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. 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