$1/N$ expansion of circular Wilson loop in $\mathcal N=2$ superconformal $SU(N)\times SU(N)$ quiver

Localization approach to $\mathcal N=2$ superconformal $SU(N) \times SU(N)$ quiver theory leads to a non-Gaussian two-matrix model representation for the expectation value of BPS circular $SU(N)$ Wilson loop $\langle\mathcal W\rangle$. We study the subleading $1/N^2$ term in the large $N$ expansion of $\langle\mathcal W\rangle$ at weak and strong coupling. We concentrate on the case of the symmetric quiver with equal gauge couplings which is equivalent to the $\mathbb Z_{2}$ orbifold of the $SU(2N)$ $\mathcal N=4$ SYM theory. This orbifold gauge theory should be dual to type IIB superstring in ${\rm AdS}_5\times (S^{5}/\mathbb Z_{2})$. We present a string theory argument suggesting that the $1/N^2$ term in $\langle\mathcal W\rangle$ in the orbifold theory should have the same strong-coupling asymptotics $ \lambda^{3/2}$ as in the $\mathcal N=4$ SYM case. We support this prediction by a numerical study of the localization matrix model on the gauge theory side. We also find a relation between the $1/N^2$ term in the Wilson loop expectation value and the derivative of the free energy of the orbifold gauge theory on 4-sphere.


Introduction and summary
Supersymmetric Wilson loop operators provide an important class of observables that shed light on the 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 represent the expectation value of a supersymmetric loop in terms of a matrix model integral.
Here we will consider a particular N " 2 supersymmetric gauge theory which is the SU pN qŜ U pN q quiver with two bi-fundamental hypermultiplets [2,3,4,5]. In the "symmetric" case when the two 't Hooft couplings λ 1 , λ 2 are equal this theory is equivalent to the Z 2 orbifold of the SU p2N q N " 4 SYM theory [6]. The orbifold theory has the same planar diagrams as the parent N " 4 SYM theory [7], i.e. the two are closely related at large N . The dual string theory should be the corresponding orbifold of the AdS 5ˆS 5 superstring, i.e. type IIB string on AdS 5ˆp S 5 {Z 2 q [8,9]. 1 For each of the two SU pN q factors of the quiver theory one may define 1 2 -BPS circular Wilson loops coupled to the corresponding gauge and scalar fields (a " 1, 2) W a " tr P exp " ¿ ds pi 9 x µ A µ a`| 9 x| Φ a q ı , (1.1) where we choose not to include the 1{N factor in front of the trace. For the orbifold theory their (normalized) expectation values are equal xW 1 y " xW 2 y " xWy orb , (1.2) and at large N coincide [4] with the famous SU pN q N " 4 SYM result [10,11,12] xWy orb N Ñ8 " xWy SYM N Ñ8 " xWy 0 , For general N the expression for xWy orb is given by a special non-Gaussian matrix model integral following from the localization approach [12]. In contrast to the N " 4 SYM case where the corresponding matrix model is Gaussian leading to the closed expression [11,12] xWy SYM " e λ 8N p1´1 N q L working out the 1{N expansion of xWy orb turns out to be a non-trivial problem. Below we will address the question about the structure of the λ-dependent coefficients in the 1{N expansion of xWy orb by considering separately the small and large λ limits. On the dual string theory side, the 1{N 2 expansion is the genus expansion, and the λ dependence of the 1{N 2p coefficient in the analog of (1.4) corresponds to the string tension dependence of the partition function with world surface of topology of a disc with p handles.
As discussed recently in [13], the strong coupling expansions of the 1 2 -BPS circular Wilson loops in N " 4 SYM and ABJM gauge theories with string duals defined on AdS 5ˆS 5 and AdS 4ˆC P 3 have a remarkably similar structure. The string counterpart of the dominant at large N (p " 0) term in (1.3), (1.4) is the open-string partition function on the disk which contains an overall factor of the inverse closed string coupling g s xWy 0 " 2π g s e 2πT e´Γ " 1`O´T´1¯ı . (1.5) In the SU pN q N " 4 SYM case 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 the 1-loop fluctuation determinants [14] appearing in the string partition function expanded near the AdS 2 minimal surface (corresponding to the circular Wilson loop). In the case of genus p surface the UV divergent part of the one-loop effective action Γ " 1 2 ř i log det ∆ i reads [13] Γ "´ζ tot p0q logpLΛq`Γ , ζ tot p0q " χ " 1´2p , where Λ is 2d cutoff, L is the AdS radius (T " L 2 2πα 1 ) and the ζ tot p0q coefficient turns out to be equal to the Euler number of the surface. The 2d UV divergence should be canceled by a universal superstring measure contribution logp ? α 1 Λq 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 1 "´χ log ?
T and thus the string partition function on a genus p surface is proportional to e´Γ fin " p ? T q χ , i.e.
xWy "  (1.4). One can also use similar considerations to predict the structure of the string theory expansions for other related observables [15]. It is important to emphasize the universality of the structure of the expansion in (1.8): it relies only on the fact that one expands near the AdS 2 minimal surface embedded into the AdS 3 part of AdS n space and should thus be valid also for the corresponding partition functions in the AdS 4ˆC P 3 and AdS 3ˆS 3ˆT 4 superstring theories [13]. It should also apply to the orbifold AdS 5ˆp S 5 {Z 2 q theory: orbifolding the S 5 should not change the above argument determining the tension dependence from the way how the AdS radius L appears in (1.7). 2 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 " 2 orbifold theory case, i.e. where the coefficients c p " cp p8πq p´1{2 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 xWy orb we shall consider the genus one term corresponding to the leading 1{N 2 correction to the planar part in (1.3). Normalizing to xWy 0 in (1.3) we have in both N " 4 SYM and N " 2 orbifold cases where the form of the function qpλq will be our main focus in what follows. In the SU pN q SYM case the expression for qpλq follows from the expansion of the exact Laguerre polynomial expression in (1.4) (I n are modified Bessel functions of the first kind) As discussed above, the leading strong-coupling behaviour of the genus one correction in xWy SYM 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.
In the case of SU pN qˆ...ˆSU pN q N " 2 quiver theory which is the Z k orbifold of the SU pkN q N " 4 SYM and should be dual to the superstring on AdS5ˆpS 5 {Z k q one has for the AdS radius L 4 " 4πkN gsα 1 and thus instead of (1.6) we get gs " where the value of the coefficient C may of course be different from 1 96 in the SYM case in (1.11). Confirming the prediction (1.13) starting from the localization matrix model representation for xWy 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 " 2 gauge theory partition function Z orb pλ; N q on S 4 and for xWy orb imply a remarkable relation between ∆qpλq ∆qpλq " q orb pλq´q SYM pλq , (1.14) and the N Ñ 8 limit of the deviation of the orbifold free energy F orb "´log Z orb from its SYM counterpart (1.16) The leading OpN 2 q term in F orb´2 F SYM cancels due to the planar equivalence between the SU pN qˆSU pN q orbifold theory and the two decoupled copies of the SU pN q N " 4 SYM theory. 3 Using (1.15) the expected strong-coupling behaviour (1.13) of qpλq translates into the following scaling for the difference of free energies in (1.16) (c 1 "´16C) In view of (1.16) this leads to a prediction about the strong coupling asymptotics of the leading 1{N 2 correction to the planar F orb pλ; 8q " 2 F SYM pλ; 8q part of the free energy of the orbifold theory.
Let us first recall that the free energy of SU pN q N " 4 SYM on S 4 should not be renormalized, i.e. should be given exactly by the familiar one-loop expression F SYM pλ; N ; Λq " 4 a " logprΛq`f 0 ‰ . Here a is the conformal anomaly coefficient, a " 1 4 pN 2´1 q, r is the radius of S 4 , Λ is a 4d UV cutoff and f 0 is a constant. Since the free energy is UV divergent, its finite part is not universal depending on a particular regularization scheme. The localization procedure [12] representing the free energy in terms of a finite matrix model integral with a simple λ-independent measure implicitly assumed a special regularization in which the renormalized SU pN q SYM free energy is given by as this expression follows simply from the Gaussian matrix model integral [16] (we drop an additive numerical constant). 3 More precisely, for the Wilson loops (1.2) the planar equivalence means that the normalized expectation value of an operator in one of the two SU pN q factors of the quiver is the same as in the SU pN q N " 4 SYM theory. In general, the planar correlators of operators from Z2 symmetric (i.e. "untwisted") sector should match between the orbifold and the parent SU p2N 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 pN qˆSU pN q orbifold theory should match those of the two copies of the SU pN q N " 4 SYM.
From the dual string theory point of view, the gauge theory free energy is expected to be given by the AdS 5ˆS 5 string partition function or (at the tree level) by the IIB string effective action S " S 0`S1`. .. " ... Evaluated on AdS 5ˆS 5 (using (1.6) and R`... "´8L´2) the leading supergravity term here is S 0 " 1 π 2 N 2 V AdS 5 where V AdS 5 is the (logarithmically) IR divergent volume of unit-radius AdS 5 . Subtracting the IR divergence in V AdS 5 using a particular AdS/CFT motivated prescription gives V AdS 5 ÑV AdS 5 " π 2 log ? λ and thus one reproduces [16] the N 2 log λ term in (1.18). 4 The´1 shift of N 2 in (1.18) should come from the 1-loop superstring correction (again proportional to V AdS 5 ): this should follow the same pattern as found for the N " 4 SYM conformal anomaly and S 3 Casimir energy in [17] (with only loops of short supergravity supermultiplets contributing). Other string α 1n tree level (e.g. α 13 R 4 , cf. [18]) and string loop corrections should vanish on maximally supersymmetric AdS 5ˆS 5 background. Let us now turn to the orbifold theory that should be dual to the superstring on AdS 5ˆp S 5 {Z 2 q. Combining (1.16),(1.17) and (1.18) we get the following prediction The leading N 2 term here is implied by the planar equivalence to the SU p2N q SYM and should follow again from the leading type IIB supergravity term evaluated on AdS 5ˆp S 5 {Z 2 q. 5 The planar equivalence also means that like in (1.18) this leading N 2 term should not get string tree level α 1 -corrections, i.e. they should still vanish when evaluated on AdS 5ˆp S 5 {Z 2 q. One may attempt to give an independent string theory explanation of the subleading λ 1{2 term in (1.19) without using the connection (1.15) to the Wilson loop. The string one-loop (genus one or order g 0 s " N 0 ) type IIB effective action is known to start with S 1 " 1 . [19,20,21]. If we conjecture that when evaluated on the orbifold AdS 5ˆp S 5 {Z 2 q it is no longer zero then on dimensional grounds it should scale as S 1 " L 2 α 1 " ? λ, reproducing the subleading term in (1.19). If a non-zero contribution comes just due to the curvature singularity then it may not be proportional to the AdS 5 volume so there will be no extra log λ factor. The remaining puzzle is why the one-loop R 4 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] (cf. the case of compactification of 6d orbifolds [23]).
Starting with the matrix model representation for xWy orb we shall first study the structure of the function q orb pλq in (1.10) or ∆qpλq in (1.14) at weak coupling. While the small λ expansion of q SYM pλq in (1.11) has only rational coefficients, the coefficients in the expansion of ∆qpλq in 4 One way to understand the origin of the log ? λ term is as follows. On AdS side the IR cutoff is measured in units of AdS radius L. On the gauge theory side viewed as originating from the flat-space open string theory the natural UV cutoff is inverse of the string length ? α 1 . Thus the two cutoffs are related by the ratio L ? α 1 " λ 1{4 . 5 To recall, the orbifold projection of SU p2N q SYM giving the SU pN qˆSU pN q theory with two bi-fundamental hypermultiplets reduces the number of degrees of freedom and thus also the leading large N term in the conformal anomaly coefficient from a " 1 4 p2N q 2 to 2ˆ1 4 N 2 which is twice the anomaly of a single copy of SU pN q SYM theory at large N . The exact expressions for the conformal anomaly coefficients of the SU pN qˆSU pN q quiver theory are a " 1 2 N 2´5 12 and c " 1 2 N 2´1 3 . On the supergravity side, replacing the N 2 coefficient in the above discussion by p2N q 2 and noting that the volume of pS 5 {Z2q is half of the volume of S 5 we end up with 2N 2 as an overall coefficient. Thus ζ n may be formally used to parametrise the deviation of the orbifold theory result from the N " 4 SYM one.
To find the strong-coupling expansion of q orb pλq requires a resummation of the weak coupling expansion. We shall study resummations of particular subclasses of terms proportional to monomials built out of ζ n . While this will not be enough to determine the correct strong coupling asymptotics of q orb pλq, this may still help to shed light on the general structure of this function. 6 Exploiting the relation (1.15) we shall compute ∆qpλq up to order Opλ 20 q and also determine the resummation of all terms with the following types of coefficients involving particular ζ n and their powers I: ζ 2n`1 , II: i.e. q orb I " ř c n,m ζ 2n`1 λ m , etc. We find that they have the following behaviour at strong coupling The difference in these asymptotics implies that to find the correct strong coupling behaviour of the full q orb one needs first to sum together different subsets of terms and only then expand the total at large λ.
Lacking an analytic method to compute q orb pλq at strong coupling we performed extensive numerical simulations of the SU pN qˆSU pN q orbifold matrix model to measure it. This required an extrapolation to large N for finite λ, followed by an analysis of large λ region. We confirmed that the deviation from the N " 4 SYM case starts only at the non-planar level. The numerical data agrees with the Padé-Borel resummation of the weak-coupling expansion up to moderate λ " 50. At larger values of the coupling λ we found that the data is compatible with the following asymptotics The power of the leading asymptotics η « 1.5 is thus consistent with the string theory prediction (1.13). 7 It is interesting to notice that the values of the coefficients C and Ca 1 are very close to the values of the corresponding coefficients in (1.11) in the SYM case up to a factor of´1 2 and`1 2 respectively. This suggests a conjecture that the exact form of the strong coupling expansion of q orb pλq is given by It remains to be seen if one can prove this analytically.
One may wonder if the coefficient of the leading λ 3{2 N 2 correction in xWy orb may be found, as in the N " 4 SYM case [27], also by considering the circular Wilson loop in k-symmetric representation which should be described (for k " 1 and k ? λ N =fixed) by a classical D3-brane solution. This does not seem possible as the D3-brane solution of [27] is restricted to AdS 5 and thus the k 3 λ 3{2 N 2 term in its action should have the same coefficient as in the N " 4 SYM case, in contradiction with (1.25), (1.26). In fact, the D3-brane solution of [27] should be related not to the SU pN q Wilson loop (1.2) of the SU pN qˆSU pN q orbifold theory but to the orbifold projection of the original circular Wilson loop in the SU p2N q SYM theory. The projection of the latter is represented by the correlator xW 1 W 2 y where W 1,2 in (1.1) correspond to the two SU pN q factors of the orbifold theory. Starting with the SU p2N q Wilson loop in k-symmetric representation one is to split it into the sum of products of the two SU pN q representations. Then the D3-brane description may apply only to a special combination of the xW 1 W 2 y correlators where W 1 and W 2 are taken in the particular representations of the SU pN q appearing in the product. 8 Assuming that k-symmetric representation may be replaced by the k-fundamental one (corresponding to multiply wrapped circle, cf. [28,29]) one would expect to get the sum of the correlators is the SU pN 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 then end up with the sum of the correlators xW 1 W 2 y of the two fundamental SU pN q Wilson loops in the SU pN qˆSU pN q quiver theory with the two 't Hooft couplings λ 1 " m 2 λ, λ 2 " pm´kq 2 λ. The resulting expression should simplify in the large k limit and one expects it to be dominated by the "diagonal" term (m " k{2) with W a in the same representation. In section 6 we shall present numerical data indicating that the 1{N 2 term in this correlator (with both Wilson loops taken in the fundamental representation of SU pN q) has a similar strong-coupling behaviour to 1 96 λ 3{2 found in the SYM case in (1.11).
Below we also computed numerically the individual SU pN q Wilson loop (1.1) expectation values xW 1 y , xW 2 y in the SU pN qˆSU pN q quiver with unequal couplings λ 1 , λ 2 starting with the localization matrix model representation. Guided by the discussion in [4] here we considered the following analog of the ratio in (1.10) 9 The strong coupling result of [4] then implies that ppλ; θqˇˇλ Ñ8 Ñ 1 for any θ ‰ 0, 2π. We confirmed this prediction numerically by considering a particular value of the ratio of the two couplings 8 We thank N. Drukker for this suggestion. 9 xW2y is found by interchanging λ1 and λ2 or θ Ñ 2π´θ. λ 2 {λ 1 " 3 (i.e. θ " π 2 ) and measuring both Wilson loops xW 1 y and xW 2 y, thus effectively probing also the value θ " 3π 2 . We found that the strong-coupling expansion of the function p has the form ppλ; θq " 1`hpθq{ ? λ`... where h has a non-trivial dependence on θ. The numerical data for the function qpλ; θq turns out to be compatible with the strong-coupling asymptotics in (1.24) with the exponent η being again close to 3{2 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 model integral representation for the Wilson loop expectation value in the quiver theory which will be our starting point. In section 3 we shall consider the weak coupling expansion of the leading non-planar term 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 qpλq and the free energy. In section 4 we shall perform a resummation of some subsets of terms and then expand them at strong coupling finding non-universal behaviour. In section 5 we shall consider the weak-coupling expansion in the case of non-symmetric quiver theory. Finally, in section 6 we shall present the results of the numerical computation of the matrix model integrals. Appendices A and B will contain some technical details of the computations in sections 3 and 4. In Appendix C we shall briefly discuss similar weak-coupling analysis of the Wilson loop in SU pN q "orientifold" N " 2 superconformal theory.
Note added in v4: The coefficient c 1 in (1.17) was recently computed exactly in [30] with the result c 1 " 1 4 (also, coefficients of several subleading terms in (1.19) were also found). Since according to (1.15) the coefficient c 1 is related to the coefficient of λ 3{2 term in q or C in (1.24) as C "´1 16 c 1 this implies that C "´1 32 (invalidating the conjecture in (1.26)). This value is substantially larger than the numerical estimate for C in (1.25). This shows that the explored range of λ values was too narrow to reach the asymptotic regime λ " 1 where the leading λ 1{2 term dominates. This is consistent with the presence of the subleading log λ correction in ∆F in (1.17) (also established in [30]) that slows down the convergence.

Matrix model representation
Our starting point will be the localization matrix model representation for the S 4 partition function and the expectation values of the circular Wilson loops (1.1) in the SU pN qˆSU pN q N " 2 superconformal quiver theory [12] (see also [2,3,4]). The partition function may be written as the integral over two sets of eigenvalues (a " 1, 2; i " 1, ..., N ) where the δ-functions reflect the fact that we are considering the SU pN q case (they may be ignored in strict planar limit) and 10 f ra 1 , a 2 s " Below we shall use x¨¨¨y 0 to denote the (normalized) expectation value in the matrix model with the Gaussian measure so that (2.1) may be written as The expectation values of the two Wilson loops (1.1) are given by where x...y is given by the same integral as in (2.1) and is normalized so that x1y " 1. We use the notation a a for the diagonal matrix a a " diagpa a1 , ..., a aN q. The two expectation values (2.5) are equal (cf. (1.2)) at the orbifold point λ 1 " λ 2 " λ.
For large N one may study the saddle points of the "effective action" in (2.1) Differentiating over a ai and introducing the densities one finds the following saddle point equations [2,3] ż µ 1 (2.10) 10 Hpxq has the following representation in terms of the Barnes function Gpxq The partition function is invariant under Hpxq Ñ Hpxq e Cx 2 [3]. Note also that we ignored the instanton factor [12] in the integrand as we will be interested in perturbative 1{N expansion (see [3]).
The large N equivalence of the orbifold theory with the N " 4 SYM follows [2,4] from the fact that for λ 1 " λ 2 " λ the equations (2.8),(2.9) admit the symmetric Ansatz ρ 1 " ρ 2 " ρ, µ 1 " µ 2 " µ and reduce to the saddle point equation for the Gaussian matrix model corresponding to the N " 4 SYM case for which . (2.11) The solution of the two integral equations (2.8),(2.9) in the large λ limit was studied in [2], showing that xW a y " e ? λ , λ " 2λ 1 λ 2 λ 1`λ2 , and more recently in [4] where it was found that (see (1.29)) W 0 is the leading large N , strong coupling term in the SYM result in (1.3).

Weak coupling expansion in the orbifold theory
Considering the orbifold theory case λ 1 " λ 2 one can work out the weak-coupling expansion of xWy orb by starting with the integral representation (2.1),(2.5) for finite N . It may be formally written as a sum of functions W ζ 3 pλ, N q, W ζ 5 pλ, N q, ..., multiplying particular products of ζ n " ζpnq values 11 Here xWy SYM is given by (1.4) so that W ζ 3 , etc., scale as 1{N at large N . For small λ one has To compute these functions starting with (2.1) let us note that using (2.2),(2.10) we get where we defined C k n`2 " p´1q k`2n`2 k˘, a a " diagpa a1 , ..., a a N q (with tr a a " 0) and we also introduced the rescaled matrices A a (appearing in the exponent in (2.1))

Direct perturbative expansion
Separating different ζ n terms we may write f in (3.3) as an expansion in λ 8π 2 N "
(3.6) Using (3.3) and computing (2.5) by first integrating out the A 2 dependence with the help of we obtain for the coefficient functions in (3.1) : ptrA 2 1 q 2 :`2 : trA 2 1 : This reduces the problem to evaluating correlators in one-matrix (A " A 1 ) Gaussian model. Computing the Wilson loop correlator with normal ordered operators using that for the SU pN q SYM case [11] xWy SYM " xtr e and applying the method described in Appendix A (see also [15]), we find (g "

Leading non-planar correction and relation to free energy
Remarkably, the dependence on λ of the leading term Op1{N q in the W functions in (3.1) follows the same pattern, i.e. is proportional to the Bessel function I 1 p ? λq that appears in the leading order term in the SYM expression (3.16) The power of λ 8π 2 is coming from the ζ 2n`1 factors in (3.3) while the extra factor of λ (from the Bessel function factor ? λ I 1 p ? λq " λ 2`¨¨¨) has its origin in the Wilson loop operator insertion into the Gaussian matrix model integral at the leading order in large N (cf. (2.5)). In particular, it comes from the A 2 1 term in (trA a " 0) Separating this SYM Bessel function factor, the expression for the leading large N term in W " xWy orb´x Wy SYM in (3.1) can be written in terms of the function qpλq defined in (1.10),(1.14) q orb pλq "q SYM pλq`∆qpλq, ∆qpλq " λ As a check, using (3.3) one can compute the leading terms in the expansion of xf y 0 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 limit of the free energy of the N " 2 orbifold theory (which is finite for N Ñ 8 after the subtraction of the planar N " 4 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 1 p ? λq would be prohibitively difficult. Using it we were able to push the calculation of the ζ n expansion of qpλq up to Opλ 20 q. In particular, the next five terms beyond (3.23) read

Resummation of particular transcendental contributions to xWy orb
In an attempt to shed light on the structure of strong coupling limit of xWy orb one may try to consider separate terms in the transcendental part of (3.1), resum their weak coupling expansion and then expand at strong coupling. As we shall see, this procedure will not give the correct strong-coupling limit of xWy orb : the strong-coupling asymptotics of different functions W ζ k n ... will be 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 2n`1 have the following coefficient functions Summing all such terms in xWy orb in (3.1), i.e. Wˇˇζ " ř 8 n"1 W ζ 2n`1 , we then get the corresponding contribution to q orb or to ∆q in (1.14), (3.26) ∆qpλqˇˇζ " We can resum this series by noting that ∆qpλqˇˇζ " λ 2 16π (4.5) Using the properties of the Mellin transform 12 we find that the large λ asymptotics of (4.4) is Similarly, we may consider all terms in (3.1) proportional to ζ 3 ζ 2n`1 with n ą 1 (see Appendix B). We get the following analog of (4.2) Summing this series as in (4.3),(4.4) we get λq, Expanding at large λ here gives a different asymptotics than in (4.6) 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 sum of all contributions involving arbitrary powers of ζ 3 and ζ 5 and also of ζ 7 ∆qpλqˇˇř n,m ζ n 3 ζ m "´λ N pt 3 , t 5 , t 7 q Dpt 3 , t 5 , t 7 qˇˇˇt 3 "ζ 3 p λ 8π 2 q 2 , t 5 "ζ 5 p λ 8π 2 q 3 , t 7 "ζ 7 p λ 8π 2 q 4 λ"1 "´5 4 λ`... , (4.14) Comparing (4.11),(4.12) and (4.13) suggests that the strong coupling limit of the sum of monomials involving powers of the first k constants ζ 3 , ζ 5 , ..., ζ 2k`1 should be (cf. 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.

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 pN qŜ U pN q quiver for unequal couplings λ 1 " λ 2 . We shall consider for definiteness xWy " xW 1 y. Setting λ 2 " ρ λ 1 , It is then straightforward to compute the large N expansion of the coefficient functions of the ζ n -monomial contributions to xWy in the analog of (3.1). The first of them that generalizes (3.17) is The planar (order N ) contribution here agrees with the N Ñ 8 part of the N " 4 SYM result in (1.3) expressed in terms of the effective coupling [33,34,31] 13 In the case of the λ1 " λ2 quiver the weak coupling expansion in the planar limit was analysed also in [35] and [25].
Note that the subleading terms in (5.3) proportional to ρ´1 are similarly captured by the SYM term if we modify (5.5) as One can also find the analog of W ζ 5 in (3.1), (3.18) and the ρ´1 terms there can be generated from the SYM expression by the replacement λ Ñ λ eff generalizing (5.6) 14 p13`3ρq N 2¯λ This suggests that some essential features of the weak coupling expansion of the Wilson loop in the non-symmetric quiver case are already captured by the orbifold case (ρ " 1) discussed above.

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 with the matrix model representation (2.1), (2.5). While this is a finite dimensional integral, the fact that are interested in the limit N " 1 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 a neighbourhood of the saddle point solution. This problem is completely analogous to the one in the lattice field theory (where one computes quantum corrections by numerical path integration with N " ´1 ) and may thus be dealt with by the same Monte Carlo (MC) methods (see, for instance, [36]).

Orbifold theory
We analysed the Wilson loop expectation value (3.1) in the orbifold case by means of a Metropolis-Hastings Monte Carlo simulation [37] of the integral (2.1), a robust approach that does not require fine tuning. 15 Given a configuration X of the eigenvalues a ai corresponding to the two SU pN q groups, the matrix integral (2.1) weights each observable OpXq, like the Wilson loop, with a positive number expp´Sq where S " SpXq corresponds to the total integrand in (2.1) including the Vandermonde factor. A Markov chain obeying detailed balance is built by making a local variation of X Ñ X 1 and accepting the new configuration if SpX 1 q ă SpXq or, in the case SpX 1 q ą SpXq, with probability e SpXq´SpX 1 q . We tuned the local changes of configuration in order to have an acceptance probability around 50-60% which is a reasonable choice. Iterating this procedure produces a sequence tX n u of configurations distributed according to expp´Sq and one can measure the quantum expectation value as the ensemble average xOy " lim nÑ8 1 n ř n m"1 OpX m q. The sequence tX n u is correlated and its autocorrelation time has been measured at each data point and taken into account in the estimate of an error in this MC evaluation of xOy. 16 For each value of λ, we ran our code at various values of N and fitted the Wilson loop measurements in order to extract the function q orb pλq (3.23) that governs the leading non-planar correction in (1.10). The procedure is illustrated in Fig. 1 (left) at the value λ " 1. Fig. 1 (right) shows the histogram of measurements of the Wilson loop at λ " 200, N " 20, as a sample point.
To provide non-trivial checks of the numerical code we considered the Wilson loop at λ " 1 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 order contributions. Then q orb p1q "´0.122p2q , where the error is an estimate of the systematic error determined by including or not the the contribution of the ζ n terms explicitly computed above. The extrapolated slope from the finite N MC simulations at λ " 1 shown in Fig. 1 (left) gives 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 ∆qpλq 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 2 with partial sums blowing up beyond that value. 17 The green line is the r7{6s Padé approximant of the Borel improved series, while the blue line is the Borel transform, which is thus in good agreement with the numerical data points.
At higher values of λ we found similar extrapolations in 1{N 2 . In Fig. 3 (left) we show the intercept of the extrapolation which is expected to be 1, see (3.20). This is a measure of the systematic error associated with the fit of the N dependence. It increases with λ and we increased the maximal N in order to keep it below the 0.2% level. 18 The resulting function q orb pλq computed for up to λ " 450 is shown in Fig. 3 (right). In the SU pN q N " 4 SYM theory, we know from (1.11) that at strong coupling q SYM pλq " 1 96 λ 3{2`. .. which is valid with high accuracy already at λ Á 20. In the orbifold theory we find that q orb pλq is negative with a clear bending at large λ suggesting an asymptotic behaviour q orb pλq "´λ η , η ą 1 . (6.3) 16 As is well known (see, for instance, [45]), denoting by x¨¨¨y MC the average over MC realisations and assuming an exponential autocorrelation for the measurements On " OpXnq, i.e. xOnOmy MC " σ 2 O e´| n´m|{τ O , the variance of the expectation value estimator 1 n ř n m"1 OpXmq is 1 showing that the effective number of decorrelated measurements is roughly n decorr " n{p2τ O`1 q which is the factor entering the standard deviation of The best fit of the blue data points in Fig. 3 (right) gives η " 1.49p2q where the conservative error estimate includes statistics as well as the systematic effects due to the choice of fitting window. We estimated the latter by dropping some of the data points at smaller values of λ. This exponent is still 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 [27] in the SU pN qˆSU pN q orbifold theory (see Introduction), we numerically computed the expectation value xW 1 W 2 y of the two SU pN q Wilson loops (1.1) and determined (using the same fitting procedure as discussed above) the associated q WW pλq function defined as in (1.10) The corresponding data points are shown in Fig. 6. They decrease to negative values with rate slower than the one observed in q orb pλq. A best fit of the form (1.24) with η fixed at 3 2 gives C WW "`0.012p2q and a 1 WW "´21p2q. The coefficient C WW has the opposite sign to the one in (1.25) and is close to the SYM value 1 96 « 0.010 in (1.11). One possible interpretation of this result is that the "diagonal" correlator xW 1 W 2 y of the two Wilson loops in the fundamental representation exhibits the (at the leading non-planar order) the strong coupling behaviour which is expected from the D3-brane description, while other terms appearing in (1.27) are less important in the large k limit.

Non-symmetric quiver
In the case of generic (non-zero) λ 1 and λ 2 the strong-coupling asymptotics of the Wilson loops is given by (2.12). We shall study the functions ppλ, θq and qpλ, θq in the ratio (1.28) of xWy 1 to the planar SYM result. We begin with the special point θ " π 2 or (see (1.29)) The numerical results are shown in Fig. 4. The left panel gives the function ppλ, π 2 q. As expected, it decreases for large λ towards 1 (this should hold for any θ, see (2.12)) and a good fit is ppλ, π 2 q " 1.00`0.23 λ 1{2`8 .2 λ´1`... . (6.6) Measurement of the second Wilson loop xWy 2 provides the information about the same functions at the complementary value of the angle θ 1 " 2π´θ " 3π 2 for which The corresponding results are shown in Fig. 5. The best fit for the ppλ, 3π 2 q is 19 ppλ, 3π 2 q " 0.99´3.2 λ 1{2`3 .4 λ´1`... . (6.8) 19 The small but not negligible deviation of the estimated asymptotic value from 1 suggests that systematic errors should be reduced by N Ñ 8 extrapolations with larger values of N . This could be related to the much large value of the correcting factor wp 3π 2 q as compared to wp π 2 q.  The function qpλ; θq at θ " π 2 and 3π 2 is shown in the right panels of Fig. 4 and Fig. 5. Our estimate for the exponent ηpθq in the analog of (6.3) is ηp π 2 q " 1.3p2q and ηp 3π 2 q " 1.6p2q. Both values appear to be similar to the one found in the orbifold case (θ " π), i.e. η « 3 2 . It would be desirable to push the MC simulation to larger values of the coupling λ, but that seems to require a dedicated analysis with a substantially increased computational power. Right: Data points for the function q orb pλq defined in (1.10), (3.23). Dashed line is the non-linear fit with the functional form q orb pλq " C λ η p1`a 1 λ´1 {2 q. The fit is performed using data points with λ ě 100 which have been determined with a relative error below 3% .  Figure 4: Functions ppλ, π 2 q (left) and qpλ, π 2 q (right) for the quiver at the point λ 2 " 3λ 1 , with λ " 2λ1λ2 λ1`λ2 " 3 2 λ 1 . Figure 5: Functions ppλ, 3π 2 q (left) and qpλ, 3π 2 q (right) for the quiver at the point λ 1 " 3λ 2 , with λ " 2λ1λ2 λ1`λ2 " 3 2 λ 2 . The angle θ " 3π 2 corresponds to the Wilson loop for the second SU pN q factor.  and a multi-trace chiral operator O may be reduced to a differential operator over the coupling constant acting on xWy 0 (see (3.12)- (3.15)). This relation is exact at finite N and is achieved by exploiting the SU pN q fusion/fission relations [51] and the associated recursion relations on the expectation values t n 1 ,n 2 ,...,nr " xtrA n 1 trA n 2¨¨¨t rA nr y 0 .
Let us consider as an example @ W : trA 6 : : Doing Wick contractions leads to a combination of "single-trace" terms that can be traded for B g differential operators acting on xWy 0 and we finally obtain This procedure can be easily coded in symbolic manipulation programs.
To prove the relation (4.11) for the contribution to ∆q of the sum of terms proportional to powers of ζ 3 one may start with the following U pN q 2-matrix pA, Bq model with the ζ 3 term in the exponent representing the corresponding contribution coming from f in (2.1) (B.6) The large N limit is found from a saddle point of the effective action S eff " pN 2´1 q logpr A r B qξ As a result, using (B.5) we find the strong-coupling asymptotics in (4.11). An alternative more rigorous and general approach is based on observing that Z in (B.4) may be represented as As a result, we get again (B.7). Similar approach can be used to derive (4.15) for the contribution of terms proportional to products of ζ 3 , ζ 5 , ..., ζ 2k`1 . For example, let us consider the ζ 3 ζ 5 terms. The new interaction term in the exponent in the analog of (B.4) will be ∆S ζ 5 "´η N 3 " 2ptrA 3 q 2`2 ptrB 3 q 2´3 trA 2 trA 4´3 trB 2 trB 4 20 In this special case there will be no difference between U pN q and SU pN q cases.
`3trA 4 trB 2`3 trA 2 trB 4´4 trA 3 trB 3 ı , η "´1 0 3 ζ 5´λ 8π 2¯3 . (B.9) In this case instead of (B.8) we will need to consider Zpξ, ηq " lim N Ñ8 exp "´ξ pB 2 2¯λ´λ 8π 2¯6`6 30ζ 2 3 ζ 5 λ´λ 8π 2¯7`¨¨¨, (B.14) which agrees with the results given in the main text. It is interesting to note that (B.10) can be computed in a closed form generalizing (B.7) Zpξ, ηq " Applying λ 2 8 d dλ to the log of (B.15) as in (B.14) then gives the exact form of ∆qpλqˇˇζ C Wilson loop in SU pN q "orientifold" N " 2 superconformal theory It is possible to give a similar discussion of the large N expansion of the Wilson loop xWy and the free energy in a particular N " 2 superconformal gauge theory involving in addition to the SU pN q N " 2 vector multiplet also two hypermultiplets -in rank-2 symmetric and antisymmetric SU pN q representations. This theory admits a regular 't Hooft large N limit and thus is similar to the quiver theory discussed above. It should be dual to the type IIB superstring on a particular orientifold AdS 5ˆS 5 {pZ orient 2ˆZ orb 2 q (see [52]). This theory is one of the five cases of N " 2 superconformal theories admitting a gauge group SU pN q with generic N [53]. The corresponding BPS circular Wilson loop is again equal to the N " 4 SYM one at the planar level. 21 Here we shall focus on the weak-coupling expansion of the first subleading 1{N 2 correction, i.e. of the corresponding function qpλq defined as in (1.10).
From the supersymmetric localization, the free energy and the Wilson loop expectation value xWy orient in this theory are described by the Hermitian one-matrix model of the similar structure as in (2.1) where instead of (3.3) now we have [54] log f " 2 8 ÿ n"1 p´1q n`1´λ 8π 2 N¯n`1 ζp2n`1q n`1 n´1 ÿ k"1ˆ2 n`2 2k`1˙t rA 2k`1 trA 2n´2k`1 . (C.1) One can then organise the expansion of xWy orient in powers of monomials of ζ 2n`1 -constants as in (3.1). One finds that, as in the orbifold theory, at the leading non-planar level all appearing ζ 2n`1monomials are multiplied by I 1 p ? λq times a power of λ (cf. (3.21)). Explicitly, for ∆q defined as in (3.23), i.e. ∆q " q orient´qSYM , we find