Generating Quantum Matrix Geometry from Gauged Quantum Mechanics

Quantum matrix geometry is the underlying geometry of M(atrix) theory. Expanding upon the idea of level projection, we propose a quantum-oriented non-commutative scheme for generating the matrix geometry of the coset space $G/H$. We employ this novel scheme to unveil unexplored matrix geometries by utilizing gauged quantum mechanics on higher dimensional spheres. The resultant matrix geometries manifest as $\it{pure}$ quantum Nambu geometries: Their non-commutative structures elude capture through the conventional commutator formalism of Lie algebra, necessitating the introduction of the quantum Nambu algebra. This matrix geometry embodies a one-dimension-lower quantum internal geometry featuring nested fuzzy structures. While the continuum limit of this quantum geometry is represented by overlapping classical manifolds, their fuzzification cannot reproduce the original quantum geometry. We demonstrate how these quantum Nambu geometries give rise to novel solutions in Yang-Mills matrix models, exhibiting distinct physical properties from the known fuzzy sphere solutions.


Introduction
It has been almost eighty years since the inception of theoretical research on quantized space-time with Snyder's first explicit model [1,2]. This research field continues to be active, contributing to a deeper understanding of space-time. Non-commutative geometry presents a promising mathematical framework for describing the microscopic nature of space-time [3]. A general mathematical framework of non-commutative geometry was set up by Connes [4]. More tangible non-commutative schemes are those such as deformation quantization, geometric quantization and Berezin-Toeplitz quantization [5]. As these ideas are rooted in the canonical quantization method of the phase space [6,7], the corresponding non-commutative schemes are concerned with the quantization of the symplectic manifolds or Poisson manifolds. However, in the investigations of M theory, physicists encountered even exotic noncommutative structures beyond the conventional quantization schemes, including odd dimensional fuzzy spheres [8,9,10,11]. From M(atrix) theory point of view [12,13], matrix geometries known as fuzzy manifolds [14,15,16,17,18,19,20,21,22,23,24,25,26] represent fundamental extended objects in the theory [27,28]. Moreover, it has been recognized that the quantum Nambu algebra [29] plays crucial roles in the formulation of M theory (see Refs. [30,31,32] as nice reviews and references therein). It may be evident that a new non-commutative scheme is required to address these extraordinary non-commutative spaces that extend beyond the conventional quantization methods based on the commutator formalism. 1 Associated with the developments of the higher-dimensional quantum Hall effect, the understanding of higher-dimensional non-commutative geometry has significantly advanced in the past twenty years (see [39,40] and references therein). We have learned that the higher dimensional non-commutative geometry on M ≃ G/H can be obtained by examining the Landau model on M in the non-Abelian monopole background [41,42,43,44,45,46,47,48,49,50,51,52]. Specifically, within the lowest Landau level, fuzzy manifolds M F were successfully realized. Nonetheless, it should be noticed that the underlying reason for the success is still missing. Furthermore, while the preceding analysis has provided a nice physical understanding of non-commutative geometries, one could argue that these analyses have not revealed unknown matrix geometries. Until now, substantial attention has been given to the geometry in the lowest Landau level; however, there is no logical reason for the exclusive presence of non-commutative geometry solely in this level. Indeed, it was demonstrated that the higher Landau levels also give rise to fuzzy geometries [53], which clearly shows that level projection to any Landau level generates non-commutativity. With regards to a two-sphere, the emergent non-commutative geometries of the higher Landau levels are the same as that of the lowest Landau level. In this sense, the geometry of higher Landau levels might not be so intriguing. Nevertheless, this does not rule out the possibility of discovering new non-commutative geometries in higher dimensional systems. Following this idea, explorations of novel quantum matrix geometries have been conducted in various Landau models, such as relativistic models and supersymmetric models [53], odd dimensional models [54] and even dimensional models [55,56,57]. It is also worthwhile to mention that quantum matrix geometries associated with the Berezin-Toeplitz quantization have been intensively studied in recent years [58,59,60,61,62,63,64].
Importantly, now the higher dimensional studies are not only relevant to theoretical interests but also to practical experiments. The idea of the synthetic dimension allows physicists to reach higher dimensional topological physics [65,66,67]. In particular, exotic topological effects of the non-Abelian monopole in higher dimension have already been observed through table top experiments very recently [68,69,70,71]. It is expected that physical consequences arising from higher dimensional quantum geometry will be observed in these experimental systems.
In the present work, with an appropriate interpretation of the emergent non-commutative geometry in the Landau models, we introduce a quantum-oriented non-commutative scheme that leverages Landau models as an effective "tool" to generate noble quantum geometries. Our approach provides a concrete prescription for generating the matrix geometry of the coset manifold M ≃ G/H. It is shown that this scheme encompasses pure quantum Nambu matrix geometry, which cannot be described by conventional non-commutative methods. We also demonstrate that these quantum Nambu matrix geometries give rise to novel classical solutions in Yang-Mills matrix models. This paper is organized as follows. In Sec.2, we revisit the derivation of the fuzzy two-sphere from the SO(3) Landau model and address the underlying reasons behind the emergent non-commutative geometry of the Landau models. Sec.3 presents explicit fuzzy four-sphere matrix coordinates in the SO(5) Landau levels. We investigate the matrix structures of fuzzy four-spheres and discuss their basic properties in Sec.4. In Sec.5, the nested internal structures of higher Landau level matrix geometries are exploited. We investigate the continuum limit and the classical geometry of the quantum matrix geometry using the coherent method and the probe brane method in Sec. 6. In Sec.7, we demonstrate that the obtained quantum matrix geometries realize unexplored solutions of Yang-Mills matrix models and clarify their physical properties. Sec.9 is devoted to summary and discussions.

Quantum-oriented non-commutative scheme
In this section, we discuss the underlying mechanism behind the emergent matrix geometry in the simple SO(3) Landau model and apply this observation to propose a prescription for generating matrix geometries of G/H.

Behind the scene of the emergent matrix geometry
The SO(3) Landau model Hamiltonian is given by where A i denotes the U (1) gauge field of monopole at the origin: The index I/2 signifies the monopole charge (In the following, we assume I to be a positive integer for simplicity). While the present system is originally investigated in [72,73], we will utilize the concise notation of [53] in this paper. The eigenvalues of the Hamiltonian (1) are obtained as and the corresponding eigenstates are where D denotes the Wigner D-function: Here, S (l) i stand for the SU (2) spin matrices with spin index l. We sandwich the coordinates on S 2 to derive the corresponding matrix coordinates: In the N th Landau level, X [N ] i are explicitly obtained as [53] X (N ) i = 2I (I + 2N )(I + 2N + 2) S which satisfy Equation (8) represents the algebra of fuzzy two-sphere [14]. Note that not only the lowest Landau level but also each of the higher Landau level matrix geometries realizes the fuzzy two-sphere matrix geometry. 2 The physical properties of (8) as a classical solution of Yang-Mills matrix models are discussed in Appendix B. We depicted the fuzzy two-sphere and the probability magnitudes of the monopole harmonics in Fig.1. One may find an apparent resemblance between the two pictures. The latitudes on the fuzzy two-sphere represent the degrees of freedom of the matrix geometry, i.e., the "points", in the fuzzy space. Obviously, each point on the fuzzy space corresponds to the monopole harmonics or each state of the SU (2) irreducible representation. Therefore, one may consider the fuzzy two-sphere to be composed of the SU (2) irreducible representation.
Reflecting the emergence of the non-commutative geometry, we can obtain the following insight.
1. About the role of global symmetry and irreducible representation: (2) is naturally transformed to the SU (2) symmetry on the matrix geometry side introducing the projective representation of SO(3). In the matrix geometry, an "uncertainty area" or a "point" corresponds to each state of the SU (2) irreducible representation. The irreducible representation is "symmetric" in the sense that, while each state of an irreducible representation is transformed, the set of states in the irreducible representation remains unchanged under any SU (2) transformation. In the language of matrix geometry, this means that fuzzy geometry also remains unchanged under SU (2) transformations, as the fuzzy two-sphere is composed of the states in the SU (2) irreducible representation. Moreover, the SU (2) group is a compact group, and its irreducible representation is a finite-dimensional set with discrete quantum numbers, which aligns with the intuitive notion that a compact non-commutative space consists of finite-dimensional discrete points. In this way, while the fuzzy sphere is a discretized space, it realizes a space symmetric under continuous SU (2) transformations, unlike the lattice space, which is symmetric only by the discrete translations corresponding to the lattice spacing. This is the specific feature of the matrix geometry composed of the irreducible representation.
2. About the role of the stabilizer group and the gauge symmetry: The stabilizer group SO(2) of S 2 ≃ SO(3)/SO(2) is a subgroup of SO(3) that does not change a point on the classical manifold S 2 [74]. A point in the classical geometry corresponds to a state of the irreducible representation on the matrix geometry side. Therefore, the stabilizer group is considered to be some transformation that does not change that state. The transformation that does not change physical state is nothing but a gauge transformation. To encapsulate, the stabilizer group represents redundant symmetry of the SO(3) group in the classical system when representing S 2 ≃ SO(3)/SO (2), and such redundancy is naturally regarded as a gauge symmetry on the quantum mechanical side. Consequently, the stabilizer group SO(2) corresponds to the U (1) ≃ SO(2) symmetry on the quantum mechanical side. It is interesting to see that while the stabilizer symmetry is an external symmetry on the classical mechanical side, it acts as the internal symmetry on the quantum mechanical side. 3 3. Reinterpretation of the Landau model: The above observations suggest that the matrix geometry corresponding to S 2 ≃ SO(3)/SO(2) is obtained by considering a quantum system with global SU (2) symmetry and U (1) gauge symmetry.
As we are dealing with the spatial manifold, the U (1) gauge symmetry introduces the U (1) vector potential whose field configuration should be compatible with the SU (2) global symmetry. This necessarily leads to the radially symmetric magnetic field of the U (1) monopole. Thus, the magnetic field is just a consequence of the gauge symmetry. In this way, we can reproduce the original SO(3) Landau system. It is important to note that the primary significance lies in the gauge symmetry itself rather than the magnetic field, although the presence of a magnetic field is commonly believed to be essential for the emergence of non-commutative geometry.
These speculations provide a natural explanation for why the fuzzy two-sphere geometry has been successfully generated through the analyses of the SO(3) Landau model.

Non-commutative scheme for generating the matrix geometry
With the above understanding, we now propose a prescription for obtaining the matrix geometry of the general coset manifold, M ≃ G/H. We will utilize the quantum mechanics as a tool for generating matrix geometries. What we need to do is simply replace the SO(3) in the above discussions with G and SO(2) with H. 4

General prescription
1. Consider quantum mechanics with gauge symmetry H on base-manifold M: 3 This suggests that the external space and the internal space should be treated on the same footing in the matrix geometry [79]. 4 While we will assume that G is a compact group with finite dimensional irreducible representations, our discussions can also be applied to non-compact groups with discrete series of infinite dimensional irreducible representations [26].
where D a = ∂ a + iA a are covariant derivatives with the gauge field A a of the gauge group H. The gauge field configuration has to be chosen to be compatible with the symmetry G of the base-manifold M.
2. Solve the eigenvalue problem of the Hamiltonian (9) to derive the degenerate eigenstates of each energy level E N : The set of degenerate eigenstates constitute an irreducible representation of G. 5 3. Derive the matrix elements of x a utilizing (10) to construct the matrix coordinates of M (N ) Notice each energy level N hosts its own matrix geometry X

[N ]
a , and distinct energy levels yield different quantum matrix coordinates in general. Consequently, multiple quantum geometries will be obtained from a single classical manifold. The flow of this procedure is depicted in Fig.2.

Advantages
Here, we will outline the advantages of the present construction.
1. The first merit is that we do need to worry about mathematical inconsistency. In the present scheme, non-commutative geometry is not postulated a priori but is what emerges in each of the energy levels.
As the original quantum system is totally physical and the existence of mathematically consistent Hilbert space behind the quantum mechanics is founded, there is no need to be concerned about mathematical inconsistencies. 6 2. Following the above simple prescription, we can mechanically derive matrix geometries for arbitrary classical manifolds of the type M ≃ G/H. Notably, odd dimensional manifolds are also within the realm of this scheme. Therefore, this scheme is not restricted to the symplectic manifolds unlike the conventional quantization methods. This suggests that the present scheme is beyond the noncommutative geometry based on the canonical commutator formalism.
3. The present non-commutative scheme is primarily based on irreducible representations of quantum mechanics. In this sense, this may be referred to as a quantum-oriented scheme. The emergent matrix geometries may even encompass pure quantum geometries that do not have their classical counterparts. We may explore quantum geometries that have eluded in the conventional non-commutative schemes.

General Properties
To examine specific properties of the present scheme, let us consider even dimensional spheres,

Covariance
We assume that the global symmetry SO(2k + 1) of S 2k is given by The stabilizer group is defined so that the condition x a = δ a,2k+1 does not change, which is the SO(2k) transformation: Transformations (13) and (14) respectively correspond to the following transformations on the quantum mechanics side: and Equation (15) stands for the global transformation, and α denote the index of the irreducible representation of the Spin(2k + 1). Similarly, Eq.(16) represents the gauge transformation, and i signify that of the gauge group Spin(2k). Under these transformations, X a behave as and The matrix coordinates thus transform as the SO(2k + 1) vector, similar to the classical coordinates on S 2k , and they are gauge invariant. Generally for M ≃ G/H, the matrix coordinates are H gauge invariant and transform under G in the same way as the classical coordinates of the original manifold M.

Beyond the commutator formalism
In the well known construction of the fuzzy 2k-sphere [16,15], the matrix coordinates are given by the totally symmetric combination of the gamma matrices , which satisfy the following commutation relations The commutators of X a yield new matrices Σ ab (19a), which are the generators of SO(5). In total, X a and Σ ab together form the SO(2k + 2) algebra. Such a matrix geometry is known to emerge in the lowest Landau level of the SO(2k + 1) Landau model [79]. The lowest Landau level matrix geometry is well described by the commutator formalism. On the other hand, for the higher Landau levels, some subtleties occur. The SO(2k + 1) angular momentum operators in the SO(2k) monopole background are constructed as [44] which satisfy the SO(2k + 1) algebra: Since the coordinates x a on S 2k transform as an SO(2k + 1) vector, the algebra associated with the SO(2k + 1) transformation is represented as Let us construct matrix coordinates for a given irreducible representation of SO(2k+1), {ψ It is important to note that the completeness relation holds for the total set of the irreducible representations: but not for each individual irreducible representation: This is the origin of the non-commutative algebra of the matrix coordinates., whereas the algebra of the original classical coordinates is commutative. From the property of the irreducible representation one may easily reproduce the lower two equations of (19) using Eqs. (21) and (22). On the other hand, unlike Eq.(26), the matrix coordinates are not completely block diagonalized, ψ β δ rr ′ (see Sec.3.1 for more details). Consequently, the first relation (19a) turns out to be questionable, Equation (19a) is not guaranteed in general. So, if the Lie algebraic geometry fails, what kind of geometry will emerge? That is the topic that we shall discuss in Sec.4 and Sec.8. The failure of Eq.(19a) implies that the present scheme is beyond the realm of the conventional commutator formalism.
Here, we also mention relationship to the Berezin-Toeplitz quantization. The Berezin-Toeplitz quantization is a method that maps a function to a finite dimensional matrix [76,62,5]. In this sense, the Berezin-Toeplitz quantization shares the same spirit with the present scheme. However, Berezin-Toeplitz quantization is primarily concerned with symplectic manifolds and is based on commutator formalism. The Kernel employed in the Berezin-Toeplitz quantization corresponds to the zero-modes of the Dirac-Landau operator whose zero-modes are essentially equivalent to the lowest Landau level eigenstates [55,53]. Therefore, the Berezin-Toeplitz quantization is thus closely related to the lowest Landau level matrix geometry and can be viewed as a special case of the present scheme. 7 We will revisit this in Sec.4.

Matrix coordinates from the SO(5) Landau model
In this section, we will directly apply the present scheme to generate quantum matrix geometries for S 4 . Using the SO(5) Landau model, we will derive the complete form of matrix coordinates in arbitrary Landau levels. This section also includes a review of Ref. [55].

The SO(5) Landau model
Since S 4 ≃ SO(5)/SO(4), we need to consider a quantum mechanics on S 4 with Spin(4) gauge degrees of freedom. For the Spin(4) gauge field configuration to respect the SO(5) global symmetry of S 4 , we place a Spin(4) monopole at the origin. While the Landau model in such a Spin(4) monopole background has been investigated [57], we will consider a simpler system by taking one SU (2) from the Spin(4) ≃ SU (2)⊗SU (2). In the following, we then consider a quantum mechanics on S 4 in the SU (2) monopole background, which was originally introduced in Refs. [77,78,41].
Let us briefly discuss such a SO(5) Landau model with a modern notation [55]. The SO(5) Landau Hamiltonian is given by where D a = ∂ a + iA a and The gauge field is chosen to be Yang's SU (2) monopole: withη i µν = ǫ µνi4 − δ µi δ ν4 + δ νi δ µ4 . The SO(5) Landau Hamiltonian is equal to the SO(5) Casimir up to a constant. Consequently, the energy eigenvalues are specified by two indices of the SO(5) Casimir, (p, q) 5 = (N + I, N ) 5 . The SO(5) Landau levels are explicitly given by The eigenstates of each of the Landau levels form an irreducible representation of SO (5) and are referred to as the SO(5) monopole harmonics in this paper 8 to emphasize its SO(5) covariance. We parameterize the coordinates of the four-sphere with a unit radius as where ξ represents the azimuthal angle and y m denote the coordinates of (normalized) S 3 -hyper-latitude: 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.
Consequently, the N th Landau level degeneracy is counted as

Matrix coordinates
The matrix coordinates have non-zero components only within the same Landau level and among adjacent Landau levels [55]: See the left of Fig.4 where the non-zero matrix elements are denoted as the shaded color regions. Under the SO(4) rotation around the fifth axis, x 5 behaves as a scalar (j, k) = (0, 0), while x µ transform as a bi-spinor (j, k) = (1/2, 1/2). From (40), we can see that the SO(4) selection rule implies that non-zero matrix coordinates exist only for take finite values. The first is (∆n, ∆s) = (±1, 0) that corresponds to the green shaded rectangles in Fig.4, representing transitions between two adjacent SO(4) lines in Fig.3. The second (∆n, ∆s) = (0, ±1) corresponds to the small purple shaded rectangles in Fig.4, signifying transitions between two adjacent SO(4) irreducible representations on same SO(4) lines in Fig.3. With this in mind, we will explicitly evaluate the matrix elements of x a . We can perform integrations of the azimuthal part and the S 3 -hyper-latitude part separately. For instance, the ortho-normal condition (36) is evaluated as where The matrix elements of X where The matrix coordinate (50) takes equally spaced discrete values specified by s = I/2, I/2 − 1, · · · , −I/2, which are regarded as the hyper-latitudes on fuzzy four-sphere. This structure is quite similar to that of the fuzzy two-sphere ( Fig.1). However notice that while the latitudes of fuzzy two-sphere are not degenerate, the hyper-latitudes of fuzzy four-sphere are degenerate, resulting in an intriguing internal structure as we shall discuss in Sec.5. Next, we turn to Here, the azimuthal part is evaluated as 9 The minus sign in (49) is not essential but added for later convenience. 10 In the derivation of (52), we used the formulas, where C The formulas of Appendix D in [55] were utilized in the derivation of Eq. (55). We thus derived the explicit form of the matrix coordinates in the SO(5) Landau levels. For a better understanding, in Appendix C, we provide the matrix coordinates for the case of (N, I) = (1, 1). Note that all of the quantities involved in the matrix coordinate calculations, such as an integral measure and S 4 coordinates, are SO(5) invariant or covariant quantities. Consequently, the obtained matrix coordinates are necessarily SO(5) covariant coordinates that transform as the SO(5) vector like the original S 4 coordinates (see Eq. (85)).
In the case of I = 0, the gauge symmetry no longer exists. Therefore, we cannot expect fuzzy geometries (recall that the gauge symmetry is crucial in the present scheme). Indeed, when I = 0, the energy eigenstates are given by the SO(5) spherical harmonics and the matrix coordinates become trivial: The corresponding dimensions of the SO(5) spherical harmonics are Therefore in these matrix dimensions, the matrix geometries do not exist. In Ref. [16], the authors argued the non-existence of five-dimensional matrix coordinates, which corresponds to the smallest dimension in Eq.(58).

Pure quantum Nambu matrix geometry
Using the explicit matrix coordinates, we now expand concrete discussions about the matrix geometries. It is shown that the matrix coordinates satisfy and where the quantum Nambu bracket denotes the totally antisymmetric combination of the four-quantities inside the bracket: (Detail discussions about the coefficients, c 1 and c 3 , will be given in Sec.4.2.) Equations (59) and (60) signify a realization of the fuzzy four-sphere [16,15]. The quantum Nambu geometry thus emerges as the matrix geometry in the SO(5) Landau levels. We delve into geometric structures hidden in the mathematics of the quantum Nambu algebra using the explicit form of the matrix coordinates.

The lowest Landau level matrix geometry
For N = 0, Eqs. (49) and (51) reproduce the lowest Landau level matrix coordinates previously obtained in [55,60]: where Γ a represent the I-fold symmetric tensor product of the SO(5) gamma matrices [16]: with We can readily check that a satisfy (59) and (60), and The radius and the non-commutative scale are derived as which implies that the ordinary four-sphere with a unit radius is reproduced in the continuum limit (Fig.5).
It should be emphasized that the algebra of X a can be described within the commutator formalism, and the quantum Nambu algebra (66) is not indispensable for the description of the lowest Landau level matrix geometry. The matrix coordinates X which is the SU (4) [20]. The quantum Nambu algebra (66) is not exactly equivalent with the SU (4) algebra (68), however, they have been treated almost synonymously thus far. This is because the known matrix realization of the fuzzy four-sphere was only the fully symmetric representation that satisfies both (66) and (68). The closed algebra (68) suggests that the natural symmetry in the lowest Landau level is the Figure 5: Fuzzy four-sphere in the lowest Landau level (the left) and its continuum limit (the right).
SU (4) rather than the original SO (5). This becomes clearer in the following discussion. The symmetric representation can be simply realized using the Schwinger boson operators. 11 : The boson number indicates the SU (2) index of Yang's monopole : One may readily check that (69) satisfy the SU (4) algebra (68) together with X The fuzzy manifold constructed from the SU (4) matrices of the SU (4) fully symmetric representation is referred to as the fuzzy CP 3 [19]. Note that the dimension of the SO(5) lowest Landau level (p, q) 5 = (I, 0) 5 , is exactly equal to that of the SU (4) fully symmetric representation: Therefore, the fuzzy S 4 is equivalent to the fuzzy CP 3 (see Ref. [81] for discussions including matrix functions on them). The commutation relations of the Schwinger boson operators correspond to the canonical quantization of the homogeneous coordinates of the symplectic manifold CP 3 . Therefore, it may be reasonable that the lowest Landau level geometry can be described within the conventional commutator formalism of the Lie algebra. The corresponding continuum limit is CP 3 , which is the coset Here, we encounter the SU (4) symmetry again. It is also worth noting that CP 3 is locally equivalent to While the original S 4 itself is not a symplectic manifold, the S 2 -fibre twisted on S 4 makes the entire bundle symplectic. 11 Historically, the Schwinger boson operators were utilized in the first construction of the fuzzy four-sphere [15]

Higher Landau level matrix geometry
From (49), we have where .
Since all of X which implies Eq. (59): One can explicitly check the validity of Eq.(78) using Eqs. (49) and (51). The radius of the fuzzy four-sphere is given by Since the matrix coordinates have two parameters, N and I, there are two different infinity limits of the radius: Equation (80a) signifies the usual commutative limit in which the fuzzy four-sphere is reduced to the continuum four-sphere with a unit radius. On the other hand, Eq.(80b) indicates the collapse of the fuzzy four-spheres at N → ∞. We will revisit this in Sec.4.3. It is demonstrated that X [N ] a satisfy the quantum Nambu algebra (60): where For instance, tr(X 5 ) = 2896 7503125 , 217 124416 , 856 5250987 for (N, I) = (1, 1), (1, 2), (2, 1). 12 The matrix coordinates of the higher Landau levels not only satisfy the quantum Nambu algebra (81) but also encompass all possible matrix realizations of that algebra, because the higher Landau level matrix geometries encompass all possible irreducible representations of SO(5). 12 In the lowest Landau level, we have and Eq. (81) reproduces Eq. (66).
It is also easy to see where Σ ab denote the SO(5) generators in the (N + I, I) 5 representation. Equation (84) is consistent with the general discussions in Sec.2.3.2. While the commutators of X a do not give rise to the SO(5) generators, X a themselves transform as an SO (5) vector: The higher Landau level geometry is thus the one that adheres to the quantum Nambu algebra but not the SU (4) algebra in contrast to the lowest Landau level matrix geometry. Let us recall again that the present scheme is beyond the conventional commutator formalism. The quantum geometry in the higher Landau levels is thus qualitatively different to that of the lowest Landau level. The algebraic structure of the higher Landau level geometry is apparent only after introducing the quantum Nambu bracket and cannot be captured by the ordinary commutator formalism. In this sense, the higher Landau level geometry is considered to be a pure quantum Nambu geometry.

Nested fuzzy four-sphere
Let us delve into the matrix structure of X The diagonal blocks in X are denoted as X (n) µ (the lower right in Fig.6), which signify the matrix coordinates on the SO(4) line (n). We will delve into the matrix structure of X (n) a that represents the fuzzy geometry on the SO(4) line (n). The sum of the squares of X (n) a is given by where with B(j, k) defined by (103). Thus, is not proportional to unit matrix 1 d(n,I) (except for the special case I = 1) 13 , and so X (n) a does not give rise to a complete fuzzy four-sphere geometry but provides a fuzzy four-sphere-like structure referred to as the quasi-fuzzy four-sphere [55]. The I + 1 diagonal blocks on the most right-hand side of Eq.(88) signify the I + 1 fuzzy hyper hyper-latitudes on the quasi-fuzzy four-sphere. Inside the matrix coordinates X The nested geometry of the N + 1 quasi-fuzzy four-spheres is depicted in Fig.7. One should not confuse the present geometry with the nested structure made of a completely reducible representation [82]: In the case of the completely reducible representation, the nested fuzzy structure originates from the direct sum of the irreducible representations, while in the present, the nested fuzzy four-sphere is constituted from a single SO(5) irreducible representation and each of the quasi-fuzzy four-spheres is not made of an SO(5) irreducible representation (but rather consists of the SO(4) representations on the SO(4) line). 14 Consequently, each quasi-fuzzy four-sphere is not regarded as an SO(5)-symmetric object. This is also evident from the right-hand side of (88), which is apparently not SO(5) invariant. The quasi-fuzzy fourspheres along with their interactions collectively form an SO(5)-symmetric fuzzy manifold. We would like to draw the analogy to benzene. Each Kekulé structure only respects the C 3 rotational symmetry, while 13 For I = 1, we have only two hyper-latitudes with the same radius, and 5 a=1 X  (N + 2) 2 (n + 2)(n + 1) quantum mechanical superposition of two Kekulé structures results in benzene, which exhibits higher C 6 symmetry. Such a structure cannot be comprehended without quantum mechanics, and benzene realizes a purely quantum mechanical structure with no classical counterpart. In a similar sense, the nested fuzzy four-sphere can be considered a pure quantum geometry. This stems from the present quantum-oriented scheme, which can encompass pure quantum geometries. The non-commutative scale differs in each of the quasi-fuzzy four-spheres (87): and the "radius" of the quasi-fuzzy four-sphere is estimated as The outer quasi-fuzzy four-spheres have wider non-commutative scales (see Fig.7). It can be confirmed that the outermost quasi-fuzzy four-sphere of n = N (92) exhibits the same behavior as the nested fuzzy four-sphere (79), as anticipated. We now provide an intuitive explanation for the previous result of the two limits (80). In the commutative limit I → ∞, while ∆X (n) ∼ 2 I (91) is reduced to zero, the number of the hyper-latitudes I goes to infinity. These two contributions are compensated to realize a continuum four-sphere with a unit radius, which simultaneous implies that all of the N + 1 quasi-fuzzy four-spheres are reduced to the single four-sphere. 15 On the other hand, in the limit N → ∞, while the number of hyper-latitudes remains unchanged, the non-commutative length (91) ∆X (n) ∼ 1 N converges to zero. This leads to the collapse of the very nested fuzzy four-sphere, R (n) → 0 (Fig.8). and the radii (the lower left), the continuum limit I → ∞ (the upper right) and the N → ∞ limit (the lower right). 15 In the commutative limit I → ∞, each point on the four-sphere is highly degenerate. Because of the SO(5) symmetry, we can count this degeneracy, for instance, at the north pole X

Internal matrix geometry
Fuzzy three-sphere geometry can be realized as a sub-manifold of the (unnested) fuzzy four-sphere. Here, we explore the generalization of this concept for the nested fuzzy four-spheres.

Fuzzy hyper-latitudes
The quasi-fuzzy four-sphere is constituted from the SO(4) irreducible representations on the SO(4) line (n). The matrix coordinates of the hyper-latitudes on the quasi-fuzzy four-sphere are readily derived from Eq. (55): which denotes a d(n, I) × d(n, I) matrix. 16 The sum of the squares of Y (n) µ is given by Note A( n 2 + I 2 , n 2 ) = 0. Equation (97) represents a block diagonal matrix, with diagonal blocks indicating the hyper-latitudes of the radius R (n,s) Y . At I → ∞ and |s| << I, we have 16 The matrix Y (n) µ has the same matrix form as X (n) µ (the lower right of Fig.6): and A(j, k) and A(k, j) in (98) are given by Around s ∼ 0, the radii of the hyper-latitudes converge to unity, as anticipated from 4 µ=1 y µ y µ = 1. The hyper-latitudes for s as the vertical axis are depicted in the left of Fig. 9. We also evaluate the radii of the hyper-latitudes within the quasi-fuzzy four-sphere. Using Eq.(51), we can derive where R with B(j, k) ≡ 4 (N + For the distributions of R (n, I 2 ) X , see the right of Fig.9. Obviously, the distribution of points of the same color forms a quasi-fuzzy four-sphere. (The distribution of R (n,s) X is illustrated in Fig. 7 with X 5 as the vertical axis.) At I → ∞ and |s| << I, we have When I is even, R .
This quantity does not depend on n, indicating that the equators of all the quasi-fuzzy four-spheres have the same radius, which is identical to the radius of the fuzzy S 4 (79) (see Fig.7 also).

Fuzzy three-sphere
The fuzzy three-sphere is naturally embedded within the geometry of the fuzzy four-sphere [18,21,23]. This sub-space is composed of SO(4) representations with s = 1/2 ⊕ −1/2. In the case of the usual (un-nested) fuzzy four-sphere, there exists only one fuzzy three-sphere around the equator of the fuzzy four-sphere. In contrast, the nested fuzzy four-sphere consists of multiple quasi-fuzzy four-spheres, each of which accommodates a fuzzy three-sphere. Consequently, the N th Landau level fuzzy four-sphere hosts N + 1 fuzzy three spheres around its equator. To extract the fuzzy three-sphere geometry, we focus on the s = 1/2 ⊕ −1/2 sub-space of the matrix coordinates Y (n) µ . For odd integer I, we can derive the fuzzy three-sphere matrix coordinates (see Fig.10) which satisfy Unlike the sum of the squares of Y where the "three bracket" with Equations (107) and (109) clearly show that Y (n) µ realize the matrix coordinates of the fuzzy three-sphere. 17 Fig.11 illustrates the behaviors of the matrix sizes and the radii (108) of fuzzy three-spheres. The qualitative 17 With Y (n) 5 ≡ I + 1 2 1 2(2n + I + 3)(2n + I + 1) features of these quantities are similar to those of the fuzzy four-sphere (Fig.8) as the fuzzy three-spheres being embedded in of the fuzzy four-sphere. Note that the radius of the fuzzy three-sphere is not equal to that of the fuzzy hyper-latitude of the same s (98), R  We also explain how the fuzzy three-sphere itself is obtained within the present non-commutative framework, without referring to the geometry of the fuzzy four-sphere. Since S 3 can be identified with SO(4)/SO (3), the stabilizer group SO(3) is interpreted as the SU (2) gauge symmetry on the quantum mechanics side. Then, we consider an SU (2) gauged quantum mechanics on S 3 , known as the SO(4) Landau model [54,50,46]. In this model, n represents the Landau level index, and s signifies the subband index. The SO(4) Landau model exhibits degeneracy due to the presence of the left-right Z 2 symmetry, in addition to the global SO(4) symmetry. The fuzzy three-sphere geometry Y (n) µ emerges in the lowest energy sub-bands with indices s = 1/2, −1/2, for arbitrary nth Landau level. The degenerate energy eigenstates that constitute the fuzzy three-sphere consist of the direct sum of irreducible representations of the global symmetry SO(4), which is an irreducible representation of the entire symmetry group SO(4) ⊗ Z 2 .

Continuum limit and the S 4 geometry
We discuss the continuum limit and the classical geometry of the nested fuzzy four-sphere. While the continuum limit of the fuzzy two-sphere is the usual classical two-sphere, this is not generally the case for other fuzzy manifolds. For instance, the continuum limit of the unnested fuzzy 2k-sphere yields the symplectic manifold SO(2k + 1)/U (k) [20], which is obviously distinct from S 2k .

The 2nd Hopf map
The Hopf maps are a key to bridge non-commutative geometry and classical geometry [39]. The second Hopf map Y µ=1,2,3,4 satisfy the orthonormal condition: Equation (109) is realized as a special case of the four-algebra, d ] = −2 √ 2 (I + 1) 3 2n + I + 2 (2n + I + 3)(2n + I + 1) provides a clear understanding of the fuzzy four-sphere geometry. The fuzzification is simply executed by replacing the components of the Hopf spinor with four annihilation operators: The "quantized" Hopf map is now given bŷ which satisfy Notice that X a (118) coincide with the lowest Landau level coordinate operators (69). The total manifold S 7 represents the classical manifold of the Hopf spinor and the S 7 modulo U (1) phase is CP 3 , which is the continuum limit of the (unnested) fuzzy four-sphere. The second Hopf map thus presents a relationship between the unnested fuzzy four-sphere and its continuum limit. Also notice that the Hopf spinor for (115) can be chosen as which satisfies 5 a=1 x a γ a ψ = +ψ.
This is the simplest version of the SO(5) spin-coherent state equation, which plays a central role in deriving the classical geometry of the fuzzy four-sphere in Sec.6.3.1.

Continuum limit
To expand a concrete discussion, let us focus on the north point of the nested fuzzy four-sphere. Since the nested fuzzy four-sphere is an SO(5) symmetric object, we can choose the north pole as a reference point without loss of generality. The north pole is represented by the index s = I/2, which correspond to the N + 1 most right edges of the oblique SO(4) lines in Fig.3: Since j and k are two independent SU (2) indices, the (j, k) 4 realizes a direct product of two fuzzy spheres specified by the SU (2) spins, j and k, in the language of the fuzzy geometry. In the continuum limit I → ∞, Eq.(122) becomes which suggests that the fuzzy structure of the north pole is well approximated by the N + 1 identical fuzzy two-spheres, each with the SU (2) spin I/2. Since every SO(4) line or quasi-fuzzy four-sphere thus accommodates a fuzzy two-sphere, each quasi-fuzzy four-sphere is described locally by S 4 × S 2 , or CP 3 in the continuum. Consequently, the nested fuzzy four-sphere is reduced to N + 1 overlapped identical CP 3 s. This is also suggested by the continuum limit of the degeneracy (44) D(N, I) where 1 6 I 3 denotes the degrees of freedom of a single fuzzy CP 3 . It should be emphasized that while the continuum limit of the nested fuzzy four-sphere is N + 1 overlapped CP 3 s, their fuzzification does not recover the original nested fuzzy four-sphere but just provides N + 1 identical fuzzy CP 3 s (or N + 1 unnested identical fuzzy four-spheres). In other words, the nested fuzzy four-sphere geometry cannot be reproduced from its corresponding continuum manifold. This agrees with the previous observation that the nested fuzzy four-sphere is a pure quantum object.

S 4 geometry
The coherent state method [88,89,90] and the probe brane method [91,92,93] are systematic methods to obtain a classical manifold corresponding to a given matrix geometry. These two methods are related but not exactly the same [60,93]. Here, we derive the S 4 geometry from the matrix coordinates using there methods.

Coherent state method
For matrix coordinates X a , the coherent method [88,89,90] adopts the following matrix Hamiltonian, We can derive classical manifold as a configuration of x a by following the three steps: First, we diagonalize the matrix Hamiltonian to derive the groundstate energy E G (x a ). Second, we examine the minimum of E G (x a ) as a function of x a to determine the vacuum manifold of x a . Last, we take the I → ∞ limit of this configuration. The matrix Hamiltonian (125) is rewritten as The cross term of Eq.(126) is diagonalized as where The maximal eigenvalue of X where Equation (131) The ground state energy is then obtained as and the corresponding eigenstates are given by (132) We thus obtained the classical S 4 geometry (x a x a = 1) from X [N ] a .

Probe brane method
The probe brane method [91,92,93] adopts the Dirac-operator matrix In this method, the classical manifold is obtained through the following two steps. First, we consider the condition for the existence of the zero-modes of the Dirac-operator matrix (139). For zero-modes to exist, x a must satisfy a certain condition which characterizes a classical manifold. Subsequently, we take I → ∞ limit of the classical manifold to derive the corresponding classical geometry. Due to the tensor product form of (139), it is rather technically intricate to derive general results unlike the case of the coherent 18 From (131) This concise transformation from matrix coordinates to classical coordinates is given in Ref. [62]. 19 The spin-coherent states (132) should not be confused with the SO(5) Landau level eigenstates (34), even though both are realized in the unitary matrix (129) [57].
Equation (140) indicates that the zero-modes exist when x a satisfy r = I 2N +I+4 , which is equal to the previous result (137). Therefore, both the coherent state method and the probe brane method yield the identical classical geometry in the present case. Meanwhile, the number of the zero-modes (141) is distinct from that of the coherent states (136).

Realization in Yang-Mills matrix models
In this section, we demonstrate that the nested fuzzy four-spheres realize new classical solutions of Yang-Mills matrix models and investigate their physical properties. In particular, we clarify distinct behaviors between the lowest Landau level matrix geometry and the newly obtained higher Landau level matrix geometries.

Basic relations
Using the explicit forms of X where c 1 and c 3 are given by (76) and (82) and the potential energy is expressed as The behaviors of the cs and the V are illustrated in Fig.12. Their behaviors are similar to those of the fuzzy two-sphere (see Fig.16). While we have utilized X [N ] a as the matrix coordinates, from an algebraic standpoint, it might be more natural to adopt "normalized" matrix coordinates that align with the quantum Nambu algebra: 20 In the lowest Landau level (N = 0), the coefficients are given by The quantities in Eq.(149) are plotted in Fig.13. The radiusR increases as N increases (Fig.13) unlike the original R (79) (Fig.8). The behaviors of the quantities in Eq.(149) are qualitative similar to those of the fuzzy two-sphere (Fig.17), except for the potential energy densities (the lower right of Fig.13) in which the order of magnitudes for I = 1, 2, 3 is reversed between the lowest Landau level (N = 0) and the higher Landau levels (N ≥ 1).

With a mass term
Let us consider Yang-Mills matrix model with a mass term [94]: 21 In the lowest Landau level (N = 0), Eq.(147) is reduced toX Under the scaling A i → √ ρA i , the parameter ρ turns to the overall scale factor of the action and does not have any physical effect. We will take ρ = 1: The equations of motion are derived as Using (142), we readily see that the nested fuzzy four-spheres realize new classical solutions: 22 where the non-commutative parameter is given by The non-commutative parameter α is a parameter-dependent quantity unlike the case of the fuzzy twosphere solution (see Appendix B.2). This brings specific physical properties to the fuzzy four-sphere solu-tions. The physical quantities (149) are evaluated as 23 action : S cl mass = (− action density : The behaviors of Eqs.(154) and (156) are shown in Fig.14. Similar to the case ofX in Sec.7.1, the action densities (the lower right of Fig.14) exhibit qualitatively distinct behaviors to the fuzzy two-sphere (Fig.17).

With a fifth-rank Chern-Simons term
We next consider the Yang-Mills matrix model with a fifth-rank Chern-Simons term [94] The coupling constant λ can be absorbed in the action when scaling A a as A a → 1 λ · A a . We then set λ = 1 and deal with the following action: The equations of motion are given by From (142), we easily obtain new classical solutions as and 24 radius : action density : (165c) Figure 15 depicts the behaviors of Eq.(165). There are three noteworthy points. First, the order of α CS for I = 1, 2, 3 is the reverse of that of α mass (the upper left in Fig.14). Second, the order of magnitudes of R CS for I = 1, 2, 3 is reversed between N = 0 and N = 2. Last, the order of magnitudes of both S cl CS (lower left in Fig.15) and S cl CS /R CS 4 (lower right in Fig.15) of N = 0 for I = 1, 2, 3 is the reverse of those of N ≥ 1. Thus, the lowest Landau level matrix geometry (N = 0) and the newly obtained higher Landau level matrix geometries (N ≥ 1) exhibit qualitatively distinct physical properties. It is rather curious that, while the matrix size D is a monotonically increasing function about I and the quantities such as R CS and S cl CS are expected to show similar behaviors the higher Landau level matrix geometries, i.e. the nested fuzzy four-spheres, do not follow this anticipation.

Even higher dimensions
We here investigate higher Landau level matrix geometries in even higher dimensions. The associated higher form gauge field and Yang-Mills matrix model are also discussed.

Landau level matrix geometries
It is known that (unnested) higher dimensional fuzzy spheres are realized as the lowest Landau level matrix geometries in higher dimensional Landau models [80,79,44]. Since S d ≃ SO(d + 1)/SO(d), the 24 In the lowest Landau level (N = 0), we have α = 2 3 which satisfies Equation (165) reproduces the results of Ref. [94] for N = 0: (γ a ⊗ 1 ⊗ · · · ⊗ 1 + 1 ⊗ γ a ⊗ · · · ⊗ 1 + · · · + 1 ⊗ 1 ⊗ · · · ⊗ γ a ) sym , which satisfy the Spin(2k + 2) Lie algebraic commutation relations together with the SO(2k) generators. The higher Landau level geometries in the SO(d + 1) Landau model have not been investigated so far. Though it is in principle possible to derive higher Landau level matrix geometries by following the present non-commutative scheme, it is rather laborious to solve the eigenvalue problem of the higher dimensional Landau Hamiltonian. Furthermore, the resulting matrix structures may be mathematically too involved to deduce useful information about the higher dimensional non-commutative geometry. Therefore, we will engage in a somewhat speculative yet more general discussion based on group theory. Let us focus on the following SO(2k + 1) irreducible representation [l 1 , l 2 , · · · , l k ] SO(2k+1) = [N + I, I, · · · , I], which corresponds to the N th Landau level eigenstates of the SO(2k +1) Landau model studied in Refs. [79,44]. From group representation theory, the corresponding degeneracy is given by Meanwhile, for the one dimension lower SO(2k) Landau model, the Landau levels consist of sub-bands [80]. The eigenstates of the s band of the nth Landau level constitute an SO(2k) irreducible representation: with degeneracy There exists an exact relation between the degeneracies (168) Equation (171) signifies a higher dimensional generalization of Eq. (44) and implies that the SO(2k + 1) irreducible representation is constructed by adding up the SO(2k) sectors from n = 0 to n = N each of which is made of s = I/2, · · · , −I/2. Since the geometric structure of the fuzzy manifold reflects on the structure of the irreducible representation, the matrix geometry of the N th Landau level is expected to exhibit N + 1 nested fuzzy structures in arbitrary dimensions, just like the nested fuzzy four-sphere. It is also reasonable to consider that fuzzy (2k − 1)-spheres are embedded within the nested fuzzy 2k-sphere.

Higher form gauge field and Yang-Mills matrix model
The lowest Landau level matrix geometry in 2k dimension is associated with a generalized Hopf maps [39] and are described by both the SO(2k + 2) Lie algebra and quantum Nambu 2k algebra. Meanwhile, the matrix coordinates in the higher Landau levels are not associated with the generalized Hopf map but are covariant under the SO(2k + 1) transformation like the lowest Landau level matrix coordinates. Therefore, the higher Landau level matrix coordinates will not conform with the Lie algebraic description but instead is described by the quantum Nambu algebra exclusively: When one adopt more general irreducible representations beyond Eq.(167), the corresponding fuzzy manifold will exhibit a more exotic quantum geometry than the nested fuzzy structure, however, due to the existence of the SO(2k + 1) covariance, the matrix coordinates will also adhere to the quantum Nambu algebra (172). Interestingly, "magnetic field" appears on the right-hand side of (172) [79], which signifies the tensor monopole field strength : The existence of the higher form gauge field behind the quantum Nambu geometry is thus glimpsed. One may wonder where such a higher gauge symmetry comes from, where as the present quantum mechanical system only has the SO(2k) gauge symmetry. Indeed, the tensor monopole gauge field is directly obtained from the SO(2k) monopole gauge field through the Chern-Simons term [79].
The (unnested) fuzzy 2k-sphere realizes a solution of [22] [[X a , X b ], which is derived from the Yang-Mills matrix model with a 2k + 1 rank Chern-Simons term, Since the equations of motion are concerned with the covariance of the matrix coordinates, it is anticipated that the nested fuzzy 2k-spheres realize classical solutions of Eq.(174). The action of for the fuzzy 2k-sphere solution is given by where While the signs of S CS (X cl a ) and V (X cl a ) are opposite for k = 1, they have the same sign for k ≥ 2, as we have seen, for k = 2, in Eq. (165b).

Summary and discussions
Based on the insight obtained from the emergent fuzzy geometry in the simple SO(3) Landau model, we proposed a novel non-commutative scheme for generating the matrix geometries for arbitrary manifolds of the coset type G/H. In the present approach, manifolds need not be either symplectic or even dimensional unlike the conventional non-commutative schemes. We explicitly derived the matrix geometries for S 4 by utilizing the SO(5) Landau model. The emergent matrix geometries in higher Landau levels realize pure quantum Nambu geometries in which matrix coordinates are not closed within the canonical formalism of the Lie algebra but are described only by introducing the quantum Nambu algebra. We also demonstrated that such pure quantum matrix geometries manifest new solutions of the Yang-Mills matrix models. The particular features of the nested quantum geometry, such as the internal matrix geometries, continuum limit, and classical counterpart, were clarified. The pure Nambu matrix geometries are common to the higher Landau levels of the Landau models in arbitrary dimensions.
While the conventional scheme is based on the spirit of quantization of classical (symplectic) manifolds, the present non-commutative scheme is largely based on the quantum mechanics from the beginning. In this sense, the present scheme is considered to be a quantum-oriented non-commutative scheme. That is the reason why we obtained the pure quantum geometry. We showed this non-commutative scheme is practically useful in deriving novel solutions of the Matrix models. As Matrix model solutions, the nested matrix geometries exhibit quantitatively distinct behaviors with the unnested fuzzy four-sphere.
The discovery of the novel quantum Nambu matrix geometries now brings various open questions, such as brane construction, relation to tachyon condensation [95,96], realization in the Nahm equation in higher energy physics. The higher form gauge field implied by the quantum Nambu algebra is closely related to the higher Berry phase [97,98] whose usefulness is getting appreciated in the very recent studies of strongly correlated many-body systems. It would be intriguing to speculate on the role of quantum Nambu geometry in condensed matter physics. We also add that the present scheme itself should be appropriately generalized to treat less symmetric fuzzy objects, while we studied highly symmetric objects in this work.
To the best of the author's knowledge, this work is the first example of quantum matrix geometry found in the analysis of the Landau models being practically applied to the solutions of the M(atrix) models. M(atrix) theory is assumed to describe the physics at the Planck scale of 10 19 Gev, while the Landau models or the quantum Hall effect are about the low temperature physics at milli-electron volt. It is rather amazing that same mathematics work in both physics with such a huge energy gap.

Acknowledgments
This work was supported by JSPS KAKENHI Grant No. 21K03542.

A Groenewold-Moyal plane from planar Landau model
We demonstrate a realization of the Groenewold-Moyal plane in higher Landau levels. Let us consider a 2D plane subject to a constant perpendicular magnetic field: We employ the gauge-independent relation (178), and so all of the following results are also gauge independent. The covariant derivatives and the center-of-mass coordinates are respectively constructed as which satisfy two independent commutation relations: We then realize two sets of creation and annihilation operators as The Hamiltonian of the planar Landau model is given by The corresponding energy Landau levels and the eigenstates are Using we readily evaluate the matrix elements of x and y: The intra-Landau level matrix coordinates are then obtained as or (188) Notice that (188) does not depend on the Landau level index N , and the matrix coordinates satisfy Obviously, the dimensionless coordinates, , satisfy the Heisenberg-Weyl algebra together with 1: We have thus confirmed the emergence of the Groenewold-Moyal plane in any Landau level.

B Two-sphere matrix coordinates in Yang-Mills matrix models
For comparison with the fuzzy four-sphere (Sec.7), we revisit matrix model the analyses of fuzzy twosphere [100,99], clarifying physical properties of the matrix coordinates in the SO(3) Landau model.

B.1 Basic properties
The N th Landau level matrix coordinates X (N ) i (7) satisfy the following relations 25 .
(193c) 25 In the literatures on matrix models, it is common to denote the variables D and I as N and n, respectively.
These cs are not independent quantities; instead, they satisfy From (193), we readily have and The behaviors of (193) and (196) are shown in Fig.16. We introduce "normalized" matrix coordinates that satisfy the SU (2) algebra Notice thatX The potential energy density is simply the twice the matrix size of the fuzzy two-sphere.

B.2 Matrix model analysis
Yang-Mills matrix models with a mass term and with a third-rank Chern-Simons term are given by and the corresponding equations of motion are respectively The fuzzy two-sphere is realized as a solution: Notice that both coefficients, α mass and α CS , are constant unlike the case of the fuzzy four-sphere solutions (see Sec.7). The classical solutions (202) have the following properties: action : action density : S cl mass R mass 2 = −α mass See Fig.17 for the behaviors of these quantities. Figure 17: Physical quantities of the fuzzy two-spheres. All quantities monotonically increase or decrease as N and I increase.