Non-Abelian Vortices with an Aharonov-Bohm Effect

The interplay of gauge dynamics and flavor symmetries often leads to remarkably subtle phenomena in the presence of soliton configurations. Non-Abelian vortices -- vortex solutions with continuous internal orientational moduli -- provide an example. Here we study the effect of weakly gauging a U(1)_R subgroup of the flavor symmetry on such BPS vortex solutions. Our prototypical setting consists of an SU(2) x U(1) gauge theory with N_f=2 sets of fundamental scalars that break the gauge symmetry to an"electromagnetic"U(1). The weak U(1)_R gauging converts the well-known CP1 orientation modulus |B| of the non-Abelian vortex into a parameter characterizing the strength of the magnetic field that is responsible for the Aharonov-Bohm effect. As the phase of B remains a genuine zero mode while the electromagnetic gauge symmetry is Higgsed in the interior of the vortex, these solutions are superconducting strings.


Introduction
Topological solitons play a fundamental role in various physical systems, especially in those characterized by gauge interactions. The interplay of the classical or quantum gauge dynamics with the global symmetries in such systems leads to remarkably rich and subtle phenomena. Of particular interest among these is a class of vortex solutions carrying continuous orientational zero modes: non-Abelian vortices [1] - [32].
These typically arise when the (complete) gauge symmetry breaking supports vortex solutions, and at the same time the vacuum is invariant under a color-flavor diagonal group so that the system is in the so-called color-flavor locked phase [33]. As the individual vortex breaks the exact global symmetry, it develops orientational zero modes which can fluctuate on the vortex world sheet. At low energies these fluctuations can be described by an appropriate 2D sigma model that has its own nontrivial, large-distance, quantum dynamics. A longstanding goal concerning non-Abelian vortices is to find solutions describing a non-Abelian vortex-monopole complex [2], [4], [29], [24] - [26].
Here we examine yet another, little studied, question regarding non-Abelian vortex systems: what happens if a part or the whole of the exact global (color-flavor diagonal) symmetry is weakly gauged? In Ref. [34], we have initiated the study of such effects by weakly gauging the entire exact flavor symmetry of these vortex solutions. In a sense this was a simpler question: the answer is that now any color-flavor rotation of a given solution is a genuine global gauge transformation, rendering all charge one vortex solutions gauge equivalent whatever their orientations. Stated differently, a mini-Higgs mechanism is at work in the vortex worldsheet, transforming the orientational zero modes into light, propagating modes. As a result of the non-Abelian nature of the gauge groups involved, these light modes are unstable and decay into massless 4D gauge bosons.
Here we turn our attention to the case where only a part of the global symmetry group is gauged. As a simple concrete model we take an SU(2) L × U(1) gauge theory with two complex scalars in the fundamental representation, in which a U(1) R ⊂ SU(2) R subgroup of the flavor symmetry is weakly gauged 1 . The SU(2) L ×U(1) and U(1) R gauge symmetries are broken by the scalar VEVs, but a U(1) em subgroup remains unbroken. These 4D massless gauge modes interact with the 2D zero modes on the vortex worldsheet. The result is rather unexpected and elegant. We find that the two-dimensional vortex moduli (partially) survive the U(1) R gauging, but the complex parameter B inherited from the original CP 1 vortex moduli acquires a new physical meaning. The modulus |B| can still be interpreted as a truly 2D vortex modulus which now characterizes the magnetic flux carried by the vortex responsible for an Aharonov-Bohm effect a particle with unit U(1) R charge wil experience [35] in going around the vortex far from the core. The phase Arg(B), on the other hand, is eaten by the 4D gauge bosons. An AB effect on vortices have been studied earlier [36]. Compared to that case, a peculiar feature of our case is that the AB phase depends on the modulus |B|.
The question of U(1) gauging was originally raised in the context of dense quark matter. QCD in the high baryon density limit is believed to be in the color-flavor locked phase with diquark condensation [33], where the SU(3) color symmetry and the SU(3) flavor symmetry of the three light quarks are spontaneously broken to the diagonal subgroup. In this context, non-Abelian vortices appear [37] - [39] which differ from local non-Abelian vortices for which the U(1) group is a global symmetry. Nevertheless, they possess normalizable orientational zero modes [40], [41]. Taking into account the U(1) electromagnetic coupling corresponds to gauging a U(1) subgroup of the flavor SU(3) symmetry, some consequences of which were studied in Refs. [42] - [44]. See Ref. [45] for a recent review.

Orientational CP 1 modes of the standard non-Abelian vortex
Before introducing the weak gauging of a part of the flavor symmetry, let us first briefly review a few salient features of the non-Abelian vortex. The simplest example is an SU(2) × U(1) gauge theory with two scalar fields transforming in the fundamental representation, Q = q 1 q 2 , written in a color-flavor mixed 2 × 2 matrix form. The action is As is often done in the recent literature, the matter content and the potential terms are chosen such that the model above can be extended to have an N = 2 supersymmetry. As a consequence, the scalar quartic couplings are set to the critical value so that the classical equations for the soliton configurations become first order differential equations, in the "selfadjoint", or BPS, form.
In the presence of a nonzero parameter ξ, the system is in a Higgs vacuum: The gauge and flavor symmetries are completely broken, but there remains an unbroken color-flavor diagonal SU(2) C+F global symmetry (i.e., it is in color-flavor locked phase). As the system possesses stable, nonsingular vortices. An individual vortex solution breaks the SU(2) C+F global symmetry to a U(1) subgroup and so it develops an orientational modulus B ∈ CP 1 = SU(2)/U(1). Indeed, the vortex solution with a generic orientation and in the regular gauge takes the form where the boundary conditions are The "reducing matrix" U has the form with the matrices X and Y defined by The vortex tension, T = 2πξ does not depend on the CP 1 coordinate B.
More generally, perturbations of these solutions can be described by promoting the modulus B to a collective coordinate which depends upon the worldsheet coordinates of the vortex. The fluctuations of B are then described by a worldsheet CP 1 sigma model: (1.11) Our main interest below is to determine the fate of these CP 1 collective coordinates in the presence of an external, weak U(1) R gauge field.

The model, BPS equations and vortex solutions 2.1 The model and BPS vortex equations
The model that we consider in this paper is the same SU(2) L × U(1) 0 gauge theory with N f = 2 flavors of scalar fields as was described above, but we weakly gauge a U(1) R ⊂ SU(2) R subgroup of the flavor symmetry. The action is then The vacuum is the same as in Eq. (1.3), 3) The U(1) 0 , SU(2) L and U(1) R gauge groups are all broken, but a combination of U τ 3 (1) ⊂ SU(2) L and U(1) R remains unbroken. A gauge boson remains massless in the bulk: we call it "electromagnetic" in analogy with the situation in the Weinberg-Salam theory. 2 All other gauge bosons, the orthogonal combination

5)
A L ± µ , and A (0) µ via the Higgs mechanism acquire a mass of order of M ∼ g L √ ξ ∼ g 0 √ ξ. The inverse of (2.4) and (2.5) is Ordinarily, this would be the end of the story: B µ , A L ± µ , A µ are massive fields and cannot propagate farther than 1/M; the long-distance physics is dominated by the massless photon A em µ . However in the presence of a vortex, a one dimensional infinitely long "hole" in the condensate, the situation is a little subtler. Of course, one knows from the standard ANO vortex that a "massive" gauge boson can become massless along the vortex core, the associated magnetic field penetrating the vortex core and giving rise to important physical effects such as the vortex tension, a magnetic flux, vortex interactions, etc. In the case of a non-Abelian vortex these features are also present, because in the simplest context of a U(N) theory the latter may be considered to be an ANO vortex embedded in a corner of the color-flavor mixed space. The interesting phenomena related to the internal orientational zero modes and their dynamics arise from the fluctuation of this embedding direction. 2 As we need several coupling constants here is the summary. The unbroken electromagnetic field is coupled with the coupling e ≡ g R g L g 2 L + g 2 R ; while the broken gauge field B µ has coupling g ′ = g 2 L + g 2 R . Throughout, we take g 0 and g L to be of the same order of magnitude, whereas g R ≪ g L hence e ≪ g L , g 0 .
In the present case, where U(1) R weak external gauge interactions break the color-flavor rotational symmetry, yet one more, perhaps less familiar, effect arises. In a vortex solution of a generic orientation, some combination of scalar fields Q vanish along the vortex core, meaning that some of the "massive" gauge bosons become massless. Part of the effect goes, as in the ANO vortex, into producing a magnetic flux and the associated vortex energy (tension). On the other hand, far from the core another combination of the gauge fields, the electromagnetic field, becomes massless allowing a nontrivial Wilson loop at infinity. In our setting we will find that the value of this Wilson loop is unrelated to the vortex tension. As there are no electrically charged condensates at infinity, the electromagnetic Wilson loop can have any value and so it is observable via an Aharonov-Bohm (AB) effect: electrically charged particles circumnavigating the vortex acquire an AB phase.
Which components of the gauge fields become massless at the vortex core depends on the particular solution considered, and it might appear that it is quite a complicated matter to disentangle these two effects. Fortunately the BPS nature of our systems allows us to determine in detail the solutions, with the asymptotic behavior of all gauge fields explicitly exhibited. Knowing them, and if we are interested only in the observable effects far from the vortex, we shall be able to describe unambiguously the new AB effect associated with the weak U(1) R gauge field and the massless A em µ . In obtaining the minimum tension vortex solutions, we will take advantage of the basic fact is that it is possible to write a BPS completion, even in the presence of the U(1) R gauge field, for static vortex configurations: where the tension depends only on the winding of U(1) 0 . The BPS equations read

BPS solution with B = 0
Several BPS solutions can be found by inspection. A vortex solution in which the scalar fields take the color-flavor diagonal form, whereas A ± i ≡ 0 (as in the "B = 0" solution of the standard non-Abelian vortex, Eq. (1.5)), can be found easily. The profile functions satisfy the boundary conditions, Eq. (1.7). The solution takes the form In other words, Eq. (2.9) and the non-Abelian part of Eq. (2.8) are identical, whereas only the combination µ is as in Eq. (1.5). In order to have a finite energy configuration the kinetic term must approach zero asymptotically. This means that non-vanishing gauge fields (Eq. (2.5)) must, in the regular gauge, approach In fact the condition is even stronger: .12)). Combining this and Eq. (2.15), one finds that at large r When a probe particle carrying a unit U(1) R charge +1 follows a large circle around the vortex, it will acquire an AB phase equal to (2.17)

|B| = 1 solution
The solution with |B| = 1 can also be found easily. Setting in Eq. (1.5)-Eq. (1.9), one finds the scalar field configuration AsQQ is orthogonal to the direction τ 3 of the right U(1) R , one sees from Eq. (2.7) that a BPS solution can be constructed by setting A R i ≡ 0 and by

BPS solution with B = ∞
Using Eq. (1.5) and Eq. (1.9) one finds that the B = ∞ vortex has a squark condensate of the form, In the model without U(1) R weak gauge interactions this corresponds to the origin of the second patch of the CP 1 vortex moduli space, the south pole. The vortex equations are symmetric under an exchange of the two flavors combined with a reflection of the moduli space CP 1 , which maps B −→ 1/B. Therefore this solution is essentially identical to the B = 0 vortex with the two flavors are interchanged. As a result the BPS solution can be constructed, in the presence of the U(1) R weak gauge fields, as in Subsection 2.2, taking into account certain sign changes (i.e., in Eqs. (2.15) and (2.16)). A probe particle carrying a unit U(1) R charge +1 going around the vortex will this time obtain an AB phase of i.e., of the same magnitude as the B = 0 vortex but with the opposite sign.

A general BPS solution
To show the existence of other BPS solutions interpolating between the B = 0, B = 1 and B = ∞ solutions found above requires a more careful analysis, to be presented in the next section. Nevertheless, it is easy to obtain the U(1) R flux of such a general solution if one assumes that it exists and is given by the deformation in g R = 0 of the unperturbed BPS vortices Eq. (1.5) -Eq. (1.10) characterized by the CP 1 coordinate B. The point is that to find the BPS solution to first order in g R it is sufficient (see Eq. (2.4)) to know A L 3 µ to the zeroth order and A R µ to first order. The former is given by Eq. (1.5) -Eq. (1.10), whereas the latter is given by the BPS equation (2.9) in which the right hand side is replaced by the scalar fields in the zeroth-order solution Eq. (1.5). One finds indeed The F R 12 flux (times g R ) is given by

Moduli matrix and the master equations
Showing that there are indeed generic BPS vortex solutions interpolating between those with B = 0, B = 1 and B = ∞ is somewhat more difficult as the color-flavor rotations are no longer exact symmetry operations. Below we will instead appeal to the powerful modulimatrix method developed in Ref. [7,11,28]. Before beginning the analysis, let us note that the index for the dimension of the BPS vortex moduli space (i.e., the number of the zero modes) [1], [21] does not get modified by the addition of the weak gauge U(1) R interactions under which the two squark fields have charge 1 2 and − 1 2 , respectively. ν is the U(1) 0 winding index depending on the gauge group considered: [20]. For G ′ = SU(N), ν = k N where k is the winding number of the vortex. As a result we expect, in our case, N = N f = 2, k = 1, the two zero modes (the complex CP 1 coordinate B) to persist somehow upon weak U(1) R gauging.

The origin of the vortex moduli
Following the familiar procedure [28] we now set where B is a complex number and z ≡ x + iy. 3 We also recall that the moduli matrix H 0 (z) and S matrices in (3.2) are defined up to a "gauge choice" of the form where the holomorphic matrices V L (z) and V R (z) belong to the complexifications of the SU(2) L × U(1) and U(1) R gauge groups respectively. In the analysis of the vortex moduli spaces as complex manifolds, these V -equivalence relations play a fundamental role.

Remarks
We continue to use here the same letter B for the complex parameter characterizing the solutions, as in the model of Subsection 1.1 without the U(1) R gauging, even though the physical meaning will be different. The normalizable CP 1 zero modes B were massless Nambu-Goldstone bosons traveling along the vortex length in the absence of U(1) R gauge fields; on the other hand, with the U(1) R weak gauging, only Arg(B) remains the true Nambu-Goldstone mode, but it is coupled with and eaten by the exact, asymptotically massless electromagnetic field which propagates in the 4D bulk, while |B| will determine the (electro-)magnetic AB flux carried by the vortex. The change in the nature of the zero modes reflects the fact that now the non-Abelian vortex is coupled to an exact 4D gauge mode whose direction is nontrivially oriented with respect to the underlying (g R = 0 theory) non-Abelian vortex orientation.
According to Eq. (3.2) the scalar field takes the form, where 4 After the SU(2) L ×U 0 (1)×U(1) R gauge freedom is used to fix the form the the S matrices, Eq. (3.3), there remains still an arbitrariness of the V R transforms, (3.4). The latter can be used to fix the value of ψ R (0), for instance, to 0. This unambiguously defines the meaning of the parameter B. The physical meaning of the parameter B can be understood as the ratio of the scalar fields of the first and second flavors at the vortex core (z = 0).
Note that the electromagnetic gauge transformation acts on Q as Keeping the functions ψ 0 , ψ 3 , ψ R real via a combination of the V transformations, B can be seen to transform under the electromagnetic gauge transformations simply as For a comparison with the physical approach of the previous section it is convenient to recall the relation between the gauge fields and various ψ functions the SU(2) L gauge fields are related to ψ's and w through Let us first consider the B = 0 solution. By setting the master equations reduce to the first two, which are exactly the equations for the familiar non-Abelian vortex with the B = 0 orientation. This solution was discussed in Subsection 2.2.
The solutions with B = 0 are more interesting. Note that by redefining Actually, this is not so. The coupled differential equations (3.12) define the solution only after a specific set of boundary conditions are imposed. The appropriate boundary conditions at large r are so that the squark fields approach the form, But this does not uniquely fix the boundary conditions for ψ 3 and ψ R separately: there remains a freedom in sharing log |z| between them. A general boundary condition is thus one has In the next section we solve Eqs. (3.12) explicitly for small g R and determine δ: the asymptotic behaviors of ψ R , ψ + , ψ 3 and ψ 0 6 .

Perturbative expansion in λ = g R /g L
We will now solve Eq. (3.12) perturbatively in λ ≡ g R /g L . At zeroth order one must solve the equations for generic B at g R = 0. Setting g R = 0 one has 20) and the functions appearing here satisfy We first review the B = 0 solution, ψ (

3.22)
To simplify the formulas somewhat, below we will set The difference between the two equations yields whereas the sum yields Taubes' equation [46] 4 ∂∂ϕ(z) = 1 r for (2ψ    The solution ϕ(r) behaves as The modified Bessel function of the second kind K 0 is exponentially damped at large r. This gives Q| B=0 . The solutions at B = 0 can be found by a color-flavor rotation of Q| B=0 , followed by an appropriate SU(2) L gauge transformation to bring Q back to the form, (3.20).
The answer is (see Appendix B):

(3.33)
Once one knows these solutions with λ = g R /g L = 0 for generic B, the solutions for λ ≪ 1 can be found by perturbation theory. Using Eq. (3.33), the fourth BPS equation (3.12) yields Next, using the Taubes' equation (3.25) for ϕ, one finds this means that where we have recalled Eq. (3.29) and have appropriately taken into account the boundary conditions for ψ R 7 .
7 Note that ϕ =ψ The large r behavior follows thus from Eq. (3.30). Recalling (Eq. (3.15), Eq. (3.16)) that ψ + = ψ 3 + ψ R ∼ log r, one finds to this order that For completeness, we get for ψ em (Eq. (3.19)) whereas the U 0 (1) gauge fields winds half (E. (3.15)): The asymptotic behavior of the gauge fields A L 3 i and A R i is then For completeness, the "broken" gauge field B µ behaves asymptotically as This leads, for a particle carrying only the unit charge with respect to U(1) R , to an AB phase as it goes around the vortex, in accordance with Eq. (2.26); one sees that the three explicit solutions we found earlier, with |B| = 0, 1, ∞, are interpolated by the modulus |B|.

Aharonov-Bohm effect
The result (3.43) is well defined and gauge invariant. A particle with a unit U(1) R charge but with no charges with respect to the SU(2) L × U 0 (1) will experience the AB effect (3.43). More generally, a particle in a definite representation of the SU(2) L × U 0 (1) × U(1) R gauge group will get a definite AB phase after encircling around the vortex. Some examples are shown in Table 1. We note that, in the case of the U(1) R charge one particle (particle K in Table 1), the AB phase is maximum (in the magnitude) for the vortices B = 0 and B = ∞, where the vortex orientation and the external gauging are aligned, whereas for the vortices |B| = 1 (the vortex solutions along the equator of CP 1 ∼ S 2 ) the orientations are orthogonal and there is no AB effect. As the U(1) R gauge symmetry is a spontaneously broken symmetry, it might be thought that any physical effect at large distances (and far from the vortex core) should be describable in terms of the coupling of the particle to the massless gauge field, A em µ . For instance, a particle carrying a unit charge with respect to U(1) R (but with no other charges) interacts through the covariant derivative It would seem natural to assume that the long-distance physics is dominated by the coupling to the massless "photon" field A em µ , the interactions with the massive B µ field providing some small corrections (the "weak interactions") calculable in perturbation theory.
In the presence of vortex, this is not quite so. As one can explicitly see from (3.42) also the massive gauge fields contribute to the AB effect to the same order, as is well known [36]. As a result, two particles with the same electromagnetic charge but with different couplings to the broken gauge field B µ such as K and ψ 1 in Table 1 (i.e., two particles belonging to different representations of SU(2) L × U 0 (1) × U(1) R ) experience different AB effect going around the vortex, however far from it.
One must also be somewhat careful in deriving the physical AB effect from the results of the calculations in the preceding sections, as one is working with a spontaneously broken SU(2) L gauge theory. The situation is somewhat analogous to that of the Weingberg-Salam theory. In the Weinberg-Salam theory the neutrino and electron are usually described by the upper and lower components of an SU L (2) doublet field; nevertheless both describe physical particles of definite mass and charge. Of course, the solution of this apparent puzzle is well known: the identification of the two leptons with the two components of a doublet is correct in a gauge in which the upper component of the Higgs doublet gets the nonvanishing VEV. More approariately, the electron and neutrino must be associated with some appropriate SU L (2) gauge-invariant composite fields involving the left-handed lepton and the Higgs scalar fields [51].
In our case, SU(2) L × U 0 (1) × U(1) R gauge symmetry is broken in the bulk by the scalar VEVs, with the massless "photon" (and B µ with the massive field, see Eqs. i are found to behave as in (3.41), (3.37) in this gauge. Note that the identification of the combination (4.3) with the masseless field in the bulk is appropriate in this gauge also 8 .
Even if choosing a gauge is unavoidable as in any other gauge theory calculations, clearly the concept of the AB phase which a particle in any definite representation of the SU(2) L × U 0 (1) × U(1) R group will acquire in encircling the vortex far from it, is a physical one. The only unusual aspect is that, as pointed out above, the AB effect does not depend only on the "electromagnetic" charge but also on the "weak" charge. The calculation can be done in any gauge, of course, but when the result is transformed back to the conventional gauge where the scalar VEV's approach a color-flavor unit matrix form, in order to relate it to the physical effect, the answer is the same.
As a further remark on the gauge invariant nature of our results, let us note that, as 't Hooft's observation for the electroweak theory [51], a particle described by a gauge variant field such as ψ 1 (in our conventional gauge) can be regarded as a gauge invariant object associated with an SU(2) L singlet composite field such asQ (1) ψ or ǫ ij Q (2) i ψ j . The physical AB phase is indeed the same in all these descriptions, see Table 1.

Low-energy effective action
When the moduli parameters B are allowed to fluctuate along the vortex length and in time, i.e., in the vortex worldsheet, the associated collective coordinates become dynamical, described by a long-wavelength effective action. In the model of Section 1 (without the U(1) R gauging) this is just a 2D CP 1 sigma model, Eq. (1.11), with Kähler potential [1] - [10], corresponding to the Fubini-Study metric.
In the presence of 4D bulk zero modes -an exact unbroken U(1) em gauge symmetry -coupled to the vortex dynamics, the straightforward approach of Appendix A cannot be applied. Although there are nontrivial mixings of A em with other broken gauge fields inside of the vortex, the vortex configurations themselves can be approximated by setting g R = e = 0 to lowest oder. One then finds [34] Table 1: In the table are some fields with their charges under the various gauge groups, and the associated AB phase (given in the unit of 2π and modulo 2π, so that − 1 2 ∼ 1 2 ). The extra factor 1 2 's in the last three columns take into account of the τ 3 /2 in front of the A L,3 µ and A R µ . The electromagnetic charge represents the coupling to A em µ in the unit of the coupling constant e; the coupling to the broken gauge field B µ is in the unit of g ′ = g 2 This form is dictated by electromagnetic gauge invariance, under which the B and A em fields transform as (see Eq. (3.9)): The current carried along the vortex is given by and is proportional to which is conserved along the vortex, Dynamical aspects of the vortex zero modes (their fluctuations) are subtle, as the 2D vortex zero modes interact nontrivially with the 4D bulk zero modes (the "photon"). Eq. (5.2) appears to give rise to the familiar Higgs mechanism, giving the photon a small nonvanishing mass, However the B condensate lives only inside the vortex, whereas the photon is massless in the 4D bulk outside the vortex. As the action is quadratic in A α the effect of integrating out the photon field can be determined. For instance, a constant time variation ∂ 0 B would give rise to the charge density along the vortex: The electromagnetic potential of an infinite string of such charge density is given by so that Substituting this into Eq. (5.2) one finds a (divergent) energy where an infrared cutoff in the transverse plane at x 2 + y 2 = Λ has been introduced.
This physical discussion explains the result which one would obtain if one were to formally apply the standard method of calculation to the effective action in the presence of a massless 4D field: one would find that the Kähler potential is given by (Appendix (A.3)) namely a finite part which coincides (in the limit g R = 0) with the CP 1 Kähler potential and a divergent part. As the above discussion shows, the latter is caused by the coupling of B with the 4D massless modes, and it is such an interaction that effectively makes the |B| mode non-normalizable. As a potential physical application, one may consider a vortex loop of finite size, as in Witten's cosmological string model [48] or a vorton [49], [50], and study the resulting finite physical effects. We shall leave the study of these issues to a separate work.

Conclusion
The AB effect found above is a result of the mismatch between the the fixed τ 3 R weak gauging direction and the generic vortex orientation B in the color-flavor SU(2) space. A particle with unit U(1) R charge, such as those in Table 1, will get an AB phase as it travels along a large circle around the vortex.
Although we restricted ourselves in this paper to the simplest non-trivial prototype model based on SU(2) L × U 0 (1) × U(1) R gauge symmetry for the sake of clarity of presentation, it is indeed quite straightforward and rather interesting, to extend our analysis to more general gauge groups and patterns of partial weak gauging (in preparation). Also, even though our derivation and the persistence of the vortex moduli space upon U(1) R gauging both depend on the BPS nature of the model considered, the occurrence of the "electromagnetic" AB effect itself has a clear physical explanation, and is independent of the BPS approximation.
As noted in the Introduction, weakly gauging a U(1) subgroup of the flavor symmetry occurs in the color-locked phase of dense quark matter. Therefore, non-Abelian vortices in such a phase should also possess AB fluxes. If such a CFL phase is realized in the cores of neutron stars, non-Abelian vortices will be created by a rapid rotation. Consequently, there may be significant AB effects on particles charged under the asymptotically unbroken gauge symmetry, which could give considerable effects on evolutions of neutron stars. In a non-BPS set-up, the tension of the vortex will generically depend on the value of the modulus |B|, and the magnitude of the AB effect will depend upon the which value of |B| corresponds to the vortex solution with lowest tension. In Ref. [44] it was shown, in the case of high density QCD, that the vortex solutions with the smallest tension are those corresponding to |B| = 0, ∞, which, as shown in this paper, have a non-trivial AB effect for the unbroken electromagnetic field.
In conclusion, as a result of the AB effect an external weak gauging G W of part of the exact color-flavor symmetry converts some of the internal orientational moduli of a non-Abelian vortex into a new observable: an AB phase associated with G W . This can be regarded as a novel physical property of non-Abelian vortices. The value of the phase depends on the particular solution considered, even though in our BPS systems the vortex tension is independent of the solution.
The extension of our results to more general gauge groups and weak gauging subgroups, including supersymmetric systems and roles of the fermions, will be discussed in a forthcoming paper.
tum Phenomena" Grant-in-Aid for Scientific Research on Innovative Areas (No. 25103720) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. W.V. is supported by the Global Engagement for Global Impact programme funded by EP-SRC, and with support from ARO grant number W911NF-12-1-0523. K.K. is supported by INFN special project, "Nonperturbative Dynamics in Gauge Theories and in String Theory (PI14)".
where we used C[A, B]C −1 = [CAC −1 , CBC −1 ]. BPS equations for gauge fields can be rewritten to where we assume that, under complexified gauge transformation with C ∈ G C , ζ is invariant as A.2 Zero modes and Gauss's law Following Ref. [13], let us now lift the moduli parameters in the moduli matrix H A 0 to chiral fields By assumption H 0 contains no anti-chiral field so that δ † α H A 0 = 0 (A.14) where δ α , δ † α are defined by We can show that with the following gauge fields parallel to the vortex configuration (α = 0, 3), We see that ∆L cancels between the gauge and scalar kinetic terms, and the effective Lagrangian can be written in the form, where ∂K core ∂φ X = d 2 xH † 0 Ω −1 ∂H 0 ∂φ X (A. 30) and where the V-transformation V B can be taken as so as to bring back S to the lower triangular form. Therefore there must be the following relation with an appropriate U(2) L gauge transformation U L that is e ψ 0 e ψ 3 e ψ 3w e ψ 3 w |w| 2 e ψ 3 + e −ψ 3 = e ψ (0) Comparing the both sides we find: