Skyrme-type Non-linear sigma Models via The Higher Dimensional Landau Models

A curious correspondence has been known between Landau models and non-linear sigma models in low dimensions: Reinterpreting the base-manifold of Landau models as a field manifold, the Landau models are transformed to non-linear sigma models with same global and local symmetries. With the idea of the dimensional hierarchy of the higher dimensional Landau models, we exploit this correspondence to present a systematic procedure for construction of non-linear sigma models in higher dimensions. We explicitly derive $O(2k+1)$ non-linear sigma models in $2k$ dimension based on the parent tensor gauge theories that originate from non-Abelian monopoles. The obtained non-linear sigma models turn out to be Skyrme-type non-linear sigma models with hidden $O(2k+1)$ local symmetries. By a dimensional reduction based on the Chern-Simons tensor field theory, we also derive Skyrme-type $O(2k)$ non-linear sigma models in $2k-1$ dimension. As a unified description, we explore Skyrme-type $O(d+1)$ non-linear sigma models and clarify their basic properties, such as stability of soliton configurations, scale invariant solutions, and topological field configurations of higher winding number.

1 Introduction the 4D quantum Hall effect are known to be membrane-like objects whose internal space is S 4 which is described by the field-manifold of the O(5) NLS model [65,66]. Thus, the Landau/NLS model correspondence is naturally generalized from 2D to 4D. Arbitrary 2kD generalization of the quantum Hall effect has been constructed in our previous works [67,68,69]. The mathematical set-up of the 2kD quantum Hall effect is the SO(2k + 1) Landau model on S 2k in the SO(2k) monopole background. The excitations are (2k −2)-dimensionally extended anyonic objects whose fractional statistics are well investigated in [70,71,72,73,74]. Besides, the effective field theory is given by a tensor-type Chern-Simons field theory coupled to the (2k − 2)-brane with S 2k internal space, which is identified with the field manifold of O(2k + 1) NLS models [68].
While NLS model solitons play crucial roles in the higher dimensional quantum Hall effect, a systematic analysis of the O(2k + 1) NLS model to host membrane excitations is still lacking. To be more precise, there are numerous possible NLS models with field-manifold being S 2k , but there is no criterion to choose better models or hopefully the best model among them. A main purpose of this paper is to provide a systematic procedure to construct appropriate NLS models based on the Landau/NLS model correspondence [ Fig.1]. The idea of the dimensional hierarchy of the higher dimensional Landau models [75,68,67] is essential in the construction, and the obtained NLS models necessarily inherit structures of the differential geometry of the Landau models. For a concrete construction of NLS model Hamiltonians, we adopt the idea originally suggested by Tchrakian [18] and recently made manifest by Adam et al. [76] where a BPS equation is firstly given and the Hamiltonian is later derived so that the Hamiltonian may satisfy the BPS equation.
The paper is organized as follows. Sec.2 reviews the differential geometry associated with non-Abelian monopoles in the higher dimensional Landau models. In Sec. 3 (2)) SO(2k) Chern number 1st 2nd kth Topological map π 1 (U (1)) ≃ Z π 3 (SU (2)) ≃ Z π 2k−1 (SO(2k)) ≃ Z Table 1: Geometric and topological features of the Landau models. The monopole gauge group SO(2k) is chosen so that it is identical to the holonomy group of the base-manifold S 2k ≃ SO(2k + 1)/SO(2k) [67]. In the SO(5) Landau model, the holonomy of S 4 is SO(4) ≃ SU (2) ⊗ SU (2) and one SU (2) is adopted as the gauge group. models and explicitly construct the O(7) NLS model and O(2k + 1) NLS model Hamiltonians in Sec.4. In Sec.5, we derive O(2k) S-NLS models using the Chern-Simons term of pure gauge fields. We explore a general construction of O(d + 1) S-NLS models and analyze their basic properties in Sec. 6. Sec.7 is devoted to summary and discussions.

Differential Geometry of the Higher Dimensional Landau Model
In this section, we review the differential geometry of the SO(2k + 1) Landau models and discuss extended objects that are realized as the O(2k + 1) NLS model solitons.

(2k)!
ǫ a 1 a 2 ···a 2k+1 r 2a+1 dr a 1 dr a 2 · · · dr a 2k = 1, where A(S 2k ) denotes the area of S 2k : 3 At r 2k+1 = 0, the SO(2k) monopole configuration, (3) or (9), is reduced to the meron configuration on R 2k [81]: which satisfies the pure Yang-Mills field equation on R 2k [82,83]: Fµν + i[Aµ, Fµν ] = 0. (12) implies that the Chern number for the monopole configuration is accounted for by the winding number (the Pontryagin index) from S 2k phys. to S 2k field : Another expression of the SO(2k) monopole gauge field is which is regular except for the north-pole. The two expressions of the monopole gauge fields, (2) and (15), are related by a gauge transformation on the S 2k−1 -equator of S 2k : where g denotes a transition function of the form g = 1 Here,r i = 1 √ r 2 −r 2k+1 2 −r 2k 2 r i (i = 1, 2, · · · , 2k − 1) represent the normalized-latitude S 2k−2 at an azimuth angle θ on the S 2k−1 -latitude. 5 Since describes a map from S 2k−1 to SO(2k) group and satisfies With g, A and A ′ are simply represented as where The kth Chern number (11) can be expressed by the transition function as [68] where A(S 2k−1 ) signifies the area of (2k − 1)-sphere: 5 θ is given by The associated topology is indicated by Substituting (17) into (22), we have which reproduces the previous result (12), as it should be. The equivalence between (12) and (25) holds for other higher dimensional representations of gauge group matrix generators [75]. We thus find that there are the two equivalent but superficially different representations of the kth Chern number for the monopole field configuration: 1. Winding number associated with π 2k (S 2k ) ≃ Z 2. Winding number associated with π 2k−1 (S 2k−1 ) ≃ Z.
We utilize the first observation in the construction of the O(2k+1) S-NLS models, while the second one in the construction of the O(2k) S-NLS models. This equivalence will also be important in the discussions of topological field configurations (Sec.6.2).

Tensor gauge fields and extended objects
The Chern number (11) can be expressed as where G 2k denotes a 2k rank tensor field strength or Here, we introduced the antisymmetric tensor field strength [18] There are [k/2] ways in the decomposition (28) in correspondence with l = 1, 2, · · · , [k/2]. 6 The antisymmetric tensor field strength (29) will play a crucial role in constructing higher dimensional NLS models in Sec.4. 6 [k/2] signifies the maximum integer that does not exceed k/2. Apparently, there exists a local degree of freedom in the decomposition [87]: For the non-Abelian monopole gauge field (9), we can evaluate (28) as [68] G 2k = 1 2 k+1 r 2k+1 ǫ a 1 a 2 ···a 2k a 2k+1 r a 2k+1 dr a 1 dr a 2 · · · dr a 2k , or which signifies the 2k-rank tensor monopole field strength in its own right [90,91], and the (2k−1)rank tensor gauge field (dC 2k−1 = G 2k ) [92] couples to (2k − 2)-dimensionally extended objects, i.e., (2k − 2)-branes. In the higher dimensional quantum Hall effect, the size of the gauge space is comparable with the size of the base-manifold S 2k [68], and the whole system is regarded as a (4k − 1)D space-time. The (2k − 2)-brane current in (4k − 1)D space-time is simply given by where n a denote the internal field coordinates of the (2k − 2)-brane, which is depicted as the blue sphere of the left-figure of Fig.1. A simple subtraction, (4k − 1) − (2k − 2) = 2k + 1, implies that the dimension of the internal space of the (2k − 2)-brane is 2kD and is naturally described by the S 2k field-manifold of O(2k + 1) NLS models. Indeed, (33) is identical to the topological current of the O(2k + 1) NLS model soliton in (4k − 1)D space-time with coordinates n a subject to 2k+1 a=1 n a n a = 1. The (2k − 2)-brane current is coupled to the (2k − 1)-rank tensor Chern-Simons field and used to describe anyonic excitations in higher dimensions. In this way, the O(2k + 1) NLS model solitons necessarily appear in the context of the higher dimensional quantum Hall effect.

1D promotion and the O(5) S-NLS model
In Sec.2, we first introduced the two monopole gauge field configurations on S 2k and later gave the transition function connecting them on the S 2k−1 -equator of S 2k . In this section, we apply the reveres process to construct the O(5) S-NLS model from the Skyrme's S 3 field-manifold.

Translation to the field manifold and 1D Promotion
Recall that the base-manifold of the SO(5) Landau model is S 4 with its equator being S 3 . We reinterpret S 4 and S 3 as field manifolds in the NLS model side.
Instead of using n m directly, we will represent the field as the SU (2) group element 7 whereq m ≡ {−q i=1,2,3 , 1} are quaternions that satisfy In a matrix representation, q i can be represented as The associated gauge field is simply a pure gauge on S 3 field : whereη i mn ≡ ǫ mni4 − δ mi δ n4 + δ m4 δ ni and we used 2k m=1 n m n m = 1. Suppose n m is a field on x α ∈ R 3 , the Skyrme's higher derivative term can be expressed as

1D promotion
Stacking S 3 field s along a virtual 5th direction, we form a virtual S 4 field (the middle figure of Fig.2), in which the radii of S field 3 s are continuously tuned as n a=1,2,3,4,5 realize the coordinates of S 4 field : 5 a=1 n a n a = 1.
This process demonstrates 1D promotion from 3D to 4D to manifest the idea of the dimensional hierarchy [75,84]. The SU (2) group element (35) now turns to We regard g as a transition function connecting two gauge fields on the S 3 field -equator of the virtual field manifold S 4 field : Figure 2: We first promote the S 3 field to S 4 field . Secondly, we construct a gauge field theory on the field manifold S 4 . Lastly, we derive O(5) S-NLS model Hamiltonian.
Such gauge fields are given by (20): (44) Let us assume that n a denote the field representing a map from x µ ∈ R 4 phys. to n a ∈ S 4 field , and then (44) becomes Notice that (45) can be regarded as field configurations on R 4 phys. : The corresponding field strengths on R 4 phys. are When n a are given by the inverse stereographic coordinates on S 4 phys. from R 4 phys. : (46) and (47) realize the BPST instanton configuration [93]: which carries unit 2nd Chern number. (49) simply corresponds to the stereographic projection of the Yang's SU (2) monopole gauge field (1) on S 4 [94] (see Appendix A for details).

From the non-Abelian gauge theory to O(5) S-NLS model
The next step is to adopt an appropriate gauge theory action to construct NLS model Hamiltonian. As the field strength is represented by the NLS field, we readily construct NLS model Hamiltonian, provided a gauge theory action was given. A natural choice is to adopt the pure Yang-Mills action The previous studies [8,13,14,15,16,17] already showed that substitution of F µν (47) into (50) yields the O(5) S-NLS model Hamiltonian One may notice that (51) is a straightforward 4D generalization of the Skyrme term (39). We reconsider this result below. [18] and [76,85,86,87,88,89] indicate a procedure to construct an action from a given BPS inequality. 8 Usually for a given system we have an action at first, and the BPS inequality is later derived, but here the process is reversed: BPS inequality is firstly given, and then an appropriate action is constructed so that the action can satisfy the given BPS inequality. We discuss how this idea works in the 4D Yang-Mills gauge theory. The BPS inequality is given by

BPS inequality and Yang-Mills action
whereF µν are defined asF The integral of the right-hand side signifies the second Chern number: and (53) implies where A(S 4 ) = 8 3 π 2 . From the special property in 4D, S 4,2 (56) "accidentally" coincides with the pure Yang-Mills action (50): In even higher dimensions, actions are no longer Yang-Mills type but higher tensor-field type as we shall see in Sec.4.

Construction of the O(5) S-NLS model
We next substitute (47) into the parent gauge theory action (58) to obtain 9 which is nothing but (51). Hereafter, [· · · ] denotes the totally antisymmetric combination only about the Latin indices. For instance, Note that the antisymmetricity of the Latin indices inherits the antisymmetricity of the Greek indices of the parent tensor field strengths. Similarly, the 2nd Chern number (55) turns to the winding number: which indicates the homotopy Since we started from the BPS inequality of the gauge field (53), the obtained O(5) S-NLS model Hamiltonian satisfies the BPS inequality: Some technical comments are added here. It is a rather laborious task to derive (59) by directly substituting (47) into (58), but fortunately there exists a much easier way. First, we temporally NLS model neglect the clumsy parts associated with n 5 in (47); F µν ∼ 1 2 η i mn σ i ∂ µ n m ∂ ν n n . With such simplified F µν , we next evaluate the Yang-Mills action tr(F 2 µν ) to have 1 2 (∂ µ n m ∂ ν n n · ∂ µ n [m ∂ ν n n] ). Lastly, we just recover n 5 -component in such a way that 1 2 (∂ µ n m ∂ ν n n ·∂ µ n [m ∂ ν n n] ) should respects the SO(5) symmetry, which is 1 ). This short-cut method will be useful in deriving S-NLS model Hamiltonians in even higher dimensions.
From (59), the equations of motion for the O(5) NLS field are derived as Here, λ denotes the Lagrange multiplier and is given by (64) is highly non-linear, but a solution is simply given by n a = r a with r a being the coordinates on S 4 phys. (48). The solution also carries the winding number N 4 = 1, which is expected from the discussions around (49).

O(2k + 1) S-NLS Models
In this section, we present a general procedure to construct S-NLS models in arbitrary even dimension and demonstrate the procedure to derive O(7) S-NLS and O(2k + 1) S-NLS model Hamiltonians, respectively ( Table 2).

General Procedure
The basic steps for the construction of higher dimensional S-NLS models are as follows.
1. Promote S 2k−1 field -coordinates n m to S 2k field -coordinates n a . First prepare a normalized field, n m=1,2,··· ,2k , representing a manifold S 2k−1 field . We assume that S 2k−1 field is realized as a latitude of a virtual S 2k field : where n m and n 2k+1 on the right-hand side denote the coordinates on S 2k field : 2k+1 a=1 n a n a = 1.
We also suppose that NLS field n a (x) represents a map from x µ ∈ R 2k phys. to n a ∈ S 2k field . Note that the dimension of the physical space is same as the dimension of the field space.
2. Derive SO(2k) gauge fields on the field-manifold S 2k field from the transition function. The Spin(2k) group element is expressed as whereḡ m denote some higher dimensional counterpart of the quaternions: which we call the g matrices in this paper. In the O(5) NLS model, γ i were given by the Pauli matrices, i.e. the SO(3) gamma matrices (37). Therefore, to take γ i (i = 1, 2, · · · , 2k − 1) as the SO(2k −1) gamma matrices will be a natural choice and is also implied by the expression of the SO(2k) transition function (17). The basic properties of the g matrices are given by [see Appendix B.1 also] g mḡn + g nḡm =ḡ m g n +ḡ n g m = 2δ mn , where either of σ mn andσ mn denote Spin(2k) matrix generators. By the 1D promotion (66), (68) becomes which acts as a transition function that connects the SO(2k) monopole gauge fields defined on the field manifold S 2k field : The gauge field is expressed as and the field strength 3. Make use of the BPS inequality to construct tensor field theory actions.
With the totally antisymmetric tensor field strength and its dual tensor field strength 10 the kth Chern number can be expressed as where Following to the idea of [18] and [76], we construct tensor gauge theory action so that the action can satisfy the BPS inequality: which is 11 According to the distinct decompositions of the kth Chern number (79), there exist [k/2] different tensor gauge theory actions. 12 From we can find that (81) has the symmetry and hence there are [k/2] independent actions S 2k,2l in accordance with (79). 11 Here, we added the coefficients in front of F 2 andF 2 for the later convenience. Recall that there exists the local degree of freedom indicated by λ(x) in (30). 12 See Appendix C for details about the tensor gauge field theory.

Express the tensor gauge theory action by the NLS field.
Substitute (74) into (81) to express S 2k,2l with the NLS field: which stands for the O(2k + 1) NLS model winding number associated with π 2k (S 2k ) ≃ Z [95]. The BPS inequality (80) is rephrased as Two important features of the tensor field gauge theory are inherited to the S-NLS models. One is the local symmetry and the other is the BPS inequality. As the tensor field strength action (81) enjoys the SO(2k) gauge symmetry, the S-NLS model Hamiltonian necessarily possesses the hidden local SO(2k) symmetry. Similarly, as the tensor gauge field action is constructed so as to satisfy the BPS inequality, the S-NLS model Hamiltonian automatically satisfies the BPS inequality.

O(7) S-NLS model
From the general procedure, we explicitly construct the O(7) S-NLS model Hamiltonian. The steps 1 and 2 are obvious. From (74), the SO(6) gauge field strength is given by where σ mn denote the Spin(6) generators, and (75) yields the totally antisymmetric four-rank tensor and its dualF The BPS inequality, introduces the tensor gauge field action: Here, we used A(S 6 ) = 16 (91) is essentially the 6D action constructed by Tchrakian [18]. 13 With (87) and the properties of the Spin (6) generators we can express the two terms of S 6,2 as and then The third Chern number c 3 also turns to the O(6) NLS model winding number associated with π 6 (S 6 ) ≃ Z: phys.
Notice that the second term of H 6,2 is the octic derivative term and is expanded as The first quartic derivative term of H 6,2 acts to shrink a soliton configuration, while the second term acts to expand the configuration just like the original Skyrme term. model.

O(2k + 1) S-NLS models
For the previous O(5) and O(7) cases, we have single S-NLS model Hamiltonian, but for O(2k +1), we have [k/2] Hamiltonians. In the following, we construct O(2k + 1) NLS model Hamiltonians for two typical cases, 2 + (2k − 2) and k + k. 13 Saclioglu constructed another 6D action [22] of a triple form of the field strengths, 1 6 f abc F a µν F b νρ F c ρµ , which is not positive definite in general. Meanwhile, S6,2 (91) only with even powers of the field strengths does not have such a problem.

2 + (2k − 2) decomposition
In 2 + (2k − 2) decomposition, the tensor gauge theory action is given by From the properties of the Spin(2k) generators the two terms of S 2k,2 (99) can be represented as and so we have Notice that the first term is a quartic derivative term while the second term is the 4(k − 1)th derivative term. Their competing scaling effect determines the size of soliton configurations (except for the scale invariant case k = 2). For k = 2 and 3, (102) indeed reproduces the previous O(5) (59) 14 and O(7) (95) NLS model Hamiltonians, respectively.
denotes k! terms of totally antisymmetric combination about the Latin indices, m 1 , m 2 , · · · , m k . The Spin(2k) matrix part of (113) can be expressed as Here, P mn signifies an operation that interchanges m and n, i.e. P mn (γ m γ n ) = γ n γ m , and in the present case, due to the antisymmetricity of ms, we can just replace (1 − P mn ) with 2. Besides the epsilon tensor part of (114) obviously has no effect in (113), and thereby (115) In the last arrow we assumed that k is even. Eventually, we obtain which implies

O(2k) S-NLS Models
In this section, based on the Chern-Simons term expression of the kth Chern number, we construct O(2k) S-NLS model Hamiltonians in (2k−1)D. The dimensional hierarchy of the Landau models [75,63] suggests that the dimensional reduction of the O(2k) NLS model may yield the O(2k + 1) NLS model (Fig.3). More specifically, the 1D reduction of H 2k,2l gives rise to two O(2k) Hamiltonians, H 2k−1,2l−1 and H 2k−1,2l . By removing duplications from the symmetry  Figure 3: The dimensional ladder of the higher dimensional Landau models and of the higher dimensional S-NLS models.

The Chern-Simons term and the action of pure gauge fields
As is well known, the Chern number (density) can be expressed by where L In low dimensions, (120) reads as We make use of the Chern-Simons field description of the Chern number to construct O(2k) S-NLS model Hamiltonians. Recall that the transition function (68) represents S 2k−1 field , and the associated gauge field is given by a pure gauge For the pure gauge (122), the Chern-Simons term (120) is reduced to where we used 1 0 dt (t − t 2 ) k−1 = ((k−1)!) 2 (2k−1)! and assumed that A is one-form on x α ∈ R 2k−1 phys. : We introduce tensor field for the pure gauge as and its dualÃ In (125), (−i) 1 2 p(p−1) is added so that A α 1 α 2 ···αp can be Hermitian. For instance, In a similar manner to Sec.4.1, we represent the Chern-Simons action as where p = 1, 2, · · · , k − 1.
In low dimensions, (129) yields From the BPS inequality we introduce an action of the pure gauge tensor field as Unlike the 2kD action S 2k,2l (81), S 2k−1,p (133) is made of the tensor gauge field itself (not of the field strength), and so S 2k,p does not have the SO(2k) gauge symmetry.

Explicit constructions
For (68), the pure gauge field (122) can be represented as whereσ mn denote the Spin(2k) matrix generators. Substituting (134) into (125), we can derive the NLS field expression of A α 1 α 2 ···αp . For instance Just as in the tensor gauge field strength in Sec.4, the antisymmetricity of the Greek indices of the parent tensor gauge field is inherited to that of the Latin indices of the NLS field. With such substitutions, the O(2k) S-NLS model Hamiltonian is obtained from S 2k−1,p : Similarly, the Chern-Simons term (129) turns to the winding number of π 2k−1 (S 2k−1 ) ≃ Z: As in the previous O(2k + 1) S-NLS models, the parent BPS inequality (132) guarantees the BPS inequality of the O(2k) S-NLS models: Since the pure gauge field actions (133) do not have gauge symmetries, the corresponding O(2k) S-NLS models do not either. This is a higher dimensional analogue of the non-existence of the gauge symmetry of the Skyrme model. In the following, we explicitly derive the O(2k) S-NLS model Hamiltonian for d = 3 and d = 5.

The Skyrme model: O(4) S-NLS model
For d = 3, the pure gauge field action is given by where A α and its dual fieldÃ α are represented as with Spin(4) matrix generators:σ From the following formula 18 we can readily show and so This O(4) S-NLS model Hamiltonian is nothing but the Skyrme Hamiltonian. As mentioned before, the anti-symmetricity of the indices of A αβ is inherited to the anti-symmetricity of the Latin indices of O(4) NLS field of the Skyrme term.

O(6) S-NLS models
Next we consider the case d = 5. There exist two distinct actions in this case: As the expansion coefficients, we introduced an SU (4)-generalized 't Hooft symbol,η A mn = tr(λ Aσmn ) (see Appendix B.2 for its detail properties). With unit matrix, the SU (4) matrix generators span the 4 × 4 matrix space, and then the product of two SU (4) Gell-Mann matrices can be expressed by a linear combination of the U (4) matrix generators. 19 Indeed, the product of two Spin(6) generators is explicitly given bȳ From this formula, the pure tensor gauge fields are expressed as where we used Substituting (148) The octic derivative term of H 5,1 is similarly given by (97) and the sextic derivative term of H 5,2 is (∂ α n [m ∂ β n n ∂ γ n p] ) 2 = 6((∂ α n m ) 2 ) 3 −18(∂ α n m ) 2 (∂ β n n ∂ γ n p ) 2 +12(∂ α n m ∂ β n m )(∂ β n n ∂ γ n n )(∂ γ n p ∂ α n p ).

O(d + 1) S-NLS Models
We demonstrate a general construction of the S-NLS models from the expression of the higher winding number. This method actually reproduces all of the S-NLS model Hamiltonians previously derived and also supplements other S-NLS models that eluded the tensor gauge theory based constructions.

General O(d + 1) S-NLS model Hamiltonians
The winding number of the O(d + 1) NLS model associated with is given by [95] where n a (x) are the O(d + 1) NLS model field on x µ ∈ R d subject to d+1 a=1 n a n a = 1 : S d .
Note that H 2,1 represents the well known O(3) NLS model Hamiltonian.

Equations of motion and the scaling arguments
From (158), it is not difficult to derive the equations of motion: where λ denotes the Lagrange multiplier The The scale parameter R can be considered as a variational parameter of the size of the field configuration. energetically favors a smaller size field configuration while while the second term favors a larger size configuration. These two competing effects determine an optimal size of the field configuration. More specifically, we take the derivative of E d,p (R) (168) with respect to R to obtain the local energy minimum, and the size is determined as . (169) The present S-NLS models thus realize field configurations with the finite size given by (169) (except for the scale invariant case).

Scale invariant solutions
Next let us consider the case (d, p) = (2k, k), in which the two Hamiltonians coincide, H 2k,k = H (2) 2k,k , and their competing effects balance to give scale invariant field solutions. The S-NLS model Hamiltonian (158) becomes When k is even, (170) is exactly equal to the former scale invariant Hamiltonian (117). The equations of motion (163) and the BPS equation (160) are reduced to ∂ µ 1 n [a 1 ∂ µ 2 n a 2 · · · ∂ µ k n a k ] − 1 k! ǫ µ 1 µ 2 ···µ 2k ǫ a 1 a 2 ···a 2k+1 n a 2k+1 ∂ µ k+1 n a k+1 ∂ µ k+2 n a k+2 · · · ∂ µ 2k n a 2k = 0. (171b) Especially for d = 4, (171a) reproduces the (d, p) = (4, 2) equation of (166). The equations of motion (171a) are highly non-linear equations, but inverse stereographic coordinate configuration realizes a simple solution of (171a) and also satisfies the BPS equation (171b). 22 (172) carries a topological configuration of unit winding number N d = 1. Since from the one-to-one correspondence between the points on R 2k and those on S 2k , it may be obvious that (172) represents a field configuration of the winding number 1. One can explicitly confirm this by (173) The energy density of (172) is also evaluated as which implies that (172) signifies a solitonic configuration localized around the origin.

Topological field configurations
Recall that the kth Chern number has two equivalent expressions, N 2k−1 and N 2k (Sec.2.1), which implies intimate relations between topological field configurations of O(2k) and O(2k + 1) S-NLS models of same winding number. In this Section, we demonstrate this idea to construct topological field configurations of higher winding numbers. The size of the topological field configurations is variationally determined by the scaling arguments.

Topological field configurations in odd d
The transition function g (17) 23 represents N 2k−1 = 1 associated with the homotopy π 2k−1 (S 2k−1 ) ≃ Z. Using (175), a map from r µ ∈ S 2k−1 phys. to n µ ∈ S 2k−1 field with arbitrary winding number N is obtained as Here, n µ is given by The argument of the trigonometric function in (177) is N · θ, meaning that when the azimuthal angle θ sweeps S 2k−1 phys. once, it wraps S 2k−1 field N times. For small N , (177) is given by Notice that the map with the winding number N is expressed by the N th polynomials of rs. N 2k−1 (153) is actually evaluated for (177) as Regarding n µ as the O(2k) NLS field, we treat (177) as topological field configuration on S 2k−1 phys.
with the the winding number N . To construct topological field configurations on R 2k−1 phys. , we apply the stereographic projection in the physical space: or Here, we took the radius of S 2k−1 phys. as R. Substituting (182) into the expressions of n µ such as (178), we obtain one-parameter family O(2k) NLS field configuration on R 2k−1 phys. : For instance, Substituting (184) into (153), one can explicitly confirm that (184) represents the topological field configurations of N 2k−1 = 1, 2, 3. Obviously, the radius of the sphere R corresponds to the size of the soliton configuration. Intuitively, when the size of the sphere becomes larger, the "concentration" of the soliton field around the origin will be thinner, and subsequently the size of the soliton becomes larger. Treating R as a variational parameter of n (R) µ (x), we consider minimal energy configuration in each topological sector. The previous scaling argument (169) indicates 2k−1,p (N ) which is the optimal size of the O(2k) NLS field configuration of the topological number N .

Topological field configurations in even d
With the idea of the dimensional hierarchy, we construct O(2k + 1) topological field configuration in 2kD from the set-up of (2k − 1)D: We add a radial direction to S 2k−1 phys. and consider 1D higher space, R 2k phys. (the left-figure of Fig.4). The original map from r µ ∈ S 2k−1 phys. to n µ ∈ S 2k−1 field is now transformed to (Fig.4) x µ ∈ R 2k phys. → h µ ≡ n µ (x) ∈ R 2k field .
The radial direction has no effect about the winding in (186), and the winding number associated with the map (187) is accounted for by the winding from S 2k−1 phys. on R 2k phys. to the S 2k−1 field on R 2k field (Fig.4), which is nothing but the previous (2k − 1)D winding, π 2k−1 (S 2k−1 ) ≃ Z (Fig.4). In correspondence with (178), we have Figure 4: The O(2k + 1) NLS field of the winding number π 2k (S 2k ) ≃ Z is constructed by the O(2k) NLS field of the winding number π 2k−1 (S 2k−1 ) ≃ Z.
To realize topological field configurations with field-manifold S 2k field , we apply the inverse stereographic projection in the field space (the right figure of Fig.4): Substituting (188) into (189), we obtain the O(2k + 1) topological field configurations on R 2k phys. : One can explicitly check that (190) describes topological field configurations of N 2k = 1, 2, 3 by (153). The scaling argument (169) determines the parameter R as 2k,p (N ) For the O(3) NLS model, soliton solutions of arbitrary topological numbers are given by the holomorphic functions on C ≃ R 2 [56,57], and the power of the complex coordinates corresponds to the winding number [97,56]. Meanwhile for the O(5) S-NLS model (of H 4,2 ), though the topological field configuration is simply obtained by the multiple of quaternionic analytic function [12,16], it is not easy to derive the soliton solutions except for N 4 = 1. Similarly as obtained above, the O(2k + 1) topological field configurations are a solution of the equations of motion (171a) of N 2k = 1 but other configurations of higher winding number do not satisfy the equations of motion.

Summary
We proposed a systematic procedure to construct higher dimensional S-NLS models based on the Landau/NLS model correspondence. Exploiting the differential geometry of the Landau models, we introduced the [k/2] distinct parent tensor gauge theories on the field manifold S 2k and subsequently derived the [k/2] O(2k + 1) S-NLS models on R 2k phys. . The gauge symmetry and the BPS inequality of the parent tensor gauge theories are necessarily inherited to the obtained O(2k + 1) S-NLS models. As a dimensional reduction from 2kD to (2k − 1)D, we adopted the Chern-Simons term description of the Chern number. Representing the transition function by O(2k) NLS field, we constructed the O(2k) S-NLS model Hamiltonians from pure tensor gauge fields, which indeed include the Skyrme model as its O(4) model. The obtained O(2k) S-NLS models do not possess gauge symmetries unlike the O(2k + 1) S-NLS models. From the NLS field expression of the higher winding number, we discussed a unified formulation of the S-NLS models. We derived the equations of motion and constructed an exact scale invariant solution with unit winding number. The topological field configurations with arbitrary winding number are also constructed by exploiting the idea of the dimensional hierarchy. The topological field configurations depend on the variational scaling parameter which is determined by the scaling arguments. A particular feature of the present construction is that the decomposition of the topological number necessarily yields two competing terms in the S-NLS model Hamiltonian to realize finite size soliton configurations.
Though we obtained the equations of motion, their explicit solutions have not been generally derived. The derivation of explicit solutions is not easy even for the original "simple" Skyrme model. 24 One apparent direction is to evaluate the field configurations by using numerical methods. Another direction will be a generalization of the S-NLS models based on different symmetries. While in this work we were focused on the O(N ) S-NLS models that are closely related to the SO(N ) Landau models, many Landau models with different symmetries, including supersymmetric generalizations [99,100], have been constructed with the developments of the higher dimensional quantum Hall effect in the past two decades. In view of the topological insulator [101], the present Landau models are categorized as A-class or AIII class. The topological table accommodates various cousins of the Landau models with different symmetries. It is tempting to construct other NLS models from such various Landau models. The NLS models not only exhibits deep mathematical structures but appear in important physical applications. As is well known, the Skyrme model plays a crucial role in the non-perturbative analysis of QCD. As the S-NLS model solitons emerge as anyonic collective excitations in the higher dimensional quantum Hall effect, their roles will be indispensable in understanding topological phases in higher dimensions.

Acknowledgments
I am glad to thank Yuki Amari for useful email correspondences and a lecture about the recent developments of NLS models. I am also grateful to Shin Sasaki and Atsushi Nakamula for fruitful discussions and arranging a seminar at Kitasato University. This work was supported by JSPS KAKENHI Grant Number 16K05334 and 16K05138.

A Stereographic projection and SO(2k) instanton configurations
Here, we review the stereographic projection from S 2k to R 2k and explore the relationship between the monopole gauge field on S 2k and the instanton field on R 2k [103,23,94,104].

A.1 Map from R 2k to S 2k
First we introduce a general map from R 2k to S 2k : where n a are subject to 2k+1 a=1 n a n a = 1.
We give gauge fields A µ on R 2k and A a on S 2k as Since dn a = ∂na ∂xµ dx µ , they are related as The SO(2k) monopole gauge field on S 2k is expressed as and the monopole field strength (196) and (197) are related to (73) and (74)

A.2 Stereographic projection and gauge theory on a sphere
We choose n a as the inverse stereographic coordinates on S d , r a = {r µ , r d+1 }: Through (195), the monopole configuration on S 2k is transformed to the "instanton" configuration on R 2k , (112) and (106): (200) represents the BPST instanton configuration for k = 2. Even for arbitrary k, in this paper we call (200) the "instanton" configuration, although (200) is not a solution of the pure Yang-Mills field equations except for k = 2 (Appendix A.4). Notice that the moduli size-parameter of the instanton (200) is identified with the radius of S 2k on which the monopole gauge field lives. Indeed, under the scale transformation to change the radius of sphere from 1 to R: Since the instanton configuration can be obtained by the stereographic projection of the monopole configuration on the sphere, it may be obvious that the size of the instanton is determined by the size of the sphere. From (199), we can obtain the tensor monopole field strength on S 2k [68]: and similarly the tensor instanton field strength on R 2k : where we used tr(σ [µ 1 µ 2 σ µ 3 µ 4 · · · σ µ 2k−1 µ 2k ] ) = 1 2 (2k)! ǫ µ 1 µ 2 µ 3 ···µ 2k .
Ĉ a 1 a 2 ···a 2k−1 and C µ 1 µ 2 ···µ 2k−1 that satisfŷ are obtained from the Chern-Simons term: In low dimensions, (207b) is expressed as For the instanton configuration (200), (208) becomes 25 (195) implies the transformation between the monopole and instanton tensor fields as which can be explicitly confirmed with the expressions of the fields. In (210), we introduced an important quantity or K µ a are known as the conformal Killing vectors [103] that satisfy the conformal Killing equations and the transversality condition r a K µ a = 0.
The conformal Killing vectors have the following properties: For more detail properties about K µ a , see [103]. We here discuss somewhat in detail about the formulation of the field theory on sphere by adding some more informations to [103,104]. Apparently, gauge fields on R 2k and on S 2k are generally related as The derivative on S 2k is constructed aŝ where Although r a are the coordinates on S d subject to d+1 a=1 r a r a = 1, we can treat r a as if they are independent parameters in using (218).∂ a are non-commutative operators and satisfy the SO(d + 1, 1) algebra with L ab : 26 [L ab , L cd ] = iδ ac L bd − iδ ad L bc + iδ bd L ac − iδ bc L ad .
The field strength on S 2k is given by 27 Notice the last term on the right-hand side of (224). For tensor fields, (224) may be generalized asĜ (224) can be easily confirmed for the monopole and instanton configurations. Substituting (217) and (218) into (224), we havê For the monopole field (199) and the instanton field (200), we can show Therefore, only the first term on the right-hand side of (226) survives to yieldF ab = K µ a K ν b F µν , which is (216).

A.3 Yang-Mills action and Chern number
With the area element of S d the Yang-Mills action is expressed as For the special case 2k = 4, the conformal factor on the right-hand side of (230) vanishes and (230) becomes which yields the equations of motion: 27 (224) is simply related to the three-rank antisymmetric field strength [104] F abc = i(L abÂc + L bcÂa asF ab = rcF abc .
Meanwhile, the kth Chern number is expressed as tr F a 1 a 2 · · ·F a 2k−1 a 2k ǫ a 1 a 2 ···a 2k+1 r a 2k+1 dΩ 2k In the third equation, we used (215) and (217). The Chern number of the instanton configuration on R 2k is exactly equal to that of the monopole configuration on S 2k . Indeed for instance, (199) and (200) yield c k = 1 in (233).

A.4 Equations of motion for the monopole fields and the instanton fields
For the monopole gauge fieldÂ a (199), the field strength is obtained from (224): where we used where we used∂ (236) is expected from the previous result (10). Meanwhile, the instanton configuration (200) satisfies where Notice that for the special case 2k = 4, the second term on the left-hand side of (238) vanishes, and the instanton configuration is a solution of the pure Yang-Mills field equation: where Spin(2k) generators are given by Several properties of g matrices are g mḡn + g nḡm =ḡ m g n +ḡ n g m = 2δ mn ,

The original 't Hooft symbol
The SO(4) gamma matrices and matrix generators are expressed as 28 where η i mn andη i mn are the 't Hooft symbols [77]: The Pauli matrices are inversely represented as The Spin(4) matrix generators satisfy the self-dual and the anti-self-dual equations, and 28 The components of σmn andσmn are The above relations are rephrased as the properties of the 't Hooft symbol: and Note that ǫ ijk = −i 1 2 tr(σ i σ j σ k ) are the structure constants of the SU (2). Except for (258c) and (258b), all relations also hold forη i mn :

C Tensor gauge field theory
Here, we review tensor gauge field theories in even dimensions mainly based on [18,21,23] .

C.2 Gauge Symmetry and covariant derivatives
Under the gauge transformation the tensor field strength (280) is transformed as The covariant derivative of the tensor field strength is introduced so as to satisfy and such a covariant derivative is simply constructed as One may easily check that (286) transforms as (285) under (283a) and (284). Note that the covariant derivative linearly acts to the original constituent 2-rank field strength of the tensor field strength. For instance, where index µ in the second term is not included in the antisymmetrization.

C.3 Bianchi Identity and equations of motion
The original Bianchi identity is readily verified from the definition of the field strength, F µν = ∂ µ A ν − ∂ ν A µ + i[A µ , A ν ]. For tensor field strength, (288) is generalized as 29 (282a) was utilized in 8D tensor gauge theory of [39] to realize a 7(+1)D Skyrmion from the Atiyah-Manton construction.
One may easily verify (289) with the linearity of the covariant derivative (287) and the original Bianchi identity (288). We introduce the (Euclidean) tensor field theory action as Since tensor field strength is originally made of the field strength, we should take a variation of S with respect to A µ to derive equations of motion: where For instance, l = 1 : G µν = F µν , l = 2 : G µν = F µνρσ F ρσ + F ρσ F µνρσ = {F µνρσ , F ρσ }, l = 3 : G µν = F µνρσκτ F ρσκτ + F ρσ F µνρσκτ F κτ + F ρσκτ F µνρσκτ .
The self-dual tensor field satisfies From (295), the self-dual tensor field realizes a solution of the equations of motion (291).