Non-Abelian Vortices with a Twist

Non-Abelian flux-tube (string) solutions carrying global currents are found in the bosonic sector of 4-dimensional N=2 super-symmetric gauge theories. The specific model considered here posseses U(2)local x SU(2)global symmetry, with two scalar doublets in the fundamental representation of SU(2). We construct string solutions that are stationary and translationally symmetric along the x3 direction, and they are characterized by a matrix phase between the two doublets, referred to as"twist". Consequently, twisted strings have nonzero (global) charge, momentum, and in some cases even angular momentum per unit length. The planar cross section of a twisted string corresponds to a rotationally symmetric, charged non-Abelian vortex, satisfying 1st order Bogomolny-type equations and 2nd order Gauss-constraints. Interestingly, depending on the nature of the matrix phase, some of these solutions even break rotational symmetry in R3. Although twisted vortices have higher energy than the untwisted ones, they are expected to be linearly stable since one can maintain their charge (or twist) fixed with respect to small perturbations.


Introduction
Vortex, resp. string-type solutions appear in many models, and as they have many applications there is an enduring interest for them. In the plane, a vortex corresponds to a cross-section of a straight string in 3 spatial dimensions in the plane orthogonal to its direction. In the context of spontaneously broken gauge field theories by scalar fields, the paradigm is the Abrikosov-Nielsen-Olesen (ANO) vortex [1] associated to the breaking of an Abelian gauge group. The ANO vortex corresponds to the planar cross-section of a static straight, infinitely long magnetic flux-tube, with quantized magnetic flux and rotational symmetry. ANO vortices have an (integer) winding number, proportional to their quantized magnetic flux, which is also responsible for their stability. For a fixed winding number, ANO solutions form a one parameter family, depending on the mass ratio β = m s /m v , where m s , resp. m v denote the mass of the scalar resp. of the vector field. In the special case β = 1, the energy of a vortex is proportional to its winding number [2,3], and surprisingly vortices of like fluxes do not interact [2]. For this special value of the coupling, minimal energy vortices satisfy a set of first order -Bogomolnyequations, which are easier to solve than the field equations [2,3].
In non-Abelian gauge theories with or without a Chern-Simons term, non-Abelian vortices (some of them with an electric charge) were first obtained in Ref. [4]. Since the seminal papers [5] started the investigation of vortex-string solutions in supersymmetric non-Abelian gauge theories, the subject continues to attract attention. A simple model containing the essential features is an U(N c ) gauge theory, coupled to N f scalar fields in the fundamental representation, where N f ≥ N c . Vortex solutions in such theories are usually referred to as non-Abelian vortices, (NAV). NAV have attracted considerable interest, since they are at the heart of intriguing relationships between 2d sigma-models and 4d gauge theories. NAV in a U(N) gauge theory possess "orientational moduli" whose low-energy dynamics is described by versions of CP N −1 sigma-models on the string worldsheet. Moreover, NAV also held the promise to be relevant to bring us closer to give a description of quark-confinement, for a review, see [6] and references therein. Static NAV are absolute minima of the energy functional in a fixed topological sector, characterized by the winding numbers associated to the Cartan subalgebra of the gauge group. Moreover NAV satisfy first order, Bogomolny-type equations admitting rather complicated, static multi-vortex solutions. Most remarkably a complete description of the NAV moduli space has been found [7,8]. An interesting application of non-Abelian vortices is illustrated by confined monopoles emerging as junctions of NAV with different moduli [9,10]. The dynamics of NAV based on the moduli approximation has been worked out in Refs. [11,12,13]. In addition to NAV and related monopoles, also domain walls have received due attention [14,6], and a number of intriguing relations between moduli spaces of monopoles and domain walls have been discovered [15].
In this paper we point out that by allowing for space-time dependent phases among the scalar fields, which we shall call "twisting", new families of charged vortex-strings arise. We shall restrict our attention to an U(2) gauge group broken by two scalar doublets with an appropriate scalar potential compatible with SUSY, but generalization to other groups should not be too difficult. It is known, that imposing the usual space-time symmetries on field configurations leading to vortex-type solutions, translational invariance in time and along the, say, x 3 or z direction, scalar fields may have a phase, with a linear dependence on (t , z) [16,17,18]. In general, the energy of stationary configurations with a nontrivial (t , z) dependent phase is bigger than that of the static ones, however this does not make them necessarily unstable [16].
In this paper we systematically investigate straight vortex-string solutions in the simplest theory admitting non-Abelian vortices, when the scalar fields, Φ, possess a (t , z) dependent phase, i.e.
where the flavours correspond to the columns of the (2×2) matrix Φ, M is a constant Hermitian matrix acting on the flavour indices, ω α is vector in (t , z) plane. As it is well known [16,17,19] when ω α is a light-like vector, the field equations in the (x 1 , x 2 ) plane decouple completely from those in the (t , z) directions. The chromo-electric components in the (t , z) plane are determined by a set of linear, 2nd order equations, (Gauss constraints), depending on the solution in the (x 1 , x 2 ) plane. Simple analysis of the Gauss constraints makes it very plausible that any NAV can be twisted by an arbitrary twisting matrix, although we have not attempted to formally prove this. There are global currents flowing in the z direction of an M-twisted vortexstring which has nonzero charge, momentum and, unless M is specially aligned in internal space, angular momentum. Since a twisted NAV has nontrivial components orthogonal to its symmetry-plane, currents flowing along its axis, etc., it has some genuine 3-dimensional structure, hence it is more conveniently thought of as a twisted string.
The simplest twisted NAV can be obtained from the rotationally symmetric "elementary" vortex solution of Ref. [5], in which case, for a general twisting matrix the problem reduces to a single 2nd order Gauss-constraint, which can be easily analyzed. In fact our twisted NAV solution turns to be the same object as the "dyonic" vortex solution found in a "mass-deformed" SUSY gauge-field theory [12]. The bulk of our paper concerns the twisting of "compositecoincident" NAV [7], corresponding to the rotationally-symmetric superimposition of elementary vortices. We characterize these superimposed NAV by a relative winding number and a moduli parameter. Interestingly we find that when the twisting matrix M, contains off-diagonal components the twisted string looses rotational symmetry in 3-dimensions.
The energy of an M-twisted string is higher than that of an untwisted one, it is given as a sum of the usual "magnetic" energy per unit length proportional to the magnetic flux, and of an "electric" contribution due to the rotating phase. The actual magnitude of the "electric" contribution depends essentially on the components of the twisting matrix, M. When M is diagonal the magnitude of the electric energy of a twisted NAV is typically much smaller than its magnetic one, in fact, it can be made arbitrarily small when the untwisted NAV is sufficiently close to a diagonal one. In the case when M contains off-diagonal components, the magnitude of the electric energy is comparable to the magnetic one.
It seems to us that all their above mentioned properties make twisted NA vortex-strings of some interest and worthy of further investigations.
We introduce in Sec. 2 the theory that we shall study and proceed to the dimensional reduction by splitting four dimensional Minkowski space into planar and temporal-longitudinal coordinates. Making an appropriate Ansatz we get a dimensionally reduced Lagrangian. Assuming that coupling constants satisfy Bogomolny conditions we present the minimal energy first order equations satisfied by untwisted solutions and write the energy as a sum of electric and magnetic contributions. We discuss how the rotational symmetry of the Ansatz, or the absence thereof, depends on the properties of the twisting. In Sec. 3 we consider twisted vortices, both elementary (Sec. 3.1) and composite (Sec. 3.2) ones, presenting the numerical study of the solutions. Finally in Sec. 4 we summarize the results and present our conclusions.

Dimensional reduction
The 3+1 dimensional theory we consider is defined by the following Lagrangian: where F µν is the Abelian field strength tensor and G a µν (a = 1, 2, 3) is the non-Abelian, SU(2) one. The two scalar doublets are encoded in the matrix (Φ) iA with i = 1 , 2 being the gauge ("colour") and A = 1 , 2 the "flavour" index; the trace is taken over the flavour indices; the scalar potential, V = V 1 + V 2 , can be written more explicitly as: where summation over repeated indices is understood, except when the contrary is indicated. For more details on notations and conventions, see Appendix A. The fields transform under the U(2) gauge symmetry as where U(x) ∈ SU(2), Λ(x) is a real function. The flavor symmetry acts from the right on the scalars as Φ → ΦV , V ∈ SU(2) .
Let us now consider stationary and translationally symmetric fields in the x 3 direction. It will turn out to be convenient to split 4 dimensional Minkowski coordinates as Since the symmetries are generated by two commuting vector fields, there exist a gauge where the symmetric gauge fields are simply independent of the coordinates x α [20]. At the same time, however, the scalar doublets, Φ, may still depend linearly on x α through an SU(2) phase, i.e. the most general symmetric Ansatz can be written as: where M is a constant Hermitian matrix, ω α is vector in (t , z) plane. Straight flux-tube/string solutions of the theory defined by Eq. (1), described by the Ansatz (6c) with a non-trivial M, will be referred to as 'twisted' strings. The form of the Ansatz, Eq. (6) also restricts the symmetries of the model. Those symmetries, which preserve the Ansatz are flavour transformations which commute with the twist matrix M, and all gauge transformations where U∂ α U † and ∂ α Λ only depends on x i and not on x α . The current generating these transformations is where and Being hermitian, the twist matrix, M, can always be diagonalized by a unitary matrix, Therefore, twisting a configuration with the matrix M, is equivalent to twisting a suitably flavour transformed (with V M ) configuration with the diagonal matrix, M D . As the theory considered is SU(2) flavour symmetric it makes no difference which form of twisting one uses.
In what follows we shall work with the non-transformed twisting matrix, M in Eq. (6c).
The Ansatz (6) yields the dimensionally reduced Lagrangian where It is now convenient to introduce basis vectors in the x α plane, We remark that in Eq. (12) only the α indices are raised or lowered by the induced Minkowskian metric in the (x 0 , x 3 ) plane, the repeated i , j-type lower indices are summed with (+, +) signature. It turns out that the dimensionally reduced Lagrangian (12), is closely related to the trivial reduction from 4 to 2 dimensions of the "mass deformed" theory considered in Refs. [12]. In the absence of twist, M ≡ 0, (12) corresponds to the trivial reduction of the theory to 2 dimensions, whose solutions are the non-Abelian vortices discussed in detail in References [8]. With respect to the ω ,ω basis the gauge field components in the (x 0 , x 3 ) plane are expressed as: The field equations can be grouped according to variations with respect toĀ ,C resp. A , C.
Consider first the variational eqs. wrtĀ ,C: where △ = ∂ i ∂ i with Euclidean metric summation. From Eqs. (14) we obtain the following integral identity: Assuming finite energy boundary conditions and global regularity, the integral of∆ is zero. For the case when ω is light-like, ω 2 = 0, one finds C a ∆ a = 0, therefore Eq. (15) enforcesĀ ≡ 0,C ≡ 0. When ω 2 < 0, Eq. (15) implies once more the vanishing ofĀ andC, since thenC a ∆ a ≤ 0. In the case of a space-like ω vector, ω 2 > 0, Eq. (15) is not sufficient to exclude the existence of non-trivial solutions of the Gauss-constraints, (14). It is consistent, however, to assumeĀ ≡ 0,C ≡ 0, even for ω 2 > 0, sinceĀ,C satisfy homogenous equations. AssumingĀ ≡ 0,C ≡ 0 the remaining field equations are: The total energy of an M-twisted string can be written as the sum of an "electric" and of a "magnetic" part: where the "electric", resp. "magnetic" densities, E 0 , E 1 are defined as: A straightforward computation shows that using the Gauss constraints, Eqs. (16a)-(16b), the "electric" density, E 0 can be expressed as: where ∆ = △ (A 2 /g 2 1 + C a C a /g 2 2 ) and Q is given by Eq. (10). In most work on non-Abelian vortices the supersymmetry induced relations between the couplings, λ 1 = g 2 1 , λ 2 = g 2 2 , have been assumed. These relations ensure that E 1 can be expressed as a sum of squares and a topological term, leading to 1st-order, Bogomolny-type equations in the (x 1 , x 2 ) plane. Then minimal energy, untwisted solutions of the 2nd order field Eqs. (16c)-(16e) are obtained by solving the following 1st-order equations: while for solutions of Eqs. (21) the "magnetic" energy density simplifies to: Eq. (22) implies that the total "magnetic" energy, E 1 , is given by the net Abelian flux through the (x 1 , x 2 ) plane. Let us quote here the actual value of the "magnetic energy" in Eq. (22), E 1 , for configurations considered in this paper, characterized by winding numbers n A , m A as in Eq.
(26) and subject to the 1st-order Eqs. (21): which of course also holds for the Ansatz (30). In the case when ω α is light-like, the field equations in the (x 1 , x 2 ) plane, Eqs. (16c)-(16e), decouple from the Gauss-type constraints, Eqs. (16a)-(16b), and become identical to those corresponding to untwisted vortices. Therefore the problem of finding twisted non-Abelian strings for ω 2 = 0 reduces to solve Eqs. (16a)-(16b) in the background of a non-Abelian vortex in the (x 1 , x 2 ) plane. In the present paper we shall consider a light-like ω vector, and concentrate on twisting vortex solutions of minimal energy satisfying the 1st order Eqs. (21). It is left for future work to clarify if for ω 2 = 0 there exists solutions analogous to the twisted vortices of Refs. [17,18]. An M-twisted string has momentum flowing along the z direction, which is easily obtained from the stress-energy tensor. The longitudinal momentum, P = T 03 , carried by a twisted string, can be recast exploiting the Gauss constraints, Eqs. (16a)-(16b), as: M-twisted strings may also have angular momentum, J, as it can be seen from the angular momentum density however, to compute J in a more explicit form, one needs a parametrization of angle-dependence of the fields. This will be presented in the next Section.

The Ansatz; rotational symmetry and its loss
Let us now impose rotational symmetry in the x 1 , x 2 plane to the fields. Denoting the usual polar coordinates in the plane as x 1 = r cos ϑ, x 2 = r sin ϑ, rotational symmetry implies that by ϑ-dependent gauge transformations one can always achieve In order to ensure consistency with the U(2) gauge and the global SU(2) flavour symmetry, the integers, n A , m A , satisfy the following relation: n 2 − n 1 = m 2 − m 1 = N, which can also be expressed on the 2 scalar doublets as where N = Diag{0, N}, which encodes the relative winding between the two flavours. Since in general the twisting matrix, M, mixes the 2 flavours, when N = 0 in Eq. (27) one can immediately see, that the right hand sides of Eqs. (16a)-(16b) depend explicitly on ϑ, breaking rotational symmetry in the x α direction, i.e.
It is easy to see that the condition to ensure rotational symmetry of solutions of the field eqs. (16) can be written as implying that the twisting matrix be diagonal. This happens whenever M does not contain terms proportional to σ 1 , σ 2 . Obviously, the anisotropy generated by twisting matrices not commuting with N, would simply rule out the possibility to consider rotationally symmetric configurations. Remarkably in the case of a light-like twist vector, due to the decoupling of the field eqs. in the (x 1 , x 2 ) plane decouple from Eqs. with x α = (x 0 , x 3 ) components, (16a)-(16b). This decoupling allows for solutions which actually break rotational symmetry in the x α direction. Whenever M does not commute with N the corresponding M-twisted strings have rotationally symmetric spatial sections in any plane orthogonal to the x 3 -axis, however, the complete configuration is not rotationally symmetric in the whole space-time.
Keeping the possibility of breaking rotational symmetry in mind, we now present our Ansatz. By singular U(2) gauge transformations (linear in ϑ) on the scalars in Eq. (27) we can achieve n 1 = m 1 = 0. Furthermore, by assuming that the functions, Φ A i (r), are all real, one reduces the number of scalar fields from 8 to 4 (minimality of the Ansatz). Then from Eq. (21b) it follows that C 2 ϑ = const, which constant can be set to zero. Finally choosing the radial gauge, our Ansatz can be written as: We can now display the angular momentum, J for our Ansatz (30) in a more explicit form. Exploiting Eqs. (16a)-(16b) we obtain: It is worthwhile to point out, thatQ is a combination of the flavour charge densities (see Eq. (56)),Q = −4nâKâ 0 /ω 0 , with the coefficients n a = Tr(Nσ a )/2, n 0 = Tr N/2.
Let us now write out the explicit form of the first order Eqs. (21) for our Ansatz: where for convenience we have chosen units such that ξ = 1. The vacuum manifold of E 1 for the Ansatz (30) corresponds to the fix-point manifold of Eqs. (32a), a curve, which can be parametrized as: 3 Twisted vortices

Twisted elementary vortices
The simplest non-Abelian vortex solution is a 'diagonal' one, with just φ 1 , ψ 2 , a, c 3 being non-trivial and subject to Eqs. (32a), while φ 2 , ψ 1 , c 1 are all zero. In this case the 'vacuum angle', α = 0. Such solutions have been thoroughly investigated in Refs. [8]. A larger family of 'elementary' vortex solutions of Eqs. (32a), with φ 2 , ψ 1 , c 1 being also nontrivial, can be obtained by a 'colour-flavour' transformation from a "diagonal" one [8]. We note that for non-diagonal NAV the parameter α is different from zero. For diagonal vortices the relative winding between the 2 doublets is always trivial, N = 0, which remains so for the general elementary vortices. The general form of the elementary solution can be written as: where n a is the 'orientational' unit vector of an elementary NAV, which can be parametrized by the 2 spherical angles as n a = (sin α cos β , sin α sin β , cos α). In this case the two moduli parameters are just the angles, α and β [6,8].
We note here that for the most general (with the 2 moduli) elementary vortex solution in (34) C 2 ϑ = 0, hence for β = 0 it is not in the form of the minimal Ansatz (30), this has no influence, however, on our twisting of this solution. It is now simple to generalize, or deform the elementary vortex solution by M. Parametrizing the twisting matrix as in Eq. (9), a short computation shows that the Ansatz A = m 0 , C a = (m·n)[1 − C(r)]n a + C(r)m a , m·n = m a n a , containing just a single radial function, C(r), reduces the Gauss constraints, Eqs. (16a)-(16b), to a single 2nd order inhomogeneous equation for C(r): Without going into more rigorous mathematics one can easily convince oneself that Eq. (36) admits a unique solution subject to boundary conditions guaranteeing regularity at r = 0 and at r → ∞. Conversely, using maximum-principle-type arguments it is not difficult to show that all globally regular solutions are necessarily of the form of the Ansatz in Eq. (35). The charge density Q, determining the energy resp. momentum, of the elementary string, is given as: As one can easily see from Eq. (31) the angular momentum density of twisted elementary vortices vanishes. As already pointed out, that this Subsection reproduces and extends previously obtained results of Ref. [12] using different methods, being our starting point rather different.

Twisted coincident composite vortices
Next we consider a vortex solution with nonzero relative winding between the 2 flavours. Assuming the minimal Ansatz, the vortex in the (x 1 , x 2 ) plane is rotationally symmetric, satisfying the 1st order eqs. (32a). Such solutions have already been analyzed in Ref. [7], where it has also been pointed out that they correspond to superimposed vortices on the top of each other. Therefore such vortices can be considered as composed of elementary ones. As we have already argued, deforming composite vortices with a general matrix, M induces a nontrivial angle-dependence in the (x 0 , x 3 ) plane. Without losing generality one can parametrize the twisting matrix as Taking into account the possible angle-dependence of the ω α -components of the gauge fields, we introduce the following decomposition: together with the conditions A + = A * − , C a + = (C a − ) * ensuring reality of the fields, moreover A 0 , A ± , and C a 0 , C a ± are functions of r. Then the corresponding Gauss constraints can be put in the form: where the SU(2) gauge fields have been split as C a = {Cā , C 2 },ā = 1, 3; ǫ 13 = −1, ǫ 31 = 1; △ r is the radial part of the two dimensional Laplacian, △ (N) = △ r − N 2 /r 2 . Furthermore we introduce the notationsâ = {0, a} and σ 0 ≡ 1, to present more compactly the often appearing combinations moreover to simplify somewhat the formulae we write η ≡ η 0 + , ηā ≡ ηā + . We have omitted the equation for C 2 0 from Eqs. (40a)-(40e), since a straightforward application of the maximum principle leads to C 2 0 ≡ 0. The reason behind C 2 0 ≡ 0 is the minimality of the Ansatz (30). As one can see, the Gauss-type Eqs. (40a)-(40e) can be decomposed into three equation-groups, one for {A 0 , Cā 0 }, and one for each {A ± , Cā ± }, decoupled from each other. In fact it is sufficient to consider only one of the set of eqs. for ±-components and impose reality on the solutions.
From Eqs. (40a)-(40e) one can easily deduce the asymptotic r → ∞ behaviour of the ω α components of the gauge potentials. We note first that for r → ∞ then one finds that for r → ∞ Quite interestingly, the equations for the angle-dependent components, (40c)-(40e) can be reduced to a quadrature, i.e. to solve a single 1st order, linear ordinary differential equation. The key observation is that A ± = 0 is a solution of Eq. (40c). This is not completely obvious at first sight, since assuming A ± = 0, Eq. (40c) leads to an algebraic relation/constraint between C 1 ± and C 3 ± . A straightforward computation shows that this constraint is compatible with the remaining two coupled 2nd order Eqs. (40d),(40e). As a matter of fact one can find yet another simple algebraic relation among the C a ± . In conclusion Eqs. (40c)-(40e) admit a globally regular solution, which can be given as: and in term of the solution of Eq. (45b), the remaining functions C 2 ± , C 3 ± can be found from the following algebraic relations: It follows from Eqs. (45-46) that Cā − = Cā + and C 2 − = −C 2 + * , which implies that Cā ± are real and C 2 ± are imaginary. Using the solution of the Gauss-constraints Eq. (46), the electric energy contribution, Q, simplifies to: where T (ϑ) = cos(Nϑ + µ). In the angular momentum density J (Eq. (31)),Q takes the form The total electric energy, longitudinal and angular momenta are given as where Q tot m,s = d 2 xQ m,s . We have plotted the radial components of twisted coincident vortices for N = −1. In Figs Table 1: Q tot m , Q tot s for g 2 = 1 and different values of g 1 , α. Note that the BPS energy is E 1 = 4π and 0.785398 ≈ π/4. resp. 4, the Fourier components of the out-of-plane gauge fields are shown. The charge density terms Q s and Q m are plotted in Fig. 5. Their integrals over the plane are given in Table 1.
In Table 1, there is a striking difference between the energy of a vortex string with s = 1, m = 0 and s = 0, m = 1, although they are counterparts in the sense that they have the same frequency. Noting that the vortex with s = 1, m = 0 is rotationally symmetric, while the one with s = 0, m = 1 is not, only the magnitude of the difference is surprising.
For an explanation of the magnitude of the abovementioned energy difference, let us apply perturbation theory, expanding the solution in powers of α (see Eq. (33)), assuming α ≪ 1. An expansion of the background vortex as  can be substituted in Eqns. (32a-g), yielding a consistent solution. Note, that in the α 0 order, the vortex is always gauge equivalent to diagonal one. If α = π/2, the configuration can be brought to a diagonal [7]. The field components A 0 , A ± , Cā 0 , C a ± are expanded as 0 , and substituting into Eq. (47) yields 2 ) 2 + . . . , 2 ) 2 + . . . .
The two orders of α between the leading terms of Q s and Q m in Eq. (52) explain the magnitude of the energy difference between the same planar vortex twisted with the same frequency, either with a diagonal or with an off-diagonal twisting matrix.
Finally, we give some arguments for the stability. The conserved charge Q is strongly localised, and therefore small perturbations cannot change its value. In the plane, we are solving BPS equations, therefore planar vortices, being absolute minima of the energy in their topological sector, are stable. If there were an instability, it would be expected to manifest itself as an energy reducing deformation along the z axis. In the case of the twisted semilocal vortices of Ref. [17], with ω α timelike, such deformations indeed exist [21], however, they correspond to the same type of instability as those of ANO vortices embedded in a two-component extended Abelian Higgs model [22]. In the present case, however, such potential instabilities are absent. The untwisted vortices are BPS saturated, and the dispersion relations of their perturbation modes are gapped. For untwisted vortices, planar and off-plane modes decouple. At least for small values of the twist, it cannot change the sign of the otherwise non-vanishing positive eigenvalues.

Conclusions
In this paper, we have constructed charged, stationary rotating non-Abelian vortex strings in an U(2) gauge × SU(2) flavor theory. The scalar fields rotate around the string axis, and they have a (matrix) phase depending linearly on x α = (t, z) as which is referred to as a twist. We consider here the case ω α ω α = 0, in which, the planar equations decouple from those of the t, z components. The energy contribution due to the twist, and the z component of the momentum are both proportional to the Noether charge corresponding to the flavor symmetry generated by the matrix M.
Adding twist to the coincident composite vortices of Ref. [7], leads to some striking phenomena. These vortex strings carry total angular momentum, unless M is purely off-diagonal. If M is non-diagonal, the vortex strings are not rotationally symmetric, although, all their planar cross sections are. This is explained by the fact, that the nontrivial realizations of rotations and z-translations act on them non-commutatively. The energy of a solution, which breaks rotational symmetry, is significally larger than that of its rotationally symmetric counterpart.
Vortex hair for charged rotating asymptotically AdS black holes has revealed interesting features particularly in the holographic context of the gauge/gravity duality [23]. The interplay between the angular momentum of vortices as those constructed here and the black hole angular momentum could give rise to interesting effects. We hope to discuss this issue in a forthcoming work. with σ a , (a = 1, 2, 3) denoting the Pauli matrices, σ a σ b = δ ab + iǫ abc σ c . For later use, the covariant derivative of adjoint representation fields,D µ Σ a = ∂ µ Σ a + ε abc C b µ Σ c . Yang-Mills equations: where the color currents are The flavor current, i.e., the Noether current corresponding to the global SU(2) flavor symmetry is where the component K 0 µ = −J 0 µ has been introduced for convenience sake, a U(1) transformation agrees with a gauge transformation with a constant (global) phase.