Two Species of Vortices in a massive Gauged Non-linear Sigma Model

Non-linear sigma models with scalar fields taking values on $\mathbb{C}\mathbb{P}^n$ complex manifolds are addressed. In the simplest $n=1$ case, where the target manifold is the $\mathbb{S}^2$ sphere, we describe the scalar fields by means of stereographic maps. In this case when the $\mathbb{U}(1)$ symmetry is gauged and Maxwell and mass terms are allowed, the model accommodates stable self-dual vortices of two kinds with different energies per unit length and where the Higgs field winds at the cores around the two opposite poles of the sphere. Allowing for dielectric functions in the magnetic field, similar and richer self-dual vortices of different species in the south and north charts can be found by slightly modifying the potential. Two different situations are envisaged: either the vacuum orbit lies on a parallel in the sphere, or one pole and the same parallel form the vacuum orbit. Besides the self-dual vortices of two species, there exist BPS domain walls in the second case. Replacing the Maxwell contribution of the gauge field to the action by the second Chern-Simons secondary class, only possible in $(2+1)$-dimensional Minkowski space-time, new BPS topological defects of two species appear. Namely, both BPS vortices and domain ribbons in the south and the north charts exist because the vacuum orbit consits of the two poles and one parallel. Formulation of the gauged $\mathbb{C}\mathbb{P}^2$ model in a Reference chart shows a self-dual structure such that BPS semi-local vortices exist. The transition functions to the second or third charts break the $\mathbb{U}(1)\times\mathbb{S}\mathbb{U}(2)$ semi-local symmetry, but there is still room for standard self-dual vortices of the second species. The same structures encompassing $N$ complex scalar fields are easily generalized to gauged $\mathbb{C}\mathbb{P}^N$ models.


Introduction
The existence of BPS equations and a BPS bound in a non-linear sigma model with the target manifold an S 2 -sphere and R 1,2 as Minkowski space-time was investigated by B.J. Schroers in the short letter [1]. In that work, the author introduced a U(1) gauge field minimally coupled to the scalar fields via covariant derivatives, whereas a Maxwell term allowed for the presence of planar vector bosons. This approach led to BPS solitons, which were found to be akin to baby skyrmyons or CP 1 -lumps existing in other (2 + 1)-dimensional models. Nevertheless, one may think that in a system of this kind, suitable for describing the dynamics of ideal charged bosonic plasmas, topological defects of the Nielsen-Olesen vortex type should exist. In the guise of two-dimensional instanton these vortices were discovered by Nitta and Vinci in the (1 + 1)-dimensional CP 1 -sigma model with extended N = (2, 2) supersymmetry, see Reference [2]. Here we shall show that in a slight modification of the Schroers scenario, self-dual topological vortices do indeed exist and enjoy quite standard properties, except that there are two species of magnetic flux tubes, i.e., we shall describe in detail the Nitta-Vinci vortices in a purely bosonic model in (3 + 1)-dimensions.
One-dimensional topological defects of the kink type have been unveiled in massive non-linear S 2 -sigma models in (1 + 1)-dimensions. In that case the search was made possible because of the Hamilton-Jacobi separability in elliptic coordinates of the Neumann problem, a solvable dynamical system which is tantamount to the search for kinks in the non-linear S 2 -sigma model with a non-degenerate mass spectrum, see [3,4]. The procedure also worked for finding kinks in a hybrid non-linear S 2sigma/Ginzburg-Landau 2D model [5]. As in this latter case, we shall see that only a very precise choice of the potential is compatible with the existence of self-dual vortices in the massive gauged non-linear S 2 -sigma model. To build the U(1) gauge theory out of the O(3) background symmetry of the S 2 -sphere, our proposal is to gauge the stereographic coordinates rather than the original fields. By doing so, the potential can be chosen in such a way that self-duality is guaranteed simultaneously in both the south-and north-charts of the sphere. Although the vorticial solutions corresponding to both charts have different energies per unit length, there is a local version of the Bogomolny bound that ensures their separate stability.
The BPS structure of this model is compatible with the modification of the magnetic field by a dielectric function. By means of this generalization and the choice of a dielectric factor that is well behaved in the two charts of the sphere, we find families of self-dual vortices carrying scalar profiles and magnetic fields that are susceptible to being modified almost at will. These new vortices belong to the class discovered in [6] but also appear in two species attached respectively to the north and south poles. It is interesting to point out that the Nitta-Vinci vortices correspond to the choice of a constant dielectric function. We also remark that the choice H(|φ|) = 1 |φ| 2 as the dielectric function is very particular: in the south chart the analogue to the self-dual vortices existing in the Abelian Chern-Simons-Higgs planar gauge theory appear, see [7], obtained in this case by replacing R 2 by S 2 as the scalar field space. The corresponding self-dual vortices in the north chart, however, are singular. The four-parametric family H(|φ|) = c 0 |φ| 2 +c 1 b 0 |φ| 2 +b 1 of dielectric functions provides an ample supply of self-dual vortices that we shall describe in detail. In particular, we shall show that self-dual vortices of two species exist which are akin to those unveiled in [8] in the commutative limit but choosing S 2 as the target manifold.
The S 2 sphere as a complex manifold is the CP 1 compactification of C, and the stereographic version of the round metric, which is the key ingredient to giving the right properties to the vortices on the sphere, becomes precisely the Fubini-Study metric of that Khäler manifold. Thus, it is natural to think that some kind of wellbehaved vortices, sharing many features with S 2 vortices, should also exist in the higher rank non-linear CP n -sigma Abelian Higgs model. In the last section of the paper we shall show that this is in fact the case. There is, however, an important novelty: in one of the CP n -charts there exist BPS semi-local topological solitons, with or without vorticity, which are the cousins of the semi-local defects discovered in the scalar sector of Electroweak Theory when the weak angle is π 2 and described in References [9,10,11]. In the n − 1 remaining charts, only purely vorticial self-dual vortices exist, all of them of the second species.
2 Construction of the non-linear S 2 -sigma model We begin with a system of three scalar fields Φ a , a = 1, 2, 3 taking values on a sphere: The respective stereographic coordinates for the south and north charts are the complex scalar fields The transformation φ = ρ 2 ψ * from the south to the north chart reverses the orientation. The reason for choosing this option is to deal with scalar fields coupled to the gauge field with identical electric charges in both charts. The massless Lagrangian L Φ describing the dynamics of the Φ fields, in terms of the south-chart field, becomes: The global U (1) symmetry is made local following the standard procedure: a gauge field A µ enters the system and supplements the local U (1) transformation φ → e ieχ(x) φ with the gauge transformation A µ → A µ + ∂ µ χ. A potential energy density yielding spontaneous symmetry breaking and the Higgs mechanism can also be introduced. All this leads to the Lagrangian of a gauged massive Abelian non-linear S 2 -sigma model of the form The covariant derivative and the electromagnetic tensor are defined in the usual way: Under the change of coordinates φ = ρ 2 ψ * , the covariant derivatives and the potential energy density in the north chart become: in such a way that the Lagrangian in this chart now reads:

Bogomolny splitting and self-dual vortices
Lagrangians such as (1) and (3) with a function of the scalar field multiplying the covariant derivative term were studied by M. A. Lohe in [12]. Other references in which models with a similar structure were investigated are, for instance, [13,14,15]. In a field theory encompassing scalar and gauge fields where the Lagrangian density is of the general form and both the function g(|ϕ| 2 ) and the potential energy density are semi-definite positive, it is possible to write the energy per unit length of static and x 3 -independent configurationsà la Bogomolny after taking the Weyl plus axial gauge: A 0 = A 3 = 0. Here, B = F 12 is the magnetic field and R = − i 2 g(|ϕ| 2 )ε ij D i ϕ * D j ϕ, i, j = 1, 2 can be rewritten as where F is a primitive of g such that F (0) = 0 in order to avoid singularities in the logarithm. Choosing a potential energy density where a 2 belongs to the range 0 < a 2 < F (∞) such that U (1) spontaneous symmetry breaking is ensured, the last two terms in the energy integral combine as a boundary contribution that is proportional to the magnetic flux Φ M . Thus, finite-energy configurations comply with the inequality E ≥ 1 2 ea 2 |Φ M |, and the Bogomolny bound is saturated if and only if the first-order self-duality equations are satisfied. For radially symmetric fields ϕ(r, θ) = f (r)e inθ , rA θ = nβ(r) in the topological sector with winding number n, the PDE system (5) becomes the ODE system of coupled equations Boundary conditions on the solutions are dictated by regularity at the origin and energy finiteness: f (0) = 0, β(0) = 0, f (∞) = v and β(∞) = 1 e . v 2 is a zero of the potential, F (v 2 ) = a 2 . The solutions are vortices and anti-vortices with magnetic flux eΦ M = 2πn and energy per unit length E = π|n|a 2 . Beyond the radially symmetric case, an index theorem calculation gives the dimension of the moduli space of solutions of (5) in each topological sector, see [12,16]. The space of linear deformations of a vortex that preserve the self-duality equations is the kernel of the differential operator while, by introducing the auxiliary operator it is not difficult to show that ker P D † = {0}, and hence ker D † = {0}. This vanishing theorem and the supersymmetric pairing of the non-zero eigenvalues of the Laplacians associated to D and D † , give the dimension of the moduli space as where the result comes from trace evaluation in the limit M 2 → ∞. The interpretation, [17,18], is that the Higgs field of the general solution with winding number n has |n| zeroes along the plane, and the 2|n| parameters of the solution correspond to the coordinates of these zeroes. The self-dual defects are thus non-interacting solitons at the verge between two superconducting regimes. Rewriting the coupling in (4) as λe 2 , one has that for λ < 1 the mass of the vector boson would be greater than that of the Higgs boson and long-range inter-vortex forces would be attractive (type I superconductivity), whereas in the opposite λ ≥ 1 case, the vortices would repel each other (type II superconductivity).

Self-dual vortices in the south chart
In the gauged non-linear sigma model on the sphere, the potential energy density, together with the metric function, are chosen to be in order to find the Lagrangian (1) at the critical self-dual point in the south chart. It is required that 0 < a 2 < 4ρ 2 whereas U S (|φ| 2 ) vanishes along the vacuum orbit see Figure 1(left). The Higgs mechanism occurs and provides identical masses to the vector and the Higgs bosons: The Bogomolny equations are and the vortex energy per unit length is E = π|n|a 2 for a winding number n. The scalar field maps the center of the vortices to the south-pole of the sphere, whereas the image of the boundary circle of the plane r → +∞ is the parallel circle |φ| = v in the south-chart of the sphere. The radially symmetric configurations φ(r, θ) = f (r)e inθ and rA θ = nβ(r) are BPS φ-vortex solutions if the functions f (r) and β(r) solve the ODE system (6) for the choice of F (|φ| 2 ) written in formula (7). The solutions have been determined numerically by a standard shooting procedure for configurations with vorticity n = 1, 2, 3, 4 and are shown in Figure 2, together with the energy density (r) and the magnetic field B(r). Figure 3 is a graphical representation of the n = 1 and n = 2 self-dual vortices using the original valued-on-the sphere field Φ, which is plotted as a unit vector on each point of the spatial plane.

Self-dual vortices in the north chart
The field redefinition ψ * = ρ 2 φ applied in formula (7), together with the subsequent changes (2), leads to the self-dual Lagrangian (3) in the ψ-chart. Thus, in this north chart we find: see Figure 1(right). The north-chart Bogomolny equations are and the BPS energy of vortices and anti-vortices per unit length is E = π|n|(4ρ 2 −a 2 ). The point of the scalar configuration space corresponding to the center of the defects is now the north-pole, whereas the fields at large distances reach the parallel in the ψ-chart. Because w 2 = ρ 4 v 2 , this is the same parallel that we found in the φ-chart and represents the global vacuum orbit of the theory. Moreover, the Higgs mechanism in this chart produces the same masses for the vector and Higgs bosons as in the south chart: The system is thus globally well defined. The vacuum orbit divides the sphere into two disjoint skullcaps, the scalar field of φ-and ψ-vortices or antivortices respectively taking values in the south and north ones. Radially symmetric configurations in this second chart ψ(r, θ) = f (r)e inθ and rA θ = n β(r) are BPS-vortex solutions living in the north skullcap if f (r) and β(r) solve the ODE system (6) for the F (|ψ| 2 ) function appearing in (9). The system is again solved numerically and the f (r) and β(r) profiles, together with the energy density (r) and the magnetic field B(r), are respectively displayed in Figure 4 for the ψ-vortices with vorticity n = 1, 2, 3, 4. Figure 5 is a graphical representation of the Φ field for these solution with n = 1 and n = 2.
In sum, there exist two species of BPS vortices whose energies are respectively E = π|n|a 2 and E = π|n|(4ρ 2 −a 2 ) for configurations with n-vorticity. From Figures 2 and  4 we observe that for a = 1 the first type of vortices supports thick profiles, whereas vortices of the second species are thin; the energy per unit length of thick/thin vortices is less/more concentrated in the plane. The difference in vortex energy density width, is more pronounced when the vacuum parallel |φ| 2 = v 2 approaches the North Pole, and disappears when the vacuum orbit is the Equator, a 2 = 2ρ 2 , and thick vortices become thin and viceversa if the vacuum orbit lies in the south hemisphere, a 2 > 2ρ 2 .
A subtle point is the following: although φ-and ψ-vortices with the same winding number live in the same topological sector, both of them are separately stable. The higher-energy per unit length vortices do not decay to the lower-energy per unit length ones. The scalar field at their centres is respectively the North or the South Pole, depending on the species. In fact, BPS vortices of different species saturate either the φ-or ψ-Bogomolny bounds, which are valid on either the south or the north charts. The circumstance that each type of vortex carries a different energy per unit length seems a bit puzzling: the BPS bounds coming from the φ-and ψ-field Bogomolny splittings are, respectively, E ≥ e 2 a 2 Φ M and E ≥ e 2 (4ρ 2 − a 2 )Φ M . Thus, if 2ρ 2 = a 2 , the energy of ψ-vortices contradicts the φ-bound, and viceversa. In fact, however, the contradiction is only apparent and is due, contrary to what happens in the usual Abelian Higgs model, to the fact that finite-energy configurations can have isolated points where the scalar field goes to infinity. Recall that the target space is CP 1 = C ∪ {∞}. A radially symmetric configuration φ(r, θ) = f (r)e in∞θ such that lim r→0 f (r) = ∞ is therefore admissible. In this case, the antisymmetric sum of double derivatives of ln φ produces a new contribution to the R term of the Bogomolny splitting of the form The φ-field Bogomolny bound in the complex plane plus the infinity point becomes In the previous formula, n 0 denotes the number of zeroes of φ in C, and therefore the total winding number is n = n 0 + n ∞ . This is the global BPS bound that encompasses the bounds in both charts. If n ∞ = 0, there are vortices only in the south chart, n = n 0 , and we find the BPS φ-bound. n 0 = 0 means that n ∞ = n, which is tantamount to the existence of ψ-vortices with vorticity n in the south chart, in perfect agreement with the BPS ψ-bound. The φand ψ-Bogomolny bounds are the two local forms of the global Bogomolny bound.
We conclude that the gauged massive non-linear S 2 -sigma model admits stable self-dual solutions in the form of either φ-or ψ-vortices. One might wonder if there are also solutions given by the symbiosis of defects of both species. When the change of variables φ = ρ 2 ψ * is applied to a configuration satisfying (8) with the plus sign, we obtain a configuration that satisfies (10) with the minus sign, and viceversa. Only a superposition of a φ-vortex with a ψ-antivortex could be a self-dual solution of the theory, but in this case the total winding number vanishes, and there is no topological obstruction to ensure stability. Notice, however, that for n 0 = 1, n ∞ = −1 the global Bogomolny bound gives an energy per unit length E = 4πρ 2 , which is the area of the sphere and coincides with the energy of a fundamental CP 1 lump in the class N = 1 of the homotopy group π 2 (S 2 ). In fact, as it is understood in [2], a different topological interpretation of these φ-vortex ψ-antivortex pairs is possible; in it, they appear as the basic constituents of CP 1 lumps. On the other hand, the superposition of a φ-vortex with a ψ-vortex could also be a non self-dual solution of the Euler-Lagrange equations, but this can only be decided by a numerical evaluation of the interaction energyà la Jacobs-Rebbi [19], which we have so far not undertaken. In any case, given that the Higgs mechanism gives mass m to the elementary particles, the interaction energy among defects located at distances of order R 1 m is small, E int e −2mR [20], and mixed configurations of widely separated φ-and ψ-defects can always be considered good approximate solutions to the Euler-Lagrange equations.

More doubly self-dual models
We have built the non-linear Abelian-Higgs sigma model using the natural round metric on the sphere, but once the gauge field has been introduced and the potential energy density has been chosen, the symmetry is reduced from the O(3) group to the U(1) subgroup of rotations around the Φ 3 -axis. There is nothing to prevent us from considering models on the sphere with other metrics as long as they respect this reduced symmetry. There is, however, an appealing feature of the round metric that we would like to preserve: this is the fact that it gives rise to doubly self-dual models. By this we mean that the self-duality equations on both charts have almost the same form, the only difference being the value of W (0). Next we shall briefly describe a possible way to produce a number of other non-linear sigma models with this type of double self-duality. Working on the south chart, the idea is to take a dimensionless function f (|φ| 2 ) such that f (0) = 0, f (|φ| 2 ) ≥ 0, ∀|φ|, and f (∞) = q 2 ≤ ∞. We then define the F part of the potential as F (|φ| 2 ) = ρ 2 f (|φ| 2 ) − f ( ρ 4 |φ| 2 ) + q 2 in such a way that F (0) = 0, as it should be, and F (∞) = 2ρ 2 q 2 is finite. Therefore, the metric and potential on the south chart are whereas the change of fields (2) gives the following metric and potential for the ψ field: Double self-duality is thus apparent. The energies per unit length of φ-and ψvortices or antivortices with winding number n are respectively E = π|n|a 2 and E = π|n|(2ρ 2 q 2 −a 2 ), and the phenomenology is analogous to what has been described for the model with the round metric. The case of the function f (|φ| 2 ) = ln ρ 2 +2|φ| 2 ρ 2 +|φ| 2 gives an interesting example where double self-duality combines with a logarithmic potential.
3 Dielectric functions and self-dual vortices of two species in the Abelian S 2 -sigma Higgs model

The Lagrangian and the Bogomolny splitting
The kinetic energy density for the gauge field in the Lagrangians (1) and (3) that we have been dealing with in the previous section is given by the standard Maxwell term. This is the canonical choice when the model is intended to describe the interactions of fundamental quanta in vacuo, but there are many physical systems in which the Abelian-Higgs model plays the rôle of an effective theory ruling the dynamics of the excitations of some background medium. For this sort of application, it may be the case that the minimal Maxwell term has to be supplemented with a dielectric function that will account for the enhancement or screening of the forces among quanta due to the polarization of the underlying condensate. The U(1) gauge symmetry requires that the dielectric factor should depend only on the scalar field modulus, and hence the Lagrangian of Section 3 changes to the form Models of this type where H(|ϕ| 2 ) is a positive definite function admit first-order Bogomolny equations, see e.g. [6,14,15,8]. The arrangement leads, together with the choice of the potential as to self-duality equations of the form whose solutions are vortices and antivortices of energy density E = π|n|a 2 for winding number n.

Hybrid vortices in the Abelian S 2 -sigma Higgs model with dielectric function
In practice, the dielectric function can be chosen in many different ways that can generate a broad variety of vortex profiles. For the non-linear sigma model on the sphere, a natural option is to use a function H(|ϕ| 2 ), leading to regular vortices with a similar structure in both the south and north charts, as indeed happened in the minimal model without dielectric function. To achieve this behaviour we choose the south chart dielectric function and self-dual potential in the form where b 0 , b 1 , c 0 , c 1 are positive real numbers, see Figure 6. The vacuum orbit occurs in these models along the circle |φ| = aρ √ 4ρ 2 −a 2 , and there is always a critical point of the potential at φ = 0 that can be tuned to be a minimum or maximum with a suitable selection of the parameters. There are two special or limiting cases: a) If c 0 = b 0 and c 1 = b 1 , the function H becomes unity and we recover the system where the Nitta-Vinci vortices arise. b) The other case appears when c 0 = b 1 = 0 and c 1 = b 0 , the dielectric function is H(|φ|) = 1 |φ| 2 and we deal with the nonlinear S 2 -sigma model version of the Chern-Simons topological topological solitons discovered in [7], see also [21]. This dielectric function prompts a BPS potential such that the critical point at |φ| = 0 becomes an absolute minimum. The vacuum orbit is the disjoint union of one point and a circle and the space of BPS topological solitons is richer than in the other cases. As in its planar counterpart, the system encompasses a Coulomb and a Higgs phase. In this subsection we shall describe hybrid selfdual vortices interpolating between Nitta-Vinci and nonlinear Chern-Simons vortices.
Changing coordinates to the north chart, the metric mutates to the known round g N (|φ| 2 ) metric on S 2 , whereas the dielectric function and potential energy density become It is clear that at the Chern-Simons limit the potential in the north chart becomes infinity at |ψ| = 0; henceforth the corresponding self-dual solitons are singular. The duality between the theories with (b 0 , b 1 , c 0 , c 1 ; a 2 ) and (b 1 , b 0 , c 1 , c 0 ; 4ρ 2 − a 2 ) is thus patently clear. The regular vortices of the south (north) chart in one theory are akin to the regular vortices appearing in the north (south) chart in the other. In particular, the energy per unit length of the vortices in the north chart is: E = π(4ρ 2 − a 2 )|n|. The same argument developed in the microscopic scenario of the previous section regarding the existence of a global Bogomolny bound works here.

Profiles of self-dual radial vortices
The ODEs giving the radially symmetric defects are of the generic form where, of course, south-or north-variables and F and H functions have to be appropriately substituted and finite-energy boundary conditions respected. The energy density per unit length and the magnetic field for these solutions are: As an illustration, here we shall present the solutions for two cases, let us call them A and B, where the parameters are chosen to be ρ = a = b 0 = b 1 = c 0 = 1, c 1 = 5 in Case A, and ρ = a = b 0 = c 0 = c 1 = 1, b 1 = 10 in Case B. The potential energy densities for these parameters are plotted in Figure 6 (blue lines), where a comparison with the potentials (red lines) giving self-duality in the microscopic, non dielectric, case is also offered. Observation of these graphics reveals to us that the potentials in case A are closer to the non-dielectric self-dual potentials than the potentials in case B in both charts. In the second model, one can observe that there is a local maximum of U at φ = 0 in the south chart, but the potential in the south chart shows a local minimum at ψ = 0.  Figures 7 to 10 show the specific scalar and vector boson field profiles, as well as the energy densities and the magnetic fields, of the radially symmetric solutions for cases A and B with winding numbers increasing from 1 to 4. Again, these magnitudes in case A are close to the field profiles and density energies of the Nitta-Vinci vortices.
Interesting new features arise in case B profiles: namely, the magnetic field presents a local minimum at ψ = 0 in the north chart even for solutions with vorticity n = 1. Thus, the maximum values of the magnetic field are attained at a ring in the plane enclosing the origin, a configuration that resembles the self-dual Chern-Simons-Higgs vortices.

BPS solitons in the massive gauged non-linear
CP 2 -sigma model The non-linear sigma model on the sphere is the simplest n = 1 representative among a hierarchy of U(1)-gauged non-linear sigma models with CP n manifolds as target spaces. In order to extend the results found in S 2 CP 1 to other members of the hierarchy, in this section we shall analyse the next case, since it turns out that CP 2  already exhibits the most relevant features arising for general n. CP 2 is a Khäler manifold of complex dimension two. A coordinate system is built from a minimal atlas with three charts. We shall call them V φ , V ψ and V ξ and shall denote the complex coordinates in each chart, respectively, as (φ 1 , φ 2 ), (ψ 1 , ψ 2 ) and (ξ 1 , ξ 2 ). The transition functions giving the change of coordinates in the intersections between charts are: The Khäler potential, expressed in the coordinates of the chart V φ , takes the form K = 4ρ 2 ln(1 + |φ 1 | 2 ρ 2 + |φ 2 | 2 ρ 2 ) and it is easy to check that it adopts an equivalent form, in terms of the respective coordinates, on the other two charts.

The gauged Abelian
To formulate an Abelian gauge theory on CP 2 let us begin by working on the V φ chart. Gauging of the scalar non-linear CP 2 sigma model gives rise to the Lagrangian where g pq = ∂ 2 K ∂φp∂φ * q is the standard Fubini-Study metric on CP 2 : The covariant derivatives are D µ φ p = ∂ µ φ p − ieA µ φ p and, to keep things simple, here we have discarded the possibility of introducing a dielectric function. We choose the potential energy density in (16) in the form that generalizes the potential function entering the S 2 -sigma model.
The Lagrangian corresponds to a semi-local theory in which the U(1) local invariance is accompanied by a global SU(2) symmetry. The vacuum orbit, the set of zeroes of It is well known that S 3 is a Hopf bundle, i.e., it is a manifold fibered on S 2 with fibre S 1 and Hopf index 1, see e.g. References [22,23]. Moreover, the winding number of the map from the S 1 ∞ circle enclosing the spatial plane to the S 1 fiber provided by the gauge field at infinity classifies the configuration space in n ∈ Z disconnected subspaces. In the n = 0 subspace, one sets a particular point of S 3 as the vacuum, for instance (v 1 = ρa √ 4ρ 2 −a 2 , v 2 = 0). The Higgs mechanism is worked out and gives mass to the physical fields. One thus finds that the mass spectrum includes, along with a degenerate couple encompassing the Higgs scalar meson and a massive vector boson, another complex Goldstone boson. The masses are: Both the Higgs and the Goldstone fields are coupled to the gauge field through the covariant derivative terms. All this refers to elementary quanta; the other perspective is about solitons living in sectors of the configuration space with n = 0. In this respect, we look at the static part of the energy per unit length. Working in the simultaneous temporal and axial gauge, and focusing on static and x 3 -independent configurations, one writes: On one hand, we have that: On the other hand, we also split the covariant derivative terms in a similar manner where the last term can be conveniently recast in the form: Here α p is the phase of φ p and M pq = g pq φ p φ * q (no sum in p and q) is a real symmetric matrix. Upon discarding total derivative terms, we write R = eε ij V ip S jp , where and we find as the final outcome of all these manipulations that: Therefore, we finish with a self-dual theory on the chart V φ where the Bogomolny bound is saturated if the Bogomolny equations are satisfied. As required by the mixing of SU(2) and U(1) symmetries, these firstorder equations are of the semi-local type. Although the equation for the magnetic field is more complicated here than in the standard examples, the results found in references lsuch as [9,10,11] give a solid guarantee that semi-local vortices enjoying stability against decay to the vacuum and filling, for winding number n, a moduli space of complex dimension 2n, will also exist in the present situation.

Radially symmetric semi-local topological solitons
The radial ansatz for the fields φ 1 (r) = f (r)e inθ , n ∈ Z + , rA θ (r) = nβ(r) φ 2 (r) = |h(r)|e i(ω+lθ) , l = 0, 1, · · · , n , ω ∈ R + , converts the PDE Bogomolny system (21) in the first-order ODE system The solutions of this system (22) To illustrate these points, in Figures 11 and 12 we show two self-dual solutions for h 0 = 0.1 and h 0 = 0.4 obtained by means of the same shooting procedure as applied before. In the first case, the soliton profiles, the energy density per unit length and the magnetic field are quite close to their counterparts in the Nitta-Vinci vortices.
For higher h 0 , we see that the profiles of the solutions, together with the energy density per unit length and magnetic field, are less concentrated and tend slowly to their vacuum values.

The gauged Abelian CP 2 -sigma model in the second chart
The next task is to write the Lagrangian (16) as an Abelian theory in the chart V ψ . In order to do so, we first observe that the Khäler potential becomes in the new chart. Thus, the Fubini-Study metric remains formally identical. The changes in the covariant derivatives seem to be more drastic but are explained as follows: the gauge transformations φ p → e ieχ φ p , p = 1, 2 of the φ-fields correspond, after the application of the transition functions on V φ ∩ V ψ , to the local phase redefinitions ψ 1 → e −ieχ ψ 1 and ψ 2 → ψ 2 for the ψ-fields. The ψ 1 field couples to the gauge field with opposite charge to the two φ p fields, but ψ 2 remains as a neutral field. Consequently, the covariant derivatives on the chart V ψ are defined as: and have the right holomorphic transformation properties under the change of chart: This is all that is needed to ensure that the Khäler structure of the kinetic term is preserved, leading us to the Lagrangian in the chart V ψ : g pq is the Fubini-Study metric defined in terms of the ψ fields, and the new potential energy density is (25) Alternatively, one can derive the Lagrangian (24) from the Lagrangian (16) by applying the transition functions in a direct but lengthy calculation. The potential energy density (25) is only invariant under the Abelian subgroup of the global SU(2) non-Abelian symmetry. Because ψ 2 is neutral the local U(1) transformation does not act on this field and the system is still gauge invariant in the V ψ chart. The transition functions break the semi-local U(1) × SU(2) invariance, leaving us with only a semi-local U(1) local × U(1) global symmetry. The vacuum orbit in this V ψ chart, the set of zeroes of U ψ , ψ 1 = w 1 , ψ 2 = w 2 , is the H 3 one-sheet hyperboloid: Note that even though the hyperboloid (26) is an open space in C 2 it is accommodated within CP 2 through the "infinite line ": CP 2 = C 2 ∪ CP 1 . We stress, however, that only points complying with (26) and living in V ψ , i.e., |w 1 | < +∞, |w 2 | < +∞, are bona fide vacua of the system. Like S 3 , H 3 is also a 3D fibered space with the S 1circle as fibre. The base, however, is one sheet in the 2D hyperboloid living in the R 3 subspace of C 2 where Im w 1 = 0. Despite the differences in charges of the fields and vacuum orbit geometry, the Higgs mechanism gives rise to a spectrum formed by a couple of massive particles, the Higgs and vector bosons, and a complex (but neutral) Goldstone boson. By choosing the vacuum, e.g., on the Equatorial circle of , w 2 = 0), we find the same mass spectrum as in the V φ chart: We remark that, despite being "neutral", the Goldstone field ψ 2 is coupled to the A µ vector boson through the mixed terms induced by the Fubini-Study metric, as can be checked from perturbiations around the vacuum chosen. The Bogomolny splitting, worked according to the standard procedure, confirms that the Lagrangian (24) is again at the self-dual point. The Bogomolny bound is in this V ψ chart: E ≥ 1 2 e(4ρ 2 − a 2 )|Φ M | where, of course, 4ρ 2 is greater that a 2 . Note that it is the same bound as in the north chart of the S 2 -sigma model. The Bogomolny equations guaranteeing that this bound is saturated are: In particular, the equation between the covariant derivatives of ψ 2 in (27) is the Cauchy-Riemann equation: the self-dual ψ 2 field solutions are holomorphic (+ sign) or antiholomorphic (− sign) functions. The energy per unit length finiteness requires, for configurations reaching a bona fide vacuum |w 2 | < +∞ at infinity, a constant ψ 2 value at long distances. Not allowing for the presence of poles, the Liouville theorem, ensuring that an entire and bounded function in C is constant, requires that ψ 2 take a constant value all along the plane. Once this constant value is substituted in the equation for the magnetic field, (27) reduces to a system of two equations with the same structure as those giving the self-dual vortices on the north chart of the sphere in Section 4. It then follows that, with very slight modifications, the S 2 solutions found in that section can be embedded in the V ψ chart, becoming self-dual vortices of the non-linear CP 2 sigma model of the second species. The situation is reminiscent of that studied on the sphere in that regular vortices on V ψ , with a zero of ψ 1 at their cores, are singular solutions on the chart V φ , whereas regular semi-local vortices on V φ , where φ 1 is nonzero at infinity, are singular on the chart V ψ . In any case, the vortices on V ψ are stable solutions, the reason being that the ψ 1 field winds at infinity around a two-dimensional throat of H 3 whose radius (4ρ 2 − a 2 )(ρ 2 + |w 2 | 2 ) is positive regardless of the value of |w 2 |. Specifically, the choice ψ 2 = 0 leads to exactly the same vortices as those existing in the north chart in the S 2 -model. Other constant values of ψ 2 merely give rise to a redefinition of the parameters in the solutions.

Final comments
The transition between the charts V φ and V ξ works similarly, and leads to couplings D µ ξ p = ∂ µ ξ p − iN p eA µ ξ p with N 1 = 0, N 2 = −1 and a self-dual Lagrangian L ξ , which adopts exactly the form (24), (25) with the substitutions ψ 1 → ξ 2 , ψ 2 → ξ 1 . The self-dual vorticial solutions in this third chart therefore belong to the same species as those found in the V ψ chart. The results described along this paper about the existence and structure of selfdual topological defects in the massive gauged CP 2 -sigma model suggest a general picture which should be valid for other higher-rank gauged CP n -sigma non-linear systems. In the CP n theory there are n scalar fields and n + 1 charts, let us call them V 0 , V 1 , . . . , V n . Starting with the same coupling for all the fields on the chart V 0 , we can construct on this chart a self-dual model with semi-local SU(n) global × U(1) local symmetry group, whereas in the other n charts the transition functions would break the global symmetry to a SU(n−1) global subgroup acting on the neutral fields, and only one field would couple to the U(1) gauge group through the covariant derivative. There are thus semi-local topological defects on the chart V 0 and one-complex-component vortices analogous to those found on the sphere on the remaining ones. The vacuum orbit is S 2n−1 for the fields on V 0 and H 2n−1 for V 1 , V 2 , . . . , V n . The stability of the solutions is guaranteed, in the case of V 0 by the known arguments for semi-local vortices, and in the other charts because the vorticity of the charged field is due to its asymptotic winding around the throat of H 2n−1 . Some modifications of this scenario could be considered for cases in which the assignation of couplings to the fields on V 0 is different. Let us assume, for instance, that there are on V 0 r fields with coupling N r e and n − r fields with coupling N n−r e. The symmetry group in this chart is SU(r) global ×SU(n−r) global ×U(1) gauge , and there are two types of semi-local vortices corresponding to the two global factors. All the other charts fit into two different classes. There are r charts such that the transition functions break the symmetry to SU(r − 1) global × SU(n − r) global × U(1) gauge , where the fields complying with the global SU(r − 1) are the neutral ones, those supporting the global SU(n − r) symmetry have coupling −(N r − N n−r )e, and the remaining field has coupling −N r e. The solutions existing in these charts are of two kinds: there exist embedded S 2 vortices, where only the field with −N r e coupling winds, and there are also defects in which this onecomplex-component field is mixed with an SU(n−r) global ×U(1) gauge semi-local vortex, with winding numbers proportional to the charges; in all these cases, the neutral fields remain frozen at constant values. In the remaining n − r charts we observe the reciprocal situation, where the symmetry is SU(r) global ×SU(n−r −1) global ×U(1) gauge and the SU(n − r − 1) corresponds to neutral fields, the charged fields have couplings −(N n−r − N r )e for the global SU(r), and there is a final charged field which couples to A µ with intensity −N n−r e.