diff --git "a/1001.0016.txt" "b/1001.0016.txt" new file mode 100644--- /dev/null +++ "b/1001.0016.txt" @@ -0,0 +1,2042 @@ +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 +sjrey@snu.ac.kr suyama@phya.snu.ac.kr +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<00)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. + = + + + + .... +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) +References +[1] J. M. Maldacena, The large N limit of superconformal field theories and superg rav- +ity, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] +[arXiv:hep-th/9711200]. +[2] S. J. Rey and J. T. Yee, Macroscopic strings as heavy quarks in large N gauge theory +and anti-de Sitter supergravity , Eur. Phys. J. C 22(2001) 379 [arXiv:hep-th/9803001]. +S. J. Rey, S. Theisen and J. T. Yee, Wilson-Polyakov loop at finite temperature in +large N gauge theory and anti-de Sitter supergravity , Nucl. Phys. B 527(1998) 171 +[arXiv:hep-th/9803135]. +[3] J. M. Maldacena, Wilson loops in large N field theories , Phys. Rev. Lett. 80(1998) 4859 +[arXiv:hep-th/9803002]. +[4] J. K. Erickson, G. W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric +Yang-Millstheory ,Nucl. Phys.B 582, 155(2000)[arXiv:hep-th/0003055]. +[5] N.DrukkerandD.J.Gross, AnexactpredictionofN=4SUSYMtheoryforstringtheory , +J.Math.Phys. 42(2001)2896[arXiv:hep-th/0010274]. +47[6] V. Pestun, Localization of gauge theory on a four-sphere and supersymm etric Wilson +loops,arXiv:0712.2824[hep-th]. +[7] S. J. Rey, T. Suyama and S. Yamaguchi, Wilson Loops in Superconformal Chern-Simons +Theory and Fundamental Strings in Anti-de Sitter Supergrav ity Dual, JHEP0903(2009) +127[arXiv:0809.3786[hep-th]]. +[8] N. Drukker, J. Plefka and D. Young, Wilson loops in 3-dimensional N=6 supersym- +metric Chern-Simons Theory and their string theory duals , JHEP0811, 019 (2008) +[arXiv:0809.2787[hep-th]]. +[9] B.ChenandJ.B.Wu, SupersymmetricWilsonLoopsinN=6SuperChern-Simons-mat ter +theory,arXiv:0809.2863[hep-th]. +[10] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, N=6 superconformal Chern- +Simons-matter theories, M2-branes and their gravity duals , JHEP0810, 091 (2008) +[arXiv:0806.1218[hep-th]]. +[11] T. Suyama, talk given at KEK String Advanced Lec- +tures Workshop (April 17, 2009, Tsukuba, Japan) +http://research.kek.jp/group/www-theory/theory center/SAL/slides/Suyama 090417.pdf +. +[12] S.J. Rey, talk given at Strings 2009 Conference (June 25 , 2009, Rome, Italy) +http://strings2009.roma2.infn.it/talks/Rey Strings09.PDF . +[13] N. Berkovits, A Ten-dimensional superYang-Mills action with off-shell s upersymmetry , +Phys.Lett.B 318,104(1993)[arXiv:hep-th/9308128]. +[14] J.M.Evans, SupersymmetryalgebrasandLorentzinvarianceford=10sup erYang-Mills , +Phys.Lett.B 334,105(1994)[arXiv:hep-th/9404190]. +[15] L.Baulieu,N.J.Berkovits,G.BossardandA.Martin, Ten-dimensionalsuper-Yang-Mills +with nine off-shell supersymmetries , Phys. Lett. B 658, 249 (2008) [arXiv:0705.2002 +[hep-th]]. +[16] D. V. Vassilevich, Heat kernel expansion: User’s manual , Phys. Rept. 388, 279 (2003) +[arXiv:hep-th/0306138]. +[17] A.Voros, Spectralfunctions,specialfunctionsandSelbergzetafun ction,Commun.Math. +Phys.110,439(1987). +48[18] I.R.KlebanovandN.A.Nekrasov, GravitydualsoffractionalbranesandlogarithmicRG +flow,Nucl. Phys.B 574(2000)263[arXiv:hep-th/9911096]. +[19] P. C. Argyres and N. Seiberg, S-duality in N=2 supersymmetric gauge theories , JHEP +0712(2007)088 [arXiv:0711.0054[hep-th]]. +[20] D. V. Boulatov and V. A. Kazakov, The Ising Model On Random Planar Lattice: The +StructureOfPhaseTransitionAndTheExactCriticalExpone nts,Phys.Lett. 186B(1987) +379. +[21] D.J. Gross, I.R. Klebanov, A.V. Matytsin and A.V. Smilg a,Screening vs. Confinement in +1+1Dimensions ,[arXiv:hep-th/9511104]. +[22] W.A.Bardeen,A.Duncan,E.EichtenandH.Thacker, Quenchedapproximationartifacts: +A Studyintwo-dimensionalQED , Phys.Rev.D 57(1998)3890. +[23] H. D. Trottier, String breaking by dynamical fermions in lattice QCD: From t hree to four +dimensions ,Phys. Rev.D 60(1999)034506[arXiv:hep-lat/9812021]. +[24] A. Duncan, E. Eichten and H. Thacker, String breaking in four dimensional lattice QCD , +Phys.Rev.D 63(2001)111501[arXiv:hep-lat/0011076]: +C. W. Bernard et al.,Zero temperature string breaking in lattice quantum chromo dynam- +ics,Phys.Rev. D 64(2001)074509[arXiv:hep-lat/0103012]: +G. S. Bali, H. Neff, T. Duessel, T. Lippert and K. Schilling [S ESAM Collabo- +ration],Observation of string breaking in QCD , Phys. Rev. D 71(2005) 114513 +[arXiv:hep-lat/0505012]. +[25] For a modern review, see I. Affleck, Conformal field theory approach to Kondo effect , +[arXiv:cond-mat/9512099]andoriginalreferences therei n. +[26] A.GiveonandD.Kutasov, Branedynamicsandgaugetheory ,Rev.Mod.Phys. 71(1999) +983[arXiv:hep-th/9802067]. +[27] S.J.Rey, Theconfiningphaseofsuperstringsandaxionicstrings ,Phys.Rev.D 43(1991) +526; +C. G. . Callan, J. A. Harvey and A. Strominger, Worldbrane actions for string solitons , +Nucl.Phys.B 367(1991)60. +[28] S. J. Rey, Axionic string instantons and their low-energy implicatio ns, in ’Workshop on +SuperstringandParticleTheory’eds.L.ClavelliandB.Har ms,pp.291-300(1989,World +Scientific, Singapore); +49C. G. . Callan, J. A. Harvey and A. Strominger, World sheet approach to heteroticinstan- +tonsand solitons ,Nucl. Phys.B 359(1991)611; +S. J. Rey, On string theory axionicstrings and instantons ,in ’APS-DPF Annual Meeting’ +eds. D. Axen, D. Bryman and M. Comyn, pp. 876-881 (1991, World Scientific, Singa- +pore); +C. G. . Callan, J. A. Harvey and A. Strominger, Supersymmetric string solitons . +arXiv:hep-th/9112030. +[29] S. Elitzur, A. Giveon,D. Kutasov,E. Rabinoviciand G. S arkissian, D-branes in theback- +groundofNSfivebranes ,JHEP0008(2000)046[arXiv:hep-th/0005052]. +[30] A.GiveonandD.Kutasov, Littlestringtheoryinadoublescalinglimit ,JHEP9910(1999) +034[arXiv:hep-th/9909110]; +A. Giveon and D. Kutasov, Comments on double scaled little string theory , JHEP0001 +(2000)023 [arXiv:hep-th/9911039]. +[31] D.Gaiotto, N=2 dualities ,arXiv:0904.2715[hep-th]. +[32] D. Gaiotto and J. Maldacena, The gravity duals of N=2 superconformal field theories , +arXiv:0904.4466[hep-th]. +[33] H. Kawai and T. Suyama, AdS/CFT Correspondence as a Consequence of Scale Invari- +ance, Nucl. Phys.B 789,209 (2008)[arXiv:0706.1163[hep-th]]. +[34] H. Kawai and T. Suyama, Some Implicationsof PerturbativeApproach to AdS/CFT Cor- +respondence ,Nucl. Phys.B 794, 1(2008)[arXiv:0708.2463[hep-th]]. +[35] N.BerkovitsandC.Vafa, TowardsaWorldsheetDerivationoftheMaldacenaConjectur e, +JHEP0803,031(2008)[AIPConf. Proc. 1031,21(2008)][arXiv:0711.1799[hep-th]]. +[36] T. Azeyanagi, M. Hanada, H. Kawai and Y. Matsuo, Worldsheet Analysis of +Gauge/GravityDualities ,arXiv:0812.1453[hep-th]. +[37] A.Kapustin,B. WillettandI.Yaakov, ExactResultsforWilsonLoopsinSuperconformal +Chern-SimonsTheories withMatter ,arXiv:0909.4559[hep-th]; +T.Suyama, On LargeN SolutionofABJM Theory ,arXiv:0912.1084[hep-th]; +N.DrukkerandD.Trancanelli, AsupermatrixmodelforN=6superChern-Simons-matter +theory,arXiv:0912.3006[hep-th]; +M. Marino and P. Putrov, Exact Results in ABJM Theory from Topological Strings , +arXiv:0912.3074[hep-th]. +50[38] D. Israel, Non-critical string duals of N = 1 quiver theories , JHEP0604(2006) 029 +[arXiv:hep-th/0512166]. +51 \ No newline at end of file