# BPS Center Vortices in NonrelativisticSU(N) Gauge Models with Adjoint Higgs Fields

Authors: L. E. Oxman
Journal: Advances in High Energy Physics
 (2015)

---

## Abstract

We propose a class ofSU(N) Yang-Mills models, with adjoint Higgs fields, that accept BPS center vortex equations. The lack of a local magnetic flux that could serve as an energy bound is circumvented by including a new term in the energy functional. This term tends to align, in the Lie algebra, the magnetic field and one of the adjoint Higgs fields. Finally, a reduced set of equations for the center vortex profile functions is obtained (for N=2,3). In particular, Z(3) BPS vortices come in three colours and three anticolours, obtained from an ansatz based on the defining representation and its conjugate.


---

## Body

## 1. Introduction

Topological solitons are present in many areas of physics. Some well known examples are kinks in polyacetylene [1], vortices in type II superconductors, skyrmions in magnetic systems [2], and skyrmions to describe baryons in flavour symmetric models [3, 4]. To gain information about these objects, it is important to identify a critical point where a BPS bound is obtained, namely, a point where the energy can be written as a sum of squares plus the topological charge of the field configuration. As continuous field deformations cannot modify this charge, setting the squares to zero leads to a set of (BPS) equations whose solutions are absolute minima in the given topological sector. In this process, the equations are reduced to first order, which facilitates analytical and numerical studies of these systems. In addition, BPS multisoliton solutions with a given total charge have the same energy, so the forces between BPS solitons vanish. For these reasons, the critical point provides a nice reference to introduce perturbations and study the soliton dynamics [5].Topological solitons are also important in effective descriptions of the strong interactions. Abelian Higgs models have been proposed to describe theqq- potential [6–8] and the interaction among three quarks [9–11]. In [12–15], center vortices were accommodated in SU(N) Yang-Mills models with N adjoint Higgs fields; these objects can describe the N-ality properties of the confining string [16]. Recently, we proposed a class of flavour symmetric models supporting not only the confining string between a qq- colourless pair of external quarks, but also other possible excited states [17]. Among them, qgq-′ hybrid mesons [18–20] are formed, for example, by a red/antigreen pair of quarks bound by an antired/green valence gluon. While the normal string is a center vortex of the effective model, the excited string is formed by a pair of center vortices interpolated by a monopole, which is identified with a confined valence gluon.The topology and classification of center vortices have been analyzed in [16], when a general compact gauge group G is broken down to its center. The roots of the Lie algebra and the weights of their representations play an important role, as occurs when characterizing non-Abelian monopoles [21]. BPS equations for non-Abelian vortices have been obtained in [22–24]; for a review, see [25, 26].In [17], we proposed a Lorentz invariant flavour symmetric model that is expected to contain center vortices, as it possesses the proper SSB pattern and topology. The problem is that the field equations are mathematically difficult. They can only be solved by following numerical methods. With the aim of exploring the usual tools to understand topological objects, in this work we will look for models accepting BPS center vortices, governed by first-order field equations. As an intermediate step, we will simplify the content of the flavour symmetric model, which is based on N2-1 adjoint Higgs fields that form a local Lie basis at the nontrivial vacua of the Higgs potential. Observing that the essential features of the Lie algebra can be captured by a reduced set of fields and conditions, labelled by the simple roots, a Lorentz invariant model that for N≥3 has a simplified field content will be obtained. The Higgs potential will be such that its minimization returns a set of conditions that essentially define a Chevalley basis. This model has the same SSB pattern and topology as the former. Next, we will make an extension to obtain a model that accepts BPS center vortices. In this process, we can anticipate some peculiarities. Generally, BPS equations are derived by working on the energy functional to obtain a bound (for an alternative approach, see [27]). For U(1) vortices, the bound is given by the magnetic flux. This is a topological term that can be written locally, by means of a flux density, so it can indeed arise by working on the energy, which is a local functional. On the other hand, for center vortices, the flux concept is replaced by the asymptotic behavior of the gauge invariant Wilson loop, a nonlocal object that may not appear in the energy calculation. Then, the search for BPS equations in SU(N)→Z(N) SSB models led us to consider the introduction of a nonrelativistic interaction term that tends to align, in the Lie algebra, the magnetic field along one of the adjoint Higgs fields. This in turn implied a different type of bound. After completing the squares, the energy is always greater than or equal to zero. Thus, BPS center vortices are nonrelativistic objects characterized by an exact compensation between the positive definite part of the energy functional (kinetic energy plus Higgs potential) and the Lie algebra alignment contribution.In this regard, two comments are in order. (i) The reason for considering the intermediate step is that, forN≥3, the direct inclusion of the alignment term in the flavour symmetric model would lead, after completing the squares, to too many (possibly incompatible) conditions to saturate the bound. (ii) Because of Lorentz symmetry breaking and rotational symmetry breaking in 3+1 dimensions, the BPS models are not directly physically relevant. However, the presence of a BPS point in the extended parameter space could serve as a check for the numerical analysis, when moving away from the physically relevant non-BPS Lorentz invariant confining models.The general BPS solution will be written in terms of a set of profile functions and a mappingR(S) in the adjoint representation of SU(N). The mapping S∈SU(N) contains information about the asymptotic Wilson loop and the possible defects at the vortex guiding centers, which determine the profile behaviours. Because of the model’s topology, a given phase S0(φ), defined close to and around a vortex guiding center, can be extended to different asymptotic phases Sa(φ), where Sa(φ+2π)=ei2πza/NSa(φ), za∈Z. The Z(N) charge is due to the fact that the different extensions are related by (za′-za)/N∈Z. For the same reason, a given Sa can be matched with different phases S0, with their respective pointlike defects. When leaving the critical point, by lowering the alignment interaction term, some of these extensions will become unstable. For example, for vanishing Z(N)-charge the defect can be avoided, and the lowest energy solution will simply correspond to a trivial regular gauge transformation of the SSB vacua. For Z(N) charge ±1, we will discuss the BPS solutions that are expected to be related to the stable Z(2) and Z(3) noncritical center vortices.The paper is organized as follows. In Section2, we construct the simplified SU(N) model and discuss the possible vacua. In Section 3, we obtain the bounds and the set of BPS equations (for N=2,3). Some properties of the field parametrization are discussed in Section 4. Section 5 is devoted to obtaining information about the BPS solutions and discussing the BPS center vortex. Finally, in Section 6, we present our conclusions.


## 2. Models withSU(N)→Z(N) SSB

In order to support center vortices, we are interested in driving a phase where the gauge symmetry is spontaneously broken down toZ(N). For example, in [17], we introduced a model displaying a flavour symmetry. That is, we considered the energy functional (we are using the inner product X,Y=Tr⁡Ad(X)†Ad(Y), where Ad(·) is a linear map into the adjoint representation):
(1)E=∫d3x12Bi2+12DiψA2+VHiggsψA,
where Bi is the chromomagnetic field, Di=∂i-ig[Ai,], and the potential for the Hermitian adjoint Higgs fields ψA, A=1,…,d=N2-1, is given by
(2)VHiggs=c+μ22ψA,ψA+κ3fABC-iψA,ψB,ψC+λ4ψA,ψB,ψA,ψB,
where fABC are structure constants of the su(N) Lie algebra. At μ2=2/9κ2/λ,κ<0, we can write
(3)VHiggs=λ4ΨAB2,ΨAB=fABCvcψC+iψA,ψB,vc=-κ2λ±κ2λ2-μ2λ=-2κ3λ,
after adjusting c, so that the potential energy for vacuum configurations vanishes. The space of vacua M is obtained from conditions ΨAB=0; that is,
(4)ψA,ψB=ifABCvcψC.
This encompasses the trivial symmetric point ψA=0, separated by a potential barrier from the nontrivial points. Of course, starting from a nontrivial point, ψA∈M, we can generate a continuum SψAS-1, S∈SU(N), that is also in M. In addition, the only transformations that leave these points invariant are S∈Z(N), so they correspond to SU(N)→Z(N) SSB vacua. For N≥3, the SSB points can be divided into a pair of distinct sets, separated by a potential barrier, corresponding to the defining representation and its conjugate:
(5)ψA=STAS-1,ψA=S-TA*S-1.
For N=2, this pair collapses into a single component, as a matrix Sc∈SU(2) exists such that -TA*=ScTASc-1, A=1,2,3.Although the model in (1) contains center vortices, we did not succeed in taking it as a starting point to obtain BPS equations (for N≥3). The presence of too many fields ultimately leads to incompatible conditions to saturate the bound. For this reason, in the next section, we will look for a simplified model. Instead of the previous N2-1 Hermitian fields, we introduce N-1 Hermitian and N-1 complex adjoint Higgs fields. For N≥3, this will result in a simpler set of fields and conditions to define the SSB vacua. This, together with the “alignment” term introduced in Section 3, will finally lead to a set of compatible BPS equations.


### 2.1. Simplified Model

Let us consider Hermitian variables,ψq, q=1,…,r=N-1, and complex variables ζα, labelled by the positive simple roots α→q(α→1<α→2⋯<α→r). The conditions,
(6)ψq,ψp=0,vcα→qζα-ψq,ζα=0,
contain most of the relevant structure of the Lie algebra. For nontrivial fields ζα, we can imply the following.(i)
The fieldsψq are nontrivial, as their sizes are fixed by the eigenvalues vcα→q. They are also linearly independent: if there is a combination γqψq=0, then using (6) we get γ→·α→=0, for every simple root, so that γ→=0.(ii)
[ζα,ζα†] is in the Cartan subalgebra generated by the fields ψq.(iii)
As the positive (negative) roots can be written as a linear combination of the simple roots, with nonnegative (nonpositive) integer coefficients, any root vector is proportional to an appropriate chain of operations of the form[ζα,ζα′]([ζα†,ζα′†]).(iv)
As the difference of a pair of positive simple roots cannot be a root, we have[ζα,ζα′†]=0.However, considering a potential whose minimization only leads to the conditions in (6) would not be the desired one. Given a nontrivial solution (ψq,ζα), the replacement ζα→tζα,t∈R, would also lead to a solution. Then, the interesting SSB initial point could be continuously moved to (ψq,0), and then ψq could be continuously moved to 0, always staying in the space of vacua M. That is, there would be no potential barrier between the interesting configurations and the trivial one. This will be corrected by including a term in the potential to avoid, after minimization, the possibility of moving the fields ζ to zero, when we start with a SSB point. For this purpose, let us consider the additional condition:
(7)∑qsqvcα→q·ψ→-ζαq,ζαq†‍=0,α→·ψ→=α→q′ψq′,
where sq takes values +1 or -1. Now, we consider a solution to (6) and recall that given linearly independent fields ψq it is always possible to introduce unique elements Hα such that
(8)Hα,ψq=vcα→q.
As is well known [28–30], these variables satisfy
(9)ζα,ζα†=ζα,ζαHα.
Using this piece of information in (7) and projecting with Hαp, we get
(10)∑qsqvcα→qq′Hαp,ψq′-ζαq,ζαqHαp,Hαq‍=∑qsqvc2α→q·α→p-ζαq,ζαqHαp,Hαq=0.‍From the Lie algebra internal product and the mappingα→→Hα, an internal product on the root space can be defined [28–30]:
(11)α→,α→′≡Hα,Hα′.
These quantities are strongly constrained. In particular,
(12)2α→,α→′α→′,α→′∈Z
are the so-called Cartan integers, which determine the geometry of the root lattice. They do not depend on the Cartan basis, coinciding with
(13)2α→·α→′α→′·α→′,
which corresponds to (12), when computed with an orthogonal basis ψq, ψq,ψp=vc2δqp. Note that, in this case, Hα=(1/vc)α→qψq. In addition, for su(N), the lengths of the roots are equal, α→,α→=c, α→·α→=1/N. Then, using this piece of information, (10) implies
(14)∑qsqvc2α→q·α→p-Ncζαq,ζαqα→q·α→p=0‍.
This is valid for any basis element α→p; that is,
(15)∑qsqvc2-Ncζαq,ζαqα→q=0,‍
and as the simple roots α→q are linearly independent, we get
(16)ζαq,ζαq=vc2Nc.
This means that if we define the space of vacua M by means of conditions (6) and (7), a nontrivial SSB point (ψq,ζα)∈M cannot be continuously moved to the trivial solution (0,0), always staying in M. In effect, starting with nontrivial fields ζα implies linearly independent fields ψq, and this in turn leads to nontrivial Hα that protect the size of ζα through (16). In other words, a potential whose minimization gives conditions (6) and (7) has a barrier between the SSB vacua and the trivial one.It is convenient to introduce a model without referring to a particular convention for the simple roots. The fieldψ→ with r=N-1 components, such that ψ→q=ψq, can be expanded in terms of either the simple roots α→q or the λ→q basis satisfying
(17)α→q·λ→p=δqp,α→qpλ→qp′=δpp′,(18)ψ→=α→qϕqwith,ϕq=λ→q·ψ→,ψ→=λ→qϕqwith,ϕq=α→q·ψ→,ψp=α→qpϕq=λ→qpϕq.
For su(N), the λ→q basis is given by
(19)λ→q=2NΛ→q,
where Λ→q are the fundamental weights (see the appendix).The relation between the different components is(20)ϕq=Aqpϕp,ϕq=Aqpϕp,Aqp=α→q·α→p,Aqp=λ→q·λ→p.
For su(N), the quantities Cqp=2NAqp are the elements of the Cartan matrix, which define the natural product in the root space.With these definitions, together withζq=ζαq, the conditions in (6) and (7) become
(21)ϕq,ϕp=0,vcδ(p)qζ(p)-ϕq,ζp=0,∑qsqvcϕq-ζq,ζq†=0,‍
which can be obtained by minimizing the Higgs potential:
(22)VHiggs=12Φ,Φ+Zq,Zq+R,
where
(23)Φ=∑qsqΦq‍,Φq=γvcϕq-ζq,ζq†,Zq=γzvcζq-ϕq,ζq,R=γr∑q≠p12ϕq,ϕp2+ϕq,ζp2.‍
Now, noting that
(24)Diψq,Diψq=AqpDiϕq,Diϕp,
we initially propose the model:
(25)E=∫d3x12Bi2+12Diϕq,Diϕq‍HHHhHH12+Diζq,Diζq+VHiggs.
Here, the space of vacua M is given by the trivial point ϕq=0, ζq=0, separated by a potential barrier from the SSB points. For N≥3, the latter can be separated into the sets,
(26)ϕq=Sα→q·H→S-1,ζq=SEαqS-1,(27)ϕq=S-α→q·H→S-1,ζq=S-EαqTS-1,
where EαT is the transpose of Eα and Hq, Eα are elements of a Cartan basis (see the appendix). For N=2, the sets in (26) and (27) are equal.


## 2.1. Simplified Model

Let us consider Hermitian variables,ψq, q=1,…,r=N-1, and complex variables ζα, labelled by the positive simple roots α→q(α→1<α→2⋯<α→r). The conditions,
(6)ψq,ψp=0,vcα→qζα-ψq,ζα=0,
contain most of the relevant structure of the Lie algebra. For nontrivial fields ζα, we can imply the following.(i)
The fieldsψq are nontrivial, as their sizes are fixed by the eigenvalues vcα→q. They are also linearly independent: if there is a combination γqψq=0, then using (6) we get γ→·α→=0, for every simple root, so that γ→=0.(ii)
[ζα,ζα†] is in the Cartan subalgebra generated by the fields ψq.(iii)
As the positive (negative) roots can be written as a linear combination of the simple roots, with nonnegative (nonpositive) integer coefficients, any root vector is proportional to an appropriate chain of operations of the form[ζα,ζα′]([ζα†,ζα′†]).(iv)
As the difference of a pair of positive simple roots cannot be a root, we have[ζα,ζα′†]=0.However, considering a potential whose minimization only leads to the conditions in (6) would not be the desired one. Given a nontrivial solution (ψq,ζα), the replacement ζα→tζα,t∈R, would also lead to a solution. Then, the interesting SSB initial point could be continuously moved to (ψq,0), and then ψq could be continuously moved to 0, always staying in the space of vacua M. That is, there would be no potential barrier between the interesting configurations and the trivial one. This will be corrected by including a term in the potential to avoid, after minimization, the possibility of moving the fields ζ to zero, when we start with a SSB point. For this purpose, let us consider the additional condition:
(7)∑qsqvcα→q·ψ→-ζαq,ζαq†‍=0,α→·ψ→=α→q′ψq′,
where sq takes values +1 or -1. Now, we consider a solution to (6) and recall that given linearly independent fields ψq it is always possible to introduce unique elements Hα such that
(8)Hα,ψq=vcα→q.
As is well known [28–30], these variables satisfy
(9)ζα,ζα†=ζα,ζαHα.
Using this piece of information in (7) and projecting with Hαp, we get
(10)∑qsqvcα→qq′Hαp,ψq′-ζαq,ζαqHαp,Hαq‍=∑qsqvc2α→q·α→p-ζαq,ζαqHαp,Hαq=0.‍From the Lie algebra internal product and the mappingα→→Hα, an internal product on the root space can be defined [28–30]:
(11)α→,α→′≡Hα,Hα′.
These quantities are strongly constrained. In particular,
(12)2α→,α→′α→′,α→′∈Z
are the so-called Cartan integers, which determine the geometry of the root lattice. They do not depend on the Cartan basis, coinciding with
(13)2α→·α→′α→′·α→′,
which corresponds to (12), when computed with an orthogonal basis ψq, ψq,ψp=vc2δqp. Note that, in this case, Hα=(1/vc)α→qψq. In addition, for su(N), the lengths of the roots are equal, α→,α→=c, α→·α→=1/N. Then, using this piece of information, (10) implies
(14)∑qsqvc2α→q·α→p-Ncζαq,ζαqα→q·α→p=0‍.
This is valid for any basis element α→p; that is,
(15)∑qsqvc2-Ncζαq,ζαqα→q=0,‍
and as the simple roots α→q are linearly independent, we get
(16)ζαq,ζαq=vc2Nc.
This means that if we define the space of vacua M by means of conditions (6) and (7), a nontrivial SSB point (ψq,ζα)∈M cannot be continuously moved to the trivial solution (0,0), always staying in M. In effect, starting with nontrivial fields ζα implies linearly independent fields ψq, and this in turn leads to nontrivial Hα that protect the size of ζα through (16). In other words, a potential whose minimization gives conditions (6) and (7) has a barrier between the SSB vacua and the trivial one.It is convenient to introduce a model without referring to a particular convention for the simple roots. The fieldψ→ with r=N-1 components, such that ψ→q=ψq, can be expanded in terms of either the simple roots α→q or the λ→q basis satisfying
(17)α→q·λ→p=δqp,α→qpλ→qp′=δpp′,(18)ψ→=α→qϕqwith,ϕq=λ→q·ψ→,ψ→=λ→qϕqwith,ϕq=α→q·ψ→,ψp=α→qpϕq=λ→qpϕq.
For su(N), the λ→q basis is given by
(19)λ→q=2NΛ→q,
where Λ→q are the fundamental weights (see the appendix).The relation between the different components is(20)ϕq=Aqpϕp,ϕq=Aqpϕp,Aqp=α→q·α→p,Aqp=λ→q·λ→p.
For su(N), the quantities Cqp=2NAqp are the elements of the Cartan matrix, which define the natural product in the root space.With these definitions, together withζq=ζαq, the conditions in (6) and (7) become
(21)ϕq,ϕp=0,vcδ(p)qζ(p)-ϕq,ζp=0,∑qsqvcϕq-ζq,ζq†=0,‍
which can be obtained by minimizing the Higgs potential:
(22)VHiggs=12Φ,Φ+Zq,Zq+R,
where
(23)Φ=∑qsqΦq‍,Φq=γvcϕq-ζq,ζq†,Zq=γzvcζq-ϕq,ζq,R=γr∑q≠p12ϕq,ϕp2+ϕq,ζp2.‍
Now, noting that
(24)Diψq,Diψq=AqpDiϕq,Diϕp,
we initially propose the model:
(25)E=∫d3x12Bi2+12Diϕq,Diϕq‍HHHhHH12+Diζq,Diζq+VHiggs.
Here, the space of vacua M is given by the trivial point ϕq=0, ζq=0, separated by a potential barrier from the SSB points. For N≥3, the latter can be separated into the sets,
(26)ϕq=Sα→q·H→S-1,ζq=SEαqS-1,(27)ϕq=S-α→q·H→S-1,ζq=S-EαqTS-1,
where EαT is the transpose of Eα and Hq, Eα are elements of a Cartan basis (see the appendix). For N=2, the sets in (26) and (27) are equal.


## 3. Nonrelativistic Models with BPS Center Vortex Equations

As is well known, center vortices are characterized by a center element(28)z=ei2πz/NI∈SU(N)
such that, for a path linking the vortex and contained in the asymptotic region, the Wilson loop gives
(29)WA=z.
The center vortex has a Z(N) charge given by z, defined as modulo N. In particular, this is the case when in an asymptotic region r>rm the gauge field is given by
(30)Ai=1g∂iφβ→·H→,ei2πβ→·H→=z,
where r and φ are polar coordinates with respect to the vortex axis. The possible magnetic weights β→ satisfy
(31)β→·α→∈Z,
for every root α→. The solutions to (31) are [14, 16, 21]
(32)β→=2Nw→,
where w→ are the weights of the different representations. The minimum charge center vortices (z=±1) can be labelled by the weights of the defining representation and its conjugate [16].In the asymptotic region, if a Higgs field takes the valueX0, X0,X0=vc2 at φ=0 then, on the circle at infinity, the non-Abelian phase will accompany the Ai behaviour in (30) as follows (we use X to denote any of the Higgs fields ϕq,ζq):
(33)X=SX0S-1,S=eiφβ→·T→.Now, we would like to propose models accepting BPS center vortex equations forSU(2) and SU(3). To simplify the discussion, let us consider planar systems, replacing d3x→d2x=dx1dx2, B3→B, and taking B1=B2=0. Initially, we note that the type of models we have discussed so far cannot accept a BPS bound. Indeed, this would be the case in any model whose energy functional only vanishes for vacuum configurations, while on the space of field configurations {A,X}z, with a given nontrivial asymptotic behaviour labelled by z, it is strictly positive. In this case, to obtain BPS center vortex equations, the energy functional should be bounded by a nonzero term verifying the following: (i) there is gauge invariance, (ii) it assumes a fixed value on the space A,Xz that only depends on z (topological), and (iii) as the bound would be derived by working on the energy density, it should have the form ∫d2xρ‍ (locality). While the Wilson loop verifies (i) and (ii), it is a nonlocal object that cannot arise in the calculation. On the other hand, while
(34)∫d2xη,B‍,B=∂1A2-∂2A1-igA1,A2,
with η an adjoint field, satisfies (i) and (iii), it does not satisfy (ii). This would be a boundary term for homogeneous η and those Abelian-like fields in A,Xz such that B=∂1A2-∂2A1 on thewhole plane. Then, the search for BPS center vortices should consider a modified class of models where configurations in A,Xz do not necessarily have strictly positive energy.For example, we will see that model (25) could be reorganized as a sum of squares plus a term of form (34). As this term does not satisfy (ii), setting the squares to zero will not produce, in a given sector A,Xz, solutions to the field equations associated with (25). Then, it is natural to try a modified model,
(35)E=∫d2x12B2+12Diϕq,Diϕqhhhhhhhh12+Diζq,Diζq+VHiggs-η,B.
Here, we have included the gauge invariant η,B-interaction that tends to align B along η in the Lie algebra. The field η will be an appropriate combination of the adjoint Higgs fields, to be determined in order for the model to accept BPS center vortex equations. At the critical point, we will see that, in spite of the last term in (35), this energy functional satisfies E≥0. For BPS solutions, the contribution originated from the positive definite terms will be exactly compensated by the energy lowering due to the Lie algebra alignment between magnetic and Higgs fields. Thus, the topologically nontrivial BPS center vortices will have vanishing energy. We note that with this term the planar model becomes nonrelativistic although it continues to be isotropic in 2+1 dimensions (in 3+1 dimensions, this type of model would also break rotation symmetry).Let us derive the fundamental property to discuss BPS bounds. Using the cyclicity of the internal product,(36)X,Y,Z=X,Z†,Y,
as Ai is Hermitian, we have
(37)DiX,Y=∂iX-igAi,X,Y=∂iX,Y-X,∂iY+igAi,X,Y=∂iX,Y-X,DiY,(38)DiX,Y+X,DiY=∂iX,Y.
Now, defining
(39)D=D1+iD2,
we note that
(40)DX,DX=D1X+iD2X,D1X+iD2X=D1X,D1X+D2X,D2X-iD2X,D1X+iD1X,D2X.
In addition, as B is Hermitian,
(41)X,B,X=X,X†,B=B,X,X†.
Using this equation, together with (37) and,
(42)Dμ,DνX=-igFμν,X,
which is obtained from the Jacobi identity, we get
(43)D2X,D1X-D1X,D2X=-X,D2D1X+X,D1D2X+∂2X,D1X-∂1X,D2X=-igX,F12,X+∂2X,D1X-∂1X,D2X=-igB,X,X†+∂2X,D1X-∂1X,D2X.
Therefore,
(44)DiX,DiX=DX,DX+gB,X,X†+∂2X,iD1X-∂1X,iD2X,
and, similarly,
(45)DiX,DiX=D-X,D-X-gB,X,X†-∂2X,iD1X+∂1X,iD2X,D-=D1-iD2.


### 3.1.SU(2) Model

ForSU(2), there is simply one component positive root, α1=1/2, and A11=1/2, A11=2. Naming ϕ1=ϕ, ζ1=ζ, the model in (35) is
(46)E=∫d2x12B2+Diϕ2+Diζ2+VHiggs-η,B.‍
The Higgs potential can be written as
(47)VHiggs=12Φ2+Z2,Φ=γvcϕ-ζ,ζ†,Z=γzvcζ-2ϕ,ζ.
Now, using
(48)B2+Φ2=Φ-B2+2Φ,B
and the property (44), for X=ζ, namely,
(49)Diζ2=Dζ2+gζ,ζ†,B+∂2ζ,iD1ζ-∂1ζ,iD2ζ,
we obtain
(50)E=∫d2x12Diϕ2+Dζ2+Z2‍hhhhhhhhh+12Φ-B2+Φ+gζ,ζ†-η,B.
Here, we have used a boundary condition at (x1,x2)→∞,
(51)Diζ⟶0forx1,x2⟶∞.Then, atγ=g2 and taking the Lie algebra element,
(52)η=gvcϕ,
we get
(53)E=∫d2xDiϕ2+Dζ2+Z2+12Φ-B2.‍
The bound is saturated when
(54)Diϕ=0,(55)vcζ-2ϕ,ζ=0,(56)Dζ=0,(57)B=gvcϕ-ζ,ζ†.
At the critical point and taking γz=g2/2, we can write
(58)VHiggs=g22vcϕ-ζ,ζ†2+vcζ-2ϕ,ζ2.
Using ϕ=α1ψ1=1/2ψ1 and defining
(59)ψ2=ζ+ζ†2,ψ3=ζ-ζ†2i,σ=vc2,
the model accepting BPS solutions is given by
(60)E=∫d2x12B2+12DiψA2+VHiggs-gvcϕ,B,‍VHiggs=g22σψ1+iψ2,ψ32+σψ2+iψ3,ψ12hhhhh+σψ3+iψ1,ψ22,
which is a modified version of the flavour symmetric model in (1), (2).


### 3.2.SU(3) Model

The Higgs potential is(61)VHiggs=12Φ,Φ+Zq,Zq+R,Φ=s1Φ1+s2Φ2,Φq=γvcϕq-ζq,ζq†,R=γrϕ1,ϕ22+ϕ1,ζ22+ϕ2,ζ12.
To obtain a set of BPS equations, we initially diagonalize the ϕq-kinetic term in (35). Note that any quantity of the form Xq,Xq can be written as
(62)Xq,Xq=12X2+X1,X2+X1+12X2-X1,X2-X1.
On the other hand, the Cartan matrix for SU(3) is
(63)C=6A=2-1-12,Cqp=Cqp, Aqp=Aqp. Therefore,
(64)X2+X1=A22+A12X2+A21+A11X1=16X2+X1,X2-X1=A22-A12X2+A21-A11X1=12X2-X1.
That is,
(65)Xq,Xq=3X+,X++X-,X-,XiqX+=X2+X1,X-=X2-X1,
and the energy functional in (35) results in
(66)E=∫d2x12B2+32Diϕ+2‍hhhhhhhh+12Diϕ-2+Diζq,Diζq+∫d2x12Φ2+Zq,Zq+R-η,B.‍
Next, similarly to the SU(2) case, using
(67)B2+Φ2=Φ-B2+2Φ,B
and properties (44), (45) for X=ζ2, ζ1, respectively,
(68)Diζ2,Diζ2=Dζ2,Dζ2+gζ2,ζ2†,B+∂3ζ2,iD2ζ2-∂2ζ2,iD3ζ2,Diζ1,Diζ1=D-ζ1,D-ζ1-gζ1,ζ1†,B-∂3ζ1,iD2ζ1+∂2ζ1,iD3ζ1,
we obtain
(69)E=∫d2x32Diϕ+2+12Diϕ-2+D-ζ12‍hhhhhhhh32+Dζ22+Zq,Zq+∫d2x12Φ-B2‍hhhhhhhhh12+Φ+gζ2,ζ2†-gζ1,ζ1†-η,B+∫d2xγrϕ1,ϕ22+ϕ1,ζ22+ϕ2,ζ12,‍
where we have used that the system is in a local vacuum at (x1,x2)→∞.Then, takings1=-1, s2=1, and γ=g2, which gives
(70)Φ=gvcϕ--gζ2,ζ2†-ζ1,ζ1†,ϕ-=ϕ2-ϕ1,
and the Lie algebra element
(71)η=gvcϕ-,
we obtain
(72)E=∫d2x32Diϕ+2+12Diϕ-2+D-ζ12‍hhhhhhhh32+Dζ22+Zq,Zq+∫d2x12Φ-B2+γrϕ1,ϕ22+ϕ1,ζ22hhhhhhhhhhhhhhhhhhhhhhh12+ϕ2,ζ12.
Therefore, the BPS equations are
(73)Diϕ-=0,Diϕ+=0,(74)vcζ2-ϕ2,ζ2=0,ϕ2=3ϕ++ϕ-,vcζ1-ϕ1,ζ1=0,ϕ1=3ϕ+-ϕ-,ϕ1,ϕ2=0,ϕ1,ζ2=0,ϕ2,ζ1=0,(75)Dζ2=0,D-ζ1=0,(76)B=gvcϕ--ζ2,ζ2†-ζ1,ζ1†.


## 3.1.SU(2) Model

ForSU(2), there is simply one component positive root, α1=1/2, and A11=1/2, A11=2. Naming ϕ1=ϕ, ζ1=ζ, the model in (35) is
(46)E=∫d2x12B2+Diϕ2+Diζ2+VHiggs-η,B.‍
The Higgs potential can be written as
(47)VHiggs=12Φ2+Z2,Φ=γvcϕ-ζ,ζ†,Z=γzvcζ-2ϕ,ζ.
Now, using
(48)B2+Φ2=Φ-B2+2Φ,B
and the property (44), for X=ζ, namely,
(49)Diζ2=Dζ2+gζ,ζ†,B+∂2ζ,iD1ζ-∂1ζ,iD2ζ,
we obtain
(50)E=∫d2x12Diϕ2+Dζ2+Z2‍hhhhhhhhh+12Φ-B2+Φ+gζ,ζ†-η,B.
Here, we have used a boundary condition at (x1,x2)→∞,
(51)Diζ⟶0forx1,x2⟶∞.Then, atγ=g2 and taking the Lie algebra element,
(52)η=gvcϕ,
we get
(53)E=∫d2xDiϕ2+Dζ2+Z2+12Φ-B2.‍
The bound is saturated when
(54)Diϕ=0,(55)vcζ-2ϕ,ζ=0,(56)Dζ=0,(57)B=gvcϕ-ζ,ζ†.
At the critical point and taking γz=g2/2, we can write
(58)VHiggs=g22vcϕ-ζ,ζ†2+vcζ-2ϕ,ζ2.
Using ϕ=α1ψ1=1/2ψ1 and defining
(59)ψ2=ζ+ζ†2,ψ3=ζ-ζ†2i,σ=vc2,
the model accepting BPS solutions is given by
(60)E=∫d2x12B2+12DiψA2+VHiggs-gvcϕ,B,‍VHiggs=g22σψ1+iψ2,ψ32+σψ2+iψ3,ψ12hhhhh+σψ3+iψ1,ψ22,
which is a modified version of the flavour symmetric model in (1), (2).


## 3.2.SU(3) Model

The Higgs potential is(61)VHiggs=12Φ,Φ+Zq,Zq+R,Φ=s1Φ1+s2Φ2,Φq=γvcϕq-ζq,ζq†,R=γrϕ1,ϕ22+ϕ1,ζ22+ϕ2,ζ12.
To obtain a set of BPS equations, we initially diagonalize the ϕq-kinetic term in (35). Note that any quantity of the form Xq,Xq can be written as
(62)Xq,Xq=12X2+X1,X2+X1+12X2-X1,X2-X1.
On the other hand, the Cartan matrix for SU(3) is
(63)C=6A=2-1-12,Cqp=Cqp, Aqp=Aqp. Therefore,
(64)X2+X1=A22+A12X2+A21+A11X1=16X2+X1,X2-X1=A22-A12X2+A21-A11X1=12X2-X1.
That is,
(65)Xq,Xq=3X+,X++X-,X-,XiqX+=X2+X1,X-=X2-X1,
and the energy functional in (35) results in
(66)E=∫d2x12B2+32Diϕ+2‍hhhhhhhh+12Diϕ-2+Diζq,Diζq+∫d2x12Φ2+Zq,Zq+R-η,B.‍
Next, similarly to the SU(2) case, using
(67)B2+Φ2=Φ-B2+2Φ,B
and properties (44), (45) for X=ζ2, ζ1, respectively,
(68)Diζ2,Diζ2=Dζ2,Dζ2+gζ2,ζ2†,B+∂3ζ2,iD2ζ2-∂2ζ2,iD3ζ2,Diζ1,Diζ1=D-ζ1,D-ζ1-gζ1,ζ1†,B-∂3ζ1,iD2ζ1+∂2ζ1,iD3ζ1,
we obtain
(69)E=∫d2x32Diϕ+2+12Diϕ-2+D-ζ12‍hhhhhhhh32+Dζ22+Zq,Zq+∫d2x12Φ-B2‍hhhhhhhhh12+Φ+gζ2,ζ2†-gζ1,ζ1†-η,B+∫d2xγrϕ1,ϕ22+ϕ1,ζ22+ϕ2,ζ12,‍
where we have used that the system is in a local vacuum at (x1,x2)→∞.Then, takings1=-1, s2=1, and γ=g2, which gives
(70)Φ=gvcϕ--gζ2,ζ2†-ζ1,ζ1†,ϕ-=ϕ2-ϕ1,
and the Lie algebra element
(71)η=gvcϕ-,
we obtain
(72)E=∫d2x32Diϕ+2+12Diϕ-2+D-ζ12‍hhhhhhhh32+Dζ22+Zq,Zq+∫d2x12Φ-B2+γrϕ1,ϕ22+ϕ1,ζ22hhhhhhhhhhhhhhhhhhhhhhh12+ϕ2,ζ12.
Therefore, the BPS equations are
(73)Diϕ-=0,Diϕ+=0,(74)vcζ2-ϕ2,ζ2=0,ϕ2=3ϕ++ϕ-,vcζ1-ϕ1,ζ1=0,ϕ1=3ϕ+-ϕ-,ϕ1,ϕ2=0,ϕ1,ζ2=0,ϕ2,ζ1=0,(75)Dζ2=0,D-ζ1=0,(76)B=gvcϕ--ζ2,ζ2†-ζ1,ζ1†.


## 4. Center Vortex Ansatz

In order to propose a center vortex ansatz, it would be useful to have a parametrization analogous to the simpleU(1) case, where evidencing the modulus and the phase of the complex Higgs field, ρeiχ, accompanied by the gauge field, a∂iχ, permits the implementation of boundary conditions. For this purpose, we could initially determine whether the asymptotic vacua are of the form given in (26) or (27) and then look for the mapping S (the non-Abelian phase) such that
(77)Sα→q·H→S-1,SEαqS-1,
respectively;
(78)S-α→q·H→S-1,S-EαqTS-1,
is the closest local basis to the field configurations ϕq, ζq. The “polar” decomposition is then
(79)ϕq=SFqS-1,ζq=SZqS-1.The notion of closest mapping can be obtained by following similar steps to those used when defining adjoint Laplacian center gauges [31]. For example, in SU(2), we can take ψ1=2ϕ1, together with ψ2, ψ3 (obtained from ζ1 using (59)), and expand these fields in the TA basis:
(80)ψA=ψABTB,ψAB=ψA,TB,A=1,2,3. The real elements ψAB form a 3×3 matrix Ψ, for which a polar decomposition exists,
(81)Ψ=QR,
where R∈SO(3) and Q is real symmetric and positive semidefinite. The closest orthogonal matrix to Ψ is R; then the closest orthonormal basis to {ψA} is given by
(82)nA=RABTB=STAS-1,
where S is defined up to a global center element. That is, for SU(2) adjoint Higgs fields, the “modulus and phase” decomposition is
(83)ψA=SQABTBS-1,
which can be translated back to ϕ1, ζ1-language.With regard to the gauge field, we note that on any simply connected region, which does not contain the pointlike defects of the local basis, the Higgs field ansatz looks as a gauge transformation. Therefore, in that region, the field equations would be simplified by representing the smoothAi as a gauge transformation of a vector field Ai. However, in the defining representation, S is in general discontinuous on some curves, as it changes by a center element when we go around a center vortex. Therefore, on R2-{pointlikedefects}, the ansatz,
(84)SAiS-1+igS∂iS-1,
cannot work, as it contains a contribution (Ii) concentrated at the points where S-1 is discontinuous. There are three equivalent possibilities to circumvent this problem.(i)
Proceed as in [32, 33], proposing the parametrization,
(85)Ai=SAiS-1+igS∂iS-1-Ii.(ii)
Proceed as in [34, 35] to write
(86)Ai=AiA-CiAnA,
where CiA only depends on the local colour frame (82).(iii)
Work with the fields mapped into the adjoint representation (the matricesMA are generators of the adjoint representation),
(87)AdAi=RAd(Ai)R-1+igR∂iR-1,Ad(Ai)=AiAMA,(88)Adψq=RAdPqR-1,Adζα=RAdPαR-1,where we used
(89)RABMB=RMAR-1.
Here, we will use the third possibility. The advantage of the second and third options is that nA and R contain at most pointlike defects, as they are always single-valued when we go around a loop. Then, the R∂iR-1 term does not introduce delta distributions concentrated on curves, and a smooth Ad(Ai) ansatz can be implemented with Ad(Ai) satisfying appropriate boundary conditions at the vortex guiding centers.It is important to underline that, in the ansatz (87), Ad(Ai) is not a gauge transformation of Ad(Ai). The magnetic field B is given by
(90)AdB=RAdBR-1+igR∂1,∂2R-1,(91)B=∂1A2-∂2A1-igA1,A2,
where the last term in (90) is concentrated at the vortex guiding centers. The profiles Ai, Fq, and Zq must be such that B and the Higgs fields be well-defined and smooth everywhere and satisfy the desired asymptotic behaviour. For a single center vortex, with charge z modulo N, we can impose on the asymptotic region
(92)Ai⟶0,S⟶eiφβ→·T→,
where β→ satisfies (30). When minimizing the energy, the extension of R=Ad(S) from the asymptotic region to the vortex core should contemplate keeping not only R along a Cartan direction but also other possibilities. In this regard, note that, for
(93)β→-β→0=2Nγ→,γ→∈ΛAdSUN,
where Λ(⋯) represents the lattice of weights of the adjoint representation (or root lattice), it is always possible to obtain a map R(r,φ) verifying
(94)Rr,φ=eiφβ→·M→,r>rmeiφβ→0·M→,r<r0,
that is smooth for r≥r0. This map can be constructed as R=eiφβ→0·M→R0, with
(95)R0r,φ=eiφ2Nγ→·M→,r>rmI,r<r0.
Note that R0 always exists as eiφ2Nγ→·T→ is a closed path in SU(N), and therefore eiφ2Nγ→·M→ is topologically trivial in Ad(SU(N)). Different magnetic weights β→0 imply different types of defect and profile function behaviours at r→0. For example, an asymptotic behaviour with z=0 is described by any β→∈2NΛ(Ad(SU(N))). All these values can be extended to β→0=0. For this choice, R(r,φ) contains no defect at the origin and the minimization process will simply return a trivial result, corresponding to a pure gauge transformation of the vacuum configuration. For z=±1, that is, β→=2Nw→+2Nγ→, where w→ is a weight of the defining representation or its conjugate, there is no manner to avoid a defect at r→0. The energy is expected to be minimized by β→0=2Nw→, as in this case some of the basis components will only give one turn when we go around a small circle centered at r=0.


## 5. BPS Center Vortices

At the critical point, to solve theSU(2) and SU(3) BPS equations, it will be enough to consider
(96)ϕq=vcSα→q·H→S-1,ζq=uSEαqS-1,
(and a similar expression for the conjugate sector). The possible non-Abelian phases S are such that R=Ad(S) behaves as in (94). As we will see, the Ai parametrization in terms of Ai and S together with the BPS equations implies
(97)Ai=ciδ→·H→,
where Hq denote the Cartan generators. Then, from (85)–(87), for r<r0, the gauge field is
(98)Ai=ciδ→·H→+1g∂iφβ→0·H→,
and in order to obtain a regular magnetic field, we must have δ→=±β→0 and ci→∓(1/g)∂iφ, when r→0.


### 5.1.su(2)

For nonzeroζ, (55) implies
(99)ϕ=vc2SH1S-1,ζ=uSEα1S-1,
where the possible magnetic weights β in (94) are β=q2, q∈Z. Now, using any of the parametrizations (85)–(87), (54) gives
(100)DiAH1=0orAi,H1=0,
whose solution is
(101)Ai=ciβ0H1
(the case δ=-β0 is discussed at the end). Similarly, (56) becomes
(102)D(A)(uEα1)=0or∂1+i∂2uEα1-iguA1+iA2,Eα1=0.
Thus, joining this piece of information, we obtain
(103)β02c1+ic2=12g∂2h-i∂1h,u=vceh/2,AdB=∂1c2-∂2c1β0AdH1=-12g∂12+∂22hAdH1,
where we have changed the variables from u to h=2ln⁡(u/vc), as is usually done in the U(1) case. Therefore, (57) and (90) imply
(104)∂12+∂22h+g2vc21-ehRAdH1R-1=i2R[∂1,∂2]R-1.
The second member is obtained from (94):
(105)iR∂1,∂2R-1=β0∂1,∂2φAdH1.
As is well-known, although for ji=∂iφ the quantity ∂2j3-∂3j2=[∂2,∂3]φ seems to vanish, it is in fact concentrated at x2=x3=0, where eiφ contains a defect. Namely,
(106)∂1j2-∂2j1=2πδ2x1,x2.
This can be checked using Stokes’ theorem. Then, we get
(107)∂12+∂22h+g2vc21-eh=2π2β0δ(2)(x1,x2).
For q even, the asymptotic behaviour S=eiq2φH1, R=eiq2φAd(H1), on the circle r→∞, can be continuously changed to a behaviour characterized by β0=0, as r is varied from ∞ to 0. The absence of defects will lead to a trivial pure gauge solution for the BPS equations. On the other hand, for q odd, the asymptotic behavior can be changed to β0=+2, as well as β0=-2. For these values, the frame components STα1S-1, STα-1S-1,
(108)Tα=12Eα+E-α,Tα-=12iEα-E-α,
rotate only once, when we go close to and around the origin. The solution to (107) is well-defined for β0=+2, while it is ill-defined for β0=-2. In the latter case, the well-defined solution is obtained using the conjugate ansatz (see a similar discussion in Section 5.2):
(109)ϕ=vc2S-H1S-1,ζ=uS-Eα1TS-1,Ai=ci(-2)H1.
In SU(2), both the vortex and its conjugate satisfy
(110)Sφ+2π=-Sφ,
so they are equivalent objects.


### 5.2.su(3)

Equations (74) imply
(111)ϕq=vcSα→q·H→S-1,ζq=uqSEαqS-1
(q=1,2) so that imposing (73), we obtain
(112)DiAα→q·H→=0orAi,α→q·H→=0.
This means that Ai is in the Cartan subalgebra. Taking δ→=+β→0,
(113)Ai=ciβ→0·H→,(75) gives
(114)∂1+i∂2u2Eα2-igc1+ic2u2β→0·H→,Eα2=0,∂1-i∂2u1Eα1-igc1-ic2u1β→0·H→,Eα1=0.
Then, we get
(115)β→0·α→2c1+ic2=12g∂2h2-i∂1h2,β→0·α→1c1-ic2=-12g∂2h1+i∂1h1,(116)B=-12g∂12+∂22hβ→0·H→β→0·α→2,
where ui=vcehi/2. In addition, (76) reads
(117)iR∂1,∂2R-1=RAd-gB+g2vc2(α→2-α→1)·H→hhhhhhh-g2vc2α→2eh2-α→1eh1·H→R-1,
while, for a single vortex, (94) implies
(118)iR∂1,∂2R-1=∂1,∂2φAdβ→0·H→=2πδ2x1,x2Adβ→0·H→.
Putting this piece of information together, we arrive at
(119)-gB+g2vc2α→2-α→1-g2vc2α→2eh2-α→1eh1=2πδ2x1,x2β→0.
Let us consider the case where β→0 is associated with a weight of the defining representation. Noting that B=(∂1c2-∂2c1)β→0·H→ and α→2-α→1=1/2β→2, in order to have a nontrivial solution, we are led to β→0=±β→2. For these cases, β→0·α→2=-β→0·α→1, (115) give h1=h2=h, ui=u=eh, and both sides of (119) turn out to be oriented along the same direction. Under these conditions, we obtain
(120)β→0·α→2-1∂12+∂22hβ→0+g2vc21-ehβ→2=4πδ2x1,x2β→0.
That is, for β→0=+β→2(β→0·α→2=+1),
(121)∂12+∂22h+g2vc21-eh=4πδ2x1,x2.
On the other hand, the choice β→0=-β→2 would imply
(122)∂12+∂22h+g2vc21-eh=-4πδ2x1,x2.
The second choice does not lead to well-defined Higgs fields. In effect, while close to the origin (121) gives h~2ln⁡r, u=eh/2~r, producing single-valued Higgs fields (and ci~-1/g∂iφ), (122) gives h~-2ln⁡r, u=eh/2~1/r. However, it is easy to see that the new ansatz obtained from (111) by the replacement,
(123)α→q·H→⟶-α→q·H→,Eαq⟶-EαqT,
solves the BPS equations with a well-defined h satisfying (121), provided that we choose β→0=-β→2. Other weights can be obtained by replacing in (111) (resp., (123)),
(124)α→q⟶α→qW,
where W is a Weyl transformation. The solutions will be characterized by the gauge field behaviour (94), with β→0→β→0W=β→2W (resp., β→0→β→0W=-β→2W) (and β→→β→W). Then, these solutions are characterized by the weights of the defining representation, β→1,β→2,β→3, and the weights of the conjugate representation, -β→1,-β→2,-β→3. As the mappings S satisfy
(125)Sφ+2π=e±i2π/3S(φ),
they correspond to center vortices with the minimum charges z=±1.


## 5.1.su(2)

For nonzeroζ, (55) implies
(99)ϕ=vc2SH1S-1,ζ=uSEα1S-1,
where the possible magnetic weights β in (94) are β=q2, q∈Z. Now, using any of the parametrizations (85)–(87), (54) gives
(100)DiAH1=0orAi,H1=0,
whose solution is
(101)Ai=ciβ0H1
(the case δ=-β0 is discussed at the end). Similarly, (56) becomes
(102)D(A)(uEα1)=0or∂1+i∂2uEα1-iguA1+iA2,Eα1=0.
Thus, joining this piece of information, we obtain
(103)β02c1+ic2=12g∂2h-i∂1h,u=vceh/2,AdB=∂1c2-∂2c1β0AdH1=-12g∂12+∂22hAdH1,
where we have changed the variables from u to h=2ln⁡(u/vc), as is usually done in the U(1) case. Therefore, (57) and (90) imply
(104)∂12+∂22h+g2vc21-ehRAdH1R-1=i2R[∂1,∂2]R-1.
The second member is obtained from (94):
(105)iR∂1,∂2R-1=β0∂1,∂2φAdH1.
As is well-known, although for ji=∂iφ the quantity ∂2j3-∂3j2=[∂2,∂3]φ seems to vanish, it is in fact concentrated at x2=x3=0, where eiφ contains a defect. Namely,
(106)∂1j2-∂2j1=2πδ2x1,x2.
This can be checked using Stokes’ theorem. Then, we get
(107)∂12+∂22h+g2vc21-eh=2π2β0δ(2)(x1,x2).
For q even, the asymptotic behaviour S=eiq2φH1, R=eiq2φAd(H1), on the circle r→∞, can be continuously changed to a behaviour characterized by β0=0, as r is varied from ∞ to 0. The absence of defects will lead to a trivial pure gauge solution for the BPS equations. On the other hand, for q odd, the asymptotic behavior can be changed to β0=+2, as well as β0=-2. For these values, the frame components STα1S-1, STα-1S-1,
(108)Tα=12Eα+E-α,Tα-=12iEα-E-α,
rotate only once, when we go close to and around the origin. The solution to (107) is well-defined for β0=+2, while it is ill-defined for β0=-2. In the latter case, the well-defined solution is obtained using the conjugate ansatz (see a similar discussion in Section 5.2):
(109)ϕ=vc2S-H1S-1,ζ=uS-Eα1TS-1,Ai=ci(-2)H1.
In SU(2), both the vortex and its conjugate satisfy
(110)Sφ+2π=-Sφ,
so they are equivalent objects.


## 5.2.su(3)

Equations (74) imply
(111)ϕq=vcSα→q·H→S-1,ζq=uqSEαqS-1
(q=1,2) so that imposing (73), we obtain
(112)DiAα→q·H→=0orAi,α→q·H→=0.
This means that Ai is in the Cartan subalgebra. Taking δ→=+β→0,
(113)Ai=ciβ→0·H→,(75) gives
(114)∂1+i∂2u2Eα2-igc1+ic2u2β→0·H→,Eα2=0,∂1-i∂2u1Eα1-igc1-ic2u1β→0·H→,Eα1=0.
Then, we get
(115)β→0·α→2c1+ic2=12g∂2h2-i∂1h2,β→0·α→1c1-ic2=-12g∂2h1+i∂1h1,(116)B=-12g∂12+∂22hβ→0·H→β→0·α→2,
where ui=vcehi/2. In addition, (76) reads
(117)iR∂1,∂2R-1=RAd-gB+g2vc2(α→2-α→1)·H→hhhhhhh-g2vc2α→2eh2-α→1eh1·H→R-1,
while, for a single vortex, (94) implies
(118)iR∂1,∂2R-1=∂1,∂2φAdβ→0·H→=2πδ2x1,x2Adβ→0·H→.
Putting this piece of information together, we arrive at
(119)-gB+g2vc2α→2-α→1-g2vc2α→2eh2-α→1eh1=2πδ2x1,x2β→0.
Let us consider the case where β→0 is associated with a weight of the defining representation. Noting that B=(∂1c2-∂2c1)β→0·H→ and α→2-α→1=1/2β→2, in order to have a nontrivial solution, we are led to β→0=±β→2. For these cases, β→0·α→2=-β→0·α→1, (115) give h1=h2=h, ui=u=eh, and both sides of (119) turn out to be oriented along the same direction. Under these conditions, we obtain
(120)β→0·α→2-1∂12+∂22hβ→0+g2vc21-ehβ→2=4πδ2x1,x2β→0.
That is, for β→0=+β→2(β→0·α→2=+1),
(121)∂12+∂22h+g2vc21-eh=4πδ2x1,x2.
On the other hand, the choice β→0=-β→2 would imply
(122)∂12+∂22h+g2vc21-eh=-4πδ2x1,x2.
The second choice does not lead to well-defined Higgs fields. In effect, while close to the origin (121) gives h~2ln⁡r, u=eh/2~r, producing single-valued Higgs fields (and ci~-1/g∂iφ), (122) gives h~-2ln⁡r, u=eh/2~1/r. However, it is easy to see that the new ansatz obtained from (111) by the replacement,
(123)α→q·H→⟶-α→q·H→,Eαq⟶-EαqT,
solves the BPS equations with a well-defined h satisfying (121), provided that we choose β→0=-β→2. Other weights can be obtained by replacing in (111) (resp., (123)),
(124)α→q⟶α→qW,
where W is a Weyl transformation. The solutions will be characterized by the gauge field behaviour (94), with β→0→β→0W=β→2W (resp., β→0→β→0W=-β→2W) (and β→→β→W). Then, these solutions are characterized by the weights of the defining representation, β→1,β→2,β→3, and the weights of the conjugate representation, -β→1,-β→2,-β→3. As the mappings S satisfy
(125)Sφ+2π=e±i2π/3S(φ),
they correspond to center vortices with the minimum charges z=±1.


## 6. Conclusions

In this paper we presented Yang-Mills-Higgs nonrelativistic models withSU(N)→Z(N) SSB pattern that accept BPS center vortex equations (for N=2,3).For this purpose, we initially proposed a class ofSU(N) Lorentz invariant models containing real and complex adjoint Higgs fields that can be labelled by the simple roots of the su(N) Lie algebra. The Higgs potential is such that its minimization returns a set of conditions that essentially define a Chevalley basis. The space of vacua also contains a trivial symmetry preserving point, where the Higgs fields vanish, separated from the SSB points by a potential barrier.Next, we introduced a nonrelativistic interaction term so as to obtain a set of BPS equations. This is a term that tends to align, in the Lie algebra, the magnetic field and one of the Higgs fields. Finally, we obtained some solutions. For example, theZ(3) vortices come in three colours (the weights of the defining representation), which are physically equivalent, and three anticolours, obtained from an ansatz based on the conjugate representation.Generally, BPS equations are derived by working on the energy functional, which is a local object, and obtaining a bound that only depends on some topological charge. ForU(1) vortices, the bound is given by the magnetic flux. This is a topological term that can be written locally, by means of a flux density. On the other hand, for center vortices, the flux concept is given by the asymptotic behaviour of the gauge invariant Wilson loop, a nonlocal object that may not arise in the calculation. For this reason, the search for BPS equations led us to consider the alignment interaction. After completing the squares, the energy is always greater than or equal to zero. Thus, BPS center vortices are characterized by an exact compensation between the positive definite part of the energy functional (kinetic energy plus Higgs potential) and the contribution originated from alignment.Similarly to the minima of the Higgs potential, the BPS equations have trivial solutions with vanishing Higgs fields (and pure gauge fields) and a sector where the asymptotic fields are in SSB vacua. Although the BPS solutions have vanishing energy, no finite energy configurations continuously interpolating the center vortex (z=±1) and the trivial configuration (z=0) exist. In other words, there is an energy barrier for the continuous deformation of one configuration into the other. The general solution to the BPS equations was written in terms of a reduced set of profile functions and a mapping R(S) in the adjoint representation of SU(N). The mapping, S∈SU(N), contains information about the asymptotic Wilson loop and the set of possible defects at the vortex guiding centers, which determine the behaviour of the profile functions.In spite of the Abelian looking profile functions obtained, we would like to underline two important differences. As the number of BPS center vortices is increased, the energy continues to vanish. This is in contrast to theU(1) case, where the energy increases linearly with the number of vortices, a property that is modified below and above the critical coupling, implying either attractive or repulsive forces. In addition, the topological properties of the adjoint representation of SU(N) modify the relation between asymptotic phases and defects. A U(1) asymptotic phase implies a unique type of pointlike defect and a unique order for the zero of the corresponding Higgs profile function. On the other hand, for an asymptotic non-Abelian phase, many extensions to reach a pointlike defect are possible, with corresponding conditions on the profile functions. For Z(N)-charge equal to ±1, a defect is always present, while for vanishing Z(N)-charge the defects in R(S) can be avoided, and the lowest energy solution simply corresponds to a regular gauge transformation of the SSB vacua. When leaving the critical point, the new energetics, topology, and field content are expected to modify the forces between center vortices, as compared with the U(1) case. This may be a possibility worth exploring.To sum up, the search for BPS bounds is among the preferred analytical tools to understand topological objects. In this paper, we showed what would be the situation in the context of center vortex(2+1)d models: they become nonrelativistic. Then, although BPS center vortices are not directly physically relevant, they could provide a useful concept when embarking on numerical simulations. The presence of a BPS point in the extended parameter space could serve as a check of the numerical analysis when moving away from the physically relevant non-BPS Lorentz invariant confining models.



---

*Source: 494931-2015-01-08.xml*