1001.0016.txt

arXiv:1001.0016v4  [hep-th]  1 Feb 2011ExactResultsandHolographyof WilsonLoops
in
N=2Superconformal(Quiver)GaugeTheories
Soo-JongReya,b, Takao Suyamaa
aSchool ofPhysicsand Astronomy&Center forTheoreticalPhy sics
Seoul NationalUniversity,Seoul 141-747 KOREA
bSchool ofNaturalSciences, InstituteforAdvancedStudy,P rinceton NJ 08540 USA
[email protected] [email protected]
ABSTRACT
Using localization, matrix model and saddle-point techniq ues, we determine exact behavior of
circularWilsonloopin N=2superconformal(quiver)gaugetheoriesinthelargenumbe rlimit
of colors. Focusing at planar and large ‘t Hooft couling limi ts, we compare its asymptotic be-
havior with well-known exponential growth of Wilson loop in N=4 super Yang-Mills theory
with respect to ‘t Hooft coupling. For theory with gauge grou p SU(N)coupled to 2 Nfunda-
mental hypermultiplets,we find that Wilson loop exhibits non-exponential growth – at most, it
can grow as a power of ‘t Hooft coupling. For theory with gauge group SU( N)×SU(N)and
bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups.
We find Wilson loop in untwisted sector grows exponentially l arge as in N=4 super Yang-
Mills theory. We then find Wilson loop in twisted sector exhib itsnon-analytic behavior with
respecttodifferenceofthetwo‘tHooftcouplingconstants . Bylettingonegaugecouplingcon-
stanthierarchically larger/smallerthan theother, wesho wthatWilsonloops inthesecond type
theory interpolate to Wilson loops in the first type theory. W e infer implications of these find-
ings from holographic dual description in terms of minimal s urface of dual string worldsheet.
We suggest intuitive interpretation that in both classes of theory holographic dual background
must involve string scale geometry even at planar and large ‘ t Hooft coupling limit and that
new resultsfound in thegaugetheorysideare attributablet o worldsheet instantonsand infinite
resummation therein. Our interpretation also indicates th at holographic dual of these gauge
theoriesis providedby certain non-critical stringtheories.1 Introduction
AdS/CFTcorrespondence[1]between N=4superYang-MillstheoryandTypeIIBstringthe-
ory onAdS5×S5has been studied extensively during the last decade. One rem arkable result
obtained from thestudy is exact computationforexpectatio n valueofWilson loopoperators at
strongcoupling[2][3]. Forahalf-BPS circularWilsonloop ,based on perturbativecalculations
at weak ‘t Hooft coupling [4], exact form of the expectation v alue was conjectured in [5], pre-
ciselyreproducingtheresultexpectedfromthestringtheo rycomputation[2],[3]andconformal
anomalytherein. Theirconjecturewas confirmed laterin[6] usingalocalizationtechnique.
Inthispaper,westudyaspectsofhalf-BPScircularWilsonl oopsin N=2supersymmetric
gaugetheories. Wefocusonaclassof N=2superconformalgaugetheories—the A1(quiver)
gaugetheory of gaugegroup SU (N)and 2Nfundamentalhypermultipletsand ˆA1quivergauge
theory of gauge group SU( N)×SU(N)and bifundamental hypermultiplets— and compute the
Wilson loop expectation value by adapting the localization technique of [6]. We then compare
the results with the N=4 super Yang-Mills theory, which is a special limit of the ˆA0quiver
gauge theory of gauge group SU( N) and an adjoint hypermultiplet. Their quiver diagrams are
depictedin Fig. 1.
(a)                                                     (b)                                                           (c)
Figure 1: Quiver diagram of N=2superconformal gauge theories under study: (a) ˆA0theory with G
= SU(N)and one adjoint hypermultiplet, (b) A1theory with G=SU(N) and2Nfundamental hypermul-
tiplets, (c) ˆA1theory with G=SU(N)×SU(N) and2Nbifundamental hypermultiplets. The A1theory
is obtainable from ˆA1theory by tuning ratio of coupling constants to 0 or ∞. See sections 3 and 4 for
explanations.
We show that, on general grounds, path integral of these N=2 superconformal gauge
theories on S4is reducible to a finite-dimensional matrix integral. The re sulting matrix model
turns out very complicated mainly because the one-loop dete rminant around the localization
fixed point is non-trivial. This is in shartp contrast to the N=4 super Yang-Mills theory,
where the one-loop determinant is absent and further evalua tionof Wilson loops or correlation
1functionsisstraightforwardmanipulationinGaussianmat rixintegral.
Nevertheless, in the N→∞planar limit, we show that expectation value of the half-BPS
circular Wilson loop is determinable provided the ’t Hooft coupling λis large. In the large λ
limit, the one-loop determinant evaluated by the zeta-func tion regularization admits a suitable
asymptotic expansion. Using this expansion, we can solve th e saddle-point equation of the
matrixmodelandobtainlarge λbehavioroftheWilsonloopexpectationvalue. In N=4super
Yang-Mills theory, it is known that the Wilson loop grows exp onentially large ∼exp(√
2λ)as
λbecomesinfinitelystrong.
InˆA0gauge theory, we find that the Wilson loop expectation value g rows exponentially,
exactly the same as the N=4 super Yang-Mills theory. The result for A1gauge theory is
surprising. We find that the Wilson loop is finite at large λ. This means that the Wilson loop
exhibitsnon-exponential growth. The ˆA1quiver gauge theory is also interesting. There are
two Wilsonloops associated witheach gaugegroups, equival ently,onein untwistedsector and
anotherin twistedsector. Wefind that theWilsonloopin untw istedsector scales exponentially
large, coincidingwith the behavior of the Wilson loop N=4 super Yang-Millstheory and the
ˆA0gauge theory. On the other hand, the Wilson loop in twisted se ctor exhibits non-analytic
behavior with respect to difference of two ‘t Hooft coupling constants. We also find that we
can interpolate the two surprising results in A1andˆA1gauge theories by tuning the two ‘t
Hooft couplings in ˆA1theory hierarchically different. In all these, we ignored p ossible non-
perturbative corrections to the Wilson loops. This is becau se, recalling the fishnet picture for
the stringy interpretation of Wilson loops, the perturbati ve contributions would be the most
relevantpart forexploringtheAdS/CFT correspondenceand theholographytherein.
We also studied how holographic dual descriptions may expla in the exact results. Expec-
tation value of the Wilson loop is described by worldsheet pa th integral of Type IIB string in
dual geometry and that, in case the dual geometry is macrosco pically large such as AdS 5×S5,
itisevaluatedbysaddle-pointsofthepathintegral–world sheetconfigurationsofextremalarea
surface. We first suggest that non-exponential growth of the A1Wilson loop arise from deli-
catecancelationamongmultiple—possiblyinfinitelymany— saddle-points. Thisimpliesthat
holographicdualgeometryofthe N=2A1gaugetheoryoughttobe(AdS 5×M2)×Mwhere
the internal space M= [S1×S2]necessarily involves a geometry of string scale. The string
worldsheet sweeps on average an extremal area surface insid e AdS5, but many nearby saddle-
point configurations whose worldsheet sweep two cycles over Mcancel among the leading,
exponential contributions of each. We next suggest that ˆA1Wilson loop in untwisted sector is
givenbyamacroscopicstringinAdS 5×S5/Z2andhencegrowsexponentiallywithaverageof
thetwo‘tHooftcouplingconstants. Intwistedsector,howe ver,itisnegligiblysmallandscales
withdifferenceofthetwo‘tHooftcouplingconstants. This isagainduetodelicatecancelation
2among multiple worldsheet instantons that sweep around col lapsed two cycles at the Z2orb-
ifold fixed point. We also demonstrate that Wilson loop expec tation values are interpolatable
between ˆA1andA1behaviors(orviceversa)bytuningNS-NS2-formpotentialo nthecollapsed
twocyclefrom 1 /2to0,1 orviceversa.
This paperis organized as follows. In section 2, we showthat evaluationof theexpectation
value of the half-BPS circular Wilson loop in a generic N=2 superconformal gauge theory
reduces to a related problem in a one-matrix model. The reduc tion procedure is based on lo-
calization technique and is parallel to [6]. Compared to [6] , our derivations are more direct
and elementary and hence makes foregoing analysis in thepla nar limitfar clearer physicswise.
In section 3, we evaluate the Wilson loop at large ‘t Hooft cou pling limit. Based on general
analysis for one-matrix model (subsection 3.1), we evaluat e the matrix model action which is
induced by the one-loop determinant (subsection 3.2). As a r esult, we obtain a saddle-point
equationwhosesolutionprovidesthelarge‘tHooftcouplin gbehavioroftheWilsonloop(sub-
section 3.3). In section 5, we discuss interpretation of the se results in holographic dual string
theory. For both A1andˆA1types, we argue contribution of worldsheet instanton effec ts can
explain non-analytic behavior of the exact gauge theory res ults. Section 7 is devoted to dis-
cussion, including a possible implication of the present re sults to our previous work [7] (see
also [8][9]) on ABJM theory [10]. We relegated several techn ical points in the appendices. In
appendixA,wesummarizeKillingspinorson S4. InappendixB, weworkoutoff-shellclosure
ofsupersymmetryalgebra. InappendixC,wepresentasympto ticexpansionoftheWilsonloop.
In appendix D, we present detailed computation of c1that arise in the evaluation of one-loop
determinant.
Results of this work were previously reported at KEK worksho p and at Strings 2009 con-
ference. Foronlineproceedings,see[11]and [12], respect ively.
2 ReductiontoOne-MatrixModel
The work [6] provided a proof for the conjecture [4, 5] that th e evaluation of the half-BPS
Wilson loop in N=4 super Yang-Mills theory [2, 3] is reduced to a related probl em in a
Gaussian Hermitian one-matrix model. In this section, we sh ow that the similarreduction also
works for N=2 superconformal gauge theories of general quiver type. The resulting matrix
model is, however, not Gaussian but includes non-trivial ve rtices due to nontrivial one-loop
determinant.
32.1 From N=4toN=2
A shortcut route to an N=2 gauge theory of general quiver type — with matters in variou s
different representations and coupling constants in diffe rent values — is to start with N=4
super Yang-Mills theory. In this section, for completeness of our treatment, we elaborate on
this route. Let Gbe the gauge group. The latter theory consists of a gauge field Amwith
m=1,2,3,4, scalar fields A0,A5,···,A9and anSO(9,1)Majorana-Weyl spinor Ψ, all in the
adjointrepresentationof G. Theaction can bewrittencompactlyas
SN=4=/integraldisplay
R4d4xTr/parenleftBig
−1
4FMNFMN−i
2ΨΓMDMΨ/parenrightBig
, (2.1)
whereM,N=0,···,9and
FMN=∂MAN−∂NAM−ig[AM,AN], (2.2)
DMΨ=∂MΨ−ig[AM,Ψ], (2.3)
ΓΨ= +Ψ. (2.4)
Note that the metric of the base manifold R4is taken in the Euclidean signature, while the
ten-dimensional’metric’ ηMNis taken Lorentzian with η00=−1. As usual in thedimensional
reduction,thederivativesotherthan ∂mare setto zero.
Theaction (2.1)is invariantunderthesupersymmetrytrans formations
δAM=−iξΓMΨ, (2.5)
δΨ=1
2FMNΓMNξ, (2.6)
whereξis a constant SO(9,1)Majorana-Weyl spinor-valued supersymmetry parameter sat is-
fying the chirality condition Γξ=+ξ. In what follows, we rewrite the action (2.1) so that the
resulting action provides a useful guide to deduce the actio n of an N=2 gauge theory with
hypermultipletfields ofarbitrary representations.
We first choose which half of the supercharges of the N=4 supersymmetry is to be pre-
served. This choice corresponds to the choice of embedding t he SU(2) R-symmetry of N=2
theory intotheSU(4)R-symmetryofthe N=4theory. Consideronesuchembeddingdefined
by thematrix
M:=
x6+ix7−(x8−ix9)
x8+ix9x6−ix7
. (2.7)
Its determinantis
detM=(x6)2+(x7)2+(x8)2+(x9)2, (2.8)
4soit isobviousthatany transformationoftheform
M→gLMgR,gL∈SU(2)L,gR∈SU(2)R (2.9)
belongs to the SO(4) transformation acting on (x6,···,x9)∈R4. Note that this transformation
preserves the embedding (2.7). In the ten-dimensional lang uage, SU(4) R-symmetry of the
N=4theoryisrealizedastherotationalsymmetrySO(6)of R6. Therefore,oneembeddingof
SU(2) R-symmetry into SU(4) is chosen by selecting SU (2)Lor SU(2)R. We choose the latter
as theR-symmetryofthe N=2 theories.
There is a U(1) subgroup of SU (2)Lgenerated by σ3. LetR(θ)be an element of this U(1).
This isθ-rotation in 67-plane and (−θ)-rotation in 89-plane. In the following, we require
that the supercharges preserved in N=2 theory should be invariant under the R(θ). For an
infinitesimal θ,R(θ)acts onthesupersymmetrytransformationparameter ξas
δθξ=−1
2θ(Γ6Γ7−Γ8Γ9)ξ. (2.10)
Therefore, ξshouldsatisfy
Γ6789ξ=−ξ, (2.11)
selectingeightcomponentsoutoftheoriginalsixteenones .
The scalar fields Aswiths=6,7,8,9 can be combined into the doublet qα(α=1,2) of
SU(2)Ras
q1:=1√
2(A6−iA7),q2:=−1√
2(A8+iA9), (2.12)
and their conjugates qα=(qα)†. Gamma matrices γα,γαare defined similarly in terms of Γs.
Theysatisfy
{γα,γβ}=2δα
β,{γα,γβ}=0={γα,γβ}. (2.13)
Notethat,forarbitrary vectors VsandWs, onehas
VsWs=VαWα+VαWα. (2.14)
The Majorana-Weyl spinor Ψis split into the chirality eigenstates with respect to Γ6789as
follows:
λ:=1
2(1−Γ6789)Ψ,η:=1
2(1+Γ6789)Ψ. (2.15)
Both fermionsareMajorana-Weyl. We furthersplit ηintoη±, which areeigenstatesof
γ:=1
2[γα,γα]=i
2(Γ6Γ7−Γ8Γ9). (2.16)
Notethat γisthegeneratorfor R(θ)and hencesatisfies
γ2=1
2(1+Γ6789),[γ,γα]=+γα,[γ,γα]=−γα. (2.17)
5Now,η±arenotMajorana-Weyl. Infact, theyarerelated by chargeconjugation
(ηA
±)∗=CηA
∓, (2.18)
whereAistheindexfortheadjointrepresentationof GandCisthecomplexconjugationmatrix.
So, weshalldenote η−byψ. Then,moduloa phasefactor, η+isψ†.
In termsof Aµ(µ=0,···,5),qα,qα,λandψ, theaction (2.1)can bewritten as
SN=4=/integraldisplay
R4d4xTr/parenleftBig
−1
4FµνFµν−DµqαDµqα−i
2λΓµDµλ−iψΓµDµψ
−gλγα[qα,ψ]−gψγα[qα,λ]−g2[qα,qβ][qβ,qα]+1
2g2[qα,qα][qβ,qβ]/parenrightBig
,(2.19)
with the understanding that the dimensional reduction sets ∂µ=0 forµ=0,5. The supersym-
metrytransformations(2.5),(2.6)can bewrittenas
δAµ=−iξΓµλ, (2.20)
δqα=−iξγαψ, (2.21)
δqα=−iψγαξ (2.22)
δλ= +1
2FµνΓµνξ−ig[qα,qβ]γα
βξ, (2.23)
δψ= +DµqαΓµγαξ. (2.24)
Again,if ξobeystheprojectioncondition(2.11), theaction (2.19)ha sN=2 supersymmetry.
At this stage, we shall be explicit of representation conten ts of(qα,ψ)fields and their con-
jugates. Let (TA)B
C=−ifAB
Cbe the generators of Lie (G)in the adjoint representation. We also
impose on ξthe projection condition (2.11). In terms of them, the actio n (2.19) can be written
as
SN=2=/integraldisplay
R4d4x/parenleftBig
−1
4tr(FµνFµν)−i
2tr(λΓµDµλ)−DµqαDµqα−iψΓµDµψ
+gλAγαqαTAψ+gψγαTAqαλA−g2(qαTAqβ)2+1
2g2(qαTAqα)2/parenrightBig
,(2.25)
wherethegaugecovariantderivativesare
Dµqα=∂µqα−iAA
µTAqα, (2.26)
Dµqα=∂µqα+iqαTAAA
µ, (2.27)
Dµψ=∂µψ−iAA
µTAψ. (2.28)
6TheN=2 supersymmetrytransformationrules are
δAµ=−iξΓµλ, (2.29)
δλA= +1
2FA
µνΓµνξ+iqαTAqβγα
βξ, (2.30)
δqα=−iξγαψ, (2.31)
δqα=−iψγαξ (2.32)
δψ= +DµqαΓµγαξ. (2.33)
Theaboveaction(2.25)isequivalenttotheoriginalaction (2.1): wehavejustrewrittentheorig-
inal action in terms of renamed component fields. The supersy mmetry transformations (2.29)-
(2.33)are also equivalentto (2.5) -(2.6) in so far as ξis projected to N=2 supersymmetryas
(2.11).
It turns out that the action (2.25) is invariant under N=2 supersymmetry transformations
(2.29)-(2.33) even for TAin a generic representation Rof the gauge group G, which can also
bereducible. Therefore, (2.25) defines an N=2 gaugetheory withmatterfields (qα,ψ)in the
representation Rand theirconjugates.
It is also possible to treat ˆAk−1quiver gauge theories on the same footing. We embed the
orbifold action Zkinto SU(2)L. In thispaper, we shall focus on ˆA1quivergaugetheory. In this
case, weshouldsubstitute
Aµ=
Aµ(1)
Aµ(2)
,λ=
λ(1)
λ(2)
,
qα=
q(1)α
q(2)α
,ψ=
ψ(1)
ψ(2)
. (2.34)
into(2.19). Notethatthe N=2supersymmetry(2.29)-(2.33)ispreservedevenwhenthega uge
couplingconstant gis replaced withthematrix-valuedone:
g=
g1I
g2I
. (2.35)
Ingeneral, g1/ne}ationslash=g2andcanbeextendedtocomplexdomain. Extensionto ˆAk(k≥2)isstraight-
forward.
72.2 Superconformal symmetryon S4
Following [6], we now define the N=2 superconformal gauge theory on S4of radius r. For
definiteness, we consider the round-sphere with the metric hmninduced through the standard
stereographicprojection. Details aresummarizedinAppen dixA.
For this purpose, it also turns out convenient to start with N=4 super Yang-Mills theory
defined on S4. To maintain conformal invariance, the scalars ought to hav e the conformal
couplingtothecurvaturescalarof S4. Theactionthusreads
SN=4=/integraldisplay
S4d4x√
hTr/parenleftBig
−1
4FMNFMN−1
r2ASAS−i
2ΨΓMDMΨ/parenrightBig
, (2.36)
whereS=0,5,6,···,9. Theactionisinvariantunderthe N=4supersymmetrytransformations
δAM=−iξΓMΨ, (2.37)
δΨ= +1
2FMNΓMNξ−2ΓSAS/tildewideξ, (2.38)
providedthat ξand/tildewideξsatisfytheconformal Killingequations:
∇mξ=Γm/tildewideξ,∇m/tildewideξ=−1
4r2Γmξ. (2.39)
Explicitform ofthesolutiontotheseequationsare givenin AppendixA.
The action of an N=2 gauge theory on S4with a hypermultiplet of representation Rcan
bededuced easilyas intheprevioussubsection. Oneobtains
SN=2=/integraldisplay
S4d4x√
h/parenleftBig
−1
4Tr(FµνFµν)−i
2Tr(λΓµDµλ)−1
r2Tr(AaAa)
−DµqαDµqα−iψΓµDµψ−2
r2qαqα
+gλAγαqαTAψ+gψγαTAqαλA−g2(qαTAqβ)2+1
2g2(qαTAqα)2/parenrightBig
,(2.40)
wherea=0,5. Theactionisinvariantunderthe N=2 superconformalsymmetry
δAµ=−iξΓµλ,
δλA= +1
2FA
µνΓµνξ+igqαTAqβγα
βξ−2ΓaAA
a/tildewideξ,
δqα=−iξγαψ,
δqα=−iψγαξ
δψ= +DµqαΓµγαξ−2γαqα/tildewideξ,
8whereξsatisfies the conformal Killing equations (2.39) in additio n to the projection condition
(2.11). We emphasize that this is the transformation of the N=2 superconformal symmetry,
not just the Poincar´ e part of it. This can be checked explici tly, for example, by examining the
commutatoroftwo transformationsonthefields.
We find it convenient to define a fermionic transformation Qcorresponding to the above
superconformal transformation δ. It is obtained easily by the replacement δ→θQandξ→θξ
withθareal Grassmannparameter. Theresultingtransformationi s
QAµ=−iξΓµλ,
QλA= +1
2FA
µνΓµνξ+igqαTAqβγα
βξ−2ΓaAA
a/tildewideξ,
Qqα=−iξγαψ,
Qqα=−iψγαξ,
Qψ= +DµqαΓµγαξ−2γαqα/tildewideξ, (2.41)
where now ξand/tildewideξarebosonicSO(9,1) Majorana-Weyl spinors satisfying N=2 projection
(2.11)andconformal Killingequation(2.39).
2.3 Localization
By extending the localization technique of [6], we now show t hat computation of Wilson loop
expectation value in N=2 superconformal gauge theory of quiver type can be reduced t o
computationofaone-matrixintegral.
LetQbe a fermionic transformation. Suppose that an action Sunder consideration is in-
variantunder Q. Then, thefollowingmodification
S(t):=S+t/integraldisplay
d4x√
hQV(x) (2.42)
does notchangethepartitionfunctionprovidedthat
/integraldisplay
d4x√
hQ2V(x)=0. (2.43)
Likewise,correlationfunctionsremainunchangedifopera torsunderconsiderationare Q-invariant.
We shall choose V(x)such that the bosonic part of QV(x)is positive semi-definite. For this
choice, since tcan be chosen to be an arbitrary value, we can take the limit t→+∞so that
9the path-integral is localized to configurations where the b osonic part of QV(x)vanishes. It
willturn out laterthat thevanishinglocusof QV(x)is parametrized by a constantmatrix. This
is why the evaluation of the expectation value of a Q-invariant operator reduces to a matrix
integral. Theaction oftheresultingmatrix modelis thesum ofSevaluatedat thevanishinglo-
cus and the one-loop determinant obtained from the quadrati c terms of QV(x)when expanded
around thevanishinglocus.
One might think that the fermionic transformation Qdefined in the previous section can be
used asQabove. In fact, Q2is asumofbosonictransformations,and therefore, (2.43)a ppears
toholdaslongas V(x)isinvariantunderthetransformations. Theproblemofthis choiceisthat
Q2is such a sum only on-shell. According to [13],[14] and [15], Qhas to be modified so that
theresulting Qclosestoasumofbosonictransformationsfor off-shell.
To this end, we introduce auxiliary fields K˙m(˙m=ˆ2,ˆ3,ˆ4),KαandKα. They transform in
the adjoint, RandRrepresentations of the gauge group G, respectively. Utilizing them, we
modifytheaction (2.40)in atrivialmanner:
SN=2=/integraldisplay
S4d4x/parenleftBig
−1
4Tr(FµνFµν)−i
2Tr(λΓµDµλ)−1
r2Tr(AaAa)
−DµqαDµqα−iψΓµDµψ−2
r2qαqα
+gλAγαqαTAψ+gψγαTAqαλA−g2(qαTAqβ)2+1
2g2(qαTAqα)2
+1
2K˙mK˙m+KαKα/parenrightBig
. (2.44)
Evidently,this action is physicallyequivalentto the orig inalone. Themodified action (2.44) is
nowinvariantunderthefollowing Qtransformations:
QAµ=−iξΓµλ,
QλA= +1
2FA
µνΓµνξ+igqαTAqβγα
βξ−2ΓaAA
a/tildewideξ+K˙mAν˙m,
Qqα=−iξγαψ,
Qqα=−iψγαξ,
Qψ= +DµqαΓµγαξ−2γαqα/tildewideξ+Kανα,
Qψ= +DµqαξγαΓµ+2/tildewideξγαqα+Kανα,
QK˙mA=−ν˙m/parenleftBig
−iΓµDµλA+gγαqαTAψ−gγαψ∗TAqα/parenrightBig
,
QKα=−να/parenleftBig
−iΓµDµψ+γβTAqβgλA/parenrightBig
,
QKα=−/parenleftBig
−iDµψΓµ−gλAγβqβTA/parenrightBig
να. (2.45)
10To makeQ2close to a sum of bosonic transformations off-shell, the spi norsν˙m,να,ναshould
be chosen appropriately out of ξ,/tildewideξ. Details on them are summarized in Appendix B. With the
correct choice, Q2closes,forexample,on λas follows:
−iQ2λ=/parenleftbigg
vm∇mλ−1
2(ξΓmn/tildewideξ)Γmnλ−ig[vµAµ,λ]/parenrightbigg
+1
2(ξΓst/tildewideξ)Γstλ.(2.46)
Thisshowsthat Q2isasumofadiffeomorphismon S4,aGgaugetransformationand aglobal
SU(2)Rtransformation. In particular, notice that ξΓst/tildewideξturns out to be independent of xm. The
actionof Q2on theauxiliaryfields isslightlydifferent. Forexample,o nK˙m, oneobtains
−iQ2K˙m=vk∇kK˙m−ig[vµAµ,K˙m]+ν˙mΓk∇kν˙nK˙n. (2.47)
Here, the index ˙ mdoes not transform as a part of the four-vector on S4. This is not a problem
sinceK˙mis contracted with ν˙minVdefined below, and not with some other four-vectors. The
Qdefined aboveis therighttransformationavailableforthel ocalizationprocedure.
We areat thepositiontochoose V. Wetake
V:=Tr(Vλλ)+Vψψ+ψVψ, (2.48)
where
Vλ=1
2FµνξΓ0Γµν+igqαTAqβtAξΓ0γα
β+2/tildewideξΓ0ΓaAa+K˙mν˙mΓ0,(2.49)
Vψ=DµqαξΓ0Γµγα+2/tildewideξΓ0γαqα+KαναΓ0, (2.50)
Vψ=DµqαγαΓµΓ0ξ−2γαqαΓ0/tildewideξ+KαΓ0να. (2.51)
Notethat Visascalarwithrespecttoaparticularcombinationofthedi ffeomorphismon S4,the
GgaugetransformationandtheglobalSU (2)Rtransformation. Thisfollowsfromtheidentities
forthespinors,forexample,
vm∇mξ−1
2(ξΓmn/tildewideξ)Γmnξ+1
2(ξΓst/tildewideξ)Γstξ=0, (2.52)
and similarones for/tildewideξandνIwhichare summarizedin AppendixA and B. Therefore, (2.43)i s
satisfiedwith thischoice, as required.
Afterstraightforward buttediousalgebra, oneobtainsthe bosonicpart of QVexpressedas
Tr(VλQλ)+VψQψ+QψVψ/vextendsingle/vextendsingle/vextendsingle
bosonic
=Tr/bracketleftBig
cos2θ
2(F+
mn+w+
mnA5)2+sin2θ
2(F−
mn+w−
mnA5)2−(K˙m−2A0ν˙m/tildewideξ)2
+DmAaDmAa−1
2g2[Aa,Ab]2+g2tAtB(2qαTAqβqβTBqα−qαTAqαqβTBqβ)/bracketrightBig
+2D0qαD0qα+2|D˙µqα+ξΓ0˙µγα
β/tildewideξqβ|2+3
2r2qαqα−2KαKα, (2.53)
11whereθisthepolarangleon S4, ˙µ=1,2,···,5and
w+
mn:=1
cos2θ
2ξΓ05Γmn1−Γˆ1ˆ2ˆ3ˆ4
2/tildewideξ, (2.54)
w−
mn:=1
sin2θ
2ξΓ05Γmn1+Γˆ1ˆ2ˆ3ˆ4
2/tildewideξ. (2.55)
Here, thehatted indicesaretheLorentzones. Theaboveexpr essionshowsthat,aftera suitable
Wick rotation for A0and the auxiliary fields, the bosonic part of QVis positive semi-definite.
Therefore, by taking the limit t→+∞, the path-integral is localized at the vanishing locus of
QV. Itturns outthat,as in[6], non-zero fields at thevanishing locusare
A0=−i
grΦ,Kˆ2=−i
gr2Φ, (2.56)
whereΦis aconstantHermitianmatrix. Thecoefficients are chosenforlaterconv enience.
Now, the path-integralis reduced to an integraloverthe Her mitian matrix Φ. The action of
the corresponding matrix model is a sum of the action (2.44) e valuated at the vanishing locus
and the one-loop determinant for the quadratic terms in QV. Note that higher-loop contribu-
tions vanish in the large tlimit since t−1plays the role of the loop-counting parameter. At the
vanishinglocus,theaction(2.44)takesthevalue
S=−/integraldisplay
S4d4x√
hTr/parenleftBig1
r2(A0)2+1
2(Kˆ2)2/parenrightBig
=4π2
g2TrΦ2. (2.57)
An importantdifference from the N=4 superYang-Millstheory isthat theone-loop determi-
nant around the vanishinglocus does not cancel and has a comp licated functional structure. In
the next section, we show that the presence of the non-trivia l one-loop determinant is crucial
fordeterminingthelarge‘t Hooftcouplingbehavioroftheh alf-BPS Wilsonloop.
Thehalf-BPS Wilsonloopof N=2 gaugetheory hasthefollowingform:
W[C]:=TrPsexp/bracketleftBig
ig/integraldisplay2π
0ds/parenleftBig
˙xmAm(x)+θaAa(x)/parenrightBig/bracketrightBig
. (2.58)
The functions xm(s),θa(s)are chosen appropriately to preserve a half of the N=2 supercon-
formal symmetry. We shall choose Cto be the great circle at the equator of S4(i.e.θ=π
2)
specified by
(x1,x2,x3,x4)=(2rcoss,2rsins,0,0), (2.59)
andθaas
θ0=r,θ5=0. (2.60)
12Forthischoice,onecan showthat
˙xmAm(x)+θaAa(x)=−rvµAµ(x), (2.61)
wherevµ=ξΓµξ. See Appendix A for theexplicit expressionsof vµ. This implies that W[C]is
invariantunder Qdueto theidentity
ξΓµξξΓµλ=0. (2.62)
Thus, we have shown that /an}bracketle{tW[C]/an}bracketri}htis calculable by a finite-dimensional matrix integral. The
operatorwhoseexpectationvaluein thematrixmodelis equa lto/an}bracketle{tW[C]/an}bracketri}htis
Trexp/parenleftBig
2πΦ/parenrightBig
. (2.63)
Noticethatitissolelygovernedbytheconstant-valued,He rmitianmatrix Φ. Thisenablesusto
compute the Wilson loops in terms of a matrix integral. This o bservation will also play a role
inidentifyingholographicdual geometrylater.
3 Wilson loopsatLarge‘t HooftCoupling
We have shown that evaluation of the Wilson loop /an}bracketle{tW[C]/an}bracketri}htis reduced to a related problem in
a one-Hermitian matrix model. Still, the matrix model is too complicated to solve exactly.
In the following, we focus our attention to either the N=2 superconformal gauge theory
ofA1type with G=U(N)coupled to 2 Nfundamental hypermultiplets and of ˆA1type with
G=U(N)×U(N), both at large Nlimit. For these theories, we show that the large ‘t Hooft
couplingbehaviorisdeterminablebyafewquantitiesextra ctedfromtheone-loopdeterminant.
This allows us to exactly evaluate the Wilson loop /an}bracketle{tW[C]/an}bracketri}htin the large Nand large ’t Hooft
couplinglimit.
3.1 General resultsin one matrixmodel
Consider a matrix model for an N×NHermitian matrix X. In the large Nlimit, expectation
valueofanyoperatorinthismodelisdeterminableintermso feigenvaluedensityfunction ρ(x)
ofthematrix X. By definition, ρ(x)isnormalizedby
/integraldisplay
dxρ(x)=1. (3.1)
13LetDdenotethesupportof ρ(x). Weassumethat1
min{D}=:b<0<a:=max{D}. (3.2)
Expectationvalueoftheoperator1
NTr(ecX) (c>0)isgivenintermsof ρ(x)as
W:=/angbracketleftbigg1
NTr(ecX)/angbracketrightbigg
=/integraldisplay
dxρ(x)ecx. (3.3)
By theassumptiononthesupport D,thevalueof Wis bounded:
ecb≤W≤eca. (3.4)
b                                               a                        x βα(a - x)
Figure2: Typical distribution of the eigenvalue density ρ.
Weareinterestedinthebehaviorof Winthelimit a→+∞. Introducingtherescaleddensity
function/tildewideρ(x)=aρ(ax),Wis writtenas
W=eca/integraldisplay1−b
a
0du/tildewideρ(1−u)e−cauwhere x=a(1−u). (3.5)
At therightedgeofthesupport D,weexpect thatthedensitycutsoffwithapower-lawtail:
/tildewideρ(1−u)=βuα+χ(u)where |χ(u)|≤Kuα+ε,u∈(0,δ) (3.6)
for a positive K,ε,δ. See figure 2. Here, α>0 signifies the leading powerof the fall-off at the
rightedge: χrefers tothesub-leadingremainder. Then,fora largeposit ivea, (3.6)leads to the
followingasymptoticbehavior:
W∼βΓ(α+1)(ca)−α−1eca, (3.7)
1IfXis traceless, the assumption is always valid since/integraltextdxρ(x)x=0 must hold. In the large Nlimit, the
contributionfromthetracepartisnegligible.
14Detailsofthederivationof(3.7)are relegatedtoAppendix C.
Wehavefoundthatthelarge abehaviorof Wisdeterminedbythefunctionalformof ρ(x)in
thevicinityoftherightedgeofits support. In particular, we foundthat theleadingexponential
part isdeterminedsolelyby thelocationoftherightedgeof theeigenvaluedistribution.
For comparison, let us recall the exact form of the Wilson loo p inN=4 super Yang-Mills
theory [4], which is a special case of the ˆA0gauge theory. In this case, the eigenvalue density
functionisgivenby
ρ(x)=4π
λ/radicalbigg
λ
2π2−x2, (3.8)
whichis thesolutionofthesaddle-pointequation
4π2
λφ=/integraldisplay
−dφ′ρ(φ′)
φ−φ′. (3.9)
TheWilsonloopisevaluatedas follows:
/an}bracketle{tW[C]/an}bracketri}ht=4π
λ/integraldisplay+√
λ/π
−√
λ/πdxe2πx/radicalbigg
λ
2π2−x2
=2√
2λI1(√
2λ)
∼/radicalbigg
2
π(2λ)−3
4e√
2λ. (3.10)
Weseethat thisasymptoticbehavioris reproduced exactlyb y(3.7)with α=1
2of(3.8)2.
3.2 One-loop determinant and zetafunction regularization
Let us return to the evaluation of /an}bracketle{tW[C]/an}bracketri}ht. To determine the eigenvalue density function ρof
the Hermitian matrix Φ, it is necessary to know the explicit functional form of the o ne-loop
determinant. However,thisisaformidabletask forageneri cN=2gaugetheory. Fortunately,
as shown in the previous subsection, the leading behavior of /an}bracketle{tW[C]/an}bracketri}htis governed by a small
numberofdataif a=max(D)islarge.
So, we shall assume that the limit λ→+∞induces indefinite growth of a. This is a rea-
sonable assumption since otherwise /an}bracketle{tW[C]/an}bracketri}htdoes not grow exponentially in the limit λ→+∞,
implying that any N=2 gauge theory with such a behavior of the Wilson loop cannot h ave
an AdS dual in the usual sense. In other words, we assume that t he rescaled density function
2Here,thedefinitionofthegaugecouplingconstant gisdifferentbythe factor2fromthatin[4]
15λγρ(λγx)has a reasonable large λlimit for a positiveγ. Under this assumption, we now show
that the large λbehavior of the Wilson loop is determined by the behavior of t he one-loop de-
terminant in the region where the eigenvalues of Φare large. The asymptoticbehavior in such
a limit is most transparently derivable from the heat-kerne l expansion for a certain differential
operatorinthezeta-functionregularizationoftheone-lo opdeterminant.
•A1gaugetheory :
Consider first the A1gauge theory. There are contributions to the one-loop effec tive action
both from the hypermultiplet and the vector multiplet. We fir st focus on the hypermultiplet
contribution. If QVis expanded around the vanishing locus (2.56), quadratic te rms of the
hypermultipletscalars become:
−qα(Δ)α
βqβ+1
r2ΦAΦBqαTATBqα, (3.11)
where
(Δ)α
β= (∇mδα
γ+Vmαγ)(∇mδγ
β+Vmγ
β)−1
4r2(3+cos2θ)δα
β, (3.12)
Vmα
β=ξΓ0mγα
β/tildewideξ. (3.13)
IfΦis diagonalizedas Φ=diag(φ1,···,φN), thenthesecond termin (3.11)can bewrittenas
2N
r2N
∑
i=1(φi)2qiαqα
i. (3.14)
Nowthequadratictermsaredecomposedintothesumoftermsf orcomponents qα
i. So,theone-
loop determinant of the hypermultiplet scalars is the produ ct of determinants for each compo-
nents. Let FB
h(Φ)denoteapartofthematrixmodelactioninducedbytheone-lo opdeterminant
forthehypermultipletscalars qα. Itscontributionto theeffectiveaction can bewrittenas
FB
h(Φ)=2NN
∑
i=1FB
h(φi), (3.15)
whereFB
h(m)is formallygivenas
FB
h(m):=logDet/parenleftBig
−Δ+m2
r2/parenrightBig
. (3.16)
Noticethat the eigenvalues φienteras masses of qα
i. Therefore, what we need to analyze is the
largembehaviorof FB
h(m).
We now evaluate the function FB
h(m)in the limit m→∞. In terms of Feynman diagram-
matics, this amounts to expanding the one-loop determinant in the background of scalar field
16(m/r)2. LetD(m)=Det(−Δ+m2/r2). The relation (3.16) is afflicted by ultraviolet infinities,
so it should be regularized appropriately. The determinant is formally defined over the space
spanned by the normalizable eigenfunctions of −Δ. Letλk(k=0,1,2,···)be eigenvalues of
−Δ:
−Δψk=λkψk. (3.17)
Then,D(m)can beformallywrittenas
D(m)=∞
∏
k=0/parenleftBig
λk+m2
r2/parenrightBig
. (3.18)
To makethisexpressionwell-defined, letus definearegulari zed function
ζ(s,m):=r−2s∞
∑
k=01
(λk+m2/r2)s, (3.19)
wheresisacomplexvariable. Thissummationmaybewell-definedfor swithsufficientlylarge
Re(s). Onecan formallydifferentiate ζ(s,m)withrespect to stoobtain
∂sζ(s,m)/vextendsingle/vextendsingle/vextendsingle
s=0=−∞
∑
k=0log(r2λk+m2)=−log[r2D(m)]. (3.20)
Since the left-hand side makes sense via a suitable analytic continuation of (3.19), it can be
regarded that the right-hand side is defined by the left-hand side. Therefore, we define the
functionFB
h(m)viathezeta-function regularization:
FB
h(m):=−∂sζ(s,m)/vextendsingle/vextendsingle/vextendsingle
s=0. (3.21)
The large mbehavior of FB
h(m)is determined as follows. For a suitable range of s,ζ(s,m)
can bewrittenas
ζ(s,m)=r−2s
Γ(s)/integraldisplay∞
0dtts−1e−m2t/r2K(t), (3.22)
where
K(t):=∞
∑
k=0e−λkt=Tr(etΔ) (3.23)
is the heat-kernel of Δ. The convergence of this sum is assumed. The asymptoticexpa nsion of
K(t)is knownas theheat-kernel expansion. Forareviewon thissu bject, seee.g. [16]. Since Δ
isadifferential operatoron S4, theheat-kernel expansionhastheform
K(t)∼∞
∑
i=0ti−2a2i(Δ) (3.24)
In theexpansion, a2i(Δ)are knownas theheat-kernel coefficients for Δ.
17Theexpression(3.22)of ζ(s,m)isonlyvalidforarangeof s,butζ(s,m)canbeanalytically
continued to theentire complex plane provided that the asym ptoticexpansion (3.24) is known.
In particular, there exists a formulafor the asymptoticexp ansion of ζ(s,m)in the large mlimit
[17]
ζ(s,m)∼∞
∑
i=0a2i(Δ)r2i−4Γ(s+i−2)
Γ(s)m−2s−2i+4, (3.25)
valid in the entire complex s-plane. Note that a2i(Δ)r2i−4are dimensionless combinations.
Differentiatingwith respect to sandsetting s=0, oneobtains
FB
h(m) =/parenleftBig1
2m4logm2−3
4m4/parenrightBig
a0(Δ)r−4−/parenleftBig
m2logm2−m2/parenrightBig
a2(Δ)r−2
+logm2a4(Δ)+O(m−2logm). (3.26)
The evaluation of the one-loop determinant for the hypermul tiplet fermions can be done
similarly. Thequadratictermsofthefermionsaregivenby
iψΓm∇mψ−i
rψΓ0ΦATAψ+i
2(ξΓµν/tildewideξ)ψΓ0Γµνψ. (3.27)
Weneed to evaluate −logDet(iD/)where
iD/:=iΓm∇m−m
riΓ0+κ
2(ξΓµν/tildewideξ)Γ0Γµν(3.28)
withκ=i. Inthefollowing,wewillevaluate −1
2logDet(iD/)2withareal κ,forwhich (iD/)2is
non-negativeand its heat-kernel is well-defined, and then s ubstituteκ=iinto the final expres-
sion. Thevalidityofthisprocedure isjustifiedbyconverge nceoftheresult.
Theexplicitform of (iD/)2isgivenby
(iD/)2=−(∇m+Vm)(∇m+Vm)−1
2Γmn[∇m,∇n]−3κ2
4r2sin2θ
−κ2
4(ξΓµν/tildewideξ)(ξΓρσ/tildewideξ)ΓµνΓρσ+iκm
r(ξΓµν/tildewideξ)Γµν+m2
r2
:=−ΔF+m2
r2. (3.29)
where
Vm=iκ(ξΓmµ/tildewideξ)Γ0Γµ. (3.30)
The fermion case is slightly different from the scalar case s ince there is a term linear in m
in−ΔF. However,theasymptoticexpansionofthezeta-function-r egularizedone-loopdetermi-
nant can be made in the fermion case as well. The part FF
h(Φ)of the matrix model action due
toψhasa similarform with FB
h(Φ),withdifferentcoefficients.
18The total one-loop contribution of hypermultiplet to the ef fective action is Fh=FB
h+FF
h.
Because ofunderlyingsupersymmetry,thetermsoforder m4andm4logm2cancel between FB
h
andFF
h. Theresultingexpressionfor Fhis
Fh=2NN
∑
i=1F(φi), (3.31)
F(m) =c1m2logm2+c2m2+c3logm2+O(m−2logm). (3.32)
The fact that c1is positive will turn out to be important later, while the exa ct values of the
coefficients are irrelevant for the large ‘t Hooft coupling b ehavior of the Wilson loop. We
presented details of computation of c1in Appendix D. Notice that, at least up to this order,
F(m)is an evenfunctionof m.
Obviously, Fhdepends on field contents. The expression for FhwhenRis the adjoint rep-
resentation can be found easily by noticing that, for exampl e, the ’mass’ term of qαcan be put
to
1
r2∑
i/ne}ationslash=j(φi−φj)2qijαqα
ji. (3.33)
In thiscase, Fhis writtenas
Fh/vextendsingle/vextendsingle/vextendsingle
adj.=∑
i/ne}ationslash=jF(φi−φj). (3.34)
Notethat F(m)here isthesamefunctionas (3.32).
Direct evaluation of the contribution from the vector multi plet, which we denote as Fv,
appears morecomplicatedsincetherearemixingtermsbetwe enAmandAa. Fortunately,itwas
shown in [6] that FvandFhcancel each other in N=4 super Yang-Mills theory. This implies
from (3.34)that
Fv=−∑
i/ne}ationslash=jF(φi−φj). (3.35)
•ˆA1gaugetheory :
We next consider the ˆA1quiver gauge theory. In this case, qαandψconsist of bi-fundamental
fields. The Φis ablock-diagonalmatrix:
Φ=
Φ(1)
Φ(2)
, (3.36)
in which Φ(1)=diag(φ(1)
1,···,φ(1)
N)andΦ(2)=diag(φ(2)
1,···,φ(2)
N), respectively. By repeating
thesimilarcomputations,onecan easilyshowthat Fhhastheform
Fh=2N
∑
i,j=1F(φ(1)
i−φ(2)
j), (3.37)
19andFvhas theform
Fv=−∑
i/ne}ationslash=jF(φ(1)
i−φ(1)
j)−∑
i/ne}ationslash=jF(φ(2)
i−φ(2)
j). (3.38)
Thetotalone-loopcontributionisthesum F=Fh+Fv.
As a consistency check of the above result, consider taking t he two nodes identical. This
reduces the number of nodes from two to one, and hence must map theˆA1gauge theory to ˆA0
one. The reduction puts Φ(1)andΦ(2)equal. Then, up to an irrelevant constant, Fvis precisely
minus of Fh. We thus see that Fvanishes identically, reproducing the known result of the ˆA0
gaugetheory.
3.3 Saddle-point equations
We can now extract the saddle-point equations for the matrix model and determine the large ‘t
HooftcouplingbehavioroftheWilsonloopfromthem.
•A1gaugetheory :
In thistheory,thesaddle-pointequationreads
8π2
λφk+2F′(φk)−2
N∑
i/ne}ationslash=kF′(φk−φi)=2
N∑
i/ne}ationslash=k1
φk−φi. (3.39)
Asexplainedbefore,weassumethat λγρ(λγφ)forapositiveγhasasensiblelarge λasymptote.
By rescaling φk→λγφk, oneobtains
8π2φk+2λ1−γF′(λγφk)−2
N∑
i/ne}ationslash=kλ1−γF′(λγ(φk−φi))=2
Nλ1−2γ∑
i/ne}ationslash=k1
φk−φi.(3.40)
Recall that F(x)∼c1x2logx2for largex. This shows that the leading-order equation for large
λisgivenby
4c1φklogφk+2(c1+c2)φk−2
N∑
i/ne}ationslash=k/bracketleftBig
2c1(φk−φi)log(φk−φi)+(c1+c2)(φk−φi)/bracketrightBig
=0.(3.41)
Differentiatingtwicewithrespect to φk, oneobtains
1
φk=1
N∑
i/ne}ationslash=k1
φk−φi. (3.42)
Notice that c1andc2dropped out. Now, this equation has no sensible solution. Th erefore, we
conclude that the scaling assumption we started with is inva lid, implying that the Wilson loop
inthistheory cannotgrowexponentiallyin thelarge‘t Hoof t couplinglimit.
20There is another way to check the finiteness of the Wilson loop . Let us rewrite the saddle-
pointequationas follows:
8π2
λφk+2F′(φk)=2
N∑
i/ne}ationslash=kF′(φk−φi)+2
N∑
i/ne}ationslash=k1
φk−φi. (3.43)
The left-hand side represents the external force acting on t he eigenvalues, whilethe right-hand
side represents the interactions among the eigenvalues. Fo r a large φk, the external force is
dominated by 2 F′(φk), which is nonzero. This implies that the large λlimit must be smooth,
and the Wilson loop expectation value approaches a finite val ue. Recall that in the case of
N=4 super Yang-Mill theory, the large λlimit renders the external force to vanish, resulting
in an indefinite spread of the eigenvalues. This is reflected i n the exponential growth of the
Wilsonloopexpectationvalue.
Implicationsofthissurprisingconclusionarefarreachin g: the N=2supersymmetricgauge
theorycoupledto2 Nfundamentalhypermultiplets,althoughsuperconformal,m usthaveaholo-
graphic dual whose geometry does not belong to the more famil iar cases such as N=4 super
Yang-Mills theory. Central to this phenomenon is that there are two ‘t Hooft coupling param-
eters whose ratio can be tuned hierarchically large or small . In particular, we can tune one of
them to be smaller than O(1), which also renders two widely separated length scales (in u nits
of string scale) in the putative gravity dual background. In the next section, we shall discuss
how nonstandard the dual geometry ought to be by using the non -exponential behavior of the
Wilsonloopas aprobe.
•ˆA1gaugetheory :
In this theory, there are two saddle-point equations corres ponding to two matrices Φ(1)and
Φ(2):
8π2
λ1φ(1)
k+2
NN
∑
i=1F′(φ(1)
k−φ(2)
i)−2
N∑
i/ne}ationslash=kF′(φ(1)
k−φ(1)
i)=2
N∑
i/ne}ationslash=k1
φ(1)
k−φ(1)
i,(3.44)
8π2
λ2φ(2)
k+2
NN
∑
i=1F′(φ(2)
k−φ(1)
i)−2
N∑
i/ne}ationslash=kF′(φ(2)
k−φ(2)
i)=2
N∑
i/ne}ationslash=k1
φ(2)
k−φ(2)
i,(3.45)
whereλ1=g2
1Nandλ2=g2
2Nare the‘t Hooftcouplingconstantsofeach gaugegroups.
Denoteρ(1)(φ),ρ(2)(φ)the eigenvalue distribution functions for the Φ(1),Φ(2)matrices,
respectively. Itis convenientto define
ρ(φ):=1
2(ρ(1)(φ)+ρ(2)(φ)), (3.46)
δρ(φ):=1
2(ρ(1)(φ)−ρ(2)(φ)). (3.47)
21In termsofthem,theabovesaddle-pointequationsaresimpl ifiedas follows:
4π2
λφ=/integraldisplay
−dφ′ρ(φ′)
φ−φ′, (3.48)
2π2/bracketleftBig1
λ1−1
λ2/bracketrightBig
φ−2/integraldisplay
−dφ′δρ(φ′)F′(φ−φ′) =/integraldisplay
−dφ′δρ(φ′)
φ−φ′, (3.49)
where
1
λ:=1
|Γ|/parenleftbigg1
λ1+1
λ2/parenrightbigg
and|Γ|=2. (3.50)
For obvious reasons, we refer these two as untwisted and twis ted saddle-point equations. By
the scaling argument, one can show that δρ(φ)is negligible compared to ρ(φ)in the large λ
limit. In particular,when λ1=λ2,itfollowsthat δρ=0is asolution,consistentwith Z2parity
exchangingthetwonodes. Therefore, thelarge λbehavioroftheWilsonloopisdeterminedby
(6.7), which is exactly the same as (3.9). Indeed, λdefined by (3.50) is exactly what is related
togsN[18].
The two Wilson loops are then obtainablefrom the one-matrix model with eigenvalueden-
sityρ±δρ:
W1=/integraldisplay
Ddxeaxρ(1)(x) =/integraldisplay
Ddxeax[ρ(x)+δρ(x)]
W2=/integraldisplay
Ddxeaxρ(2)(x) =/integraldisplay
Ddxeax[ρ(x)−δρ(x)]. (3.51)
Weseethat theuntwistedandthetwistedWilsonloopsare giv enby
W(0):=1
2(W1+W2)=/integraldisplay
Ddxeaxρ(x)
W(1):=1
2(W1−W2)=/integraldisplay
Ddxeaxδρ(x). (3.52)
Inferring from the saddle-point equations (3.48, 3.49), we see that these Wilson loops are di-
rectly related to the average and difference of the two gauge coupling constants. It also shows
thatthetwistedWilsonloopwillhavenonzero expectationv alueoncethetwogaugecouplings
are set different. In the next section, we shall see that they descend from moduli parameters of
six-dimensionaltwistedsectors at theorbifoldsingulari tyin theholographicdual description.
We have found the following result for the Wilson loop in ˆA1quiver gauge theory. The
two Wilson loops, corresponding to the two quiver gauge grou ps, have exponentially growing
behavior at large ‘t Hooft coupling limit. Its functional fo rm is exactly the same as the one
exhibitedby theWilsonloopin N=4superYang-Millstheory.
223.4 Interpolationamongthe quivers
Withthesaddle-pointequationsathand,wenowdiscussvari ousinterpolationsamong ˆA0,A1,ˆA1
theories and learn about the gauge dynamics. Our starting po int is the ˆA1theory, whose quiver
diagramhas twonodes. Seefigure 1.
•Considerthesymmetricquiverforwhichthetwo‘tHooftcoup lingconstantstaketheratio
λ1/λ2=1. Then the twisted saddle-point equation (3.49) asserts th atδρ=0 is the solution. It
follows that /an}bracketle{tW1/an}bracketri}ht−/an}bracketle{tW2/an}bracketri}ht=0, viz. the Wilson loop in the twisted sector vanishes identi cally.
Intuitively,the two gauge interactions are of equal streng th, so the two Wilson loops are indis-
tinguishable. Moreover,fromtheuntwistedsaddle-pointe quation(3.48),weseethattheWilson
loopintheuntwistedsectorbehavesexactlythesameastheo neinˆA0theoryand,inparticular,
N=4 superYang-Millstheory:
W(0)=1
2/parenleftBig
/an}bracketle{tW1/an}bracketri}ht+/an}bracketle{tW2/an}bracketri}ht/parenrightBig
=1√
2λI1(√
2λ). (3.53)
It follows that the Wilson loop grows exponentially at large ‘t Hooft coupling limit, much the
sameway asthe ˆA0theory does.
•Considertheasymmetricquiverwherethe ratio λ1/λ2/ne}ationslash=1 but finite. Thetwisted saddle-
pointequation(3.49)can berecast as
1
λ/parenleftbigg
B−1
2/parenrightbigg/integraldisplay
−dφ′ρ(φ′)
φ−φ′=/integraldisplay
−dφ′δρ(φ′)/bracketleftbigg1
21
φ−φ′+F′(φ−φ′)/bracketrightbigg
. (3.54)
Here, weparametrized thedifferenceoftwoinverse‘tHooft couplingsas
/parenleftbigg
B−1
2/parenrightbigg
:=1
2/parenleftbigg1
λ1−1
λ2/parenrightbigg/slashBig/parenleftbigg1
λ1+1
λ2/parenrightbigg
. (3.55)
Obviously, taking into account the Z2exchange symmetry between the two quiver nodes, B
ranges overtheinterval [0,+1]. Thesymmetricquiverconsidered abovecorresponds to B=1
2.
Solvingfirst ρfrom(3.48)andsubstitutingthesolutionto(3.54),onesol vesδρasafunctionof
B. Weseefrom(3.54)that δρoughttobea linearfunctionof Bthroughouttheinterval [0,+1].
Equivalently, extending the range of Bto(−∞,+∞), we see that δρis a sawtooth function,
piecewiselinearovereach unitintervalof B. Inparticular,itisdiscontinuousacross B=0(and
across all other nonzero integer values). This is depicted i n figure 3. Therefore, we conclude
that the Wilson loops W1,W2at strong ‘t Hooft coupling limit are nonanalytic not only in λ
but also in B. In fact, as we shall recall in the next section, B=0 is a special point where
thespacetimegaugesymmetryisenhancedandtheworldsheet conformalfieldtheorybecomes
23singular. Nevertheless,the Wilsonloopin theuntwistedse ctorbehaves exactlythesameas the
symmetric quiver, viz. (3.53). We conclude that the untwist ed Wilson loop is independent of
strengthofthegaugeinteractions.
-1               -1/2                 0                +1/2               +1   B  tW
Figure 3: Dependence of twisted sector Wilson loops on the parameter B. It shows discontinuity at
B=0,resulting in non-analytic behavior of the Wilson loops tob oth gauge couplings.
•Consider an extreme limit of the asymmetric quiver where the ratioλ1/λ2→0, equiva-
lently,λ2/λ1→∞,viz. thetwo‘tHooftcouplingsarehierarchicallyseparat ed. Inthiscase,one
gauge group is infinitely stronger than the other gauge group and theˆA1quiver gauge theory
ought to become the A1gauge theory . This can be seen as follows. In the ˆA1saddle-point
equations (3.45), we see that φ(1)→0 solves the first equation. Plugging this into the second
equation, we see it is reduced to the A1saddle-point equation (3.43). This reduction poses a
very interesting physics since from the above consideratio ns the Wilson expectation value in-
terpolates from the exponential growth of the ˆA1quiver gauge theory to the non-exponential
behavior of the A1gauge theory. In the next section, we shall argue that this is a clear demon-
stration (as probed by the Wilson loops) that holographic du al of theA1gauge theory ought to
haveinternalgeometryof stringscale size.
Wecanalsounderstandtheinterpolationdirectlyintermso ftheWilsonloop. Consider,for
example, λ2/λ1→∞. From the ˆA1Wilson loops, using the fact that ρ(1)(x),ρ(2)(x)are strictly
positive-definite,wehave
/an}bracketle{tW2/an}bracketri}ht=/integraldisplay
dλρ(2)(λ)eλ
24≤2/integraldisplay
dλ1
2[ρ(1)(λ)+ρ(2)(λ)]eλ
=4√
2λI1(√
2λ). (3.56)
Sinceλ∼λ1→0, the Wilson loop is bounded from above by a constant. Note th at the limit
λ1→0 can besafely taken: thesaddle-pointequation(3.48)isin fact exact in λ.
•Considerthelimit λ1,λ2→0. In thislimit,
λ=2λ1λ2
λ1+λ2→0,κ:=λ2
λ1=fixed (3.57)
and theexact result(3.53)isexpandablein powerseries of λandκ:
W(0)/vextendsingle/vextendsingle/vextendsingle
exact=1
2/parenleftBig
/an}bracketle{tW1/an}bracketri}ht+/an}bracketle{tW2/an}bracketri}ht/parenrightBig
=1
2∞
∑
ℓ=0∞
∑
m=0(−)m(ℓ+m−1)!
(ℓ−1)!ℓ!(ℓ+1)!λℓ
1κℓ+m. (3.58)
Here, the exact result (3.53) is symmetric under λ1↔λ2, so we assumed in (3.58) that κ<1.
Ontheotherhand,fromstandpointofthequivergaugetheory ,theWilsonloopinthefixed-order
perturbationtheoryis givenby powerseries in λ1orλ2:
W(0)/vextendsingle/vextendsingle/vextendsingle
pert=∞
∑
ℓ=0∞
∑
m=0Wℓ,mλℓ
1λm
2=∞
∑
ℓ=1∞
∑
m=1Wℓ,mλℓ+m
1κm. (3.59)
Weseethattheexactresult(3.58)andtheperturbativeresu lt(3.59)donotagreeeachother.
Recallthatbothresultsareobtainedatplanarlimit N→andoughttobeabsolutelyconvergentin
(λ,B)andin(λ1,λ2),respectively. Thereasonmaybethatthetwosetsofcouplin gconstantsare
notanalyticin C2complexplane. Infact,from(3.57),weseethat λ(λ1,λ2)hasacodimension-1
singularityat λ1+λ2=0. Anexceptionalsituationiswhen λ1=λ2. Inthiscase,thesingularity
disappearsand,withthesamepowerseriesexpansion,weexp ecttheexactresult(3.58)andthe
perturbativeresult (3.59)are thesame.
We should note that the change of variables is well-defined at strong coupling regime. In
thisregime,powerseriesexpansionsin1 /λ1and1/λ2isrelatedunambiguouslytopowerseries
expansionsin 1 /λandB. In fact, thechangeofvariables
/parenleftBig1
λ1,1
λ2/parenrightBig
−→/parenleftBig1
λ,B/parenrightBig
(3.60)
isanalyticanddoesnotintroduceanysingularityaround λ1,λ2=∞. Infact,aswewillrecapit-
ulate,theseare thevariablesnaturallyintroducedin theg ravitydualdescription.
WeremarkthattheanalyticstructureoftheWilsonloopsinq uivergaugetheoriesissimilar
totheIsingmodelinamagneticfieldonaplanarrandomlattic e[20]. Thelatterisdefined bya
25matrix modelinvolvingtwo interactingHermitian matrices and involvestwo couplingparame-
ters: average‘tHooftcouplingandmagneticfield. Hereagai n,byturningonthemagneticfield,
one can scale two independent ‘t Hooft coupling parameters d ifferently. In light of our results,
it would be extremely interesting to study this system in the limit the magnetic field is sent to
infinity.
4 IntuitiveUnderstandingofNon-Analyticity
In the last section, the distinguishingfeature of the A1theory from the ˆA0,ˆA1theories was that
growth of the Wilson loop expectation value was less than exp onential. Yet, these theories are
connected one another by continuously deforming gauge coup ling parameters. How can then
suchanon-analyticbehaviorcomeabout?3In thissection,weofferanintuitiveunderstanding
ofthis in termsof competitionbetween screening and over-s creeningof colorcharges and also
draw analogytotheKondoeffect ofmagneticimpurityinamet al.
•screeningversusanti-screening :
Consider first the weak coupling regime. The representation contents of these N=2 quiver
gaugetheoriesaresuchthatthe ˆA0theorycontainsfieldcontentsinadjointrepresentationso nly,
while the ˆA1and theA1theories contain additional field contents in bi-fundament al or funda-
mental representations, respectively. The A1theory contains additional massless multiplets in
fundamental representation, so we see immediately that the theory is capable of screening an
external color charge sourced by the Wilson loop for any repr esentations. Since the theory is
conformal, the screening length ought to be infinite (zero is also compatible with conformal
symmetry, but it just means there is no screening) and impedi ng creation of an excitation en-
ergyabovethegroundstate. Evenmoreso,‘tension’oftheco lorfluxtubewouldgotozero. In
other words, once a static color charge is introduced to the t heory, massless hypermultipletsin
fundamental representation will immediately screen out th e charge to arbitrary long distances.
Though this intuitive picture is based on weak coupling dyna mics, due to conformal symme-
try, it fits well with the non-exponential growth of the Wilso n loop in the A1theory, which we
derivedintheprevioussectionin theplanarlimit.
We stress that the screening has nothingto do with supersymm etrybut is a consequence of
elementary consideration of gauge dynamics with massless m atter in complex representations.
Thisisclearlyillustratedbythewellknowntwo-dimension alSchwingermodel. Generalization
of this Schwinger mechanism to nonabelian gauge theories sh owed that massless fermions in
arbitrarycomplexrepresentationscreenstheheavyprobechargeinth efundamentalrepresenta-
3ThisquestionwasraisedtousbyJuanMaldacena.
26tion[21]. The screening and consequentstring breakingby t hedynamical masslessmatterwas
observedconvincinglyinbothtwo-dimensionalQED[22]and three-dimensionalQCD[23]. In
four-dimensional lattice QCD, the static quark potential V(R)awas computed ( adenotes the
lattice spacing) for fermions in both quenched and dynamica l simulations [24]. For quenched
simulation,thepotentialscaledlinearlywith R/a,indicatingconfinementbehavior. Fordynam-
ical simulation, the potential exhibited flattening over a w ide range of the separation distance
R/a.
             (a)                                                                              (b)
Figure4: Responseofgaugetheoriestoexternalcolorchargesource. (a)ForA1theory,anexternalcolor
charge infundamental representation ofthegaugegroupiss creened bythe Nf=2Ncflavorsofmassless
matter fields, which are in fundamental representation (blu e arrow). (b) For ˆA1theory, an external color
charge in fundamental representation of the first gauge grou p is screened by the massless matter fields.
As the matter fields are in bi-fundamental representations ( black and white arrows), color charge in the
secondgaugegroupisregenerated andanti-screened. Thepr ocessrepeatsbetweenthetwogaugegroups
and leads thetheory to exhibit Coulomb behavior.
The case of ˆA1theory is more interesting. Having two gauge groups associa ted with each
nodes,considerintroducingastaticcolorcharge oftherep resentation Rfor, say,thefirst gauge
groupinSU (N)×SU(N). Thehypermultipletstransformingin (N,N)and(N,N)areindefining
representations with respect to the first gauge group, so the y will rearrange their ground-state
configuration to screen out the color charge. But then, as the se hypermultipletsare in defining
representationwithrespecttothesecondgaugegroupaswel l,acompletescreeningwithrespect
to the first gauge group will reassemble the resulting configu ration to be in the representation
27Rof the second gauge group in SU (N)×SU(N). This configuration is essentially the same as
thestartingconfigurationexceptthatthetwogaugegroupsa reinterchanged(alongwithcharge
conjugation). The hypermultiplets may opt to rearrange the ir ground-state configuration to
screenoutthecolorchargeofthesecondgaugegroup,butthe ntheprocesswillrepeatitselfand
returns back to the original static color charge of the first g auge group — in ˆA1theory, perfect
screeningofthefirstgaugegroupisaccompaniedbyperfecta nti-screeningofthesecondgauge
group and vice versa. This is depicted in figure 4. Consequent ly, a complete screening never
takes place for bothgauge groups simultaneously. Instead, the external color c harge excites
the ground-state to a conformally invariant configuration w ith the Coulomb energy. Again, we
formulated this intuitive picture from weak coupling regim e, but the picture fits well with the
exponentialgrowthoftheWilsonloopexpectationvalueof ˆA1theorywederivedintheprevious
sectionat planarlimit.
•AnalogytoKondoeffect :
It is interesting to observe that the screening vs. anti-scr eening process described above is
reminiscentofthemulti-channelKondoeffectinametal[25 ]. There,astaticmagneticimpurity
carrying aspin Sinteracts withconductionelectronsand profoundlyaffect s electrical transport
propertyatlongdistances. Supposeinametalthereare Nfflavorsofconductionbandelectrons.
Thus,thereare Nfchannels and theyare mutuallynon-interacting. Theantife rromagneticspin-
spin interaction between the impurity and the conduction el ectrons leads at weak coupling to
screening of the impurity spin StoSren= (S−Nf/2). We see that the system with Nf<2S
is under-screened, leading to an asymptotic screening of th e impurity spin and that the system
withNf>2Sisover-screened,leadingtoanasymptoticanti-screening oftheimpurityspin. The
marginallyscreenedcase, Nf=2S,isattheborderbetweenthescreeningandtheanti-screeni ng:
thespinSof themagneticimpurityis intact underrenormalization by the conductionelectrons
(modulo overall flip of the spin orientation, which is a symme try of the system). We thus
observe that the Coulomb behavior of the external color sour ce inˆA1theory is tantalizingly
parallel tothemarginallyscreened caseofthemulti-chann elKondoeffect.
•Interpretationviabraneconfigurations :
We can also understand the screening-Coulomb transition fr om the brane configurations de-
scribingˆA1andA1theories4. Consider Type IIA string theory on R8,1×S1, where the circle
direction is along x9and have circumference L. We set up thebrane configuration by introduc-
ing two NS5-branes stretched along (012345)directions and Nstack of D4-branes stretched
along(01239)directions on intervals between the two NS5-branes. Generi cally, the two NS5-
4Fora comprehensivereviewofbraneconfigurations,see [26] .
28branes are located at separate position on S1and this corresponds to the ˆA1theory. The gauge
couplings 1 /g2
1and 1/g2
2of the two quiver gauge groups are proportional to the length of the
twox9-intervalsoftheD4-branes. WhenthetwoNS5-branesareloc atedatdiagonallyopposite
points,say,at x9=0,L/2, thetwogaugecouplingsofthe ˆA1theoryare equal. Thisis depicted
in figure 5(a). By approaching one NS5-brane to another, say, atx9=0, we can obtain the
configurationin figure5(b). Thiscorrespondsto A1theory sincethegaugecouplingoftheD4-
branes encircling the S1becomes arbitrarily weak compared to that of the D4-branes s tretched
infinitesimallybetweenthetwooverlappingNS5-branes.
NS5                                                NS5                                                                         NS5-NS5
(a)                                                                                          (b) F1                                F1                                              F1                                    F1
Figure5: SemiclassicalWilsonloopinbraneconfigurationof N=2superconformal gaugetheoriesun-
der study: (a) ˆA1theory with G=SU(N)×SU(N) and2Nbifundamental hypermultiplets. ND4-branes
stretch between twowidely separated NS5-branes on acircle . TheF1(fundamental string) ending on or
emanating from D4-brane represent static charges. On D4-br anes, having finite gauge coupling, conser-
vation of the F1 flux is manifestly. (b) A1theory with G=SU(N) and2Nfundamental hypermultiplets.
TheA1theory is obtained from ˆA1in (a) by approaching the two NS5-branes. The flux is leaked in to
the coincident NS5-branes and run along their worldvolumes . On D4-branes, having vanishing gauge
coupling, conservation of the F1fluxisnot manifest.
We now introduce external color charge to the D4-branes and e xamine fate of the color
fluxes. Theexternalcolorsourcesareprovidedbyamacrosco picIIAfundamentalstringending
on the stacked D4-branes. Consider first the configuration of theˆA1theory. The color charge
is an endpoint of the fundamental string on one stack of the D4 -branes, viz. one of the two
quiver gauge groups. Along the D4-branes, the endpoint sour ces color Coulomb field. The
color field will sink at another external color charge locate d at a finite distance from the first
external charge. See figure 2(a). We see that the color flux is c onserved on the first stack of
D4-branes. Wealsoseethat,atweakcouplingregime,effect softheNS5-branesarenegligible.
Considernexttheconfigurationofthe A1theory. Basedontheconsiderationsoftheprevious
section,weconsideranexternalcolorchargetothestackof D4-branesencirclingthe S1. Inthis
29configuration, the two NS5-branes are coincident and this op ens up a new possible color flux
configuration. To understand this, we recall the situation o f stack of D1-D5 branes, which is
related to the macroscopic IIA string and stack of NS5-brane s. In the D1-D5 system, it is well
known that there are threshold bound states of D1-branes on D 5-branesprovided two or more
D5-branes are stacked. For a single D5-brane, the D1-brane b ound-state does not exist. This
suggestsinthebraneconfigurationofthe A1theorythatthecolorfluxmaynowbepulledtoand
smear out along the two coincident NS5-branes. From theview pointof stack of the D4-branes
encircling S1, thecolorflux appears not conserved.
5 HolographicDual
Theexactresultsofthe N=2Wilsonloopsatstrong‘tHooftcouplinglimitweobtainedi nthe
previoussectionrevealedmanyintriguingaspects. Inpart icular,comparedtothemorefamiliar,
exponentialgrowthbehaviorofthe N=4Wilsonloops,wefoundthefollowingdistinguishing
features and consequences:
•InA1gauge theory, the Wilson loop /an}bracketle{tW/an}bracketri}htdoesnotexhibit the exponential growth. Re-
placing 2 Nfundamental representation hypermultiplets by single adj oint representation
hypermultipletrestores the exponential growth, since the latter is nothing but the N=4
counterpart. This suggests that /an}bracketle{tW/an}bracketri}htinˆA1gauge theory has (possibly infinitely) many
saddle points and potential leading exponential growth is c anceled upon summing over
the saddle points. We stress that, in this case, the ratio of t wo ‘t Hooft coupling goes
to zero, equivalently, infinite. The limit decouples dynami cs of the two quiver gauge
groupsandrendertheglobalgaugesymmetryasanewlyemerge ntflavorsymmetry. The
non-exponential behavior of the Wilson loop originates fro m the decoupling, as can be
understoodintuitivelyfrom thescreening phenomenon.
•InˆA1quivergaugetheory,thetwoWilsonloops /an}bracketle{tW1/an}bracketri}ht,/an}bracketle{tW2/an}bracketri}htassociatedwiththetwoquiver
nodes exhibit the same exponential growth as the N=4 counterpart. The exponents
depend not only on the largest edge of the eigenvalue distrib ution but also on the two ‘t
Hooftcouplingconstants, λ1,λ2, equivalently, λ,B.
•InˆA1quiver gauge theory, in case the two ‘t Hooft couplings are th e same, so are the
two Wilson loops. If the two ‘t Hooft couplings differ butremain finite, the two Wilson
loops will also differ. As such, /an}bracketle{tW1/an}bracketri}ht−/an}bracketle{tW2/an}bracketri}htis an order parameter of the Z2parity ex-
changing the two quiver nodes. It scales linearly with Band shows non-analyticity over
thefundamentaldomain [−1
2,+1
2].
30In this section, we pose these features from holographic dua l viewpoint and extract several
new perspectives. Much of success of the AdS/CFT correspond ence was based on the obser-
vation that holographic dual geometry is macroscopically l arge compared to the string scale.
In this limit, string scale effects are suppressed and physi cal observables and correlators are
computable in saddle-point, supergravity approximation. For example, the AdS 5×S5dual to
theN=4superYang-Millstheory has thesize R2=O(√
λ):
ds2=R2ds2(AdS5)+R2dΩ2
5(S5), (5.1)
growing arbitrarily large at strong ‘t Hooft coupling. Many other examples of the AdS/CFT
correspondence share essentially the same behavior. In suc h a background, expectation value
oftheWilsonloop /an}bracketle{tW/an}bracketri}htisevaluatedbythePolyakovpathintegralofafundamentals tringinthe
holographicdualbackground:
/an}bracketle{tW/an}bracketri}ht:=/integraldisplay
C[DXDh]⊥exp(iSws[X∗g]) (5.2)
withaprescribedboundaryconditionalongthecontour CoftheWilsonloopattimelikeinfinity.
The worldsheet coupling parameter is set by the pull-back of the spacetime metric, and hence
byR2. AsRgrows large at strong ‘t Hooft coupling, the path integral is dominatedby a saddle
point and /an}bracketle{tW/an}bracketri}htexhibits exponential growth whose Euclidean geometry is th e minimal surface
Acl:
/an}bracketle{tW/an}bracketri}ht ≃eAclwhere Acl≃O(R2). (5.3)
NotethattheminimalsurfaceoftheWilsonloopsweepsoutan AdS3foliationinsidetheAdS 5.
Thisexplainsthe R2growthoftheareaoftheminimalsurface atstrong‘t Hooftco upling.
Central to our discussionswill consist of re-examination o n global geometry of the gravity
dualto N=2superconformalgaugetheoriesincomparisonto N=4superYang-Millstheory.
5.1 Holographic dualof A1gauge theory
At present, gravity dual to the A1gauge theory is not known. Still, it is not difficult to guess
whatthedualtheorywouldbe. Ingeneral, N=2gaugetheoryisdefinedinperturbationtheory
by threecouplingparameters:
λ,g2
c:=1
N2,go:=Nf
N, (5.4)
associated ‘t Hooftcoupling,closedsurface couplingasso ciatedwith adjointvectorand hyper-
multiplets, and open puncture coupling associated with fun damental hypermultiplets. For A1
31gauge theory, go=2∼O(1)and it indicates that dual string theory is described by the w orld-
sheet with proliferating open boundaries. Moreover, as we s tudied in earlier sections, the A1
gaugetheoryis related to the ˆA1quivergaugetheory as thelimitwhere oneofthetwo ‘t Hooft
coupling constants is sent to zero while the other is held fini te. Equivalently, in the large N
limit,oneofthetwo‘tHooftcouplingconstantsisdialedin finitelystrongerthantheother. This
hierarchical scaling limit of the two ‘t Hooft coupling cons tants, along with the PSU (2,2|2)
superconformal symmetry and the SU(2) ×U(1) R-symmetry imply that the gravity dual is a
noncritical superstring theory involving AdS 5andS2×S1space. One thus expects that the
gravitydual of A1gaugetheory hasthelocalgeometry oftheform:
(AdS5×M2)×[S1×S2]. (5.5)
By local geometry, we mean that the internal space consists o fS1andS2, possibly fibered or
warped over an appropriate 2-dimensionalbase-space M25. The curvature scales of AdS 5and
ofM2are equal and are set by R∼λ1/4, much as in the N=4 super Yang-Mills theory. The
remaining internal geometry [S1×S2]involves geometry of string scale, and is describable in
termsofa(singular)superconformalfieldtheory. Inpartic ular,theinternalspace [S1×S2]may
havecollapsed2-cycles. Therefore, theten-dimensionalg eometryis schematicallygivenby
ds2=R2(ds2(AdS5)+ds2(M2))+r2ds2([S1×S2]) (5.6)
whereR,rare the curvature radii that are hierarchically different, r≪R(measured in string
scale). Inparticular, rcanbecomesmallerthan O(1)intheregimethatthetwo‘tHooftcoupling
constantsaretaken hierarchically disparate.
Consider now evaluatingthe Wilson loop /an}bracketle{tW[C]/an}bracketri}htin thegravity dual (5.5). As well-known,
the Wilson loop is holographically computed by free energy o f a macroscopic string whose
endpoint sweeps the contour C. From the viewpointof evaluatingit in terms ofa minimalare a
worldsheet, since the internal space has nontrivial 2-cycl es, there will not be just one saddle-
point but infinitely many. These saddle-point configuration s are approximately a combination
of minimal surface of area Aswinside the AdS 5and surfaces of area a(i)
swwrapping 2-cycles
insidetheinternalspacemultipletimes. Notethat Aswhastheareaoforder O(r2)≫1instring
unit anda(i)
swhas the area of order O(1)since the 2-cycles are collapsed. Therefore, all these
configurations have nearly degenerate total worldsheet are a and correspond to infinitely many,
5The expected gravity dual (5.5) may be anticipated from the A rgyres-Seiberg S-duality [19]. At finite N, S-
duality maps an infinite coupling N=2 superconformal gauge theory to a weak coupling N=2 gauge theory
combined with strongly interacting, isolated conformal fie ld theory. The presence of the strongly interacting,
isolated conformal field theory suggests that putative holo graphic dual ought to involve a string geometry whose
size istypicallyoforder O(1)instringunit.
32nearbysaddlepoints. Ineffect,thesurfacesofarea a(i)
swwrappingthecollapsed2-cyclemultiple
timesproducesizableworldsheetinstantoneffects. Wethu shave
/an}bracketle{tW/an}bracketri}ht=∑
i=saddlescaexp/parenleftBig
Asw+a(i)
sw+···/parenrightBig
≃/bracketleftBig
∑
i=saddlescaexp(a(i)
sw)/bracketrightBig
·exp(Asw), (5.7)
wherecadenotes calculablecoefficients of each saddle-point,incl udingone-loop stringworld-
sheet determinants and integrals over moduli parameters, i f present. This is depicted in figure
6. Since we do not have exact worldsheet result for each saddl e point configurations available,
we can only guess what must happen in order for the final result to yield the exact result we
derived from the gauge theory side. In the last expression of (5.7), even though contribution
of individual saddle point is same order, summing up infinite ly many of them could produce
an exponentially small effect of order O(exp(−Asw)). What then happens is that summing
up infinitely many worldsheet instantons over the internal s pace cancels against the leading
O(exp(Asw))contributionfromtheworldsheetinsidetheAdS 5. Afterthecancelation,thelead-
ing nonzero contribution is of the same order as the pre-expo nential contribution. It scales as
Rνforsome finitevalueoftheexponent νat strong‘t Hooftcoupling.
<W>    =                             +                                  +                                  +                                 +  ....
Figure 6: Schematic view of holographic computation of Wilson loop ex pectation value in instanton
expansion. Each hemisphere represents minimal surface of s emiclassical string in AdS spacetime. In-
stantons are string worldsheets P1’s stretched into the internal space X5. Their sizes are of string scale,
and hence of order O(1)for any number of instantons. The gauge theory computations indicate that
these worldsheet instantons ought to proliferate and lead t o delicate cancelations of the leading-order
result (the first term) upon resummation.
At the orbifold fixed point, there are in general torsion comp onents of the NS-NS 2-form
potential B2, whoseintegralovera2-cycleisdenotedby B:
Ba:=/contintegraldisplay
CaB2
2π,Ba=[0,1) (5.8)
33TheA1theory has the global flavor symmetry Gf=U(Nf)=U(2N). For a well-defined con-
formal field theory of the internal geometry, Bamust take the value 1 /2. But then, the string
worldsheetwrappingthe2-cycle Canatimespicksup thephasefactor
∞
∏
a=1exp(2πiBana)=∞
∏
a=1(−)na, (5.9)
givingriseto ±relativesignsamongvariousworldsheetinstantoncontrib utionstotheminimal
surface dualtotheWilsonloop.
5.2 Holographic dualof ˆA1quiver gauge theory
Considernextholographicdescriptionof the ˆA1quivergaugetheory. It is knownthattheholo-
graphic dual is provided by the AdS 5×S5/Z2orbifold, where the Z2acts onC2⊂C3of the
coveringspaceof S5. Locally,thespacetimegeometryisexactly thesameas AdS 5×S5:
ds2=R2ds2(AdS5)+R2dΩ2
5(S5). (5.10)
Thesizeof boththe AdS5and theS5/Z2isR, which growsas (λ)1/4at large ‘t Hooft coupling
limit.
Located at the orbifold fixed point is a twisted sector. The ma ssless fields of the twisted
sector consists of a tensor multiplet of (5+1)-dimensional (2,0) chiral supersymmetry. The
multiplet contains five massless scalars. Three of them are a ssociated with S2replacing the
orbifoldfixed point,and theothertwo areassociated with
B=/contintegraldisplay
S2B2
2πandC=/contintegraldisplay
S2C2
2π, (5.11)
whereB2,C2are NS-NS and R-R 2-form potentials. Both of them are periodi c, ranging over
B,C=[0,1)6. Thesetwomasslessmoduliarewell-definedeveninthelimit thattheotherthree
modulivanish, viz. S2shrinks back to theorbifold singularity. Along withthe typ eIIB dilaton
andaxionoftheuntwistedsector, thesetwotwistedscalarfi elds arerelatedtothegaugetheory
parameters. In particular,wehave
1
gs=1
g2
1+1
g2
2;1
gs(B−1
2)=1
g2
1−1
g2
2. (5.12)
The other moduli field Cis related to the theta angles. This can be seen by uplifting t he brane
configuration to M-theory. There, the theta angle is nothing but the M-theory circle. It would
varyifweturn onC-potentialon twocycles.
6The periodicitycan be seen from the T-dual, brane configurat ionas well. Consider the moduli B. The quiver
gauge theories are mapped to D4 branes connecting adjacent N S5 branes on a circle in two different directions.
Thesumovergaugecouplingsisthenrelatedtocirclesize,w hilethedifferencebetweenadjacentgaugecouplings
isgivenbythelengthofeachinterval. Evidently,theinter valcannotbelongerthanthe circumference.
34Consider now computation of the Wilson loop expectation val ue from the Polyakov path
integral(5.2). Again,asthecontour CoftheWilsonloopliesattheboundaryofAdS 3foliation
insideAdS 5, theTypeIIB stringworldsheetwouldsweep a minimalsurfac ein AdS 3. Thearea
isoforder O(R2). Ontheotherhand,theTypeIIBstringmaysweepoverthevani shingS2atthe
orbifold fixed point. As the area of the cycle vanishes, the co rresponding worldsheet instanton
effect is of order O(1)and unsuppressed. Thus, the situation is similar to the A1case. In the
ˆA1case, however, we have a new direction of turning on the twist ed moduli associated with B.
From (5.12), we see that this amounts to turning on the two gau ge couplings asymmetrically.
Now, for the worldsheet instanton configuration, the Type II B string worldsheet couples to the
B2field. Therefore, theWilsonloopwillget contributionsofe xp(±2πiB)oncethemoduli Bis
turnedon.
There is another reason why infinitely many worldsheet insta ntons needs to be resummed.
We proved that the twisted sector Wilson loop is proportiona l to|B|. AsBranges over the in-
terval[−1
2,+1
2],weseethattheWilsonloophasnonanalyticbehaviorat B=0. Ingravitydual,
we argued that the Wilson loop depends on Bthrough the string worldsheet sweeping vanish-
ing two-cycle at the orbifold fixed point. The ninstanton effect is proportional to exp (2πinB)
forn=±1,±2,···. It shows that Bhas the periodicity over [−1
2,+1
2]and effect of individual
instantonis analytic overthe period. Obviously,in order t o exhibitnon-analyticity such as |B|,
infinitelymanyinstantoneffects needsto beresummed.
5.3 CommentsonWilsonloopsin Higgsphase
Startingfrom the ˆA1quivergaugetheory,wehaveanotherlimitwecan take. Consi dernowthe
D3-branesdisplacedawayfromtheorbifoldsingularity. If allthebranesaremovedtoasmooth
point,thenthequivergaugesymmetry Gisbroken tothediagonalsubgroup GD:
G=U(N)×U(N)→GD=UD(N) (5.13)
modulo center-of-mass U(1) group. Of the two bifundamental hypermultiplets, one of them is
Higgsed away and the other forms a hypermultiplet transform ing in adjoint representation of
the diagonal subgroup. This theory flows in the infrared belo w the Higgs scale to the N=4
superconformal Yang-Mills theory, as expected since the ND3-branes are stacked now at a
smoothpoint.
We should be able to understand the two Wilson loops of the ˆA1quiver gauge theory in
this limit. Obviously, the two Wilson loops W1,W2are independent and distinguishable at an
energy above the Higgs scale, while they are reduced to one an d the same Wilson loop at an
energy below the Higgs scale. Noting that Higgs scale is set b y the location of the D3-branes
from the orbifold singularity, we therefore see that the min imal surface of the macroscopic
35string worldsheet must exhibita crossover. How this crosso vertakes place is a very interesting
problemleft forthefuture.
Theaboveconsiderationisalsogeneralizableto variouspa rtialbreaking patternssuchas
SU(2N)×SU(2N)→SU(N)×SU(N)×SUD(N). (5.14)
Now,thereareseveraltypesofstrings. Therearestringsco rrespondingtoWilsonloopsofthree
SU(N)’s. There are also W-bosons that connect diagonal SU( N) to either of the two SU( N)’s.
The fields now transform as (N,N;1),(N,N;1)and(1,1,N2−1). As the theory is Higgsed,
localization method we relied on is no longer valid. Still, N evertheless, taking holographic
geometry of the conformal points of quiver gauge theories as the starting point, the gravity
dual is expected to be a certain class of multi-centered defo rmations. We expect that one can
stilllearn a lot of (quiver)gaugetheory dynamics by taking suitableapproximategravity duals
and then computing Wilson loop expectation values and compa ring them with weak ‘t Hooft
couplingperturbativeresults.
6 Generalizationto ˆAk−1QuiverGaugeTheories
So far, we were mainly concerned with A1andˆA1ofN=2 (quiver) gauge theories. These
are the simplest two within a series of ˆAk−1type. These quiver gauge theories are obtainable
fromD3-branessittingattheorbifoldsingularity C×(C2/Zk). Thereare (k−1)orbifoldfixed
pointswhoseblow-upconsistsof S2
i(i=1,···,k−1). ThetwistedsectoroftheTypeIIBstring
theory includes (k−1)tensor multiplets of (5+1)-dimensional (2,0) chiral supersymmetry.
Two setsof (k−1)scalarfields areassociated with
Bi=/contintegraldisplay
S2
iB2
2πandCi=/contintegraldisplay
S2
iC2
2π(i=1,···,k−1). (6.1)
Again,afterT-dualitytoTypeIIA stringtheory,weobtaint heˆAk−1braneconfiguration. Asfor
k=2, we first partially compactify the orbifold to S1of a fixed asymptotic radius and resolve
theˆAk−1singularities. This results in a hyperk¨ ahler space where t heS1is fibered over the
base space R3. The manifold is known as k-centered Taub-NUT space. There are 3 (k−1)
geometricmoduliassociatedwith (k−1)degenerationcenters(wherethe S1fiberdegenerates)
which, along with the 2 (k−1)moduli in (6.1), constitute5 scalar fields of the aforementi oned
(k−1)tensor multiplets. Now, T-dualizing along the S1fiber, we obtain Type IIA background
involving kNS5-branes,whichsourcenontrivialdilatonandNS-NS H3fieldstrength,sittingat
36the degeneration centers on the base space R3and at various positions on the T-dual circle /tildewideS1
set bythe Bi’sin(6.1).
In the Type IIA brane configuration, there are various limits where global symmetries are
enhanced. Atgenericdistributionof kNS5-branesonthedualcircle /hatwideS1,theglobalsymmetryis
givenbySU (2)×U(1)associatedwiththebasespace R3andthedualcircle /hatwideS1. When(fraction
of)NS5-branesallcoalescetogether,thespacetransverse totheNS5-branesapproaches C2very
close to them and the U (1)symmetry is enhanced to SU(2). In this limit, (a subset of) ga uge
couplings of D4-branes become zero and we have global symmet ry enhancement. It is well
known that k-stack of NS5-branes, which source the dilation and the NS-N SH3field strength,
generate the near-horizon geometry of linear dilaton [27]. In string frame, the geometry is the
exact conformalfield theory[28]
R5,1×/parenleftBig
Rφ,Q×SU(2)k/parenrightBig
where Q=/radicalbigg
2
k. (6.2)
Modulo the center of mass part, the worldvolume dynamics on D 4-branes stretched between
various NS5-branes can be described in terms of various boun dary states [29], representing
localized andextendedstates inthebulk.
Thestringtheoryinthisbackgroundbreaksdownatthelocat ionofNS5-branes,asthestring
couplingbecomesinfinitelystrong. Toregularizethegeome tryand definethestringtheory,we
maytake Cinsidetheaforementionednear-horizon C2,splitthecoincident kNS5-branesatthe
centerandarraythemonaconcentriccircleofanonzeroradi us. Thestringcouplingisthencut
off at a value set by the radius. The resulting worldsheet the ory is the N=2 supersymmetric
Liouvilletheory.
In the regime we are interested in, ktakes values larger than 2, k=3,4,···. In this regime,
theN=2 Liouville theory (6.2) is strongly coupled. By the supersy mmetric extension of the
Fateev-Zamolodchikov-Zamolodchikov(FZZ) duality, we ca n turn the N=2 supersymmetric
Liouville theory to Kazama-Suzuki coset theory. To do so, we T-dualize along the angular
direction of the arrayed NS5-branes. Conserved winding mod es around the angular direction
is mapped to conserved momentum modes and the resulting Type IIB background is given by
anotherexactconformal field theory
R5,1×/parenleftBigSL(2;R)k
U(1)×SU(2)k
U(1)/parenrightBig
(6.3)
moduloZkorbifolding. Forlarge k,theconformalfieldtheoryisweaklycoupledanddescribes
thewell-knowncigargeometry[30].
In the large (finite or infinite) k, what do we expect for the Wilson loop expectation value
and,fromtheexpectationvalues,whatinformationcanweex tractfortheholographicgeometry
37of gravity dual? Here, we shall remark several essential poi nts that are extendible straightfor-
wardly from the results of ˆA1and relegate further aspects in a separate work. For ˆAk−1quiver
gaugetheories,thereare knodesofgaugegroupsU( N). Associatedwiththemare kindependent
Wilsonloops:
W(i)[C]:=Tr(i)Psexp/bracketleftBig
ig/integraldisplay
Cd/parenleftBig
˙xmA(i)
m(x)+θaA(i)
a(x)/parenrightBig/bracketrightBig
(i=1,···,k).(6.4)
From these,wecan constructtheWilsonloopin untwistedand twistedsectors. Explicitly,they
are
W0=1
k/parenleftBig
W(1)+W(2)+···+W(k−1)+W(k)/parenrightBig
(6.5)
fortheuntwistedsectorWilsonloopand
W1=W(1)+ωW(2)+···+ωk−1W(k)
W2=W(1)+ω2W(2)+···+ω2(k−1)W(k)
···
Wk−1=W(1)+ωk−1W(2)+···+ω(k−1)2W(k)(6.6)
for the(k−1)independent twisted sector Wilson loops. They are simply knormal modes
of Wilson loops constructed from {ωn|n=0,···,k−1}Fourier series of Zkover thekquiver
nodes. Considernowtheplanarlimit N→∞. TheWilsonloops W(i)areallsame. Equivalently,
all the twisted Wilson loops vanish. Furthermore, as in ˆA1quiver gauge theory, the untwisted
Wilsonloopwillshowexponentialgrowthat large‘t Hooft co upling.
It isnot difficult to extendthegaugetheory results to ˆAk−1case. Aftertaking large Nlimit,
thesaddlepointequationsnowread
4π2
λφ=/integraldisplay
−dφ′ρ(φ′)
φ−φ′, (6.7)
2π2
λaφ−(1−ω)/integraldisplay
−dφ′δaρ(φ′)F′(φ−φ′) =/integraldisplay
−dφ′δaρ(φ′)
φ−φ′,(a=1,···,k−1)
(6.8)
where
ρ:=1
k/parenleftBig
ρ(1)+···+ρ(k)/parenrightBig
δaρ:=1
kk
∑
i=1ωi−1ρ(i)(a=1,2,···,k−1), (6.9)
38and
1
λ:=1
k/parenleftBig1
λ(1)+···+1
λ(k)/parenrightBig
1
λa:=1
kk
∑
i=1ωi−11
λ(i)(a=1,2,···,k−1). (6.10)
It isevidentthat δaρisproportionalto 1 /λalinearly,and henceexhibits non-analytic behavior.
BytheAdS/CFTcorrespondence,theWilsonloopsaremappedt omacroscopicfundamental
TypeIIBstringinthegeometryAdS 5×S5/Zk. Thereare (k−1)2-cyclesofvanishingvolume.
As in the ˆA1case,nworldsheet instanton picks up a phase factor exp (2πiBn). Again, since
B=1/2 for the exact conformal field theory, the phase factor is giv en by(−)n. As (fraction
of)thegaugecouplingsaretunedtozero,weagainseefrom(6 .8)thattwistedWilsonloopsare
suppressedbytheworldsheetinstantoneffects. Thisisthe effect ofthescreening weexplained
intheprevioussection,butnowextendedtothe ˆAk−1quivertheories. Thesuppression,however,
is less significant as kbecomes large since the one-loop contribution in (6.8) is hi erarchically
small compared to the classical contribution. We see this as a manifestation of the fact we
recalled abovethat,at k→∞, theworldsheet conformalfield theory isweakly coupled in T ype
IIB setupand theholographicdual geometry,thecigargeome try,becomes weaklycurved.
It is also illuminating to understand the above Wilson loops from the viewpoint of the
brane configuration. For the brane configuration, we start fr om the Type IIA theory on a
compact spatial circle of circumference L. We place kNS5-branes on the circle on intervals
La,(a=1,2,···,k)such that L1+L2+···+Lk=Land then stretch ND4-branes on each in-
terval. The low-energy dynamics of these D4-branes is then d escribed by N=2 quivergauge
theory of ˆAk−1type. In this setup, the W(a)Wilson loop is represented by a semi-infinite,
macroscopic string emanating from a-th D4-brane to infinity. Since there are kdifferent states
for identical macroscopic strings, we can also form linear c ombinations of them. There are k
different normal modes: the untwisted Wilson loop W0is the lowest normal mode obtained by
algebraic average of the kstrings,W1is the next lowest normal mode obtained by discrete lat-
ticetranslation ωforadjacentstrings, ···,andtheWk−1isthehighestnormalmodeobtainedby
discretelatticetranslation ωk−1(whichis thesameas theconfigurationwithlatticemomentum
ωby theUnklappprocess)foradjacent strings.
If the intervals are all equal, L1=L2=···=Lk=(L/k), then the brane configuration has
cyclicpermutationsymmetry. Thissymmetrythenensuresth atalltwistedWilsonloopsvanish.
If the intervals are different, (someof) the twisted Wilson loops are non-vanishing. If (fraction
of) NS5-branes become coalescing, the geometry and the worl dvolume global symmetries get
enhanced. We see that fundamental strings ending on the weak ly coupled D4-branes will be
pulled to the coalescing NS5-branes. The difference from th eA1theory is that, effect of other
39NS5-branes away from the coalescing ones becomes larger as kgets larger. This is the brane
configuration counterpart of the suppression of twisted Wil son loop expectation value which
wereattributedearlier totheweak curvatureofthehologra phicgeometry(6.3)inthislimit.
7 Discussion
In this paper, we investigated aspects of four-dimensional N=2 superconformal gauge theo-
ries. Utilizingthe localization technique, we showed that thepath integralof these theories are
reducedtoafinite-dimensionalmatrixintegral,muchasfor theN=4superYang-Millstheory.
The resulting matrix model is, however, non-Gaussian. Expe ctation value of half-BPS Wilson
loops in these theories can also be evaluated using the matri x model techniques. We studied
two theories in detail: A1gauge theory with gauge group U (N)and 2Nfundamental hyper-
multiplets and ˆA1quivergauge theory with gauge group U (N)×U(N)and two bi-fundamental
hypermultiplets.
In the planar limit, N→∞, we determined exactly the leading asymptotes of the circul ar
Wilson loops as the ‘t Hooft coupling becomes strong, λ→∞and then compared it to the
exponentialgrowth ∼exp(√
λ)seeninthe N=4superYang-Millstheory. Inthe A1theory,we
found the Wilson loop exhibits non-exponential growth: it is bounded from above in the large
λlimit. In the ˆA1theory, there are two Wilson loops, corresponding to the two U(N)gauge
groups. WefoundthattheuntwistedWilsonloopexhibitsexp onentialgrowth,exactlythesame
leading behavior as the Wilson loop in N=4 super Yang-Millstheory, but the twisted Wilson
loopexhibitsanew non-analytic behaviorindifference ofthetwogaugecouplingconstants.
Wealsostudiedholographicdualofthese N=2theoriesandmacroscopicstringconfigura-
tionsrepresentingtheWilsonloops. Wearguedthatboththe non-exponential behaviorofthe A1
Wilsonloop and the non-analytic behaviorofthe ˆA1Wilson loopsare indicativeofstringscale
geometriesofthegravitydual. Forgravitydualof A1theory,thereareinfinitelymanyvanishing
2-cyclesaroundwhichthemacroscopicstringwrapsarounda ndproduceworldsheetinstantons.
These different saddle-points interfere among themselves , canceling out the would-be leading
exponentialgrowth. What remains thereafter thenyields an on-exponentialbehavior, matching
with the exact gauge theory results. For gravity dual of ˆA1theory, there is again a vanishing
2-cycle at the Z2orbifold singularity. On the 2-cycle, NS-NS 2-form potenti al can be turned
on and it is set by asymmetry between the two gauge coupling co nstants. The macroscopic
string wraps around and each worldsheet instanton is weight ed by exp (2πiB). Again, since the
2-cycle has a vanishing area, infinite number of worldsheet i nstantons needs to be resummed.
The resummation can then yield a non-analytic dependence on B, and this fits well with the
40exact gaugetheoryresult.
A key lesson drawn from the present work is that holographic d ual of these N=2 super-
conformal gauge theories must involvegeometry of string sc ale. ForA1theory, suppression of
exponential growth of Wilson loop expectation value hints t hat the holographic duals must be
a noncritical string theory. In the brane construction view point, this arose because the two co-
inciding NS5-branes generates the well-known linear dilat on background near the horizon and
macroscopicstring is pulled to theNS5-branes. In theholog raphicdual gravity viewpoint,this
arosebecauseworldsheetofmacroscopicstringrepresenti ngtheWilsonloopisnotpeakedtoa
semiclassicalsaddle-pointbutisaffectedbyproliferati ngworldsheetinstantons. Wearguedthat
delicate cancelation among the instanton sums lead to non-e xponential behavior of the Wilson
loop.
It should be possible to extend the analysis in this paper to g eneral N=2 superconformal
gauge theories. Recently, various quiver constructions we re put forward [31] and some of its
gravity duals were studied [32]. Main focus of this line of re search were on quivergeneraliza-
tion of the Argyres-Seiberg S-duality, which does not commu te with the large Nlimit. Aim of
the present work was to characterize behavior of the Wilson l oop in large Nlimit in terms of
representationcontentsofmatterfieldsand,fromtheresul ts,infertheholographicgeometryof
gravityduals. Wealsoremarkedthatourapproachiscomplem entarytotheresearchesbasedon
variousworldsheetformulations[33][34][35][36].
Recently, localization in the N=6 superconformal Chern-Simons theory was obtained
and Wilson loops therein was studied in detail [37]. It shoul d also be possible to extend the
analysis to the superconformal (quiver) Chern-Simons theo ries. In particular, given that these
twotypesoftheoriesarerelatedroughlyspeakingbypartia llycompactifyingon S1andflowing
intoinfrared,understandingsimilaritiesanddifference sbetweenquivergaugetheoriesin(3+1)
dimensionsandin(2+1)dimensionswouldbeextremelyusefu lforelucidatingfurtherrelations
ingaugeandstringdynamics.
Finally, it should be possible to extend the analysis in this work to N=1 superconformal
quiver gauge theories and study implications to the Seiberg duality. Candidate non-critical
stringdualsofthesegaugetheorieswere proposedby[38].
Wearecurrentlyinvestigatingtheseissuesbutwillrelega tereportingourfindingstofollow-
up publications.
41Acknowledgments
WearegratefultoZoltanBajnok,DongsuBak,DavidGrossand JuanMaldacenaforusefuldis-
cussionsontopicsrelatedtothisworkandcomments. SJRtha nksKavliInstituteforTheoretical
Physics for hospitality during this work. TS thanks KEK Theo ry Group, Institute for Physics
andMathematicsoftheUniverseandAsia-PacificCenterforT heoreticalPhysicsforhospitality
duringthiswork. ThisworkwassupportedinpartbytheNatio nalScienceFoundationofKorea
Grants 2005-084-C00003, 2009-008-0372, 2010-220-C00003 , EU-FP Marie Curie Research
& Training Networks HPRN-CT-2006-035863 (2009-06318) and U.S. Department of Energy
Grant DE-FG02-90ER40542.
A Killingspinoron S4
TheKillingspinorson S4aredefinedasfollows. Let ya(a=1,···,5)becoordinatesof R5. We
embedS4intoR5bythehypersurface
(ya+za)2=r2,za=(0,···,0,r). (A.1)
Eachpointon S4canbemappedtoapointonafour-dimensionalhyperplane R4,y5=0,tangent
totheNorthPolethrough
ya=−2za+eΩ(xa+2za),eΩ=/parenleftbigg
1+x2
4r2/parenrightbigg−1
, (A.2)
wherexa=(xm,x5=0). Thisdescribes aprojectionon R4from theSouthPoleof S4. Accord-
ingly,theinducedmetricon S4isgivenby
ds2=hmndxmdxn
=e2Ωδmndxmdxn. (A.3)
Letθbe the polar angle measured from the North Pole, viz. the orig in of theR4. Then, for a
fixedθ,thecoordinates xmsatisfy
4
∑
m=1(xm)2=4r2tan2θ
2. (A.4)
Wealso denoteorthonormalframecoordinatesas xˆm,(ˆm=ˆ1,···,ˆ4)withvierbein eˆm
m=δˆm
meΩ.
42It isstraightforward toshowthatthespinors
ξ=e1
2Ω(ξs+xˆmΓˆmξc), (A.5)
/tildewideξ=e1
2Ω(ξc−1
4r2xˆmΓˆmξs), (A.6)
whereξsandξcare arbitrary constant Majorana-Weyl spinors, satisfy the conformal Killing
spinorequations
∇mξ=Γm/tildewideξ,∇m/tildewideξ=−1
4r2Γmξ. (A.7)
We furtherimposeanti-chiralitycondition:
Γˆ1ˆ2ˆ3ˆ4ξs=−ξs,ξc=1
2rΓ0ˆ1ˆ2ξs. (A.8)
Theseequationsimply
ξ/tildewideξ=0,ξΓ05/tildewideξ=0. (A.9)
Onecan showthatthecomponentsof vM=ξΓMξhavethefollowingexplicitforms:
v1=x2
r,v2=−x1
r, (A.10)
v3=x4
r,v4=−x3
r, (A.11)
v0=−1,v5=cosθ, (A.12)
v6,7,8,9=0, (A.13)
wherewenormalized ξssuchthat ξsΓ0ξs=−1.
Theexpression(A.5) can berewrittenas follows:
ξ=e1
2Ωξs+1
2e−1
2ΩvˆmΓˆmΓ5ξs. (A.14)
Wedefine
nˆm:=vˆm
sinθ(A.15)
sothat
(nˆmΓˆmΓ5)2=−1. (A.16)
Then, itiseasy to showthat theconformal Killingspinorise xpressibleas
ξ(x) =/parenleftbigg
cosθ
2+sinθ
2nˆm(x)ΓˆmΓ5/parenrightbigg
ξs
=exp/parenleftbiggθ
2nˆm(x)ΓˆmΓ5/parenrightbigg
ξs. (A.17)
43Theconformal Killingspinors ξand/tildewideξsatisfythefollowingidentities:
vm∇mξ−1
2(ξΓmn/tildewideξ)Γmnξ+1
2(ξΓst/tildewideξ)Γstξ=0, (A.18)
vm∇m/tildewideξ−1
2(ξΓmn/tildewideξ)Γmn/tildewideξ+1
2(ξΓst/tildewideξ)Γst/tildewideξ=0. (A.19)
B Spinorsfor off-shellclosure
Wedefine
ν˙m
0:=Γ˙mΓˆ1ξs,νs
0:=ΓsΓˆ1ξs, (B.1)
where ˙m=ˆ2,ˆ3,ˆ4. LetI=(˙m,s). Itcan beshownthat
ξsΓMνI
0=0, (B.2)
νI
0ΓMνJ
0=δIJξsΓMξs, (B.3)
1
2vM
sΓM=ξsξs+νI
0ν0I (B.4)
hold,where vM
s=ξsΓMξs. Sinceξisobtainedfrom ξsthrougharotation,ifwe define
νI:=exp/parenleftBigθ
2nˆmΓ5Γˆm/parenrightBig
νI
0, (B.5)
thenthefollowingrelationsfollow:
ξΓMνI=0, (B.6)
νIΓMνJ=δIJξΓMξ, (B.7)
1
2vMΓM=ξξ+νIνI (B.8)
Ifthelastequationis projectedontothespaceof λ,onefinds
1
2vMΓM=ξξ+ν˙mν˙m, (B.9)
whilein thespace of ψ, itbecomes
1
2vMΓM=νανα. (B.10)
44Thespinorssatisfythefollowingidentities:
vm∇mν˙k−1
2(ξΓmn/tildewideξ)Γmnν˙k+1
2(ξΓst/tildewideξ)Γstν˙k+(ν˙kΓm∇mν˙n)ν˙n=0,(B.11)
vm∇mνα−1
2(ξΓmn/tildewideξ)Γmnνα−νβνβΓm∇mνα=0.(B.12)
Dueto theabovechoiceofspinors, Q2closes onfields as follows:
−iQ2Am=vn∇nAm+∇mvnAn−ig[vµAµ,Am]−∇m(vµAµ), (B.13)
−iQ2Aa=vm∇mAa−ig[vµAµ,Aa], (B.14)
−iQ2qα=vm∇mqα−ig(vµAA
µ)TAqα+2ξγα
β/tildewideξqβ, (B.15)
−iQ2qα=vm∇mqα+ig(vµAA
µ)qαTA−2qβξγβα/tildewideξ, (B.16)
−iQ2λ=vm∇mλ−1
2(ξΓmn/tildewideξ)Γmnλ−ig[vµAµ,λ]+1
2(ξΓst/tildewideξ)Γstλ,(B.17)
−iQ2ψ=vm∇mψ−1
2(ξΓmn/tildewideξ)Γmnψ−ig(vµAA
µ)TAψ, (B.18)
−iQ2ψ=vm∇mψ+1
2(ξΓmn/tildewideξ)ψΓmn+ig(vµAA
µ)ψTA, (B.19)
−iQ2K˙m=vk∇kK˙m−ig[vµAµ,K˙m]+ν˙mΓk∇kν˙nK˙n, (B.20)
−iQ2Kα=vm∇mKα−ig(vµAA
µ)TAKα+ναΓm∇mνβKβ, (B.21)
−iQ2Kα=vm∇mKα+ig(vµAA
µ)KαTA−KβνβΓm∇mνα. (B.22)
C AsymptoticexpansionofWilson loop
Inthisappendix,weprovidedetailsoftheasymptoticexpan sionoftheWilsonloopinthelarge
alimit.
We firstestimatethefollowingintegral:
I(α,a):=/integraldisplay∞
δdu uαe−au, (C.1)
wherea,α,δ>0. Thissatisfiestherelation
I(α,a)=δα
ae−δa+α
aI(α−1,a). (C.2)
45There exists an integer Kfor which α−K+1>0 andα−K<0. Then, repeating integration
by parts,I(α,a)can bewrittenas
I(α,a)=K−1
∑
n=0δα−n
an+1Γ(α+1)
Γ(α+1−n)e−δa+1
aKΓ(α+1)
Γ(α+1−K)I(α−K,a).(C.3)
I(α−K,a)isestimatedas follows:
I(α−K,a)≤δα−K/integraldisplay∞
δdue−au=δα−K
ae−δa. (C.4)
Therefore, forlarge a,I(α,a)is estimatedto be
I(α,a)=O(a−1e−δa). (C.5)
With the above result, we now estimate W. With the assumed behavior of rescaled density
function/tildewideρinsection3, onecan write e−caWas
/integraldisplay1−a
b
0du/tildewideρ(1−u)e−cau=β/integraldisplayδ
0duuαe−cau+/integraldisplayδ
0duχ(u)e−cau+/integraldisplay1−a
b
δdu/tildewideρ(1−u)e−cau.(C.6)
Thefirst termoftheright-handsideis
β/integraldisplayδ
0duuαe−cau=β/integraldisplay∞
0du uαe−cau−βI(α,ca)
=βΓ(α+1)(ca)−α−1+O((ca)−1e−δca). (C.7)
The second term can be evaluated similarly, and it turns out t o be negligible compared to the
first term. Thethirdterm is
/integraldisplay1−a
b
δdu/tildewideρ(1−u)e−cau≤e−δca/integraldisplay1−a
b
δdu/tildewideρ(u−1)≤e−δca. (C.8)
Thiscompletestheproofoftheproclaimedestimate(3.7)in thelargealimit.
D Coefficient c1
In this appendix, we elaborate detailed calculation of the c oefficient c1of the leading term in
theone-loopdeterminant. Theheat-kernel coefficient a2(Δ)is
a2(Δ)=1
(4π)2/integraldisplay
S4d4x√
htrB/bracketleftBig
−1
4r2(3+cos2θ)+1
6R/bracketrightBig
, (D.1)
46where tr Bis the trace over the indices α,β. The second term is canceled by the fermionic
contribution. Thefirst termyields −5
12r2.
Thecoefficient a2(ΔF)forthefermionsis
a2(ΔF)=1
(4π)2/integraldisplay
S4d4x√
htrF/bracketleftBig3κ2
r3+κ2
4(ξΓµν/tildewideξ)(ξΓρσ/tildewideξ)ΓµνΓρσ+1
6R/bracketrightBig
,(D.2)
where tr Fis the trace over the subspace of the spinor corresponding to ψ. One can show that
thefirst twotermscancel each other.
As the−ΔFhas the term linear in m,a4(ΔF)also contribute to c1. The relevant part of the
coefficient a4(ΔF)is
1
(4π)2/integraldisplay
S4d4x√
htrF/bracketleftBig1
2/parenleftBig
iκm
r(ξΓµν/tildewideξ)Γµν/parenrightBig2/bracketrightBig
=−2
3m2. (D.3)
As aresult,it followsthat
c1=−/parenleftBig
−5
12/parenrightBig
−1
2/parenleftBig
−2
3/parenrightBig
=3
4. (D.4)
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