$SU(3)$ Knot Solitons: Hopfions in the $F_2$ Skyrme-Faddeev-Niemi model

We discuss the existence of knot solitons (Hopfions) in a Skryme-Faddeev-Niemi-type model on the target space $SU(3)/U(1)^2$, which can be viewed as an effective theory of both the $SU(3)$ Yang-Mills theory and the $SU(3)$ anti-ferromagnetic Heisenberg model. We derive the knot solitons with two different types of ansatz: the first is a trivial embedding configuration of $SU(2)$ into $SU(3)$, and the second is a non-embedding configuration that can be generated through the B\"{a}cklund transformation. The resulting Euler-Lagrange equations for both ansatz reduce exactly to those of the $CP^1$ Skyrme-Faddeev-Niemi model. We also examine some quantum aspects of the solutions using the collective coordinate zero-mode quantization method.


I. INTRODUCTION
It is of great importance to consider SU (3) generalizations of the O(3) NLσ model, because they may possibly play a crucial role in a relevant limit of fundamental theories like low energy limit of the SU (3) pure Yang-Mills theory or continuum limit of the SU (3) Heisenberg models. The main achievement of the present paper is that we successfully construct novel soliton solutions which are called as Hopfions on the flag manifold F2 = SU (3)/U (1) 2 . Hopfions are topological solitons with knotted structure characterized by a Hopf invariant. It has been pointed out that the knotted structures appear in various branches of physics: QCD [1][2][3], BEC [4,5], superconductor [6], liquid crystal [7] and so on.
A typical theory realizing Hopfions is the Skyrme-Faddeev-Niemi (SFN) model [2,8], which is an O(3) nonlinear σ-model in (3 + 1)-dimensional Minkowski space-time, i.e., the scalar field theory defined by the Lagrangian density where M has dimension of mass, e is dimensionless coupling constant and n is a three components vector with unit length, i.e., n · n = 1. The second term of the right-hand side in (1), the Skyrme term, was introduced by Faddeev [8] so as to make the theory meet the Derrick's criteria for the existence of stable soliton solutions. The solutions of toroidal shape which are possessed of a lower Hopf number H = 1, 2 were firstly found under an axial symmetric ansatz by Gladikowski and Hellmund [9], and also by Faddeev and Niemi [1]. After that, the higher charge Hopfions including the form of twisted tori, linked loops and knots were constructed by means of full 3D energy minimization [10][11][12][13][14]. Faddeev and Niemi discussed in detail that, by means of the Cho-Faddeev-Niemi-Shabanov decomposition, the SFM model (1) can be derived as an effective theory which describes the confinement phase of the SU (2) pure Yang-Mills theory [2]. From the point of view, the Hopfions are considered as a natural candidate of glueballs which can be interpreted as closed fluxtubes. The model is sometimes referred to as the CP 1 SFN model, based on a formula that the Lagrangian can be described in terms of a complex scalar field via the stereographic projection S 2 → CP 1 , i.e., where u is a complex scalar field and ∆ = 1 + |u| 2 . For finite energy configuration, the field n has to approach a constant vector at space infinity. This makes the points at the infinity to be identical and the space R 3 is compactified to S 3 . The field n defines a mapping S 3 → S 2 and field configurations are characterized by an integer, Hopf invariant, the element of π3(S 2 ) = Z. Since the invariant is nonlocal, an integral form of the invariant cannot be written by n nor u, and in order to define it, we need introduce the complex vector Z = (Z0, Z1) T with | Z| 2 = 1 which satisfies u ≡ Z1/Z0. Then, the Hopf invariant can be defined as In this paper, we construct Hopfions in a generalization of the SFN model to the case of SU (3), the gauge group of QCD. In the SU (N + 1) case where N ≥ 2, there are several possibilities for the field decomposition associated to dynamical symmetry breaking patterns. In particular, the two options, the maximal case SU (N + 1) → U (1) N [3] and the minimal case SU (3) → U (2) [15,16], have been well studied. According to the options, the SFN type models on the relevant target spaces FN = SU (N + 1)/U (1) N and CP N = SU (N + 1)/U (N ) were proposed in [3] and [17] respectively. Note that CP 1 = F1 = SU (2)/U (1) is equivalent to S 2 , the target space of the standard SFN model. Unfortunately, the CP N model cannot possess knot solitons as a static stable solution because of the relevant homotopy group being trivial, i.e., π3(CP N ) = 0 for N ≥ 2. In 2(N + 1) dimensional space-time, however, higher dimensional Hopfions associated to π2N+1 CP N are discussed [18]. In addition, if N is odd, 3D time-dependent nontopological solitons, Q-balls and Q-shells, are obtained in the CP N model with V-shaped potential [19].
Contrary to the case of CP 2 , the third homotopy group of the flag manifold is nontrivial, i.e., π3(F2) = Z. Thus it is expected that there exist Hopfions in the F2 SFN model, which is composed of the F2 nonlinear σ-model and quadratic terms in derivatives. The main purpose of the present paper is to confirm the existence of the F2 Hopfion and understand their detailed structure. It is recently found that the 2-dimensional F2 nonlinear σ-model possesses vortex-like solutions (2D instantons) both of embedding type [20] and of genuine (nonembedding) type [21,22]. The Hopfions considered in this paper may be the vortices with knot structure. Note that in [21] the so-called Kalb-Ramond field is introduced with specific coefficients so that the model gets integrable [23,24]. Though the Kalb-Ramon field naturally appears for some continuum limit of the SU (3) antiferromagnetic spin chain, for the moment we do not consider the field. The reason is that if one derives nonlinear σ-model from other fundamental theories, it is not so clear whether the field can naturally appear. The genuine solutions are constructed by the CP 2 Din-Zakrzewski tower generated by the Bäcklund transformation [25] and it implies the solution is a composite of solitons and anti-solitons.
The F2 nonlinear σ-model are derived from the SU (3) antiferromagnetic Heisenberg model on square lattice [26], triangular lattice [27] and 1D chain [28] as an effective model and has been supposed to describe some phenomena of a quantum spin-nematic [27] and a color superconductor in high dense quark matter [29]. This work will be able to be applied to such condensed matter physics as well as QCD.
The paper is organized as follows: in the Sec.II we introduce the F2 SFN model and some important quantities, in the Sec.III we will give a parametrization which is convenient to solve the equations of motion. In the Sec.IV, the formal Euler-Lagrange equation is derived and we solve the equation with some ansatzes. A belief analysis of some quantum aspects of the solutions is given in the Sec.V. We conclude with the Sec.VI.

II. THE MODEL
The fundamental degrees of freedom of F2 nonlinear σ-models can be given by the su(3) valued fields, called color direction fields in the context of QCD, defined as where U is an element of SU (3) and the matrices ha are the Cartan generators in su (3). The F2 SFN model is defined by the Lagrangian where the angle bracket denotes the inner product on su (3), i.e. A, B = Tr A † B for A, B ∈ su(3). The second rank tensors are defined as and the 2-forms F a = 1 2 F a µν dx µ ∧ dx ν are called the Kirillov-Kostant (KK) symplectic forms. The Lagrangian (5) is invariant under the left global SU (3) transformation such that U → gU, g ∈ SU (3), and the local U (1) 2 transformation such that U → U k, k ∈ U (1) 2 . According to the symmetries, one can understand that the target space of this model is the coset space SU (3)/U (1) 2 equivalent to the flag manifold F2. For simplicity, we employ the length unit (M e) −1 and the energy unit 4M/e. Then the static energy functional associated with (5) is given by Since the energy consists of both quadratic and quartic term, three dimensional particle-like configurations evidently evade the Derrick's no-go theorem. We reformulate the energy functional (7) into a more tractable form which is given in terms of only the off-diagonal components of the Maurer-Cartan form U † ∂µU . To perform the reformulation, we decompose the Maurer-Cartan form in terms of the SU (3) Cartan-Weyl basis as where we use the basis of the form Since the basis are orthonormalized, one can write where A a µ are real and J −p µ = J p µ * . Notice that the KK forms can be written as F a = dA a where A a = A a µ dx µ , so that the 2forms are closed. Under the gauge transformation U → U k with k = exp (iθ a ha), A a µ transforms as a gauge field and J p µ a charged particle, i.e., where α p a is an a-th component of the root vector corresponding to ep and J p = J p µ dx µ . Now the root vectors are given by It is known that the quadratic term in (5), the nonlinear σ-model, can be written by only the off-diagonal components J p µ . In addition, one can write the KK forms in terms of the off-diagonal components as F a = −i p α p a J p ∧ J −p . Thus the energy of a static configuration can be written in terms of the off-diagonal components as is pure imaginary, and then the energy functional is positive definite. It is worth to note that, similar to the CP 1 case [30], the energy functional (12) can be interpreted as a gauge fixing functional for a nonlinear maximal Abelian gauge, without making the Abelian subgroup components fixed.
For finiteness of the energy functional the fields na should approach to constant matrices at space infinity, so that the space R 3 is topologically compactified to S 3 and the fields na define a map S 3 → F2 = SU (3)/U (1) 2 . Consequently, the finite energy configurations can be characterized by elements of the homotopy group π3 SU (3)/U (1) 2 = Z. The corresponding topological charge, Hopf invariant, is given by where The Hopf invariant (13) is nonlocal since A a µ cannot be written in terms of the fields na, and therefore (13) does not possess the local U (1) 2 symmetry. Moreover, in general, the each constitution, i.e., the Abelian Chern-Simons (CS) terms or Γ, is non-topological. The Hopf invariant can be constructed by means of the Novikov's procedure [31] via the isomorphism between π3 SU (3)/U (1) 2 and π3 (SU (3)) which indicates 3 is the winding number of the map U : S 3 → SU (3). Note that since the winding number is equivalent to the Chern-Simons term for the SU (3) flat connection U † dU , even the SU (3) case the Hopf term is given by the non-Abelian Chern-Simons term as discussed in the SU (2) case [30,32]. For non-symmetric manifold, unlike Hermitian symmetric spaces, the Kähler form λ is not closed, i.e., dλ = 0, in general. For the flag manifold, the Kähler form can be defined as where the coefficients Bp are real constants [23]. The so-called skew torsion T = dλ is given by the form Under the local U (1) 2 transformation, the Kähler form is invariant and the torsion too. Note that in the 2-dim F2 nonlinear σ-model, the solutions of the Euler-Lagrange equation make the Kähler form topological, and then the torsion disappears. By analogy, in this paper we consider a class of configuration that satisfies the torsion-free condition T = 0.

III. PARAMETRIZATION
In order to make the analysis transparent, let us parametrize the SU (3) matrix U in terms of complex scalar fields which equivalent to the local coordinates of the target space F2. As a result of the Iwasawa decomposition (see e.g. [33]), one can construct the SU (3) matrix from a triangular matrix in SL(3, C). Therefore we begin the parametrization with the 3 × 3 lower triangular matrix with the determinant of unity: where χi are complex functions with χ1 and χ4 being finite. In the 2-dim F2 nonlinear σ-model, it is enough to consider a triangular matrix with all diagonal components being unity. Note that, however, in this case we employ the general matrix (17) in order to derive the degrees of freedom of the full SU (3) rather than the target space F2, because the relevant Hopf invariant should be given by an arbitrary SU (3) matrix due to the nonlocalness. We write X in terms of column vectors as X = ( c1, c2, c3). The SU (3) matrix can be obtained by the Gram-Schmidt orthonormalization for the vectors ci. For simplicity, we normalize the orthogonalized vector vj, which are constructed from ci, by imposing the normalizing conditions | vj| = 1 which reads Actually, we have the five complex functions and the two constraints (18), namely 8 degrees of freedom. Therefore the matrix U = ( v1, v2, v3) can describe an arbitrary SU (3) matrix.

IV. EQUATION OF MOTION AND HOPFIONS
Firstly we derive the formal Euler-Lagrange equation, and then implement two classes of configuration that satisfy the torsion-free condition T = 0. The Euler-Lagrange equation is equivalent to the conservation of the Noether current Jµ associated with the global SU (3) transformation, i.e. ∂µJ µ = 0. The current takes the form If we factorize the current as Jµ = W BµW † , the equations of motion can be written as We again decompose the Maurer-Cartan form W † ∂µW as The composite vector fields C a µ is a gauge transformed version of Aµ and K p µ is of J p µ . Then, one finds Bµ consists of the only off-diagonal components as For short notation, we introduce R p µ = a α p a C a µ and G p µν = a α p a F a µν . Then, the equation (24) can explicitly be written as for ∀q ≡ 1, 2, 3 (mod 3) and their complex conjugations. Since the equations (27) are very complicated and highly nonlinear, we shall introduce two different ansatzes that satisfy the torsionfree condition T = 0 as mentioned earlier.

A. Trivial CP 1 reduction
The first class is a trivially embedded configuration an F1 = CP 1 Hopfion into F2 space. It can be implemented by imposing two of the three scalar fields trivial. Here we set u1 = u3 = 0 and also write u2 = u without loss of generality. Then, the complex vectors Za are written by the function u(x) as where ∆ = 1 + |u| 2 . The currents K p µ are given by (29) and the skew torsion T is thus vanished. It is directly to see that the equations of motion (27) for q ≡ 1, 3 are automatically satisfied and that for q ≡ 2 reduces to where for convenience we introduced Rµ and Gµν as The static energy for the configuration (28) is given by Both the equation of motion (30) and the energy (33) are exactly same as those of the CP 1 SFN model (1). Further, by a definition (13) coincides with the CP 1 version (3), i.e., Form of the static energy (33) and the Hopf charge (34) are perfectly same as the CP 1 model. Apparently this coincidence is induced because the trivial embedding configuration (28) corresponds to a CP 1 submanifold in F2. Next we examine another class of configuration which may exhibit more nontrivial nature and see what happens for the equations and others.

B. Nontrivial CP 1 reduction
For the trivial embedding (28), we observed that two pairs of the currents K p µ vanished, i.e., K ±1 µ = K ±3 µ = 0. Here we examine the case of just one pair of the current K ±2 µ = 0, while K ±1 µ , K ±3 µ remain finite. This setting automatically satisfies the torsion-free condition. Note that the result is independent for the choice of the components; for a different pair one just repeat the same prescription by permuting the vectors Za. The condition K ±2 µ = 0 reads which is satisfied if u2 is a function of u1, i.e. u2 = f (u1), and u3 is given by u3 = f (u1) where the prime stands for the derivative in u1. It means the independent field is only u1, so that the Euler-Lagrange equation seems to be an overdetermined system. In order to resolve the overdeterminedness, we consider the case that the Euler-Lagrange equations for q ≡ 1, 3 are proportional to each other. It is realized when the ratio ∆1/∆2 is a constant Note that we leave the equation for q ≡ ±2 intact because q ≡ 2 is now special due to the constraint K 2 µ = 0. By comparing the order of u1 in ∆i's, it implies that Since we are not interested in embedding solutions now, we omit the case where u1 is a constant and we get where ϕ ∈ [0, 2π] is a constant. Note that due to U (1) symmetries the constant ϕ can take an arbitrary value. For simplicity, we choose ϕ = π and write u1 = √ 2u. Then, the triplet vectors can be written as It is worth to note that the three vectors are linked by the Bäcklund transformation, i.e., where P+Za = ∂uZa − Z † a ∂uZa Za. Such relation between the triplet vector are observed in the nonembedding solutions of the two dimensional F2 nonlinear σ-model [21,22]. The currents K p µ are given by the form It implies that the Euler-Lagrange equation (27) for q ≡ 2 is automatically satisfied. In addition, we obtain R 1 µ = R 3 µ = −Rµ and G 1 µν = G 3 µν = −Gµν , then one can easily observe that the equations (27) for both q ≡ 1, 3 are reduced to the complex conjugation of (30). To see this, one can use the fact that Rµ and Gµν are real. Now we got somewhat surprising observations; These results clearly mean that it is not only in the trivial embedding case but also in the nonembedding case that all the known Hopfion solutions u in the CP 1 SFN model are also solutions. However, we should remark with emphasis that their structure is perfectly different because of the parametrizations (38) and (28). The configuration (38) possesses the static energy which is exactly four times greater than (28). For the evaluation of the relevant Hopf number, we parametrize u = Z1/Z0, e iϑ 1 = Z 2 0 /|Z0| 2 , and ϑ4 = 0. Then, we obtain 6A, and therefore Note that (34) is given by only the sum of the Abelian CS terms, which now become topological, because the configuration (28) satisfies Γ = 0 as well as the torsion-free condition T = 0.
The F2 nontrivial Hopfion with Hnontri = 4n ( n is an integer) might be interpreted as a molecule state of some lower charged ones with the Hopf number H = n such as the trivial Hopfion, sitting on top of each other with no binding energy. It is not true, however. Note that it does not imply there exists no interaction between the constituents and it is not possible to remove one of them from others because the form of the Hopf charge (42) is defined by the different field variables (38), not the (28). Such situation has been observed in an SU (N ) Skyrme model [34].

V. ISO-SPINNING HOPFIONS
We have seen that the EL eq. in both the trivial embedding and nonembedding ansatz solved by same function u and their energy functional or Hopf invariant proportional like (41) and (42). Their structure is clearly different, but there might be someone who considers these solutions have almost same features. Now we shall show that their quantum natures are quite different. In this section, we give a brief analysis to demonstrate noticeable differences in the quantum aspects based on the collective coordinate quantization of the zero modes. We consider an adiabatic iso-rotation associated to the SU (3) global symmetry, i.e., the time dependent transformation The Lagrangian can be written as where Ecl is the static energy of the Hopfion, the dot denotes the time derivative, i.e.,β = dβ/dt, and The energy collection depends on the scale length unit r0 = (M e) −1 .
In order for the integral in (43) to be finite, β †β and ma should commute to each other at space infinity. Since the fields ma( x) approach constant elements of u(1) × u(1) as x goes to infinity, β †β should also be in u(1) × u(1) and therefore can be written as where ωa denote the angular velocity in the iso-space. We chose the coefficients in (45) consistent with the definition of the SU (3) Euler angle [35]. The quantum Lagrangian (43) can be written as the quadratic form of the angular velocities where ω T = (ω1, ω2) and The moment of inertia (m.o.i) are explicitly obtained as following: • the non-trivial reduction case • the trivial reduction case By the Legendre transformation of the Lagrangian (43), the Hamiltonian are derived as H = ωiPi − L with the canonical momentum defined by In the nontrivial reduction case, the Hamiltonian are straightforwardly obtained as where we wrote I11 = I. Therefore the Hamiltonian are obtained as Consequently, the Hopfions of the embedding type inherit quantum property of the CP 1 Hopfions, possessing only one quantum number. The quantum property of the two types of the Hopfion solutions seems quite different, at least qualitatively, as a reflection of their symmetries.

VI. CONCLUSION
We have studied Hopfions in the SFN model on the target space F2 = SU (3)/U (1) 2 which is an SU (3) generalization of the standard SFN model whose target space is CP 1 = SU (2)/U (1). By analogy of the 2-dim F2 nonlinear σ-model , we introduced two classes of configuration which satisfy the torsion-free condition, i.e., the trivial embedding of the CP 1 Hopfions and the SU (3) genuine one which can be constructed through the Bäcklund transformation.
For both the cases, the Euler-Lagrange equation reduces to that of the CP 1 SFN model. In addition, though the Hopf invariant is equivalent to the Chern-Simons term for the SU (3) flat connection, it is shown that the invariant is given by the Chern-Simons terms for Abelian components of the flat connection if one substitutes the configuration into the Hopf invariant.
The most important open problem is probably stability of the genuine solutions. Their energy is four times greater than that of the embeddings, comparing the configurations given by the same scalar function. It is also of importance to understand mathematical implication of the torsion-free condition in detail and to confirm whether there exist Hopfions outside the condition.
We examined the quantum aspect of the Hopfions based on the collective coordinate quantization and found that their aspects are quite different for the choice of the ansatz.
For the estimation of the physical spectrum of the glueball, we need to perform more complete analysis of the collective coordinate quantization, including the space rotational modes and also the discussion of their statistical property. The analysis of this subject is now in progress and the results will be reported in a subsequent paper.