Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD

1.9 MB·English

by  Mohamed M. Anber

Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD
Mohamed M. Anber,1 Erich Poppitz,2 and Tin Sulejmanpaˇsi´c3
1Institute de Th´eorie des Ph´enomen`es Physiques,
E´cole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland
2Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada
3Department of Physics, North Carolina State University, Raleigh, NC 27695, USA
(Dated: July 7, 2015)
WestudystringsbetweenstaticquarksinQCDwithn adjointfermions,includingN =1Super
f
Yang-Mills (SYM), in the calculable regime on R3×S1, which shares many features with the XY-
spin model. We find that they have many qualitatively new features not previously known. The
differencefromotherrealizationsofabelianconfinementisduetothecompositenatureofmagnetic
bions,whoseDiracquantumwithfundamentalquarksistwo,andtotheunbrokenpartoftheWeyl
group. In particular we show that strings are composed of two domain walls, that quarks are not
confined on domain walls, that strings can end on domain walls, and that “Y” or “∆” baryons can
5 form. By similar argumentation, liberation of vortices on domain walls in the condensed matter
1 counterparts may have important implications in the physics of transport. In the gauge theory we
0 briefly discuss the lightest modes of strings and the decompactification limit.
2

l
u While ubiquitous in nature, color confinement is one clidean) Lagrangian is:
J oftheleast-understoodfeaturesofYang-Mills(YM)the-
6 ory. Theoretically controlled approaches usually involve (cid:20) m2
= M (∂ σ)2+(∂ φ)2+ (cosh2φ cos2σ)
modelsthatdiffer,invariousways,fromreal-worldQCD. LB µ µ 2 −
] Nonetheless,one’shopeisthattheirstudywillrevealfea-
h + (n 1)V (φ)] . (1)
f pert.
t turesofconfinementthattranscendtheparticularmodel. −
-
p Araretheoreticallaboratorywhereconfinementisun- Thescalesandfieldsin(1)areasfollows. ThescaleM is
e der theoretical control within field theory is offered by oforderg2/L,whereg2 istheweak4dgaugecouplingat
h QCD(adj): an SU(N ) YM theory with a strong scale thescale1/L. Thescalem exp( 4π2/g2)M isnonper-
c
[ Λ and n Weyl fermions in the adjoint representation, turbative (4π2/g2 is the act∼ion of −a monopole-instanton)
f
4 compactifiedonR1,2×S1L withfermionsperiodicaround andexponentiallysmall. Withexponential-onlyaccuracy
v the spatial S1 of size L. U¨nsal showed [1] that for LN Λ (for pre-exponential factors, see [8, 22]) one can think of
L c
3 1 confinement is due to the proliferation of topolog- M as the cutoff scale of our effective theory and of m as
7 (cid:28)ical molecules—the magnetic bions. These are non-self- the mass scale of infrared physics.
7
dualcorrelatedtunnelingeventscomposedofvariousfun- The long-distance theory (1) has two bosonic fields.
6
damental and twisted [2, 3] monopole-instantons. For The field φ describes the deviation of the trace of the
0
1. small but finite LNcΛ, magnetic bion confinement ex- Wilson line around S1L from its center symmetric value.
tends the 3d Polyakov mechanism of confinement [4] to Equivalently, φ is the radial mode of the adjoint Higgs
0
locally-4dtheoriesqualitativelydifferentfrom3dtheories (the Wilson line) breaking SU(2) U(1). The field σ
5 →
1 with fermions where confinement is lost [5]. In passing, is dual to the photon in the unbroken Cartan subalge-
: we note that QCD(adj), bions and their constituents are bra (the τ(3) direction) of SU(2). In Minkowski space
v
studied in connection with deconfinement, resurgence, M∂ σ F(3) isthemagneticfield, andM(cid:15) ∂ σ E (3)
Xi theta-dependence and volume independence, e.g. [6–18]. is th0e e∼lect1r2ic field (where (cid:15) = (cid:15) and i,ijj=j 1,∼2).i
ij ji
−
r The goal of this paper is to study confining strings in As is clear from the discussion of scales, the terms in
a
QCD(adj)/SYM in the calculable regime. We find that (1) proportional only to M are perturbative. We shall
they have an interesting structure, to the best of our notneedtheexplicitexpression[1,8]fortheperturbative
knowledge not previously discussed in confining strings potential V (φ). This term is absent for SYM (n =1).
pert f
studies. We shall see that, while still abelian in na- Forn >1,V stabilizesφatthecentersymmetricvalue
f pert
ture, QCD(adj) strings retain more features expected in and gives it mass of order M, hence φ can be integrated
the nonabelian theory compared to other theories with out. The relative normalization between the potentials
abelian confinement. In addition the entire discussion for φ and σ given in (1) is for SYM.
relies only on the effective Lagrangians (1) or (3) below. Of most interest to us is the origin of the nonpertur-
Since these have been related to the condensed matter bative terms in (1). The nonperturbative potential for
systems [6, 19–21], most of our observations may have φ, cosh2φ, is due to neutral bions [8, 23] and will not
∼
directimportantconsequencesinthephysicsofthesesys- play an important role here (except for being the only
tems. sourcestabilizingφatthecenter-symmetricvalueφ=0in
At distances L massless QCD(adj) on R3 S1 dy- SYM).Theotherterm, cos2σ,isofutmostimportance
(cid:29) × L ∼
namically abelianizes [1]. Ignoring fermions for the mo- tous,asitcapturestheeffectofthemagneticbions—the
ment, for SU(2) gauge group the effective bosonic (Eu- leadingcauseofconfinementinQCD(adj). Thefactorof
2
Kink
1.0
e2i CA(3)
H
0.8
� winds by 2⇡
0.6
R
0.4
FIG. 1: Left: the Wilson loop and the monodromy of σ.
Right: Sketchoftheconfiningstringconfigurationσ¯ withthe
0.2
correct monodromy, composed of two domain walls. The dot
and cross represent probe quarks a distance R apart. The
maximum distance between the walls, of thickness 1/m, is d.
FIG.2: Theactiondensityoftheconfiningstringσ¯ obtained
by numerically minimizing, via Gauss-Seidel relaxation, the
twointheargumentofthecosinereflectstheircomposite action (1) with the correct monodromies. The lattice has
nature: they have magnetic charge two while fundamen- spacing 1/M, size 100×100, and M/m = 20. The classical
tal monopole-instantons have unit charge. This term is logR growth of the transverse separation from the model of
responsibleforthegenerationofmassgapforgaugefluc- Fig.1isalsoseentoholduponstudyingdifferentsizestrings.
tuations (mass m for the dual photon σ) and for the
confinement of electric charges. The theory (1) has two
vacuaσ=0,π,bothwithφ=0,correspondingtothespon-
Now,theσ¯-fieldconfigurationwiththerightmonodromy
taneousbreakingoftheanomaly-freediscretechiralsym-
has to be more complicated than a single domain wall.
metry (the R-symmetry in SYM).
To study it, we keep the time (y) extent of C infinite
ConfinementisdetectedbythearealawfortheWilson
and consider a finite spatial (x) extent R. As the σ¯ con-
loop in a representation , taken along a closed contour
R(cid:72) figuration has monodromy 2π across C, in this simple
C, W(C, ) Tr P exp(i A). For an SU(2) funda-
mental repRre≡sentaRtion, we nCeed to compute the expecta- one-field case it is clear that (since the periodicity of the
tion value of W(C,1) exp(i (cid:72) A(3))=exp(i (cid:82) B(3)). cos2σ potential is π) the string has to be composed of
2 ∼ 2 C 2 S two domain walls. To get a picture of the extremal con-
Here A(3) is the (electric) gauge field in the Cartan di-
figuration, consider Fig. 1, with parameters R,d defined
rection, B(3)=dA(3) is its field strength, and S is a sur-
in the caption. A sketch of a two-domain wall configura-
face spanning C (the omitted second contribution to the
tionisshown,withthesecondinfiniteworldvolumedirec-
trace of the fundamental Wilson loop gives an identical
tion (the time y) perpendicular to the page. The action
area law).
hastwoparts,excludingcontributionsfromthejunctions
Insertion of the Wilson loop in the dual language of
(subleading at large R): the tension of the two domain
the σ field (recalling that σ σ+2π) amounts to the fol-
∼ walls, proportional to twice their area (we take T(R+d)
lowing instruction [4]: erase the contour from the space,
as the area) and the wall-wall long-distance repulsion
and have σ wind by 2π for any contour which has link-
( e md). Thus, S MmT(R+d)+MmTRe md, up
ing number one with the Wilson loop—a 2π monodromy ∼ − ∼ −
to numerical factors. Extremizing with respect to d, we
(see left panel of Fig. 1). Take a rectangular contour in
find md logmR, a logarithmic growth of the trans-
the y−x-plane (y is Euclidean time) with span T (R) verse siz∗e∼of the confining string configuration with the
in the y (x)-direction. For infinite R and T, σ jumps
separation between the probe charges.
by 2π upon crossing the y x plane. If the potential in
− Remarkably, the above simple model captures the be-
(1) was—as in Polyakov’s original 3d SU(2) gauge the-
havior of the actual extremum of (1), shown on Fig. 2,
ory with an adjoint Higgs field—cosσ, the field configu-
including the logR growth of the transverse size. Our
ration extremizing the action (1) with the correct mon-
remarks so far also hold for deformed-YM theory [25],
odromy, which we denote σ¯, would be equivalent [4] to a
where,forθ=π[26]thesinglemonopolecontributionvan-
domainwallwithy x-planeworldvolume,whereσ¯would
− ishes.
change by 2π as z varies between . We would have
W(C,21) ∼ e−ΣstrRT, with string±te∞nsion Σstr propor- tanThcoemadpjooninenttfserhmavioenasnweeffree,ctsiovefaLr,agigrnaonrgeidan. T[1h]eir Car-
tionaltothedomainwalltension(forarecentreviewsee
[24]). (cid:104) mcosσ (cid:105)
The physical difference between monopole-instanton LF =M iλ¯σ¯µ∂µλ+ 2Mnf−1 [(λλ)nf +h.c.] . (2)
confinement in the 3d Polyakov model and QCD(adj)
on R3 S1—the fact that the magnetic bions have mag- We omitted, for brevity, a summation over the n flavor
f
×
netic charge two—is reflected in the cos2σ potential (1). indices in the kinetic term and a product over the flavor
3
indices in the interaction term (the ’t Hooft determinant
in the monopole-instanton background). The field φ is
also set to its vanishing vev. For SYM, apart from omit-
ting φ, (2) has correct normalization. It is, in fact, the
effect of the fermions on the confining string where the
differencebetweenSYMandQCD(adj)withn >1shows
f
up most profoundly.
InSYM,thefermionsaremassiveintheσ=0,π vacua.
They have exact zero modes in a single domain wall
background, with exponential fall off away from the 1) 2) 3)
wall. Because of the gap m in the bulk, the fermion
induced wall-wall interaction is expected to be exponen- FIG. 3: A sketch of how a qq¯ pair can fuse into the DW
tially suppressed, m2e cmd, c 1 (a calculation of the (from left to right). The shaded and white regions represent
−
determinant, requi∼ring some mi≥ld background modeling distinct vacua of the theory. The solid black line represents
even for parallel walls, yields attraction with c>1). The theBPS1DW,whilethedashedlinerepresentstheanti-BPS2
DW, while the arrows represent their electric fluxes. The
fermion-inducedexponentialinteractionatlargedisfur-
theraccompaniedbyan“(cid:126)” m loopsuppressionfactor, black dots are the quark and the anti-quark. The inlay in
∼M theupperleftcornershowsafundamentalstringendingona
hence the classical bosonic repulsion between the walls
DW.
Mme md dominates. Thus, in SYM the logarithmic
−
∼
growthofthetransversestringsizeisnotaffectedbythe
fermions. The logR growth of the string transverse size is a new scale on the string worldsheet, well below the
is reminiscent of the behavior of magnetic strings (ANO “glueball”—the bulk mass gap m for gauge fluctuations.
vortices) which confine monopoles on the Higgs branch The fact that the strings are composed out of domain
of N=2 SQCD [27]. However, the underlying semiclassi- walls(DW)–asituationoppositetowhatwassuggested
cal physics is different; in particular, in contrast to [27], in [28] – has drastic implications on how the fundamen-
our strings in SYM obey the usual area law with tension tal quarks interact with DWs. For SU(2) there are two
Mm. typesofDWs,whichwelabelBPS andBPS , andtheir
∼ 1 2
IncontrasttoSYM,innon-supersymmetricQCD(adj) anti-walls. The distinction is in the electric fluxes which
with n >1 the Cartan components of the n Weyl ad- they carry, but they both satisfy the same BPS equa-
f f
joints are massless, due to the unbroken SU(n ) chiral tion, e.g. [29]. The fundamental string is made out of
f
symmetry. Thus, despite the fact that their interaction the BPS and an anti-BPS , where each carries 1/2 of
1 2
with the wall in (2) is highly suppressed, they induce a the fundamental electric flux. If a quark anti-quark (qq¯)
power-law force competing with the exponential repul- pair is in the vicinity of the DW, however, the DW flux
sion at large d. The leading effect of the fermions oc- can cancel part of the flux of a qq¯ pair, and absorb it
curs at 2n 1 loop order; its calculation, of which we into its worldsheet, see Fig. 3. The qq¯pair on the DW
f
−
just give the result, is similar in spirit to Casimir en- would then be liberated, as all the tension of the pair
ergy calculations. Fermion loops are found to generate has been absorbed into the DW tension. This leads to
a wall-wall attraction at large d. Per unit volume, it is deconfinement in the DW worldsheet. This is reminis-
p∼u−lsimon2(cid:0)MmM(cid:1)4nmfe(mmdd)−a4tnfla+r4g,eddo.mTinhaetinexgptrheessbioonsofnoirctrhee- cweonrtldosfhteheteiDsWinCloocualloimzabtiopnh,asweh,esorethaatthqeuorayrkisnatrheeliDbeWr-
−
∼ ated[30]. WealsonotethatinacertainHiggsvacuumof
actionofourtoymodel,withfermionattractionincluded,
is S = T(R+d)Mm+RTMme md RTm2(cid:0)m(cid:1)4nf / 4dtheories, monopole–anti-monopolepairshavesupport
− − M on stable non-abelian (magnetic) strings [31, 32]. In this
(md)4nf−4. The extremum condition (to which the area work the strings are genuine electric strings.
term does not contribute for large T) is now e md
− ∼ Deconfinement of quarks on the DW also implies that
e−4π2(4nf+1)/g2/(md)4nf−3. At small g2, we thus have strings can end on DWs (see inlay of Fig. 3). In MQCD,
md 4π2(4n +1)/g2, a stable wall-wall separation
f SYM strings have been argued to end on DWs and a
par∗am≈etrically large compared to the single domain wall
heuristic explanation by S.-J. Rey [33], using the vac-
width. Numerical confirmation of the stabilized trans-
uum structure and ideas about confinement, is given in
versesized ofthestringischallenging,butourestimate
[34]. The phenomenon was subsequently explored from
ofthesizes∗tabilizationisreliableatsmallg andlargeR.
modeling the effective actions of the Polyakov loop and
Asaconsequenceofthestabilizedtransversesizeofthe gaugino condensates [35]. Here, we found—for the first
confining string in nf>1 QCD(adj), the second transla- time, to the best of our knowledge—an explicit realiza-
tional Goldstone mode, the “breather” mode of the two tion of this phenomenon in a field theory setting where
walls, is now gapped even at infinite R. The gap for the confining dynamics is understood [41].
this mode, mbr, can be estimated by taking the sec- OurdiscussionofconfiningstringsinQCD(adj)gener-
ond derivative of the wall-wall interaction potential at alizes to the higher-rank case. We shall focus only on a
d , mbr me−4π22nf/g2. The breather mode mass mbr few salient points. All fields in (1) become Nc 1 dimen-
∗ ∼ −
4
mental quarks. From the P symmetry, it suffices to take
(cid:126)σ monodromy 2πw(cid:126) , appropriate to the highest weight
} } 1
WW-pairs 2M of the fundamental (the Nc 1 fundamental weights w(cid:126)k
} W mbr obey α(cid:126)p∗·w(cid:126)k=δkp, p=1,...,Nc−−1). We shall argue that
these strings are also composed of two domain walls.
⇤ To this end, recall [8] that SU(N ) QCD(adj)/SYM has
c
L Nc vacua, (cid:104)(cid:126)σ(cid:105)=2Nπckρ(cid:126), k=1,...Nc, related by the broken
⇤�1 ZNc(⊂Z2Ncnf) chiral symmetry. Here, ρ(cid:126)=(cid:80)kN=c−11w(cid:126)k is
the Weyl vector and the dual photons’ periodicity is
⇠
(cid:126)σ (cid:126)σ+2πw(cid:126) . An “elementary” domain wall between the
FIG.4: Asketchoftheabelianstringspectrum, correspond- k-(cid:39)th and (kk+1)-th vacua then has monodromy 2πρ(cid:126). To
ing to the tower of WW-bosons pairs attached to the double Nc
constructaconfigurationof2πw(cid:126) monodromy, wenotice
string, and the breather mode excitations mbr. 1
the identity 2πw(cid:126) =2πρ(cid:126) 2πPρ(cid:126). A (cid:126)σ monodromy 2πw(cid:126)
1 Nc − Nc 1
can now be engineered from an elementary domain wall
sionalvectors,describingthelightdegreesoffreedomleft andaP-transformedanti-domainwall,asinFig.1. Anu-
after SU(Nc) U(1)Nc−1 breaking. It suffices to study merical minimization of (3) confirms that, indeed, this is
→
theoperatorW(C,λ)=ei(cid:126)λ· CA(cid:126)(3),with(cid:126)λ—aweightof thestringconfigurationinnonsupersymmetricQCD(adj)
(avectorofU(1)Nc−1electri(cid:72)ccharges),asthetraceofthRe withNc=3,4(theactiondensityplotissimilartoFig.2).
We also note that, contrary to Seiberg-Witten theory
Wilson loop is obtained by summing over all weights of
. As in (1), semiclassically W(C,λ) e−Sclass[σ¯(C)], whereonlylinearbaryonsexist[37],inQCD(adj)baryons
R (cid:104) (cid:105) ∼ in “Y” or “∆” configurations arise naturally. The affine
with the magnetic bion potential
monopole-instanton and the unbroken part of the Weyl
(cid:88)Nc (cid:104) (cid:105) symmetry are, again, crucial for this. The combinatorics
Lbion =−m2M cos (α(cid:126)i∗−α(cid:126)i∗+1(modNc))·(cid:126)σ , (3) of such a construction follows from the above string pic-
i=1 ture. Weshallnotdiscusstheenergeticsdeterminingthe
preferred configuration here.
replacingtheonein(1). Hereα(cid:126) labelthesimple(i<N ),
i∗ c ForN >2SYM,thechallengeistoincludethenowrel-
affine (i=N ) coroots (α(cid:126) 2=2); M and m are, up to c
c | i∗| evant φ(cid:126)-(cid:126)σ coupling (φ and σ decouple only in SU(2) at
irrelevant factors, as in (1). The fields φ(cid:126) are set to their
g 1[21]); fornow,wenotethatcandidatestringconfig-
vev φ(cid:126)=0; the full Eq. (3) is in [8] for nf>1 and [21] for u(cid:28)rations with the right monodromies can be engineered
SYM (to get back (1), use α1∗=−α2∗=√2, λ=1/√2 and from appropriate BPS and anti-BPS walls.
redefine m,M,σ). Clearly, a string between quarks with Similar observations to the ones in gauge theories are
charges(cid:126)λ should have 2π(cid:126)λ monodromy of (cid:126)σ around C. stilltruefordomainwallsinXY-modelswiththep-clock
An important fact, with crucial consequences for the deformation which are dual to thermal gauge theories
string spectrum, is that, due to the existence of the [6]. There, vortices would be liberated on the domain
twisted(affine)monopole-instanton[2]andthepreserved wall, which might have important consequences for the
center symmetry, a Z subgroup of the Weyl group, physics of transport as well as thermodynamics of these
Nc
cyclically permuting the N roots in (3), is unbroken in walls(e.g. heatcapacity,magneticpermeabilityandcon-
c
QCD(adj). Denoting by P the generator of the cyclic ductivity).
Weyl group, using an N -dimensional basis for the roots A very interesting question is how our QCD(adj)
c
(one linear combination of the N σ ’s decouples [8]), its strings behave upon decompactification to R4. In SYM,
c k
action is: Pσ =σ , or Pα(cid:126) =α(cid:126) . The no phase transition occurs and the transition to R4
k k+1(modNc) k k+1(modNc)
P symmetry ensures that strings confining quarks in should be smooth. For n >1, the SU(n ) chiral sym-
f f
R
ofSU(N )haveequaltensionforallweightsof thatlie metry is expected to break, at least for sufficiently small
c
R
in the same orbit of the cyclic Weyl subgroup. Since P n [38] (since fermions play crucial role in both mag-
f
permutes the N weights of the fundamental representa- neticbionformationandinstabilizingthestringsize,one
c
tion,stringsconfininganycomponentofthefundamental mightexpectinterestinginterplaybetweenchiralsymme-
quarkshaveequaltension. ThisisdifferentfromSeiberg- try breaking and confinement).
Witten theory where the Weyl group is completely bro- On R4, not much is known about strings in SYM or
ken [36]. Still, the multiplicity of meson Regge trajec- QCD(adj)fromfieldtheoryalone. Anexceptionissoftly-
tories in the calculable regime of QCD(adj) is different broken Seiberg-Witten theory [36] (not pure SYM). In
from that expected in the full nonabelian theory with MQCD,thetransitionfromsoftly-brokenSeiberg-Witten
unbroken Weyl group. Further, for higher N-ality repre- theory to pure SYM was studied in [37]. It was found
sentations, there are different “P-orbits” of “k-strings” that pure SYM strings on R4 conform, at least in the
(both previous statements hold without accounting for MQCDregime,tothebehaviorexpectedfromnonabelian
screening by heavy “W bosons”). strings,withfullyunbrokenWeylgroupandN-ality-only
We leave a full taxonomy of “k-strings” in QCD(adj) dependent tensions. The transition from the different
for the future and briefly study strings between funda- abelian behaviors, found here and in [36], to the non-
5
abelianoneshouldclearlyinvolvetheW-bosons(asthey excitation spectrum would correspond to the exponen-
become light upon increasing L). Their inclusion can tiallysmallbreathermodem ,andatowerofW-bosons.
br
modify both the vacuum configurations and the confin- Then,upondecompactificationitisreasonabletoexpect
ing strings themselves (a pure YM theory scenario, re- the abelian string-spectrum to go into the non-abelian
lating monopoles, W-bosons, and center vortices is in spectrum (see Fig. 4). Needless to say, all of these ques-
Ch. 8 of [39]). The difficulty in pursuing this transition tionscaninprinciplebeaddressedinlatticesimulations.
is, not surprisingly, the loss of theoretical control upon
de-abelianization. Acknowledgements We thank Soo-Jong Rey, Mithat
It is, however, tempting to speculate, at least in SYM U¨nsal and Alyosha Yung for discussions. EP acknowl-
where continuity is guaranteed, that the gapped modes edges support by NSERC and hospitality of the Perime-
due to the double string will be responsible for the truly ter Institute in the Fall of 2014. MA acknowledges the
non-abelian structure of the string in the decompacti- SwissNationalScienceFoundation. TSacknowledgesthe
fication limit. In the abelian regime the “non-abelian” support of the DOE grant de-sc0013036.
[1] M. Unsal, Phys.Rev. D80, 065001 (2009), 0709.3269. [22] M. M. Anber and E. Poppitz, JHEP 1106, 136 (2011),
[2] K.-M. Lee and P. Yi, Phys.Rev. D56, 3711 (1997), hep- 1105.0940.
th/9702107. [23] E. Poppitz and M. Unsal, JHEP 1107, 082 (2011),
[3] T. C. Kraan and P. van Baal, Nucl.Phys. B533, 627 1105.3969.
(1998), hep-th/9805168. [24] M. M. Anber, Annals Phys. 341, 21 (2014), 1308.0027.
[4] A. M. Polyakov, Nucl.Phys. B120, 429 (1977). [25] M.UnsalandL.G.Yaffe,Phys.Rev.D78,065035(2008),
[5] I.Affleck,J.A.Harvey,andE.Witten,Nucl.Phys.B206, 0803.0344.
413 (1982). [26] M. Unsal, Phys.Rev. D86, 105012 (2012), 1201.6426.
[6] M. M. Anber, E. Poppitz, and M. Unsal, JHEP 1204, [27] A.Yung,Nucl.Phys.B562,191(1999),hep-th/9906243.
040 (2012), 1112.6389. [28] A. Armoni and M. Shifman, Nucl.Phys. B664, 233
[7] E. Poppitz, T. Schafer, and M. Unsal, JHEP 1210, 115 (2003), hep-th/0304127.
(2012), 1205.0290. [29] K. Hori, A. Iqbal, and C. Vafa (2000), hep-th/0005247.
[8] P. C. Argyres and M. Unsal, JHEP 1208, 063 (2012), [30] G.DvaliandM.A.Shifman,Phys.Lett.B396,64(1997),
1206.1890. hep-th/9612128.
[9] G. Basar, A. Cherman, D. Dorigoni, and M. Unsal, [31] M.ShifmanandA.Yung,Phys.Rev.D86,025001(2012),
Phys.Rev.Lett. 111, 121601 (2013), 1306.2960. 1204.4165.
[10] M.M.Anber,Phys.Rev.D88,085003(2013),1302.2641. [32] M.ShifmanandA.Yung,Int.J.Mod.Phys.A29,1430064
[11] E.ShuryakandT.Sulejmanpasic,Phys.Lett.B726,257 (2014), 1410.2900.
(2013), 1305.0796. [33] S.-J. Rey, unpublished work (1997).
[12] E. Poppitz and T. Sulejmanpasic, JHEP 1309, 128 [34] E. Witten, Nucl.Phys. B507, 658 (1997), hep-
(2013), 1307.1317. th/9706109.
[13] M. M. Anber and T. Sulejmanpasic (2014), 1410.0121. [35] A. Campos, K. Holland, and U. Wiese, Phys.Rev.Lett.
[14] F. Bruckmann and T. Sulejmanpasic, Phys.Rev. D90, 81, 2420 (1998), hep-th/9805086.
105010 (2014), 1408.2229. [36] M.R.DouglasandS.H.Shenker,Nucl.Phys.B447,271
[15] G. Bergner and S. Piemonte, JHEP 1412, 133 (2014), (1995), hep-th/9503163.
1410.3668. [37] A.Hanany,M.J.Strassler,andA.Zaffaroni,Nucl.Phys.
[16] T. Misumi and T. Kanazawa, JHEP 1406, 181 (2014), B513, 87 (1998), hep-th/9707244.
1405.3113. [38] E. Poppitz and M. Unsal, JHEP 0909, 050 (2009),
[17] T. Misumi, M. Nitta, and N. Sakai (2014), 1409.3444. 0906.5156.
[18] R. Larsen and E. Shuryak (2014), 1408.6563. [39] J. Greensite, Lect.Notes Phys. 821, 1 (2011).
[19] M. M. Anber, S. Collier, and E. Poppitz, JHEP 1301, [40] M.ShifmanandA.Yung,Phys.Rev.D67,125007(2003),
126 (2013), 1211.2824. hep-th/0212293.
[20] M. M. Anber, S. Collier, E. Poppitz, S. Strimas-Mackey, [41] That magnetic strings can end on DWs has been known
and B. Teeple, JHEP 1311, 142 (2013), 1310.3522. for some time, however [40].
[21] M. M. Anber, E. Poppitz, and B. Teeple, JHEP 1409,
040 (2014), 1406.1199.