JHEP01(2016)075 1 − a,b N CP Springer D , January 13, 2016 December 5, 2015 : November 26, 2015 : September 24, 2015 : : D Kenichi Konishi, 2 interacting quantum - Yang-Mills modes and c,b D , 10.1007/JHEP01(2016)075 D Revised D -2 Accepted Published Received 4 D doi: ) model with scalar fields in a N ( [email protected] r Jarah Evslin, , Published for SISSA by SU b,a × effects such as the nonAbelian Aharonov- ) ) factor gauge groups. D N N ( l vortex moduli modes are intimately correlated, SU D × a [email protected] , (1) 0 [email protected] , . Chandrasekhar Chatterjee, 3 and Luigi Seveso a,b 1509.04061 a,b,d The Authors. Spontaneous Symmetry Breaking, Sigma Models, Field Theories in Lower c

Geometric and dynamical aspects of a coupled 4 nonAbelian vortex zeromodes excite the massless 4 , [email protected] D Moreover, all sorts of global, topological 4 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585,E-mail: Japan [email protected] [email protected] INFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C,Institute 56127 of Pisa, Modern Italy Physics,NanChangLu Chinese 509, Academy of 730000 Sciences, Lanzhou, China Osaka City University Advanced Mathematical Institute, Department of Physics “E.Largo Fermi”, Pontecorvo, University 3, of Ed. Pisa, C, 56127 Pisa, Italy b c d a Open Access Article funded by SCOAP Keywords: Dimensions, Field Theories in Higher Dimensions ArXiv ePrint: Bohm effect come into play. Theseassociated topological with global features the and fluctuation theas of dynamical properties shown the 2 concretely herebifundamental representation in of a the two U SU( Abstract: field theory — theinternal gauged nonAbelian 2 vortex —in are investigated. general The give fluctuations risezeromodes of to associated the with divergent a energies. nonAbelian vortex This become means nonnormalizable. that the well-known 2 Stefano Bolognesi, Keisuke Ohashi Geometry and dynamics ofquantum a field coupled theory JHEP01(2016)075 14 34 31 16 = 0) r g 25 3 7 9 8 15 – 1 – 6 17 5 20 r ψ 11 η 12 and 3 and l 18 24 ρ ψ = 0) nonAbelian vortex r 1 ]. Typically they occur in a system in the color-flavor locked phase, i.e., g 5 28 — have been extensively investigated in the last decade, revealing many ]–[ } 1 i B { 5.3 Divergences of5.4 the energy Infrared cutoff 4.2 Global ( 5.1 Partial-waves and5.2 asymptote Exact solution 3.1 Topological obstructions3.2 in general Topological obstruction3.3 in our case? Geometric obstruction 4.1 Equations for 2.1 Vortex solutions 2.2 The Aharonov-Bohm2.3 effect Vortex zeromodes (degeneracy) romodes interesting features arisingsymmetries from [ the solitonsystems and in gauge which the dynamics, gaugeever, topology, leave symmetry a is and color-flavor broken diagonal global by symmetry a intact. set Color-flavor of locked scalar systems condensates appear that, to how- 1 Introduction NonAbelian vortices — vortex solutions carrying nonAbelian continuous orientational ze- A Excitation energy in the standardB nonAbelian vortex Gauge ( fixing for 6 Origin of the non-integer power7 divergences Discussion 5 Solution of Gauss’s equations and vortex excitation energy 4 Zeromode excitations 3 Topological and geometric obstructions Contents 1 Introduction 2 The model, vortex solutions and AB effect JHEP01(2016)075 ], ]. 8 15 , , and 1 14 (vortex ) gauge ) gauge ]. They 6 N 6= N D ( r ) orientational SU bulk, and their D G/H × ( D ) 1 π N ( l SU × , with H (1) 0 interactions are asymptotically → 1 − are expected to affect significantly G N } = 2 supersymmetric theories softly i B CP N { The 1 . – 2 – 1 − ), etc. ) gauge theory. Similar models, involving color-flavor locked N zeromodes N N D U( CP / SU( ) fluctuates strongly at large distances. As a result, all of × N } i (1) . These phenomena have been investigated in various gen- B 0 } { i ) gauge symmetry, yield sigma models with target Hermitian symmetric B N ), USp(2 { N ]. In particular they could shed light on the mysteries of nonAbelian U( ] with gauge symmetry breaking, / 7 ) 13 N massless gauge modes and their couplings to the massless 2 ) or USp(2 ]–[ ]. 9 N D gauge interactions on the gauged nonAbelian vortex collective dynamics in = 1, and may carry important hints about the mechanism responsible for 16 D N ]–[ vortex worldsheet. A nontrivial task is that of disentangling the effects of the 14 D It is the purpose of the present article to examine more thoroughly the effects of the Let us remind ourselves that a characteristic feature of a color-flavor locked vacuum is A sort of paradox or dilemma seems to arise, however. One of the most important, Other fascinating aspects associated with these vortex solutions appear when further This occurs in a model with U 1 effects of the 4 modes. These problems will be worked out systematically below. vacua with SO(2 spaces, such as SO(2 the dynamics of the latter. Some preliminary studies of these issues haveunbroken been 4 done [ the 2 extra gauge fields on the static vortex configuration itself from those of residual dynamical scalars or gauge bosons surviveare in those the of bulk. the vortex Incitation, orientational confined the zeromodes, within vortex which the sector, are vortex theinstead, a worldsheet. only some kind In massless combinations of the of modes Nambu-Goldstone case the ex- ofcoupling gauge gauged to bosons nonAbelian the remain vortices, fluctuations massless of in the the 4 2 free: the vortex orientation the global topological effects mentioned above would be washedthe away. fact that all masslessall Nambu-Goldstone of particles are which eaten become by the massive, broken gauge maintaining fields, mass degeneracy among them. No massless groups [ characteristic features of nonAbeliantuations vortices is of their the collective vortexworldsheet) dynamics. internal sigma Quantum models, orientational fluc- such modes as are described by various 2 exchange of parallel vortices, Cheshirethe charges, vortex etc. orientations, make theireral appearance, contexts depending [ on more recently, in the contexttheory, of concrete with model, scalar e.g., in fields a in U the bifundamental representation of the two SU( which may well be realized in the interiors ofgauge neutron stars. fields are introduced,metry. coupled to We part or shallAll all refer sorts of to the of these color-flavor global diagonalobstruction effects, systems global to such as sym- as part “gauged nonAbelian of nonAbelian Aharonov-Bohm the phases vortices” “unbroken” and in gauge scattering, this symmetry, an nonAbelian paper. statistics under the teristic features of thisoccur sort, in as the canbroken be infrared to seen effective in theories aquark of hidden-symmetry confinement many [ perspective [ monopoles. They are realized in high-density QCD in the color superconductor phase [ be quite ubiquitous in Nature. Standard QCD at zero temperature exhibits some charac- JHEP01(2016)075 ) 6 R N (2.1) (2.2) (2.3) (2.4) ) used 4.4 , where ) . Section Q 5 µ → ∞ D † R Q µ D , ). The matter sector 2 . ) N ) and section 0 and r Q ( µ 4 † + Tr( → Q a , QA r µν t r ) = SU( g f ) r ig ( µ R µν (Tr , f + A G 2 r l 2 g R 1 4 g Q the model and the main properties = . G µ − + − , a 2 L . Vortex fluctuations and the induced 5 )), is decoupled. It is argued that it is ) 2 ) 0 × l N ) ( µ G 3 † N L ig 1 µν A ( ) l ξ r G r − ( g QQ ( p × F (1). a Q 2 ) t 2 0 ) l – 3 – r l = g ( ( µ µν (1) i A F 0 of the zeromode excitations of the vortex. Finally, + (Tr l = 2 supersymmetric theory. Q 1 and section ) (or SU 2 l 2 r h 2 ig g g Tr( 4 N N in the bifundamental representation of the two SU( = U ( − 1 2 q − ) gauge group unbroken. The fields r G Q 2 Q − N ) = µ 2 ) 0 ∂ µ v dynamics µν A ) = l − ( Q Q F µ † ) l ( µν Q D Yang-Mills modes are worked out in section F (Tr D Tr( 2 0 = 0 case. Appendix B proves the uniqueness of the Ansatz eq. ( 2 g 1 2 and the covariant derivative is r g − − Nξ = we examine the apparent discontinuity in physics in the limit in which one of 7 L = 2 0 v This paper is organized as follows. In section This is, after adding the appropriate adjoint scalar fields (not relevant for the vortex solution hence set 2 to zero), the truncated bosonic sector of a The scalar-field condensate in the vacuum takes the form leaving a left-right diagonal SU( where factors, with unit charge with respect to2.1 U Vortex solutions We shall work with a BPS-saturated action Even though ourthe study type can be generalized to the casewe of shall an choose, arbitrary forconsists gauge concreteness, of a to group complex work of scalar with field the particular to solve Gauss’s equations in section 2 The model, vortex solutions and AB effect gauge-vortex system and the in section the nonAbelian gauge factors,essentially SU due to theis noncommutativity an of infrared the cutoff twofluctuations. introduced limits, to Appendix regularize A deals the with energy the divergences peculiarity caused of by the the solution of vortex Gauss’s equations in of the vortex solutioneffects. are The presented closely together related withgenerally question a of and brief topological in and review our geometricexcitations of of obstructions concrete the the is model, 4 associated reviewed, global inis section dedicated to clarifying the connection between the global, topological aspects of the JHEP01(2016)075 (2.8) (2.9) (2.6) (2.7) (2.13) (2.14) (2.15) (2.10) ), one 1 − N . 1 , . ,  2 12
) , a ) = 0 ) = 0 1) N ξ f Q Qt 0 † a Nξ − t ): g Q y † − + N 1 Q ( 2 Tr( Q | , † N a and Q Tr( t 2 Z Q 2 r x a ; (2.12) ) (2.5) t , g p = ) 1 l , iD r (Tr − g ( µ ! − 1  0 ≡ 2 + 1 ) ) − A g + N r 2 r − N ( 12 ) T N Q g + l N N ( 1 ( F 12 T 1 . R − ; (2.11) 12 D NA N  ) | 2 r ) f l r f g NA ( µ N SU ( f ) 2 − ,C A NC − + N l × 1 Q − + Tr g N 2 + Tr l 1 NC ! ) ( ( j g 2 1 ) 0  j 1 R
x 2 2 l r 2 N j x 2 − – 4 – ) ( r q ( g
+ ij r x 2 1 ij ) N  L L r  0 ij + ≡ 1 Q 2 Q Nξ l  1 0 0 0 a 2 r 1 ,,F f g − SU SU g g 2 iθ t 3 r g g 0 − e g † g 1 0 − − × q Q Q = 0

† − 0 = = = ) = 0 = (1) Q i ) ) Q = a l Tr( 0 N r 2 a ( i µ ( i a U Q Qt t A B

A (Tr † iD l  0 g 1 N Q g + π C + + Q ) Tr( l 1 ≡ ( a 12 12 , become massive. t 1 D f F µ r − a 2 g 2 1 + N −  T ) x r 2 ( 12 d F Z = T The nontrivial first homotopy group We consider the minimal winding vortex below; the generalization of the formulas below to higher- 3 and winding solutions is straightforward. where whereas the nonAbelian and Abelian gauge fields can be written in the diagonal form For a minimal vortexcan with take a the fixed scalar orientation field Ansatz in to color-flavor, be for example (1 The BPS equations are accordingly: configurations depending only on the transverse coordinates means that the systemvortex solutions admits can stable be found vortices, by which the are BPS completion our of main the expression interest for below. the tension (for The and the U(1) field remain massless in the bulk, whereas the orthogonal combination JHEP01(2016)075 ) r Q ( A )) (2.20) (2.23) (2.16) (2.17) (2.19) (2.18) 2.7 and ) l ( , , , A . field condenses ; (2.22) = 0 = 0 = 0 Q  !! 2 2  , transforms in the 1 !! Q 0 f − , Nξ ξ . 1 2 − N + ) − − p 0 2 1 1 N are now shared between N 2 2 1 0 Q 1 N NA ) = 0 2.13 Q 2 f 1 0 N ) = − g = 0, except for the fact that ∞ − 1) − ( )–( ), which is the identity, so ∞ l 1 N . The corresponding AB phase  ( − g − 2 2 1 R ∂Q/∂θ 0 N NA ) − Q f Q N 2.11 2.23 0 r 0 NA 1 is now quite straightforward. For − N f − 1 ; (2.21)

+ ( 0 2 ) = ) = G = 0 or 2 1 2

2  r 0 ∞ ∞ r g g Q 2 N 2 l ( ( 0 g  1 g 1 g πi N Q 2 πξ . 2 πi 0 πi N 2 g 2 −  = 2 – 5 – − ) divided by ( 0 . The vortex tension is given by (see eq. ( T r f , rQ ξ basis. Of course the two answers must agree. To use 1 √ 2.22 , f B = exp = exp = exp , which is of charge 1 under U(1) = 0 ∼ Q     ) ) a l and r ( ( f ) and ( A (0) = 1 I A A ) then show that the profile functions satisfy + 0 I NA I ig 2.9 l 2.21 f NA r  f ig ig N )–( ) fields. In other words, the static vortex profile remains basically and the antifundamental of SU(  1)  l N ) exp 2.8 − (0) = N exp f exp N →∞ ( lim r  →∞ lim →∞ lim r 1 r Q − 0 1 rQ The AB phase can be calculated using the gauge fields in the ultraviolet These Wilson loops are easily calculated from eqs. ( is simply the productdoes of not transform. ( Thisand is any an condensate important must consistency be check single because valued. the basis or in the mass eigenstate To obtain the ABexample, phase consider given the scalar afundamental field representation of SU( of Whenever the (untraced) Wilsonvortex loop is in some not representation equalwhen of to encircling a the unity, gauge particles vortex. fieldThis around belonging is The the called to transformation the that is Aharonov-Bohm representation (AB) given effect. by are the transformed (untraced) Wilson loop. vortex), with a well-defined width, 2.2 The Aharonov-Bohm effect which can be solvedearlier by for numerical the methods. global Thesethe nonAbelian equations vortex, gauge are i.e., fields identical for compensating tothe the left those scalar and found winding right SU( energy unmodified as compared to the standard nonAbelian vortex (and for that matter, the ANO The BPS equations ( The boundary conditions are JHEP01(2016)075 ) N ( r (2.33) (2.25) (2.28) (2.29) (2.30) (2.31) (2.32) (2.24) (2.26) (2.27) + l charge, it . Therefore A . = SU B , † ! 1 H , − ) BB . N B 0 0 1 + ( is invariant. The in- † 1 1 N U − Q − ) N . r ( i 1 1 , , N . ! A † ) 1 ) transformations, 1 ≡ 0 and also under ) and U(1) − , − U B , − N 0 ( B N ( } 1 Q, B N N † ( ) r 1) µ r 0 0

+ U ( i U a l − CP 0 1 0 = ! 0 = πi ,A ig −
) 1 ∼ N ) l ) ( i − r is rotated by the exponential of −

B,Y ( = 2 1) † N r, B A ( i
Q Q 1 πi l B ) − = U( µ ) g Q, A r ( r B { e 0 ( N + H A 2 1 + U r ) ig ) = 2 l U( g Q – 6 – ( → ≡ B ,A / ) by global SU 0 − ) 0 A ) ) ) − g r r } → ) Q ( g N B l N ) 1 ( + µ ( r 2.10 0 ( i † ∂ A a Q l 0 I U has charge 1 under U(1) g = iθ ,A g ) ,X r ) l ( e 2 l ( i i = SU( g 0 Q = SU( ( i Q l ! g A I g µ e 1 2 I H ) ) i i H D − B B 1 2 Q, A ( ( H/ Y { − →∞ lim † →∞ →∞ U U r r r B = = = lim = lim − ) ) B B A ) further breaks the unbroken color-flavor diagonal BY vanishes implies that the AB phase is independent of the ( 1 2 ) at spatial infinity as l, B ) is integral. This in fact is the quantization condition on the nonAbelian 1 2 ( i − Q I I A − 0 A 2.2 X r 2.10 0 H X 2.27 ig g , which is necessary for the condensate to be well-defined, imposes that the l g g

Q i →∞ = lim r U →∞ lim r Rewriting ( where the rotation matrix i.e., implying the existence of degenerate solutions, corresponding to the coset, The general solutions are related to ( 2.3 Vortex zeromodesThe (degeneracy) solution ( group as This exponential is unityvariance and of so againmatrix we in eq. recover ( the factvortex charge. that In theof low energy a gauge Higgsed theory itdeformation gauge is of field the the is topological vortex. condition quantized. that the In flux particular, it is preserved by any continuous one sees that inupon the circumnavigating mass a basis vortex at large radius, The fact that only depends on the charge under the massive gauge field the second basis, one needs the corresponding Wilson loops, which are the exponentials of JHEP01(2016)075 ]. , the 17 , (2.34) B ]. 16 , 21 are related , } ]–[ 13 i , ψ 19 B . Generalizing Γ 12 , { } † i ) → . down to a smaller B B )-independent parts { ψ (    G orientations of multi- † U ) in our model has been z, t B 1    ( , various closed paths en- − 1 } U − N . The orthogonal parts of the N    t relative ,... 1 1 2 CP ) obviously does not depend on 1 N − 2 N 2 and l ,B 0 g 1 g 1 2.20 z , the 1)-component complex vector 1 B N πi } i 2 { 2 − 2 r 0 − g B g e { N πi 1 , it will experience an AB effect similar to 2 − e N is restored. There are natural gauges, such G N 1 2 2 l − 0 H N – 7 – g g N 2 πi 2 r 0 2 g g . The tension ( e 1 πi − 2    N ) − ), will experience various nonAbelian Aharonov-Bohm e B ( N    CP ( ) R ) gauge transformations. The moduli can be promoted to ,U . B } πi N i ( N ] by studying these vortex solutions in the large-winding limit, SU 2 B U e { × 17    , ) 16 N ( ) = L B ), particles carrying (for instance) fundamental charges with respect to ]. . In the presence of a vortex the symmetry is broken yet further. However, SU Γ( 17 , H 2.23 = 0), here the gauged vortex solutions with distinct moduli × l g 16 )–( , (1) 0 13 2.21 , If there are multiple vortices with orientations Even in the case of vortices with constant In contrast with the standard nonAbelian vortices (which appear in theories with That the vortex (internal) moduli space is exactly a 12 = U = 0 or r 3 Topological and geometric obstructions In the vacuum ourgauge group scalar condensate breaksfar the from gauge the vortex, symmetry as the the symmetry regular gauge group used above, in which the scalar condensate is position-dependent and rienced by a particleThese depend and on the many orderin other in [ beautiful which various features vortices related are circumnavigated. to such systems have been discussed If a particle isthe in above a but generic with representation appropriate of charge and generators. circling these vortices give rise to nonAbelian AB effects: the gauge transformations expe- Consider two oreqs. more ( parallel vorticesG with different orientations, (AB) effects (“gauge transformations”) when encircling one of them, of the fields, indo gauge-invariant correspond combinations. to Asmechanism. physical a degrees result of the freedom, corresponding oscillations which are eatenple in vortex a systems kind do of have mini-Higgs observable effects and hence are physical [ can be determined by following the analysis `ala Nielsen-Olesen-Ambjorng [ by global partcollective of coordinates the if they SU( arecollective allowed to coordinates depend on can combine with each other, or with the ( inhomogeneous coordinates of the internal orientation verified recently in [ where vortex configurations can be analytically determined, and accordingly all zero modes (known as the reducing matrix) depends on the ( JHEP01(2016)075 ] is 25 , G 24 can be bundle ] that it H H 23 , . Again the has the very 1 . In this case . This means 22 2 S , as the gauge H H 2 S S may not exist for ). However when . The obstruction 2 H S H ( 1 π bundle on is broken to H on this G H bundle. However the G and these hemispheres are related gauge bundle on 2 bundle that gives ’t Hooft-Polyakov S is, in many gauges, position dependent G which is a subgroup of the ultraviolet H G can be extended over the H in and so the high energy theory is described H 1 H do not generally commute with the AB phase. S – 8 – H . The AB effect potentially makes the embedding bundle is not principle. Nonprinciple U(1) bundles subbundle of the G which links a monopole. It is known [ H 2 H in S . θ H and the low energy theory by an 1 S ) and so it is entirely characterized by the transition function when π 2 , symmetry exists. Colored dyons would be charged under such a global (0 ∈ H . Consider a sphere . bundle on θ π G . The fact that the embedding of , and so one obtains an G H H = 2 θ yet more complicated, as elements of under which they transform, the The Alice string is particularly exotic because a particle encircling the Alice string A similar obstruction can occur in the case of vortices, as was discovered in ref. [ This nontrivial bundle poses no problem for the existence of the monopole. Indeed there are many well-known cases where such an obstruction is known to exist. is nonAbelian, this gauge transformation acts nontrivially on any set of generators of symmetry, and so the topological obstruction to a global definition of bundle is nontrivial. As the base space of the bundle is just a circle, the bundle can be and so implies that no set of generators of = U(1). The vortex is linked by a circle H H consists of multiplication byGroup elements multiplication by in a U(1)the fixed and U(1) element the of bundle inversion U(1) isthe of yields a modulus the an of torus AB this element and torus effect.other the in is hand, not The multiplication U(1). the purely total by Alice imaginary, space nonetheless a string the of corresponds fixed bundle to element is the trivial. simply bundle means On in the that which the U(1) fiber is inverted, passing negates its electric charge. Whenever particlesof encircling a vortex change the representation have transition functions in Iso(U(1)), thecorresponds group of to isometries an of U(1). element of So each Iso(U(1)). U(1) bundle This is essentially an infinite dihedral group, it which introduced the Alice string.H Again a symmetry group by a trivial H trivialized on H that no global H physical consequence that no colored dyons exist. monopoles their topological charge. defined on northern and southernby hemispheres a of the gauge transformationH which corresponds to the generator of is not possible to continuouslyarises define as a follows. set offield generators The is of connection continuous defines andbroken a globally to trivial defined insideis of nontrivial. the Indeed, sphere. it is The the gauge nontriviality symmetry of the means that such aall global, values continuous of definition the of azimuthal angle the generators of For example, consider nonAbelianwhich preserve monopoles. a nonAbeliangauge These symmetry group are group ’t Hooft-Polyakov monopoles of 3.1 Topological obstructionsIn in general various contexts itsymmetry is useful to define a global symmetry corresponding to the gauge therefore so is the embedding of JHEP01(2016)075 r ). ), × ) are . N H and and N r N ( 1 l H 1 up to 1 Z ) ? The S π / S SU( r and also N H H ) , in other r × . However ) l N r r N ) ) SU( N Z N N ∈ × . Therefore our ), the choice of . Therefore one SU( l N H H SU( /N × Z ij/N ) th roots of unity. If 0 × bundle over e bundles over k . l − N ) H j are classified by H ( H i N 2 e , transforms in the funda- with S 0 SU( by Q . All three gauge fields leave × makes the story slightly more Q Q . In particular, if the AB phase 0 H . This means that SU( except for the r in the same connected component of ) bundles over Q , paying particular attention to torsion N H H H diagonal symmetry of are invariant under U(1) N l bundle although the converse is not necessarily Z and ) ’s. If one acts on – 9 – N H N G . Including U(1) and the gluons carrying the left and right SU( Z is charge one under U(1) cannot be globally defined, since the generator of k 0 . Similarly the right gauge fields are invariant under Q l H = ) ) then there is no topological obstruction to a global j N H ( 0 π SU( ⊂ obstruction we only are interested in an element of generally acts freely on l N 0 Z . changes sign when circumnavigating the circle. r N Z H and the antifundamental of SU( ), the set of connected components of . then the total effect will be to multiply except for the central l are in one to one correspondence with elements of × ) H H topological r l ( 1 ) Q ) 0 N S π N N . However the scalar field l N Z SU( SU( / l invariant. The left gauge fields SU( × ∈ ) ) which appear as monodromies are multiplication by elements of 0 0 If these were the only fields, the total gauge symmetry would be SU( Similarly in the case of monopoles, nontrivial Now for a How can we classify such topological obstructions to the global definition of N H are related by a continuous deformation and so topologically ik/N acts freely on e must quotient the gaugewords symmetry by by the the elementscomplicated. for Again which U(1) U(1) under their own center U(1) SU( mental of SU( gauge fields, the photon carryingThus the U(1) gauge group couldthe in gauge principle be symmetries as whichso, large leave to as U(1) all avoid confusion, of shouldkinds the be of fields fields. quotiented invariant out The have of three no the gauge total physical fields gauge effect and group. and the There scalar are field four 3.2 Topological obstructionTo in determine our whether case? orneed not a there more is carefulelements, a global which topological are definition often obstruction of responsible in for our topological obstructions. case, There we are will three kinds first of construction of although, as the originalis ’t not Hooft-Polyakov sufficient monopole for case athat illustrates, topological the obstruction a transition to nontrivial functions the bundle act existence nontrivially of on a the dyon, it generators is of also the necessary Lie algebra of a continuous deformation.H Any two elements of classified by represents the trivial class in indeed an obstruction implies a nontrivial true. These bundlesmonodromy are when classified winding around geometricallyno the by vortex. particles an In change element theIso( representations of case when of Iso( gauged encircling nonAbelianbundles the vortices, over vortex so the only elements of total space of thebundle bundle is is the not Klein trivial.the bottle, Lie it In algebra is particular, of not homeomorphic to the torusobstruction and the was a consequence of the nontrivial topology of the corresponding to the negation of electric charge, when circumnavigating the vortex. The JHEP01(2016)075 . r k Q ) as H Q = r N (3.4) (3.5) (3.6) (3.1) (3.2) (3.3) ) j is the N needs to H . as follows. or SU( G SU( and let the θ l H ) × must preserve M l N ) H N is included then this r ) = 0. Therefore no ), such that the corre- ) at arbitrary . H N (  3.2 H 0 1 π − N 2 . . In either case, so long as all N r r ) + NC transformation be l elements must be equal, leaving a N is eliminated. Furthermore, as ) iθT , l and so need to be quotiented out, ) r 0  N ) ! and/or SU( Q N . , given by ( SU( 1 G l N vanishes. Elements of N. k ) r /N − exp ) × 0 + N k B l N † l ) ) 1 − = SU( N j consists of transformations such that 2 N mod M N N + iθ Z times the first factor in the right hand side Z 0 r H  k e ), one can construct an element of m + 1 ) reduces to ( and SU( l i Q )

SU( − = SU( N – 10 – l e N 2 ) ξ j N × N = 2.10 N p 0 + NC and so −→ H there exists a , which leaves all of the fields in the theory invariant. is path connected and so iθT SU( = of SU( denominator in m 2 N Q G 2 N as well as the central elements of SU( because ⊂ − H U(1) Q Z 2 N M Z 0  invariant. In conclusion, the total gauge group N Z Q = ∈ Z Q G ) U(1) , where eq. ( charge, = exp r ∈ j, m r 0 u should be quotiented, while with the inclusion of both there would N im/N ? This is the symmetry group left invariant by a vacuum value of Z e H . The central by r transformation be trivial. Let the SU( + . Given an element l 0 Q ) Q . N is only invariant if H transformation be = 0 so they lie in the Q r ) is proportional to the identity. ) m In fact, it is not difficult to explicitly construct elements of What about is no longer quotiented out of N 3.6 N This transformation preserves of ( corresponding to the gauge in whichthis the form of modulus Let the U(1) SU( defining symmetry group at large Note that if matter transformingZ under only SU( fields have integral U(1) bundle over the circle can be nontrivial, so there is no topological obstruction to globally is proportional to the identity,single the SU( SU( and leaving the projective unitary group If in addition onethen includes only matter a transforming single inbe the no fundamental quotient of at SU( all. This is not invariant under any rotation, so the U(1) sponding gauge action leaves be quotiented by this unphysical We then conclude above, then and so Therefore for any pair ( one acts on JHEP01(2016)075 ) H ( (3.7) (3.8) (3.9) 0 at all (3.10) π charges H 0 . , — the little 1 1 − − ) ) H 2 2 U U ( ( N N : T T gauge bundles over G N N but its nonAbelian dependent, and as a θ θ 4 . H NC NC θ † H r r U iζ iζ e 1 − · e − 2 . 1 N 1 · ) is determined by integrating ) = 1 θ becomes θ ( − ) ( UT 2 u U r to be finite, the gauge field must G ( N u , . = (0) = is abelian we do not expect this to T 2 | u , 1 H N 0 , θ − (0) dl Q ) · i 2 i U G NC A ( , l N D Q θ | 0 1 h ∈ iζ T R ) − e ) = 0 implies that all ) i ) R 2 θ U e θ ( θ ( N N H ( u i ( T P u 0 N – 11 – π ξ e θ ) = NC . ) = p θ l 2 r θ ( ) and traced. g ( iζ i 2 r = , with ) for the various simple factors in + e g u 2 l Q 1 3.6 g H h − 3.7 ≡ ) = U → asymptotically. That is, θ r . However if we allow matter with nonintegral U(1) ( ζ 1 ! l ), is more adequate for a discussion valid in general gauge G , H u − 2 r 1 ) g 3.8 2 l θ + g iθ ( 2 l ) generator is given by e g u ) , ) as N θ

≡ θ ( N l i u ζ 3.7 i e ξ U . However the nonAbelian Aharonov-Bohm effect yields a closely related i∂ θ p − ) = will change, becoming covers of the definitions above. In such a case ) — inside the original symmetry group θ θ ( = ( 1 ) = H i i u θ ( Q generators described in ( A ) can be read off from ( i h θ ( and Q H u h means that the color electric charge of a perturbed vortex solution is well defined. which are covariantly constant with respect to the gauge field. G In such systems, in order for the energy Let us rewrite ( A covariantly constant embedding of the unbroken symmetry group Recall that the scalar fields in the regular gauge have a nontrivial winding at spatial What does this all mean physically? In the monopole case, the fact that the ’t Hooft- Summarizing, we have observed that We also recall that H 4 H approach the gauge field where the integral is computed along a circle at radial infinity. This rewriting, equation ( symmetry breaking systems, result some of the generators are not globally defined. where where the rotated SU( group of phenomenon: a geometric obstruction.of It is not possible to globally constructinfinity generators by the 3.3 Geometric obstruction We have seen that thereazimuthal angles is no topological obstruction to the global definition of Polyakov monopole hasgeneralization no does obstruction implies that to thethe former a latter can cannot. global be modified In definition to theof present carry of case, the the electric lack charge of but This a topological color obstruction electric to charge the definition can be calculated by simply integrating the current multiplied a circle are trivial.exist to Therefore a global ours definition must of then be trivial and sois no no topological longer necessarily obstructionlead trivial. can to a Nonetheless topological as obstruction U(1) in the definition of JHEP01(2016)075 (4.3) (4.4) (4.1) (4.2) varies (3.11) (3.12) (3.13) (3.14) G derivatives are allowed y } ). By a basis i b inside B T { ). In such a basis a and † H T v x , − 6= 0 are not. α  v a V a (  r ξ i modulation of the vortex . η Q a } QT ) := Tr( + , b † T b ) and † α z, t ) T † ,T Q { . W v a Q α ab 1 r (0) = 0 to fluctuate. If α T i D − ρ + O ( ) } D v Q i θ ( = ) containing the h . = ( − B a ) a u 1 − { r a T T 2.9 ( α Q − a ) Q α T , i.e., ) = 0 are globally well-defined and generate π α † )–( D θ πiξ v † a D ( 2 (2 ξ a e u ). Define the Lie algebra automorphism u Q 2.8 is real its eigenvalues come in conjugate pairs a a T G † and T T O – 12 – ) ) =  ) = Q v θ  π π ( ,A Tr a . The generators for which (2 (2 Tr α a a u T H a V T ) = 0 if l T r T θ η l → ⊂ ( ) = ig i a + ig π e H − Q T α h (2 = ) = a W ) θ , l l ) ( T ( iα r ρ a ( iα F the field equations ( i T F = 0 = i } D ) α l D ( α a can always be diagonalized (every orthogonal matrix is unitarily diag- z, t A { O ). ). The diagonal basis will involve in general complex linear combinations ) C H x, y be a basis of generators for Lie( = a i and corresponding eigenvectors must be a real orthogonal matrix to preserve Hermiticity and normalization of the ; T ∗ O : λ z, t θ It turns out that the form of the new gauge field components consistent with these Let = and α equations is given by the following Ansatz and ( however are modified, andinto in account the order correct to response describeorientation. of the The the new gauge zero fields gauge modeSavart to components, effects, excitations the induced satisfy one by the must the following the take equations nonAbelian of Gauss motion: and Biot- 4 Zeromode excitations Let us now allow theto depend vortex orientation only zeromodes on and the corresponding gauge fields components are unmodified. The other field equations The covariantly constant generators for which the unbroken symmetry group generators with respect tochange the in invariant Lie( inner productonalizable ( over of the original generators.λ However, since In general, after a full circuit where A covariantly constant embeddingwith of the unbroken symmetry group JHEP01(2016)075 ]. 5 [ † (4.9) (4.6) (4.7) (4.8) (4.5) UQU = 0), it was . l ) below). The g  ). 2 | 4.18 . Q 4.2 α ! D 1 | = 0 or − , N , r † ) and ( ) } ). 1 g U U − ( 4.1 ⊥ ) + Tr 4.2 ) T 1 UT , α α 2 ] yields α ) U i ) ∂ r ), ( ∂ α ) ( † 26 ∂ U ) contains an additional term that ≡ = F † ( 4.1 ) r T α U ]. For the gauge nonAbelian vortex ( 4.4 TU iα ( α ) is also richer than in the case of the 1 F ∂ − 4.4 i U U Tr( α which would generate higher derivative terms ∂ = 2 ]. Below we shall work in the regular gauge, † )2 ) + Tr( ) U α – 13 – 16 U dz dt l/r ( i { ) ( l αβ 1 2 ( T ,V Z ) F ) ), as we are going to see next, satisfy the equations F U U ) I ≡ ( ( l ( iα r, θ T T ⊥ = ( ] on the Nambu-Goldstone direction in the tangent space. ) F ) α η ,T U U ( † eff 18 i ∂ α Tr( S T ∂  α = † x α ) and U i ∂ UTU 3 ( W ≡ = r, θ dtd ( ) α ρ U Z ( W ). We shall also see that result obtained for the ungauged vortex in the T = eff 2.13 ) means that the massless excitations correspond to the Nambu-Goldstone S 4.6 )–eq. ( 2.10 To lowest order the excitations above the static, constant tension vortex are described The price that we pay for using the regular gauge is that the equations of motion take a For the standard, ungauged, nonAbelian vortices (with We neglect here the terms coming from in the effective action; we shallBy come back inserting later the to discuss Ansatz, the a validity of straightforward this calculation approximation. [ by the effective action The more general formarises of from the the gaugestrongest gauge field justification transformation Ansatz for the to ( Ansatz,the the however, color comes regular from structure gauge the of fact the (see it equations eq. allows in one ( a to closed resolve form ( is the Delduc-Valent projection [ The form ( modes, whose direction must be kept orthogonal to the “rotating background”, angle. The colorstandard structure nonAbelian of vortices, the where gauge the fields only color ( structure (in the singular gauge) was where gauge develops Dirac sheeteq. singularities ( [ singular gauge is correctly reproduced. more complicated form as the background and quantum fields depend upon the azimuthal that follow from the insertion of the Ansatz intoconvenient equations to ( develop the effectivethe action one using the in vortexinstead which solution the the in use the scalar of singular fields gauge, the do singular not gauge wind is somewhat [ subtle, as the gauge fields in such a The profile functions where JHEP01(2016)075 ) = r NA ) . , ρ l and f α θ ρ , θ ρ ∂ (4.13) (4.12) (4.10) (4.11) W 0 r = 0 = 0 ∂ f l η l r r r η r , ρ l η η σ g 2 2 , g r r 2 θ 1 2 g g  θ θ Q − − sin l r NA l  g . f η 2 η sin r cos 2 2 θ l l θ ) Q 2 ∂ g − 2 g r ∂ 1 l r  η = 0 Q 1 Q σ σ r 2 Q 1 1 ( ( B ) α g  r α Q 2 V Q   + ρ ∂ + α  r l 1 + l − 2 l g η 0 NA NA − η W g r 2 2 l 2 2 g f f r r ρ − g Y Q r † − −  l Tr g ρ 1 1 2 B l + + ( θ − g α 2 2 1 2 2 the other functions ∂ ) )   1  r Q r cos + ( 2 2 2 l η l r − ρ 0 0 ) θ g 2 g g r 2 l g g − B, NA ) ∂ g η l α Q f ( r   θ r η given 1 ∂ η 2 + ∂ 2 1 1 , r − − r ( Q l l  g − ) ) + 2 1 ρ η dz dt X 1 l 2 2 l r r + Y + − g  η η † , NA l 2 l 2 2 l r 4 Z f ) + 0 ρ  l r η B g g r and l ) g 2 2 g 2 α η − r g – 14 – ) l 2 ∂ r l (1 + ( Q σ = 8 ρ 1 1   + 4 + 4 l ∂ η − , r 2 η 2 − θ θ + 2 2 l r  l r ) l ∂ 2 . Alternatively they can be derived directly from the ) σ σ g 2 1 ) η η + ( ( ( r  θ U I , Q 2 2 2 cos cos ( ρ − 2 l ∂ r + ( ) 0 NX ). r   2 2 l g T ρ r g 2 g − σ α ρ ) Q Q NA l r appear in various contributions, the total action is simply 4.2 θ  1 1 ∂ NA NA θ = 4 2 f ρ η ∂ f f ) r r r η θ r Q Q ( 2 − α U g − ∂ 2 , due to the following identities: 2 − − 4 4 1 + 2 Tr( l ( cos ( 1 ( V 2 r 1 1 ρ 1 θ ) 2 2 T − + 2 θ 1 ≡ ) 2 r α   0 ∂ 2 2 2 2 and ) and ( + Q 2 sigma model action. The equations for the profile functions U r 2 2 ∂ g 1 ( sin 2 Q Q σ +   1 dz dt ρ = Tr ) T Q 4.1 NA l 2 , 2 2 r = − 2 + 0 0 2 l l α 2 2 g + + f r α ) ρ l Z Q g g l N g ρ r g 4 ∂ α r 2 1 2 1 l ρ 1 2 2 ρ − W − ∂ V with respect to g r Q Q Q ∆ 2 2   1 ∂ CP 2 I l Tr ( 2 1 Q + + + ( + + + 0 g 2  g and + 1 + 2 ) 2 1 − U ≡ l ( Q l η T dθ dr r σ − ) ∆ U 2 l ( Z 1 Note that, in spite of the fact that terms with two different color structures g T = α can be determined by minimizing                            I fixed, one finds the following four coupled equations 4.1 Equations for By minimizing equations of motion ( i∂ proportional to Tr( and is the standard η where ( JHEP01(2016)075 , . . , = 0, r (4.20) (4.15) (4.14) (4.16) (4.17) (4.18) (4.19) = 0 = 0 = 0 = 0 ρ l l l r ) is just g ρ η η η σ ρ l l 2 2 = θ 2 θ g g g g r  θ θ 2 2 4.17 2 2 η cos sin , Q Q r ,( NA 2 g sin α 2 cos f 2 2 r + 2 + r r . V Q 2 Q 2 g g 1 2 1 2 1 ) ) − 1 iθ/N ) Q Q U e Q Q s Q 1 1 Q 1 2 ( ( 2 = 0 and 4 − Q  T Q − − r 2 − θ σ g g + r  + + 2 2 η θ 2 l r 0 ], it is necessary to gauge Q 2 2 N ρ g g l 5 (sin 1 Q g sin + g  1 θ 2 ]–[ 2 2 2 1 2 2 + 1 . − Q − Q N 2 1 α + 2 1 cos 2 V Q r  α , − N Q 2 g † ρ W − Q NA 2 U . = + 1 r and ) f (1) transformation .  1) N r η η 1 ρ ! Q 0 α 2 2 α N − − ) 1 − 2   NA ≡ U + W − 1 ) W /NC ( f r l N 0 s 1 r ) N ( NC  ρ T s NA − NA − ) ρ ( 1 2 4 f f 0 U 1 2 2 r r r ρ ( N θ σ g g iθ Q  0 − − 2 and e = – 15 – iθT 1 1 2 + ) − , e η −

s (cos   2 1 ( α 2 r 2 2 T g = ≡  U 1 A η Q 2 −  l θ u + 2 r η = ρ ∂ 0 − NA = g θ g , f N r u ∂ θ 1 g NA 1  − − 2 f 1 2 ≡ NA r − cos 2 l f − uu  r r 2 − N g α ρ 1 g − 2 1 Q ∂ θ 2 1 2 N 1 0 i ∂ g 2 g 2 Q − r = . Apart from an irrelevant U 2 − g ) NA ) nonAbelian vortex − η − s 4 1 2 1 f θ ( − r η N r − − ∂ ρ 2 2 2 − = 0 u gρ ∆ ) N ∆ 1 Q r s NA NC ( α T 2 2 g r 2 ) reduce to ( f 1 0 r + 2 g g 1 + 2 − 2 1 uA 1 4.13 + Q ≡ = r ) − s η α ( + 2 A ∆ σ ρ 2 r is given by 1 ∆ It can be verified that the same equations follow directly from the Gauss equations, g ) s                  ( α                            which winds the scalar fields once. A useful relation is where a gauge transformation The gauge transformation from the singular to regular gauge is achieved by That is: In order to comparetransform them to form the the equations singularA studied to earlier the [ regular gauge. In the singular gauge the gauge field 4.2 Global ( A nontrivial check ofungauged the vortex case. equations After found setting theequations above right ( gauge is coupling to provided zero by the consideration of the after factoring out the common color structures JHEP01(2016)075 in ψ (5.1) found (4.26) (4.27) (4.25) (4.28) (4.21) (4.23) (4.24) ) s ) already ( gρ . ρ, η . = 0 ψ = 0 ) 2 2  = 1 + 2 . The result for 2 . Q ) . ) , Q s s ) ( ; (4.22) NA 1 ) ( + r l f σ ρ 1 2 Q ρ gρ gρ l r , Q g g − = ( ) 1) and 2 ) s 2 g ( s . ( f − ψ ) ) , ψ ) − s = 1 + 2 = 1 + 2 2 2 s 2 ( ( = 1 + 2 = 1 + 2 ) iθ Q Q . s l e , r ) ( θ σ , 2 θ σ s σ 2 σ + 1 2 ( 1 ( ( Q ) and to obtain the associated excitation 2 1 . ψ Q Q Q 1 1 2 Q iθ (cos (sin Q e =  Q Q g g 2 4.14 1 1 2 ) 2 2 g = s g iθ ( e + = = σ ψ − – 16 – = 0 theory) known from the earlier studies. We ) ρ η = ψ as a single complex equation: s = r 2 ( → ) and ( ) g ψ  ) → ψ ψ → s , s ( 2 , ( ) r l s ψ  4.13 η NA ( ψ η r f l ρ r r NA ig ig (= 0) − f ) igη , σ s 1 (  , σ + 2 + 2 η )  l r s + 2 − ( σ σ        −
σ σ ) ) in terms of , this time both for the left and right fields: s ≡ ≡ ψ ( ≡ = θ l r ψ ∂ ) ψ ) into this equation one gets, after some simple algebra, ψ ψ s 4.2 4.15 r ( NA ψ r∂ 2 f 4.24 r r − ∂ 1 r 1 i ] in the singular gauge, whose solution is given by 2 5 − ) then becomes simply ]–[ 1 ψ ∆ 4.21 At this point it is a simple matter to verify that our equations in the regular gauge energy, it is convenient to use theintroduced complex in combination of section the profile functions ( 5 Solution of Gauss’sTo equations study the and solutions of vortex equations excitation ( energy as can be shownthe by regular using gauge the is BPS then equations for This is preciselyearlier the [ equation for the profile function write the equations ( By substituting ( Eq. ( give the same profile function (for the complex combination of the profile functions This gauge transformation can be expressed in a more elegant form by introducing the The profile functions are accordingly transformed as JHEP01(2016)075 , ξ , . √ (5.6) (5.7) (5.8) (5.9) (5.2) (5.3) (5.4) (5.5) (5.10) (5.11) = 0 = 0 → 2   l r decoupled ψ ψ ,Q } 1 iθ ) iθ , , r e − Q ( 2 e 2 m r Q = 0 = 0 1 Q ). exponentially fast 1 , φ α   Q ) a Q m r . r m l m r r ( − φ , where φ φ − l r∂ 2 2 m l ( . r ψ r φ θ , → Q Q 2 2 ∂ ψ { 1 1 ) = 1 1) 1 r 2 2 . → ∞ Q m l − Q Q ∞ Q φ = 0 r = . m ( 2 + ( − − l α i 2 , = 0 + ζ m r 2 1 a e l m r m l r φ 2 1 ζ Q dependence in fact drops out of m l m r 2 = φ φ ) for is Q φ φ 2 2 2 2   6= θ 2 2 l r m  Z m r ,A Q Q 5.5  g ζ 2 r ∈ φ l m g 2 2 X ζ + + m − if 2 1 2 1 l − 1 + r r − = 1. = ψ r Q Q r ) for 2 r − ψ m ζ (0) = 0    2 2 2 r l ) ,  m 5.6  + g g ) l ) l  ζ − – 17 – ζ NA − − const. if − f NA − m l , ψ , m r m r m l f ), one can prove that ,A ( φ − ) φ φ m r r )  r  2 2 imθ − φ r r 5.6 r (1 e  ( l   ∂ ) since ln( ζ (1 m l r ∝ A r 1 A r φ NA ∂ l r ) reduce to two complex equations  ζ ∝ 5.8 ζ ζ f m l r 1 Z + 5 r  φ − ∈ m l 2 − − r ) and ( + X ) r φ m 4.14 − ∂ l 1 + 2 r m )  5.5 ∂ = iψ r −  ( l  θ ) = 1 ) and one gets an infinite tower of pairs of iψ ψ m ∂ r − ( ( θ , is ) and (  5.3 m l ∂ A m r NA φ − 2 f φ r r NA 4.13 m r 2 f ∆ − → r φ 1 r ) and ( − l m l ζ 1 ∆ φ 2 5.2 r ζ − l denotes the radial part of the two dimensional Laplacian: ∆ + 2 ψ 1 and r r ∆ ψ ∆ 5 → ∆ which is the same as equation ( or Similarly, the asymptotic form of equation ( A This equation has two independent solutions: with Taking the difference ofat ( spatial infinity. The asymptotic form of equation ( from each other and satisfying Note the particular way thatreflecting the left the and minimum right fields windingequations ( are of correlated in the angular vortex. momentum, The 5.1 Partial-waves and asymptote These equations can be solved via the partial wave decomposition: Then equations ( JHEP01(2016)075 (5.17) (5.18) (5.12) (5.21) (5.19) (5.20) (5.13) (5.14) (5.15) (5.16) defined in 2 , . Q , 1 = 0 = 0 l Q r , ψ . ψ 2 iθ ) behaves as 1 iθ NA e − f Q . ]. 2 e ) can be rewritten as rQ 5.5 , ) 2 Q 2 2 27 f 1 Q = 5.3 Q 1 Q ϕ 2 Q . . . − 0 − , 2 6= 0. 0 g 2 1 0, eq. ( r l , g e ζ ψ m Q r = 0 = 0 = 0 . With , ( ζ . ϕ → r + r 2 ) m l 0 6= 0 ζ l 2 m m r + A g φ e ) and eq. ( ψ 6= 0 r 2 r , e , ζ m − ψ   ∗ = r ) 5.2 m at the origin is excluded by the regularity if ψ r  r 2 2 = m ( ) r if 0 2 2 Q 0 and at large )  Q ) + log r + − → rϕ r 2 1 1) ( + ( ), eq. ( 1 2 m l , ψ ←→ – 18 – − r log 1 2 1 Q m const. if const. if l φ Q ( Q m  e Q ( ψ 2 '  r 5.18 ( ) 0 ϕ l  r , r , r 2 l g 2 ϕ r 0 l ∂ ζ ∝ , ζ ζ g , 2 l r − 1 0 r log log ∝ e m l ψ ζ ϕ − ) + log have been studied in detail in [ ( + φ m r ∝ ∝ r = 2 2 − = r ( φ ) ϕ l 0 1 ∂ r m l m 2 ψ ) ϕ φ φ Q 0  l = 0), where ) as follows: NA ζ ϕ r f r ( r ( r log ζ ϕ − ( − ) can be solved exactly once the vortex profile functions are known. 1 θ symmetry = − ) satisfies the differential equations 5.3 2 i∂ ≡ − θ r ϕ 0 ): ( Z ϕ r i∂ ϕ ϕ 0 l = 0 and behaves as 5.6 r ζ ϕ ), r 2 r ) and ( ζ − 5.2 2.19 = 0 wave, the solution behaving as log ∆ )–(  m ∆ + 2  2.17 By using the first equation of eq. ( Near the vortex core ( is regular at In order to simplify the equations we set Note the following The properties of the function ϕ eq. ( 5.2 Exact solution Equations ( Define the function or For the requirement, and so are the negative-power solutions for or Similarly, from ( This equation has two independent solutions: JHEP01(2016)075 (5.28) (5.30) (5.25) (5.26) (5.27) (5.29) (5.22) (5.23) (5.24) , see the next l , they behave as r < g ξ , r g p , . ). r e ψ = 5.4 z that satisfy 2 = 0 = 0 , , l r Q r = ). At large r e e e ψ ψ z ψ l ψ , ( , 1 e . , l l ψ z ) iθ − ) e χ 2 2 ) = 0 ψ , z e z z ; (¯ r ) (¯ 2 1 l Q . , l 2 ϕ 2 ' l ¯ χ ) ¯ Q r 2 ) = 0 l χ g − = 1 wave for ) 2 log z r ¯ z ϕ g z ϕ (¯ z g 2 (¯ ) and l l + ϕ (¯ l ζ − and z l − + m ¯ r l χ ξ re ( e χ + ζ e ¯ χ e l 2 r r ϕ ψ ζ r − l ( g p χ ζ − r e e  ψ r ) + 2 2 ) then become iθ ≡ , =  z (1 →∞ Q − lim ( 2 1 2 r 1 ) + 2 0 0 2 l e r ) + 2 z Q g g – 19 – Q 5.19 m ( z = 0 2 r r ( symmetry, we find another set of solutions. The ) is therefore given by the linear combinations z χ g − = r r ) + χ = e ϕ 2 ¯ z z ψ χ 2 , 0 l − 0 ( ϕ 5.3 ¯ z ζ ∂ Z r ) r r ¯ z ; ζ ¯ ζ − (1) coupling, r z 0 ∂ χ r Ng 0 e e 0 rϕ = 0 ) = const. ; ¯ ¯ r r z 2 ( ϕ ζ z iθ 0 rϕ 0 l = = ( r e r 1 ζ r l ϕ r , ∂ − rϕ 4 r χ ψ ' ' ψ ζ 0 . l − ξ r → 2 0 ) and eq. ( ψ lim = 0 r ψ = 1 ∆ ) in this case coincides with the solution of Taubes’ equation l f r e 5.2  ψ Ng ( ). Using the ∆ + 4 ¯ z ¯ z = ϕ ∂  ∂ = 2 z = ∂ NA ξ f 2 ϕ 0 ∝ g ≡ 2 0 ) (∆ m 5.22 For the special choice of the U with and with and the function eqs. (2.17)–(2.19) become simply subsection) corresponds to the particular choice above, The holomorphic and antiholomorphicgular terms momenta correspond respectively to in the the positive partial and wave decomposition, negative ( an- The solution of the minimum excitation energy ( general solution to eq. ( in terms of two arbitrary holomorphic functions It can now be easily seen that the holomorphic functions solve ( and we use a complex formulation. Eqs. ( JHEP01(2016)075 , the (5.33) (5.34) (5.35) (5.36) (5.31) (5.32) r ψ ! ) and r ! . l ψ , ) θ l ψ ) # ∂ † ), one sees that ! ψ , ∗ r m r 2 θ φ ) ψ ∂ 4.5  ∗ l m l m r 2 = 0. Going back to UTU ψ φ φ
Im( )–( ! | (  2 r )( r 2 2 2 2 ) which appears in front † ) ψ .
Q A ψ  r Im( 2 | 4.4 | 1 l m l r r # r 2 ) ζ = φ ψ Q A r 4.10 | ( , . l iθ l Aζ l Aζ 2 UTU ζ − − ( ψ + 2  2 e Aζ in ( l
α r 2 ∗ r = 0 2 | − | 1 + ∂ − ) r I r ψ 2 ) = 0 r Aζ l 2 | − ) † − ψ i m r g ψ | is homogeneous in l r ) | ψ l θ 2 it simplifies considerably: φ − 2 ψ U ∂ ψ m ( | θ I becomes the sum of different angular r |  − m 2 ∂ = 0, at r T Re(  ψ + ( | 2 UTU I  α = r 1  2 ( 2 r l + 6= 0 in what follows. I ∂ ) 2 Aζ ) α 1 Q r r ), r + 2 r 2 m l ( α + 1 and  Aζ ) D 2 ψ φ Tr( 2 l ) Q + m r ( | 5.4  + r ψ → φ 2 m l – 20 – 2 | r − 2 2 + ψ | l φ r ∂ r r
6= 0, 2 ψ Q ( | ∂ dz dt 2 =0) ∂ ψ r ,A l = 0, and thus, from eqs. ( l | | ( θ 8

∂ B ) + r ψ ψ r |  ( ∂ † Z 2 r

θ | ψ 2 1 η 2 r |  g 1 Q 2 ∂ l 2 I g 1 | 1 4 Q 2 l g α 1 r = 4 2 + g 1 4 ∂ 1 = l r 4 + + 2 UTU + " η | +  l 2 eff )( + | ψ † m ). In terms of 0 = S | r 2 X | and l ψ 2 2 r )). As the integrand of 4.9 ψ r z ∂ r Q g dθ dr r | dr r UTU ∂ 2 ( | 8

+ / α Z Z Im( 2 r

1 2 1 gauge form, ∂ , g 1 2 π l l ) − 4 Q r g g 1 i z + 4 2 I ≥ l ), we see that such minimum corresponds to constant profile functions = + + = 2 ) " action in ( − r Re( I 4.4 N ρ 1 = , − l ) l N g ( α | ≥ 2 dθ dr r . Therefore we shall assume A z / | 1 CP Z πξ − Inserting the partial wave expansion ( = = = 2 I l T momentum excitations: In addition, as it should. Our interest is in excitations of the above static vortex configuration, with is a pure SU( (using absolute minimum of theour integral Ansatz is ( givenρ by First of all, we note that this expression is positive semidefinite: To study the excitation energyof we the consider the integral 5.3 Divergences of the energy JHEP01(2016)075  ), 2 ) = 1) m r 5.38 (5.41) (5.42) (5.37) (5.38) (5.39) (5.40) φ ( m 2  r Aζ . r , . 1+ ! 2 and the second by − ) = 0 = 0 2 l m m l g   φ  l ( r 2 φ + φ 2  2 2 l ) Q Q ζ 1 (dropping the index 1 m r , r − Q φ plane, one gets r Q | r l ! φ corresponding to the minimum ζ ∂ − m − m r ( xy l r ,  = 1 for which the divergence is  − φ m φ φ 2 2 . r and m l 1 2 2 2 2 / r + g m m ζ φ l | 1 ), we have assumed implicitly ( 2 Q Q 2 2 φ ) , + | = > R 2 2 Q l + +  m l ζ 1 2 5.39 2 r 2 1 2 1 φ = ) − r g Q Q Q ) m with m l 2 ∂ 2 l l | ( ζ + 2 g φ   r ( − − 2 2 l r 2 l

R , s φ 2 g g g
– 21 – s  2 2(1 r l m = ) − − X = R l r , divide the first equation by m r r ∼ I ∼ l Aζ φ φ φ ψ m r r ). This can however be shown as follows. Multiply 2 2 m r − , φ dr r  I ∼  > ζ φ +( iθ m 2 A e 1 A Z 5.15 l ) ∼ r l r ∞ 0  ζ )

ζ m l r φ ) by l m l + )–( ζ m r φ −  φ ( 2 = φ 5.6 1 ) − r l − 2 2 l m  π 5.13 ψ r (1 φ 2 r ∂ Q l φ r − ζ r 2 + m r ∂ and ( l 2 0 ( 2 1 φ ∆ φ has the growing power law behavior. This does not necessarily follow g  r 2 m l r Q 4 1 2 l g φ 1 ∆ g m l,r + φ we have +

         l I ∼ ∞ 0 ) by

dr r . this becomes m l < g 5.5 ∞ I φ r . is an infrared cutoff. As the contributions to the energy of the various modes r 0 r s g 2 π Z R r ∂ = = In concluding that the energy diverges as ( For m l and sum the two equations. Integrating this over the φ 2 l 2 r 1 g Namely, we have reproduced thewith well the known standard result kinetic that term, any action computed quadratic at in its the minimum, fields is a total derivative and thus that is that the from the general resultequation ( ( g The solution is givensatisfying by the system value of and thus the mode with least energy is the one with is therefore where are decoupled, the excitation of minimum energy has The contribution to the integral from a single mode For large JHEP01(2016)075 1. 0  g and R l = (5.45) (5.46) (5.43) (5.47) (5.44) g ξ = 0 mode, ). For m . One has also 5.44 1) this minimum- → ∞ r → , as r ) for which the profile l R , is plotted. The integral ζ r at ζ . ). I . 5.27 ) l 0, 14 ζ R 0

r γr r ( ξR → 4.28 φ ) corresponds to the particular r , ∼ l r ζ √ 0 r r .  g 12 ζ φ 2 1 r ∂ − < g ∞ r e Q Q 0 ( ) = 0 r ∼ φ = g z l  2 r (¯ O r l φ 10 g → π ¯ = must be regular) and as the right hand 2 χ + , r r r r g m r + ζ ζ 2 2 φ for the particular solution ( . We take the parameters normalized to 8 R 0
2

R , , 0 0, and l R g m l φ ξ R – 22 – r φ , φ ) are plotted for several different values of the planar density in → = ) is then r I ∼ = 1 wave for p 6 ζ 2 3 r 0 0 r ∂ ) L g l r 5.44 m H L 5.41 φ 1 Ng γr r H 2 l ( Q 2 l f l g π near g ) = const. ζ π cannot vanish at ), up to a certain infrared cutoff 4 = 2 2 z L  r r ( H m r 1 2 r 2, so the excitation energy is minimum for the ' φ NA / ∼ 5.42 χ f Q Q ) . In figure ) = 1 r 2 ∼  R given by ( 2 R φ ( < ( ). For this set of solutions, the vortex profile functions are given as a function of r m l = I l I φ φ ζ l 4 φ 5.28 and 0.4 0.2 1.0 0.8 0.6 l , in figure φ 0 instead, g const. and l ∼ l ). The exact solution of ( > g φ . Also, r is a constant. Note that in the limit . Vortex profile functions for 2 g 1 it can be approximated, up to exponential vanishing terms, by 5.24 γ ξ For We present below some numerical solutions for the vortex profile functions and the The minimum excitation energy ( √ 0 for constant can be expressed by using ( /g r for which in this case This is plotted in1 figure corresponding zero modes. Wefunctions consider simplify for to simplicity ( the casein figure ( g I where energy solution smoothly approaches the knownnonAbelian profile function vortex, for brought the to standard, the ungauged regular gauge form, eq. ( choice in eq. ( is determined by thelower boundary limit values vanish of on theside the fields is left only. positive hand Since definite, side the ( contributions from the Figure 1 JHEP01(2016)075 r 14 12 10 r ), for various values of R 8 2  5.44 Π 6 14 15 ). 4 12 5.45 5) (from the bottom to the top). The . 2 is given here for the same set of values of 10 4; 0 r . 10 r 0 following from eq. ( 3; 0 Φ . 8 3.0 2.5 2.0 1.5 1.0 0.5 r (0) = 1. r φ r 2; 0 φ . – 23 – ) as defined in ( 6 R and 1; 0 ( . l 14 I φ . 5 (0; 0 4 · such that ) as a function of . 12 2 γ 2 π/ 5.36 ( 2 = Figure 4 10 I θ L 0 8 R density 0 H 8 6 4 2 2.0 1.5 1.0 0.5 10 I I ) with θ 6 sin θ, 4 . The numerical solutions for . The integrand of 2 ) = (cos r l , g l 0 Φ g the coupling constants as in figure Figure 3 ( normalization is fixed by choosing 0.5 Figure 2 2.5 2.0 1.5 1.0 3.0 JHEP01(2016)075 r φ (5.52) (5.51) (5.48) (5.50) (5.49) and l φ : . gauge fields R ) 0 r + T l with , . α ) ∂ 0 iθ N − T = 0 = 0 e sigma model we are β . r ∂   r 0 φ l ζ r D 2 T φ φ ) 2 α = 2 ) the terms coming from 2 ∂ Q λ 0 r Q 1 2 ξ R l 1 ) unbroken gauge theory is 4.8 ψ T g Q √ β Q N 0 are interchanged. ∂ g − − ( r ) plane, to regularize the integral and − l r D φ 0 l 4 φ φ 1  φ 2 2 2 2 T x, y Λ β
Q Q and ∂ = 1 0  2 2 l l + + T − φ λ ψ 2 1 2 1 β λ . r ∂ Q Q in the ( ζ 0  . At the same time, by definition of the vortex    T − R r 2 2 l r α D r ζ r g g 4 ∂ ζ R ζ – 24 – 0 2 R √ Λ − − ) T √ / l 4 r α ) in which  φ λ ∂ φ ξ R 2 2 r 2 ξ g  √  0 2 l Tr( 5.41 √ 1 A g g A r 2 r l ( r
g ζ ζ 1 r 2  ). The solution is − R − ) − 1 2 2 r l | g 5.40  φ r l/r ) we need (neglecting some irrelevant coefficients) + − ψ ∆ 2 | l r g 4.9 φ ( r . To summarize, one must assume 2 l/r ξ 1 ∆ g √ 8 0          is the typical wavelength of the fluctuation of the 2 ) can be applied. First of all we neglected in ( = /g λ 2 4.10 ). The divergences arise due to the presence of the massless SU( l/r αβ F )–( where 5.45 and we have to check when this approximation is reliable. The extra term that we are Tr Another restriction comes from the fact that the SU( 4.8 4 in ( l/r αβ /λ vortex size, 1 for the validity of our analysis. asymptotically free (unless,coupled e.g., at large one distances. Our introduces lowest-ordermuch calculations less many are than thus the fermions) valid confinement only length, and for 1 effective wavelengths becomes action, one strongly is calculating the fluctuations at length scales much larger than the This is thus a low-derivative condition, but related to the choice of the cutoff considering. In order for thisthe terms effective to action be ( negligible with respect to the two-derivative term This is a1 higher derivative correction to the effective action, it is thus proportional to I in the bulk. There arein certain ( limitations to the parameterF space where the effective action neglecting in the action is These are the same as equations ( 5.4 Infrared cutoff We have introduced above an infrared cutoff satisfying which is the same as in ( JHEP01(2016)075 ) θ /g ( 1 u 6= 1. (6.5) (6.2) (6.3) (6.4) (6.1) : the 0 and ) , with → H θ Q ) vortex → position- a ⊂ r condenses, . G/H G A ( e H 1 are Q G/H , ⊂ π ( θ 6= 0 representation, 1 , , a π H α Q = 0 into H is taken to depend . Our theory has α = 0 H 3 U → α G in the ,A ) , G ,A , which implies ) ) t, z U Q ( ( ) ( S,A S U ) V ) ( → ∞ r r ( S ( r − coupled to a scalar field ) θ θ r 2 a a | ( . For a nontrivial G θ Q = 0 as = a H µ θ . θ = D | iθS | → θ e Q i ) + . Then, z generates the internal zeromodes. are the generators of D ,A ,A ) = µν | – 25 – a θ e F ( H ,A T u = 0 and = 0 µν r t H/ r F = 0 to a subgroup r Tr( 0. Since the vacuum of the theory leaves an unbroken 2. The scalar potential is chosen so that G , where 1 2 / a → ,A − ,A ab T ) δ ) a µ r . One may take the following Ansatz r = r ,A ( S ( A ) 0 0 L r Q = ( Q ) = ) 0 as 0 , one obtains a vortex with a generic orientation, ) b µ θ θ Q ( T ]. Consider a generic Lagrangian H ( A ) a . In the spirit of the moduli space approximation, u → approaches a uniform vacuum configuration at spatial infinity and θ † T ( of 10 θ Uu 0 , = u a 9 and ) Q U = Q µ . USU . Therefore, the boundary conditions on the profile function t, z Q ( S and = z, t igA ) U ) is nonAbelian, a given vortex configuration further breaks − U = /g ( = µ H α S (1 ∂ Q → ∞ Let us summarize these features in a model independent fashion following the reasoning We recall also that the presence of the vortex makes the embedding = → r µ θ where on the string worldsheet coordinates A as symmetry group, which howeverorientational moduli, can describing act its embedding nontriviallytransformation in on the our Lie algebra. Ansatz, Through the a vortex global gauge carries The Ansatz for the gauge fieldsIn is particular, determined a by necessary the condition requirement is of that finiteness of the energy. algebra that we denote as where describes the asymptotic winding of the scalar condensate, which describes aD gauge theory withnormalized gauge as group Tr( breaking the fullconfigurations gauge exist. group The elementary vortex is oriented in some fixed direction in the Lie explain the precise nature of the non-integer, powerlike divergencesof encountered. Alford et al. [ dependent in the regularof gauge. the As generators aBohm become consequence, scattering multivalued. in of a gauge Multivaluedzero covariantly bosons modes symmetries constant at lead basis give to large some rise nontrivial distances effects, to from like the Aharonov- cosmic string vortex. color superconductivity. The They corresponding also In this sectionintimately it related will to be thevortex shown solutions geometric that as obstruction a the discussedWhen consequence non-integer in of power the section divergences bulkaction symmetry just of breaking found the unbroken are generators in 6 Origin of the non-integer power divergences JHEP01(2016)075 . (6.9) (6.6) (6.7) (6.8) piece. (6.11) (6.12) (6.13) (6.14) (6.10) i = 0. At A r α approaches ∂ A static gauge α A . and α . fields are induced, ], which at spatial 6= 0 A · i α α ]] ) are D S, α A [ i + h.c. = 6.1 − , Q S, A iα [ θ α α F ∂ . left unbroken by the vacuum D A S, . S,A [ α , 2 a θ ) α 2 θ = 2 , G A r T , D † r a ( A 2 † ] ) θ 2 2 θ 2
⊂ 1 ) can be written as = 0 θ α r ] Q
g a 0, which implies that ( ] · a u · H + − = , iθS S, 6.12 ] 0) S, A [ α S, − θ s [ α | → igT α θ [ i i r, A Q A ( r = θ α Me isψ α − iga iga ), which is the nonAbelian Gauss law. Sub- θ D iα A | ), going to what we shall call the = − ) iθS − ∂ F = ∆ S, ∂ 6.8 ,A e θ i θ [ s α α ( – 26 –
α 2 ∂ θ 2 t, z ] D 1 ψ u r A ( r a · A i † θ 2 i = 0 1 ∂ isθ r ig S, U ∆ r D e [ reduce to the ordinary derivatives since 2 i i = ' − + , r D = − α − † α iα s α α θ = A F A r ψ ∂ ∂V A r ∂Q ,A iα ∆ ) − F i r ( = ∆ = ∆ , = D 0 r iα ). Q Q ) F µ in eigenstates of the operator ∆ i θ ∞ ( ( D D α 0 u µ . Passing to the static gauge allowed us to eliminate the A Q iα D S = ) ) F θ r is taken to fluctuate along the string, nontrivial ( Q ( i u a U = i transform of) the gauge fields belonging to the right hand side of Gauss’s equation vanishes as the covariant derivative of the being a real contant. The eigenfunction in ( A r s U We may now gauge transform by with using the fact that One may expand infinity commutes with ∆ where the covariant derivatives in large scalar field approaches zero. One obtains thus a Laplace-type equation This rewriting also follows from the fact that in static gauge We focus now on thestituting second it of into our equations Ansatz, ( the left hand side becomes where The equations of motion following from the Lagrangian density ( This gauge choice makes computation somewhat simpler. The new Ansatz is One obtains for whose precise forminfinity, is however, is dictated fixed by by the(the requirement the equations ofconfiguration motion. Their behavior at spatial As soon as JHEP01(2016)075 a Z is S a ∈ S (6.22) (6.16) (6.17) (6.19) (6.20) (6.21) (6.15) (6.18) 6= 0 it . If s α θ m ψ . Related 5 , at a , then S and H 4 ⊂ e H , . , 1 . ! − α ) ) 2 | 1 U V then implies that ( ) describes the embedding N r − m θ T . If, on the contrary, s α η ( − N , Z N . s ψ ) u | 1 + θ s 2 U ∈ −  ( NC α R l r , ( T s 1 a iζ W α r e ∝ S , ∂ ρ a a s α ≡ ∼ O S ≡ = ψ = πiξ ) 2 ) α θ = . are excited can only be done by studying e r ( a ( α l † α ξ u ) = ⇒ A π ) theory, it was found that the gauge fields † − = Energy ) ). (2 implies that N Z π θ ( u ( , – 27 – s r α a ,V ,A and requiring that the color matrix structures of (2 ∈ s † α a ) ψ ) S u α ψ s S r U , ) θ a SU ( V 2 2 ( ) π = 0 is oriented along some generator l l | S s r T η u ) × ) θ m (2 ) l If a parallel transport by ,T π ) U . ' − u ( α . The single valuedness of + ( † s r a | at 6 (2 N s A α α T . for all ξ cannot be determined by a general argument based on the ( ) u R l α ψ α θ ( W s r M l ( A l . Allowing for a nonvanishing angular momenta UTU i ∂ ρ | ∆ SU u s | = ≡ 2 ∼ O = × → ) ) α R α l ) U ( α l ( A ( α W (1) A 0 T A ), one finds the asymptotic behavior, 6.11 )), , we see that if ), one gets in general 3.9 G ) in ( 6.11 independent matrix θ 6.13 ), ( inside In our specific U Such a general argument, however, carries us only this far. In particular, the determi- One can now see the consequences of the requirement of single valuedness of 6.9 This is simply the condition of parallel transport of a constant. H 6 Knowing this, andvortex knowing (see how ( these gauge fields are parallel-transported around the with asymptotic behaviors of thethe fields vortex alone. configuration at Both finite of these features carryexcited by information the about vortex modulation lie in the directions nation of the generators along whichthe the nonAbelian fields Gauss equation the left and right handto sides this, match precisely, the as non-integer has power been done in sections The energy growsin like ( Using ( and the single-valuedness conditionnot for globally defined, with some non integer constant of is oriented along the rotated generator is covariantly constant and belongs to the globally defined subgroup for any JHEP01(2016)075 ) × D D D N and = 0 ( r (1) l r gauge (6.25) (6.23) (6.24) 0 g πiζ D 2 − e ) theory) on N ) with this specific SU( × 6.19 ) are thus , , ! (1) α 0 spacetime. Another venue, α 6.23 α iW and its Hermitian conjugate V W D coupled quantum field theory 2 ) − U D ( ! -2 T ) ) . ] = l α l ) D , just by starting with a given 2 ∂ 2 r U πζ πζ g ( ) is given concretely by + 2 l ) + g system in 4 ,T U 2 l α 6.16 ( g symmetric (“color-flavor” locked) vacuum. D V in the case of U T [ ) cos(2 ab initio ) r  l ) sin(2 1 l U + l ( − = πζ ) or ( ) T πζ – 28 – l N , ζ α N ∂ α  3.14 simply decouples from the vortex system. sin(2 CP iV = cos(2 = − D v a ξ

] = 2 ) → U ( ! and the dynamical properties of the vortex excitation explored ,T α 3 α in equation ( α variables are the fluctuations of the vortex collective coordinates, ], i.e., embedding the 2 V W a W sigma model ( ξ indeed agree with the general expectations ( [ D

30 , 5 D 29 coupled quantum field theory, in the context of the underlying U ), one finds that upon encircling the vortex ) degrees of freedom are part of the original Yang-Mills fields, the coupling D r ) gauge theory in an SU( ( α -2 reduces to an integer. Thus no geometric obstruction, no nonAbelian AB D A N D a . In particular, let us note that for the standard nonAbelian vortex ( ξ )) Yang-Mills system in 4 5 SU( dynamics becomes fully operative, and at the same time the massless, free SU N × ( l 1 ) = 0) − , with respective eigenvectors l In a standard nonAbelian vortex, the modulation of the vortex internal orientation This is a new type of interacting quantum field theory, in which field variables living These discussions clearly illustrate the connection between the global features of the l N system. Here the 2 N g . Thus in our model πiζ † D 2 in different dimensions interactan nontrivially. existence proof Our that waysystem of such is a approaching a consistent theory the local can problem gauge be gives theory. consistently defined,is as described the by underlying a 4 2 which we have opted to followwhich here, emerges is to as study a such4 low-energy a effective 4 description ofwhereas the the vortex 4 sector ofamong the them underlying being fixed by the local gauge symmetries of the underlying theory. interacting 4 SU( Of course, such asystem system possessing can some be globalfields coupled symmetry, considered to and it [ making it local, by introducing 4 (or SU 7 Discussion In this paper we discussed in some detail the topological and dynamical features of an system discussed in section in section or effects, no non-integer-powerCP growth of vortex excitation energy, etc., occur. The 2 Our results in section index. have been used.e The eigenvalues ofv the monodromy matrix ( where the commutation relations (similarly for JHEP01(2016)075 0. D D -4 (7.1) → D r g ) gauge N ( r + the divergent l r  2 | Q = 1 wave, one finds α ) indeed shows that . D m | R 5.40 ) + Tr ) space α , the screening of the massive 0 first, for the vortex effective ) i r x, y gauge coupling which was frozen ( → F ) Yang-Mills system, now decoupled D r ) , and in fact, the non-integer power → ∞ r g } ( N iα i ( R F B r . The renormalization group flow of two { ξ The expression ( √ 7  ) + Tr( α 2 – 29 – i ) ) l ) ( U ( F model in two dimensions is asymptotically free and ) T nonetheless the result of the interactions with the 2 l ( iα α 1 ) come from the dominant, massless SU ∂ F − do not commute, pointing clearly to the origin of the 0 limit is that most of these divergent effects smoothly are N Tr( 5.40 Tr( , i.e., the contribution of the partial →  CP r x , into the infrared, showing a remarkable realization of 2 → ∞ 3 g ξ dz dt √ R )–eq. ( dtd Z I Z 5.38 = = is turned on, however, the physics changes immediately, and drastically. 0 and 0, if the spacetime is finite. r eff ]. g S → → 28 , r r g g 4 , minimum energy cost action divergent and the vortex fluctuations become nonnormalizable. They be- 2 1 = 0. of the vortex in the Lie algebra fluctuates wildly, and the AB effects are washed away. − Summarizing, if one first takes the IR cutoff What happens in the Let us look into the nature of this discontinuity more carefully. At large One wonders however how such a discontinuous change of physics is possible at As soon as When the orientational mode dynamics becomes strongly coupled, the orientation r N } Here the relevant dimension is the extension of the transverse ( i g 7 B gauge fields impliesscalings. that the On massless theaction gauge other of hand, fields if dominate, one leading takes to the the limit power law does not necessarily mean that they are related to the vortex physics. reflect the details of the vortex configurations near thetransit vortex into core. the continuum spectrum offrom the the free vortex. SU The factaction that they appear in the coefficient in front of the vortex effective effects such as eq.field ( excitations (the massivesuppressed fields contributions). in They the bulkvortex Higgs collective coordinates, mechanism give theand only fluctuations the of exponentially precise direction in color space in which the massless gauge fields are excited, both the limits discontinuity. Stated differently, it is becauseextended one spacetime is in accustomed relativistic-field-theory to applications thinking that aboutat physics infinitely looks discontinuous come infinitely costly to excite.and At the nonlocal same effects time, emerge. the AB effects and all sorts ofAs topological in the phenomena ofchange phase at transitions, physics cannot really exhibit such a qualitative { This is consistent withGoldstone the bosons fact in that 2D: there the spectrum is has no a spontaneous massThe symmetry gap. presence breaking of and massless no CP fields in the bulk makes the coefficient in front of the effective becomes strongly coupled at lowdimensional energies model Λ “carries on”at the the evolution vortex of mass the scale duality 4 [ the vortex worldsheet. The JHEP01(2016)075 ). D N U(1) (7.2) (7.3) (7.4) × remains in gets smaller, D ] which is the /R 15 , which should play ξ √ ) field in 4 quantum effects. N ], based on a U(1) D 31 worldsheet modes. The first massless gauge fields will re- . D D
decreases. For the gauged non- ξR µ √ l as the cutoff scale 1  g − D e 2 simultaneously down to some infrared µ Λ . } O i  2 l B + g π { 2 2 l increasing g π = 2 – 30 – model, for the ungauged nonAbelian vortex, is is ) = log I gauge invariance. By integrating out the coupled 1 fields I µ − ( ) = G I model coupled to the 4 N D R 0 limit clarified here, should provide us with a useful π ( 1 ⊂ 8 I − → CP 0 N H r → g r ), due to the interaction with the massless unbroken SU( lim g CP would not be washed away by the 2 } 5.46 i limit we recover the physics of the standard nonAbelian vortices B logarithmically as the RG scale { coupling 1 → ∞ − R N gauge fields and the 2 , and varying it, one would find an appropriate RG flow towards the infrared. CP decreasing in the Higgs phase, at least as long as the massless SU( would consist of two pieces: the full 4D action for the massless gauge fields µ D ) that is ξ orientational modes in the vortex worldsheet. √ I 1 5.46 frozen = − , which is a sort of Nambu-Goldstone bosons living in the vortex worldsheet. Their Although such a renormalization group program is still to be properly set up and be Let us end with a few remarks on other types of vortex systems in which certain bulk To compute the effective action in this paper, we have integrated out all gauge fields, The renormalization of the zero modes exist and are coupled nontrivially with the 2 N } i UV D B 4 is the famous superconducting cosmicgauge string theory. model by Our Witten model [ model. can There in is factimmediate an be predecessor even regarded of closer as the connection a present with nonAbelian paper, the generalization where model of the discussed that “right” in gauging was [ made only in cutoff scale worked out, we believedecoupling/transitions to that the the numberstarting point of for results such a established task. and the subtle issues of Λ living in the bulk and{ the 2D action describingcoupling the fluctuations is of dictated the by internalmassless the orientation, 4 full including the massless gauge fieldsthe renormalization in group point the of bulk. view.first, This One and should is obtain integrate formally away the the effective not massivethe action gauge quite fields defined role consistent at of from the an vortex mass UV scale cutoff for the study of further quantum effects. The effective action at main the Coulomb phase. Note thatmodes Coleman’s in theorem two-dimensional on spacetime the would absencemassless not of apply the gauge here Nambu-Goldstone fields. due to Thisthe the would vortex interaction of orientation mean the that 4 the global and topological effects related to namely Abelian vortex, we find anotherthis effect: time as a powerWe law may ( thus conjecture the Taking the limits in thisCP order, the only massless degrees of freedom thatgiven remains by are the Taking then the with a finite from ( JHEP01(2016)075 ]. D 35 (A.2) (A.1) ]–[ 32 dynamical system (such = 0) D r . g ], the divergences ! massless degrees of 1 l 15 bulk, coupled to “2 φ ! 2 2 D ) D Q m l 2 1 φ ( Q 2  − A  r − + 1 m 2 ; )  = 0 and discuss how the analysis of m l + r imθ φ ( g e 2 ) m l m m l φ X φ action caused by the 4 r ,  m ∂ 2 2 – 31 – X D ( Q

= = 1 8 l + r m ψ 2 1 ψ X Q 2 l g 1 ]) is quite distinct from, and new as compared to, those + 4

10 , 9 dr r Z π effective action, as is described in detail in this paper. = 2 I D gets modified. By partial wave expansion 5 Thus the nature of the non-integer-power behaved infrared divergences analyzed in The second is the case of the so-called semilocal vortices, where due to a larger flavor bulk, and interact nontrivially with the vortex orientational zeromodes. Interesting D one finds A Excitation energy in theIn standard this nonAbelian appendix vortexsection we ( consider the special case Simone Giacomelli for interestingand discussions H. and Itoyama for forCity information. warm University. hospitality KK KO during thanks thanks severalUniversity. H. M. JE visits is Itoyama Nitta to supported for Keio bythe all University NSFC INFN the and grant special 11375201. help Osaka project given This GAST research to (“Gauge work him and is at String supported Theories”). the by Osaka City aspects of the 2 Acknowledgments The authors thank Muneto Nitta for useful discussions at an earlier stage of the work and occurring in these other systems.the presence The of effects an discussed unbrokenmassless here Yang-Mills gauge matter” reflect interactions in describing in an the theand essential 4 vortex manner elegant internal connections excitations. betweenas These the the bring global, nonAbelian in topological Aharonov-Bohm rather features effects, subtle nonAbelian of statistics, the etc.) 4 and the features of the vortex effectiveAgain, action the in divergences these systems infreedom have are the been always found effective investigated to in 2 beNo [ logarithmic, topological as effects the are latter is present. due to massless scalar particles. the present paper (and in [ found are logarithmic; therepresent are work. no nontrivial topological effects such as discussedsymmetry (as in compared the to4 color), some massless Nambu-Goldstone bosons survive in the a U(1) subgroup of the flavor group. Both in Witten’s model and in [ JHEP01(2016)075 ξ √ (A.6) (A.7) (A.8) (A.9) (A.3) (A.4) (A.5) → action (A.10) 2 1 . − N # ,Q s . 1 σ # Q 2 CP ( Q 2 1 + 1) Q 2 φ → ∞ − ( r 2 2
= 1 term and dropping Q m + 1 8 + 2 2 1 6= 1. By dropping the indices ) in front of the φ . s Q 2 l φ , σ m I g 2 ( = 0). As ) one has , + 2 2 ) (hence the color-flavor diagonal µ m l l m 2 2 Q , N 0) this becomes ! m ∼ φ φ , ( 2 Q , ]: by making use of the equations for 2 + iθ R φ φ µr 5 0 ≡ → 8 r + 2 e 2 1 2 for − s  µ 2 1 φ e ]–[ ∼ Q A r σ . 1 2 = 1 there exist physical solutions. Substi- Q A + φ c 2 r + ≡ g π r  log φ − + m √ + 2 φ 2 2 iθ , 2 ), keeping only the 2 e ! m  r µr m =  2 – 32 – 1 1 e   A.2 φ A I 1 − s − r c + const. const. and r σ = − ∆ 2 r l m ) ∼ ∼  m NA ψ → ) in ( φ  φ found earlier [ f r φ 2  ∂ ∼  I ( Q A.1 = φ

+ is (or 0, φ 2 l 2 ∆ φ g 1 ) m ∆ s 4 → ), i.e.,  σ " r 1 r N ( ∂ Q ∼ ( L dr r ) becomes for the angular momentum (

φ 2 Z m g 1 = 0 ( A.7 is the W-boson mass. The solution is a combination of modified Bessel 4 π r " ξ = 2 √ l ) and for the SU g , it can be shown that at its minimum I } N l 1), eq. ( ( ≡ dθ dr r )). The vortex orientational zeromodes fluctuate strongly in the infrared in the L NA → µ Z )) is a global symmetry, there are no divergences in 4.9 , f A 2 Consider instead the contribution of a single wave N , = 1 I for SU Q where functions of the first and second kind, i.e., at large which has solutions and Near the origin The equation of motion for (eq. ( vortex worldsheet. l Thus for the standard nonAbelianSU( vortex, where SU This is a familiar expression{ for tuting accordingly the Ansatzthe ( index one gets It will be shown below that actually only for JHEP01(2016)075 φ = 1 and φ m (A.14) (A.15) (A.16) (A.17) (A.11) (A.12) (A.13) ) by # . φ ) is the  2 A.7 φ . Q 2 2 ) gives 1 A.6 0 = 1. The normal- Q Q 1 > m Q − A.13 2 l  2 g . φ 2 0, ( 2 − l Q + 1) ) we know that actually g 2 1 2 2 2 → φ Q φ 2 2 Q r ( 2 l A.13 Q g 2 2 2 + ), multiply eq. ( | 2 Q 2 1 + − m | 2 1 Q 8 + φ ). At r 2 2 Q 2 1 , + 2 , instead, l . Q Q ∼ 2 g 1) A.13 φ 2 + r µr + φ 2 + φ , − 2 1 −  √ 2 2 → ∞ ), ( e ! 1 φ r Q φ 2 6= 0. ( A r 2 2 l 2 φ − 1 g  ∼ r 2 − µ c 1 A.12 – 33 –  A φ + = m . At ∼ A ∆ φ r − r φ  2 r ) before yields this time φ −  ∆ ∼ m + 1 A 2  φ  ) A.11 r φ − + + r = 0 theory, the only physical mode in eq. ( 2 1 ∂ 2 ) r ( )  . From the exact solution of ( φ g  φ ). Even though these latter solutions are nonnormalizable also, r r r µr = ∂ ∂  . To show that the exponentially growing term is necessarily ( √ ( e 2 φ  , dr r 5.47 1 c

) plane, to get ∆ c 2 l Z g 1 dr r x, y 4 1 + = is " Z ), eq. ( ∼ ∞ 0 I | = φ φ dr r r 5.40 ∞ 0 | Z φ r π φ r ∂ = 2 φ r ∂ We conclude that in the Such an exponentially growing behavior for the gaugeIt fields is must amusing be to regarded see as what un- makes the difference in the special case I wave. This is tovortex, be eqs. contrasted ( with thethey powerlike divergent represent modes the for continuous thephysical spectrum general excitation of gauged modes. the theory and as such are to be regarded as Even though the l.h.s. vanisheslonger in positive this definite, case thus also, no the contradiction integrand arises. of the right hand side is no which implies behaves as The procedure which has led to ( whose regular solution behaves as The crucial difference is the last term in ( and the equation of motion is This would lead to a contradiction, unless physical (not an acceptable quantum state), and we discardization them. integral present for theintegrate solution over regular the ( at the origin ( with some constants JHEP01(2016)075 ] (B.5) (B.6) (B.7) (B.8) (B.1) (B.2) (B.3) (B.4) ): it does not θ hep-th/0307287 [ , , , . α , ) ) V ) r ! l r B ρ 1 ρ ( η l r − † (2003) 187 g g ), or the expression for the + N U , ) 1 α ), 5.3 Non-Abelian superconductors: . ! U − ( r W 1 4.5 T 1 r ψ − B 673 α = 1 + 2 = 1 + 2 N iβ iρ ∂ . )–( l } r e 1 ) σ } ) and ( σ ) = ≡ = ). In other words, the apparent extra θ ( ) ( 4.4 r ( ) → β ( α r 5.2 2 ( α r ,Q + 4.24 Q r θ ψ ) 0 ( )–( Nucl. Phys. r iβ ( , 1 ,Q – 34 – , ψ , e ,A r 0 ,V l l Q 4.19 ) α ψ ψ , ψ U iθ V , l ( iβ e l SQCD , iβ r ψ r T e η l e η ) ,T { {

η ψ † r l U + ) ( = 2 → ig ig α B ), which permits any use, distribution and reproduction in l T ( N α ψ W ∂ UTU + 2 U and + 2 l l r l iρ ≡ = = σ σ ψ ) ) α = U = = B ( ( ) W l l r α ( T Q ψ ψ CC-BY 4.0 A This article is distributed under the terms of the Creative Commons ]. ), are clearly invariant under a common phase rotation, 5.31 SPIRE IN vortices and confinement in [ R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, [1] References zeromode corresponds torepresent the a space physical zeromode. rotation (the shiftOpen of Access. theAttribution origin License of ( any medium, provided the original author(s) and source are credited. are gauge equivalent to The algebra neededthe is singular the to same regular as gauge, the see eqs. one ( involved in the gauge transformation from one can however show that the scalar and gauge fields By recalling the definitions One wonders whether this isfields also which necessary, i.e., solve if it. thereenergy are ( The any other equations choice of for motion the gauge ( solve the Gauss equation, given the vortex configuration We have seen in the main text that the Ansatzes ( where B Gauge fixing for JHEP01(2016)075 , ]. ]. SPIRE ] σ , IN SPIRE ][ IN ][ , ]. (1989) 606 ]. The interactions Zero modes of ]. , , SPIRE (2003) 037 Cheshire charge SPIRE hep-th/0005076 IN (2014) 101 SPIRE B 315 [ IN ][ IN 07 hep-th/9112038 ][ 10 color ][ [ , hep-ph/9804403 ]. [ Vortex zero modes, large flux (1990) 1632 ]. Quantum field theory of JHEP Non-Abelian vortices with an ]. SU(3) (2000) 37 ]. ]. JHEP 64 , , Nucl. Phys. SPIRE , SPIRE IN (1992) 251 [ IN SPIRE SPIRE SPIRE (1999) 443 ]. ]. B 590 IN ][ IN IN hep-th/0412082 ]. ][ ][ ][ arXiv:1503.00517 [ arXiv:1310.1224 B 384 [ Non-Abelian vortices, large winding limits and [ Color flavor locking and chiral symmetry SPIRE SPIRE B 537 Dynamical symmetry breaking in supersymmetric IN IN Non-Abelian vortex dynamics: effective world-sheet SPIRE (1991) 414 – 35 – ][ ][ Phys. Rev. Lett. ]. IN ]. , cosmic strings and [ Non-Abelian Meissner effect in Yang-Mills theories at Nucl. Phys. , Supersymmetry breaking on gauged non-Abelian vortices (2015) 143 B 349 (2014) 086 SPIRE Nucl. Phys. hep-th/0403149 SPIRE (2005) 045010 , IN [ 04 Nucl. Phys. IN hep-ph/9310262 hep-th/9306006 01 [ arXiv:1007.2116 , [ [ ]. An unstable Yang-Mills field mode [ (1985) 494 ]. On electroweak magnetism Non-Abelian string junctions as confined monopoles D 71 Classical and quantum structure of the compact K¨ahlerian Vortices, instantons and branes Non-Abelian vortices and non-Abelian statistics JHEP JHEP , gauge theories SPIRE arXiv:1206.4546 arXiv:1010.4105 , Nucl. Phys. SPIRE [ [ ) IN B 253 , c (1978) 376 IN ][ n (1990) 668] [ (2010) 012 (1994) 4167 (1993) 4821 (2004) 045004 ][ Vector mesons and an interpretation of Seiberg duality Phys. Rev. 65 08 , USp(2 B 144 D 49 D 48 D 70 (2012) 014 (2011) 019 ]. ]. Nucl. Phys. and JHEP 09 02 , ) , c n SPIRE SPIRE IN arXiv:1408.1572 Erratum ibid. IN hep-th/0306150 models Nucl. Phys. [ limit and Ambjørn-Nielsen-Olesen magnetic[ instabilities Aharonov-Bohm effects JHEP Aharonov-Bohm effect non-Abelian strings and vortices Phys. Rev. Phys. Rev. and excitations of non-Abelian[ vortices non-Abelian vortices SU( [ breaking in high density QCD action JHEP [ Phys. Rev. weak coupling N.K. Nielsen and P. Olesen, J. Ambjørn and P. Olesen, S. Bolognesi, C. Chatterjee and K. Konishi, F. Delduc and G. Valent, K. Konishi, M. Nitta and W. Vinci, J. Evslin, K. Konishi, M. Nitta, K. Ohashi and W.S. Vinci, Bolognesi, C. Chatterjee, S.B. Gudnason and K. Konishi, H.-K. Lo and J. Preskill, M. Bucher and A. Goldhaber, SO(10) M.G. Alford, K. Benson, S.R. Coleman, J. March-Russell and F.M.G. Wilczek, Alford, K.-M. Lee, J. March-Russell and J. Preskill, M.G. Alford, K. Rajagopal and F. Wilczek, M.G. Alford, K. Benson, S.R. Coleman, J. March-Russell and F. Wilczek, S.B. Gudnason, Y. Jiang and K. Konishi, Z. Komargodski, G. Carlino, K. Konishi and H. Murayama, M. Shifman and A. Yung, A. Gorsky, M. Shifman and A. Yung, A. Hanany and D. Tong, [8] [9] [5] [6] [7] [3] [4] [2] [19] [20] [17] [18] [14] [15] [16] [12] [13] [10] [11] JHEP01(2016)075 ]. ] , ]. SPIRE quantum IN D ][ 4 (2009) 254 - (2013) 070 SPIRE D IN 2 09 ]. , [ arXiv:1104.2077 B 678 [ , SPIRE , JHEP , IN models and their gauging supersymmetric QCD [ (1990) 193 , σ , arXiv:1107.3779 ]. = 2 [ ]. Phys. Lett. ]. ]. , N (2011) 125017 B 330 SPIRE (1985) 557 SPIRE IN SPIRE Nonlinear IN SPIRE [ IN IN ][ D 83 ][ ][ Quantum dynamics of low-energy theory on B 249 (2011) 065018 Nucl. Phys. (1986) 1 , D 84 ]. – 36 – ]. Phys. Rev. ]. ]. , Surface defects and resolvents B 266 Effective world-sheet theory for non-Abelian semilocal Nucl. Phys. SPIRE hep-th/9806056 , SPIRE [ SPIRE SPIRE IN Grand unification and mirror particles arXiv:0704.2218 hep-th/0603134 IN [ IN IN [ [ [ Phys. Rev. Global color is not always defined [ [ , ]. ]. A condensate solution of the electroweak theory which interpolates Non-Abelian semilocal strings in Monopole topology and the problem of color Nucl. Phys. , (1998) 005 SPIRE SPIRE IN IN (1983) 1553 (1983) 943 (1982) 141 (1982) 427 11 ][ ][ (2007) 105002 (2006) 125012 supersymmetric QCD 50 50 Field theories with no local conservation of the electric charge On the moduli space of semilocal strings and lumps Multiple layer structure of non-Abelian vortex = 2 JHEP B 208 B 209 D 76 D 73 Superconducting strings , , Master Thesis, University of Pisa, Pisa Italy (2015). Gauged non-Abelian vortices: topology and dynamics of a coupled The BPS spectra of two-dimensional supersymmetric gauge theories with twisted N ]. SPIRE IN arXiv:1307.2578 arXiv:0903.1518 strings in [ semilocal non-Abelian strings Phys. Rev. Phys. Rev. in and out of superspace [ field theory [ mass terms Nucl. Phys. Nucl. Phys. between the broken and the symmetric phase Phys. Rev. Lett. Phys. Rev. Lett. P. Koroteev, M. Shifman, W. Vinci and A. Yung, M. Shifman and A. Yung, M. Eto et al., M. Shifman, W. Vinci and A. Yung, C.M. Hull, A. Karlhede, U. Lindstr¨omand M. Roˇcek, D. Gaiotto, S. Gukov and N. Seiberg, E. Witten, M. Eto et al., N. Dorey, A.S. Schwarz, A.S. Schwarz and Y.S. Tyupkin, L. Seveso, P.C. Nelson and A. Manohar, A.P. Balachandran et al., J. Ambjørn and P. Olesen, [35] [32] [33] [34] [29] [30] [31] [27] [28] [24] [25] [26] [22] [23] [21]