Physics and Mathematics of non-Abelian vortices

38 Pages·2011·6.72 MB·English

M i n n e a p o l i s 1 3 / 0 5 / 2 0 1 1
Physics and Mathematics of
non-Abelian vortices
-
a key to the mysteries of the
non-Abelian monopoles and confinement
K . K o n i s h i
U n i V . P i s a / I N F N P i s a
Tuesday, May 31, 2011
Plan
I. Non-Abelian vortices

Effective worldsheet action and GNOW duality
Higher-winding vortices: Group theory
II. Non-Abelian monopole-vortex complex
Monopole-vortex complex with orientational moduli
Gudnason, Jiang, Konishi
JHEP 2010
Eto, Fujimori, Gudnason, Jiang, Konishi, Ohashi, Nitta
JHEP 2010
Cipriani, Dorigoni, Fujimori, Gudnason, Konishi, Michelini ’11
Tuesday, May 31, 2011
I. Non-Abelian vortices
Tuesday, May 31, 2011
Non-Abelian vortices
Hanany-Tong, ‘03
Auzzi-Bolognesi-Evslin-Konishi-Yung.
Def: Vortex solutions with continuous (non-Abelian) moduli
Natural generalizations of ANO vortex
Shifman-Yung, ... (Minnesota).
N.B. • H ⇒


1 with ∏ (H)≠ 1 Eto-Nitta-Ohashi-Sakai- ... (TiTech, Tokyo).
1 Tong, (Cambridge).
Pisa group, ‘03-’11

with H: non-Abelian not sufficient
e.g., H= SU(N)/Z (cid:15482) Z vortex ! ( ∏ (H) = Z )
N N 1 N
•
Global (flavor) symmetry: e.g. U(N) theory with N = N “squarks”
f Fujimori’s talk
Sakai’s talk
Koroteev’s talk
• “
Color-flavor locked” phase
Cui’s talk
〈q〉= v
1NxN flavor
•
Local gauge symmetry broken (Higgs)
(1) (2) (N)
q q q
➯ vortex solutions 1 1 1
···
. .
 q(1) q(2) .. .. 
• 2 2
Global symmetry GF = GC+F =SU(N) unbroken (q)iα =  ... ... ... ... 
•  . . . 
Individual vortex breaks it  . . . 
. . .
 
 
(1) (2) (N)
➯ Orientational zeromodes in SU(N)/ U(N-1) =CPN-1  q q 0 q 
 N N ··· N  color
 
➯ They can fluctuate in (z,t)
Tuesday, May 31, 2011
1 Introduction
The last several years have witnessed quite an unforeseen progress in our understanding of
non-Abelian vortices, i.e. soliton vortex solutions in four (or three-) dimensional gauge theo-
ries possessing exact, continuous non-Abelian moduli. These continuous zero-modes arise from
the breaking (by the soliton vortex) of an exact color-flavor diagonal symmetry of the system
under consideration. The structure of their moduli, the varieties and group-theoretic properties
of these modes as well as their dynamics, and the dependence of all these on the details of the
theory such as the matter content and gauge groups, etc. turn out to be surprisingly rich. In
spite of quite an impressive progress made in the last several years, the full implication of these
theoretical developments is as yet to be seen.
In the present work we turn our attention to the low-energy vortex dynamics. In particular our
aim is to construct the low-energy effective action describing the fluctuations of the orientational
moduli parameters on the vortex worldsheet, generalizing the results found several years ago in
the context of U(N) models [1]-[3]. For concreteness and for simplicity, we start our discussion
with the case of the SO(2N) U(1) and USp(2N) U(1) theories, although our method
× ×
is quite general. In the case of the SU(N) U(1) theory our result exactly reduces to the
×
one found earlier; furthermore we shall obtain the effective action for a few other cases with
higher-winding vortices in U(N) and SO(2N) theories.
2 Self-dual vortex solutions and the orientational moduli
The models
(with G’ x U(1), G’ = SU(N), SO(N), USp(2N),... gauge groups)
Our system is a simple generalization of the Abelian Higgs model with quartic scalar potentials
1 1 e2 v2 2 g2 2
= F0 F0µν Fa Faµν + ( q )† µq q† t0q q† taq ,
L −4e2 µν − 4g2 µν Dµ f D f − 2 f f − √4N − 2 f f
! !
! !
! !
! (1) !
! !
! !
! !
to a•g eneral class of gauge groups G! U(1) where G! is any simple LΠie1(gGro’ uxp U. (T1o))c ≠on c1retize
Natural generalization of BPS Abelian Higgs model
×
our idea let us consider two classes of theories G! = SO(2N), USp(2N) with any N 1.
• ≥
Bosonic sector of N=2 SQCD (control with Seiberg-Witten soln)
The repeated indices are summed: a = 1, . . . , dim(G!) labels the generators of G!, 0 indicates
•
the AbUel(iNan) gcaausgee sfiteulddi,efd =ext1e,n.s.i.v,eNly l➯ab e lEsxtahme mplaetst eorf fl aSvUo(r2s )(x“Usc(a1la)r quark” fields), all of
f
them in the fundamental representation of G!.1 The covariant derivatives and the field tensors
•
Generalizations (General gauge groups and vortex moduli space -- non-trivial complex
1
We adopt the convention where the metric η = diag(+, , , ).
µν
manifold; semi-local vortices; fractional vortices, no−n −BP−S vortices, interactions and stability,
higher-winding vortices, ... )
1
‘03-’11
Not
going to discuss all this
• Here:
Nature of the orientational zero modes
How they transform (GNO duality !); higher-winding cases subtle
How they fluctuate (Worldsheet effective action)
Tuesday, May 31, 2011
Methods of analysis
•
The standard field equations of motion

(most physical, and intuitive; standard differential eqs, Taubes eqs., existence, stability analysis )
•
The moduli-matrix

(vortex moduli as complex manifolds; transformation properties )
•
The Kähler-quotient

(group-theoretic aspects)
Tuesday, May 31, 2011
Effective worldsheet action
Gudnason, Jiang, Konishi 2010

A global continuous symmetry broken spontaneously
➯ a massless (
Nambu-Goldstone) particle
2 2
V ( φ ξ)
∝ | | −
Im Φ Nambu-Goldstone modes ( Cfr. π mesons
of SU(2)×SU(2) /SU(2) )
Re Φ “vacuum”
z
Orientational zeromodes are sort of Nambu-Goldstone modes propagating along the
vortex axis only (no symmetry breaking in the bulk)
t
Vortex worldsheet
t
z
⏎
Tuesday, May 31, 2011
BTW
A background melody of these studies ...
Rich structures from
degeneracy (moduli) &
symmetry breaking
in particular:
Double structure
of
vortex moduli
over
vacuum moduli
(NOT the main theme of this talk)
Fig. 1: Vacuum moduli , fiber F over it, and possible singularities
M
Ansatz, as is well known [15, 16], see Eqs. (A.5), (A.6). The other equations (the gauge field
Tuesday, May 31, 2011
equations) reduce, in the strong coupling limit or anyway sufficiently far from the vortex center,
to the vacuum equations for the scalar fields. In other words, the vortex solutions tend to sigma
model lumps.
3.1 Structures of the vacuum moduli
Let us first consider what we regard as the global aspect of our vortices. More precisely, our first
concern is the vacuum moduli on each point of which the vortex solutions are defined. Let
M
the symmetry group of the underlying system be
K = L G , (3.1)
F
⊗
where L is the local gauge group, while G is the global symmetry group. LetM be the manifold
F
of the minima of the scalar potential, the vacuum configuration M = q q†TIq = ξI . The
i
{ | }
vacuum moduli is given by the points
M
p = M/F , (3.2)
∈ M
where the fiber F is the sum of the gauge orbits of a point in M
f F = qg qg = gq , g L/L , (3.3)
0
∈ { | } ∈
4
1 Introduction
The last several years have witnessed quite an unforeseen progress in our understanding of
non-Abelian vortices, i.e. soliton vortex solutions in four (or three-) dimensional gauge theo-
ries possessing exact, continuous non-Abelian moduli. These continuous zero-modes arise from
the breaking (by the soliton vortex) of an exact color-flavor diagonal symmetry of the system
under consideration. The structure of their moduli, the varieties and group-theoretic properties
of these modes as well as their dynamics, and the dependence of all these on the details of the
theory such as the matter content and gauge groups, etc. turn out to be surprisingly rich. In
spite of quite an impressive progress made in the last several years, the full implication of these
theoretical developments is as yet to be seen.
1 Introduction
1 Introduction
In the present work we turn our attention to the low-energy vortex dynamics. In particular our
aim is to construct the low-energy effective action describing the fluctuations of the orientational
The last several yeaTrhsehlaasvtesewvietrnalesyseeadrsqhuaivtee waintneusnsefdorqeuseiteenanprougnfroersesseienn opruorgruesnsdienrsotuarnduinndgersotfanding of
1 Introduction
non-Abelian vortices, i.e. soliton vortex solutions in four (or three-) dimensional gauge theo-
non-Abelian vortices, i.e. soliton vortex solutions in four (or three-) dimensional gauge theo-
moduli parameters on the vortex worldsheet, generalizing the results found several years ago in
ries possessing exact, continuous non-Abelian moduli. These continuous zero-modes arise from
ries possessing exact, continuous non-Abelian moduli. These continuous zero-modes arise from
The last several years have witnessed quite an unforeseen progress in our understanding of
the context of U(N) models [1]-[3]. For concreteness and for simplicity, we start our discussion
the breaking (by the soliton vortex) of an exact color-flavor diagonal symmetry of the system
the breaking (by the soliton vortex) of an exact color-flavor diagonal symmetry of the system
non-Abelian vortices, i.e. soliton vortex solutions in four (or three-) dimensional gauge theo-
under consideration. The structure of their moduli, the varieties and group-theoretic properties
with the case of the SO(2N) U(1) and USp(2N) U(1) theories, although our method
under consideration. The structure of their moduli, the varieties and group-theoretic properties
ries possessing exact, continuous non-Abelian moduli. These continuouosf ztehreose-mmooddeess aasriwseellfraosmtheir dynamics, and×the dependence of all these on the de×tails of the
of these modes as well as their dynamics, and the dependence of all these on the details of the
is tqheuoriytesucgheans ethreaml.attIerncotnhteent canadsgeauogef gtrohueps,SetUc. t(urNn o)ut to bUe su(r1pr)isintghlyeroicrhy. Ion ur result exactly reduces to the
the breaking (by the soliton vortex) of an exact color-flavor diagonal symmetry of the system
×
theory such as the mspaitteteorf qcuointeteanntimanpdresgsaivuegperoggrroeusspms,aedteci.nttuhrenlaosut tsetvoerableyesaurrsp, rthiseinfuglllyimripclhic.atiIonn of these
under consideration. The structure of their moduli, the varieties and group-theoretic properties
one found earlier; furthermore we shall obtain the effective action for a few other cases with
spite of quite an imptrheesosrievteicaplrdoegvreelosspmmeantdseisinasthyeetltaostbesesveeenr.al years, the full implication of these
of these modes as well as their dynamics, and the dependence of all these on the details of the
theoretical develohpmigehntIesnrti-shweapsirneysdeetnitntowgobrekvwoseerettnui.rcneosuriantteUntio(nNto t)healonwd-enSergOy v(or2teNx dy)natmhices.oIrnipeasr.ticular our
theory such as the matter content and gauge groups, etc. turn out to be surprisingly rich. In
aim is to construct the low-energy effective action describing the fluctuations of the orientational
spite of quite an impressive progress made in the last sIenvtehraelpyreeasresn,ttwheorfkulwl eimtuprlincaotuiornatotfenthtieosne to the low-energy vortex dynamics. In particular our
moduli parameters on the vortex worldsheet, generalizing the results found several years ago in
theoretical developments is as yet to be seen. aim is to construct the low-energy effective action describing the fluctuations of the orientational
the context of U(N) models [1]-[3]. For concreteness and for simplicity, we start our discussion
moduli parameter2s on theSvoertlefx-wdorludshaeelt, gvenoerarlitzinegxthe sreosulltsufotunidosenvesral ayeanrsdagotinhe orientational moduli
In the present work we turn our attention to the low-energy vortex dynwaimthictsh.e IcnaspeaorftitchuelaSrOo(u2rN) U(1) and USp(2N) U(1) theories, although our method
× ×
the context of U(Ni)s mquoitdeelgsen[1er]-a[l3. ].InFtohreccoansceroeftetnheesSsUan(Nd )for sUim(1p)licthiteyo,ryweousrtarerstuoltuerxdacistclyusrseidounces to the
aim is to construct the low-energy effective action describing the fluctuations of the orientational
×
with the case of theoSneOf(o2unNd)earlieUr;(f1u)rtahnerdmUoreSwpe(2shNal)l obtUain(1th)etheffeeocrtiievse,aaclttiohnoufogrhaouferwmotehtehrodcases with
moduli parameters on the vortex worldsheet, generalizing the results found several years ago in
are defined in the standard manner
× ×
higher-winding vortices in U(N) and SO(2N) theories.
is quite general. In the case of the SU(N) U(1) theory our result exactly reduces to the
the context of U(N) models [1]-[3]. For concreteness and for simplicity, we start our discussion
×
with the case of the SO(2N) Uq(1)=and∂UqSp+(2oNnieA)fouqnUd,(1ea)Frtliheero;r=ifeusr∂,thaelAtrhmoourgehw∂oeurAshmaeltl+hoobidt[aAin t,hAe eff]ec,tiveAacti=onAfo0rta0 f+ewAotahtear ,cases with
µ f µ f µ µν µ ν ν µ µ ν µ
× × µ µ
D −
higher-winding voOrtiu2cers sinySsUetl(efN-md)uaisandlaSvsOoimr(2tpNexl)estghoeelonurietersi.oalnizsaatniodntohfethoerieAnbtealtiaionnHaligmgsodmuoldi el with quartic scalar potentials
is quite general. In the case of the SU(N) U(1) theory our result exactly reduces to the
(2)
×
one found earlier; furthermore we shall obtain the effective action for a few other cases with
2
with the normalization as follows 1 1 e2 v2 g2 2
higher-winding vortices in U(N) and SO(2N) theories.
0 0µν a aµν † µ † 0 † a
2 Self-dual =vortex sFoluFtions and tFheForien+ta(tionqal) modquli q t q q t q ,
µ f f f f
µν µν f f
LOur s−y1ste4mei2s a simple gene1ral−izati4ongo2f the Abelian Higgs moDdel with quaDrtic scalar−poten2tials − √4N − 2
2N ! !
a b ab 0
Tr t t = δ , t . (3) ! !
! !
2 1 ≡ 1√4N e2 v2 2 g2 2 ! (1) !
2 Self-dual vortex solutions and the orientat=ionaFl 0mF0oµνduli Fa Faµν + ( q )† µq q† t0q q† ta!q , ! ! !
! "L −4e2 µν − 4g2 µν Dµ f D f − 2 f f − √4N − 2 f !f !
Construction of L ! !
! !
To allow the system to possess a vacuumeffwith the maximally color-flavor locked sy!mmetry, we !
! (1) !
Our system is a sitmoplae ggeennerearliazalticolnaosfsthoefAgbaeuliagneHgigrgosumposdeGl w!ith quUa!r(ti1c )scawlahreproete!nGtia!lsis!any s!imple Lie group. To concretize
! !
assume that number of matter flavors is N = 2N. The squark fields q can t×hen conveniently
to a genefral class of gauge groups G! U(1) where G! is any simple Lie group. To concretize
2
1 our id1ea let us consider tw×o clae2sses of theovr2ies G!g2= SO(22N), USp(2N) with any N 1.
be represented as a c=olor-flavFor0 mF 0ixµνeodurmideaatrlFeitxauoFsfcaodµnνismi+deern(tswiooqncla2)s†sNes oµfqth2eoNrie,s Gthq!e†=tc0oSqlOo(r2N(fl)a,vUoSr)p(i2nNd)exwiqth† atnayqN , 1.
Our system is a simple generalization of the AbeliaLn Hig−gs4meo2deµlνwith qu−ar4tigc2scaµlνar potentiaDlsµ f D ×f − 2 f f − √4N − 2 f f ≥ ≥
running vertically (horizontally).BTyThcThhehoeeorrsveeinpapegcaeuttaehudetmeipndlduiicnsienssiwagdrnheicifscouehrmsamwlaleerdo:efwatsho=uerkmU1,(im.1n.).2eN,ddis!!i:mca(GhGfa!=!=)rf1aal,a2ccb,t1 t.oe.e.l,r sr s2.it,zNh.oeend.gee,!!nfibdenyrdaitmstohrase(so!oGflGu!t!)i,o0nliano!bdfitcehaltesesfotrhme3 generators of G!, 0 indicates
2 ! (1) !
= 1 F 0 F 0µν 1 F a F aµν + ( q )† µq e2 q† t0q the Avb2elian gaugge2 fiqel†d,tafq=21,,. . . , Nf labels t!!⊂he matter flavors (“s!!calar qua!rk” fields)!, all of
L −4e2 µν − 4g2 sµqνuark vacuDuµmfexDpteocatfag−teinoen2rav!lacflluasest(fohVf−teEghaeV√Amug)4bieneNegitθlrh!φioeauf−npu(rnsd)g2Ga1ma! uenfgtaUel r(fie1f0p)erelwsdehn,etarfteioGn=e!oifθisφG1a!,.n(1ry.T).sh+i.me ,cpφoNlvea(rLrfia)ienltagdrbeoreuivplas.teivTitθeohsφeacno(drnmc)trhaeettifiφtzeeeldr(rtefl)nasovrsors (“scalar quark” fields), all of
1 N! ! 1 (Also f2or SU(N); higher- 1 2
By choosing the plus sign for all of the U(1)N ! G! facqto=rs, one!finds a s!o×lut(1io)n!of the f=orm3 1 + − T ,
our idea let!!⊂us cotnhsiedme1rW!teiwandoop!!tctlhatv0hseesecfosnuvo!enfBntdtyiohφancehmwo(ohrroieee!r)sesin1ntGthgea"t!mhl=eetrrpieclSupηsOµrνsei(g=2snNedfinoa)rg2t,(aa+Ullt,wSoiifon,ptdn(hi,n2egNoU )v.f2o()N1rGtw)iNcie!t.sh1) aGnTy! fhaNcet2ocrso, v1ona.erfiiandnsta sdoelurtiiovnaotfitvheesforamn3 d the field tensors
2 N − − −
q = 1 . ⊂ (4) ≥
2N
otouraigdeenaelreatl uclsascsoqnosf=idgea!rugteewiθogφrco1l0u(arpsss)e1GsNo! f×tφhUe(o(r0r1i)e)1swBGh"y!ertTc==ehhheoGeSeoA!rsiOθeibinpφse(gel21aia(antNthrnyee))d2sg,+piaimUnluudpφgSsile2cepAse(ifi(Lgsri%2en1i)al=NedW1rf&,ego2)12rerNfsowu"aua+im=lipjldt√hx.rmooe12fjTpe2ai,tθd[ontNh.(φ:y.1ect1a.ohNU(−n,er=Nc()rf2c1e−1o()tqrN,inlz1a.)φv=e.b).⊂e2e.1n(l,s2rtGeNdi)tioi!hθTm+nφfea1,(cmw((Gt1raoh)!tr−)e1tsrNe,learfobNnflteAehal1sv(efiortn0rmh)ds)ese(Tgt“aer]snsiccoe,a=lrluηaatertµioioqνθrnφus=a1oo(rffrktGd)”h+i!efi,aef0gφoldr2(ismn(+)rd,3)i,ac1−al2ltNe,osf+−,ei−θφ)1.(r(1)0−) φ2(r)T ,
Performing2 a BNogomol’nyi completion one obtains th≥ef BPS! (or s0elf-duaφl) (erq)u1at"ions 2 2
The repeated indices ar1e sumxmj ed: a = 1, . . . , dimqt(hG=em!) lianebitθehφlse1t(fwhurehn)ed1greaNenmeernattoarl0sroepf rGes!e, n=0tainetidioθincφa1ot(efrsG)!1+.1 φTx2hj(er)co1varia+n2tediθeφriN1v(arti)ve−s φan2d(rt)hTe fi,eld tensors
2N
A = " [(1 f(r)) 1 + (1 f (r)) T ] , (10)
the Abelian gaugeifield2, fij=r21, . .−. , Nf labe2lsNthe m1aW−tt!eeraNdfloAapvto0trhse (c“osncvaenlφatir2on(qruw)ah1rekNre”"tfiheelmdse)t,riacAlηlio=f =2¯2d"qiiaj=gr(2+[0(,1,−, f, (r)).) 12N + (21 − fNA(r1)) T(5] ), (10)
µν
them in the fundamental representation of G!.1 The cov1arianxtj derivatives and the field tensors DT = dia−g −(1N−,{q, A1}N le)av,es U(N) (11)
−
where Ai = "ij [(1 f(r))e122N + (1wherfeNA(r)) T ] , invariant (10)
2 r2 − −
1We adopt the convention where the metric η = diag(+, , , ). F 0 Tr(qq†) v21 = 0 , (6)
µν − − − and12z,−r, θ√are cylindrical coo−rdinates. The appropriate boundary conditions are
4N
T = diag (1 , 1 ) , (11)
whTer=e diag (1 , 1 ) , (11) N N
N N −
− g2 v ! "
1 Fφa ta( ) = qq† , Jf†((qq)†)=TJf =( 0) ,= 0 , φ (0) = 0 , ∂ φ (0(7))= 0 , f(0) = f (0) = 1 .
1,2 and z, r, θ arNeAcylindrical coordin1ates. The approrpri2ate boundary conditions aNrAe
and z, r, θ are cylindrical coordinatebso. uTndhaeryappropr1i2ate b∞−oun4da√ryT2cNo=n−dditiiaongs(∞1are, 1 ) ∞, (11)
N N
−
conditions ! "v (12)
v where 2 ¯ + i and z x1 + ix2 is theφs1t,a2(nda)r=d compl,exfc(oor)d=infaNteA(in )th=e 0tr,anφsv1e(0rs)e= 0 , ∂rφ2(0) = 0 , f(0) = fNA(0) = 1 .
φ ( ) = , f( ) =afnd1z(, r,)θ=2ar0e c,ylinφdr(i0ca)l=co0or,dina∂teφs. (T0h)e=ap∞0pr,oprfi√a(t02e)Nb=oufndar∞(y0c)o=ndi1ti.on∞s are
1,2 D ≡ DNA D ≡1 r 2 NA
∞ √2N ∞ ∞ By going to singular gauge,
plane. A glance at Eq. (1) reveals that the BPS-saturated tension [4] (12)
v C-S Lin and Y. Yang ’10
(12)
φ ( ) = , f( ) = f ( ) = 0 , φ (0) = 0 , ∂ φ (0) = 0 , f(0) = f (0) = 1 .
1,2 ∞ √2N ∞ NA ∞By going tqo s1ingduliaarggaeu−gei,θ1r ,21 q , NA (13)
N N
Tuesday, May 31, 2011 2 Z →
By going to singular gauge, T = πv k , k , (8)
+ (12)
∈ # $
−iθ
q diag e 1 , 1 q , (13)
the vortex takes the form N N
→
−iθ
qBy godiinaggtoesing1ula,r1gaugqe,, (13)
is related to the U(1) windiNng oNnly. # $
→
φ (rth)1e vortex t0akes the formφ (r) + φ (r) φ (r) φ (r)
1 N 1 2 1 2
# $ q = = 1 + − T ,
2N
the vortex takes the fTorhmis last fact shows that a minimalqvo!rtdexiag0soleu−tiiθoφ1n (,cr1a)n1 b"qe ,construc2ted [5] by letting th(21e3)
N2 NN φ (r)1 0 φ (r) + φ (r) φ (r) φ (r)
→ 1 N 1 2 1 2
q = = 1 + − T ,
scalar field wind (far from the vortex axis) b1y anx#joverall U($1!) ph0ase rφot(art)i1on"with half2angle 2N 2
φ (r)1 0 φ (r) + φ (r) A = φ (r") φ[f((rr)) 1 + f (r) T]2; N (14)
1 N the vortex tak1es the for2m i 1 ij 2 2N NA
q = = 12N + −2 −r2 T , 1 xj
!(π),0and coφm(prl)e1tin"g (or cance2ling) it by a half wind2ing (+Aπ =or π" ) in[efa(crh) 1and+afll of(rth) Te ]C;artan (14)
2 N φin(rt)h1is gauge 0the whole φtop(orl)og+iicaφl −s(−trr2)ucitjurr2e arφise(sr2f)NromφtNh(Aer)gauge-field singularity along the
1 N 1 2 1 2
s1ubgrxojups U(1)N Gq!=. Depending on which si=gns are chosen in1 the+N U(1−) factorTs,,we find
2N
Ai = −2"ij r2 [f(r) 12N +⊂fNA(r) T!] ;vor0tex axiφs.2(Trh)e1BNP"Sienqtuhaitsiognasu2g(e5)t-h(e7)wfh(o1orl4et)hteopporloofigilceaflu2sntrcutciotnurseaarreisgeisvefrnom(inthbeogthauggaeu-figeelsd)sbinygularity along the
N
2 distinct solutions.
1 xj vortex axis. The BPS equations (5)-(7) for the profile functions are given (in both gauges) by
1 1
in this gauge the wh2oSleee tSoupbosleocg.ic3a.3l bsterluowcAt.uire=arises"ifjrom[fth(∂er)gφa1u2Ng=e-+fiefld(NfAsi(+nrg)ufTla]ri)t;yφal,ong the ∂ φ = (f(14)f ) φ , (15)
−2 r2 r 1 NA 1 1 r 2 NA 21
2r 2r −
∂ φ = (f + f ) φ , ∂ φ = (f f ) φ , (15)
vortex axis. The BPS equations (5)-(7) for the profile functions are given (in both grau1ges) by NA 1 r 2 NA 2
1 e2 v2 2r 1 g2 2r −
in this gauge the whole topological structure arises from the gauge-field singularity along the
2 2 2 2
∂rf = φ + φ1 e2, v2∂rfNA = φ1 φ . g2 (16)
1 r 1 2 1 2∂−fN= φ2 + φ2 r , 2 1∂−f 2 = φ2 φ2 . (16)
∂ φ = (f + f )voφrte,x axis. The BPS equa∂tioφns =(5)-(7)(ff%or thfe pr)roφfirle,fun&c2t(io1n5s)1are g2iv−enN(in both gauges) rbyr NA 2 1 − 2
r 1 NA 1 r 2 NA 2 # $
2r 2r − % &
# $
1 e2 v2 1 T1he above is ga2particular vortex solution with a1fixed (++. . . +) orientation. As the system
∂ f = φ2 + φ2 ,∂ φ = (f ∂+ff )=φ , φ2 Tφhe2abo.ve is a p∂a(r1tφi6c)ul=ar vorte(xfsoluftion w) φith a, fixed((1+5)+. . . +) orientation. As the system
r r 2 1 2 − N r 1 2hras arn erxaNNcAAt SO1(22N)1 − o2r USp(2N)r 2 col2orr-flavo−r diNagAonal2(global) symmetry, respected
% & 2 hCa+sFan exact SO(2NC)+F or USp(2N) color-flavor diagonal (global) symmetry, respected
1 e2 v2 1 C+F g2 C+F
# $
∂ f = 3 φ2 + φ2 , ∂ f = φ2 φ2 . (16)
It is convenient to work with the skew-diagonal basis for the SO(2N) group, i.e. the invariant tensors are
The above is a particular vortex solutiorn wr ith a fi2xed (1++.2. .−+N) orienta3tItioisn.coAnvsentihenert stoyrswtoeNrmkAwith t2he skew1-d−iagon2al basis for the SO(2N) group, i.e. the invariant tensors are
% &
taken as
taken as # $
has an exact SO(2N) or USp(2N) color-flavor diagonal (global) symmetry, respected
C+F C+F
The above is a particular vortex solution with a fixed (++.0. . +1)Norientation. 0As t1hNe system
3 J = , J = , (9) (9)
It is convenient to work with the skew-diagonal basis for the SO(2N) group, i.e. the invariant tensors are
has an exact SO(2N) or USp(2N) color-flavor dia!g"o1nNal (g0lo"bal) sym!m"1eNtry,0re"spected
C+F C+F
taken as
3 where " = for SO(2Nw) haenrde "U=Sp(2foNr S) Ogr(o2uNps),arnedspUecStipv(e2lNy. ) groups, respectively.
It is convenient to work with±the skew-diagonal basis±for the SO(2N) group, i.e. the invariant tensors are
0 1
N
takenJas= , (9)
3
!"1 0 " 3
N
0 1
N
J = , (9)
where " = for SO(2N) and USp(2N) groups, respectively.
± !"1 0 "
N
where " = for S3O(2N) and USp(2N) groups, respectively.
±
3
by the vacuum (4), which is broken by such a minimum vortex, the latter develops “orientational”
by the vacuum (4), which is broken by such a minimum vortex, the latter develops “orientational”
zero-modes. Dezgeernoe-rmatoedevso.rteDxegsoelnuetrioantse cvaonrtienxdeseodlubteiognesnecraantedinbdyeecdolbore-flgaevnoerraStOed(2bNy )co(loorr-flavor SO(2N) (or
USp(2N)) traUnsSbfoyprmt(h2aeNtvioa)nc)usutmran(4s)f,owrmhiachtioisnbsroken by such a minimum vortex, the latter develops “orientational”
zero-modes. Degenerate vortex solutions can indeed be generated by color-flavor SO(2N) (or
−1 −1
q U q U , A UA −U1 , −1 (17)
USp(2N)) transformations q i U q Ui , A UA U , (17)
→ → i i
→ →
as by the vacuum (4), whiqch isUbrqokUe−n1b,y sAuch a mUiAnimUu−m1 ,vortex, the latter devel(o1p7)s “orientational”
as by the vacuum (4), which is broken by such ai minimuim vortex, the latter develops “orientational”
→ →
In a similar spirit, we study in a later section certain subclasses of vortices among given
zero-modes. Degenerate vortex solutions can indeed be generated by color-flavor SO(2N) (or
φ (r)1zero-mo0des. Degeneratφe v(orr)t+exφso(lurt)ions canφin(dre)ed bφe(gre)nerated by color-flavor SO(2N) (or
q = U 1 asN φ (r)U1 −1 = 01 2 1 φ+(r)1+ φ−(r)2 UTUφ−(1r,) φ (r)
windiInng-anusmimb!ielarrsos0lpuiqtrUiiUo=tSn,SspφUw,p(2te2((r2rNas)Nnt1)1usN))fdo)"tyrrtamrniaNnisnnfogsafromalrcaamcttoeiaorrtdniossinen2cgstiUtoon−s1ocm=ere2tNadi1enfinsuitbecilrar22sesdeusc1oib2fNlevo+rretpirce1essenatm−atoinong2 ogfivUenT U−1 ,
! 0 φ (r)1 " 2 2
φ (r)1 2 0 N φ (r) + φ (r) φ (r) φ (r)
thwein(dduinagl -onfutmh1eb)ercxojsloolru-qfltia=ovnoUsr, gtrrao1nuspf.orNming accordUin−g1t=o so1me defin2 ite 1irred+uci1ble r−epr2esenUtaTtUio−n1o,f
−1 2N
the (Adiu=al −of2t"hije)r2cAofilo(=rr-)fl1a2v1No"!r+ijgxfrjNo0uAfp(r(.)rU)φ1T22(UNr)1+N"fq,NA→q(→rUi)=UqUTU1q,U−2U1−.−21, 1 ,,Ai A→ii →U=A1Ui,U2A−.i1U2,−(118,) (18()17) (17)
# −21 rx2j $
Vortex of generic orientation
Actually, the full SAO(2=N) ("or USfp((r2)N1)) g+rofup d(ore)sU(sniTnogtUual−acr1t goa,nugteh)eis=olu1t,io2n., as the latter (18)
i ij # 2N NA $
aass −2 r2
4 Vortex Amctuoaldly,utlhie flfulul ScOtu(2aNt)io(onr Us:Spt(h2Ne))wgroourplddosehs enoettacat contithoensolution, as the latter
remains invariant under U(N) S#O(2N) (or USp(2N)). On$ly the coset SO(2N)/U(N)
Actually, the f⊂ull SO(2N) (or USp(2N)) group does not act on the solution, as the latter
4 (or UVSopr(2tNe)x/rUemm(Nain)os) daincvtusarnlioiann-tflφtruφiuv(niradc()lerl1tyr)u1Uona(iNtt, a)ino0dn0thSsuOs:g(e2tnNehr)aet(eoswφrphU(yoφrsSir)cp(al+r(ldl2)yφN+sdih(s)φtr)i.en)c(Oetrnts)olyluatthicoφentsc.i(o4rosφe)nt(SrφO) ((2rNφ) )(/rU) (N)
remainqsq=in=vUaUriant 1un1derNUN(N) ⊂SO(2NU−)U1(o−=r1U=S1p(21N)). 2Only2 1the c1+oset +S1O(21−N)/2−U(N2U) TUU−T1 U, −1 ,
2N 2N 4
As tAhen oapripernotpartiaioten(oaprla(oUrmramSUodepSte(rpsi2(zN2actNoi)no)/n!s/Ui!dUofe((rNNteh0de))0))cianaoccsteEtstsφ,qnno.2vφon(a(⊂n-r2l1ti-()dr8tri1rv)iN)iinva1r"lieNlaaypl"lrcoyeonsooiertndn,itiantna,edtxaetnahpcdutastthNgcheu2ansmiengr2cabeltnuueeds-rGipnaUhogt yel s⊂dstih csp eatSholOlayyn(bs2edioN-civsla)iet/kliUlneyc(Ntzd2s)ei,so rotlouirn- t2icotnss.o4lutions.
modseosl,utnioonth, ihnags bcaeAennnpkarnpeovpwernnoptfortrihasteoe1mm1pefartrxoiammjxmejefl(tcuraiclzltaeudtaiotthninegorfeidntuhcteihnecgosmspeatat,creivx-at)lii[md6,ei4,n]f,raomcooorndeinpaotUieSnptp(2atNtoc)ha/UniNon)tc h l uedr,ing the above
As the orientationAanl mapoprdoepsriactoenpsaidraemreedtrizinatiEonq.of(1th8e) croespetr,evsaelnidt ie−nx1aacctooNrdainmabteup-Gatochldisntcolundei-nlgiktehezearboo-ve
A = " f(r) 1 + f (r) UTU −,1 i = 1, 2 . (18)
Ai = i"j f(r)21N +NfA (r) UT U , i = 1, 2 . (18)
solution, ihas−b2een kirjn2own for som2Ne timeN(Acalled the reducing matrix) [6, 4],
wmithodaens,arnboitthrainriglycsa1msnolaupltlrieoeBnvx,ep†nheatns−dtbhXiet2eeu−mnr1ekfnorroof2wme0nnefflorurgcsyot.1muHaetotiwinm0geevie(ncra,tllthehdeetysXhpae−arrc1eeedn-utocitminBggee,†mnYfaurto−irnmi1xe)No[6na,em4]pb, oui-nGtotlodsatnonoteher,
N 2 # N 2 $ 2
U = − # = − $ , (19)
mwoditehs,anasartbhietrwvaarh!icleuy0rusemAmAcatc1iulttNlusaeelallx"lyflp,y!ie,tsnhtedsh01y1iefNtmNuuflumrleYlSleot−BOSBfr12i†Oe(c†"2n(Ne!u2rnNXgB)dXye−(.)ro−21Hr1(21SoNoUrOw"0SUe(vp02Se(Nr!2p,NB()1t2ChNX)Ne+)1y−FNg)a120r)oroegurrp0noUYoudtSpo−gep21dseX(non2−euo"Ns12itnXnea)oCc−Ntt+12BaaoFmcn†:tYbto−uthhnBe-21Ges†ytooYhlleaud−rtssei21tooonlun,teaiosnt,haeslatthteerlatter
remaiUnUs =in=variant −−under U(N) SO(2N) (or U=Sp(=2N)). O−nly th−e cose,t SO((1,29)N)/(U19()N)
mmasoswdivheeesr,ematohsdetemhsaetinrvicaetcshueXurem4ma-ndadiitimnsYeselnfians!rvii!eosadn0r0esaifiyalnnmestd11mpuNNabencytd"er"-eit!cr!imUu0en(0Ndbeu⊂r)YlkSY−. 12O−"ST(12O!2"heN(B!y2N)Bp1r)Nop("1oarNgoaUr"t!eSUBpfSXr(!ep2−eBN(l12y2X)No)−n.)Y12lOy−na21llyo:Ynt"hgt−eh21techyoes"eatreSO(2N)/U(N)
⊂ C+F C+F 4
(or USp(2N)/U(N)) acts non-trivially on it, and thus generates physically distinct solutions.
4
vortex-axis and in tim(eo.r TUoSspt(u2dNy t)h/eUse(Nex)c)itaecdtsmnoodne-striwveiaslleytotnheit,maondduUtl=ih u“prseadrguaceimnnge ertaeDtreeslsduBpc,hVaytleosnitc b’a8e5lly distinct solutions.
massive modesawninhdwerthehereteht4eh-emdmiamatrteirncicesesisoXXnalaannsddpYaYceaa-retriedmedfieenfiebnduebdlky.by They propagatmeatrfixr”eely only along the
An apXpropr1iate+pBar†aBm,etriYzation1 of+thBeBco†s,et, valid in a coordinate(2p0a)tch including the above
(quantum) fields of theAfnorampp≡roprNiate parametriz≡atioNn of the coset, valid in a coordinate patch including the above
vortex-axis and in time. To study these excited modes we set the moduli parameters B to be
solution, has been knoXwn fo1r s+omBe†tBim,e (Ycalled1 th+eB =rBe adBnut†icsyi,nmgm mNaxtNr i xfo)r [ 6SO, 4(2]N, ); (20)
N † N †
solution, has been kXno≡wn 1for +somBeBtim, e (≡Ycalled1the+reBdBucin,g matrix) [6, 4], (20)
in terms oAf allnowN theN zecormomploexdmesa ttroix flαBu,cbtueiantge:anNαtisymme3tric 0for SO(2NN ) saynmdms NymxNm efotrr i cUfSopr(2N);
(quantum) fields of the forBm= B(x ) , ≡ x = (x , x ) . ≡ (28)
×
USp(2N). Noteitnhtaetrmthseomf aantrNix (19N1)NicnodmeepdBlesx†amtisafitreXisxt−Bh21e, bdeeifin0nginagntpisryo1mpNemrteiters0icthf1oxer NtSw-O1o (fgo2rXrNo Su−U)p(12saNn)d symBm†Yetr−ic21for
When this expresisnioUtneSrmips(s2souNfba)s.ntUBiNtNUuo=t=te×e=×dt!hBNai0tn(1txctNohoαem−t)m1hp−N,aeletBxra"ixcm†t!(ia1xot9nαr)0iXxin=dB−de(e4,21Ydxxb3s−eLa,i12ntx,i0"gsfi0hae)!osnw.tBtihesevy1emdNr1e,mfiNnoe"intn0regi=cpimrf!oo=pmrBeSreXtdOieis−a(Xt2t12ehN−ley−12t)wnaoYongt−d−eros21B(suy2pm†8sY)"m−e,t21ric f,o(r19) (19)
that buUt Sthpe(n2N). NoUte−t1h=a!tUt0h†e, mU1aNtTriJx"U(!1=9)J0in,deedYs−a12ti"sfi!esBthe d1eNfin"ing p!roB(p2Xe1r)t−ie12s the tYw−o 21grou"ps
!
When this expression is substituted into the action d4x , however, one immediately notes
where the matrices X andUY−1ar=e dUe†fi,nedUbTyJU = J , (21)
2N L
1
with the respective iwnvhaerriaentthteenmsoartr(i9c)e.s TXheamndatrY−ix1aBrepdaer†afimneetdribzTeys the “Nambu-Goldstone”
that ∂ q 2 +U = U!F, U2 J,U ➞= ∞J ,energy (29) (21)
α f iα
with the respective |invaria|nt tensor (92).gT2|he†ma|trix B parametrizes th†e “Nambu-Goldstone”
modes of symmetry breaking (b#y the vortex) X 1 + B B ,$ Y 1 + BB , (20)
α"=0,3 f"=12N i"≡=1,2N 1 † ≡ N † ⏎
witmhodtheseorfessypmecmteivtrey ibnrvea∂arkiianqngt(b2tyeX+ntshoervo(19rtN)e.x+)ThBeFmBa2t,rix,YB pa1raNm+etrBizeBs th,e “Namb(u2-9G)oldstone” (20)
leads to an infinite excitation energy, wαherfeas on≡e knows thiaαt the sy≡stem must be excitable
| | 2g2| |
SOin(2teNrm) s ofUan(#NN) , N ocormpleUxSmpa(t2rNix )B, bUein(Ng$a)n,tisymmetric for S(2O2)(2N) and symmetric for
modes of syα→"m=m05,3etrfy"=b1×reaking (by tih"=e1v,2ortex)→
without mass gap (classically). SO(2N) U(N) , or USp(2N) U(N) , (22)
leads4AtTsouewsadaansy,s Mtiunadyfii 3en1d,i 2tin0e1U1dieneSxtatcpieli(tri2amnNtRsioe)ofn..f [Na4e]nno, tetNehregtyhv,oartNtwe→txhhcmeeorommedauapsltilresoixpxnacm(ee1a9kint)nrSiionxOwdB(se2eN,dthb)saea(tiotnrisgtUfihaeSenspts(t→i2yhsNysetm)de)memtfihenetmoirnriiugecsspftiosrrobapSeerOetxi(ec2siNttahb)elatewnod gsyromumpsetric for
×
4
Tnhone-twrivaiyal hcoomwpltehx emsaynUiAsfotsSledwm,pars(er2qsetuNauirdciin)te5gsd.SatNinOtooldte(teah2etsaeNtitl2shNip)ana−tRc1teeh(f-.oUter[i4m2(m],NNeta)htd)elroeiv,cxpoarlet(cen1oxd9omr)eodnoriindntuadctliehesenapUdenaicggSseheabipntooi(rfsSh2fiotONohe(dse2s)Ntm(hp)aoet(dcodUhrueelUsfi(i)NSn.ppiTan()rh2gaeN,mp)r)eottpheeerorsrt,ieisesistahe tw(o22g)roups
without mass gap (classically).
non-trivial complex manifold, req→uiring at lea−st12N−1 (†or 2N) lTocal coordinate→neighborhoods (patches). The
moduli space structure is actually richer, as these vortices posseUss semi=-locUal m,oduUli (reJlaUted=to tJhe,size and shape (21)
can be found by an 4appropriate generalization of the procedure adopted earlier for the vortices
mAodsuwliaspsascteusdtriuecdtuirne idseatcatuilalilny rRicehfe.r,[a4s],thtehsee vvoorrtitceexs pmososdesuslsiemspi-alcoecalinmoSdOuli(2(rNela)ted(otro UtheSspiz(e2aNnd))shathpeeories is a
moduli) as well, besides the orientational moduli under consideratio−n1here, ev†en with Tthe minimum number of
The way how the system reacts to the space-timUe dep=endUen,t chUangJeUof=theJm,oduli parameters, (21)
nonm-tordiuvliia)lacsowmeplll,ebxesmidaensitfhoeldo,rireenqtuatirioinnagl amtoldeualsitun2dNer−c1on(soirde2raNti)onlohcearle,ceovoerndiwniathtethneeimghinbiomruhmoondusm(bpeartocfhes). The
in Ufl(aNvor)s ntehedeeodrfioersa. cAolowrk-ifletahyvotorhbleoscekrreedvsapptehicaosteni,vie[n1ci]no-[nv3tar]raisiatsnttototthieennotsrriogorinda(ul9cU)e.(NnTo)hnme-otmdreiavl.tiHrailexrgeBawuegpcoeanrsfiaidmeelrdeotnrciloyzemtshpetohnee“nNtsa,mbu-Goldstone”
can be found by aflnavaorpspnereodpedrfioartaecogloern-flearvaorlilzocakteidopnhaosef, itnhceonptrraostcteodthueroerigaindaol pUt(eNd) meaodrelil.eHrefroerwetchoensivdoerrotniclyetshe
moduli space structure is actually richer, as these vortices possess semi-local moduli (related to the size and shape
orientational moduli related to the exact symmetry of the system.
A , to cancel the largmewoiedtxhecsittohafetsiyormensmpeenectetrriygvyebrfiernaovkmairni(ga2n(9tb)y.tetAnhsenovar¨oıv(r9tee)xg. )uTeshsewmoualtdrixbeB parametrizes the “Nambu-Goldstone”
α orientational moduli related to the exact symmetry of the system.
in U(N) theorimeos.duAli) kaseyweollb, sbeersivdaestitohne o[1ri]e-n[t3a]tiiosnatlomiondturloi duundceer ncoonnsi-dterriavtiiaonl ghaerue,geevefinelwdithcotmhepmoinneimnutms, number of
modes of symmetry breaking (by the vortex)
flavors needed for aAcolo=r-flavoirρlo(crke)dUph−a1se∂, inUcon,trast to the original U(N) model. Here(w3e0c)onsider only the
4
A , to cancel the large excitatioαn eSnOer(g2yNfr)om (U29()N.α)A,na¨ıveorguessUwSopu(l2dNb)e U(N) , (22)
α − 4
orientational moduli related to the ex→act symmetry of the system. →
with U of Eq. (19) and4Asosmweaspsrtoudfiileed fiunSndOcetti(ao2ilnNinρ).ReTf.h[U4i−s],1(htNhoew)ve,ovreterxdmoooerdsulniostpUawcSeoprink(.2SNOT(h)2eNp)r(Uoobr(leUNmS)pi(,s2N)) theories is a (22)
A = i ρ(r) U ∂ U , (30)
→ →
α α
non-trivial complex manif−old, requiring at least 2N−1 (or 2N) local coordinate neighborhoods (patches). The
that even though 4As was studied in detail in Ref. [4], the v4ortex moduli space in SO(2N) (or USp(2N)) theories is a
with U of Eq. (19)maonddulissopmaceesptrruoctfiulree ifsuanctcutailolynriρch.erT, ahsitshehseowvoervtiecers pdoosesesssnsoemt iw-loocrakl m. oTduhlie(rpelraotebdletmo thiessize and shape
Xn−on1-Btriv†∂ial cBomXpl−ex1 man∂ifolXd, 1reXqu−iri1ng at least 2NX−1−(o1r∂2NB)†lYoca−l1coordinate neighborhoods (patches). The
mod2uli) as αwell, besi2des theαorien2tation2al moduli under cons2ideαration her2e, even with the minimum number of
thiaUt −ev1e∂n Utho=ugih − − ,
α moduli space structure is actually richer, as these vortices possess semi-local moduli (related to the size and shape
% flavors needYed−fo12r∂aαcBoloXr-fl−av12or locked phaseY, in−c12oBntr∂aαstBto†tYhe−or12iginal∂Uα(YN12)Ymo−d21el&. Here we consider only the
modu1li) as well, besid1es the orien1tationa1l moduli under cons1id−eration her1e, even with the minimum number of
X− B†∂ BX− ∂ X X− X− ∂ B†Y −
orienta2tional mαoduli rela2ted to tαhe ex2act sy2mmetry of the system2. α 2
i U−1∂ U = i − − (31),
flavors needed for a color-flavor locked phase, in contrast to the original U(N) model. Here we consider only the
α
1 1 1 1 1 1
% Y −2 ∂αBX−2 Y −2B∂αB†Y −2 ∂αY 2Y −2 &
5 orientational moduli related to the exact symmetry of the system. −
Whereas in the far infrared, we expect that either the world-sheet effective sigma model will by quantum
4
(31)
C N−1
effects develop a dynamic mass gap (as the P model) or end up in a conformal vacuum – a possibility for
5 4
SO, UWShpertehaesoriinest[h1e4].far infrared, we expect that either the world-sheet effective sigma model will by quantum
C N−1
effects develop a dynamic mass gap (as the P model) or end up in a conformal vacuum – a possibility for
7
SO, USp theories [14].
7