New Results on Non-Abelian Vortices - further insights into monopole, vortex and confinement
aa r X i v : . [ h e p - t h ] M a r IFUP-TH/2010-10
New Results on Non-Abelian Vortices– further insights into monopole, vortex andconfinement K. KONISHI ∗ Department of Physics, “E. Fermi”, University of Pisa, andINFN, Sezione di PisaLargo Pontecorvo, 3, 56127, Pisa, Italy ∗ ∼ konishi/ ABSTRACT
We discuss some of the latest results concerning the non-Abelian vor-tices. The first concerns the construction of non-Abelian BPS vorticesbased on general gauge groups of the form G = G ′ × U (1). In particulardetailed results about the vortex moduli space have been obtained for G ′ = SO ( N ) or U Sp (2 N ). The second result is about the “fractionalvortices”, i.e., vortices of the minimum winding but having substruc-tures in the tension (or flux) density in the transverse plane. Thirdly, wediscuss briefly the monopole-vortex complex. Talk presented at the 2009 International Workshop on “Strong Coupling Gauge The-ories in LHC Era” [SCGT 09], December 8-11, 2009, Nagoya University, Japan
Introduction
The last few years have witnessed a remarkable progress in our understandingof the non-Abelian vortices and their relation to monopoles, both of which arethirty-year old problems in theoretical physics, and which can bear importantimplications to some deep issues such as quark confinement. The plan of thistalk is: (i) a very brief review of non-Abelian monopoles; (ii) a brief reviewof ’03-’07 results on non-Abelian vortices; (iii) a new result on non-Abelianvortices based on general gauge groups; (iv) the fractional vortices; and (v)a brief discussion on the monopole-vortex complex and non-Abelian duality.It has become customary to think of quark confinement as a dual super-conductor, in which (chromo-) electric charges are confined in a medium inwhich a magnetic charge is condensed. The original suggestion by ’t Hooftand Mandelstam is essentially Abelian: the effective low-energy degrees offreedom are magnetic monopoles arising from the Yang-Mills gauge fields.It is however possible that the dual superconductor relevant to quark con-finement is of a non-Abelian kind, in which case we must better understandthe quantum mechanical properties of these degrees of freedom. We wouldlike to understand how the ’t Hooft-Polyakov monopoles [1] (arising from agauge symmetry breaking, G → H ) and Abrikosov-Nielsen-Olesen vortices[2] (of a broken gauge theory H → ) are generalized in situations where therelevant gauge group H is non-Abelian.The key developments which allowed us a qualitatively better under-standing of these solitons are the following. First, the Seiberg-Witten so-lutions of N = 2 supersymmetric gauge theories [3, 4] revealed the quantum-mechanical behavior of the magnetic monopoles in an unprecedented fash-ion. In the presence of matter fields (quarks and squarks) these theorieshave, typically, vacua with non-Abelian dual gauge symmetry in the infrared[5, 6]. Thus in these systems non-Abelian monopoles do exist and play acentral role in confinement and dynamical symmetry breaking. Second, thediscovery of non-Abelian vortex solutions [7, 8], i.e., soliton vortices withcontinuous, non-Abelian moduli, has triggered an intense research activityon the classical and quantum properties of these solitons, leading to a richvariety of new interesting results [9, 10, 11].2 Monopoles
When the gauge symmetry is spontaneously broken G h φ i6 =0 −→ H (1)where H is some non-Abelian subgroup of G , the system possesses a setof regular magnetic monopole solutions in the semi-classical approximation.They are natural generalizations of the Abelian ’t Hooft-Polyakov monopoles[1], found originally in the G = SO (3) theory broken to H = U (1) by a Higgsmechanism. The gauge field looks asymptotically as F ij = ǫ ijk B k = ǫ ijk r k r ( β · H ) , (2)in an appropriate gauge, where H are the diagonal generators of H in theCartan subalgebra. A straightforward generalization of the Dirac’s quanti-zation condition leads to [12] 2 β · α ∈ Z (3)where α are the root vectors of H . In the simplest such case, G = SU (3), H = SU (2) × U (1) / Z ∼ U (2), a straightforward idea that the degeneratemonopole solutions to be a doublet of the unbroken SU (2) leads however tothe well-known difficulties [14, 15].On the other hand, the quantization condition Eq. (3) implies that themonopoles should transform, if any, under the dual of U (2): the individ-ual solutions are labelled by β which live in the weight vector space of ˜ H ,generated by the dual roots, α ∗ = αα · α . (4)As transformation groups of fields H and ˜ H are relatively non-local, thesought-for transformations of monopoles must look as non-local field trans-formations from the point of view of the original theory [16].But the most significant fact is that fully quantum mechanical monopolesappears in the low-energy dual description of a wide class of N = 2 super-symmetric QCD [5, 6]. There must be ways to understand the physics ofnon-Abelian monopoles starting from a more familiar, semiclassical solitonpicture. 3 Vortices
Attempts to understand the semi-classical origin of the non-Abelian monopolesappearing in the so-called r -vacua of the N = 2 supersymmetric SU ( N )gauge theory, has eventually led to the discovery of the non-Abelian vor-tices [7, 8]. They are natural generalizations of the Abrikosov-Nielsen-Olesen(ANO) vortex. Unlike the ANO vortex, however, the non-Abelian vorticescarry continuous zeromodes, i.e., it has a nontrivial moduli.The simplest model in which these vortices appear is an SU ( N ) × U (1)gauge theory with N f = N flavors of squarks in the fundamental representa-tion. The secret of the non-Abelian vortices lies in the so-called color-flavorlocked phase, in which the squark fields (written as N × N color-flavor mixedmatrix) takes the VEV of the form, h q ( x ) i = v N × N . (5)The SU ( N ) gauge symmetry is completely broken, but the color-flavor mixeddiagonal symmetry remains unbroken.The vortex configuration in this vacuum involves scalar fields of the form, q ( x ) = v e iφ f ( ρ ) 0 0 00 g ( ρ ) 0 00 0 . . . 00 0 0 g ( ρ ) (6)where ρ, φ, z (the static vortex does not depend on z ) are the cylindricalcoordinates. The gauge fields take appropriate form, in order to ensure thatthe kinetic term tends to zero asymptotically, D q ( x ) →
0. In Eq. (6) thefirst flavor of the squark winds, but the full solution A i , q ( x ) can be rotatedin the color flavor SU ( N ) transformations, A i , → U ( A i + i∂ i ) U † , q ( x ) → U q ( x ) U † , leaving the tension invariant.In other words, individual vortices break the exact SU ( N ) C + F symme-try of the system, developing therefore non-Abelian orientational zeromodes.Its nature is seen from the fact that the vortex Eq. (6) breaks the globalsymmetry as SU ( N ) → SU ( N − × U (1) / Z N − ; (7)4he vortex moduli is given by CP N − ∼ SU ( N ) SU ( N − × U (1) / Z N − . (8)They are Nambu-Goldstone modes, which however can propagate only insidethe vortex: far from it they are massive.The quantum properties of the non-Abelian orientational modes (the ef-fective CP N − sigma model), the study of non-Abelian vortices of higherwinding numbers, the generalization to the cases of larger number of flavorsand the study of the resulting, much richer vortex moduli spaces, the questionof vortex stability in the presence of small non-BPS corrections, extensionto more general class of gauge theories, etc. have been the subjects of anintense research activity recently. One of the new results by us [17] is the construction of non-Abelian vortexsolutions based on a general gauge group G ′ × U (1), where G ′ = SU ( N ) ,SO ( N ) , U Sp (2 N ), etc. As in models based on SU ( N ) gauge groups studiedextensively in the last few years, we work with simple models which havethe structure of the bosonic sector of N = 2 supersymmetric models. Themodel contains a FI (Fayet-Iliopoulos)-like term in the U (1) sector, allowingthe system to develop stable vortices. A crucial aspect is that we work in acomplete Higg vacuum, but with an unbroken color-flavor diagonal symmetry.We take as our model system L = Tr c h − e F µν F µν − g ˆ F µν ˆ F µν + D µ H ( D µ H ) † − e (cid:12)(cid:12) X t − ξt (cid:12)(cid:12) − g | X a t a | i , with the field strength, gauge fields and covariant derivative denoted as F µν = F µν t , F µν = ∂ µ A ν − ∂ ν A µ , ˆ F µν = ∂ µ A ν − ∂ ν A µ + i [ A µ , A ν ] ,A µ = A aµ t a , D µ = ∂ µ + iA µ t + iA aµ t a , µ is the gauge field associated with U (1) and A aµ are the gauge fields of G ′ .The matter scalar fields are written as an N × N F complex color (vertical)– flavor (horizontal) mixed matrix H . It can be expanded as X = HH † = X t + X a t a + X α t α , X = 2 Tr c (cid:0) HH † t (cid:1) , X a = 2 Tr c (cid:0) HH † t a (cid:1) . t and t a stand for the U (1) and G ′ generators, respectively, and finally, t α ∈ g ′⊥ ,where g ′⊥ is the orthogonal complement of the Lie algebra g ′ in su ( N ). Thetraces with subscript c are over the color indices. e and g are the U (1) and G ′ coupling constants, respectively, while ξ is a real constant.We choose the maximally “color-flavor-locked” vacuum of the system, h H i = v √ N N , ξ = v √ N . (9)We have taken N F = N which is the minimal number of flavors allowing forsuch a vacuum. Note that, unlike the U ( N ) model studied extensively in thelast several years, the vacuum is not unique in these cases (i.e., with a generalgauge group), even with such a minimum choice of the flavor multiplicity.This difference may be traced to the fact that groups such as SO ( N ) × U (1)and U Sp ( N ) × U (1) form strictly smaller subgroups of U ( N ).The existence of a continuous vacuum degeneracy implies the emergenceof vortices of semi-local type; this aspect will be crucial in the discussion ofthe fractional vortices in the second part of this talk. However, for now, westick to the particular vacuum Eq. (9) and consider vortices and their moduliin this theory. The standard Bogomol’nyi completion reads T = Z d x Tr c h e (cid:12)(cid:12)(cid:12)(cid:12) F − e (cid:0) X t − ξt (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + 1 g (cid:12)(cid:12)(cid:12)(cid:12) ˆ F − g X a t a (cid:12)(cid:12)(cid:12)(cid:12) +4 (cid:12)(cid:12) ¯ D H (cid:12)(cid:12) − ξF t i ≥ − ξ Z d x F , (10)where ¯ D ≡ D + i D , z = x + ix . In the BPS limit one has T = 2 √ N πξν = 2 πv ν , ν = − π √ N Z d x F , (11)where ν is the U (1) winding number of the vortex. This leads immediatelyto the vortex BPS equations¯ D H = ¯ ∂H + i ¯ AH = 0 , (12) F = e (cid:2) Tr c (cid:0) HH † t (cid:1) − ξ (cid:3) , F a = g Tr c (cid:0) HH † t a (cid:1) . (13)6he matter BPS equation (12) can be solved by the Ansatz H = S − ( z, ¯ z ) H ( z ) , ¯ A = − iS − ( z, ¯ z ) ¯ ∂S ( z, ¯ z ) , (14)where S belongs to the complexification of the gauge group, S ∈ C ∗ × G ′ C . H ( z ), holomorphic in z , is the moduli matrix , which contains all moduliparameters of the vortices.A gauge invariant object can be constructed from S as Ω = SS † . Thiscan be conveniently split into the U (1) part and the G ′ part, so that S = s S ′ and analogously Ω = ω Ω ′ , ω = | s | , Ω ′ = S ′ S ′† . The tension (11) can berewritten as T = 2 πv ν = 2 v Z d x ∂ ¯ ∂ log ω , ν = 1 π Z d x ∂ ¯ ∂ log ω , (15)and ν determines the asymptotic behavior of the Abelian field as ω = ss † ∼ | z | ν , for | z | → ∞ . The minimal vortex solutions can then be written down by making useof the holomorphic invariants for the gauge group G ′ made of H , which wedenote as I iG ′ ( H ). If the U (1) charge of the i -th invariant is n i , I iG ′ ( H )satisfies I iG ′ ( H ) = I iG ′ (cid:16) s − S ′− H (cid:17) = s − n i I iG ′ ( H ( z )) , while the boundary condition is I iG ′ ( H ) (cid:12)(cid:12)(cid:12)(cid:12) | z |→∞ = I i vev e iνn i θ , where ν n i isthe number of the zeros of I iG ′ . This leads then to the following asymptoticbehavior I iG ′ ( H ) = s n i I iG ′ ( H ) | z |→∞ −→ I i vev z νn i . It shows that I iG ′ ( H ( z )), being holomorphic in z , are actually polynomials.Therefore ν n i must be positive integers for all i : ν n i ∈ Z + → ν = kn , k ∈ Z + , with n ≡ gcd { n i | I i vev = 0 } . The U (1) gauge transformation e πi/n leaves I iG ′ ( H ) invariant and thus the true gauge group is G = [ U (1) × G ′ ] / Z n , Z n is the center of the group G ′ . The minimal winding in U (1) foundhere, n , corresponds to the minimal element of π ( G ) = Z : it representsa minimal loop in our group manifold G . As a result we find the followingnon-trivial constraints for H I iG ′ ( H ) = I i vev z knin + O (cid:16) z knin − (cid:17) . Certain special solutions of a given theory can be found readily, as follows.It turns out that each such solution is characterized by a weight vector of thedual group , and is parametrized by a set of integers ν a ( a = 1 , · · · , rank( G ′ )) H ( z ) = z ν N + ν a H a ∈ U (1) C × G ′ C , (16)where ν = k/n is the U (1) winding number and H a are the generators ofthe Cartan subalgebra of g ′ . H must be holomorphic in z and single-valued ,which gives the constraints for a set of integers ν a ( ν N + ν a H a ) ll ∈ Z ≥ ∀ l . Suppose that we now consider scalar fields in an r -representation of G ′ . Theconstraint is equivalent to ν + ν a µ ( i ) a ∈ Z ≥ ∀ i , (17)where ~µ ( i ) = µ ( i ) a ( i = 1 , , · · · , dim( r )) are the weight vectors for the r -representation of G ′ . Subtracting pairs of adjacent weight vectors, one arrivesat the quantization condition ~ν · ~α ∈ Z , (18)for every root vector α of G ′ .Now Eq. (18) is formally identical to the well-known GNO monopole quantization condition [12], as well as to the na¨ıve vortex flux quantizationrule [13]. There is however a crucial difference here, from these earlier results.Because of an exact flavor (color-flavor diagonal G C+F ) symmetry, brokenby individual vortex solutions, our vortices possess continuous orientationalmoduli. These zero modes are normalizable, unlike those encountered in theearlier attempts to define quantum “non-Abelian monopoles”.8 ′ ˜ G ′ SU ( N ) SU ( N ) / Z N U ( N ) U ( N ) SO (2 M ) SO (2 M ) U Sp (2 M ) SO (2 M + 1) SO (2 M + 1) U Sp (2 M )Table 1: Some pairs of dual groupsThese non-Abelian modes of our vortices—they are a kind of Nambu-Goldstone modes—can fluctuate and propagate along the vortex length. Insystems with a hierarchical symmetry breaking, G → G = G ′ × U (1) → , where our G = G ′ × U (1) model might emerge as the low-energy approxi-mation, these orientational zero modes get absorbed by massive monopolesat the vortex extremities. This process endows the monopoles with fullyquantum-mechanical non-Abelian (GNO-dual) charges, as has been suggestedby the author and others in several occasions [16], but we shall not dwell onthis subject further here.The solution of the quantization condition (18) is that˜ ~µ ≡ ~ν/ , is any of the weight vectors of the dual group of G ′ . The dual group, denotedas ˜ G ′ , is defined by the dual root vectors [12] ~α ∗ = ~α/ ( ~α · ~α ) . Examples ofdual pairs of groups G ′ , ˜ G ′ , are shown in Table 1. Note that (17) is actuallystronger than (18), the l.h.s. must be a nonnegative integer. This positivequantization condition allows for a few weight vectors only. For concreteness,let us consider scalar fields in the fundamental representation, and choose abasis where the Cartan generators of G ′ = SO (2 M ) , SO (2 M + 1) , U Sp (2 M )are given by H a = diag (cid:16) , · · · , | {z } a − , , , · · · , | {z } M − , − , , · · · , (cid:17) , (19)with a = 1 , · · · , M . In this basis, special solutions H have the form for9 ′ = SO (2 M ) and U Sp (2 M ) H (˜ µ , ··· , ˜ µ M )0 = diag (cid:16) z k +1 , · · · , z k + M , z k − , · · · , z k − M (cid:17) , (20)while for SO (2 M + 1) H (˜ µ , ··· , ˜ µ M )0 = diag (cid:16) z k +1 , · · · , z k + M , z k − , · · · , z k − M , z k (cid:17) , (21)where k ± a = ν ± ˜ µ a .For example, in the cases of G ′ = SO (4) , U Sp (4) with a ν = 1 / ~ ˜ µ = ( , ) , ( , − ) , ( − , ) , ( − , − ) H ( , )0 = diag( z, z, ,
1) = z +1 ·H +1 ·H , (22) H ( , − )0 = diag( z, , , z ) = z +1 ·H − ·H , (23) H ( − , )0 = diag(1 , z, z,
1) = z − ·H +1 ·H , (24) H ( − , − )0 = diag(1 , , z, z ) = z − ·H − ·H . (25)These four vectors are the same as the weight vectors of two Weyl spinorrepresentations ⊕ ′ of ˜ G ′ = SO (4) for G ′ = SO (4), and the same as thoseof the Dirac spinor representation of ˜ G ′ = Spin (5) for G ′ = U Sp (4).The weight vectors corresponding to the k = 1 vortex in various gaugegroups are shown in Fig. 1. In all cases the results found are consistent withthe GNO duality. Another exciting recent result concerns the fractional vortex and lumps.[18]We have pointed out above that in a general class of gauge theories thevacuum is not unique, even if the Fayet-Iliopoulos term is present and even ifthe number of the flavors is the minimum possible for a “color-flavor” lockedvacuum to exist. In other words, there is a nontrivial vacuum degeneracy, orthe vacuum moduli M . In the first part of the talk, we were mainly interestedin the vortex moduli V , in a particular, maximally color-flavor locked vacuum.Here we are going to consider all possible vortices—the vortex moduli V —onall possible points of the vacuum moduli M at the same time.10 O (2) − − SO (3)0 USp (2) − SO (4) ( , ) SO (5) (1 , USp (4) ( , ) (1 , , SO (6) ( , , ) USp (6)( , , ) Figure 1: The special points for the k = 1 vortex.There are in fact two crucial ingredients for the fractional vortex: the vacuum degeneracy and the BPS saturated nature of the vortices. The firstpoint was emphasized just above: the situation is schematically illustrated inFig. 2. Even if we restrict ourselves to the minimally winding vortex solutionsonly, the vortices represent non-trivial fiber bundles over the vacuum moduli M .The BPS-saturated nature of the vortices, on the other hand, implies thatthe vortex equations are reduced to the first-order equations. The matterequations of motion are solved by the moduli-matrix Ansatz. The otherequations–the gauge field equations–reduce, in the strong coupling limit or,anyway, sufficiently far from the vortex center, to the vacuum equations forthe scalar fields. In other words, the vortex solutions approximate the sigmamodel lumps. 11igure 2: Vacuum moduli M , fiber F over it, and possible singularities Let M be the manifold of the minima of the scalar potential, the vacuumconfiguration M = { q i | q † T I q = ξ I } . The vacuum moduli M is given bythe points p ∈ M = M/F , (26)where the fiber F is the sum of the gauge orbits of a point in M . A vortexsolution is defined on each point of M , in the sense that the scalar configu-ration along a sufficiently large circle ( S ) surrounding it traces a non-trivialclosed orbit in the fiber F (hence a point in M ). The existence of a vortexsolution requires that π ( F, f ) = , (27)where f is a point in M . The field configuration on a disk D encircled by S traces M , apart from points at finite radius where it goes off M (hencefrom M ). In other words it represents an element of π ( M , p ), where p isthe gauge orbit containing f , or p = π ( f ): π is the projection of the fiberonto a point of the base space M . The exact sequence of homotopy groupsfor the fiber bundle reads · · · → π ( M, f ) → π ( M , p ) → π ( F, f ) → π ( M, f ) → π ( M , p ) → · · · where π ( M/F, f ) ∼ π ( M , p ). See Fig. 3.Given the points f, p and the space M , the vortex solution is still notunique. Any exact symmetry of the system broken by an individual vortex12igure 3:solution gives rise to vortex zero modes (moduli), V . Our main interest herehowever is the vortex moduli which arises from the non-trivial vacuum moduli M itself. Due to the BPS nature of our vortices, the gauge field equation F I = g I (cid:0) q † T I q − ξ I (cid:1) , reduces, in the strong-coupling limit (or in any case, sufficiently far fromthe vortex center), to the vacuum equation defining M . This means thata vortex configuration can be approximately seen as a non-linear σ -model(NL σ M) lump with target space M , as was already anticipated. Variousdistinct maps S
7→ M of the same homotopy class correspond to physicallyinequivalent solutions; each of these corresponds to a vortex with the equaltension T min = − ξ I Z d x F I > , by their BPS nature. They represent non-trivial vortex moduli .The semi-local vortices of the so-called extended-Abelian Higgs (EAH)model arise precisely this way. In an Abelian Higgs model with N flavors of(scalar) electrons, M = S N − , F = S , M = S N − /S = C P N − , and theexact homotopy sequence tells us that π ( C P N − ) and π ( S ) are isomorphic:each (i.e. minimum) vortex solution corresponds to a minimal σ -model lumpsolution.In most cases discussed in our paper [18], however, the base space M willbe various kinds of singular manifolds : a manifold with singularities, unlikein the EAH model. The nature of the singularities depends on the systemand on the particular point(s) of M . Our BPS degenerate vortices represent(generalized) fiber bundles with the singular manifolds M as the base space.13 .2 Two mechanisms for fractional vortex–lump There are two distinct mechanisms leading to the appearance of a fractionalvortex. The first is related to the presence of orbifold singularities in M . Forexample, let us consider a Z point p such as the one appearing in a simple U (1) model with two scalars, one of which has charge 2. At this singularity,both π ( M , p ) and π ( F, f ) make a discontinuous change. The minimumelement of π ( F , f ) is half of that of π ( F, f ) defined off the singularity,and similarly for π ( M , p ) with respect to π ( M , p ), p = p . Even thoughthe exact homotopy sequence continues to hold on and off the orbifold point,the vortex defined near such a point will look like a doubly-wound vortex,with two centers (if the vortex moduli parameters are chosen appropriately).Analogous multi-peak vortex solution occurs near a Z N orbifold point of M .Another cause for the appearance of fractional peaks is simple and verygeneral: a deformed sigma model geometry. This phenomenon can be bestseen by considering our system in the strong coupling limit. Even if the basepoint p is a perfectly generic, regular point of M , not close to any singularity,the field configurations in the transverse plane ( S ) trace the whole vacuummoduli space M . The energy distribution reflects the nontrivial structure of M as the volume of the target space is mapped into the transverse plane, C E = 2 Z C ∂ K∂φ I ∂φ † ¯ J ∂φ I ¯ ∂φ † ¯ J = 2 Z C ¯ ∂∂K . The field configuration may hit for instance one of the singularities (conic ornot), or simply the regions of large scalar curvature. Such phenomena thusoccur very generally if the underlying sigma model has a deformed geometry . At such points the energy density will show a peak, not necessarily at thevortex center. Even at finite coupling, the vortex tension density will exhibita similar substructure. A simple model showing the fractional vortex is an extended Abelian Higgsmodel, with two scalar fields A and B with charges 2 and 1, respectively.Depending on the point of M (which is C P ) the minimum vortex showsdoubly-peaked substructure clearly, see Fig. 4. The fractional vortex struc-ture in this model nicely illustrates the first mechanism discussed above: the Basically the same phenomenon was found also by Collie and Tong. [19] B = 0 is a Z orbifold point, since there the only nonvanishing field, A , having charge 2, must wind only half of the U (1) to be single-valued. (jAvevj2;jBvevj2) = (0;1) (jAvevj2;jBvevj2) = (1=4;1=2) (jAvevj2;jBvevj2) = (1=2;1=10000) Figure 4:
The energy (the left-most and the 2nd left panels) and the magnetic flux (the2nd right panels) density, together with the boundary values (
A, B ) (the right-most panel)for the minimal vortex.
Another interesting model is a U (1) × U (1) gauge theory with threeflavors of scalar electrons H = ( A, B, C ) with charges Q = (2 , ,
1) for U (1) and Q = (0 , , −
1) for U (1) . Even though the model has the same C P as the vacuum moduli M as the first model, the vortex properties arequite different. This model turns out to provide a good example of fractionalvortex of the second type (deformed sigma-model geometry).Fractional vortex occurs also in non-Abelian gauge theories, such as theone with gauge group G = SO ( N ) × U (1). An illustrative example of frac-tional vortex in an SO (6) × U (1) model is shown in Fig. 6. A more recent work of our research group concerns the monopole-vortex com-plex solitons occurring in systems with hierarchical gauge symmetry break-15 jAvevj2;jBvevj2;jCvevj2) = (0;1;1) (jAvevj2;jBvevj2;jCvevj2) = (1=2;1=2;1=2)(jAvevj2;jBvevj2;jCvevj2) = (0:9;0:1;0:1)
Figure 5:
The energy density (left-most) and the magnetic flux density F (1)12 (2nd fromthe left), F (1)12 (2nd from the right) and the boundary condition (right-most). Figure 6:
The energy density of three fractional vortices (lumps) in the U (1) × SO (6)model in the strong coupling approximation. The positions are z = −√ i √ , z = −√ − i √ , z = 2. The two figures correspond to two different choices of certain sizemoduli parameters. G h φ i6 =0 −→ H h φ i6 =0 −→ , |h φ i| ≫ |h φ i| . (28)The homotopy-group sequence · · · → π ( G ) → π ( G/H ) → π ( H ) → π ( G ) → · · · . (29)tells us that the properties of the regular monopoles arising from the breaking G → H are related to the vortices of the low-energy system. In particular,the fact that π ( G ) = for any group G , implies that π ( G/H ) ∼ π ( H ) /π ( G ) . (30)For instance, in the case of the symmetry breaking, SU ( N + 1) → SU ( N ) × U (1) / Z N the first set of checks (on the Abelian and non-Abelian magneticflux matching) have been done [20] soon after the discovery of the non-Abelian vortex in the U ( N ) theory. We wish to study more carefully themonopole-vortex configurations, taking into account a small non-BPS cor-rection term.For instance one might study the model based on hierarchically brokengauge symmetry, SU (3) → SU (2) × U (1) → , with the Hamiltonian, H = Z d x h g ( F ij ) + 14 g ( F ij ) + 1 g |D i φ a | + 1 g |D i φ | + |D i q | + g | µφ + √ Q † t Q | + g | µφ + √ Q † t Q | + 2 Q † λ † λQ i . (31)describing the system after the first breaking. Such a low-energy theory is ofthe type studied in our original work on non-Abelian vortex [8], except forsmall terms involving φ ( x ) and φ ( x ) (which were set to their constant VEVin that paper). In fact, the model is the same as the one studied by Auzziet. al. recently [21]. The system has unbroken, exact color-flavor diagonal SU (2) C + F symmetry. Neglecting the fields which get mass of the order of thehigher symmetry breaking scale, and the fields which go to zero exponentiallyoutside the monopole size, one makes an Ansatz (in the monopole and vortexsingular gauge): A φ = t A φ ( ρ, z ) + t A φ ( ρ, z ); A φ = − ρ f ( ρ, z ) , A φ = −√ ρ f ( ρ, z ) , ( r ) = v v
00 0 − v + λ ( ρ, z ) , λ ( ρ, z ) = t λ ( ρ, z ) + t λ ( ρ, z ) .q ( x ) = (cid:18) w ( ρ, z ) 00 w ( ρ, z ) (cid:19) , with appropriate boundary conditions. The equations for the profile func-tions f , f , w w , λ , λ may be studied numerically. Some qualitative fea-tures can be read off from the structure of these equations. (i) The Dirac string of the monopole is hidden deep in the vortex core;the zero of the squark field at the vortex core makes the singularityharmless. (ii)
The whole monopole-vortex complex breaks SU (2) C + F : the orienta-tional zeromodes develops which live on SU (2) /U (1) ∼ CP . (iii) The degeneracy between the monopole of the broken “ U spin” and themonopole of the broken “ V -spin”, which are na¨ıvely related by theunbroken SU (2) of the high-mass-scale breaking SU (3) → SU (2) × U (1), is explicitly broken in the vacuum with small squark VEV. (iv) Nevertheless, there is a new, exact continuous degeneracy among themonopole-vortex complex configurations, related by the color-flavorsymmetry ( CP moduli). It is possible that such an exact non-Abeliansymmetry possessed by the monopole is at the origin of the non-Abeliandual gauge symmetry which emerges at low energies of the softly broken N = 2 supersymmetric QCD [6]. Acknowledgments
The main new results reported here are the fruit of a collaboration withM. Eto, T. Fujimori, S. B. Gudnason, T. Nagashima, M. Nitta, K. Ohashi,and W. Vinci. The last part on unpublished work on the monopole-vortexcomplex is based on a collaboration with S. B. Gudnason, D. Dorigoni, A.Michelini and M. Cipriani. I thank them all. I wish to thank also theorganizers of the Nagoya 2009 International Workshop on “Strong CouplingGauge Theories in LHC Era” [SCGT 09] where this talk was presented, fora stimulating atmosphere. 18 eferences [1] G. ’t Hooft, Nucl. Phys. B , 817 (1974), A.M. Polyakov, JETP Lett. , 194 (1974).[2] A. Abrikosov, Sov. Phys. JETP , 1442 (1957); H. Nielsen, P. Olesen,Nucl. Phys. B , 45 (1973).[3] N. Seiberg, E. Witten, Nucl. Phys. B , 19 (1994); Erratum ibid. B , 485 (1994).[4] N. Seiberg, E. Witten, Nucl. Phys. B , 484 (1994).[5] P. C. Argyres, M. R. Plesser, N. Seiberg, Nucl. Phys. B , 159(1996); P.C. Argyres, M.R. Plesser, A.D. Shapere, Nucl. Phys. B ,172 (1997); K. Hori, H. Ooguri, Y. Oz, Adv. Theor. Math. Phys. , 1(1998).[6] G. Carlino, K. Konishi, H. Murayama, JHEP , 004 (2000); Nucl.Phys. B , 37 (2000).[7] A. Hanany and D. Tong, JHEP , 037 (2003)[arXiv:hep-th/0306150].[8] R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, Nucl. Phys.B , 187 (2003) [arXiv:hep-th/0307287].[9] D. Tong, “TASI lectures on solitons,” arXiv:hep-th/0509216.[10] M. Shifman and A. Yung, Rev. Mod. Phys. , 1139 (2007),[arXiv:hep-th/0703267].[11] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, J. Phys. A ,R315 (2006), [arXiv:hep-th/0602170].[12] P. Goddard, J. Nuyts and D. Olive, Nucl. Phys. B (1977) 1;F. A. Bais, Phys. Rev. D (1978) 1206; B. J. Schroers and F. A. Bais,Nucl. Phys. B (1998) 250, hep-th/9708004; Nucl. Phys. B (1998) 197, hep-th/9805163; E. J. Weinberg, Nucl. Phys. B (1980)500; Nucl. Phys. B (1982) 445.1913] K. Konishi and L. Spanu, Int. J. Mod. Phys. A18 (2003) 249, arXiv:hep-th/0106175.[14] A. Abouelsaood, Nucl. Phys. B , 309 (1983); P. Nelson, A.Manohar, Phys. Rev. Lett. , 943 (1983); A. Balachandran, G.Marmo, M. Mukunda, J. Nilsson, E. Sudarshan, F. Zaccaria, Phys.Rev. Lett. , 1553 (1983); P. Nelson, S. Coleman, Nucl. Phys. B ,1 (1984)[15] N. Dorey, C. Fraser, T.J. Hollowood, M.A.C. Kneipp, “NonAbelian du-ality in N=4 supersymmetric gauge theories,” [arXiv: hep-th/9512116];Phys.Lett. B , 422 (1996)[16] M. Eto, L. Ferretti, K. Konishi, G. Marmorini, M. Nitta, K. Ohashi,W. Vinci, N. Yokoi, Nucl.Phys. B780 , 161-187 (2007), [arXiv:hep-th/0611313].[17] M. Eto, T. Fujimori, S. B. Gudnason, K. Konishi, T. Nagashima, M.Nitta, K. Ohashi, W. Vinci, JHEP 0906:004 (2009), arXiv:0903.4471[hep-th]; M. Eto, T. Fujimori, S. B. Gudnason, K. Konishi, M. Nitta,K. Ohashi, W. Vinci, Phys. Lett. B669: 98-101 (2008), arXiv:0802.1020[hep-th].[18] M. Eto, T. Fujimori, S. B. Gudnason, K. Konishi, T. Nagashima,M. Nitta, K. Ohashi, W. Vinci, Phys. Rev. D80:045018,2009,arXiv:0905.3540 [hep-th].[19] B. Collie, D. Tong, JHEP 0908:006,2009, e-Print: arXiv:0905.2267[hep-th].[20] R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, Nucl. Phys. B686