$SO(5)$ Landau Models and Nested Nambu Matrix Geometry

The $SO(5)$ Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the $SO(5)$ Landau models and the associated matrix geometries through the Landau level projection. With the $SO(5)$ monopole harmonics, we explicitly derive matrix geometry of a four-sphere in an arbitrary Landau level: In the lowest Landau level the matrix coordinates are given by the generalized $SO(5)$ gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau level the matrix geometry becomes a nested fuzzy structure with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of a $SO(4)$ Pauli-Schr\"odinger model and the $SO(4)$ Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the $SO(5)$ Landau model are also investigated and relation to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions.


Introduction
More than forty years ago, Yang proposed a SU (2) generalization [1] of the Dirac's monopole [2]. The set-up behind the Yang's SU (2) monopole stems from a beautiful mathematical concept of the 2nd Hopf map associated with the generalization of complex numbers to quaternions [3,4]. The Yang's monopole field configuration on S 4 is conformally equivalent to the BPST instanton configuration on R 4 [5] and possesses the SO(5) global rotational symmetry. Yang also succeeded to construct generalized monopole harmonics in the SU (2) monopole background [6]. This set-up was used in the context of the Zhang and Hu's SO (5) Landau model and 4D quantum Hall effect [7] that realize natural higher dimensional counterparts of the Wu and Yang's SO(3) Landau model [8] and the Haldane's 2D quantum Hall effect on a two-sphere [9]. 1 The non-commutative geometry is the emergent geometry of the Landau models and governs the dynamics of the quantum Hall effect [12,13]. The Landau level projection truncates the whole quantum mechanical Hilbert space to a sub-space and provides a physical set-up where the non-commutative geometry naturally appears. Along this line, the fuzzy four-sphere geometry has been discussed in the context of the SO(5) Landau model [7,14,15]. It is known that the fuzzy four-sphere exhibits intriguing mathematical structure not observed in the fuzzy two-sphere: While the algebra of the fuzzy two-sphere is given by the SU (2) algebra [16,17], the five coordinates of the fuzzy four-sphere [18,19] are not closed by themselves within the Lie algebra but bring extra non-commutative coordinates constituting "internal" fuzzy structure [20,21,22]. Such a peculiar structure makes the studies of higher dimensional non-commutative geometry more interesting and attractive. There are two ways to represent the fuzzy four-sphere algebraically: (i) Lie algebra [20,21,22]: the enlarged algebra of the fuzzy four-sphere is the SO(6) ≃ SU (4) giving rise to fuzzy fibre space ( Fig.1): (ii) Four-Lie bracket [23,24] : With the quantum Nambu bracket [25,26], the fuzzy four-sphere coordinates are closed by themselves without introducing extra fuzzy coordinates. The internal structure is implicit, and the internal geometry reflects its existence in the degeneracy of (fuzzy) three-sphere latitudes ( Fig.1): [X a , X b , X c , X d ] = (I + 2)α 3 ǫ abcde X e .
(2) Figure 1: Two geometric pictures of the fuzzy four-sphere. In the left figure, X ab (1) span the fuzzy S 2 -fibre on the original fuzzy manifold "S 4 F ". In the right figure, the internal geometric structure is accounted for by the degeneracy of the fuzzy three-sphere latitudes on S 4 F .
In the previous studies [27,28,29], we demonstrated that the quantum Nambu geometry actually appear in the higher dimensional Landau models, and is elegantly intertwined with exotic ideas of differential topology, quantum anomaly, and string theory. However, the deduction of the non-commutative geometry from the Landau models has been rather heuristic and the obtained results are justified in the thermodynamic limit. 2 A rigorous way to derive the non-commutative geometry is accomplished by the Landau level projection not resorting to any approximation, and the results will capture every detail of the emergent non-commutative geometry. The Landau level projection method can also be applied to an arbitrary Landau level (not limited to the lowest Landau level) whose non-commutative geometry has rarely been investigated, in contrast to the Berezin-Toeplitz quantization focused on zero-modes. The practical procedure of the Landau level projection is quite straightforward: We just sandwich coordinates of interest by Landau level basis states to derive their matrix-valued counterparts in a given Landau level. Since the total Hilbert space of the Landau model is mathematically well-defined, the truncated subspace of the Landau level necessarily provides a sound formulation of non-commutative geometry. Based on this observation, we derived matrix geometries of the SO(3) Landau models [31] and the SO(4) Landau models [32]. We extend this project to the SO(5) Landau models. Not just rendering the similar analysis, we integrate the previous results with new SO(5) results to present a comprehensive view of the emergent fuzzy geometry of the Landau models. We unveil hidden relations between the background topological field configurations of the Landau models, and also discuss the matrix geometry of the Landau models in an arbitrary dimension.
This paper is organized as follows. In Sec.2, we review the SU (2) monopole and SO(5) Landau problem in a modern terminology. Using the SO(5) Landau level eigenstates, we derive the matrix geometry of the SO(5) Landau model in Sec.3. Sec.4 discusses the internal fuzzy three-sphere structure with emphasis on its relation to the SO(4) Landau model. We also clarify relations among the background topological field configurations in low dimensional Landau models. Relativistic version of the SO(5) Landau model and its associated zero-modes are analyzed in Sec.5. In Sec.6, we extend the matrix geometry analysis to even higher dimensions. Sec.7 is devoted to summary and discussions.
2 Review of the Yang's SU (2) monopole system In this section, we review the Yang's SU (2) monopole system [1,6] and the Zhang and Hu's SO(5) Landau model [7] adding some more informations.

SU(2) monopole and SO(5) angular momentum operators
With stereographic coordinates of S 3 -latitude on S 4 , Yang gave an expression of the SU (2) monopole gauge field [1]. However, the original expression is rather cumbersome to handle and we then adopt the Zhang and Hu's concise notation of the SU (2) (anti-)monopole gauge field [7]: where S i (i = 1, 2, 3) denote the SU (2) matrix of the spin I/2 representation: The field strength, , is given by The non-trivial homotopy for the SU (2) monopole field configuration on S 4 is guaranteed by 2 See [30,7,10] also. and the second Chern number associated with (3) is evaluated as where F = 1 2 F ab dx a ∧ dx b . We construct the covariant angular momentum operators, Λ ab , as with and the total SO(5) angular momentum operators as In detail, where L (0) ab denote the SO(5) free angular momentum operators:

The SO(5) Casimir operator and SO(5) monopole harmonics
In usual textbook derivation of the spherical harmonics, the polar coordinates are adopted to represent the SO(3) Casimir. In a similar manner, we decompose the SO(5) Casimir operator to the SO(4) part and the remaining azimuthal angle part. We introduce the polar coordinates of a four-sphere (with unit radius) as x 1 = sin ξ sin χ sin θ cos φ, x 2 = sin ξ sin χ sin θ sin φ, x 3 = sin ξ sin χ cos θ, where The SO(5) Casimir is expressed by the sum of the SU (2) L ⊕ SU (2) R Casimir parts and x 5 -part [6] 5 a<b=1 L ab where J i and K i are the SU (2) L and SU (2) R operators given by Here, η i mn andη i mn denote the 't Hooft symbols: Note that the SU (2) (anti-)monopole gauge field does not act to the SU (2) L operators but acts to the SU (2) R operators only (16b), as if the right SU (2) angular momentum acquires additional SU (2) spin angular momentum.
Here, C l+1 p−l denote the Gegenbauer polynomials, and Y lm (θ, φ) stand for the SO(3) spherical harmonics: Since the (anti-)monopole gauge field only contributes to the SU (2) R angular momentum operator, in (20) the original SU (2) R index j of the SO(4) spherical harmonics is contracted with the gauge spin index I/2 to form the SU (2) R composite spin k just as the usual SU (2) angular momentum composition rule. Therefore, k takes s signifies the difference between the left and right SU (2) quantum numbers, and hence the name the chirality parameter [28,29]. Though k and j are two independent SU (2) group indices, in the present 3 The SO(4) spherical harmonics (23) satisfy The dimension of the SO(4) spherical harmonics is given by system the range of k is not arbitrary but restricted as (26) with a given j. In (I + 1)-component vectorlike notation, the SO(4) spinor spherical harmonics (19) is expressed as From two indices j and k, we introduce the SO(4) Landau level index n: n essentially denotes the sum of two SU (2) quantum numbers. With n and s, j and k are represented as Notice that while the (anti-)monopole only acts to the SU (2) R operator, with a given n j and k are totally equivalent in the sense that either of j and k starts from n/2 and ends at I/2 + n/2 (see Fig.2 also).

Azimuthal part eigenvalue problem
The SO(5) Casimir operator was decomposed to the azimuthal part ξ and the hyper-latitude S 3 part (15). To solve the differential equation of the SO(5) Casimir operator, Yang adopted the method of separation of variables [6]: Here, Y j,k (Ω 3 ) denote the SO(4) monopole harmonics (19) with the constraint The SO(5) eigenvalue problem gives the eigenvalue equation for G(ξ): Yang showed that the difference of two Casimir indices is exactly equal to the SU (2) monopole index I [6]: Therefore, when we identify q with the SO(5) Landau level index N (= 0, 1, 2, · · · ), the SO(5) monopole harmonics correspond to the irreducible representation with the indices (31). The SO(5) Casimir eigenvalues are readily obtained as and the corresponding degeneracy is D(I, N ) = 1 6 (p + 2)(q + 1)(p + q + 3)(p − q + 1) = 1 6 (N + 1)(I + 1)(I + N + 2)(I + 2N + 3), which is equal to (35). The normalized SO(5) monopole harmonics are derived as where [14] G N,j,k (ξ) = N + Here d l,m,g (ξ) denotes the Wigner's small d-function 5 and the three indices are identified with (l, m, g) = (N + I 2 + 1, s, −n − I 2 − 1) in (45). In the small d-function d l,m,g (ξ), its two magnetic indices, m and g, generally take (half-)integer values between −l and l, while in the present case m = s and the range of s (27) is restricted to |s| ≤ I 2 which is smaller than l = N + I 2 + 1 (except for N = 0). We find that a subset of d-function is utilized in (45). The orthonormal relation for (44) is given by For instance, the SO(5) spinor representation (N, I) = (0, 1) is obtained as and 5 The small d-function can also be expressed as where S (l) y denotes y-component of the SU (2) spin matrix with spin magnitude l: the behavior of Ψ N ;j,k can be read off as At I = 0, (44) is reduced to the SO(5) spherical harmonics as expected (see Appendix C).

The SO(5) Landau model
The SO(5) Landau model [7] is a Landau model on a four-sphere in the SU (2) monopole background. With the covariant derivatives D a (9), the SO(5) Landau Hamiltonian is given by which can be rewritten as where we used Λ ab F ab = F ab Λ ab = 0. From (5), we can readily derive and the SO(5) Landau Hamiltonian is diagonalized as with the Landau level degeneracy (43). In particular, the lowest Landau level degeneracy is given by The Landau level eigenstates are given by the SO(5) monopole harmonics (44). The lowest Landau level degeneracy (57) is simply understood as the number of the fully symmetric representation [7], where m 1 , m 2 , m 3 , m 4 are non-negative integers subject to and ψs denote the components of the 2nd Hopf spinor 6 We can see equivalence between the fully symmetric representation (58) and the SO ( 6 We will discuss the 2nd Hopf map in Sec.3.2. (58) can be expanded as From the expansion coefficients on the right-hand side of (62), we can construct the (I + 1)-component "vector" as which is exactly equal to the N = 0 SO(5) monopole harmonics (44) under the identification 3 Four-sphere matrix geometry In this section, we investigate the matrix geometry of the SO(5) Landau model. First, we discuss a general structure of the matrix geometry deduced from the SO(5) irreducible decomposition rule. Next, we discuss the lowest Landau level matrix geometry at the quantum limit I = 1 and at the classical limit I >> 1. Finally, through the Landau level projection, we explicitly derive the matrix geometry in an arbitrary Landau level.

General form of matrix coordinates
In this section, we utilize the notation [[N, I]] to specify the SO(5) irreducible representation instead of (p, q) SO(5) = (N + I, N ). First, let us see a general structure of the matrix elements of the four-sphere coordinates: Here |N, I 2 is the abbreviation of the N th Landau level eigenstates (44), and the SO(5) vector x a carries the SO(5) index [ [1,0]], and hence the SO(5) index of x a |N, I 2 is given by which is irreducibly decomposed as [35,36,37] [ for several examples of (67). The corresponding dimension-counting is given by With a SU (2) monopole background fixed I, (67) implies that the Landau level transition, if occurred, only takes place between the adjacent Landau levels: Consequently, the matrix elements only have finite values between the adjacent inter Landau levels and intra Landau levels: as depicted in Fig.3. The red shaded squares denote the matrix elements in intra Landau levels, while the blue shaded rectangles represent the matrix elements between inter Landau levels. 7 (72a) is a special case of more general formula or

The 2nd
Hopf map and Bloch four-sphere (quantum limit: I = 1) The Yang's SU (2) monopole is closely related to the 2nd Hopf map [3,4,7]. Using quaternions q m (m = 1, 2, 3, 4), the 2nd Hopf map, S 7 S 3 → S 4 , is realized as ψ which we refer to as the 2nd Hopf spinor is a two-component quaternionic spinor ψ = (ψ 1 ψ 2 ) t (ψ 1 and ψ 2 are quaternions) subject to and signifies the total manifold S 7 . x a (76) satisfy the normalization condition 5 a=1 x a 2 = (ψ † ψ) 2 = 1 and are regarded as the coordinates on the base-manifold S 4 . The S 3 -fibre part of S 7 is projected out in the map (76). The four-sphere associated with the 2nd Hopf map can be considered as a 4D version of the Bloch sphere [38]. Due to the algebras of the quaternions, it is shown that γ a satisfy and act as the SO(5) gamma matrices. This will be more transparent when we introduce a matrix realization of the quaternions: Substituting (80) to (77), γ a now become the familiar SO(5) 4 × 4 gamma matrices, and the corresponding SO(5) generators are obtained as where The 2nd Hopf spinor ψ is also promoted to a 4 × 2 matrix Ψ subject to The S 3 -fibre part represents the SU (2) gauge degrees of freedom that acts to Ψ as Interestingly, (85) consists of the N = 0 SO(5) spinor multiplet for I = 1 (49): This implies that the 2nd Hopf map encodes informations of the lowest Landau level of the minimum SU (2) monopole index I = 1, which we call the quantum limit. For the SO(5) spinors, the SU (2) gauge transformation (84) acts as and the gauge field is given by which is exactly equal to the Yang's monopole gauge field (3) for I = 1. Under the gauge transformation (84), the gauge field is transformed as expected: Including the SU (2) gauge degrees of freedom, the 2nd Hopf spinor is generally given by Here Ψ(Ω 4 ) signifies the base-manifold S 4 and (φ 1 φ 2 ) t denotes a normalized SU (2) spinor taking its value on the S 3 -fibre. With some appropriate inner product, we orthonormalize φ i (i = 1, 2) as and the normalization condition of ψ is restated as With this simple set-up, we discuss the SU (2) gauge invariance and the SO(5) covariance of the matrix geometry. The SU (2) gauge transformation (84) can be reinterpreted as the transformation of the φ-part: while the SO(5) global transformation acts to Ψ-part as 8 where U ≡ e i a<b ω ab σ ab .
We define the matrix elements of observable O(Ω 4 ) as 9 (100) The final expression implies that the matrix elements are evaluated through the integration of the operator sandwiched by the SO(5) spinors. In particular, the matrix elements of x a are given by or where P denotes a 4 × 4 projection matrix From (93) P 2 = P , and P is invariant under the SU (2) gauge transformation (84). Therefore, X a (102) are obviously gauge invariant quantities. From the 2nd Hopf map x a = Ψ † γ a Ψ, X a = dΩ 4 Ψx a Ψ † can also be represented as With the following formulas we can easily evaluate (104) as 10 Thus in the quantum limit, the lowest Landau level matrix geometry is given by the SO(5) gamma matrices (77) up to a proportional factor. Under the SO(5) global transformation (97), X a are transformed as where we used the SO(5) covariance of the gamma matrices (107) indicates that the matrix coordinates transform as a SO(5) vector as expected. 9 The factor in front of the integration is introduced for the normalization 3.3 Heuristic derivation of the fuzzy geometry (classical limit: I >> 1) Next, we consider the opposite limit I >> 1, which we refer to as the classical limit by the analogy of quantum spin model S >> 1. Refining the heuristic discussions of [27], we will show how the noncommutative geometry takes place in this limit.
At I >> 1, the field strength term becomes dominant in the angular momentum L ab (10): The coordinates x a can be extracted from the SU (2) field strength (5) as [27] 1 Here c 2 (I) denotes the 2nd Chern number (7): Replacing F ab with L ab in (110), we have Since L ab are the SO(5) operators, the coordinates now become the operators given by (112). tr in (110) is taken in the "internal" fuzzy space S 2 F with dimension (I + 1) [27,15], and so tr(1 internal space ) = I + 1.
In the lowest Landau level, we may replace the SO(5) operators L ab with the SO(5) matrices Σ ab of the fully symmetric irreducible representation: and (112) turns into Since in the fully symmetric representation Σ ab satisfy 11 (115) is greatly simplified as 11 The gamma matrices in the fully symmetric representatoin are constructed as which satisfy In this paper, we will drop (I) on the shoulder of Γ (I) a for brevity otherwise stated.
Therefore in the classical limit, the lowest Landau level matrix coordinates are given by the SO(5) gamma matrices in the fully symmetric representation.
and around the north-pole which realizes the non-commutative algebra of Zhang and Hu [7].

Landau level projection and matrix geometry (arbitrary I and N)
We have obtained the matrix geometry at the quantum limit and the classical limit. Here, we apply the Landau level projection to derive more general results. The explicit form of the SO(5) monopole harmonics is crucial in the present analysis.

Landau level matrix elements
We perform integrations in the azimuthal part and the S 3 -latitude part separately. The S 4 -coordinates are decomposed to the azimuthal part and the S 3 -latitude part: where x m are expressed by the product of the radius of S 3 -latitude and the (normalized) S 3 -coordinates: y 1 = sin χ sin θ cos φ, y 2 = sin χ sin θ sin φ, y 3 = sin χ cos θ, y 4 = cos χ.
The area element of S 4 is expressed as with the S 3 area element dΩ 3 = sin 2 χ sin θ dχ dθ dφ.
For instance, an integration on S 4 is carried out as As discussed in Sec.2.2, the N th SO(5) Landau level consists of inner SO(4) Landau levels with n = 0, 1, 2, · · · N . In the SO(4) language, x m acts as a vector with the SO(4) ≃ SU (2) L ⊗ SU (2) R index (j, k) = (1/2, 1/2) and x 5 acts as a scalar with (j, k) = (0, 0). For the SO(4) Landau level index n (38) and the chirality parameter s (27), differences are given by ∆n = ∆j + ∆k and ∆s = ∆j − ∆k. The SO(4) selection rule tells that the matrix coordinates have non-zero values only for the cases  • Matrix coordinates for x 5 The matrix elements of x 5 are diagonalized as with where we used (45) and a formula for the small d-function. 13 The matrix coordinate (129) takes equally spaced discrete values specified by the chiral parameter s = I/2, I/2 − 1, · · · , −I/2, which are regarded as latitudes of a fuzzy four-sphere. Such a structure is very similar to that of the fuzzy two-sphere [31], but while the latitudes of fuzzy two-sphere are not degenerate, the latitudes of fuzzy four-sphere are degenerate giving rise to the internal structure.
represents transition between the two adjacent sub-bands specified by s inside a SO(4) Landau level (two adjacent dots on an identical SO(4) line in Fig.2) corresponding to the small purple shaded regions in Fig.4. In the following, we focus on the second case, which in the language of the SU (2) L ⊗ SU (2) R corresponds to Under the condition (132), we have where are in the relation of Hermitian conjugate. We can evaluate the S 3 -radius part of (133) as In the last equation, we used another formula of the small d-function. 14 Next, we turn to the unit-S 3 part (134). Notice that y m can be expanded by the SO(4) spherical harmonics (20): With an integration formula for the SO(4) spherical harmonics, a bit of calculation (see Appendix D.1)

Fuzzy four-sphere in the lowest Landau level
With the general results above, it is easy to derive the lowest Landau level (N = n = 0) matrix coordinates: 16 Since the matrix geometry of x 5 is given by the diagonal matrix with eigenvalues of equal spacing (141b), the present geometry can be regarded as a stacking of the matrix-valued three-spheres along x 5 -axis with equal spacing [see Fig.1]. The right-hand side of (141) are identical to the SO(5) gamma matrices in the fully symmetric representation (p, q) = (I, 0), so we have 15 Similarly, (j + m j )(k + m k + 1)). (138) 16 In the special case s = 1 2 σ, (141a) becomes Y (σ,−σ) m (j, k) realizes the matrix for the fuzzy three-sphere [32]. The matrix geometry (142) realizes the quantum Nambu geometry of the fuzzy four-sphere [23,24]: where [· · · ] of (143b) signifies the quantum Nambu bracket [25,26], In the thermodynamic limit I → ∞, (143a) is reduced the condition of a four-sphere with unit radius. As discussed above, the stacking of the matrix-valued three-sphere latitudes along x 5 -axis constitutes the fuzzy four-sphere geometry. One may wonder if the stacking along the x 5 -axis breaks the SO(5) symmetry of the four-sphere. However, this is not the case. Recall that we have adopted x 5 as a special axis. If we had chosen x 1 as a special axis, we would have had the stack along the x 1 -axis. Therefore, the picture of the stack along x 5 -axis is a kind of "gauge-artifact" by choosing x a as a special axis in R 5 , and the fuzzy four-sphere certainly respects the SO(5) symmetry.

Nested matrix geometry in higher Landau levels
Let us consider the matrix geometry in higher SO(5) Landau levels. With a given SO(5) Landau level N , there are N + 1 inner SO(4) Landau levels indexed by n = 0, 1, 2, · · · , N , and further in each of the SO(4) Landau levels there are I + 1 sub-bands indexed by the chiral parameter s. Each sub-band s realizes the matrix-valued S 3 -latitude, and a stack of such (I + 1) matrix-valued S 3 -latitudes along the x 5 -axis constitute a quasi-fuzzy four-sphere geometry in the SO(4) Landau levels. Therefore inside the N th SO(5) Landau level, there are N + 1 quasi-fuzzy spheres that form a nested structure as a whole [ Fig.6]. Recall that the range of the chiral parameter s is restricted to |s| = I 2 and does not cover the whole range of the matrix size specified by j + k = n + I 2 (except for n = 0). This implies that the corresponding matrix geometry is not a complete fuzzy four-sphere but a fuzzy four-sphere like geometry with removed north and south "caps" due to the uncovered parameter regions of s. We referred to this geometry as quasi-fuzzy four-sphere. Each SO(4) Landau level accommodates a quasi-fuzzy four-sphere geometry, and so N th SO(5) Landau level realizes N + 1 quasi-fuzzy four-spheres with different matrix size depending on the SO(4) index n. In this way, N + 1 quasi-fuzzy four-spheres exhibit a concentric nested structure in the N th SO(5) Landau level as depicted in Fig.6. The lowest Landau level (N = 0) is exceptional, because the nested structure no longer exists and only a fuzzy four-sphere geometry remains.
As the quantum states of the nested fuzzy geometry are given by a SO(5) irreducible representation (or the SO(5) monopole harmonics), the nested fuzzy geometry has the SO(5) covariance. Meanwhile each quasi-fuzzy four sphere does not possess the SO(5) covariance, since it is solely constructed by the SO(4) irreducible representations. There exist non-vanishing off-diagonal matrix elements between the adjacent SO(4) Landau levels (as represented by the green shaded rectangular blocks in Fig.4). Borrowing the string theory interpretation that the off-diagonal parts signify interactions between the fuzzy objects represented by the diagonal block matrices, one may say that the quasi-fuzzy four-spheres of the adjacent SO(4) Landau levels interact and conspire to maintain the SO(5) covariance of the nested fuzzy geometry. Furthermore, the nested fuzzy geometry has the SO(5) symmetry. Apparently as a classical geometry the nested structure [ Fig.6] does not have the SO(5) symmetry, but it does in a quantum mechanical sense. The reason is essentially same as of the discussion below Eq.(144). We had chosen x 5 as a special axis, and we obtained the x 5 -axis preferred picture like Fig.6, but if we had chosen the x 1 axis, we would have had a similar nested structure along the x 1 -axis. Actually we can adopt any axis in R 5 , and then the nested structure has to have the SO(5) symmetry. Therefore, the nested fuzzy geometry is a SO(5) symmetric quantum geometry that does not have its counterpart in classical geometry.

Internal fuzzy structure and the SO(4) Landau models
We discuss a physical model that realizes the matrix-valued three-sphere geometry inside the SO(5) Landau model. We also clarify relations among Landau models in different dimensions.

SU(2) meron gauge field and SO(4) Pauli-Schrödinger Hamiltonian
We first construct a physical model whose eigenstates are given by the SO(4) spinor spherical harmonics. The expression of the SO(4) part of the SO(5) free angular momentum operators are exactly equal to the SO(4) free angular momentum operators (see Appendix C): The SO(4) angular momentum L mn (11) can also be represented only in terms of the S 3 -coordinates (123): Therefore, the SO(4) analysis in Sec.2.2.1 can be rewritten entirely in the language of S 3 without resorting to any information of the original manifold S 4 . We then explore the SO(4) problem as an independent problem defined on S 3 , and just utilize the S 3 -coordinates, y m=1,2,3,4 , in this section. Interestingly, (146) can be realized as the SO(4) angular momentum operators in the meron gauge field introduced by Alfaro, Fubini and Furlan as a solution of pure Yang-Mills field equation [39,40]: where r = √ y m y m . The meron gauge field is simply obtained by the dimensional reduction of the Yang's SU (2) monopole gauge field: Notice that the Yang's monopole has the string-like singularity, while the meron only has the point-like singularity at the origin. The corresponding field strength is given by 17 and the total angular momentum operator is where With the replacement of 1 2 σ i with higher SU (2) spin matrix S i , (152) turns to the SO(4) angular momentum (146). The SO(4) Casimir is given by where J and K are the following SU (2) L and SU (2) R operators 17 The associated 2nd Chern number is evaluated leading to the name "meron". For the meron filed configuration with geneneral spin S (I/2) i , the 2nd Chern number is evaluated as Q = − 1 12 I(I + 1)(I + 2).
and the SU (2) L and SU (2) R Casimir eigenvalues are given by with Their simultaneous eigenstates are given by the SO(4) spinor spherical harmonics (28).
In the meron field background, we introduce a SO(4) Landau-like Hamiltonian As usual, (159) can be rewritten as where we used m<n Λ mn F mn = m<n F mn Λ mn = 0 and m<n F mn 2 = S 2 . (159) can also be expressed where n denotes the SO(4) Landau levels and s denotes the sub-bands in the SO(4) Landau levels [ Fig.7]. The SO(4) Landau level eigenstates are actually the SO(4) spinor spherical harmonics Y j,mj ; k,m k with (157), and so the previous three-sphere matrix geometry (134) is considered to be realized in the SO(4) Landau level. In this way, we can reformulate the SO(4) part of the SO(5) Landau model with the SO(4) Pauli-Schrödinger model. In other words, the SO(5) Landau model accommodates the SO(4) Pauli-Schrödinger model as its internal model.

Singular gauge transformation and SO(4) matrix geometry
Curiously, the energy levels (162) are exactly equal to the Landau levels of the SO(4) Landau Hamiltonian proposed by Nair and Daemi [42]. This coincidence implies a hidden relation between the SO(4) Pauli-Schrödinger model and the SO(4) Landau model. In the following, we adopt the notation of [28,32]. The SO(4) Landau Hamiltonian is given by where Λ mn = −iy m ( ∂ ∂y n + iA ND n ) + iy n ( with the Nair-Daemi SU (2) gauge field 18 Obviously, the Nair-Daemi SU (2) gauge field has a Dirac string-like singularity. The corresponding field strength is derived as The eigenvalues of the SO(4) Landau Hamiltonian (163) are given by (162) and the corresponding eigenstates, i.e., the SO(4) monopole harmonics (in the Dirac gauge), are given by [32,42] 18 The Nari-Daemi SU (2) monopole gauge field is equivalent to the spin connection of S 3 . 19 (168) constitutes an orthonormal set: 20g (θ, φ) is a gauge function that relates the SO(4) monopole harmonics in the Dirac gauge and the Schwinger gauge [32]. and Φ j,mj ;k,m k (Ω 3 ) A = (2j + 1)(2k + 1) 2π 2 (I + 1) with the Wigner's D-function With the preparation, we now discuss a relation between the SO(4) Pauli-Schrödinger model and the SO(4) Landau model. We have seen that the meron gauge field has the point-like singularity, while the Nair-Daemi's SU (2) monopole has the string-like singularity. It is well known a similar situation occurs in a lower dimension. In 3D the Wu-Yang SU (2) monopole [43] has a point-like singularity, while the Dirac monopole has the stringlike singularity. In this sense the meron is a 4D generalization of the Wu-Yang SU (2) monopole, while the Nair-Daemi SU (2) monopole is a 4D generalization of the Dirac monopole. To find a relation between the meron and the Nair-Daemi monopole gauge field, let us first recall the singular transformation that relates the Wu-Yang monopole and the Dirac monopole configurations [44,45]. With the R 3 coordinates the Wu-Yang monopole and the Dirac monopole in R 3 are respectively given by which are related by the singular transformation where 21 g(θ, φ) = e −iθ(ẑ2Sx−ẑ1Sy) = e −iφSz e iθSy e iφSz , with S 1 -latitude coordinatesẑ 1 ≡ cos φ,ẑ 2 ≡ sin φ. A bit of consideration tells that the SU (2) monopole and the meron gauge fields are also related by the following SU (2) singular transformation: where Hereg(θ, φ) is given by (169), andŷ i are the coordinates on S 2 -latitude parameterized aŝ y i=1,2,3 = (sin θ cos φ, sin θ sin φ, cos θ).
The SO(4) Pauli-Schrödinger model is transformed to the SO(4) Landau model by the singular gauge transformation (177). Indeed, one can also confirm that the SO(4) monopole harmonics and the SO(4) spinor spherical harmonics are related as Φ j,mj ; k,m k (χ, θ, φ) = (−1) Consequently, the matrix elements are related as where we used (−1) I+s ′ +s = −(−1) I+2s = −1. In Appendix D, we rigorously evaluate both sides of (181) and explicitly check its validity. Therefore, the matrix geometry of the SO(4) Pauli-Schrödinger model is exactly equal to the matrix geometry of the SO(4) Landau model, and hence the SO(4) Landau model describes the internal fuzzy geometry of the SO(5) Landau model. This demonstrates the idea of the dimensional hierarchy [29,15] relating the Landau physics in different dimensions. In Fig.8, we summarize the relations among the Landau models in various dimensions. For a better understanding of this section, Figure 8: Landau models and their background topological field configurations for the fuzzy sphere geometries. There exist singular gauge transformations and dimensional ladders connecting the Landau models.

Relativistic SO(5) Landau models
We explore relativistic version of the SO(5) Landau model and clarify relation to the matrix geometry of the Berezin-Toeplitz quantization [46].

Spinor SO(5) Landau model
We consider a relativistic spinor particle on S 4 , which feels the connection of the base-manifold S 4 as well as the external SU (2) monopole gauge field. In other words, the relativistic particle interacts with the synthetic gauge field of the SO(4) connection (187) and the SU (2) monopole field (3) 23 For the SO(4) ≃ SU (2) L ⊗ SU (2) R gauge group, the synthetic gauge field is irreducibly decomposed as ((1/2, 0) ⊕ (0, 1/2)) ⊗ (0, I/2) = (0, I/2 + 1/2) ⊕ (0, and their corresponding dimensions are The field strength is given by (Appendix B.4). The SO(5) angular momentum in the synthetic gauge field is given by 22 We choose the numbering of the vierbein as (184) so that the SO(5) Dirac-Landau operator is reduced to the SO(4) Dirac-Landau operator of [32] at ξ = π 2 (see (212)). The area of S 4 is calculated as with Λ ab being the covariant angular momentum operator We introduce the spinor SO(5) Landau Hamiltonian as The decomposition (190) implies that, with some appropriate unitary transformation, the spinor SO(5) Landau Hamiltonian is transformed as Here H (

SO(5) Dirac-Landau operator and zero-modes
The Dirac-Landau operator on S 4 , which we call the SO(5) Dirac-Landau operator, is constructed as whereD µ (µ = ξ, χ, θ, φ) are newly introduced covariant derivatives including the contribution of the spin connection: The second terms on the right-hand sides of (199) are attributed to the spin connections ω µ . We adopt the SU (2) gauge field in (200) as or more explicitly,

Dimensional reduction to the SO(4) Dirac-Landau operator
On the equator ξ = π/2, the SU (2) gauge field (202) is reduced to the SU (2) gauge field of the SO(4) Landau model [32]: (198) can be decomposed as When we take the gamma matrices (77) as 24 where −i˜ D S 3 is given by 24 The choice (208) is different from the previous one (77). We adopt (208) so that the SO(4) Dirac-Landau operator (211) coincides with the expression of [32]. 25 One may readily check that in the absence of the SU (2) monopole gauge field, (209) is reduced the free Dirac operator [47]. −i D S 3 signifies the SO(4) Dirac-Landau operator on S 3 [32]: On the equator of S 4 (ξ = π 2 ), −i˜ D S 3 is reduced to −i D S 3 , and so is the SO(5) Dirac-Landau operator: The relativistic SO(5) Landau model thus embeds the relativistic SO(4) Landau model on the equator as the non-relativistic Landau model does. The fuzzy three-sphere geometry is realized in the SO(4) relativistic Landau model [32], and then the SO(5) relativistic Landau model accommodates a fuzzy threesphere geometry as its sub-geometry on the equator, which suggests existence of the fuzzy four-sphere as a whole geometry.

Zero-modes and the matrix geometry
The square of the Dirac-Landau operator (198) and the SO(5) Casimir (193) are related as [48,27] ( Here, we used 5 a<b=1 F ab 2 = m<n (η i mn S (I/2) i ) 2 = 2 S (I/2) 2 = 2 · I 2 ( I 2 + 1) and R S 4 = d(d − 1)| d=4 = 12 (313). The square of the Dirac-Landau operator respects the SO(5) rotational symmetry and the chiral symmetry as well: Consequently, the eigenvalues of (−i D S 4 ) 2 generally have two kinds of degeneracies coming from the SO(5) rotational symmetry and the chiral symmetry. The zero-modes, however, do not have the degeneracy from the chiral symmetry, and only have the degeneracy of the SO(5) rotational symmetry. Since the square of the Dirac-Landau operator shares the same SO(5) Casimir 5 a<b=1 L ab 2 with the spinor Landau Hamiltonian (195), the eigenvalue problem of (213) is essentially equivalent to that of the spinor Landau Hamiltonian. We will focus on the case (p, q) SO(5) = (N + 2J, N ) The Atiyah-Singer index theorem also gives the same result about the number of zero-modes, −c 2 (I) = D(I − 1, N )| N =0 = 1 6 I(I + 1)(I + 2). In [46], the fuzzy four-sphere geometry was derived in the Berezin-Toeplitz method by taking matrix elements sandwiched by the Dirac-Landau operator zero-modes. 26 Since the zero-modes are identical to the non-relativistic lowest Landau level eigenstates, the matrix geometry obtained in the the non-relativistic analysis (142) exactly coincides with that of the Berezin-Toeplitz quantization.

Even higher dimensional Landau model and matrix geometry
We extend the discussions of Sec.3 to even higher dimensions and investigate the matrix geometry in the SO(2k + 1) Landau model.

Classical limit
Next, we consider the classical limit I >> 1, in which L ab (224) is reduced to The coordinates x a can be extracted from the field strength as [27] 1 where c k (I) denotes the kth Chern number for the SO(2k) gauge field: 27 With (224) of I = 1, we can show, where (227) 28 The coefficient in front of the integration of (230) is added to be accounted for by the normalization of Ψ: Substituting (234) to (235), we have Since L ab are the SO(2k + 1) operators, X a (237) also become operators. tr in (235) (and (236)) is taken for the "internal fuzzy space" S 2k−2 F with dimension [27,15] and then tr(1 internal space ) = D k−1 (I).
In the lowest Landau level, the SO(2k +1) operators can be replaced with the SO(2k +1) matrix generators in the fully symmetric irreducible representation: and so (237) becomes where in the second equation the Atiyah-Singer index theorem was used [27,48] c k (I) = −D k (I − 1).
Since the fully symmetric representation SO(2k + 1) matrices satisfy (241) finally takes a concise form

Even higher dimensional matrix geometry
The results in the two limits, (233) and (244), suggest that the matrix coordinates for general I take the form Since the SO(2k + 1) gamma matrices in the fully symmetric representation satisfy 2k+1 a=1 Γ a Γ a = I(I + 2k)1 D k (I) , X a (245) satisfy the quantum Nambu geometry of the fuzzy 2k-sphere [23,24]: with The quantum Nambu geometry thus naturally emerges as the lowest Landau level matrix geometry of the SO(2k + 1) Landau model. The matrix geometry (245) will also be obtained by the Berezin-Toeplitz quantization, since the zero-modes of the Dirac-Landau operator are equal to the lowest Landau level eigenstates [27] and the Atiyah-Singer theorem also hold in arbitrary even dimension. Further, when we take into account the low dimensional results including odd dimensions [31,32,29] S 2 F of SO(3) Landau model : (245) may be generalized to for the SO(d + 1) Landau model.

Summary
In this work, we performed a comprehensive study of the SO(5) Landau models and their matrix geometries. With SO(5) monopole harmonics in a full form, we completely derived the matrix coordinates of four-sphere in an arbitrary Landau level. In the lowest Landau level, the matrix geometry is given by the generalized SO(5) gamma matrices realizing the quantum Nambu geometry. We showed that the matrix geometry obtained by the Landau level projection actually interpolates the matrix geometries of the quantum limit and the classical limit. In higher Landau level, the matrix geometry exhibits a nested fuzzy structure. The N th SO(5) Landau level accommodates N + 1 inner SO(4) Landau levels each of which realizes quasi-fuzzy four-sphere geometry. As a whole, there are N + 1 quasi-fuzzy four-spheres constituting a N + 1 concentric nested structure with SO(5) symmetry. The nested matrix geometry denotes a pure quantum geometry having no counterpart in classical geometry. We introduced a SO(4) Pauli-Schrödinger model with meron gauge field background that realizes the inner SO(4) part of the SO(5) Landau model. We established a singular gauge transformation between the SO(4) Pauli-Schrödinger model and the SO(4) Landau model, and demonstrated the internal fuzzy geometry is identical to the SO(4) Landau model matrix geometry. Explicit relations among other low dimensional Landau models with fuzzy geometries were summarized too. We also analyzed the relativistic SO(5) Landau models and clarified relation to the matrix geometry of the Berezin-Toeplitz quantization. Finally, we investigated even higher dimensional Landau model and discussed the associated quantum Nambu geometry in an arbitrary dimension.
The SO(5) Landau model and four-dimensional quantum Hall effect have opened a window to a research field of topological phases in higher dimension. This is not just rendered to be a theoretical issue. Recent technologies of quantum photonics in ultra cold atom have made experimental explorations possible with the idea of synthetic dimension [49]. The present analysis will be useful not only for the non-commutative geometry but also for the practical analysis of higher dimensional topological phases such as quantum Hall effect and Weyl semi-metal [50,51]. Acknowledgement I would like to thank Goro Ishiki for useful discussions and inviting me to Tsukuba University for a seminar. This work was supported by JSPS KAKENHI Grant Number 16K05334 and 16K05138.
The SO(5) irreducible representation is decomposed to the SO(4) irreducible representation as where For other examples of the irreducible decomposition of the tensor product of SO(5) ≃ U Sp (4), one may consult [37] for instance.

B The Dirac gauge and the Schwinger gauge for S 4
We introduce the Dirac gauge and the Schwinger gauge for S 4 and derive a gauge transformation between them.
It is straightforward to check that (290) satisfy the Cartan structure equation: with the vierbein in the Schwinger gauge 31 e S 1 = sin ξ sin χdθ, e S 2 = sin ξ sin χ sin θdφ, e S 3 = sin ξdχ, e S 4 = dξ.
Similarly, we have with and

B.3 Gauge transformation and vierbein in the Dirac gauge
From the relation (280), we have and so (273) and (295) are related as or We then find that the SO(4) matrix-valued spin connections 31 The numbering of the vierbein here (292) is different from that of (184).
It is straightforward to show that (276) and (306) satisfy the Cartan structure equation: We thus successfully obtained the vierbein in the Dirac gauge from the relation (305). On the other hand, it may be a formidable task to derive the vierbein in the Dirac gauge from the Cartan structure equation (307) with the spin connection (276).

B.4 Curvature
With (300), the curvature is readily obtained as f D/S mn = e D/S m ∧ e D/S n .