Chiral non-Abelian vortices and their confinement in three flavor dense QCD

We find chiral non-Abelian vortices having windings only in one of the diquark condensations of left-handed and right-handed quarks in the color-flavor locked phase of dense QCD. They are the minimum vortices carrying half color magnetic fluxes of those of non-Abelian semi-superfluid vortices (color magnetic flux tubes) and 1/6 quantized superfluid circulations of Abelian superfluid vortices. These vortices carry ${\mathbb C}P^2$ orientational moduli in the internal space corresponding to their fluxes. The ${\mathbb C}P^2$ moduli of two chiral non-Abelian vortices with chiralities opposite to each other are energetically favored to be aligned while those of a vortex and anti-vortex to be orthogonal, and then these vortices attract each other. They are attached by chiral domain walls in the presence of the mass and axial anomaly terms explicitly breaking axial and chiral symmetries. We numerically show that two chiral non-Abelian vortices with chiralities opposite to each other are connected by a chiral domain wall, consisting a mesonic bound state which is nothing but a non-Abelian semi-superfluid vortex. We also show that Abelian and non-Abelian axial vortices attached by chiral domain walls are all unstable to decay into a set of chiral non-Abelian vortices. Furthermore, we find that chiral non-Abelian vortices exhibit unique features: one is the so-called topological obstruction implying that unbroken symmetry generators in the bulk are not defined globally around the vortices, and the other is color non-singlet Aharonov-Bohm (AB) phases implying that quarks encircling these vortices can detect the colors of magnetic fluxes of them at infinite distances.


Introduction
What are states of matter at extreme conditions is one of the challenging problem in modern physics. The ground state of the cold QCD matter at high densities is expected to exhibit color superconductivity, which may be realized in cores of neutron stars [1]. Various phases have been proposed for color superconductivity; the color-flavor locked (CFL) phase [2] in three-flavor symmetric matter is realized extremely high density limit, while the two-flavor superconducting (2SC) phase [3,4] was also proposed for two-flavor symmetric matter. If a color superconductor is realized in the core of neutron stars, there must appear quantum vortices, i. e. vortices with quantized circulations, because of rapid rotations. In colorsuperconducting quark matter, quantum vortices or color magnetic flux tubes appear, as reviewed in Ref. [5]. In the CFL phase, Abelian superfluid vortices are created by rotations [6,7], which are dynamically unstable to decay into more stable vortices [8][9][10]. The most stable vortices are non-Abelian semi-superfluid vortices carrying color magnetic fluxes and 1/3 circulation of the Abelian superfluid vortices [5,8,[11][12][13], which are analogous to non-Abelian vortices in supersymmetric QCD [14][15][16][17][18][19] (see Refs. [20][21][22][23] as a review) and two-Higgs doublet models [24][25][26][27][28]. A non-Abelian vortex confines massless particles in its core; One is bosonic Nambu-Goldstone CP 2 modes originated from spontaneous breaking of the CFL symmetry in its core [5,8,29,30], and the other is gapless Majorana fermions with more topological origin [31][32][33]. Under a rapid rotation, there appear a huge number of vortices (about 10 19 for typical neutron stars). They will form a vortex lattice [34] that behaves as a polarizer of photons [35]. One of the most recent progress is vortices penetrating through crossover between the CFL phase and hyperon nuclear matter within a quark-hadron continuity [9,[36][37][38][39][40][41][42]. While it was suggested that one superfluid vortex in the hyperon nuclear matter is connected to one non-Abelian vortex in the CFL phase [36], it was proved in Refs. [37,38] that three superfluid vortices meet three non-Abelian vortices at a point called a Boojum [9]. The CFL phase is characterized by the two diquark condensations of left and righthanded quarks q L,R , (Φ L,R ) αa ∼ αβγ abc q βb L,R q γc L,R . with the color indices α, β, γ = r, g, b and the flavor indices a, b, c = u, d, s. In the ground states, they both develop VEVs as Φ L = −Φ R . Thus, with defining Φ ≡ Φ L = −Φ R , one has discussed non-Abelian vortices in terms of Φ and gauge fields. However, since the relation Φ L = −Φ R holds only in the ground state, we do not have to assume it for excited states such as vortices. In fact, a similar situation can be found in two-component condensed matter systems: two-gap superconductors [43][44][45][46] and two-component Bose-Einstein condensates (BECs) [47][48][49][50][51][52][53]. In these systems, there are two condensations Φ 1 and Φ 2 . For a singly quantized vortex both fields have the winding Φ 1 = Φ 2 ∼ e iϕ with azimuthal angle ϕ. However, they also admit so-called half-quantum vortices (Φ 1 , Φ 2 ) ∼ (e iϕ , 1) or (Φ 1 , Φ 2 ) ∼ (1, e iϕ ), denoted as (1, 0) or (0, 1), respectively. It is called half-quantum since it carries a half magnetic flux in superconductors or half circulation in BECs. 1 When the system contains an interaction term Φ * 1 Φ 2 + c.c. known as a Josephson term in superconductors or Rabi coupling in 1 Strictly speaking, they are half quantized when the VEVs of Φ1 and Φ2 are the same. When their VEVs Φ1 = v1, Φ2 = v2 are different, they are fractionally quantized as v 2 1 /(v 2 1 + v 2 2 ) and v 2 2 /(v 2 1 + v 2 2 ).
BECs, (1,0) and (0, 1) vortices are connected by a sine-Gordon soliton [44,45,47] 2 ; they are confined to form a singly quantized vortex. When there are no such terms, they are deconfined and are weakly interacting each other. Thus, one may wonder if the same can be considered for vortices in the CFL phase.
In this paper, we investigate non-Abelian vortices having windings only in left Φ L or right Φ R condensation, while the previously known non-Abelian vortices have windings in the both components simultaneously. We call them "chiral non-Abelian vortices" in the sense that quarks of only left or right chirality participate in the vortices. A single non-Abelian semi-superfluid vortex can be decomposed into chiral non-Abelian vortices of chiralities opposite to each other. Chiral non-Abelian vortices carry half color magnetic fluxes of those of non-Abelian semi-superfluid vortices and 1/6 quantized superfluid circulations of Abelian superfluid vortices. We find that a single chiral non-Abelian vortex carries CP 2 orientational moduli in the internal space corresponding to its color magnetic flux. As the case of the Josephson or Rabi term for two-component condensed matter systems, a chiral non-Abelian vortex is attached by a chiral domain wall [5,54] in the presence of mass and axial anomaly terms explicitly breaking axial and chiral symmetries. We also study energetics of two chiral vortices in the absence of mass and axial anomaly terms. In the coexistence of a set of two chiral vortices with opposite chiralities, their CP 2 moduli must be aligned energetically and they attract each other. On the other hand, in the coexistence of a chiral vortex and an anti-chiral vortex, their CP 2 moduli are energetically orthogonal to each other, and they attract each other. We show that in the presence of mass or axial anomaly term, chiral non-Abelian vortices of opposite chirality are connected by a chiral domain wall and are linearly confined.
Other interesting features that we find in this paper are so-called topological obstruction (see Appendix A.4) [55][56][57][58][59][60][61][62] and non-Abelian Aharanov-Bohm (AB) phases. First, the topological obstruction that chiral non-Abelian vortices exhibit implies that generators of the unbroken symmetry in the ground state are not globally defined around the vortices. Second, the chiral non-Abelian vortices exhibit color non-singlet (generalized) AB phases so that the quarks can detect the colors of magnetic fluxes of these vortices at large distances. The bound state of two chiral non-Abelian vortices with the opposite chiralities, equivalent to a single non-Abelian semi-superfluid vortex at large distance, exhibits only color singlet (generalized) AB phases so that the quarks cannot detect the color magnetic flux of such a bound state at large distances.
Finally, we will point out that chiral non-Abelian vortices are the most fundamental elements among topological solitons formed during the chiral symmetry breaking. The CFL phase is accompanied by spontaneous breaking of the chiral symmetry since the diquark condensations Φ L,R of left and right-handed quarks q L,R both develop VEVs. This breaking admits several vortices without color magnetic fluxes. When the U (1) A axial symmetry is spontaneously broken, it admits an axial vortex winding around U (1) A . This vortex is attached by 2N domain walls because of the anomaly term explicitly breaking the U (1) A axial symmetry, in contrast to an analogous axial string in the linear sigma model for chiral symmetry breaking which is attached by N domain walls [63]. The U (N ) L × U (N ) R chiral symmetry breaking admits a non-Abelian axial string, which is attached by one (or two) chiral domain wall(s) depending on the form of mass terms, analogous to one in the linear sigma model for chiral symmetry breaking [12,[64][65][66][67]. The U (1) A Abelian axial vortex mentioned above is dynamically split into N non-Abelian axial strings by domain wall tensions, where each non-Abelian string is attached by one (or two) chiral domain wall(s), as an analogous decay was studied in the linear sigma model for chiral symmetry breaking [5,54]. We find that a single non-Abelian axial string decays into a pair of a chiral non-Abelian vortex and an anti-chiral non-Abelian vortex, while a single Abelian axial string decays into a set of N chiral non-Abelian vortices and N anti-chiral non-Abelian vortices. Thus, chiral non-Abelian vortices are the most fundamental strings. 3 This paper is organized as follows. In Sec. 2, we review the Ginzburg-Landau (GL) theory paying attention to symmetries, and give the order parameter manifold (OPM) which is a new result. In Sec. 3, we review superfluid vortices: Abelian U (1) B superfluid vortices and non-Abelian semi-superfluid vortices (color flux tubes). In Sec. 4, we discuss Abelian and non-Abelian axial vortices. In Sec. 5, we construct chiral non-Abelian vortices in the absence of chiral symmetry breaking terms, and show that they exhibit the topological obstruction and (generalized) AB phases of quarks encircling them. In Sec. 6, we discuss energetics of a single chiral non-Abelian vortex, non-Abelian semi-superfluid vortex, non-Abelian axial vortex, and more general composite vortices. We find that the CP 2 orientations of two chiral vortices with the opposite chiralities are energetically favored to be aligned to each other and then they attract each other, while those of chiral vortex and anti-vortex with the opposite chiralities are energetically favored to be orthogonal to each other and then they attract each other. In Sec. 7, we show that Abelian and non-Abelian axial vortices are attached by chiral domain walls in the presence of axial and chiral symmetry breaking terms, and discuss decay of these vortices. In Sec. 8, we construct a mesonic bound state of two chiral non-Abelian vortices with the opposite chiralities. Sec. 9 is devoted to a summary and discussion. In Appendix A, the terminologies used in this paper are summarized. In Appendix B, we give detailed discussions on symmetry breakings in the CFL phase, and determine associated OPMs. In Appendix C, chiral non-Abelian vortices in the CFL phase are compared with non-Abelian Alice strings [68][69][70] in the 2SC + dd phase of two-flavor quark matter proposed recently [71,72].
2 Color-flavor locked phase of three flavor quark matter In this section, after we review the color-flavor locked phase of dense QCD, we give OPMs and their topology as a new result.
The (approximate) symmetry of N flavor quark matter is (up to discrete groups) where SU (N ) C is the color gauge group, and the rests are global symmetries: U (1) B , U (1) A and SU (N ) L × SU (N ) R are baryon number, axial, and chiral symmetries, respectively. See Appendix B for more precise description including discrete groups. The light quarks q L,R = (q L,R ) αa with α = 1, 2, · · · , N (α = r, g, b for N = 3), a = 1, 2, · · · , N (a = u, d, s for N = 3) and heavy quarks Q L,R = (Q L,R ) α transform under G as where we have not introduced heavy quark flavor symmetry, and have assigned no U (1) B and U (1) A charges on the heavy quarks. 4 The case of N = 3 corresponds to the CFL phase of dense QCD, in which case the light quarks constitute diquark condensations as (Φ L,R ) αa ∼ αβγ abc q βb L,R q γc L,R . Hereafter, we mostly consider the condensates (Φ L,R ) αa as N by N matrices of complex scalar fields on which the symmetries G act as The vector symmetry SU (N ) L+R given by U L = U R is a subgroup of the chiral symmetry SU (N ) L × SU (N ) R , and the rest of generators outside SU (N ) L+R defines the coset space [SU (N ) L × SU (N ) R ]/SU (N ) L+R SU (N ) which we sometimes denote SU (N ) L−R although this does not form a group.
In this paper, we use the static GL free energy for studying vortices. The GL Lagrangian for the CFL phase was obtained as [73][74][75] where the GL coefficients depending on the temperature, density and so on can be found in Refs. [73][74][75]. Among the global symmetries, the axial U (1) A and chiral symmetries are explicitly broken in the presence of the last terms for γ 1,2, The anti-fundamental representation * of quarks for the color group is a convention to make the representation of the condensations ΦL,R introduced below to be fundamental in Eq. (2.3), below. The situation in our mind is that the only light quarks are condensed by forming diquark pairs while the heavy quarks are not. Thus, we use the terminology "U (1)B baryon symmetry" for the spontaneously broken baryon symmetry associated only with the light quarks with no charges for the heavy quarks, while the heavy quarks are also charged under the conventional baron symmetry.
remaining exaxt. The GL theory is valid only near the transition temperature T c . Beyond the GL theory, we need Bogoliubov-de Gennes (BdG) formulation [31,32]. The ground state is given by for small γ's. The symmetry G is spontaneously broken down to the CFL symmetry given by (2.7) The chiral symmetry, U (1) A and U (1) B symmetries are spontaneously broken. According to Appendix B, the full OPM for the symmetry breaking can be written, with taking into account discrete groups, as where (Z 2 ) A+B is generated by (−1, −1) ∈ U (1) B × U (1) A , and F B denotes a fiber bundle with a fiber F over a base manifold B. Here, we have defined the sub-OPMs for the vector symmetry breaking and for the axial and chiral symmetry breakings by respectively, with coset spaces For the details of derivation, see Appendix B. The (Z 2 ) A+B in the denominator of Eq. (2.8) was not recognized before (see Eq. (2.26) of Ref. [5]), and this is a key point to understand chiral non-Abelian vortices found in this paper. The nontrivial first homotopy groups of the sub-OPMs support non-Abelian semi-superfluid vortices (Sec. 3.2) and non-Abelian axial vortices (Sec. 4.2), respectively, but they are not the minimum vortices. On the contrary, the nontrivial first homotopy group of the full OPM supports chiral non-Abelian vortices as the minimum vortices.
Here, one comment is in order. Considering N = 1 in the full OPM in Eq. (2.8), we obtain the OPM for two-component BECs or superconductors, see Ref. [76], allowing halfquantized vortices. Thus, our case is a non-Abelian generalization of such two-component condensed matter systems.
For later conveniences, we define gauge invariants Here, Σ is the chiral symmetry breaking order parameter. These gauge invariants transform under the flavor symmetry as (2.14) In the following sections, we classify various vortices in the CFL phase as summarized in Table 1. To this end, let us introduce labels of vortices by with winding numbers n and m of the gauge invariants det Φ L and det Φ R , respectively.
Here ϕ is the angle coordinate of the polar coordinates in two dimensional space perpendicular to the vortex.

Superfluid vortices and color magnetic flux tubes
In this section, we review superfluid vortices in the CFL phase: Abelian superfluid vortices and non-Abelian semi-superfluid vortices.
In this notation of Eq. (2.15), the Abelian superfluid vortex is labeled by (N, N ) because of det Φ L ∼ det Φ R ∼ e N iϕ . The gauge invariant Σ is Σ = f 2 1 N having no winding. pure color magnetic flux tube Table 1. A summary table for various vortices and color magnetic flux tubes. The N = 3 case corresponds to those in the CFL phase of dense QCD. NA denotes "non-Abelian." "OPM" implies the sub-OPM that vortices are supported by nontrivial first homotopy groups π 1 (OPM), except for color flux tubes which are topologically trivial: π 1 (SU (N ) C ) = 0. M V , M A and M are the OPM for vector symmetry breaking, OPM for axial and chiral symmetry breakings, and full OPM defined in Eqs. (2.9) and (2.8). See Appendix B for details of these OPMs. "Chiral circulation" would imply an amount of magnetic fluxes if the chiral symmetry is gauged, where the normalization is taken such that a closed loop in SU (N ) L−R gives a unit flux.

Non-Abelian semi-superfluid vortices
In this subsection, we review non-Abelian semi-superfluid vortices (color magnetic flux tubes) for comparison with chiral non-Abelian vortices introduced in Sec. 5. The ansatz for a single non-Abelian semi-superfluid vortex winding around the sub-OPM M V U (N ) C−(L+R)+B for the vector symmetry breaking is given in the polar coordinates (r, ϕ) by with the boundary condition for the profile functions f, g and h Explicit numerical solutions were constructed in Ref. [13]. This carries a 1/N U (1) B circulation compared with a unit circulation of an Abelian superfluid vortex given in Eq. (3.1), and a color magnetic flux which is 1/N of that of a pure color flux tube generated by a closed loop in the SU (N ) C gauge group [77]. The latter is unstable to decay into the ground state due to the trivial first homotopy group π 1 [SU (N ) C ] = 0. In terms of the gauge invariants, the non-Abelian semi-superfluid vortex is labeled by (1,1) More generally, the SU (N ) C+L+R transformation on the ansatz in Eq. (3.2) yields a continuous family of solutions. They are characterized by the moduli space [5,8,12] .

Abelian and non-Abelian axial vortices
In this section, we discuss Abelian and non-Abelian axial vortices, which are global vortices without any color fluxes.

Abelian axial vortices
First, let us turn off the axial and chiral symmetry breaking terms by γ 1 = γ 2 = γ 3 = 0. A single Abelian axial vortex winding around U (1) A is given by with e iθ A (ϕ) = e iϕ and the profile function f with the boundary conditions f (r = 0) = 0 In the presence of the axial and chiral symmetry breaking terms, γ 1,2,3 = 0, domain walls are attached to the vortex. To see this, we consider an infinitely large circle with the spatial angle ϕ encircling the axial vortex. Then, let us substitute the ansatz in Eq. (4.1) to the potential V in Eq. (2.5), with replacing the spatial angle ϕ by a function φ(ϕ) depending on the angle ϕ with the boundary condition φ(ϕ = 0) = 0 and φ(ϕ = 2π) = 2π. Then, the potential can be evaluated at spatial infinities as Along the large circle at infinity encircling the axial vortex, there is also the gradient term. Thus, the effective energy for φ at the large circle becomes This is a variant of an N -ple sine-Gordon model. In the case of (γ 1 , γ 2 , γ 3 ) = (0, 0, γ 3 ), the Abelian axial vortex is attached by 2N domain walls. In this case, this vortex is unstable to decay into N non-Abelian axial vortices introduced in Sec. 4.2, each of which is attached by two domain walls. See discussion in Sec. 7.2.
If we turn on γ 1,2 , these 2N domain walls would constitute a composite wall in general. It is an open question whether the decay is suppressed or not in such a case.
Before closing this subsection, let us mention a relation to analogous axial vortices in the context of chiral symmetry breaking at low density. In that case, the axial vortex is attached by N domain walls [63], and it decays into non-Abelian global strings each of which is attached by one domain wall [5,54]. To compare these two cases, it is convenient to see the gauge invariant Σ in Eq. (2.13). In terms of this gauge invariant, the axial vortex in Eq. (4.1) can be rewritten as Thus, one can see that the minimum winding of the axial vortex in the CFL phase corresponds to the double winding of the axial vortex at low density [5,54,63].

Non-Abelian axial vortices
Here we discuss non-Abelian axial vortices winding in the sub-OPM M A = U (N ) L−R+A for the axial and chiral symmetry breakings. An analogue of this at low density was discussed in linear sigma models in Refs. [12,[64][65][66][67]. First, let us turn off the axial and chiral symmetry breaking terms by γ 1 = γ 2 = γ 3 = 0. The ansatz for a single non-Abelian axial vortex is given in the polar coordinates (r, ϕ) by The boundary condition is Explicit numerical solutions can be found in Refs. [65,67]. This vortex is a purely global vortex without any color magnetic flux, carrying a U (1) A winding number which is 1/N of that of the Abelian axial vortex in Eq. (4.1). The non-Abelian axial vortex is labeled by A set of solutions has CP N −1 moduli, which are non-normalizable since the SU (N ) L+R transformation changes the boundary.
In the presence of the axial and chiral symmetry breaking terms, γ 1,2,3 = 0, domain walls are attached to the vortex. In order to understand domain walls attached to the non-Abelian axial vortex, let us substitute the ansatz in Eq. (4.4) to the potential V in (2.5), with replacing the spatial angle ϕ by a function φ(ϕ) depending on the angle ϕ with the boundary condition φ(ϕ = 0) = 0 and φ(ϕ = 2π) = 2π. Then, the potential can be evaluated at spatial infinities as V = 2(γ 1 + γ 3 ) cos(2φ(ϕ)) + 2γ 2 cos(4φ(ϕ)). (4.7) Together with the gradient term, the effective energy for φ on the large circle at infinity encircling the non-Abelian axial vortex becomes E eff = v 2 (∂ ϕ φ) 2 + V . This is the double sine-Gordon model with a half periodicity π instead of the usual case of 2π.
In the case of (γ 1 , γ 2 , γ 3 ) = (γ 1 , 0, γ 3 ), one non-Abelian axial vortex is attached by two domain walls. These two domain walls are attached from the opposite sides of the vortex. This vortex is unstable to decay into two chiral non-Abelian vortices introduced in Sec. 5, each of which is attached by one domain wall. 5 See discussion in Sec. 7.2.
If we turn on γ 2 , how these two domain walls attach to the vortex depends on the parameters γ 1 , γ 2 as classified in Refs. [24,25] in the context of two-Higgs doublet models. In some case, these two domain walls constitute a composite wall. It is an open question whether the decay is suppressed or not in this case.
In terms of the gauge invariant Σ in Eq. (2.13), the ansatz in Eq. (4.4) can be rewritten as with F ≡ f 2 and G = g 2 , and U and θ A in Eq. (4.5). It is obvious that this vortex has a double-winding compared with the corresponding one in the linear sigma model. In fact, the one with unit winding in Σ discussed in Sec. 10 of the review paper [5] corresponds to the chiral non-Abelian vortex introduced in the next section.

Chiral non-Abelian vortices
In this section, we introduce a novel vortex of non-Abelian kind, that is, chiral non-Abelian vortices. Here, we restrict ourselves to the case in the absence of the chiral symmetry breaking terms: γ 1 = γ 2 = γ 3 = 0 in which the axial and chiral symmetries become exact. We then discuss the topological obstruction and AB phases around these vortices.

Solutions of chiral non-Abelian vortices
Chiral non-Abelian vortices introduced in this section are the minimum vortices in the CFL phase. There are two kinds of chiral non-Abelian vortices, namely of left and right chiralities, given by respectively. In order to construct these configurations, we note the relations for labels These imply that a chiral non-Abelian vortex can be constructed as a sum of a half non-Abelian semi-superfluid vortex and a half non-Abelian axial vortex. We thus reach the ansatz for a chiral non-Abelian vortex of the left chirality (1, 0), given in the polar coordi-nates (r, ϕ) by The equations of motions for the profile functions are given by The boundary condition is Numerical solutions for several typical parameter combinations in the case of N = 3 are plotted in Fig. 1 .
Likewise, we also can construct a vortex of the right chirality (0, 1) winding in Φ R in the same way: with F ≡ f c, G ≡ gd, and U (ϕ) and e 2iθ A (ϕ) in Eq. (5.4), while the one of the right chirality (0, 1) in Eq. (5.12) can be rewritten as with F ≡ f c, G ≡ gd, and U (ϕ) and e 2iθ A (ϕ) in Eq. (5.13). These two just look like an anti-vortex to each other. In other words, vortices labeled by (1, 0) and (0, −1) have the same form of Σ. However, the vortices (1, 0) and (0, −1) are distinct because the color magnetic fluxes that they carry are opposite to each other, which are invisible in Σ.
With this regards, vortices in the linear sigma model in terms of Σ discussed in Sec. 10 of the review paper [5] actually describe chiral non-Abelian vortices discussed in this section and should carry magnetic fluxes (invisible in the linear sigma model), although this fact was not recognized in Ref. [5].

Topological obstruction
Here we discuss the so-called topological obstruction (see Appendix A.4 for its definition) common for the vortices with the left and right chiralities in Eqs. (5.3) and (5.12). If we encircle the vortex, the generators T A (A = 1, · · · , N 2 − 1) of the SU (N ) C gauge group transform accordingly as with i, j = 2, · · · , N . After complete encirclement (ϕ = 2π), these become implying that the off-diagonal blocks are not single-valued around the vortex. Those offdiagonal blocks correspond to the broken generators of the CP N −1 moduli in Eq. (5.11). This phenomenon is known as the topological obstruction. More precisely, the obstruction is present for the CFL symmetry in Eq. (2.7) rather than the gauge symmetry itself. The two complete encirclements give This also implies that there is no obstruction around the usual non-Abelian semi-superfluid vortex in Eq. (3.2). For vortices with different color magnetic fluxes corresponding to the CP N −1 moduli in Eq. (5.11), corresponding broken generators have the obstruction.

Generalized Aharonov-Bohm phases
AB phases around the usual non-Abelian semi-superfluid vortices were studied for the electromagnetism [79], and for color gauge field [37][38][39]. Here, we do not consider electromagnetism. Let us discuss generalized AB phases around a single chiral non-Abelian vortex. In the CFL phase (N = 3), the light quarks q and heavy quarks Q receive the following transformations from Eqs. The heavy quarks not participating condensations receive ordinary AB phases contributed only from the gauge symmetry as After complete encirclement (ϕ = 2π), these phases become with is the 2N -th root of the unity, These form a Z 2N group. This is a color non-singlet, implying that heavy quarks can detect the color of the magnetic flux of the vortex from infinite distances. Note that, after two successive encirclements (ϕ = 4π), they become Thus, even numbers of manipulations give a Z N group, which is a color singlet.
On the other hand, the light quarks participate condensations, thus receiving generalized AB phases consisting of two contributions from the vortex winding and AB phases purely coming from the color gauge group, as was studied for usual non-Abelian semisuperfluid vortices in the CFL phase [37,38] as well as non-Abelian Alice strings in the 2SC+ dd phase [68][69][70] (see Appendix C). In our case, generalized AB phases around a chiral non-Abelian vortex are with i, j = 2, · · · , N . After complete encirclement (ϕ = 2π), they become For the case of N = 3, even numbers of encirclements give a trivial action.

Energetics of vortices
In this section, we calculate the leading contributions to tensions of vortices, in particular of chiral non-Abelian vortices (1, 0) and (0, ±1), non-Abelian semi-superfluid vortices (1, 1), and non-Abelian axial vortices (1, −1). We also calculate the tension of a composite state of two chiral non-Abelian vortices (1, 0) and (0, 1) with different CP N −1 orientations, to show that these orientations are energetically favored to be aligned to each other, while CP N −1 orientations of two chiral non-Abelian vortices (1, 0) and (0, −1) are energetically favored to be orthogonal to each other.

Two vortices with parallel CP N −1 orientations
Since all vortices discussed in this model have global U (1) B windings, the leading contributions to their tensions are logarithmically divergent with coming from the kinetic term of Φ R,L , as usual for global vortices. We consider the following asymptotic configuration characterized by a set of two integers {k L , k R }, as r → ∞. We have taken the CP N −1 moduli of the (1, 0) vortex of the left chirality to be oriented to the first component without loss of generality, and we assume that of the (0, 1) vortex of the right chirality to be aligned to the (1, 0) vortex in this subsection. The case that they are not aligned is discussed in the next subsection. The gauge fields should be chosen in such a way that the kinetic energy of Φ L,R is minimized: Then, the scalar kinetic energy reads where F is given by We thus find that the leading term of the tension is given by where Λ is an IR cutoff parameter, or the size of the system. We have F N (1, 0) = F N (0, 1) = 1 for a single chiral non-Abelian vortex (1, 0) or (0, 1), and F N (1, 1) = 4/(N +1) for a single non-Abelian semi-superfluid vortex (1, 1). Comparing these two, we find F N (1, 0) < F N (1, 1) for N ≤ 2, F N (1, 0) = F N (1, 1) for N = 3 (relevant to QCD), and F N (1, 0) > F N (1, 1) for N ≥ 4.
By using these results, we can discuss whether two separated chiral non-Abelian vortices with the opposite chiralities (1, 0) and (0, 1) are energetically bound to a single non-Abelian semi-superfluid vortex (1, 1) or not. To this end, we note that when the vortices (1, 0) and (0, 1) are infinitely separated, the tension of such a composite state is proportional to a sum of the tensions of individual vortices: F N (1, 0) + F N (0, 1) = 2. Comparing this with F N (1, 1) of a single non-Abelian semi-superfluid vortex, we find This implies that two chiral non-Abelian vortices with the opposite chiralities (1, 0) and (0, 1) attract each other for N ≥ 2, while there is no force between them at this order for N = 1. The latter corresponds to two-component BECs in which the absence of the leading order interaction is in fact known [76].

Two vortices with orthogonal CP N −1 orientations
In this subsection, we take the CP N −1 moduli of the (1, 0) and (0, 1) vortices to be orthogonal to each other, which is possible for N ≥ 2. To this end, without loss of generality, we consider the following asymptotic configuration characterized by the set of two integers {k L , k R }, Φ R → v diag 1, e ik R ϕ , · · · , 1 , (6.10) as r → ∞. In this case, the gauge fields should be chosen in such a way that the kinetic energy of Φ L,R is minimized: 11) with T N = diag(−1, N − 1, −1, · · · , −1). Then, the scalar kinetic energy reads with G defined by Thus, the leading contribution to the tension of the composite state can be calculated, to give 14) The tension of a set of two non-Abelian chiral vortices (1, 0) and (0, 1) with the CP N −1 orientations orthogonal to each other is thus found to be proportional to G N (1, 1) = 2 + 1/(N + 1). Since we have an inequality for all N (≥ 2), the chiral non-Abelian vortices with aligned CP N −1 orientations are energetically more favored than those with orthogonal orientations, implying that their CP N −1 moduli attract each other, to be aligned. Again, in a similar way, we can discuss the case of two chiral non-Abelian vortices with the opposite chiralities (1, 0) and (0, −1). In this case, we have an inequality for all N (≥ 2). Thus, two chiral non-Abelian vortices (1, 0) and (0, −1) with orthogonal CP N −1 orientations are energetically more favored than those with aligned orientations, implying that their CP N −1 moduli repel each other. If we separate them infinitely, the tension becomes G N (1, 0) + G N (0, −1) = 2. The inequality implies that the two chiral non-Abelian vortices (1, 0) and (0, −1) with orthogonal CP N −1 orientations attract each other, forming a bound state. It is, however, a highly nontrivial dynamical question remaining as a future problem.

Vortex-domain wall composites
We consider the case of γ 1 , γ 2 , γ 3 = 0 in which the axial and chiral symmetries are explicitly broken. This breaking induces domain walls attached to the vortices.

Chiral non-Abelian vortices attached by chiral domain walls
Let us turn on γ 1 , γ 2 , γ 3 = 0 to see their effects on vortices. In this subsection, we consider a chiral non-Abelian vortex. In Fig. 2, we present numerical simulations in the case that either of γ 1 , γ 2 , γ 3 is nonzero. We can clearly see that one chiral non-Abelian vortex is attached by one or two domain walls. The three columns correspond from the left to the right to (γ 1 , γ 2 , γ 3 ) = ( * , 0, 0), (0, * , 0), (0, 0, * ), respectively. In the middle column γ 2 = 0, the vortex is attached by the two domain walls with the same tension, and thus the configuration is stable. In left-most and right most cases, the vortex is attached by one domain wall from one side. The wall pulls the vortex and the configuration is unstable, but it is static in the comoving frame. To see why this happens, we consider an infinitely large circle parametrized by the spatial angle ϕ encircling vortices. Let us substitute the chiral non-Abelian vortex ansatz of either the left chirality in Eq. (5.3) or the right chirality in Eq. (5.12) to the potential term in Eq. (2.5), with replacing the spatial angle ϕ by a function φ(ϕ) depending on the angle ϕ with the boundary condition φ(ϕ = 0) = 0 and φ(ϕ = 2π) = 2π. It can be evaluated at spatial infinities as V = 2(γ 1 + γ 3 ) cos φ(ϕ) + 2γ 2 cos(2φ(ϕ)). (7.1) Together with the gradient term, the effective energy for φ on the large circle at infinity encircling the chiral non-Abelian axial vortex becomes E eff = v 2 (∂ ϕ φ) 2 + V . This is the double sine-Gordon model. Note that the periodicity is 2π in contrast to the case of non-Abelian axial vortices in Eq. (4.7) in which the periodicity was π.
First, let us consider the absence of γ 3 (γ 3 = 0). The case of (γ 1 , γ 2 , γ 3 ) = (γ 1 , 0, 0) corresponds to the sine-Gordon model, the case of (γ 1 , γ 2 , γ 3 ) = (0, γ 2 , 0) does to the sine-Gordon model with a half periodicity, and the case of (γ 1 , γ 2 , γ 3 ) = (γ 1 , γ 2 , 0) to the double sine-Gordon model. In the first case, a single chiral non-Abelian vortex is attached by a single sine-Gordon soliton, and thus is confined as shown in Fig. 2(a1)-(a7). In the second case, it is attached by two sine-Gordon solitons (or domain walls) of the same tension from the opposite sides, and therefore the composite state is stable, see Fig. 2(b1)-(b7). This case is a non-Abelian generalization of chiral P-wave superconductors, for which the GL theory is described by a U (1) gauge theory coupled with two complex scalar fields Φ 1 and Φ 2 with a potential term V ∼ (Φ * 1 ) 2 (Φ 2 ) 2 + c.c. [80,81]. In the third case, it is attached by two sine-Gordon solitons (or domain walls), but how they attach depends on the parameters γ 1 , γ 2 as classified in Refs. [24,25] in the context of two Higgs doublet models. In these cases, the domain walls attached to the chiral non-Abelian vortex are non-Abelian sine-Gordon solitons carrying CP N −1 moduli [82,83]. This can be clearly seen in Fig. 2  When only γ 3 is present, (γ 1 , γ 2 , γ 3 ) = (0, 0, γ 3 ), one sine-Gordon soliton is attached to one chiral non-Abelian vortex as shown in Fig. 2(c1)-(c7). This is Abelian, carrying no moduli. Indeed, the profile functions |Φ (i,i) L,R | 2 behave almost the same along the domain wall far from the vortex core. This implies that Φ L,R are proportional to the identity, and so no symmetries are broken by the domain wall.
When all γ 1,2,3 are present, there appears either attraction or repulsion among the domain walls attached to the vortex, depending on its sign. If it is attraction, the domain walls form a composite domain wall [5,54], thus confining the chiral non-Abelian vortex. If repulsion, the chiral non-Abelian vortex is attached by two domain walls with different tensions from opposite sides. Such details of the domain wall structure are worth to study on their own, but are not relevant in the following subsections for vortex molecules, as explained below.
If we do the same for the usual non-Abelian semi-superfluid vortex in Eq. (3.2), there is no potential term, implying that no domain wall is attached to the usual non-Abelian semi-superfluid vortex.

Decay of Abelian and non-Abelian axial vortices
Here we discuss that Abelian and non-Abelian axial vortices are all unstable to decay into a set of chiral non-Abelian vortices once the axial and chiral symmetry breaking terms are turned on.
One non-Abelian axial vortex discussed in Sec. 4.2 is attached from the opposite sides by two (or four for (γ 1 , γ 2 , γ 3 ) = (0, γ 2 , 0)) domain walls extending to infinities, and thus decays into two chiral non-Abelian vortices each of which is attached by one (or two) chiral domain wall(s) as in Fig. 3(a); one of the left chirality and the other of the right chirality with the opposite winding. This decay process can be written as (1, −1) → (1, 0) + (0, −1).

(7.2)
It is interesting to observe that there was no flux in the initial state while the final states contain fluxes. Similarly, the Abelian axial vortex is also unstable to decay as Another example is a doubly-wound chiral non-Abelian vortex with the same chirality, say left. This is also attached by two (or four for (γ 1 , γ 2 , γ 3 ) = (0, γ 2 , 0)) chiral domain walls extending to infinities (but vortices of the same chirality are placed at the both L and R), and therefore it is also unstable against decay into two chiral non-Abelian vortices each of which is attached by one (or two) chiral domain wall(s) as in Fig. 3(a). This decay process can be written as (2, 0) → 2(1, 0). (7.4) Similarly, an Abelian axial string having the minimum unit winding in Σ can decay as (N, 0) → N (1, 0). (7.5) This decay was numerically simulated in the linear sigma model [5,54]. 8 Non-Abelian vortex molecules

Structure of chiral non-Abelian vortex molecules
Before discussing the effect of explicit breaking terms γ 1,2,3 = 0 for general case, let us make a comment on the interaction between chiral non-Abelian vortices for γ 1,2,3 = 0 in the case of N = 1, in which the system reduces to two-component BECs. In this case, the interaction energy between (1, 0) and (±1, 0) vortices at distance R is well known E int ∼ ± log R. Thus, a vortex and (anti-)vortex repel (attract) each other as usual for single component global (superfluid) vortices. On the other hand, the interaction energy between (1, 0) and (0, 1) vortices at distance R vanishes at the leading order, to be consistent with Eq. (6.8), and the next leading order is E int ∼ λ log R/R 2 [76] with λ being λ 3 and/or λ 4 in Eq. (2.5) (reducing the same term for N = 1). Thus, it can be either repulsive (λ > 0) or attractive (λ < 0). Once we introduce the explicit breaking terms γ 1,2,3 = 0, a pair of (1, 0) and (0, 1) vortices forms a molecule in which constituents are separated at finite distance, when they are repulsive (λ > 0) [48,51]. They collapse to form an Abelian vortex (1, 1) when they are attractive (λ < 0). Here, we show that the chiral non-Abelian vortices (1, 0) and (0, 1) can form a molecule. They have the same color magnetic fluxes. Now we put a (1, 0)-vortex on the left at "L" and a (0, 1)-vortex on the right at "R" in Fig. 3 (b). We assume that the CP N −1 orientations of these vortices are the same. respectively, where we have parametrized the path r by the coordinate y, and R and A are monotonically increasing functions. Therefore, along the path r, there appears a (composite) domain wall stretching between the (1, 0)-and (0, 1)-vortices once γ 1,2,3 are turned on. The internal structure of the domain wall depends on the values of γ 1,2,3 , as discussed in Sec. 7.1. In the presence of only γ 1,3 , there exists one domain wall between the vortices, while there are two domain walls in the presence of γ 2 . Nevertheless, all domain walls must be stretched between the two vortices since there is no wall along the path b 1 + b 2 encircling the whole configuration. Dynamically, the domain wall tension pulls these chiral non-Abelian vortices and combine them to a single non-Abelian semi-superfluid vortex. Thus, these chiral non-Abelian vortices are confined to a "mesonic" configuration, which is nothing but a non-Abelian semi-superfluid vortex. We can express this confining process by (1, 0) + (0, 1) → (1, 1). (8.5) In Fig. 4, we present numerical simulations of a pair of chiral non-Abelian vortices (1, 0) and (0, 1) separated at finite distance. Clearly one can see that the domain walls are stretched between them. In the case of (γ 1 , γ 2 , γ 3 ) = ( * , 0, 0) (the left column), a single non-Abelian sine-Gordon kink is stretched between them. In the case of (γ 1 , γ 2 , γ 3 ) = (0, * , 0) (the middle column), two domain walls are stretched between them forming a ring, like vortex molecules in a chiral P-wave superconductor, see footnote 7.1. Finally, in the case of (γ 1 , γ 2 , γ 3 ) = (0, 0, * ) (the right column), a single Abelian sine-Gordon kink is stretched between them.
The two vortices are linearly confined and these configurations are on a way to collapse. It is an open question whether, as the case of two-component BECs (N = 1), these constituents can be separated at finite distance and an internal structure of the molecule is visible in certain parameter region. Possibly, it may occur when λ 4 (> 0) is large enough for which the (1, 0) and (0, 1) vortices would repel each other at short distances. Explicitly constructing numerical solutions remains a future problem.

Generalized Aharonov-Bohm phases
Let us discuss generalized AB phases around a vortex molecule in Fig. 3(b). We restrict to N = 3 relevant for the CFL phase.
When the light quarks encircle the (1,0)-vortex at L along the path b L − r, they receive the generalized AB phases These constitute a Z 2 group, which is a color non-singlet. On the other hand, when they encircle the (0,1)-vortex at R along the path b R + r, they receive the generalized AB phases constituting a Z 2 group, which is a color non-singlet.
Thus, when they encircle the both vortices along the large circle b L + b R , they receive the generalized AB phases constituting a Z 2 group. This is a color singlet. These generalized AB phases are precisely those of a single non-Abelian semi-superfluid vortex [31,32]. Interestingly, the light quarks can detect the color of fluxes of the chiral non-Abelian vortices (1, 0) and (0, 1) at the large distance by the generalized AB phases in Eqs. (8.6) and (8.7) which are color non-singlets, but they cannot do that of the whole molecule by the generalized AB phase in Eq. (8.8) which is a color singlet. As for heavy quarks, they detect only gauge fields. Thus, they do not distinguish the (1,0)-and (0,1)-vortices unlike the light quarks, since the gauge structures are identical between these two vortices carrying exactly the same color magnetic fluxes. Therefore, when they encircle either of the (1,0)-and (0,1)-vortices, they receive the AB phases in Eq. (5.20) which is a color non-singlet, while when they encircle the both of them along the path b L + b R , they receive the AB phases in Eq. (5.22) which is a color singlet. The latter forming a color singlet Z 3 group are precisely those of a single non-Abelian vortex [37].
In summary, chiral non-Abelian vortices are not confined and can exist alone when γ 1,2,3 = 0, while they are confined when γ 1,2,3 = 0. In the deconfined phase, chiral non-Abelian vortices exhibit color non-singlet (generalized) AB phases so that the light/heavy quarks can detect the colors of magnetic fluxes of these vortices at large distances. In the confined phase, chiral non-Abelian vortices exhibit only color singlet (generalized) AB phases so that the light/heavy quarks cannot detect the colors of magnetic fluxes of these vortices at large distances. Thus, stable states exhibit color-singlet (generalized) AB phases.
The opposite is not always true. Not all states with color-singlet (generalized) AB phases can exist stably in the confined phase. For instance, the two examples (1, −1) and (2, 0) in Sec. 7.2 exhibit color-singlet (generalized) AB phases; the (2, 0) made of two (1, 0) with the same color magnetic fluxes exhibits the singlet AB phase for heavy quarks in Eq. (5.22) and the trivial phases for light quarks obtained from two successive phases of Eq. (5.24); (Q L , Q R , q L , q R ) → ( 2 Q L , 2 Q R , q L , q R ), which are color singlet. Nevertheless, they are attached by the two chiral domain walls extending to infinities as in Fig. 3(a), and are unstable against decay into two chiral non-Abelian vortices each of which is attached by one (or two) chiral domain wall(s): (1, −1) → (1, 0) + (0, −1) and (2, 0) → 2(1, 0). The (N, 0) vortex made of N (1, 0) vortices with all different color magnetic fluxes give generalized AB phases as (Q L , Q R , q L , q R ) → (Q L , Q R , −q L , q R ), which are color singlet. It is, however, broken as (N, 0) → N (1, 0).

Summary and discussion
In the CFL phase of dense QCD, we have found chiral non-Abelian vortices winding only around either of left or right diquark condensation Φ L or Φ R labeled by (1, 0) and (0, 1), respectively. As can be expected from (1, 0) = 1 2 [(1, 1) + (1, −1)] and (0, 1) = 1 2 [(1, 1) − (1, −1)], they carry half color magnetic fluxes and half U (1) B circulation of those of a non-Abelian semi-superfluid vortex labeled by (1,1), and half U (1) A winding and half chiral circulation (around the sub-OPM M A = U (N ) L−R+A ) of a non-Abelian axial vortex labeled by (1, −1). A single chiral non-Abelian vortex carries CP N −1 orientational moduli in the internal space corresponding to its color magnetic flux. We have discussed the energetics of vortices and have found that CP N −1 orientations of two chiral non-Abelian vortices (1, 0) and (0, 1) are energetically aligned, while those with a chiral vortex (1, 0) and anti-vortex (0, −1) are energetically orthogonal to each other. Then, the two chiral non-Abelian vortices attract each other forming bound states. We have shown that chiral non-Abelian vortices exhibit the topological obstruction implying that the unbroken symmetry generators are not defined globally around the vortices, and color non-singlet (generalized) AB phases implying that quarks at large distances can detect the colors of magnetic fluxes by encircling these vortices. In the presence of the axial and chiral symmetry breaking terms γ 1,2,3 = 0, these vortices are confined by chiral domain walls, while they are deconfined in the absence of those terms. In the confined phase, two chiral non-Abelian vortices (1, 0) and (0, 1) with chiralities opposite to each other are connected by a chiral domain wall, constituting a mesonic bound state (1, 1) which is nothing but a non-Abelian semisuperfluid vortex, exhibiting only color singlet (generalized) AB phases implying that the quarks cannot detect the color of magnetic flux of such a bound state at large distances. We also have shown that the Abelian axial vortices (N, −N ) and non-Abelian axial vortices (1, −1) attached by chiral domain walls are both unstable to decay into a set of chiral non-Abelian vortices.
Before closing this paper, let us address several discussions and future directions. The confinement does not imply that the mesonic bound state neccesary collapses to a axisymmetric non-Abelian semi-superfluid vortex. It remians as a futre problem to numerically construct solutions of vortex molecules in certain parameter regions, in which the constituent (1, 0) and (0, 1) vortices are separated at finite distances At least the axial and chiral symmetry breaking terms should be relatively small. At finite temperature, it does not have to be the case at least in 2+1 dimensions because of the Berezinskii-Kosterlitz-Thouless (BKT) transition. The BKT transition was explicitly shown in Ref. [53] by numerical simulations for the Abelian case N = 1.
In this paper, we have constructed numerical solutions for single chiral non-Abelian vortices (1, 0) or (0, 1), in the absence of the axial and chiral symmetry breaking terms: γ 1,2,3 = 0. In the presence of these terms, chiral domain walls are attached to them. We have constructed solutions in the case that only one of γ 1,2,3 is nozero. In particular, if we turn on all γ 1,2,3 's, the situation is close to the two-Higgs doublet models [24,25]. Explicitly constructing numerical solutions of such domain-wall vortex composites in general cases remains as one of future problems. A particularly important problem is to construct a vortex molecule (1, 0) + (0, 1). This would reduce to a single non-Abelian semi-superfluid vortex in the most parameter region because of the domain wall tension, but we should examine whether these two constituents can be separated in some parameter region particularly for small γ 1,2,3 's and/or small gauge coupling g s for which there is a repulsion between constituent vortices. This problem is important in a relation with higher-form symmetries discussed in the next paragraph. Finally, we also should numerically verify decays of axial and chiral vortices as discussed in Sec. 7.2, such as a non-Abelian axial vortex (1, −1) → (1, 0) + (0, −1) and an Abelian axial vortex (N, −N ) → N (1, 0) + N (0, 1), as we did a similar problem in the linear sigma models [5,54]. In particular, in the presence of the mass terms γ 1,2 = 0, 2N domain walls attached to one Abelian axial vortex (N, −N ) constitute a composite wall as can be expected from Eq. (4.2), and it is an open question whether this fact suppresses the decay. To perform simulations, we may do either a relaxation method or real time dynamics. For the latter, we need a time-dependent GL theory.
Higher-form symmetries [84] related with a linking between Wilson loops and vortices are an indispensable tool to study phases of matter such as the so-called topological order. Higher form symmetries in the presence of non-Abelian semi-superfluid vortices and the absence or presence of a topological order of the CFL phase were studied in Refs. [39][40][41][42]85]. In this case, a linking between a Wilson loop and a non-Abelian semi-superfluid vortex is rather trivial in the sense that AB phases are color singlets. Contrary to this, a linking between a Wilson loop and a chiral non-Abelian vortex is non-trivial because AB phases are color non-singlets as we have seen in Sec. 5.3. Thus, the phase separating a non-Abelian semi-superfluid vortex (1, 1) into two chiral non-Abelian vortices (1, 0) and (0, 1) may be characterized in terms of a higher-form symmetry.
As mentioned in introduction, in the context of quark-hadron continuity, vortices penetrate through the CFL phase and hyperon nuclear matter [9,[36][37][38][39][40][41][42]. In particular, from the AB phases of quarks around vortices, one can conclude the existence of a boojum at which three hyperon vortices and three non-Abelian semi-superfluid vortices must meet [9,37,38]. This structure is modified if the deconfined phase is realized in the CFL phase. In fact, this situation is similar to two-flavor quark matter (see Appendix C).
Beyond the GL description, we could study fermion structure by the BdG formulation. In fact, fermion zero modes were studied for non-Abelian semi-superfluid vortices in the BdG equation [31][32][33], in which triplet Majorana fermion zero modes were found. Such Majorana fermions endow these vortices a non-Abelian exchange statistics in d = 2 + 1, turning them into non-Abelian anyons [86,87]. Apparently, it is a very interesting question whether fermion zero modes exist on chiral non-Abelian vortices and if so what is their exchange statistics.
The CP 2 modes of the chiral non-Abelian vortex are probably non-normalizable, unlike those of a single non-Abelian semi-superfluid vortex [5,30]. However, around a constituent of a vortex molecule (1, 1), these modes may be normalizable because of a cut-off introduced by the presence of the other. The CP 2 modes are normalizable on the chiral domain wall [82,83] that connects the (1, 0) and (0, 1). This fact together with a fact that the CP 2 modes are normalizable on a single non-Abelian semi-superfluid vortex [5,30] may suggest that these modes are still normalizable around the vortex molecule (1, 1).
In this paper, we have turned off the electro-magnetic interaction and the strange quark mass. Turning them on can be incorporated in the CP 2 effective world-sheet Lagrangian of a single non-Abelian semi-superfluid vortex in Refs. [88] and [30], respectively. This method may be applied to the case of a chiral non-Abelian vortex as well.
Finally, there are some interesting directions for studying chiral domain walls. One is a decay of chiral domain walls by quantum or thermal tunneling. In this case, a hole created on the domain wall world-volume is surrounded by an axial vortex (see Sec. 10.5 of the review paper [5]). For the minimum element of a chiral domain wall, a hole should be surrounded by a chiral non-Abelian vortex studied in this paper. The other direction is given by is the chiral non-Abelian semi-superfluid vortices under magnetic field background which would be also interesting in connection with the chiral anomaly. The domain wall connecting (1, 0) and (0, 1) is made of the η meson related to U (1) A , and η nontrivially changes along the direction perpendicular to the η domain wall. Therefore, under the presence of magnetic field, the domain wall should be magnetized as found in Refs. [89,90], see also Sec. 10.6 of Ref. [5]. Physical consequences of these domain walls are interesting to explore. this language since holonomies are Z N (N = 3 for dense QCD and N = 2 for two-Higgs doublet models). Chiral non-Abelian vortices found in this paper are accompanied by non-Abelian holonomies, and thus they are non-Abelian in this language as well.

A.2 Superfluid/semi-superfluid
Abelian superfluid vortices have integer windings around U (1) B . They do not carry any color magnetic fluxes.
Semi-superfluid vortices have fractional windings around U (1) B . For single-valuedness, they must be accompanied with color gauge transformation for single-valuedness of fields, and thus they are inevitably non-Abelian.

A.4 Topological obstruction
Here, we explain the topological obstruction [55][56][57][58][59][60][61][62]. When a symmetry G is spontaneously broken down to its subgroup H, the OPS is a coset space G/H. Note that the unbroken symmetry H is not unique. When the VEV v = φ of a field φ is transformed to v = gv with some group element g ∈ G, the unbroken symmetry H is also transformed to H = gHg −1 .

B Order parameter manifolds
Let us describe the full OPM in this Appendix. To this end, we neglect explicit breaking terms, γ 1,2,3 = 0, thus axial and chiral symmetries becoming exact.
The symmetry G acts on the condensates Φ L,R , which are N by N matrices of complex scalar fields, as With taking into account discrete groups, G can be faithfully written as Here, the discrete groups Z N are defined in Table 2, and the two U (1) groups can be explicitly written as Let us rewrite the two U (1) groups in Eq. (B.4) by the baryon and axial U (1) groups as where the relation is given by where (Z 2 ) A+B generated by (−1, −1) ∈ U (1) B ×U (1) A is redundant and must be removed. Then, the symmetry G acting on the condensates as can be rewritten as with the discrete groups in the denominator, defined in Table 2 where F B denotes a fiber bundle with a fiber F over a base manifold B. 6 We can further rewrite it as (B.13) 6 In general, when a Lie group G is spontaneously broken down to H, the OPS parametrized by Nambu-Goldstone modes is a coset space G/H. In this situation, the original group G can be regarded as a Here, note that G/H is not endowed with a product but isomorphic to a Lie group that we denote by SU (N )L−R.
Here, we have defined the OPMs for the vector symmetry breaking and for the axial and chiral symmetry breakings by C Similarities and differences with vortices in two-flavor dense QCD Let us make comments on possible similarities with recently found non-Abelian Alice strings in two-flavor dense QCD.
Recently, two-flavor dense QCD relevant for quark-hadron continuity was proposed [71,72], consisting of the 2SC condensation of up and down quarks in addition to a P-wave condensation of down quarks. This phase is referred as the 2SC + dd phase and is further classified into deconfined and confined phases of vortices. In the deconfined phase, the most stable vortices are non-Abelian Alice strings which are superfluid vortices carrying color magnetic fluxes [68][69][70]. The amount of these color magnetic fluxes are half of those of non-Abelian semi-superfluid vortices in the CFL phase.
One of the characteristic features of non-Abelian Alice stings is that unbroken symmetry generators are not globally defined around the strings, and in general they are multi-valued (topological obstruction). Another characteristic feature, which is more important, is that particles encircling these strings can detect the colors of the strings from infinite distances by color non-singlet AB phases.
In the confined phase, non-Abelian Alice strings are confined by the so-called AB defects [93][94][95] appearing to compensate a discontinuity originated from non-trivial AB phases of the 2SC condensation [70]. As a result of vortex confinement, there can exist only baryonic and mesonic bound states of the Alice strings, which exhibit color singlet AB phases of particles encircling them; The baryonic bound state consists of three Alice strings with different (red, blue, green) color magnetic fluxes with total color canceled out, which are connected by a domain wall junction resulting in a single Abelian superfluid vortex, while the mesonic bound state consists of two Alice strings with the same color magnetic fluxes, which are connected by a single domain wall resulting in a doubly-wound non-Abelian string. Although the latter carries a color magnetic flux, it can exist because of color-singlet AB phases, that is, the color cannot be detected from infinite distance by AB phases of particles encircling it. The amount of the color magnetic flux that the mesonic bound state of the Alice string (or doubly-wound non-Abelian vortex) in two-flavor quark matter is the same with that of a non-Abelian string in the CFL phase.
Moreover, both of a mesonic bound state of the Alice string (or doubly-wound non-Abelian vortex) in two-flavor quark matter and a non-Abelian string in the CFL phase exhibit Z 3 color-singlet AB phases of heavy quarks, and Z 2 color-singlet generalized AB phases of light quarks. However, a crucial difference between them is that non-Abelian Alice strings are confined by the AB defects spontaneously appearing in the formation of the 2SC condensate while chiral non-Abelian vortices are confined by chiral domain walls existing due to the explicit breaking (mass and anomaly terms) of the axial and chiral symmetries. Thus, we can summarize that a salient distinction is whether the appearance of the domain walls confining the vortices is due to spontaneous (the 2SC + dd phase) or explicit (the CFL phase) breaking.