# Non-commutative geometry of 4-dimensional quantum Hall droplet

###### Abstract

We develop the description of non-commutative geometry of the 4-dimensional quantum Hall fluid’s theory proposed recently by Zhang and Hu. The non-commutative structure of fuzzy , which is the base of the bundle obtained by the second Hopf fibration, i.e., , appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in the non-commutative algebra of functions on , are explicitly constructed and their obeying the matrix algebra is obtained. This matrix algebra is associative. We also propose a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and determine the fusion rule in such fusion scheme. By products, we provide some essential ingredients of the theory of angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of angular momentum, are given. We also discuss some possible applications of our results to the 4-dimensional quantum Hall system and the matrix brane construction in M-theory.

Keywords: 4-dimensional quantum Hall system, fuzzy monopole harmonics, non-commutative geometry.

## 1 Introduction

The planar coordinates of quantum particles in the lowest Landau level of a constant magnetic field provide a well-known and natural realization of non-commutative space [1].The physics of electrons in the lowest Landau level exhibits many fascinating properties. In particular, when the electron density lies in at certain rational fractions of the density corresponding to a fully filled lowest Landau level, the electrons condensed into special incompressible fluid states whose excitations exhibit such unusual phenomena as fractional charge and fractional statistics. For the filling fractions , the physics of these states is accurately described by certain wave functions proposed by Laughlin [2], and more general wave functions may be used to describing the various types of excitations about the Laughlin states.

There has recently appeared an interesting connection between quantum Hall effect and non-commutative field theory. In particular, Susskind [3] proposed that non-commutative Chern-Simons theory on the plane may provide a description of the (fractional) quantum Hall fluid, and specifically of the Laughlin states. Susskind’s non-commutative Chern-Simons theory on the plane describes a spatially infinite quantum Hall system. It, i.e., does the Laughlin states at filling fractions for a system of an infinite number of electrons confined in the lowest Landau level. The fields of this theory are infinite matrices which act on an infinite Hilbert space, appropriate to account for an infinite number of electrons. Subsequently, Polychronakos [4] proposed a matrix regularized version of Susskind’s non-commutative Chern-Simons theory in an effort to describe finite systems with a finite number of electrons in the limited spatial extent. This matrix model was shown to reproduce the basic properties of the quantum Hall droplets and two special types of excitations of them. The first type of excitations is arbitrary area-preserving boundary excitations of the droplet. The another type of excitations are the analogs of quasi-particle and quasi-hole states. These quasi-particle and quasi-hole states can be regarded as non-perturbative boundary excitations of the droplet. Furthermore, it was shown that there exists a complete minimal basis of exact wave functions for the matrix regularized version of non-commutative Chern-Simons theory at arbitrary level and rank , and that those are one to one correspondence with Laughlin wave functions describing excitations of a quantum Hall droplet composed of electrons at filling fraction [5]. It is believed that the matrix regularized version of non-commutative Chern-Simons theory is precisely equivalent to the theory of composite fermions in the lowest Landau level, and should provide an accurate description of fractional quantum Hall state. However, it does appear an interesting conclusion that they are agreement on the long distance behavior, but the short distance behavior is different [6].

In the matrix regularized version of non-commutative Chern-Simons theory, a confining harmonic potential must be added to the action of this matrix model to keep the particles near the origin. In fact, there has been a translationally invariant version of Laughlin quantum Hall fluid for the two-dimensional electron gas in which it is not necessary to add any confining potential. Such model is Haldane’ that of fractional quantum Hall effect based on the spherical geometry [7]. Haldane’s model is set up by a two-dimensional electron gas of particles on a spherical surface in radial monopole magnetic field. A Dirac’s monopole is at the center of two-dimensional sphere. This compact sphere space can be mapped to the flat Euclidean space by standard stereographical mapping. In fixed limit, the connection between this model and non-commutative Chern-Simons theory can be exhibited clearly. Exactly, the non-commutative property of particle’s coordinates in Haldane’s model should be described in terms of fuzzy two-sphere.

In this paper, we do not plan to discuss such description of Haldane’s model in detail, but want to exhibit the character of non-commutative geometry of 4-dimensional generalization of Haldane’s model, proposed recently by Zhang and Hu [8]. The 4-dimensional generalization of the quantum Hall system is composed of many particles moving in four dimensional space under a gauge field. Instead of the two-sphere geometry in Haldane’s model, Zhang and Hu considered particles on a four-sphere surface in radial Yang’s monopole gauge field [9], which replaces the Dirac’s monopole field of Haldane’s model. This Yang’s monopole gauge potential defined on four-sphere can be transformed to the instanton potential of the Yang-Mills theory [10] upon a conformal transformation from four-sphere to the 4-dimensional Euclidean space. Zhang and Hu had shown that at appropriate integer and fractional filling fractions the generalization of system forms an incompressible quantum fluid. They [12] also investigated collective excitations at the boundary of the 4-dimensional quantum Hall droplet proposed by them. In their discussion, an non-commutative algebraic relation between the coordinates of particle moving on four-sphere plays the key role. According to our understanding about Haldane’s and Zhang et al works, we think that fuzzy sphere structures in their models is the geometrical origin of non-commutative algebraic relations of the particle coordinates. We shall clarify this idea in this paper.

Non-commutative spheres have found a variety of physical applications [14, 15, 16, 17, 13, 19, 20]. The description of fuzzy two-sphere [14] was discovered in early attempts to quantize the super-membrane[16]. The fuzzy four-sphere appeared in [15, 17]. The connection of non-commutative second Hopf bundle with the fuzzy four-sphere has been investigated [19] from quantum group. The fuzzy four-sphere was used [21] in the context of the matrix theory of BFSS [22] to described time-dependent 4-brane solutions constructed from zero-brane degrees of freedom. Furthermore, the non-commutative descriptions of spheres also arise in various contexts in the physics of D-branes. The descriptions of them, e.g., were used to exhibit the non-commutative properties and dielectric effects of D-branes [23]. Recently, Ho and Ramgoolam [24, 25] had studied the matrix descriptions of higher dimensional fuzzy spherical branes in the matrix theory. They have found that the finite matrix algebras associated with the various fuzzy spheres have a natural basis which falls in correspondence with tensor constructions of irreducible representations of the corresponding orthogonal groups. In their formalism, they gave the connection between various fuzzy spheres and matrix algebras by introducing a projection from matrix algebra to fuzzy spherical harmonics. Their fuzzy spheres obey non-associative algebras because of the non-associativity induced by the projected multiplication. The complication of projection makes their constructions of fuzzy spherical harmonics formal.

The goal of this paper is to explore the character of non-commutative geometry of 4-dimensional quantum Hall system proposed recently by Zhang and Hu. Recently, Fabinger [26] had pointed that there exists a connection of the fuzzy with Zhang and Hu’s quantum Hall model of . The string theory and brane matrix theory related with such non-commutative structure of fuzzy are discussed by [26, 27, 28]. However, the structure of non-commutative algebra of functions on is not still clear. It is known that the key idea of non-commutative geometry is in replacement of commutative algebra of functions on a smooth manifold by a non-commutative deformation of it [20]. We shall explore the structure the character of non-commutative geometry of 4-dimensional quantum Hall system to find non-commutative algebra of functions on .

We should emphasize that the non-commutative structure of fuzzy of Zhang and Hu’s quantum Hall model is different with those commonly considered by people. In fact, the of Zhang and Hu’s quantum Hall model is the base of the Hopf bundle obtained by the second Hopf fibration, i.e., , from the connection between the second Hopf map and Yang’s [29, 30]. Equivalently, . The bundle can be parametered (see section 2 in detail). By further smearing out the common gauge symmetry parametered by , we can obtain the bundle or by solving the eigenfunctions of Zhang and Hu’s model. The eigenfunctions of the LLL consist of the space on which non-commutative algebra of functions on act. . Furthermore, we can find the sections of the bundle and its fibre since

Since at specialfilling factors, the quantum disordered ground state of 4-dimensional quantum Hall effect is separated from all excited states by a finite energy gap, the lowest energy excitations are quasi-particle or quasi-hole excitations near the lowest Landau level. The quantum disordered ground state of 4-dimensional quantum Hall effect is the state composed coupling by particles lying in the lowest Landau level state. At appropriate integer and fractional filling fractions, the system forms an incompressible quantum liquid, which is called as a 4-dimensional quantum Hall droplet. In fact, the spherical harmonic operators for fuzzy four-sphere are related with quasi-particle’s or quasi-hole’s creators of 4-dimensional quantum Hall effect. These operators is composed of a complete set of the matrices with the fixed dimensionality. Focusing on the space of particle’s lowest Landau level state, we shall construct explicitly these fuzzy spherical harmonics, also called the fuzzy monopole harmonics, and discuss the nontrivial algebraic relation between them. Furthermore, we shall clarify the physical implications of them.

This paper is organized as follows. Section two introduces the 4-dimensional quantum Hall model proposed by Zhang and Hu, and analyzes the property of Hilbert space and the symmetrical structures of this 4-dimensional quantum Hall system. We shall emphasize the intrinsic properties of the Yang’s monopole included in this system, and give the explicit forms of normalized wave functions of this system, which is given by (8) in our paper. The wave functions corresponding to the irreducible representation of are those in the LLL. They are the sections of the bundle over with fibre parametered by . These degeneracy wave functions consist of the LLL Hilbert space which the non-commutative algebra of functions on acts on. The elements of this algebra are constructed in the following section. Section three describes the elements of non-commutative algebra of functions on which is related with the fuzzy four-sphere from the geometrical and symmetrical structures of 4-dimensional quantum Hall droplet. We find the matrix forms and the symbols of the elements, given by (15) and (13) respectively. We give the explicit construction of complete set of this matrix algebra, determined by (19). In section four, we find the system of algebraic equations satisfied by the generators of the matrix algebra. The results are given by the equations (30) and (31). For the matrix forms of the elements, this non-commutative algebra should be understood as the algebraic relation of matrix mulitiplication, and for the symbols of the elements, it should be done as that of Moyal product. It can be seen from these results and the complete set (19) that the non-commutative algebra is closed. The associativity of this algebra is shown by the relation (33). Furthermore, a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and its fusion rule are established in this section, which are given by the relations (40) and (41). Section five includes discussions about the physical interpretations of the results and remarks on some physical applications of them.

## 2 The Hilbert space of 4-dimensional quantum Hall system

The 4-dimensional quantum Hall system is composed of many particles moving in four dimensional space under a gauge field. The Hamiltonian of a single particle moving on four-sphere is read as

(1) |

where is the inertia mass and the radius of . The symmetry group of is . Because the particle is coupling with a gauge field , in Eq.(1) is the dynamical angular momentum given by . From the covariant derivative , one can calculate the gauge field strength from the definition . does not satisfy the commutation relations of generators. Similar to in Dirac’s monopole field, the angular momentum of a particle in Yang’s monopole field can be defined as , which indeed obey the commutation relations. Yang[11] proved that can generate all irreducible representations.

In general, the representations of can be put in one-to-one correspondence with Young diagrams, labelled by the row lengths , which obey the constraints . For such a representation, the eigenvalue of Casimir operator is given by , and its dimensionality is

(2) |

The gauge field is valued in the Lie algebra . The value of this Casimir operator specifies the dimension of the representation in the monopole potential. is an important parameter of generating all irreducible representations and the Hamiltonian Eq.(1). In fact, for a given , if one deals with the eigenvalues and eigenfunctions the operator of angular momentum , it can be found that the irreducible representations, which the eigenfunctions called by Yang [11] as monopole harmonics belong to, are restricted. Such irreducible representations are labelled by the integers , and . Based on the expressions of monopole potentials given by Yang [11], or by Zhang and Hu [8], one can show that by straightforwardly evaluating. This implies that the eigenvalues and eigenfunctions of the Hamiltonian Eq.(1) can be read off from those of the operator . Hence, for a given , the energy eigenvalues of the Hamiltonian Eq.(1) are read as

(3) |

The degeneracy of energy level is given by the dimensionality of the corresponding irreducible representation .

The ground state of the Hamiltonian Eq.(1) plays a key role in the procedure of construction of many-body wave function and the discussion of incompressibility of 4-dimensional quantum Hall system. This ground state, also called the lowest Landau level (LLL) state, is described by the least admissble irreducible representation of , i.e., labelled by for a given . The LLL state is fold degenerate, and its energy eigenvalue is . Zhang and Hu [8] found the explicit form of the ground state wave function in the spinor coordinates. This wave function is read as

(4) |

with integers . The orbital coordinate , which is defined by the coordinate point of the 4-dimensional sphere , is related with the spinor coordinates with by the relations and . The five Dirac matrices with satisfy the Clifford algebra . The isospin coordinates with are given by an arbitrary two-component complex spinor satisfying . Zhang and Hu gave the explicit solution of the spinor coordinate with respect to the orbital coordinate as following

(5) |

By computing the geometric connection, one can get a non-Abelian gauge potential , which is just the gauge potential of a Yang monopole defined on 4-dimensional sphere [9, 8]. Since we do not need the explicit form of it here, we do not write out that of it.

The description of the 4-dimensional quantum Hall liquid involves the quantum many-body problem of particle’s moving on the 4-dimensional sphere in the Yang’s monopole field lying in center of the sphere . The wave functions of many particles can be constructed by the nontrivial product of the single particle wavefunctions, among which every single particle wave function is given by the LLL wavefunction Eq.(4). In the case of integer filling, the many-particle wave function is simply the Slater determent composed of single-particle wave functions. For the fractional filling fractions, the many particle wave function cannot be expressed as the Laughlin form of a single product. But the amplitude of the many-particle wave function can also be interpreted as the Boltzmann weight for a classical fluid. One can see that it describes an incompressible liquid by means of plasma analogy. Therefore, at the integer or fractional filling fractions, the 4-dimensional system of the generalizing quantum Hall effect forms an incompressible quantum liquid [8]. We shall call this 4-dimensional system as a 4-dimensional quantum Hall droplet.

The space of the degenerate states in the LLL is very important not only for the description of the 4-dimensional quantum Hall droplet but also for that of edge excitations and quasi-particle or quasi-hole excitations of the droplet. In fact, this space of the degenerate states is the space which we shall construct the matrix algebra acting on in the following section. In order to construct the complete set of matrix algebra of fuzzy , we need to know the explicit forms of the wave functions associated with all irreducible representations of . Although Yang had found the wave functions for all the states, the form of his parameterizing the four sphere is not convenient for our purpose. Following Hu and Zhang[12], we can parameterize the four sphere by the following coordinate system

(6) |

where and . The direction of the isospin is specified by and .

As the above explanation, we can get the eigenfunctions of the Hamiltonian Eq.(1) from the eigenfunctions of the operator . The angular momentum operators consists of an orbital part with and an isospin part involving the monopole field. The angular momentum operators generate the rotation in the subspace , and satisfy the commutation relations of generators. They can be decomposed into two algebras: . Therefore, if one would use the operators and to generate the blocks of the irreducible representations, he can not obtain all irreducible representations of . However, because of the coupling to the Yang’s monopole potential, these orbital generators are modified into , which are decomposed into and . Indeed, the generators and can be used to generate the block states of all irreducible representations [11]. Such block states can be labelled by the quantum numbers and . The and are the magnetic quantum numbers of two algebras. , where . They satisfy the identity of operator and

Applying the operators to the quantum states described by the Hamiltonian Eq.(1) in the irreducible representations of , one can see that the complete set of quantum observables of the system is composed of the operators and . Thus, there exist the simultaneous eigenfunctions of those operators, which are just the wave functions of energy eigenvalue being . Noticing that the isospin operators are coupling with the operator into the angular momentum operators , we should also introduce the quantum number labelling the isospin parameter , which is written as are abbreviated to . The wave functions corresponding to the irreducible representation of are denoted as . By using the equation of parameterizing and the parameters of the isospin direction, and following Yang [11], one can obtain the wave functions obeying the system of equations as following . The parameters of group

(7) |

The first line and second line of the equations tell us that we can use two D-functions to realize the wave function with respect to the dependence of the group parameters and . Furthermore, we also should consider the coupling relations , and to make the wave function obey the third line of the equations. The final line is the equation to determine the dependence of the wave function, which had been solved by Yang [11]. Now, we can write the explicit solution form of the normalized wavefunction as

(8) | |||||

where

(9) | |||||

and is the Jacobi polynomial.

Although is the d-function of rotation group, it should be emphasized that here the quantum numbers of the subgroup of have replaced the usual magnetic quantum number of . Practically, is the d-function of rotation group in the special case. The D-function is the standard representation matrix for the Euler angles . They are the rotation matrix elements generated rotationally by the operators . Similarly, are those generated by the isospin operators . The coupling coefficients , i.e., the Clebsch-Gordon coefficients, show the coupling behavior of the angular momentums , and by means of .

The isospin direction can be normally specified by two angles and . However, the wave function depends on the Eular angles , and of the isospin space. In fact, in the picture, the dependence of is given by the phase factor , where is simply an gauge index. Therefore, different values of correspond to the same physical state. One can smear the dependence of the wave function by the gauge choice. Such gauge choice can be fixed by taking or . If such gauge choice is taken at every step in all calculations, we call this choice as taking the physical gauge. Analogous to doing usually in the field theory, there exists another gauge choice, which such gauge choice is taken at the end of calculations. The latter gauge choice are called as taking the covariant gauge. We shall take the covariant gauge in this paper, which this thick was also used in the reference [12].

The wave functions are the monopole harmonics. Exactly, they are the spherical harmonics on the coset space , which is locally isomorphic to the sphere . By smearing the degree of freedom of the dependence, i.e., taking the physical gauge, they can be viewed as the spherical harmonics on the coset space , which is locally isomorphic to the sphere . Globally, is a bundle over the with fibre . The wave functions are the cross sections in this nontrivial fibre bundle. This implies that there exists a stabilizer group of the wave function solutions , and the action of on the state generates a space of the wave function solutions which is the space of cross sections in . If we take the covariant gauge, the monopole harmonics should be regarded as the cross sections in the nontrivial fibre bundle . Then, the stabilizer group of the wave function solutions is . doing the state generates that of cross sections in . The procedure of parameterizing the four sphere Eq.(6) and building up the isomorphic relation of the group manifold and the three sphere is just that of smearing the stabilizer subgroup . Of course, if one want to take the physical gauge, he can smear the by further smearing the gauge subgroup. Such globally geometrical structure and symmetrical structure can guide us to develop some techniques which we need in this paper.

Let us introduce the state vector , which belongs to the Hilbert space composed of the monopole harmonics . Of course, is an element of the Hilbert space . In general, taking a fixed vector in the Hilbert space, one can use the unitary irreducible representation of an arbitrary Lie group acting in the Hilbert space to produce the coherent state for this Lie group. Now, we are interesting to the coherent state corresponding to the coset space . The is the isotropic subgroup of for the state since is the stabilizer group of the wave function solutions. If we use the unitary irreducible representations of acting on the state to produce the coherent states, the coherent state vectors belonging to a left coset class of with respect to the subgroup differ only in a phase factor and so determine the same state. Consequently, the coherent state vectors depend only on the group parameters parameterizing the coset space . Thus, we can now introduce the following coherent state vector

(10) |

where the star stands for the complex conjugate. In order to realize the covariant gauge, we have added the isospin frame to the monopole harmonics . In fact, we can use the finite rotation

The wave functions in the coherent state picture are given by

(11) |

The l.h.s. of Eq.(11) is not with the label since it naturely appears in the label , which corresponds to the monopole harmonics. It should be emphasized that the above wave functions become the wave functions in the physical gauge only if one smears the isospin frame of them and projects back to the and angles. In the sense of the finite rotation of , the explicit forms of wave function solutions given here are the wave function solutions of the 4-dimensional spherically symmetrical top with the self-rotating in the isospin direction. The Yang’s monopole harmonics [11] can be interpret as the wave functions in the coherent state picture in the physical gauge.

Based on the orthogonality and completeness of the state vectors belonging to an irreducible representation of , we can give the completeness condition of the coherent state vectors

(12) |

where . Every irreducible representation of corresponds to a complete set of the coherent state vectors. The coherent states of the different irreducible representations are orthogonal each other. The coherent state corresponding to the LLL states is very important for the description of non-commutative geometry of 4-dimensional quantum Hall droplet.

In order to avoid the label of the irreducible representation of the LLL states confusing with the parameter of the model, we denote the irreducible representation as . The LLL degeneracy states consist of the Hilbert space . Because of the LLL states are the lowest energy states of particle’s living in, for a given , the smallest admissible irreducible representation of is . Therefore, the irreducible representations of are truncated since there exists an Yang’s monopole. If we focus on the Hilbert space of the LLL states, we can determine the matrix forms of tensor operator for the monopole harmonics, which are the matrices. The coupling relation between the tensor operators provides a truncated parameter for the tensor operator for the monopole harmonics. The number of independent operators is , which we shall explain in more detail in the next section. Therefore, we should replace the functions by the matrices on the fuzzy . Exactly, the cross sections in the fibre bundle , which is a bundle over with fibre , are replaced by the matrices on the fuzzy . Thus, the algebra on the fuzzy becomes non-commutative. The direct product of single-particle Hilbert spaces can be used to build up the Hilbert space of 4-dimensional quantum Hall droplet. Hence, the fuzzy appears naturally in the description of particle’s moving on the with the Yang’s monopole at the center of the sphere. Consequently, the fuzzy is the description of non-commutative geometry of 4-dimensional quantum Hall droplet.

## 3 Fuzzy monopole harmonics and Matrix operators of fuzzy

The construction of fuzzy is to replace the functions on by the non-commutative algebra taken in the irreducible representations of . This is a full matrix algebra which is generated by the fuzzy monopole harmonics. These fuzzy monopole harmonics consist of a complete basis of the matrix space. In this section, we shall find the explicit forms of such fuzzy monopole harmonics by means of the expressions of the monopole harmonics given in the previous section. Our main task of this section is to construct the operators corresponding to the monopole harmonics, and to give the matrix elements of these operators acting on the LLL states.

Although for a given , the monopole harmonics of the smallest admissible irreducible representations of can be regarded as the single-particle wave functions of the LLL in 4-dimensional quantum Hall system, the monopole harmonics smaller than the smallest admissible irreducible representations of are useful for us to construct the fuzzy monopole harmonics on . Such monopole harmonics can be read off from the expressions obtained in the previous section by the changing of the parameter . If we replace by , the irreducible representation of the monopole harmonics becomes . Equivalently, we can use and monopole harmonics are generally expressed as , which can be obtained by replacing with in the equation (8). We can find their corresponding coherent states by the same replacement. to label them. Thus, the

By using the standard techniques of the generalized coherent state [31], we can now construct the operator corresponding to the monopole harmonics . Noticing that this operator is an operator of acting in the LLL Hilbert space and is regarded as the basic function, we can express it as the following form

(13) |

The matrix elements of this operator in the LLL Hilbert space are read as

(14) | |||||

The above integral can be analytically performed by making use of the integral formulae about the Jacobi polynomial and the product of three D-functions, and the properties of D-function. The result is

(15) | |||||

where

(16) |

The another part is given by the integrated part of . It is read as

(17) | |||||