψ(χ) is the SU (2) group element in the Dirac gauge (40). Also from other exercises, such as Φ (0,0,1) , we can deduce a general formula for the lowest Landau level eigenstate: where l L and l R are given by (97) and I/2, A|l L , m ′ L ; l R , m ′ R denotes the Clebsch-Gordan coefficient. Replacing the relation (97) of the lowest Landau level with that of the higher Landau level (74), we may expect that (103) realizes the higher Landau level basis states. (This expectation turns out to be true as we shall see below.) We will refer to (103) as the SO(4) monopole harmonics. 10 Nair and Randjbar-Daemi gave the first derivation of the Landau level eigenstates [30], in which harmonic expansion on coset space [52] was applied (see [53,54] also). On the coset Inserting the complete basis relation to (105), we have 11 where D is the Wigner's D function Ψ (l) (χ) corresponds to D (l) (χ) in the Dirac gauge (Appendix D), and so we find (103) is equivalent to (107). By binding the left magnetic quantum numbers (m ′ L and m ′ R ) of two D functions with the Clebsch-Gordan coefficients, we can construct the SO(4) Landau level basis states with internal magnetic quantum number (A) as in (107). The condition determines the normalization constant as Using (107), we can construct a vector-like notation of the SO(4) monopole harmonics: which satisfies Since the D-functions depend on the SO(4) Casimir indices determined only through n+ I 2 (and s), different n and I/2 can give rise to same D functions if n + I 2 is fixed. The Clebsch-Gordan coefficients account for their difference.

Gauge fixing analysis
With gauge fixing, we further pursue the properties of the SO(4) Landau Hamiltonian eigenstates.

Dirac gauge and Schwinger gauge
We first establish relations between the Dirac and the Schwinger gauges. In the Schwinger gauge, the D-function is given by which is related to Ψ where As in the case of the Dirac gauge (43), the SU (2) gauge field in the Schwinger gauge can be expressed as where We can read off the components of the gauge field from (116) as For I = 1, (118) is reduced to the SU (2) spin connection (17). From (114), one may find Actually this is a generalization of (19) for arbitrary spin magnitude. With (117), in the Schwinger gauge the SO(4) monopole harmonics are constructed as 12 For instance, for (n, I/2, s) = (1, 1/2, 1/2), the SU (2) L ⊗ SI(2) R indices are given by (l L , l R ) = (1, 1/2) and the dimension of the multiplet is (2l L + 1)(2l R + 1)| (lL,lR)=(1,1/2) = 6. 13 As the gauge field undergoes the gauge transformation (119), the SO(4) monopole harmonics in the Dirac and the Schwinger gauges should be related as It is easy to verify (122) from (114) and the property of the Clebsch-Gordan coefficient

Properties of the SO(4) monopole harmonics
From the properties of the D function and the Clebsch-Gordan coefficients the complex conjugate of the SO(4) monopole harmonics is given by 12 Unlike the Dirac gauge Ψ In this case, the SO(4) monopole harmonics (120) are explicitly given by , Notice that the complex conjugate flips both of the SO(4) magnetic quantum numbers, m L and m R .
Integration of the product of three SO(4) monopole harmonics is given by (see Appendix C) and where and {· · · } on the right-hand side is the 6-j symbol : The integration formula (126) will be crucial to derive the matrix geometry of the SO(4) Landau level in Sec.6.

SO(4) covariance
The SO(4) covariance of the SO(4) monopole harmonics is essential for the monopole harmonics to be the eigenstates of the SO(4) Landau Hamiltonian. The gauge fixing indeed allows us to demonstrate the covariance of the SO(4) monopole harmonics.
In the Dirac gauge, the angular momentum operators are represented as (76) and the SO(4) monopole harmonics are (111) with (103). Using these, we can explicitly show 14 or more concisely, which manifests that the SO(4) monopole harmonics are the irreducible representation of the SU (2) L and and then Obviously, Φ (n,s, I 2 ) m ′ L ,mR are the Landau level eigenstates with the SO(4) index (128).

Reduction to the SO(4) spherical harmonics
We can check that the SO(4) monopole harmonics are reduced to the SO(4) spherical harmonics in the free SU (2) background limit. In literature [42,43,44], the SO(4) spherical harmonics is usually given by 14 We checked the validity of (133) for low dimensional representations by using formula manipulation system, Mathematica.

Spinor Landau Model
Before going to the analysis of relativistic Landau models, we investigate the spinor SO(4) Landau model whose Hamiltonian is the square of a relativistic Landau operator. The analysis of the spinor Landau model is a preliminary step to more complicated relativistic Landau models, but the spinor Landau model has importance in its own right. The spinor Landau model includes a synthesized connection of the spin and the gauge connections just like the relativistic Landau models. In Sec.4.1 we discuss special properties of the synthesized connection to present a basic idea to solve the eigenvalue problem. Based on the observation, we introduce the SO(4) angular momentum operators in Sec.4.2 and explicitly solve the eigenvalue problem in Sec.4.3. We discuss the physics described by the spinor SO(4) Landau model in Sec.4.4.

Synthesized connection
As mentioned in Sec.3, the components of the spin connection and the SU (2) gauge field are exactly equivalent and their difference is just the representation of the SU (2) generators: The synthesized connection of the spin connection and the gauge field is: 17 Notice that in the present model, the common factor ω i µ can be extracted in front of the synthesized SU (2) generators. We can then decompose the SU (2) representations to two direct sum of the two irreducible representations. The generators of the synthesized SU (2) gauge group are which are irreducibly decomposed as or more explicitly U denotes 2(I + 1) × 2(I + 1) unitary matrix constructed by the Clebsch-Gordan coefficients: The column is specified by the index (σ, m) (σ = +, −, m = I/2, I/2 − 1, · · · , −I/2) and the row by 18 C + and C − represent 2(I + 1) × (I + 2) 17 In the local coordinates on S 3 , the synthesized connection is given by 18 For instance I/2 = 1/2, we have and 2(I + 1) × I rectangular matrices, respectively. The unitary transformation (157) decomposes the synthesized SU (2) connection space to the direct sum of the two SU (2) spaces of spin J + and J − , such as As usual, we construct the covariant derivatives for the synthesized connection: and the field strength: 19 where They are also block diagonalized as The spinor Landau model is thus decomposed to the direct sum of two SO(4) Landau models of SU (2) gauge fields with SU (2) index J + and J − , and the eigenvalue problem is boiled down to those of the SO(4) Landau models with J + and J − sectors. Therefore, to solve the eigenvalue problem, what we need to do is just to apply the method of Sec.3 to each of the sectors.

Eigenvalue Problem
With the synthesized connection, we adopt the following as the spinor Landau Hamiltonian 20 Using the explicit form of C + and C − (157), the components of (173) can be expressed as where σ = +1, −1, m = I/2, I/2 − 1, · · · , −I/2, and P J ± is the projection operator to the Hilbert space of the SU (2) index J ± : The constant matrix term on the right-hand side was added for the spinor Landau Hamiltonian to be the precise square of the Weyl-Landau operator (see Sec.5). From (167) and we find that (176) is essentially the SO(4) Casimir made of L µν : The spinor Landau Hamiltonian is invariant under the SO(4) rotations. Recalling the results in Sec.4.2, we can derive the energy eigenvalues, E + n (s) for J + -sector and E − n (s) for J − -sector. For E ± n (s), the range of the indices are n = 0, 1, 2, · · · , s = −J ± , −J ± + 1, · · · , J ± , and hence we have E n (s) = n(I + n + 1) + s 2 with n = 0, 1, 2, · · · and s = −J + , −J + + 1, · · · , J + . In detail, where n = 1, 2, 3, · · · , s = − The schematic picture of the spectrum is given by Fig.2. Note that J − = I 2 − 1 2 ≥ 0 is not well defined for I = 0, and then the energy levels (179b) and (179c) vanish in the free background limit. In (181c), s = 0 is special: which is equal to the degeneracy (2j + 1) 2 (j = 0, 1/2, 1, 3/2, · · · ) of two particles with identical spin j. The explicit forms of the eigenstates are given by

Interactions in the spinor Landau Hamiltonian
We discuss interactions that the spinor Landau model describes. In (59), we saw that the non-relativistic SO(4) Landau Hamiltonian includes the (gauge) spin-orbit interaction. Since the synthesized connection consists of two kinds of spins coming from the holonomy and gauge groups, the spinor Landau Hamiltonian is expected to represent their interactions also.
We first decompose the SO(4) synthesized angular momentum operator as where L µν contain the gauge spin and the holonomy spin s µν : The square of SO(4) total angular momentum is derived as where we used L µν 2 is given by (59). The second term on the right-hand side of (186) contains the holonomy spin-orbit interaction and the holonomy gauge spin-spin interaction. Indeed in the Dirac gauge the holonomy part (185b) is simply represented as and the second term of (186) becomes where both L D i (77) and K i (79) contain the gauge spin S (I/2) i . The last term on the right-hand side of (186) can be expressed as Consequently, the spinor Landau Hamiltonian (178) is represented as The second, third and fourth terms respectively stand for the gauge spin-orbit, holonomy spin-orbit, and holonomy gauge spin-spin interactions. With special combination of such interactions, the spinor Landau Hamiltonian (192) respects the SO(4) symmetry.

Relativistic Landau Models
Behind the SO(4) Landau models, we implicitly assumed (3 + 1) space-time (whose spacial manifold is S 3 ), and we refer to 2×2 Dirac operator as the Weyl operator and 4×4 Dirac operator as the Dirac operator simply. The previous results of the spinor Landau models are applied to solve the eigenvalue problems of the relativistic Landau models. In Sec.5.1, we utilize the spin connection to construct the Weyl-Landau Hamiltonian on S 3 and analyze the eigenvalue problem, in which verification of the SO(4) invariance of the Dirac-Landau operator and an explicit form of the eigenstates are derived. We also account for the existence of the zero-modes from the non-commutative geometry point of view. It is shown that the obtained results are reduced to the known formulae of the free Weyl operator in the free background limit. In Sec.5.2, we make use of the results of the Weyl-Landau model to solve the eigenvalue problem of the massive Dirac-Landau model. The supersymmetric Landau operator made of the square of the Dirac-Landau operator is also analyzed.
A convenient gauge to express the relativistic operators on S 3 is the Schwinger gauge, which we will adopt in this section.

SO(4) Weyl-Landau model
Let us begin with the construction of the Weyl-Landau operator It is a (2 · (I + 1)) × (2 · (I + 1)) matrix-valued differential operator. From (13), the covariant derivatives are given by and the Weyl-Landau operator is expressed as whereD χ ≡ ∂ χ + cot χ = D χ + cot χ, The last terms ofD χ andD θ are non-hermitian terms that come from the spin connection (13) (as in the case of the Dirac-Landau operator on S 2 [1]); cot χ inD χ is from ω 1 2 and ω 3 1 , and 1 2 cot θ inD θ from ω 2 3 21 . In the previous study of the SO(3) Landau model [1], we showed that the Dirac-Landau operator is invariant under the SO (3) The first equation of (201) can be checked rigorously from the Weyl-Landau operator (197) and the SO(4) angular momentum operators. 22 With the results in Sec.4.3, we can readily obtain the Weyl-Landau operator eigenvalues as ±λ n (s) = ± n(n + I + 1) + s 2 with n = 0, 1, 2, · · · and s = −J + , −J + +1, · · · , J + , or where n = 1, 2, 3, · · · , s = − (In the free background limit I = 0, there do not exist the eigenstates for (204b) and (204c), as in the case of the spinor Landau model.) Notice that the zeroth Landau level (204c) does not explicitly depend on the monopole charge and remains in low energy region even in I → ∞ limit. The schematic picture of the Weyl-Landau operator is given by Fig.3. The degeneracies for (204) are respectively given by 21 The spin connection ω = e α a γ a ωα is given by Dirac gauge : ω D = i tan( χ 2 ) (sin θ cos φ σx + sin θ sin φ σy + cos θ σz) = i tan( 22 (201) can also be derived from a general formula [55,37,38] ( with where α s and β s are the coefficients (subject to α s 2 + β s 2 = 1) that are determined so that (207b) be the eigenstates of the Weyl-Landau operator with the eigenvalues (204b). In Appendix E, we explicitly derive α s and β s and construct the Weyl-Landau operator eigenstates for several cases. As mentioned above, the Weyl-Landau operator eigenstates (207) are the SO(4) ≃ SU (2) L ⊗ SU (2) R irreducible representations with the indices (L, R): ( 1 2 (n − 1), 1 2 (n + I)), ( 1 2 (n + I), 1 2 (n − 1)), For s = 0, two representations of (208c) coincide to be (L, R) = ( 1 4 (I − 1), 1 4 (I − 1)). (207b) consists of both the (n − 1)th Landau level basis states in J + -sector and the nth Landau level in J − -sector, because n−1+J + = n+J − . This feature is similar to that of the SO(3) Dirac-Landau model on S 2 [1]. Meanwhile, (207a) comes only from the non-relativistic nth Landau level with replacement of I/2 with J + , and this is a new feature in the SO(4) model. Notice that the zeroth Landau level (204c) comes only from the lowest Landau level of J − -sector, and the relativistic zeroth Landau level is exactly equal to that of the non-relativistic lowest Landau level with replacement of I/2 with J − = (I − 1)/2. This property is also observed in the SO(3) model on S 2 [1]. For the zeroth Landau level, the spectrum of the subbands is exactly equal to the corresponding chirality parameter: Therefore the reflection symmetry between the positive and negative eigenvalues in the zeroth Landau level is identical to the LR symmetry: Thus in the relativistic Landau model, the left-right symmetry of the non-relativistic Landau model is realized as the chiral symmetry (of the zeroth Landau level).

The lowest Landau level and the zero-modes
In [37], we showed that the total degeneracy of the SO(4) zeroth Landau level is equal to the 2nd Chern number of the SO(5) Landau model: as a manifestation of the dimensional ladder of anomaly. The total degeneracy of the zeroth Landau level thus finds its topological origin in one dimension higher space. In even dimensions, the existence of Dirac-Landau operator zero-modes is accounted for by the Atiyah-Singer index theorem [55,15]. Though constructed on three-sphere, the SO(4) relativistic Landau model also accommodates the zero-modes for odd I. Here we discuss the origin of such zero-modes. For odd I = 2q − 1 (q = 1, 2, 3, · · · ), the explicit form of the zero-modes is given by Ψ (n=0,s=0,J − ) ML,MR 1 2π S 2F = q. Therefore, the zero-mode degeneracy of the Weyl-Landau operator can be expressed by the 1st Chern-number: In this sense, the zero-modes on S 3 originates from the topological quantity in one dimension lower 2D space. From the viewpoint of the non-commutative geometry, the fuzzy three-sphere is represented as [35,30] 23 23 (217) is naturally induced from the chiral Hopf map (29): meaning that the fuzzy three-sphere is essentially the product of two independent fuzzy two-spheres. As the zero-mode degeneracy is equal to the dimension of fuzzy sphere [15], it may be natural that the zero-modes on S 3 is given by the product of the zero-modes on two S 2 s.
In arbitrary odd dimension, (217) is generalized as [35], Figure 4: The free Weyl operator spectrum. The special subbands with s = ±J + (denoted by blue in Fig.3) survive in the free limit.

SO(4) Dirac-Landau model
Next we analyze the Weyl-Landau operator to the Dirac-Landau operator. From SO(4) gamma matrices with γ a (16), the massive Dirac-Landau Hamiltonian is constructed as The mass term appears as the off-diagonal blocks. The SO(4) angular momentum operators are similarly given by with L µν (165) on the right-hand side. The chirality matrix is The massive Dirac Hamiltonian respects both of the SO(4) symmetry and the chiral symmetry The chiral symmetry guarantees the reflection symmetry of the positive and negative energy levels with respect to the zero-energy. The Dirac-Landau operator (224) can then be regarded as a Hamiltonian of chiral topological insulator. (229) suggests that the chiral transformation is equivalent to the sign flips of the Dirac-Landau operator and the mass parameter: In particular for the zeroth Landau level, the chiral transformation corresponds to s → − s.

Supersymmetric Landau model
The square of the Dirac-Landau operator yields a supersymmetric quantum mechanical Hamiltonian: where Q and Q † are supercharges that satisfy The SUSY Hamiltonian (240) consists of two identical spinor Landau Hamiltonians, and its eigenvalues are λ n (s) 2 = n(n + I + 1) + s 2 with the degenerate eigenstates The Witten parity is given by Ψ ±λ 0 belongs to the Witten parity + ("fermionic") sector, while 0 Ψ ±λ the Witten parity − ("bosonic") sector. They are the superpartner related by the SUSY transformation: For odd I, the ground-state of H SUSY is given by SUSY invariant zero-energy state (good SUSY), while for even I, the ground-state is not zero-energy state and does not respect the SUSY (broken SUSY). We can also consider the square of the massive Dirac-Landau Hamiltonian The mass term just shifts the zero-energy of the spectrum of H SUSY .

Matrix Geometry
In the above, we introduced the various SO(4) Landau models and solved their eigenvalue problem. With the developed technologies and results, we are now ready to evaluate the matrix geometry of the SO(4) Landau levels. We concretely derive the matrix elements of S 3 coordinates in each of the Landau levels based on the level projection method to appreciate the emergent non-commutative geometry. We first derive the fuzzy three-sphere geometry of the SO(4) non-relativistic Landau model in Sec.6.1, and next we explore fuzzy geometry of the spinor Landau model and relativistic Landau models in Sec.6.2 and Sec.6.3. We point out that the massive Dirac-Landau model accommodates the two fuzzy three-spheres whose interaction is induced by the mass term (Sec.6.3).
We switch the notation from Φ (n,s,I/2) mL,mR to Φ [lL,lR,I/2] mL,mR in this section.
For even I, l L l R I/2 l L l R 1/2 = 0, (I = 0, 2, 4, · · · ) (258) and then Y µ = 0. Meanwhile for odd I, and, as mentioned above, non-zero matrix elements appear only for and from (259) we obtain wherê are the off-diagonal blocks of the SO(4) gamma matrices in the symmetric representation: 24 which satisfy [35,33]  . (I = 1, 3, 5, · · · ) (265) 24 For instance, (255) is now obtained as X µ (n) = I + 1 (2n + I + 1)(2n + I + 3) To summarize, the non-trivial matrix geometry appears only in the subband |s| = 1/2 for odd I: In particular, for the lowest Landau level (n = 0), X µ (267) satisfy the relation with The relation (269) is invariant under the SO(4) rotations and is a non-commutative counterpart of the definition of three-sphere. We thus find that X µ denote the matrix coordinates of fuzzy three-sphere. The radius decreases as the Landau level increases. A similar behavior is observed in the fuzzy two-sphere of the SO(3) Landau model [1]. In each Landau level, only the |s| = 1/2 subband realizes the fuzzy three-sphere geometry and each Landau level accommodates just one fuzzy three-sphere. The fuzzy three-sphere geometry of the SO(4) Landau model is naturally understood as a subspace embedded in the one dimension higher fuzzy four-sphere of the SO(5) Landau model [36,33]. If we regard the chirality parameter s as an extra fifth coordinate x 5 , X µ can be interpreted as the coordinates of two S 3 -latitudes with x 5 = ±s on the virtual fuzzy S 4 (Fig.6), as suggested by the equation where d n=0 (s) (94) and d

SO(5) n=0
respectively denote the lowest Landau level degeneracy of the SO(4) and SO(5) Landau models.

Relativistic Landau models
The Weyl-Landau model is a "square root" of the spinor Landau model and the non-trivial matrix geometry can occur for even I. In the spectrum of the Weyl-Landau model, the degeneracies of s = 1/2 and −1/2 corresponding to (273a) and (273c) are completely lifted (see the left-figure of Fig.3), and only the half of the degeneracies in (273b) survives in the Weyl-Landau model to generate the fuzzy three-sphere. From the left figure of Fig.3, we can see I = 2, 4, · · · , ± n = ±1, ±2, · · · → S 3 F .
The square of X µ is derived as where we interchanged the second and fourth columns and rows. In the massless limit, the off-diagonal blocks of X µ (M ) vanish to yield X µ (M )

M→0
−→ X µ 0 0 X µ that actually represents the two identical noninteracting fuzzy three-spheres. When the mass term is turned on, the off-diagonal block matrices appear to bring interactions between the two fuzzy three-spheres. M/Λ n can be interpreted as the coupling of the interaction. Meanwhile, for each of the cases of (238), the degenerate subbands with s = ±1/2 appear for even I, and the fuzzy three-spheres are respectively realized as In the massless limit, the fuzzy geometries collapse in either case.
In the supersymmetric Landau model, the Hamiltonian (240) consists of two independent spinor Landau Hamiltonians and accommodates twice the number of fuzzy three-spheres of the spinor Landau model. The mass term in the SUSY Landau model does not bring any particular effect to fuzzy geometry, as the mass term just shifts the energy levels uniformly (247).

Summary and Discussions
We throughly investigated the SO(4) Landau models based on the Dirac and the Schwinger gauges. The gauge fixing enabled us to elaborate the previous works about the SO(4) Landau models and bring the new observations, such as the properties of the SO(4) monopole harmonics and the SO(4) symmetry of the relativistic operators. In the present analysis, we took into account the spin connection of three-sphere to construct the relativistic Landau operators. With the synthesized connection of the spin connection and gauge field, we solved the eigenvalue problem of the spinor Landau operator and of the Weyl-Landau operator also. The obtained results are confirmed to reproduce the known results of the SO(4) spherical harmonics and the free Dirac operator in the free background limit. The eigenvalue problems of the massive Dirac-Landau and the supersymmetric Landau model are analyzed too. We applied the level projection method to the SO(4) Landau models and derived the odd dimensional matrix geometry for the first time. It was shown that, for each of the (non-relativistic) Landau levels, the fuzzy three-sphere geometry appears only in the lowest energy |s| = 1/2 subband, and the size of fuzzy three-sphere depends on the Landau level. We also clarified realizations of the fuzzy three-sphere geometry in the relativistic Landau models. In particular, we designated that the mass term of the Dirac-Landau model induces interaction between two fuzzy spheres realized in the relativistic Landau level.
As the SO(3) Landau model has a wide range of applications, we expect that the SO(4) Landau model may also find its playing fields in many branches of physics. Even if limited to condensed matter physics, one may conceive its possible applications to Weyl/Dirac semi-metal, three-dimensional quantum Hall effect and chiral topological insulator. We have clarified the emergent fuzzy three-sphere geometry in the Landau physics. It is interesting to see that such an exotic geometry realizes "inside" the physical models, and the dynamics of the fuzzy spaces can be controlled by a physical parameter which in principle can be controlled by experiment.

Acknowledgment
This work was supported by JSPS KAKENHI Grant No. 16K05334 and No. 16K05138.
A Geometric quantities of three-sphere: component method

A.2 Spin connection
From e a = e a α dx α , we can read off the components of the dreibein (6): The inverse is The components of the spin-connection are obtained by the formula [49] ω abα = −e bβ ∇ α e β a = −e bβ (∂ α e β a + Γ β αγ e γ a ), as which are consistent with (13). The matrix form of the spin connection is given by with ω χ = 0, ω θ = − cos χσ 12 , ω φ = sin θ cos χσ 31 − cos θσ 23 , or where we used The curvature is obtained as B The SO(4) spherical harmonics and free Dirac operator

B.2 The free Weyl operator and eigenstates
The eigenvalue problems of free Dirac operators on arbitrary dimensional spheres are generally solved in [45,46,47]. Here, we apply the results to the S 3 case. For spinor particle, the covariant derivatives on S 3 are given by 27 From the general formula, we have µ<ν l (0) µν 2 = 2(L(L + 1) + R(R + 1)) L=R= n 2 = n(n + 2).

B.3 Square of the free Weyl operator
The square of the free Weyl operator on S 3 (311) is derived as or Using (310), we can show where (333a) and (333b) are simply related as 30 The eigenvalues of the SO(4) Casimir (333b) can be obtained from the non-relativistic result (75) for I/2 = |s| = 1/2: 30 Also from the general formula in [55,37], we have and then 31 C Integral of the product of three SO(4) monopole harmonics From the explicit form of the SO(4) monopole harmonics (110), we can evaluate the integral of the product of three SO(4) monopole harmonics as This is a special formula of Eq.(3.11) in [53,54]. For Φ where, in the third equation, we used (340) is consistent with (112). 31 (337) is consistent with (179a) with the replacement (n, I) → (n + 1, 0).
Using the dreibeins, (6) and (7), they are concisely represented as f S and f D are related by the SU (2) transformation: where in the third equation we used the SO(3) invariance of the Levi-Civita tensor and in the last equation (23) .

D.3 Gauge covariance of the Weyl-Landau operator
From the relations e a D = O ab e b S and g γ a g † = γ b O ba (see Sec.2), we have g (1/2) (e a S γ a ) g (1/2) † = e a D γ a .
From (375) and (376), we easily see that the Weyl-Landau operator satisfies and then GΨ S = Ψ D .

E Examples of the Weyl-Landau operator eigenstates
For several Weyl-Landau operators, we explicitly derive a coordinate representation of the eigenstates based on the general procedure presented in Sec.5.1.
In the following, we use the notation: where Φ (−,n,s) and Φ (+,n,s) respectively denote the one-component (S = 0) and three component (S = 1) irreducible representation of the SU (2) gauge group. The unitary matrix for the irreducible decomposition, 1/2 ⊗ 1/2 = 1 ⊕ 0, is given by It is straightforward to construct the eigenstates of the spinor Landau model by acting (384) to (383). Derivation of the Weyl-Landau operator eigenstates is not difficult. We act the Weyl-Landau operator to linear combination of the eigenstates and determine the coefficients of the linear combination (α and β in (207b)) so that the linear combination to be the eigenstate of the Weyl-Landau operator. The results are as follows.