BPS Sphalerons in the $F_2$ Non-Linear Sigma Model

We construct static and also time-dependent solutions in a non-linear sigma model with target space being the flag manifold $F_2=SU(3)/U(1)^2$ on the four dimensional Minkowski space-time by analytically solving the second order Euler-Lagrange equation. We show the static solutions saturate an energy lower bound and can be derived from coupled first order equations though they are saddle point solutions. We also discuss basic properties of the time-dependent solutions.


I. INTRODUCTION
Non-linear sigma (NLσ) models have been extensively studied in various branch of physics. Specifically, the O(3) NLσ model in two dimensions has been identified as a good toy model of the SU (2) pure Yang-Mills theory in four dimensions, because they share many important features such as conformal invariance, dynamical mass gap and existence of the instanton solution. Moreover, the O(3) NLσ model (with higher differential terms) can be derived as a low energy effective theory for the SU (2) Yang-Mills theory [1,2], Heisenberg model [3] etc.. The topological soliton solutions like the instantons and the Hopfions play a crucial role for the nonperturbative aspects in the theories [4,5].
It is of great importance to consider SU (N ) generalizations of the O(3) NLσ model, because they share some basic properties with the SU (N ) Yang-Mills theory and may possibly be derived from fundamental theories like the SU (N ) Yang-Mills theory or the SU (N ) Heisenberg model. There may be several variants, and in particular two feasible cases have widely been examined; the CP N−1 and FN−1 NLσ model. The models are NLσ models whose target spaces are the complex projective space CP N−1 = SU (N )/U (N −1) and the flag space FN−1 = SU (N )/U (1) N−1 , respectively 1 . Note that the CP 1 (= F1) NLσ model is equivalent to the O(3) NLσ model at least in classical level. Indeed, the CP N−1 models arose as an effective theory from the SU (N ) Yang-Mills theory with the minimal case of Maximal Abelian gauge (MAG) [6], ferromagnetic SU (N ) Heisenberg model [7,8] and so on [9][10][11][12][13]. On the other hand, the FN−1 model was considered from the SU (N ) Yang-Mills theory by means of the maximal case of MAG [6,14]. Moreover, the F2 model was derived from the SU (3) anti-ferromagnetic Heisenberg model in 1+1 dimensions [15,16] and in 2+1 dimensions [17,18].
In this paper, we study classical solutions of the F2 model. There are quite few studies for classical solutions of the FN−1 models, even in the N = 3 case, though ones of the CP N−1 models have been studied for all N since the late 70s [19,20]. To our knowledge two seminal papers about classical solutions of the F2 model are the ones. In [21], it was shown that the BPS equation is solved by the F1 instantons embedded into the target space F2. In [22], the author studied classical solutions of the F2 model with an additional term called Kalb-Ramond field which is non topological term and then contributes to the Euler-Lagrange equation. In some special choices of the coupling constant, the general static solutions were discussed in context of the integrability [23]. Note that the Kalb-Ramond field can naturally appear when one derives the F2 model from the SU (3) spin chain [15,16]. In contrast, from the SU (3) Yang-Mills theory or the SU (3) Heisenberg model on lattice, the term may not be provided.
Main goal of this paper is to find genuine (which means nonembedding) solutions in the F2 NLσ model without the Kalb-Ramond term. The model in (3+1)-dimensional Minkowski space-time is defined by the Lagrangian density Note that, the double vertical line norm for (composite) vector fields denotes the Minkowski norm with the metric gµν = (−, +, +, +), thus the square of the norm is not positive definite. Here, the fields Za (a = A, B, C) are three components complex vectors which form a complete basis. Therefore we can construct SU (3) matrix from U (3) matrix U = (ZA, ZB, ZC) and the fields satisfy the orthonormal condition and the completeness condition i.e.
where 13 is the 3 × 3 identity matrix. The Lagrangian is invariant under the left global U (3) transformation such as U → gU, g ∈ U (3). In addition, it is also invariant under the local U (1) 3 transformation i.e. three U (1) phase rotations Za → Z ′ a = e iΘa Za, a = A, B, C . The paper is organized as follows. In Sec.II, we briefly review the previous study of an embedding solution of the model. In Sec.III, the basic information for constructing the solutions including the parametrization, the topological charges are given. Sec.IV presents both the static and the time-dependent solutions of the model. Some properties of the energy, for example an energy bound for the static solutions, are devoted in Sec.V. Summary and discussion are in Sec.VI.

II. EMBEDDING SOLUTIONS
As we mentioned, the authors of [21] found a BPS solution which is just an F1(= CP 1 ) instanton embedded into the target space F2. Let us begin with a brief review of their work with introduction of some notations.
The discussion is based on an analogy with the BPS sector in the CP 2 model. For sake the reader's understanding, we introduce the covariant derivative and write the static energy per unit length such as Each term in (5) can be viewed as the static energy of the CP 2 model linked by the orthonormal condition (2). Thus, by simply applying the BPS bound of the CP 2 model to the three terms respectively, an energy lower bound may be derived as where Na is the topological degrees of the map Za : Note that there is a constraint among the topological degrees of the form NA + NB + NC = 0.
The equality in (6) is satisfied if the following (implicitly) coupled first derivative equations are simultaneously satisfied: Unfortunately, the solutions of (9) are given by only a very limited class of configurations, i.e. an F1 instantons embedded into the target space F2, for which one of the vectors Za is fixed as a trivial one. By performing a proper global transformation, the solutions can generally be written as with ∆ ≡ 1 + |p(z)| 2 /|q(z)| 2 . Here, we have introduced the standard complex variables and the functions p(z) and q(z) which are polynomials of z with no common roots. The solution (10) has apparently NC = 0, and therefore, the energy is given by where NA(= −NB) is given the highest power of z in either p(z) or q(z). The embedding solutions are unique possibility of solutions in the equation (9). For, (9) requires all the vectors Za are holomorphic or anti-holomorphic, but there is no pair of three components (anti-) holomorphic vectors without constant vectors. If one wants to get more general solution, obviously different strategy has to be implemented. Our approach for the problem is to solve the Euler-Lagrange equation, without touching first-order equations. For the solutions, of course, the energy bound (6) is no longer saturated.
The notable thing is that the nonembedding solutions satisfy a new coupled first-order equations different from (9). As a result, we obtain new energy lower bound which of course is different from (6). Note that the energy is higher than that of embedding solutions and the new energy bound contains a non-topological term. The detailed analysis will be thoroughly discussed in Sec.V. Existence of such solutions will give us deeper understanding of the BPS structure of NLσ models whose target space is non-symmetric space.

III. PRELIMINARIES
This section is preliminaries for analyzing the Euler-Lagrange equations and some properties of the energy functional.

A. Parametrization
In order to simplify the analysis, we parametrize the vectors Za in terms of complex scalar fields [24]. It can be realized by means of the isomorphism where B+ is the Borel subgroup of the upper triangular matrices with determinant being equal to one. In geometrical point of view, the complex scalar fields mean the local coordinates of the manifold F2. The coordinates can be introduced though a mapping SU (3)/U (1) 2 → SL(3, C)/B+, which is an extension of the stereographic projection i.e. S 2 → R 2 . By a formal result of the map, one can write an element of F2 in terms of local coordinates ui, i = 1, 2, 3 as the lower triangular matrix Since det X = 1, X is an element of SL(3, C). Note that, however, it is not necessary for X to be an unitary matrix. Next, in order to obtain the SU (3) matrix U or the vectors Za parametrized in terms of the scalar fields, we consider the inverse of the mapping, i.e. SL(3, Then, one can construct U from X by means of the so-called Iwasawa decomposition: any element of SL(3, C) may be factorized into an element of SU (3) and B+ in a unique fashion, up to torus elements of U (1) 2 . The procedure is as following. We regard X as element of SL(3, C) which can be expressed in terms of the column vectors such as In this case, the Iwasawa decomposition can be proved by the Gramm-Schmit orthogonalization process for (16). Then one obtains mutually orthogonal vectors where the inner product is defined as By normalizing the vectors (17), one obtains an orthonormal basis. Therefore, we can define Za = 1 √ (ea,ea) ea, a = A, B, C, and they can explicitly be written as For later convenience, we express the off-diagonal components of the one-form Z † a dZ b as

B. Topological properties
We consider static configurations in two space dimensions (x 1 , x 2 ) and also with wave components traveling along x 3 axis with the speed of light. The vectors (19) provide a mapping from R 2 (or R 2 × R 2 ) into F2. However, for finiteness of the energy, the vectors should go to a constant at infinity on the x 1 x 2 plane. Thus, the plane can be identified as S 2 . Since both the static and time-dependent configurations localize only on the plane, they are topologically classified into homotopy classes of a mapping S 2 → F2 , i.e. π2(F2).
In addition, there exists an useful theorem (see e.g. [19]); π2(G/H) = π1(H)G where π1(H)G is the subset of π1(H) formed by closed paths in H which can be contracted to a point in G. Therefore, these configurations are characterized by the second homotopy group which guarantees that the solutions are classified by two independent integers. Note that there are three topological degrees defined in (7), but independent degrees are just two due to the identity (8).
We can also define a topological invariant by means of Kähler structure of the flag space F2, though F2 is not Kähler manifold. Note that the non-symmetric manifolds, unlike symmetric manifolds, possesses a family of invariant metrics including both Kähler and non-Kähler metrics. In order to simplify the discussion, we consider the Kähler case. Then the corresponding Kähler potential can be defined as where Kj = mj log ∆j , j = 1, 2, with mj being a positive constant. Since the manifold is now the Kähler one, it possesses the closed two-form Ω i.e. dΩ = 0. Here, while the operator ∂ and∂ are called the Dolbeault operators. The condition dΩ = 0 is equivalent to If one divides the two-form into two parts Ωj = i∂∂Kj ,they are given by The topological invariant Q is defined in terms of the integral of the Kähler two-form The relation of the topological invariant and the winding numbers is directly calculable by the integration Consequently, after an appropriate normalization, i.e. m1 = m2 = 1 2π , the topological invariant is characterized just by two integers NA and NC as vi.
which of course is consistent with the formal argument (23).
There are exact expressions of the topological invariants corresponding to (23), However, the field configurations in this model are characterized by two integers and it is not so straightforward to find proper form of the topological charge interpreted as the net number of solitons. The definition (31) seems suitable for the net number of solitons for the parametrization (19).
As we shall see in Sec.V, the static energy of solutions is proportional to (31) like standard BPS type solutions and then we call (31) as the topological charge.

IV. SOLUTIONS OF THE EULER-LAGRANGE EQUATION
For finding genuine solutions, we solve the Euler-Lagrange equation of the Lagrangian (1), instead of first order coupled equations. The Euler-Lagrange equation is given by Here Jµ is the Noether current associated to the global SU (3) symmetry. The equation (32) can be rewritten as where Aµ is a flat connection defined by From (34) with (33) we obtain the coupled equations and also their complex conjugations. Plugging the explicit form of the vectors (19), (20) and the one-form (21) into the equations, we finally obtain the equations for the scalar fields where the explicit form of ∆1, ∆2 and Pij i, j = 1, 2, 3 are given in (20) and (22), respectively. Since the equations are quite complicated and highly nonlinear, for the moment we introduce the simplified ansatz; all the complex scalars ui depend only on z and y+ (y± is defined It is easy to see that the ansatz satisfies One can directly check that the second equation (38) is automatically satisfied by the ansatz (40). (37) and (39) are certainly simplified Therefore, if we find the field configurations satisfying the intriguing case where all the scalars aren't constants, the solutions can generally be written as where pi(z) and qi(z) are irreducible polynomials of z and u1 isn't proportional to u2. The winding number of the scalar field ui is equal to number of the poles of the scalar field including those at infinity and thereby According to the derivation of the topological charge in the CP N model [19], the topological degrees NA and NC are given by pair of the winding numbers ni as NA = max(0, n1, n2) − min(0, n1, n2) , (47) NC = max(−n2, −n3, 0) − min(−n2, −n3, 0) .
As we summarize in Table I, the topological degrees (NA, NC) are thus defined by pair of the winding numbers with opposite signs.
The most simple choice of the static solution (45) seems where ci are non-zero complex constants. In Fig.1, we plot the topological charge density of (49). In the case of (n1, n2) = (1, 2), it is lump shaped of which the peak locates at the origin, while for higher winding numbers, the solutions always exhibit the crater like structure. Anisotropic configuration can also be examined. The solution FIG. 3. Energy surface for u1 = 1−a 1 a 1 z 3 , u2 = 1−a 2 a 2 z 4 , u3 = 1−a 3 a 3 z. From left to right a3 = 0.5, a2 = 0.5, a1 = 0.5 exhibits non circular shape of the density (see Fig.2). For growing the parameter c1, the lumps collide and scatter with the right angle. Following to the prescription for construction of the CP N vortices [25][26][27], we examine a class of time-dependent solutions which is made by a product of localizing component on the x 1 x 2 plane and traveling wave being parallel to x 3 with speed of light, i.e. ansatz of the form ui = fi(z)wi(y+), i = 1, 2, 3. Again, the nonembedding time-dependent solutions are not allowed for the case that one or more of the functions fi and wi remain constant. When all fi and wi are the functions, by substituting the ansatz into (44) one easily obtain the conditions µ ZB is to ZC. Consequently, the vectors correspond to the tower of the general solution in the CP 2 model generated through the Bäcklund transformation [20,28]. It implies that for the solution (45) and (53), ZA can be recognized as an instanton, ZC as an anti-instanton, and ZB as an instanton-anti instanton bound state. Similar types of solutions are already discussed in [22], but the model the author employed is different from ours because it contains a non-zero Kalb-Ramond field which contributes to the Euler-Lagrange equation. We have shown that the extra term has no effect at least for existence of the analytical solutions. Further, as far as we know that any time-dependent solutions of the F2 model have not been discussed previously.

V. PROPERTY OF THE ENERGY
In this section, we discuss the energy of the static solutions (45) and also of the wave solution (53). We show that the static solutions saturate to an energy lower bound different from (6). It is proved that the solutions (45) are obtained through coupled first derivative equations which correspond to the saturation condition. For the traveling wave solution (53), the energy can be written by the topological charge plus Noether charges which are relevant to some U (1) symmetries.
Using the completeness condition (3), the static energy per unit length (5) can be written as We again emphasize that the first and second term in (54) are the static energy of the CP 2 model. Applying the BPS bound of the CP 2 model to the two terms, we obtain the energy bound According to the discussion in the Since the maximum of the non-topological term (without minus sign) gives the equivalence between the two bounds (6) and (55), the value of the term runs One can find from (21) that Z † C ∂iZA is proportional to u3∂iu1 − ∂iu2. Therefore, for the solutions of (44), this nontopological term identically vanishes (then takes the lower bound of the inequality of (56)). Consequently, the static energy of the nonembedding solutions (45) is given by On the other hand, for the embedding solutions which u1 and u3 are constants, the term satisfies the lower bound of the inequality (56) and then it corresponds to the energy bound (6) with NB = 0. Thus, we find that the static energy is twice greater than that of the embedding solution which belong to the same topological sector. This is consistent with the interpretation that the nonembedding solutions include instanton-anti instanton bound state which described by the harmonic map solutions in the CP 2 model. Note that, from the saturation conditions of the energy bound (55) and the lower inequality in (56), we obtain the following coupled first-order equations corresponding to the solutions (45) : and The equations (58) are equivalent to the Cauchy-Riemann equations for the complex scalar fields ui and (59) exactly is (44). This clearly means that the solutions (45) of the Euler-Lagrange equations (36) can be obtained by the first-order equations (58) and (59). Note that even when the saturation of the energy bound (55) is attained, the energy does not take a stationary point due to the presence of the non topological term. According to the above discussion, however, if both (58) and (59) are satisfied, the energy is stationary, i.e. maximal energy in the bound (55). Therefore we conclude that the solutions (45), which satisfy the both saturation conditions, are saddle-point. The solutions (45) possess basic properties as BPS solutions, such that they saturate an energy lower bound and are the solutions of first order equations. They seem to be unstable, because they are not of the global minima in a given topological sector, which is fulfilled by the embedding solutions. It is worth to check behavior of the energy for changing the parameters of the configuration and confirm the stability nature of our solutions. In Fig.3 we plot the energy per unit length for the configuration with the parameters ai ∈ [0, 1]. The nonembedding solutions locates on the ridge line of the surface, while the embedding solutions are on the left front corner. There is a zero mode along the ridge, but they essentially be unstable. A natural question occurs where the nonembedding solutions might eventually decay into the embedding ones. In order to see this in detail, we plot the cuts of the energy surface at several values of a1 for fixed a2 = 0.5 in Fig.4. For the configuration (60), unless u1u3 = u2 or u2 = 0, at infinities of the x 1 x 2 plane the fields Za approach the value up to phase factors. The configuration (60) with u1u3 = u2 or u2 = 0 do not support the boundary condition (61) and then, they belong to a different topological sector from the nonembedding and embedding solutions. It clearly indicates that the solutions cannot continuously deform into the configuration which satisfy u1u3 = u2 or u2 = 0 with keeping finiteness of the energy. One can find the topological barrier associated with u1u3 = u2 lies between the nonembedding and the embedding solutions. It is a common feature among all topological classification in Table I that such a barrier lies between nonembedding and embedding solutions. For configurations corresponding to the domain i. or ii. in the Table, as the above example, the barrier exists on the line corresponding to u1u3 = u2 which do not obey the boundary condition of the solutions at infinity. For the domain iv. or v., the condition u1u3 = u2 does not satisfy the boundary condition of solutions at the origin. In this case, one can also easily prove the condition u1u3 = u2 lies between nonembedding and embedding solutions. The situation for the domain iii. or vi. are little more complicate than for the others. Nonembedding solutions in the domain exist in region c1c2c3 < 0 and, on the other hand, embedding solutions are derived as c1, c2 → ∞ with keeping the condition c1 = c2c3, which gives c1c2c3 > 0. Therefore during deformation from a nonembedding solution to an embedding solution, at least one of the coefficient have to change the sign. At the point ci = 0, i = 1, 2, 3, the boundary condition of the solutions are broken and therefore the barrier appears between solutions. This observation indicates the nonembedding solution cannot decay into the embeddings in classical level, at least, through a deformation described by variations of the parameter in (60). However, quantum mechanically the tunneling between these solutions may occurs and then the decay process may be able to achieve. Some readers may expect the other possibility. One can find lower energy region behind the ridges in Fig.4, but it is a fake effect from fixing one of the parameters that the ridges divide the region from an embedding solution. If we vary all three parameters, all configuration may finally decay into the embedding solutions through the tunneling effect. Next we consider the energy of the traveling wave solution (53). From simple calculation, one can get the energy as Ewave = 2πQ + 8πk 2 {I(n1, n2, c1, c2) with I(n, m, a, b) where n = m and a, b = 0. By counting the powers of r, one finds the integral (63) is converge if We represent combinations of winding numbers for (in)finite energy solutions (62) in Fig.5. The energy (62) is associated with Noether charges. The model (1) possesses two U (1) symmetries corresponding to the transformations (u1, u2, u3) → (e iα 1 u1, u2, e −iα 1 u3), (u1, u2, u3) → (u1, e iα 2 u2, e iα 2 u3) where αi, i = 1, 2 are constant parameters. The corresponding Noether currents are given by The Noether charges per unit length are obtained as Then for the time-dependent solutions (53), the energy (62) coincides with It is worth to comment on scaling property like Derrick's theorem for the traveling wave solutions (53). Because the traveling wave solutions localize only on the x1x2 plane, we consider the rescaling xi → λxi, i = 1, 2. Then, the energy contribution from the x3-derivative term is scaled like a potential term. On the other hand, as discussed in the Q-lump case [29], the contribution from the time-derivative term behaves like a quadratic term in xi-derivatives. The scaling property of the terms are opposite, but when λ = 1, the contribution from the x3-derivative term and the time-derivative term are equal. Therefore, the traveling wave solutions satisfy the Derrick type condition.

VI. SUMMARY AND DISCUSSION
In this paper, we have constructed both static and timedependent solutions of codimension two in the F2 NLσ model by analytically solving the second order Euler-Lagrange equation. The static solutions possess novel features; the solutions saturate an energy lower bound, and satisfy coupled first order differential equations corresponding the saturation condition, then in this sense they are BPS solutions. On the other hand, caused by the fact that a non-topological term is contained in the energy bound, they are saddle point solutions, i.e. sphalerons. As far as we know, our solutions are a rare example of "BPS sphalerons". The time-dependent solutions are given by a product of localizing component on the x 1 x 2 plane and traveling wave being parallel to x 3 with speed of light. The energy of the wave solutions are given by the topological charge and the Noether charges associated to two U (1) transformations. Both the static and wave solutions consist of a triple in the tower of the CP 2 solutions generated through the Bäcklund transformations, i.e. a holomorphic, anti-holomorphic and harmonic map, and therefore can be recognized as a complex of the CP 2 instantons and anti-instantons. We are currently investigating the generalization to the FN case. Moreover, the static solution may be able to generalize the F2 model on the cylinder R × S 1 with twisted boundary conditions by means of a conformal mapping technique [30].
Since we have found several analytical solutions of the model, it is certainly worth to discuss such existence in the context of integrable theories in higher dimensions based on discussion of the integrable submodels possessing an infinite number of local conservation laws [31,32] as a generalization of the previous work [23,33], which study integrability of two dimensional NLσ models on homogeneous spaces.
Existence and stability of the solutions which we obtained implies that instanton-anti instanton creation/annihilation may occur, through the quantum tunneling effect, in condensed matters described by the SU (3) antiferromagnetic Heisenberg model such as cold atoms in an optical lattice [34], and rotating dense quark matter [35]. In addition, we hope this work gives a hint for finding finite energy nonembedding solutions of the SU (3) pure Yang-Mills theory on the four dimensional Euclidean space.
Apart from the BPS and also the analytical study, it is also important to discuss solutions of a model which breaks the scale invariance by a higher derivative order term and a potential term. Such baby-skyrmion solutions may be more easy to find in several physical applications. We will report results of these issues in forthcoming article.