1/N expansion of circular Wilson loop in \mathcal N=2 superconformal SU(N)\times SU(N) quiver
IImperial-TP-AT-2021-01 { N expansion of circular Wilson loopin N “ superconformal SU p N q ˆ SU p N q quiver M. Beccaria a and A.A. Tseytlin b, a Università del Salento, Dipartimento di Matematica e Fisica
Ennio De Giorgi ,and I.N.F.N. - sezione di Lecce, Via Arnesano, I-73100 Lecce, Italy b Blackett Laboratory, Imperial College London SW7 2AZ, U.K.
E-mail: [email protected], [email protected]
Abstract
Localization approach to N “ superconformal SU p N q ˆ SU p N q quiver theory leads toa non-Gaussian two-matrix model representation for the expectation value of BPS circular SU p N q Wilson loop x W y . We study the subleading { N term in the large N expansion of x W y at weak and strong coupling. We concentrate on the case of the symmetric quiver with equalgauge couplings which is equivalent to the Z orbifold of the SU p N q N “ SYM theory. Thisorbifold gauge theory should be dual to type IIB superstring in
AdS ˆ p S { Z q . We presenta string theory argument suggesting that the { N term in x W y in the orbifold theory shouldhave the same strong-coupling asymptotics λ { as in the N “ SYM case. We support thisprediction on the gauge theory side by a numerical study of the localization matrix model.We also find a relation between the { N term in the Wilson loop expectation value and thederivative of the free energy of the orbifold gauge theory on 4-sphere. Also at the Institute of Theoretical and Mathematical Physics, MSU and Lebedev Institute, Moscow. a r X i v : . [ h e p - t h ] F e b ontents x W y orb
145 Weak coupling expansion for non-symmetric quiver 166 Numerical analysis of the quiver matrix model 17
A Multi-trace SU p N q recursion relations and x W O y
22B Coefficient functions of ζ -terms in x W y orb
23C Wilson loop in SU p N q “orientifold” N “ superconformal theory 25 Supersymmetric Wilson loop operators provide an important class of observables that shed light onthe intricate structure of weak-strong coupling interpolation in the context of AdS/CFT duality.In special cases with extended supersymmetry the localization method [1] allows one to representthe expectation value of a supersymmetric loop in terms of a matrix model integral.Here we will consider a particular N “ supersymmetric gauge theory which is the SU p N q ˆ SU p N q quiver with two bi-fundamental hypermultiplets [2, 3, 4, 5]. In the “symmetric” case whenthe two ’t Hooft couplings λ , λ are equal this theory is equivalent to the Z orbifold of the SU p N q N “ SYM theory [6]. The orbifold theory has the same planar diagrams as the parent N “ SYM theory [7], i.e. the two are closely related at large N . The dual string theory shouldbe the corresponding orbifold of the AdS ˆ S superstring, i.e. type IIB string on AdS ˆ p S { Z q [8, 9]. For each of the two SU p N q factors of the quiver theory one may define -BPS circular Wilsonloops coupled to the corresponding gauge and scalar fields ( a “ , ) W a “ tr P exp ” ¿ ds p i x µ A µ a ` | x | Φ a q ı , (1.1) Z acts by flipping 4 of the 6 embedding coordinates of the 5-sphere, reflecting the 2+4 split of the N “ SYMscalars between the vector multiplets and the hypermultiplets of the N “ theory. { N factor in front of the trace. For the orbifold theory their(normalized) expectation values are equal x W y “ x W y ” x W y orb , (1.2)and at large N coincide [4] with the famous SU p N q N “ SYM result [10, 11, 12] x W y orb N Ñ8 “ x W y SYM N Ñ8 “ x W y , x W y “ N ? λ I p? λ q λ " “ c π N λ ´ { e ? λ ” ` O ´ ? λ ¯ı . (1.3)For general N the expression for x W y orb is given by a special non-Gaussian matrix model integralfollowing from the localization approach [12]. In contrast to the N “ SYM case where thecorresponding matrix model is Gaussian leading to the closed expression [11, 12] x W y SYM “ e λ N p ´ N q L p q N ´ ´ ´ λ N ¯ “ N e ? λ ÿ p “ ? p ? π p ! λ p ´ N p ” ` O ´ ? λ ¯ı , (1.4)working out the { N expansion of x W y orb turns out to be a non-trivial problem. Below we willaddress the question about the structure of the λ -dependent coefficients in the { N expansion of x W y orb by considering separately the small and large λ limits.On the dual string theory side, the { N expansion is the genus expansion, and the λ dependenceof the { N p coefficient in the analog of (1.4) corresponds to the string tension dependence of thepartition function with world surface of topology of a disc with p handles.As discussed recently in [13], the strong coupling expansions of the -BPS circular Wilson loopsin N “ SYM and ABJM gauge theories with string duals defined on
AdS ˆ S and AdS ˆ CP have a remarkably similar structure. The string counterpart of the dominant at large N ( p “ )term in (1.3),(1.4) is the open-string partition function on the disk which contains an overall factorof the inverse closed string coupling g s x W y “ ? T π g s e πT e ´ ¯Γ ” ` O ´ T ´ ¯ı . (1.5)In the SU p N q N “ SYM case g s “ g π “ λ πN , T “ L πα “ ? λ π , ¯Γ “
12 log p π q , (1.6)so that the leading term in (1.5) is the same as in (1.3).In general, the presence of the universal ? T prefactor in (1.5) follows from the structure of the1-loop fluctuation determinants [14] appearing in the string partition function expanded near theAdS minimal surface (corresponding to the circular Wilson loop). In the case of genus p surfacethe UV divergent part of the one-loop effective action Γ “ ř i log det ∆ i reads [13] Γ “ ´ ζ tot p q log p L Λ q ` ¯Γ , ζ tot p q “ χ “ ´ p , (1.7)where Λ is 2d cutoff, L is the AdS radius ( T “ L πα ) and the ζ tot p q coefficient turns out to beequal to the Euler number of the surface. The 2d UV divergence should be canceled by a universal2uperstring measure contribution log p? α Λ q involving only the string scale and not the AdS radius.Then the finite part of Γ depends on T through the term ´ χ log L ? α “ ´ χ log ? T and thus thestring partition function on a genus p surface is proportional to e ´ Γ fin „ p? T q χ , i.e. x W y “ ÿ p “ x W y p “ e π T ÿ p “ c p ´ g s ? T ¯ p ´ ” ` O ´ T ´ ¯ı . (1.8)Written in terms of N and λ in (1.6) this matches the structure of the { N expansion of the exact N “ SYM result in (1.4). One can also use similar considerations to predict the structure of thestring theory expansions for other related observables [15].It is important to emphasize the universality of the structure of the expansion in (1.8): itrelies only on the fact that one expands near the AdS minimal surface embedded into the AdS part of AdS n space and should thus be valid also for the corresponding partition functions in the AdS ˆ CP and AdS ˆ S ˆ T superstring theories [13]. It should also apply to the orbifold AdS ˆ p S { Z q theory: orbifolding the S should not change the above argument determining thetension dependence from the way how the AdS radius L appears in (1.7). We thus conjecture that the same form of the large N , strong coupling expansion (1.8) or (1.4)should also apply appear in the N “ orbifold theory case, i.e. x W y orb “ N e ? λ ÿ p “ c p λ p ´ N p ” ` O ´ ? λ ¯ı , (1.9)where the coefficients c p “ c p p π q p ´ { will be different from the ones in (1.4).In order to check the prediction (1.8),(1.9) for the large N , strong-coupling expansion of x W y orb we shall consider the genus one term corresponding to the leading { N correction to the planarpart in (1.3). Normalizing to x W y in (1.3) we have in both N “ SYM and N “ orbifold cases x W yx W y “ ` N q p λ q ` O ´ N ¯ , x W y “ N ? λ I p? λ q , (1.10)where the form of the function q p λ q will be our main focus in what follows. In the SU p N q SYMcase the expression for q p λ q follows from the expansion of the exact Laguerre polynomial expressionin (1.4) ( I n are modified Bessel functions of the first kind) q SYM p λ q “ λ ” ? λ I p? λ q I p? λ q ´ ı “ ´ λ ` λ ` O p λ q , λ ! , λ { ´ λ ` λ { ` O p q , λ " . (1.11)As discussed above, the leading strong-coupling behaviour of the genus one correction in x W y SYM N q SYM p λ q λ " „ λ { N „ g s T (1.12)is consistent with the universal form of the string theory expansion in (1.4). Then according to(3.15) the same should be true also in the orbifold theory case, i.e. q orb p λ q λ " “ C λ { ` O p λ q . (1.13) In the case of SU p N q ˆ ... ˆ SU p N q N “ quiver theory which is the Z k orbifold of the SU p kN q N “ SYM andshould be dual to the superstring on
AdS ˆ p S { Z k q one has for the AdS radius L “ πkNg s α and thus insteadof (1.6) we get g s “ g π “ λ πkN , T “ L πα “ ? λ π . C may of course be different from in the SYM case in (1.11).Confirming the prediction (1.13) starting from the localization matrix model representation for x W y orb will be one of the aims of the present paper. Summary of the results
As we shall see below, the matrix model representations for the orbifold N “ gauge theorypartition function Z orb p λ ; N q on S and for x W y orb imply a remarkable relation between ∆ q p λ q ∆ q p λ q ” q orb p λ q ´ q SYM p λ q , (1.14)and the N Ñ 8 limit of the deviation of the orbifold free energy F orb “ ´ log Z orb from its SYMcounterpart ∆ q p λ q “ ´ λ ddλ ∆ F p λ q , (1.15) ∆ F p λ q ” lim N Ñ8 “ F orb p λ ; N q ´ F SYM p λ ; N q ‰ “ ´ lim N Ñ8 log Z orb p λ ; N qr Z SYM p λ ; N qs . (1.16)The leading O p N q term in F orb ´ F SYM cancels due to the planar equivalence between the SU p N q ˆ SU p N q orbifold theory and the two decoupled copies of the SU p N q N “ SYM theory. Using (1.15) the expected strong-coupling behaviour (1.13) of q p λ q translates into the followingscaling for the difference of free energies in (1.16) ( c “ ´ C ) ∆ F p λ q λ " “ c λ { ` ... . (1.17)In view of (1.16) this leads to a prediction about the strong coupling asymptotics of the leading { N correction to the planar F orb p λ ;
8q “ F SYM p λ ; part of the free energy of the orbifoldtheory.Let us first recall that the free energy of SU p N q N “ SYM on S should not be renormalized,i.e. should be given exactly by the familiar one-loop expression F SYM p λ ; N ; Λ q “ “ log p rΛ q` f ‰ .Here a is the conformal anomaly coefficient, a “ p N ´ q , r is the radius of S , Λ is a 4d UV cutoffand f is a constant. Since the free energy is UV divergent, its finite part is not universal dependingon a particular regularization scheme. The localization procedure [12] representing the free energyin terms of a finite matrix model integral with a simple λ -independent measure implicitly assumeda special regularization in which the renormalized SU p N q SYM free energy is given by F SYM p λ ; N q “ ´
2a log λ “ ´ p N ´ q log λ (1.18)as this expression follows simply from the Gaussian matrix model integral [16] (we drop an additivenumerical constant). More precisely, for the Wilson loops (1.2) the planar equivalence means that the normalized expectation valueof an operator in one of the two SU p N q factors of the quiver is the same as in the SU p N q N “ SYM theory. Ingeneral, the planar correlators of operators from Z symmetric (i.e. “untwisted”) sector should match between theorbifold and the parent SU p N q SYM theory. For “extensive” quantities like the free energy (or conformal anomalies,correlators of total stress tensor, etc. ) the N Ñ 8 results in the SU p N q ˆ SU p N q orbifold theory should matchthose of the two copies of the SU p N q N “ SYM.
AdS ˆ S string partition or (more precisely, at the tree level) by the IIB stringeffective action S “ S ` S ` ... “ p π q g s α ş d x ? G “ p R ` ... q ` α R ` ... ‰ ` O p g s q ` ... .Evaluated on AdS ˆ S (using (1.6) and R ` ... “ ´ L ´ ) the leading supergravity term hereis S “ π N V AdS where V AdS is the (logarithmically) IR divergent volume of unit-radius AdS .Subtracting the IR divergence in V AdS using a particular AdS/CFT motivated prescription gives V AdS Ñ ¯ V AdS “ ´ π log ? λ and thus one reproduces [16] the N log λ term in (1.18). The ´ shift of N in (1.18) should come from the 1-loop superstring correction (again pro-portional to V AdS ): this should follow the same pattern as found for the N “ SYM conformalanomaly and S Casimir energy in [17] (with only loops of short supergravity supermultiplets con-tributing). Other string α n tree level (e.g. α R , cf. [18]) and string loop corrections shouldvanish on maximally supersymmetric AdS ˆ S background.Let us now turn to the orbifold theory that should be dual to the superstring on AdS ˆp S { Z q .Combining (1.16),(1.17) and (1.18) we get the following prediction F orb p λ ; N q λ " “ ´ N log λ ` “ c λ { ` O p log λ q ‰ ` O ´ N ¯ . (1.19)The leading N term here is implied by the planar equivalence to the SU p N q SYM and shouldfollow again from the leading type IIB supergravity term evaluated on
AdS ˆ p S { Z q . Theplanar equivalence also means that like in (1.18) this leading N term should not get string treelevel α -corrections, i.e. they should still vanish when evaluated on AdS ˆ p S { Z q .One may attempt to give an independent string theory explanation of the subleading λ { termin (1.19) without using the connection (1.15) to the Wilson loop. The string one-loop (genus oneor order g s „ N ) type IIB effective action is known to start with S „ α ş d x ? G R ` ... [19, 20, 21]. If we conjecture that when evaluated on the orbifold AdS ˆ p S { Z q it is no longerzero then on dimensional grounds it should scale as S „ L α „ ? λ , reproducing the subleadingterm in (1.19). If a non-zero contribution comes just due to the curvature singularity then it maynot be proportional to the AdS volume so there will be no extra log λ factor. The remainingpuzzle is why the one-loop R term may contribute to F orb while the tree-level one should not,even though the two invariants have the same structure in type IIB string theory [22].Starting with the matrix model representation for x W y orb we shall first study the structure ofthe function q orb p λ q in (1.10) or ∆ q p λ q in (1.14) at weak coupling. While the small λ expansionof q SYM p λ q in (1.11) has only rational coefficients, the coefficients in the expansion of ∆ q p λ q inpowers of λ are transcendental – proportional to the values ζ n ” ζ p n q of the Riemann ζ -function π ∆ q p λ q “ ´ ζ ´ λ π ¯ ` ζ ´ λ π ¯ ` ” ζ ´ ζ ı ´ λ π ¯ ` O p λ q , (1.20) One way to understand the origin of the log ? λ term is as follows. On AdS side the IR cutoff is measured inunits of AdS radius L . On the gauge theory side viewed as originating from the flat-space open string theory thenatural UV cutoff is inverse of the string length ? α . Thus the two cutoffs are related by the ratio L ? α “ λ { . To recall, the orbifold projection of SU p N q SYM giving the SU p N q ˆ SU p N q theory with two bi-fundamentalhypermultiplets reduces the number of degrees of freedom and thus also the leading term in the conformal anomalycoefficient from a “ rp N q ´ s to ˆ p N ´ q which is twice the anomaly of a single copy of SU p N q SYMtheory. On the supergravity side, replacing the N coefficient in the above discussion by p N q and noting that thevolume of p S { Z q is half of the volume of S we end up with N as an overall coefficient. ζ n may be formally used to parametrise the deviation of the orbifold theory result from the N “ SYM one.To find the strong-coupling expansion of q orb p λ q requires a resummation of the weak couplingexpansion. We shall study resummations of particular subclasses of terms proportional to mono-mials built out of ζ n . While this will not be enough to determine the correct strong couplingasymptotics of q orb p λ q , this may still help to shed light on the general structure of this function. Exploiting the relation (1.15) we shall compute ∆ q p λ q up to order O p λ q and also determinethe resummation of all terms with the following types of coefficients involving particular ζ n andtheir powers I: ζ n ` , II: ζ ζ n ` , III: ζ p , IV: ζ p ζ q , V: ζ p ζ q ζ r , (1.21)i.e. q orbI “ ř c n,m ζ n ` λ m , etc. We find that they have the following behaviour at strong coupling q orbI p λ q λ " “ ´ π λ ` O p λ { q , q orbII p λ q λ " “ ´ ζ π λ ` O p λ q , (1.22) q orbIII , IV , V p λ q λ " “ ´ p k ` qp k ´ q λ ` O p q , k “ , , . (1.23)The difference in these asymptotics implies that to find the correct strong coupling behaviour ofthe full q orb one needs first to sum together different subsets of terms and only then expand thetotal at large λ .Lacking an analytic method to compute q orb p λ q at strong coupling we performed extensivenumerical simulations of the SU p N q ˆ SU p N q orbifold matrix model to measure it. This requiredan extrapolation to large N for finite λ , followed by an analysis of large λ region. We confirmedthat the deviation from the N “ SYM case starts only at the non-planar level. The numericaldata agrees with the Padé-Borel resummation of the weak-coupling expansion up to moderate λ „ . At larger values of the coupling λ we found that the data is compatible with the followingasymptotics q orb p λ q λ " “ C λ η ” ` a λ ´ { ` ... ı , (1.24) η “ . p q , C » ´ . p q , a » . p q . (1.25)The power of the leading asymptotics η « . is thus consistent with the string theory prediction(1.13). It is interesting to notice that the values of the coefficients C and Ca are very close tothe values of the corresponding coefficients in (1.11) in the SYM case up to a factor of ´ and ` respectively. This suggests a conjecture that the exact form of the strong coupling expansion of q orb p λ q is given by q orb p λ q λ " “ ´ λ { ´ λ ` O p λ { q , (1.26)It remains to be seen if one can prove this analytically.One may wonder if the coefficient of the leading λ { N correction in x W y orb may be found, as in the N “ SYM case [26], also by considering the circular Wilson loop in k -symmetric representation A similar approach was applied [23] to SU p N q superconformal N “ theories admitting a large N limit. Also,the use of sufficiently many terms in the perturbative series as a guide towards some non-perturbative features likesingularities was emphasised in [24, 25]. If η is set to be exactly { , then the best fit value of the coefficient C slightly changes to ´ . ¨ ´ . k " and k ? λN =fixed) by a classical D3-brane solution. This doesnot seem possible as the D3-brane solution of [26] is restricted to AdS and thus the k λ { N termin its action should have the same coefficient as in the N “ SYM case, in contradiction with(1.25),(1.26). In fact, the D3-brane solution of [26] should be related not to the SU p N q Wilsonloop (1.2) of the SU p N q ˆ SU p N q orbifold theory but to the orbifold projection of the originalcircular Wilson loop in the SU p N q SYM theory. The projection of the latter is represented bythe correlator x W W y where W , in (1.1) correspond to the two SU p N q factors of the orbifoldtheory. Starting with the SU p N q Wilson loop in k -symmetric representation one is to split itinto the sum of products of the two SU p N q representations. Then the D3-brane description mayapply only to a special combination of the x W W y correlators where W and W are taken in theparticular representations of the SU p N q appearing in the product. Assuming that k -symmetricrepresentation may be replaced by the k -fundamental one (corresponding to multiply wrappedcircle, cf. [27, 28]) one would expect to get the sum of the correlators k ÿ m “ ˆ km ˙ x W p m q W p k ´ m q y (1.27)where W p m q a is the SU p N q Wilson loop in m -fundamental representation. Rescaling the fields in(1.2) (or the corresponding matrices in the matrix model representation as in [11]) one would thenend up with the sum of the correlators x W W y of the two fundamental SU p N q Wilson loops inthe SU p N q ˆ SU p N q quiver theory with the two ’t Hooft couplings λ “ m λ, λ “ p m ´ k q λ .The resulting expression should simplify in the large k limit and one expects it to be dominated bythe “diagonal” term ( m “ k { ) with W a in the same representation. In section 6 we shall presentnumerical data indicating that the { N term in this correlator (with both Wilson loops taken inthe fundamental representation of SU p N q ) has a similar strong-coupling behaviour to λ { foundin the SYM case in (1.11).Below we also computed numerically the individual SU p N q Wilson loop (1.1) expectation val-ues x W y , x W y in the SU p N q ˆ SU p N q quiver with unequal couplings λ , λ starting with thelocalization matrix model representation. Guided by the discussion in [4] here we considered thefollowing analog of the ratio in (1.10) x W y w p θ q x W y “ p p λ ; θ q ` N q p λ ; θ q ` O ´ N ¯ , (1.28)where λ “ λ λ λ ` λ , θ “ π λ λ ` λ , w p θ q “ ´ θ cot θ sin θ , (1.29)The strong coupling result of [4] then implies that p p λ ; θ q ˇˇ λ Ñ8 Ñ for any θ ‰ , π . We confirmedthis prediction numerically by considering a particular value of the ratio of the two couplings λ { λ “ (i.e. θ “ π ) and measuring both Wilson loops x W y and x W y , thus effectively probingalso the value θ “ π . We found that the strong-coupling expansion of the function p has the form p p λ ; θ q “ ` h p θ q{? λ ` ... where h has a non-trivial dependence on θ . The numerical data for the We thank N. Drukker for this suggestion. x W y is found by interchanging λ and λ or θ Ñ π ´ θ . q p λ ; θ q turns out to be compatible with the strong-coupling asymptotics in (1.24) with theexponent η being again close to { independently of θ , i.e. for both Wilson loops.The rest of this paper is organized as follows. In section 2 we shall present the matrix modelintegral representation for the Wilson loop expectation value in the quiver theory which will be ourstarting point. In section 3 we shall consider the weak coupling expansion of the leading non-planarterm the case of the orbifold theory organising it in terms of monomials of transcendental ζ n factors.We shall also derive the relation between the function q p λ q and the free energy. In section 4 weshall perform a resummation of some subsets of terms and then expand them at strong couplingfinding non-universal behaviour. In section 5 we shall consider the weak-coupling expansion inthe case of non-symmetric quiver theory. Finally, in section 6 we shall present the results of thenumerical computation of the matrix model integrals. Appendices A and B will contain sometechnical details of the computations in sections 3 and 4. In Appendix C we shall briefly discusssimilar weak-coupling analysis of the Wilson loop in SU p N q “orientifold” N “ superconformaltheory. Our starting point will be the localization matrix model representation for the S partition functionand the expectation values of the circular Wilson loops (1.1) in the SU p N q ˆ SU p N q N “ superconformal quiver theory [12] (see also [2, 3, 4]). The partition function may be written as theintegral over two sets of eigenvalues ( a “ , i “ , ..., N ) Z “ ż ź a “ ” N ź i “ da a i δ p ÿ i a a i q N ź i ă j p a a i ´ a a j q e ´ π Nλ a ř i a i ı f r a , a s , (2.1)where the δ -functions reflect the fact that we are considering the SU p N q case (they may be ignoredin strict planar limit) and f r a , a s “ ś a ś i ă j H p a a i ´ a a j q ś i,j H p a i ´ a j q , H p x q ” ź n “ ´ ` x n ¯ n e ´ x n . (2.2)Below we shall use x¨ ¨ ¨y to denote the (normalized) expectation value in the matrix model withthe Gaussian measure so that (2.1) may be written as Z “ Z x f y , Z “ Z SYM p λ ; N q Z SYM p λ ; N q , (2.3) Z “ ż ź a “ ” N ź i “ da a i δ p ÿ i a a i q N ź i ă j p a ai ´ a aj q e ´ π Nλ a ř i a i ı . (2.4)Here Z SYM p λ ; N q is the SU p N q SYM partition function on S . H p x q has the following representation in terms of the Barnes function G p x q log H p x q “ log “ G p ` ix q G p ´ ix q ‰ ´ p ` γ E q x . The partition function is invariant under H p x q Ñ H p x q e Cx [3]. Note also that we ignored the instanton factor [12]in the integrand as we will be interested in perturbative { N expansion (see [3]). x W a y “ x N ÿ i “ e πa a i y “ x f tr e πa a y x f y , (2.5)where x ... y is given by the same integral as in (2.1) and is normalized so that x y “ . We use thenotation a a for the diagonal matrix a a “ diag p a a1 , ..., a a N q . The two expectation values (2.5) areequal (cf. (1.2)) at the orbifold point λ “ λ “ λ .For large N one may study the saddle points of the “effective action” in (2.1) S r a , a s “ N ÿ a “ π λ a ÿ i a i ´ log f r a , a s . (2.6)Differentiating over a a i and introducing the densities ρ a p x q “ x N N ÿ i “ δ p x ´ a a i q y , (2.7)one finds the following saddle point equations [2, 3] ż µ ´ µ dy ρ p y q ´ x ´ y ´ K p x ´ y q ¯ ` ż µ ´ µ dy ρ p y q K p x ´ y q “ π λ x, (2.8) ż µ ´ µ dy ρ p y q ´ x ´ y ´ K p x ´ y q ¯ ` ż µ ´ µ dy ρ p y q K p x ´ y q “ π λ x, (2.9) K p x q ” ´ H p x q H p x q “ x “ ψ p ` ix q ` ψ p ´ ix q ` γ E ‰ “ ´ ÿ n “ p´ q n ζ n ` x n ` . (2.10)The large N equivalence of the orbifold theory with the N “ SYM follows [2, 4] from the fact thatfor λ “ λ “ λ the equations (2.8),(2.9) admit the symmetric Ansatz ρ “ ρ “ ρ , µ “ µ “ µ and reduce to the saddle point equation for the Gaussian matrix model corresponding to the N “ SYM case for which ρ p x q “ πµ a µ ´ x , µ “ ? λ π . (2.11)The solution of the two integral equations (2.8),(2.9) in the large λ limit was studied in [2], showingthat x W a y „ e ? λ , λ “ λ λ λ ` λ , and more recently in [4] where it was found that (see (1.29)) x W y λ " “ w p θ q W , x W y λ " “ w p π ´ θ q W , W “ N c π λ ´ { e ? λ . (2.12) W is the leading large N , strong coupling term in the SYM result in (1.3). Considering the orbifold theory case λ “ λ one can work out the weak-coupling expansion of x W y orb by starting with the integral representation (2.1),(2.5) for finite N . It may be formally9ritten as a sum of functions W ζ p λ, N q , W ζ p λ, N q , ..., multiplying particular products of ζ n “ ζ p n q values x W y orb “ x W y SYM ` W p λ, N q , W “ ζ W ζ ` ζ W ζ ` ζ W ζ ` ζ W ζ ` ¨ ¨ ¨ . (3.1)Here x W y SYM is given by (1.4) so that W ζ , etc., scale as { N at large N . For small λ one has W ζ “ O p λ q , W ζ “ O p λ q , W ζ “ O p λ q , W ζ “ O p λ q , etc . (3.2)To compute these functions starting with (2.1) let us note that using (2.2),(2.10) we get log f “ ÿ i,j ” ÿ a log H p a a i ´ a a j q ´ log H p a i ´ a j q ı (3.3) “ ÿ n “ ´ λ π N ¯ n ` p´ q n n ` ζ n ` n ` ÿ k “ C kn ` ” ÿ a tr A k a tr A n ` ´ k a ´ tr A k tr A n ` ´ k ı , where we defined C kn ` ” p´ q k ` n ` k ˘ , a a “ diag p a a1 , ..., a a N q (with tr a a “ ) and we alsointroduced the rescaled matrices A a (appearing in the exponent in (2.1)) A a ” c π Nλ a a . (3.4) Separating different ζ n terms we may write f in (3.3) as an expansion in λ π N “ g π f “ ´ ζ ´ λ π N ¯ ` T p q , ` T p q , ´ T p q T p q ˘ ` ζ ´ λ π N ¯ ` T p q , ´ T p q , ´ T p q T p q ´ T p q T p q ` T p q , ´ T p q , ˘ ` ¨ ¨ ¨ , (3.5) T p a q n ,n ,...,n r ” tr A n a tr A n a ¨ ¨ ¨ tr A n r a . (3.6)Using (3.3) and computing (2.5) by first integrating out the A dependence with the help of @ tr A D “ N ´ , @ p tr A q D “ N ´ , ¨ ¨ ¨ , (3.7)we obtain for the coefficient functions in (3.1) W ζ “ ´ ´ λ π N ¯ A tr e b λ N A “ : p tr A q : ` A : ‰E , (3.8) W ζ “ N ´ λ π N ¯ A tr e b λ N A ” p´ ` N q : tr A : ` N p ` N q ´ N ` N : tr A : ` p´ ` N q : p tr A q : ` N : tr A tr A : ´ N : p tr A q : ıE , ... , (3.9)where : tr A : “ tr A ´ N ´ , @ : tr A : D “ , etc . (3.10) Similar expansions are found in other similar N “ models, cf. [29, 30]. A “ A ) Gaussian model.Computing the Wilson loop correlator with normal ordered operators using that for the SU p N q SYM case [11] x W y SYM “ x tr e b λ N A y “ e λ N ` ´ N ˘ L N ´ ´ ´ λ N ¯ , (3.11)and applying the method described in Appendix A (see also [15]), we find ( g ” g YM “ b λN ) A tr e ? λ N A : tr A : E “ g B g x W y SYM “ λ B λ x W y SYM , (3.12) A tr e ? λ N A : p tr A q : E “ ´ g B g ´ g B g ¯ x W y SYM “ λ B λ x W y SYM , A tr e ? λ N A : tr A : E “ ” λ p´ ` N qp ` N q N ` λ p λ ` N ´ N q N B λ ´ λ p λ ´ N q N B λ ` λ N B λ ı x W y SYM , (3.13) A tr e ? λ N A : tr A tr A : E “ ” ´ λ p´ ` N qp ` N q N ` λ p´ ` N qp ` N qp λ ` N q N B λ ` λ p λ ` λN ´ N ´ λN q N B λ ´ λ p λ ´ N q N B λ ` λ N B λ ı x W y SYM , (3.14) A tr e ? λ N A : p tr A q : E “ ” λ p´ ` N qp ` N qp´ ` N ´ N ` N q N p ´ N ` N q ` λ p´ λ ´ N ` λN ` N ´ λN ´ N ` λN ` N q N p ´ N ` N q B λ ` λ p λ ´ λN ´ λ N ´ N ´ λN ` λ N ` N ` λN ´ N ´ λN q N p ´ N ` N q B λ ´ λ p λ ` N ´ λN ´ N ` λN ´ N q N p ´ N ` N q B λ ` λ N B λ ı x W y SYM . (3.15) Since according to (3.11) x W y SYM “ N ? λ I ` N λ ´ I ´ ? λ I ¯ ` O ´ N ¯ , I n ” I n p? λ q , (3.16)we find for (3.8),(3.9) W ζ “ ´ ´ λ π N ¯ ” N ? λ I ` N ´ λ p λ ´ q I ´ λ { I ¯ ` O ´ N ¯ı , (3.17) W ζ “ ´ λ π N ¯ ” N ? λ I ` λ p´ ` λ q I ´ ? λ p ` λ q I ` O ´ N ¯ı . (3.18)The expressions (3.17) and (3.18) generate all terms proportional to ζ and ζ at leading andsubleading order in { N in (3.1) within the weak coupling expansion. As expected, these functionsscale as { N for large N , i.e. W ζ “ ´ N ´ λ π ¯ ? λ I p? λ q ` O ´ N ¯ , W ζ “ N ´ λ π ¯ ? λ I p? λ q ` O ´ N ¯ . (3.19) Similarly, the leading large N terms in other coefficient functions in (3.1) are given by W ζ “ N ´ λ π ¯ ” ´ ? λ I ` N ´ ´ p´ ` λ q λI ` ? λ p ` λ q I ¯ ` ¨ ¨ ¨ ı ,W ζ “ N ´ λ π ¯ ” ? λ I ` N ´ λ p ` λ q I ´ p´ ` λ q? λI ¯ ` ¨ ¨ ¨ ı ,W ζ ζ “ N ´ λ π ¯ ” ´ ? λ I ` N ´ ´ λ p ` λ q I ` p´ ` λ q? λI ¯ ` ¨ ¨ ¨ ı ,W ζ “ N ´ λ π ¯ ” ? λ I ` N ´ λ p´ ` λ q I ´ ? λ p ` λ q I ¯ ` ¨ ¨ ¨ ı , ζ “ N ´ λ π ¯ ” ´ ? λ I ` N ´ ´ λ p ` λ q I ´ ? λ p ` λ q I ¯ ` ¨ ¨ ¨ ı , (3.20) W ζ “ N ´ λ π ¯ ” ? λ I ` N ´ λ p ` λ q I ` ? λ p ` λ q I ¯ ` ¨ ¨ ¨ ı ,W ζ ζ “ N ´ λ π ¯ ” ? λ I ` N ´ λ p ` λ q I ` ? λ p ` λ q I ¯ ` ¨ ¨ ¨ ı ,W ζ “ N ´ λ π ¯ ” ´ ? λ I ` N ´ ´ λ p´ ` λ q I ` ? λI ¯ ` ¨ ¨ ¨ ı . Remarkably, the dependence on λ of the leading term O p { N q in the W functions in (3.1) followsthe same pattern, i.e. is proportional to the Bessel function I p? λ q that appears in the leadingorder term in the SYM expression (3.16) W ś rn “ ζ kn n ` “ c k ...k r N ´ λ π ¯ ř rn “ k n p n ` q ? λ I p? λ q ` O ´ N ¯ . (3.21)The power of λ π is coming from the ζ n ` factors in (3.3) while the extra factor of λ (from theBessel function factor ? λ I p? λ q “ λ ` ¨ ¨ ¨ ) has its origin in the Wilson loop operator insertioninto the Gaussian matrix model integral at the leading order in large N (cf. (2.5)). In particular,it comes from the A term in ( tr A a “ ) tr e b λ N A “ N ` λ N tr A ` ¨ ¨ ¨ . (3.22)Separating this SYM Bessel function factor, the expression for the leading large N term in W “x W y orb ´ x W y SYM in (3.1) can be written in terms of the function q p λ q defined in (1.10),(1.14) q orb p λ q “ q SYM p λ q ` ∆ q p λ q , ∆ q p λ q “ λ ÿ n “ d n ´ λ π ¯ n . (3.23)Here q SYM p λ q was given in (1.11) and the coefficients d n are found to be (cf. (3.20)) d “ ´ ζ , d “ ζ d “ ´ ζ ` ζ , d “ ´ ζ ζ ` ζ ,d “ ´ ζ ` ζ ` ζ ζ ´ ζ , d “ ζ ζ ´ ζ ζ ´ ζ ζ ` ζ ,d “ ζ ´ ζ ζ ´ ζ ζ ` ζ ` ζ ζ ` ζ ζ ´ ζ , (3.24) d “ ´ ζ ζ ` ζ ` ζ ζ ζ ` ζ ζ ´ ζ ζ ´ ζ ζ ´ ζ ζ ` ζ . In general, starting from the definition (2.5) of the Wilson loop expectation value (2.5) (e.g. for a “ ), plugging in the expansion of f in (3.5) and taking N large we get, using that the inte-gration over the “decoupled” variable A gives an extra SU p N q SYM factor (cf. (1.10),(1.14) and(2.3),(3.4),(3.11)) N Ñ 8 : x W y orb “ x W y SYM ” ` ? λ N I @ : tr A : f D x f y N ? λ I ` O ´ N ¯ı , (3.25)12 q p λ q “ lim N Ñ8 λ @ : tr A : f D x f y “ lim N Ñ8 λ ddλ log x f y . (3.26)We used (2.2),(3.3) to represent ∆ q p λ q in (1.10),(1.14),(3.23) in terms of the Gaussian matrix modelexpectation value, traded the insertion of : tr A ` tr A : for the application of λ B λ and used the A Ø A symmetry of the integration measure. The expression (3.26) is equivalent to (1.15),(1.16)representing ∆ q p λ q in terms of the free energy F “ ´ log Z of the orbifold theory (see (2.3)).As a check, using (3.3) one can compute the leading terms in the expansion of x f y x f y “ Z orb Z “ ´ p N ´ q N ζ ´ λ π ¯ ` p N ´ qp N ´ q N ζ ´ λ π ¯ ` ” p N ´ qp N ` q N ζ ´ p N ´ qp N ´ N ` q N ζ ı´ λ π ¯ ` ¨ ¨ ¨ N Ñ8 “ ´ ζ ´ λ π ¯ ` ζ ´ λ π ¯ ` ” ζ ´ ζ ı´ λ π ¯ ` ¨ ¨ ¨ . (3.27)Hence, for (3.26) we get ∆ q “ λ ddλ lim N Ñ8 log x f y “ ´ ζ λ ´ λ π ¯ ` ζ λ ´ λ π ¯ ` ” ζ ´ ζ ı λ ´ λ π ¯ ` ¨ ¨ ¨ , (3.28)which is in agreement with (3.23),(3.24).Thus the problem of computing ∆ q in (3.23) is reduced to the calculation of the large N limitof the free energy of the N “ orbifold theory (which is finite for N Ñ 8 after the subtractionof the planar N “ SYM term). This method is rather efficient as the direct computation of ∆ q to higher orders without exploiting the resummation of the λ dependence in the factor ? λI p? λ q would be prohibitively difficult. Using it we were able to push the calculation of the ζ n expansionof q p λ q up to O p λ q . In particular, the next five terms beyond (3.23) read d “ ´ ζ ` ζ ζ ` ζ ζ ´ ζ ζ ´ ζ ζ ´ ζ ζ ζ ` ζ ´ ζ ζ ` ζ ζ ` ζ ζ ` ζ ζ ´ ζ ,d “ ζ ζ ´ ζ ζ ´ ζ ζ ζ ` ζ ζ ´ ζ ζ ` ζ ζ ` ζ ζ ζ ` ζ ζ ζ ´ ζ ζ ` ζ ζ ´ ζ ζ ´ ζ ζ ´ ζ ζ ` ζ ,d “ ζ ´ ζ ζ ` ζ ´ ζ ζ ` ζ ζ ζ ` ζ ζ ´ ζ ` ζ ζ ζ ´ ζ ζ ζ ´ ζ ζ ` ζ ζ ´ ζ ζ ´ ζ ζ ζ ` ζ ´ ζ ζ ζ ` ζ ζ ´ ζ ζ ` ζ ζ ` ζ ζ ` ζ ζ ´ ζ ,d “ ´ ζ ζ ` ζ ζ ` ζ ζ ζ ´ ζ ζ ´ ζ ζ ζ ` ζ ζ ´ ζ ζ ζ ´ ζ ζ ζ ` ζ ζ ` ζ ζ ´ ζ ζ ζ ` ζ ζ ζ ` ζ ζ ζ ´ ζ ζ ` ζ ζ ` ζ ζ ζ ´ ζ ζ ` ζ ζ ζ ´ ζ ζ ` ζ ζ ´ ζ ζ ´ ζ ζ ´ ζ ζ ` ζ ,d “ ´ ζ ` ζ ζ ´ ζ ζ ` ζ ζ ´ ζ ζ ζ ´ ζ ζ ` ζ ζ ` ζ ζ ´ ζ ζ ζ ` ζ ζ ` ζ ζ ζ ζ ` ζ ζ ´ ζ ζ ´ ζ ζ ` ζ ζ ζ ζ ζ ζ ´ ζ ζ ´ ζ ζ ζ ´ ζ ζ ` ζ ζ ζ ´ ζ ζ ζ ´ ζ ζ ζ ` ζ ` ζ ζ ´ ζ ζ ´ ζ ζ ζ ` ζ ζ ´ ζ ζ ζ ` ζ ζ ´ ζ ζ ` ζ ζ ` ζ ζ ` ζ ζ ´ ζ . (3.29) x W y orb In an attempt to shed light on the structure of strong coupling limit of x W y orb one may try toconsider separate terms in the transcendental part of (3.1), resum their weak coupling expansionand then expand at strong coupling. As we shall see, this procedure will not give the correctstrong-coupling limit of x W y orb : the strong-coupling asymptotics of different functions W ζ kn ... willbe different. That means that all such terms should first be summed up before taking the large λ limit.As we shall show in Appendix B the terms in (3.1),(3.21) which are proportional to the single ζ n ` have the following coefficient functions W ζ n ` “ N ´ λ π ¯ n ` p´ q n n ` n ´ p n ` q Γ p n ` qp n ` q π Γ p n q Γ p n ` q ? λ I p? λ q ` O ´ N ¯ . (4.1)Summing all such terms in x W y orb in (3.1), i.e. W ˇˇ ζ “ ř n “ W ζ n ` , we then get the correspondingcontribution to q orb or to ∆ q in (1.14),(3.26) ∆ q p λ q ˇˇˇ ζ “ ÿ n “ p´ q n λ ´ λ π ¯ n ` n ´ p n ` q Γ p n ` q π Γ p n q Γ p n ` q ζ n ` . (4.2)We can resum this series by noting that ζ n ` ” ζ p n ` q “ p n q ! ż dt g p t q t n , g p t q “ e t ´ . (4.3)This gives ( J n are Bessel functions) ∆ q p λ q ˇˇˇ ζ “ λ π ż dt g p t q f p t ? λ q , (4.4) f p t q “ r J p t qs ´ J p t q J p t q t ´ p t ´ q r J p t qs t . (4.5)Using the properties of the Mellin transform we find that the large λ asymptotics of (4.4) is ∆ q p λ q ˇˇˇ ζ λ " “ λ π ” ´ ` λ ´ { ´ ζ λ ´ { ` ¨ ¨ ¨ ı . (4.6) Defining the Mellin transform r f p s q “ ş dx x s ´ f p x q and considering the convolution p f ‹ g qp x q “ ş dt f p t x q g p t q we have p Ć f ‹ g qp s q “ r f p s q r g p ´ s q . Let α ă s ă β be the fundamental strip of analyticity of r f p s q . The asymptotic expansion of f p x q for x Ñ 8 is obtained by looking at the poles of r f p s q in the region s ě β .Then the pole p s ´ s q N in the Mellin transform leads to the term p´ q N p N ´ q ! 1 x s log N ´ x in the original function. ζ ζ n ` with n ą (see Appendix B).We get the following analog of (4.2) ∆ q p λ q ˇˇˇ ζ ζ “ ´ ÿ n “ p´ q n π n ` n r Γ p n ` qs Γ p ` n q Γ p ` n q λ ´ λ π ¯ n ` ζ ζ n ` . (4.7)Summing this series as in (4.3),(4.4) we get ∆ q p λ q ˇˇˇ ζ ζ “ ´ ζ π λ ` ζ π λ ż dte πt ´ f p t ? λ q , (4.8) ˆ f p t q “ ´r J p t qs ` J p t q J p t q t ` p t ´ qr J p t qs t . (4.9)Expanding at large λ here gives a different asymptotics than in (4.6) ∆ q p λ q ˇˇˇ ζ ζ λ " “ ´ ζ π λ ` ζ π λ ” ´ π λ ´ ` ζ λ ´ { ` ¨ ¨ ¨ ı (4.10)As another example one may consider all the terms in W involving only powers of ζ . The resultingcontribution takes a simple form (see Appendix B) ∆ q p λ q ˇˇˇ ř n ζ n “ ÿ n “ p´ q n n ´ n λ ´ λ π ¯ n ζ n “ ´ λ ζ p π ` λ ζ q λ " “ ´ λ ` ¨ ¨ ¨ , (4.11)with the large λ asymptotics being again different from (4.6) and (4.10).Using the general method described in Appendix B one is able to generalize (4.11) to the sumof all contributions involving arbitrary powers of ζ and ζ and also of ζ ∆ q p λ q ˇˇˇ ř n,m ζ n ζ m “ λ p´ t ` t ´ t t ` t ` t q p ` t qp ` t ´ t ´ t q ˇˇˇ t “ ζ p λ π q , t “ ζ p λ π q λ " “ ´ λ ` ... (4.12) ∆ q p λ q ˇˇˇ ř n,m,k ζ n ζ m ζ k “ ´ λ N p t , t , t q D p t , t , t q ˇˇˇ t “ ζ p λ π q , t “ ζ p λ π q , t “ ζ p λ π q λ " “ ´ λ ` ... , (4.13) N p t , t , t q “ ´ p t ´ t ` t t ´ t ´ t ` t ´ t t ` t t ` t t t ` t t ´ t ´ t t ´ t t ` t t ` t ´ t t ´ t t ` t ` t q ,D p t , t , t q “ p ` t ´ t ´ t qp´ ´ t ` t ` t ´ t ´ t t ´ t t ` t ` t q . (4.14)Comparing (4.11),(4.12) and (4.13) suggests that the strong coupling limit of the sum of monomialsinvolving powers of the first k constants ζ , ζ , ..., ζ k ` should be (cf. (4.11),(4.12),(4.13)) ∆ q p λ q ˇˇˇ ř n ,...,nk ζ n ...ζ nk k ` λ " “ ´ p k ` qp k ´ q λ ` ... , k “ , , , ... (4.15)Since the coefficient in (4.15) grows with k , summing up such contributions after taking the large λ limit would not give a meaningful result. 15 Weak coupling expansion for non-symmetric quiver
Let us now discuss the expectation value of the Wilson loops (1.1),(2.5) in the case of the SU p N q ˆ SU p N q quiver for unequal couplings λ “ λ . We shall consider for definiteness x W y ” x W y .Setting λ “ ρ λ , (5.1)the generalization of (3.3) will read log f “ ÿ n “ ´ λ π N ¯ n ` p´ q n n ` ζ n ` n ` ÿ k “ p´ q k ˆ n ` k ˙ ˆ ” ÿ a tr A k tr A n ` ´ k ` ρ n `
12 tr A k tr A n ` ´ k ´ ρ n ` ´ k tr A k tr A n ` ´ k ı . (5.2)It is then straightforward to compute the large N expansion of the coefficient functions of the ζ n -monomial contributions to x W y in the analog of (3.1). The first of them that generalizes (3.17)is W ζ “ ´ λ π N ¯ ” N p ρ ´ q I p a λ q ´ N ´ p ρ ´ qp ` λ q I p a λ q´ p ` λ qp ρ ´ q ´ λ ? λ I p a λ q ¯ ` O ` N ˘ı . (5.3)The planar (order N ) contribution here agrees with the N Ñ 8 part of the N “ SYM result in(1.3) expressed in terms of the effective coupling [31, 32, 29] ( g there is λ π ) N ? λ eff I p? λ eff q “ N ? λ I p? λ q ` N ζ p ρ ´ q ´ λ π ¯ I p? λ q ` ¨ ¨ ¨ , (5.4) λ eff “ λ ` ζ p ρ ´ q λ p π q ` ¨ ¨ ¨ . (5.5)Note that the subleading terms in (5.3) proportional to ρ ´ are similarly captured by the SYMterm if we modify (5.5) as λ eff “ λ ` ζ p ρ ´ q ´ ´ N ¯ λ p π q ` ... (5.6)One can also find the analog of W ζ in (3.1),(3.18) and the ρ ´ terms there can be generated fromthe SYM expression by the replacement λ Ñ λ eff generalizing (5.6) λ eff “ λ ` ζ p ρ ´ q ´ ´ N ¯ λ p π q ´ ζ p ρ ´ q ” p ρ ` q ´ ´ N ¯ λ p π q ` π ´ ´ p ` ρ q N ¯ λ p π q ` π ´ ` ` ρ N ¯ λ p π q ` O p λ q ı ` ... . (5.7)This suggests that some essential features of the weak coupling expansion of the Wilson loop inthe non-symmetric quiver case are already captured by the orbifold case ( ρ “ ) discussed above. In the case of the λ “ λ quiver the weak coupling expansion in the planar limit was analysed also in [33] and[24]. The N Ñ 8 limit of this expression is in agreement with Eq. (35) in [29]. Numerical analysis of the quiver matrix model
One may try to compute the Wilson loop (1.1) numerically at finite N and λ by starting withthe matrix model representation (2.1),(2.5). While this is a finite dimensional integral, the factthat are interested in the limit N " makes the numerical integration hard. At the same time,we expect that, in the large N limit, the relevant subset of the integration domain reduces to aneighbourhood of the saddle point solution. This problem is completely analogous to the one in thelattice field theory (where one computes quantum corrections by numerical path integration with N „ (cid:126) ´ ) and may thus be dealt with by the same Monte Carlo (MC) methods (see, for instance,[34]). We analysed the Wilson loop expectation value (3.1) in the orbifold case by means of a Metropolis-Hastings Monte Carlo simulation [35] of the integral (2.1), a robust approach that does not requirefine tuning. Given a configuration X of the eigenvalues a a i corresponding to the two SU p N q groups, thematrix integral (2.1) weights each observable O p X q , like the Wilson loop, with a positive number exp p´ S q where S “ S p X q corresponds to the total integrand in (2.1) including the Vandermondefactor. A Markov chain obeying detailed balance is built by making a local variation of X Ñ X andaccepting the new configuration if S p X q ă S p X q or, in the case S p X q ą S p X q , with probability e S p X q´ S p X q . We tuned the local changes of configuration in order to have an acceptance probabilityaround 50-60% which is a reasonable choice. Iterating this procedure produces a sequence t X n u of configurations distributed according to exp p´ S q and one can measure the quantum expectationvalue as the ensemble average x O y “ lim n Ñ8 n ř nm “ O p X m q . The sequence t X n u is correlatedand its autocorrelation time has been measured at each data point and taken into account in theestimate of an error in this MC evaluation of x O y . For each value of λ , we ran our code at various values of N and fitted the Wilson loop measure-ments in order to extract the function q orb p λ q (3.23) that governs the leading non-planar correctionin (1.10). The procedure is illustrated in Fig. 1 (left) at the value λ “ . Fig. 1 (right) shows thehistogram of measurements of the Wilson loop at λ “ , N “ , as a sample point.To provide non-trivial checks of the numerical code we considered the Wilson loop at λ “ which is a relatively weak coupling. From (3.23),(3.24) we see that for this value the leading ζ n contributions are negligible and we may assume that the same is true also for higher ordercontributions. Then q orb p q “ ´ . p q , (6.1) For other papers using MC methods in matrix models see, e.g., [36, 37, 38, 39, 40, 41] and section 6.5 of [42]. Inthe explored region of parameter space the Metropolis-Hastings algorithm turned out to be faster than the HybridMonte Carlo one, taking into account autocorrelation. The latter algorithm is expected to be preferable at higher N and possibly more efficient for large-scale simulations which are beyond the scope of the present analysis. As is well known (see, for instance, [43]), denoting by x¨ ¨ ¨y MC the average over MC realisations and assuming anexponential autocorrelation for the measurements O n “ O p X n q , i.e. x O n O m y MC “ σ O e ´| n ´ m |{ τ O , the variance of theexpectation value estimator n ř nm “ O p X m q is n ř nm,k x O m O k y MC „ τ O ` n σ O showing that the effective number ofdecorrelated measurements is roughly n decorr “ n {p τ O ` q which is the factor entering the standard deviation ofmeasurements σ “ σ O {? n decorr . ζ n terms explicitly computed above. The extrapolated slope from the finite N MC simulations at λ “ shown in Fig. 1 (left) givesMC : q orb p q “ ´ . p q , (6.2)which is thus consistent with the analytic estimate (6.1).To compare results at higher values of λ we need to resum the perturbative expansion of ∆ q p λ q in (3.23),(3.24),(3.29). We performed a Borel-Padé resummation for values of λ up to 50, see Fig. 2.The red line there is the perturbative series which is expected to converge for | λ | ă π with partialsums blowing up beyond that value. The green line is the r { s Padé approximant of the Borelimproved series, while the blue line is the Borel transform, which is thus in good agreement withthe numerical data points.At higher values of λ we found similar extrapolations in { N . In Fig. 3 (left) we show theintercept of the extrapolation which is expected to be 1, see (3.20). This is a measure of thesystematic error associated with the fit of the N dependence. It increases with λ and we increasedthe maximal N in order to keep it below the 0.2% level. The resulting function q orb p λ q computed for up to λ “ is shown in Fig. 3 (right). In the SU p N q N “ SYM theory, we know from (1.11) that at strong coupling q SYM p λ q “ λ { ` ... which is valid with high accuracy already at λ Á . In the orbifold theory we find that q orb p λ q isnegative with a clear bending at large λ suggesting an asymptotic behaviour q orb p λ q „ ´ λ η , η ą . (6.3)The best fit of the blue data points in Fig. 3 (right) gives η “ . p q where the conservative errorestimate includes statistics as well as the systematic effects due to the choice of fitting window. Weestimated the latter by dropping some of the data points at smaller values of λ . This exponent isstill to be taken with some caution since it is hard to say whether we are already in the asymptotic λ Ñ 8 region but it appears to match the string theory prediction in (1.13) (see also (1.24),(1.25)).Finally, motivated by the discussion of the possible role of the D3-brane solution of [26] in the SU p N qˆ SU p N q orbifold theory (see Introduction), we numerically computed the expectation value x W W y of the two SU p N q Wilson loops (1.1) and determined (using the same fitting procedureas discussed above) the associated q WW p λ q function defined as in (1.10) x W W yx W y “ ` N q WW p λ q ` O ´ N ¯ . (6.4)The corresponding data points are shown in Fig. 6. They decrease to negative values with rateslower than the one observed in q orb p λ q . A best fit of the form (1.24) with η fixed at gives C WW “ ` . p q and a WW “ ´ p q . The coefficient C WW has the opposite sign to the one in This is the radius of convergence of perturbative expansion in SYM theory in the planar limit. Its origin maybe attributed to the form of the single-magnon dispersion relation, which follows from superconformal symmetry[44, 45] and it may also be found using the quantum algebraic curve approach [46]. That such a singularity is alsopresent in the N “ theories was first noticed in the mass-deformed N “ ˚ theory case in [47]. Let us note that our numerical analysis is in a region of values of p λ, N q expected to be free from the instantoncorrections which are weighted by the typical exp p´ π N { λ q factors, at least up to instanton moduli space volumecorrections [48]. « . in (1.11). One possible interpretation of this resultis that the “diagonal” correlator x W W y of the two Wilson loops in the fundamental representationexhibits the (at the leading non-planar order) the strong coupling behaviour which is expected fromthe D3-brane description, while other terms appearing in (1.27) are less important in the large k limit. In the case of generic (non-zero) λ and λ the strong-coupling asymptotics of the Wilson loops isgiven by (2.12). We shall study the functions p p λ, θ q and q p λ, θ q in the ratio (1.28) of x W y to theplanar SYM result. We begin with the special point θ “ π or (see (1.29)) λ “ λ : λ “ λ , w p π q “ ´ π “ . . . . . (6.5)The numerical results are shown in Fig. 4. The left panel gives the function p p λ, π q . As expected,it decreases for large λ towards 1 (this should hold for any θ , see (2.12)) and a good fit is p p λ, π q “ . ` . λ { ` . λ ´ ` ... . (6.6)Measurement of the second Wilson loop x W y provides the information about the same functionsat the complementary value of the angle θ “ π ´ θ “ π for which λ “ λ : λ “ λ , w p π q “ ` π “ . . . . . (6.7)The corresponding results are shown in Fig. 5. The best fit for the p p λ, π q is p p λ, π q “ . ´ . λ { ` . λ ´ ` ... . (6.8)The function q p λ ; θ q at θ “ π and π is shown in the right panels of Fig. 4 and Fig. 5. Ourestimate for the exponent η p θ q in the analog of (6.3) is η p π q “ . p q and η p π q “ . p q . Bothvalues appear to be similar to the one found in the orbifold case ( θ “ π ), i.e. η « . It would bedesirable to push the MC simulation to larger values of the coupling λ , but that seems to requirea dedicated analysis with a substantially increased computational power. Acknowledgments
We would like to thank N. Drukker, S. Giombi, J. Russo, and K. Zarembo for useful discussionsand comments on the draft. MB was supported by the INFN grant GSS (Gauge Theories, Stringsand Supergravity). AAT was supported by the STFC grant ST/T000791/1. The small but not negligible deviation of the estimated asymptotic value from 1 suggests that systematic errorsshould be reduced by N Ñ 8 extrapolations with larger values of N . This could be related to the much large valueof the correcting factor w p π q as compared to w p π q . ����� ������ ������ ������ ������ ������ ������� / � � ����������������������������������������� λ = � �� �� �� �� ������ ������ ������ ������ � × �� � �������������������� ��� Figure 1:
Left:
Fit of the ratio x W y orb ´x W y SYM x W y (see (1.3),(3.1)) with a linear function of { N for λ “ . The three data points correspond to N “ , , . It is not necessary to take larger valuessince the intercept is already very close to the expected value 1. Right : Histogram of the Monte Carlomeasurements of the orbifold Wilson loop from simulation at λ “ , N “ . For each (uncorrelated)Monte Carlo step, one records the measured value of W orb and the plot shows the binned relative frequencies.The best statistical estimator for the quantum expectation value x W y is the mean value of this empiricaldistribution. �� �� �� �� �� λ - � - � - � - � - � - � - ��� ( λ ) Figure 2:
Borel-Padé resummation of the perturbative expansion of q orb p λ q . The red line is the pertur-bative expansion (3.24), (3.29) rapidly breaking down around λ “ π . The green line is its [7/6] Pade’approximant already close to data, while the blue line is its numerical Borel transform. ��� ��� ��� ��� λ ������������������������������������������ ����� ��� ��� ��� ��� λ - �� - �� - �� - ���� ( λ ) Figure 3:
Left : Intercept in the large N extrapolation which should be equal to 1 due to the planarequivalence with the N “ SYM. The deviation is a measure of the systematic error which can be reducedat the price of increasing the maximal N used in the simulations and in the extrapolation to N “ 8 . Right:
Data points for the function q orb p λ q defined in (1.10),(3.23). Dashed line is the non-linear fit withthe functional form q orb p λ q “ C λ η p ` a λ ´ { q . The fit is performed using data points with λ ě whichhave been determined with a relative error below 3% . � + ���� λ - � / � + ��� λ - � � ��� ��� ��� ��� ��� ��� ��� λ ��������������������������������� ( λ � π / � ) ��� ��� ��� ��� ��� ��� ��� λ - �� - �� - �� - ���� ( λ � π / � ) Figure 4:
Functions p p λ, π q (left) and q p λ, π q (right) for the quiver at the point λ “ λ , with λ “ λ λ λ ` λ “ λ . ���� - ��� λ - � / � + ��� λ - � � ��� ��� ��� ��� ��� λ ���������������� ( λ � � π / � ) ��� ��� ��� ��� ��� λ - ��� - ��� - ��� - ���� ( λ � � π / � ) Figure 5:
Functions p p λ, π q (left) and q p λ, π q (right) for the quiver at the point λ “ λ , with λ “ λ λ λ ` λ “ λ . The angle θ “ π corresponds to the Wilson loop for the second SU p N q factor. ���� ( � ) λ � / � - ���� ( � ) λ �� ��� ��� ��� λ - �� - �� - �� � ( λ ) Figure 6:
Data for the q WW p λ q function in (6.4) controlling the { N correction to x W W y in the orbifoldtheory. A Multi-trace SU p N q recursion relations and x W O y The correlators x W O y in a Gaussian one-matrix model of the Wilson loop operator W “ tr e b λ N A and a multi-trace chiral operator O may be reduced to a differential operator over the couplingconstant acting on x W y (see (3.12)–(3.15)). This relation is exact at finite N and is achievedby exploiting the SU p N q fusion/fission relations [49] and the associated recursion relations on theexpectation values t n ,n ,...,n r ” x tr A n tr A n ¨ ¨ ¨ tr A n r y . (A.1)Let us consider as an example @ W : tr A : D . We find ( g “ b λN ) : tr A : “ tr A ´ N tr A ` N p tr A q ´ A tr A ´ p tr A q ` p N ` q tr A ´ p N ` N q , @ W : tr A : D “ ÿ k “ g k k k ! ´ t k, ´ N t k, ` p N ` q t k, ´ p N ` N q t k ` N t k, , ´ t k, , ´ t k, , ¯ . (A.2)Doing Wick contractions leads to a combination of “single-trace” terms that can be traded for B g differential operators acting on x W y and we finally obtain @ W : tr A : D “ ÿ k “ g k k k ! ” k t k ` ´ kN t k ` ´ k p´ ` k ´ N q t k ` k p k ´ q N t k ´ ı “ ÿ k “ g k k k ! ” k p k ´ qp k ´ qp k ´ qp k ´ q g ´ k p´ ` k ´ N q ´ Nk p k ´ qp k ´ q g ` g N ı t k “ ” g B g ` p N ` q g B g ´ p g B g q ´ N g B g ` N g ı x W y . (A.3)This procedure can be easily coded in symbolic manipulation programs.22 Coefficient functions of ζ -terms in x W y orb Here we shall provide some details of the weak-coupling computation of the coefficient functions W ζ in (3.1). Generalizing the calculation in (3.27), the contribution proportional to a single ζ n ` to the expectation value x f y is given by x f y “ ` ÿ n “ ´ λ π N ¯ n ` p´ q n n ` ζ n ` n ` ÿ k “ p´ q k ˆ n ` k ˙ @ tr A k tr A n ` ´ k D , c ` O p ζ q“ ` ÿ n “ ´ λ π N ¯ n ` p´ q n n ` ζ n ` n ` ÿ k “ ˆ n ` k ˙ A tr A k tr A p n ´ k ` q E , c ´ ÿ n “ ´ λ π N ¯ n ` p´ q n n ` ζ n ` n ÿ k “ ˆ n ` k ` ˙ @ tr A k ` tr A n ´ k ` D , c ` O p ζ q . (B.1) Using that the connected correlators are given by [23] A tr A k tr A k E , c “ N k ` k k ` k Γ p k ` q Γ p k ` q π p k ` k q Γ p k q Γ p k q ` O p N k ` k ´ q , A tr A k ` tr A k ` E , c “ N k ` k ` k ` k ` k k Γ p k ` q Γ p k ` q π p k ` k ` q Γ p k ` q Γ p k ` q ` O p N k ` k ´ q , (B.2)we get x f y N Ñ8 “ ` ÿ n “ ´ λ π ¯ n ` p´ q n n ` ζ n ` n ` ÿ k “ ˆ n ` k ˙ n ` Γ p k ` q Γ p n ` ´ k ` q π p n ` q Γ p k q Γ p n ` ´ k q´ ÿ n “ ´ λ π ¯ n ` p´ q n n ` ζ n ` n ÿ k “ ˆ n ` k ` ˙ n ` k p n ´ k q Γ p k ` q Γ p n ´ k ` q π p n ` q Γ p k ` q Γ p n ´ k ` q ` O p ζ q“ ` ÿ n “ ´ λ π ¯ n ` p´ q n n ` ζ n ` π n ` Γ p n ` q Γ p n ` q π Γ p n q Γ p n ` q ` O p ζ q . (B.3) This leads to (4.2) after using (3.26).To prove the relation (4.11) for the contribution to ∆ q of the sum of terms proportional topowers of ζ one may start with the following U p N q p A, B q model with the ζ term in theexponent representing the corresponding contribution coming from f in (2.1),(3.3) Z “ lim N Ñ8 Z N , Z N p ξ q “ ż r dAdB s e ´ tr A ´ tr B ´ ξN p tr A ´ tr B q , ξ ” ζ ´ λ π ¯ . (B.4)Then according to (3.26), ∆ q p λ q ˇˇˇ ř n ζ n “ λ ddλ log Z . (B.5)Since the integrand in Z N depends only on tr A and tr B , introducing the radial coordinates r A , r B we get (ignoring irrelevant constant factor) Z N p x q “ ż dr A dr B r N ´ ` N p N ´ q A r N ´ ` N p N ´ q B e ´ r A ´ r B ´ ξN p r A ´ r B q . (B.6) In this special case there will be no difference between U p N q and SU p N q cases. N limit is found from a saddle point of the effective action S eff “ p N ´ q log p r A r B q ´ ξN p r A ´ r B q ´ r A ´ r B . Choosing the symmetric saddle with r A “ r B “ b N ´ and integratingover the fluctuations gives Z p ξ q “ ? ` ξ . (B.7)As a result, using (B.5) we find the strong-coupling asymptotics in (4.11). An alternative morerigorous and general approach is based on observing that Z in (B.4) may be represented as Z p ξ q “ lim N Ñ8 e ´ ξ pB x `B y ´ B xy q Z p x q Z p y q ˇˇˇ x “ y “ ,Z p x q ” ż r dA s e ´ tr A ` xN tr A “ ´ ´ xN ¯ ´ N ´ “ e ´ N x ` x ` O p { N q . (B.8)As a result, we get again (B.7).Similar approach can be used to derive (4.15) for the contribution of terms proportional toproducts of ζ , ζ , ..., ζ k ` . For example, let us consider the ζ ζ terms. The new interaction termin the exponent in the analog of (B.4) will be ∆ S ζ “ ´ ηN ” p tr A q ` p tr B q ´ A tr A ´ B tr B ` A tr B ` A tr B ´ A tr B ı , η “ ´ ζ ´ λ π ¯ . (B.9)In this case instead of (B.8) we will need to consider Z p ξ, η q “ lim N Ñ8 exp ” ´ ξ pB x ` B y ´ B x y q ´ η p B x ` B y ´ B x x ´ B y y ` B x y ` B x y ´ B x y q ı Z p x , x , x q Z p y , y , y q ˇˇˇ x i “ y i “ , (B.10) Z p x , x , x q “ ż r dA s e ´ tr A ` x N tr A ` x N { tr A ` x N tr A . (B.11)Expanding (B.11), taking log and sending N Ñ 8 we find Z p x , x , x q “ exp ” N p x ` x q ` x ` x ` x x ` x ` O p { N q ı . (B.12)Using this in (B.10) gives Z p ξ, η q “ ´ ζ ´ λ π ¯ ` ζ ´ λ π ¯ ` ζ ´ λ π ¯ ´ ζ ζ ´ λ π ¯ ` ´ ´ ζ ` ζ ¯´ λ π ¯ ` ζ ζ ´ λ π ¯ ` . . . (B.13)As a result ∆ q p λ q ˇˇˇ ζ ,ζ “ λ ddλ log Z “ ´ ζ λ ´ λ π ¯ ` ζ λ ´ λ π ¯ ` ζ p q λ ´ λ π ¯ ´ ζ ζ λ ´ λ π ¯ ` ´ ´ ζ ` ζ ¯ λ ´ λ π ¯ ` ζ ζ λ ´ λ π ¯ ` ¨ ¨ ¨ , (B.14)which agrees with the results given in the main text.24t is interesting to note that (B.10) can be computed in a closed form generalizing (B.7) Z p ξ, η q “ ” ` ξ ´ η ` ξη ´ η ´ η ı ´ { . (B.15)Applying λ ddλ to the log of (B.15) as in (B.14) then gives the exact form of ∆ q p λ q ˇˇˇ ζ ,ζ . C Wilson loop in SU p N q “orientifold” N “ superconformal theory It is possible to give a similar discussion of the large N expansion of the Wilson loop x W y andthe free energy in a particular N “ superconformal gauge theory involving in addition to the SU p N q N “ vector multiplet also two hypermultiplets – in rank-2 symmetric and antisymmetric SU p N q representations. This theory admits a regular ’t Hooft large N limit and thus is similar tothe quiver theory discussed above. It should be dual to the type IIB superstring on a particularorientifold AdS ˆ S {p Z orient2 ˆ Z orb2 q (see [50]).This theory is one of the five cases of N “ superconformal theories admitting a gauge group SU p N q with generic N [51]. The corresponding BPS circular Wilson loop is again equal to the N “ SYM one at the planar level. Here we shall focus on the weak-coupling expansion of thefirst subleading { N correction, i.e. of the corresponding function q p λ q defined as in (1.10).From the supersymmetric localization, the free energy and the Wilson loop expectation value x W y orient in this theory are described by the Hermitian one-matrix model of the similar structureas in (2.1) where instead of (3.3) now we have [52] log f “ ÿ n “ p´ q n ` ´ λ π N ¯ n ` ζ p n ` q n ` n ´ ÿ k “ ˆ n ` k ` ˙ tr A k ` tr A n ´ k ` . (C.1)One can then organise the expansion of x W y orient in powers of monomials of ζ n ` -constants as in(3.1). One finds that, as in the orbifold theory, at the leading non-planar level all appearing ζ n ` -monomials are multiplied by I p? λ q times a power of λ (cf. (3.21)). 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