BPS Wilson loop in $\mathcal N=2$ superconformal $SU(N)$"orientifold"gauge theory and weak-strong coupling interpolation

We consider the expectation value $\langle \cal W \rangle$ of the circular BPS Wilson loop in ${\cal N}=2$ superconformal $SU(N)$ gauge theory containing a vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. This model admits a regular large $N$ expansion, is planar-equivalent to ${\cal N}=4$ SYM theory and is expected to be dual to a certain orbifold/orientifold projection of AdS$_5\times S^5$ superstring theory. On the string theory side $\langle\cal W \rangle $ is represented by the path integral expanded near the same AdS$_2$ minimal surface as in the maximally supersymmetric case. Following the string theory argument in arXiv:2007.08512, we suggest that as in the ${\cal N}=4$ SYM case and in the ${\cal N}=2$ $SU(N) \times SU(N)$ superconformal quiver theory discussed in arXiv:2102.07696, the coefficient of the leading non-planar $1/N^2$ correction in $\langle\cal W \rangle $ should have the universal $\lambda^{3/2}$ scaling at large 't Hooft coupling. We confirm this prediction by starting with the localization matrix model representation for $\langle\cal W \rangle $. We complement the analytic derivation of the $\lambda^{3/2}$ scaling by a numerical high-precision resummation and extrapolation of the weak-coupling expansion using conformal mapping improved Pad\'e analysis.


Introduction
Wilson loops are an important class of observables in gauge and string theory that, in particular, help clarifying the interpolation between the weak and strong coupling regimes from the AdS/CFT perspective. In several supersymmetric gauge theories it is possible to compute expectation values of BPS Wilson loops using localization in terms of matrix model integrals (see, e.g., [1]).
In the maximally supersymmetric N " 4 SU pN q gauge theory the corresponding matrix model can be solved for any gauge coupling and gauge group rank [2]. 1 This allows, in particular, to study the strong coupling limit of the coefficients in the 1{N expansion, providing a possibility to compare to the large tension limit of the coefficients in the expansion in powers of string coupling (genus) on the dual AdS 5ˆS 5 string theory side, and thus leading to highly non-trivial checks of AdS/CFT duality [5,6,7].
Localization method applies also to a large class of gauge theories with reduced N " 2 supersymmetry, but the associated matrix models have non-polynomial potentials and are not directly solvable. While developing the small λ expansion is straightforward, extracting the strong coupling limit of the gauge theory observables is a non-trivial problem. At leading order in the large N expansion this requires a Wiener-Hopf analysis of the matrix model as first exploited in [8] for N " 2 SU pN q SYM with N F " 2N fundamental hypermultiplets, and later generalized to other superconformal Lagrangian models admitting a large N limit [9,10,11,12,13,14,15]. The methods used at leading planar level are not, however, applicable to the analysis of the higher 1{N corrections.
An interesting class of models where 1{N corrections happen to be more tractable is that of N " 2 superconformal gauge theories with SU pN q gauge group whose leading large N limit is equivalent (in a particular "common" sector) to that of the N " 4 SYM theory. One such example is the SU pN qˆSU pN q quiver gauge theory with bi-fundamental hypermultiplets and equal gauge couplings. This N " 2 theory may be interpreted as a Z 2 orbifold of the N " 4 SU p2N q SYM and is dual to superstring theory on AdS 5ˆp S 5 {Z 2 q [16].
Let us first review some basic results about the expectation value xWy of circular 1 2 -BPS Wilson loop in N " 4 SYM theory. Its planar limit is given by is [17,2,18] The relative weight of the leading 1{N correction to xWy with respect to the planar result may be represented as xWy xWy 0 " 1`1 N 2 qpλq`O´1 N 4¯, (1.2) where the function qpλq (and its analogs in the N " 2 models with planar equivalence to N " 4 SYM) will be our main interest below. Starting with the general (Laguerre polynomial) expression for xWy in N " 4 SU pN q SYM [2] one finds for the leading terms in qpλq at weak and strong coupling 2 q N"4 pλq " λ 96 The λ 3{2 scaling of q N"4 at λ " 1 has a string interpretation. The string coupling and tension for the dual string theory on AdS 5ˆS 5 are defined as . (1.4) As was argued in [5], the leading large T dependence of the string theory expectation value for xWy at each order in g s is also controlled by the Euler number, χ " 1´2p, (or genus p) of the string world sheet, i.e. 1`O´1 ? λ¯ı , (1.6) and thus matches the structure of the 1{N expansion of the exact N " 4 SYM result [2]. In particular, comparing to (1.2), we have, in agreement with (1.3), The discussion in [5] leading to (1.5) relied only on the fact that one expands near the AdS 2 minimal surface embedded into the AdS 3 part of AdS 5 space; thus it should apply not only to AdS 5ˆS 5 superstring but also to its closely related orbifold and orientifold modifications based on AdS 5ˆS 15 where S 15 is locally a 5-sphere. Indeed, since the fluctuations of string world sheet fields related to S 15 remain "massless", the reasoning [5] determining the tension dependence from the way how the AdS 5 radius appears in the 1-loop (leading large T ) string partition function should not change.
In [19] it was argued that this should apply, in particular, to the orbifold AdS 5ˆp S 5 {Z 2 q theory and evidence for the validity of (1.5), (1.6) was provided at the first non-trivial 1{N 2 order. To recall, in the SU pN qˆSU pN q orbifold model, for each of the two SU pN q factors, it is possible to define the 1 2 -BPS circular Wilson loops coupled to the associated gauge and scalar fields and xW 1 y " xW 2 y " xWy orb . At the leading planar level one has xWy orb N Ñ8 " xWy N"4 N Ñ8 " xWy 0 [20,21]. 3 Starting from the corresponding localization matrix model representation for xWy orb , a numerical analysis of the q orb pλq function, defined as in (1.3), extrapolated to large λ values gave the following estimate [19] q orb pλq The value of the asymptotic exponent η is thus quite consistent with the string theory expectation 3/2 in (1.7).
In this paper we shall consider another N " 2 superconformal model where the structure of the large N , strong coupling expansion should be of the same universal form as in (1.5). We shall confirm this expectation with an analytic argument for the strong-coupling scaling in (1.7), in addition to numerical evidence based on high-precision extrapolation from the weak to strong coupling regimes.
This theory is the SU pN q gauge theory with N " 2 vector multiplet coupled to two hypermultiplets -in rank-2 symmetric and antisymmetric SU pN q representations. 4 This model admits a regular 't Hooft large N expansion and its string theory dual is expected to be a particular orientifold of AdS 5ˆS 5 type IIB superstring theory [27,28]. For that reason in what follows we shall refer to this N " 2 gauge theory as the "orientifold theory". To recall, in N " 2 gauge theories the β-function for the gauge coupling has only the 1-loop contribution: in a model with N F hypermultiplets in the fundamental, N S in rank-2 symmetric, and N A in rank-2 antisymmetric representations one has β 1´loop " 2N´N F´NS pN`2q´N A pN´2q. It thus vanishes for the orientifold theory where N F " 0, N A " N S " 1. Let us also mention for completeness that the 4d Weyl anomaly coefficients a and c for an N " 2 theory with n V vector and n H hyper multiplets are given by a " 5 24 n V`1 24 n H , c " 1 6 n V`1 12 n H , so that in the present case with n V " N 2´1 and n H " 1 2 N pN`1q`1 2 N pN´1q " N 2 , we get a " 1 4 N 2´5 24 , c " 1 4 N 2´1 6 . Thus a and c are equal at the leading N 2 order which is consistent with the existence of a well defined holographic dual. 5 Our aim will be to consider the expectation value of the 1 2 -BPS circular Wilson loop in the orientifold theory. At the leading large N order it is the same as in the N " 4 SYM theory in (1.1). 6 The main focus will be on the leading non-planar correction represented by the function q orient pλq defined as in (1.2).
The string dual of this N " 2 model is the type IIB superstring theory defined on the orientifold AdS 5ˆS 5 {G orient [28]. Here G orient " Z orb 2ˆZ orient 2 , where Z orient 2 in addition to the target space coordinate inversions (in directions transverse to the original D3-branes) involves the product worldsheet parity operator Ω and p´1q F L . The compact part of the 10d space S 15 " S 5 {G orient is different from S 5 only by special identifications of the angular coordinates [28]: The dual string theory description of the circular Wilson loop is based again on the string partition function expanded near the AdS 2 minimal surface embedded in AdS 5 . As the UV divergent part of the 1-loop fluctuation determinants [32] near this minimal surface should not be sensitive to the global identifications in the S 15 part of the orientifold geometry, the argument in [5] leading to the universal structure of strong-coupling expansion (1.5),(1.6) should apply not only to the original AdS 5ˆS 5 or orbifold theory considered in [19] but also to this orientifold theory as well. Below we shall provide evidence for this, i.e. for the validity of (1.5),(1.6), on the dual orientifold gauge theory side by showing that the localization matrix model representation for the circular BPS Wilson loop implies that the 1{N 2 term in (1.2) indeed scales as in (1.7) at the leading order at strong coupling, i.e. q orient pλq Let us briefly summarize our main results. The aim will be to present a detailed study of the 1{N 2 coefficient in (1.2) in the orientifold theory, i.e. of q orient pλq. As in the orbifold theory discussed in [19], from the matrix model representation for the Wilson loop in the orientifold theory one can relate the difference between q orient pλq and q N"4 pλq ∆qpλq " q orient pλq´q N"4 pλq , (1.10) 5 Let us also note that it should be possible to reproduce the subleading terms in a and c on the dual orientifold string theory side by summing up the 1-loop contributions of the "massless" D " 10 supergravity fields (corresponding to short multiplets represented by towers of Kaluza-Klein modes) similarly to how that was done in the case of the N " 4 SU pN q SYM theory (where a=c" 1 4 N 2´1 4 ) [29] and some orbifold theories [30]. 6 This is a manifestation of the planar equivalence between the orientifold theory and the N " 4 SYM in the "untwisted" sector. For a detailed discussion of planar equivalence violations in "odd" sectors see [31]. to the N Ñ 8 limit of the difference of the corresponding free energies 7 ∆qpλq "´λ ∆F pλ; N q , (1.11) ∆F pλ; N q " F orient pλ; N q´F N"4 pλ; N q "´log Z orient pλ; N q Z N"4 pλ; N q . (1.12) Here Z orient and Z N"4 are the corresponding partition functions on S 4 . The function ∆F pλq turns out to have the following weak coupling expansion ∆F pλq "5ζ 5λ 3´105 2 ζ 7λ 4`4 41ζ 9λ 5´p 25ζ 2 5`3 465ζ 11 qλ 6`5 .6355 Extracting the strong coupling expansion is much harder. Since in the N " 4 SYM theory the matrix model representation implies that F N"4 "´1 2 pN 2´1 q log λ [34] (we ignore λ-independent but N -dependent constant, cf. [35]), combining (1.11),(1.12) and (1.9) we get, as in the orbifold theory case [19], the following prediction for F orient (1.14) The leading OpN 2 q term in (1.14) is implied by the planar equivalence to the SU pN q SYM theory and should follow from the leading type IIB supergravity term evaluated on AdS 5ˆp S 5 {G orient q. 8 Let us note that as one can show from the localization matrix model, large N expansion of free energy F orient pλ; N q and of the corresponding Wilson loop will contain only even powers of 1{N . This may be somewhat surprising from the dual string theory point of view where in the orientifold case one should in general get cross-cup contributions with odd powers of g s . Below we will analytically derive the ? λ term in (1.14) and thus in ∆F finding that ∆F pλq where we used (1.11). We will also confirm this result by a high-precision resummation and extrapolation analysis of the weak-coupling expansion by numerical methods including a conformalmapping improved Padé analysis. Let us note that while the coefficient in strong-coupling limit of q in N " 4 SYM case is positive, q N"4 " 1 96 λ 3{2`. .. (see (1.3)), it was found [19] to be negative in the N " 2 orbifold theory (1.8). The result in (1.15) implies that it is also negative in the orientifold theory, q orient " q N"4`∆ q " p 1 96´1 16π qλ 3{2`. .. «´0.00948λ 3{2`. ... It would be interesting to understand the reason for this sign change on the dual string theory side where q should be expressed in terms of the string partition function on the disc with one handle. 7 Note that while the individual free energies on S 4 are, in general, scheme-dependent, their difference ∆F is scheme-independent. Earlier discussion of leading terms in perturbative expansion in Wilson loop and free energy in this theory was in [33]. 8 Planar equivalence also implies that like in the N " 4 SYM theory this leading N 2 term should not get string The rest of the paper is organised as follows. In section 2 we shall describe the localization matrix model representation for the expectation value of the 1 2 -BPS Wilson loop in the orientifold gauge theory. We shall then present the derivation of the relation (1.11) between the coefficient ∆q of the 1{N 2 term in the ratio of the orientifold and N " 4 SYM Wilson loops and the large N limit ∆F pλq of the difference of the corresponding free energies. This reduces the problem of determining the strong coupling limit of ∆q to finding that of ∆F .
In section 3 we shall first study ∆F pλq at weak coupling and then find its explicit representation (3.25), i.e. ∆F " 1 2 log detp1`M q "´ř 8 n"1 1 n p´1q n tr M n , in terms of an infinite-dimensional matrix M (3.26). Each tr M n term in ∆F turns out to be of fixed order n in products of ζ-function values when written in the weak-coupling expansion. In section 4 we shall study the first two tr M and tr M 2 terms finding that at large λ one has tr M n " λ n .
The derivation of the strong coupling asymptotics (1.15) of the total ∆F implying ∆q " λ 3{2 is given in section 5. In section 6 we shall independently test this ∆F " λ 1{2 scaling by two different numerical methods. Some technical details are delegated to appendices.
The methods used here may be applicable to other similar N " 2 models. One candidate is the N " 2 superconformal SU pN q gauge theory with N F " 4 fundamental and N A " 2 rank-2 antisymmetric hypermultiplets. In this case the dual string theory is expected to be again a IIB orientifold of AdS 5ˆS 5 where S 5 is modded out by a Z 4 that mixes non-trivially the orbifold and orientifold twists [28]. 9 However, the presence of fundamentals means that here the large N expansion will go in powers of 1{N rather than 1{N 2 and thus will be different in structure from (1.5),(1.6).
Note added in v3: The exact value of the coefficient C in (1.8) in orbifold theory was recently found in [39]: C "´1 32 . Ref. [39] also found the exact values of several leading coefficients in strong coupling expansion of free energy of the orientifold theory (1.14), in particular, of the coefficient c 1 of the leading λ 1{2 term: c 1 " 1 8 . In view of (1.11) this determines also the exact value of the coefficient of the leading λ 3{2 term in ∆q. Thus instead of the result (1.15) obtained below by making a bold assumption that one can interchange the large λ expansion with computing determinant of a infinite matrix one actually gets ∆F pλq Note that the coefficients in (1.15) still give a good approximation to the ones in (1.16); this is consistent with the fact that (1.15) was also supported by numerical evaluation methods. The value c 1 " 1 8 is also in agreement with refined numerical estimates in [40], see discussion in section 4.2 of [39]. There value c 1 " 1 8 was also established by a more precise numerical method in recent paper [41]. 9 In this model the Weyl anomaly coefficients are a " 1 4 N 2`1 8 N´5 24 and c " 1 4 N 2`1 4 N´1 6 . The OpN q terms in a and c should be possible to derive on the dual string theory side as in [36] (see also [37,38]) using that here the background involves D7-branes wrapping AdS5 and S 3 of S 15 with R 2 terms in the effective 8-dimensional world-volume theory.

Matrix model representation and 1{N correction to Wilson loop
The field content of the orientifold theory is represented by the adjoint N " 2 vector multiplet (gauge vector A µ , a complex scalar ϕ, and two Weyl fermions) and rank-2 symmetric and antisymmetric hypermultiplets (each containing two complex scalars and two Weyl fermions). The 1 2 -BPS Wilson loop is defined in terms of the fields of the vector multiplet as where the contour x µ psq represents a circle of unit radius. The supersymmetric localization implies that the partition function of this gauge theory on a sphere S 4 of unit radius admits a representation in terms of an integral over the eigenvalues In the case of N " 4 SYM theory S int " 0 and the matrix model is Gaussian. As we shall discuss the 1{N expansion, we can neglect the instanton contribution term in S int so that 10 where H is expressed in terms of the Barnes G-function Here and below the constants ζ 2n`1 " ζp2n`1q are the Riemann ζ-function values. The normalized expectation value of the Wilson loop (2.1) can be computed as the matrix model average of tr e 2πm , i.e.
In the leading planar approximation S int is effectively suppressed and thus we get the same result as in the N " 4 SYM: xWy orient " xWy N"4`O p 1 N q. Here we will be interested in the leading non-planar correction in the ratio (cf. (1.2),(1.10)) xWy orient Let us show that ∆qpλq can be represented as a λ derivative (1.11) of the difference of the free energies (1.12). The Wilson loop ratio (2.9) may be represented in general as xWy orient where Dm is the normalized measure, i.e. ş Dm e´S 0 pmq ... " At large N the correlators in (2.10) factorize and the ratio goes to 1. The non-planar correction is given by the large N limit of the "connected" part of @ e´S int tr e 2πm D 0 . The leading N Ñ 8 contribution should come from the first non-trivial term in the expansion of tr e 2πm in powers of m (tr 1 " N, tr m " 0): The insertion of a factor of tr m 2 is the same as the insertion of the free action in (2.3) and thus it can be obtained by differentiating the partition function (2.2) over λ. As shown in Appendix A, taking the large N limit we then find that ∆q in (2.9) can be represented as in (1.11), i.e.
∆q "´λ This is essentially the same relation as was observed to hold in the N " 2 orbifold model in [19] (up to factor of 2 due to the SU pN qˆSU pN q instead of SU pN q gauge group).
3 Large N limit of free energy difference ∆F " F orient´F N"4 Since the leading non-planar correction to the Wilson loop can be expressed (2.13) in terms of ∆F pλq, in what follows we shall concentrate on the study of its structure both at weak and strong coupling. Redefining the matrix model variable m Ñ a as a " we can represent ∆F pλ; N q in (2.11) as where Da is the standard integration measure for the traceless matrix a, normalized so that ş Da e´t r a 2 " 1. This measure is same as in (2.4) when written in terms of the eigenvalues and dropping the "angular" part that cancels in expectation values of relevant correlators (functions of traces of matrix a).
Using (2.5),(2.7) we can write S int in (2.5) as a weak coupling expansion

Weak coupling expansion
The weak coupling (λ ! 1) expansion of ∆F in (3.2) is easily worked out by expanding e´S int and doing the Gaussian integrations. The result has a finite limit for N Ñ 8 since the leading N 2 terms present in both the N " 4 SYM and N " 2 partition functions cancel out in ∆F as a manifestation of the planar equivalence of the two models. For the leading large N contribution ∆F pλq defined in (2.13) we obtain the following expansion (cf. (1.13)) .  A check of the general relation (2.13) may be given by the direct comparison of the independent weak-coupling expansions for ∆F and ∆q (see Appendix A). The weak coupling expansion of ∆qpλq " q orient pλq´q N"4 pλq is found to be which is indeed consistent with (3.4) and (2.13).

Explicit representation for ∆F
It is possible to derive a remarkable closed expression for ∆F pλq in (2.13) as a log det of an infinitedimensional matrix. Let us start with representing S int paq in (3.3) as an infinite double sum of 1 ? N -normalized traces of odd powers of the matrix a with coefficients C ij that depend only on λ S int paq " Next, let us define the generating function Using (3.6) can then represent e´S int paq and thus the integral over a in (3.2) as 11 (3.10) 11 Here and below we assume summation over repeated indices i, j.
The large N limit of ∆F pλ; N q is thus directly related to that of Xpηq. Expanding e V in (3.8) in powers of a, computing the Gaussian integrals over a, and then rearranging the result back into the exponential form gives the following expression for the leading large N part of Xpηq where Qpηq is the following quadratic form (3.12) The closed form of the infinite matrix Q ij can be found from the results in [31] Q ij " 1 π As a result, we get from (3.9),(3.11) . (3.14) To evaluate (3.14), let us first change the variables as η " (3.15) Introducing the notation we then have (here Here we used an auxiliary Gaussian integral over y i and that e´A ij B i B j e x¨yˇx "0 " e´A ij y i y j . Thus we find the following exact representation for ∆F pλq in terms of the infinite matrix Ă M ∆F "   That way the weak-coupling expansion in (3.18) reproduces the ζ k , ζ k ζ n , ζ k ζ n ζ m , ... terms in the expansion of ∆F in (3.4). 12 The explicit form of the matrix Ă M in (3.16) appearing in (3.18) is . (3.20) Motivated by the analysis in [31], let us introduce the following matrix Remarkably, Ă M in (3.20) and M in (3.21) happen to be related by a similarity transformation, The advantage of this form of ∆F is that the matrix M in (3.21) admits the following Bessel function representation which will prove to be useful in the analysis of the strong-coupling limit.

Contributions to ∆F of finite degree in ζ-function values
The weak coupling expansion of ∆F in (3.4) can be represented as 12 Keeping only a finite number of ζ k constants in the matrix Ă Mij and using the first equality in (3.18) gives immediately the resummation of all monomials involving those ζ k . For example, the terms with only ζ5 and ζ7 come from the expansion of the exact expression ∆F ζ 5 ,ζ 7 " 1 2 logp1`10ζ5λ 3´1 05ζ7λ 4´735 4 ζ 2 7λ 8 q, and so on. 13 Notice that U is a lower triangular matrix due to the argument of the first Γ function in the denominator being non-positive integer for j ą i.
where ∆F pnq is the total contribution of terms that are products of a fixed number n of the Riemann ζ-function values. Equivalently, ∆F pnq represents the contribution of the tr M n term in (3.25) (cf. (3.19)).
Here we will study ∆F pnq , computing, in particular, its leading strong coupling asymptotic expansion. We shall focus in detail on the n " 1 term and then discuss the n " 2 one. We will find that ∆F pnq pλq In the next section 5 we will compute all the coefficients C n in (4.2) and then evaluate the sum of all ∆F pnq thus determining the strong coupling asymptotics of ∆F .
Defining Gpt, t 1 q " 8 which has also the following integral form [44] Gpt, t 1 q " 2 t t 1 we may use the Bessel function representation of the matrix M (3.26) to represent the traces of M n as the iterated integrals tr M " We remark that (4.3) coincides with the Tracy-Widom kernel [44] upon the change of variables t " ?
x. It remains to be clarified whether this is a coincidence or there is some deeper relation to eigenvalue statistics. This Bessel kernel also appears in the BES equation [45] and seems prevalent in integrable equations/models.

Term linear in ζ n
∆F p1q or tr M is just a single integral (4.5) and may be treated exactly. Using (4.3) we have From the identity we obtain ∆F p1q " where we used integration by parts. This expression is exact and may be expanded at weak or strong coupling.
Weak coupling expansion: Using and expanding in λ we recover from (4.10) the first line in (3.19). One can find the following all-order result 14 By the standard ratio test this shows that the radius of convergence is π 2 , as could be expected. Indeed, λ " π 2 is the radius of convergence of perturbative expansion in N " 4 SYM theory in the planar limit (as suggested by the single-magnon dispersion relation, fixed by the superconformal symmetry [46,45], or by the quantum algebraic curve approach [47]). The same is expected to apply also to the N " 2 superconformal theories (as was first observed in the mass-deformed N " 2t heory [48], and recently found also in the orbifold theory case [19]). Expanding the exponentials in the integral in (4.10) gives an alternative representation in terms of a sum of elliptic integrals ∆F p1q " 2 8 ÿ n"1 n "´1´8 π 2 n 2´λ 2πλ E´´λ π 2 n 2¯`8 π 2 n 2`7 λ 2πλ K´´λ π 2 n 2¯ı . (4.13) Here one sees explicit singularities at λ "´π 2 n 2 where the argument of K becomes unity.
Strong coupling expansion: The strong coupling (asymptotic) expansion of ∆F p1q may be computed by Mellin transform methods [49,50]. Defining the Mellin transform r f psq " ş 8 0 dx x s´1 f pxq and considering the convolution pf ‹ gqpxq " we have p Ć f ‹ gqpsq " r f psq r gp1´sq. Let α ă s ă β be the fundamental strip of analyticity of r f psq. The asymptotic expansion of f pxq for x Ñ 8 is obtained by looking at the poles of r f psq in the 14 This remarkably simple form of the coefficients follows from the relation x s 0 log N´1 x in the original function. In our case, we can compare the right hand side of (4.10) with (4.14) as The Mellin transform is then 16) and the asymptotic expansion at strong coupling can be extracted from the poles at s " 0, 1, 2, . . . . This gives The infinite sum in (4.17) has zero radius of convergence, with factorially divergent coefficients. 15 The leading order λ term corresponds to the n " 1 case of the general pattern (4.2). The leading term in (4.17) can be derived more directly. We can expand the integrand in (4.5) at large λ and read off the coefficient of a suitable power of λ from a convergent integral 16 tr M "

Term quadratic in ζ n
In the case of ∆F p2q "´1 4 tr M 2 in (4.1) we can obtain an all-order weak coupling expansion in almost-closed form. Although it is not as explicit as (4.12) for ∆F p1q , it may be used to generate a very large number of terms. Here we will present the final result, with details given in Appendix B. Let us define the polynomials 15 Let us note that replacing the ζ-values by the integral using (4.11) and doing the sum, we obtain another representation This integral has a logarithmic singularity at t " 1 on the t integration contour, and so should be understood as an average above and below the cut. 16 This procedure works for the leading order; at subleading orders one gets divergent integrals requiring a more careful treatment.
where P pα,βq n pxq are Jacobi polynomials. We may write d in the form d pxq " x ÿ m"m 0 ∆m"2 a p q m px m`x´m q, (4.20) where m 0 " 0{1 if is even/odd and m varies in steps of 2. The weak coupling expansion of tr M 2 can then be written in terms of sums with coefficients a p q m that are easily computed from (4.19),(4.20) (4.21) Leading term at strong coupling: The expansion (4.21) may not be used directly at strong coupling. Nevertheless, we succeed in applying the manipulation we exploited in (4.18). Indeed, we have The integrand is symmetric so we write Doing first the integral over t, we get This strong-coupling asymptotics follows again the general pattern (4.2). A numerical test of this prediction will be discussed in section 6.1.

Strong coupling limit of ∆F : analytic derivation
Let us now generalize the derivation of strong-coupling limit to the full ∆F . The starting point will be the explicit form of the large λ expansion of the matrix M in (3.26). It can be found as in (4.14)-(4.17) using the Mellin transform. We have Evaluating the Mellin transforms and taking residues, we get the asymptotic expansion of N ij Then the leading strong-coupling part of M may be written as where S is a symmetric three-diagonal infinite-dimensional matrix. As a result, we get The explicit values of the coefficients b n (related to C n in (4.2) as C n " p´1q n`1 np2π 2 q n b n ) are given in Appendix C.
Remarkably, S in (5.5) is essentially the same (up to 1/2) as the matrix appearing in Eq. (2.7) of [51]. It follows from the analysis in [51] that in the infinite matrix limit the eigenvalues ts 1 , where j 1,k are the zeroes of the Bessel function J 1 pxq. Hence, we get the following remarkable relation 17 As a result, we get for ∆F in (3.25) Eq.(5.9) implies that c 1 in (1.14) is equal to 1 2π . Then using (1.11) we obtain the following expression (1.15) for the strong-coupling limit of ∆q ∆qpλq λ"1 "´λ 3{2 16π`¨¨¨. (5.10) 6 Numerical evaluation of ∆F : interpolation from small to large λ In this final section we present various approaches to test the analytical result (5.9) for the strong coupling limit of ∆F by numerical methods. We will first consider the approach based on Padé approximants using as an input many terms in the weak coupling expansion of ∆F . Then, we will discuss a method based on a direct evaluation of ∆F " 1 2 tr logp1`M q where the large λ limit of the infinite matrix M is first replaced by its finite-size truncation. 17 This follows from the Weierstrass infinite product representation of the Bessel function in terms of its zeroes: Jν pzq " pz{2q ν Γpν`1q ś 8 n"1 p1´z

Padé-conformal method
We begin with the small λ expansion of ∆F : The particular definition ofλ is chosen so that the radius of convergence of the series in (6.1) is as close as possible to 1. This is helpful for the numerical analysis, as it avoids the appearance of very large or very small coefficients at high order. The technical goal is to extrapolate from small to large λ, starting from a finite number of terms in the weak coupling expansion. Optimal and near-optimal methods for such an extrapolation have been analyzed recently in [53,54,55]. The key information is some knowledge, either analytic or numerical, of the singularity structure of the function ∆F pλq. This information can be extracted numerically by suitable combinations of ratio tests, Padé approximants, and conformal maps.
The magnitude of the leading singularity is equal to the radius of convergence R, which can be found by a simple ratio test:ˇˇˇˇc The convergence of this ratio of successive coefficients to the inverse radius can be accelerated using Richardson acceleration [56] (for example, for the tr M case see the left hand panel of Figure 1 below). This permits an extremely precise numerical estimate of the radius of convergence, if it is not known analytically. For ∆F " 1 2 tr logp1`M q, we will see that the leading singularity is atλ «´1, i.e. λ «´π 2 . By studying the subleading corrections to this ratio test limit one can determine the nature of the leading singularity, using Darboux's theorem, see Appendix D. For this orientifold model the small λ expansion indicates that the leading singularity is logarithmic (see the right hand panel of Figure 1 below). This is consistent with the exact analytical structure of individual tr M n terms for finite n, see section 5.
A closely related method, which also yields information about the singularity structure is based on the use of a Padé approximant [56,57]. Here one matches the finite number K of terms of the expansion to the expansion of a ratio of polynomials R L and Q M : Since it is an approximation in terms of rational functions, Padé only has poles as singularities, which are the zeros of the denominator polynomial Q M . If the truncated series is that of a function with branch point singularities, then Padé produces arcs of poles accumulating at the branch points. 18 The practical implication of this is that if one has enough expansion terms one can frequently distinguish between an isolated pole and a branch point simply by looking at the poles of a Padé approximant. Indeed, the left panel of Figure 2 shows a line of Padé poles accumulating to the branch point atλ "´1.
However, this reveals a fundamental problem with Padé, because these accumulating poles, which are trying to represent a branch cut, obscure possible higher singularities which may be physical. This problem can be resolved by making a conformal map before making the Padé approximation [53,54,55]. Based on the leading branch cut p8,´1s on the negative realλ axis, as suggested by the Padé approximation in this case (see the left hand panel of Figure 2), one maps the expansion into the unit disk |z| ď 1: We re-expand ∆F´4 z p1´zq 2¯i n powers of z to the same order K, and then construct a Padé approximant in terms of z. 19 Inside the unit disk this expansion is convergent by construction, but further singularities along the lineλ P p8,´1s will appear as singularities on the unit circle. If these are branch points they will appear as the accumulation points of arcs of Padé poles.
The advantage of the conformal map is that collinear singularities in theλ plane (which may be hidden under a line of accumulating poles) are separated to different points on the unit circle. See for example the right panel of Figure 2, which shows the leading singularity at z "´1, the conformal map image ofλ "´1, but also clearly shows further singularities at the conformal map images ofλ "´4, atλ "´9, and so on. This numerical evidence suggests that the singularities are: singularites`∆F pλq˘"´l 2 π 2 , l " 1, 2, 3, . . . .
The source of these singularities can be understood analytically from the study of tr M n for finite n, and the singularity structure appears to be inherited by tr logp1`M q.
A further advantage of the conformal map is that it enhances the precision of the subsequent Padé extrapolation. To construct the Padé-conformal extrapolation 20 we make a Padé approximant in terms of z and then evaluate it on the inverse map in (6.4). This introduces square roots; thus we are representing the function not just by rational approximations, but in a much wider class of functions. For branch point singularities the increase in precision can be quantified precisely using the asymptotics of orthogonal polynomials [54] and is quite dramatic, as is illustrated in Figures 3  and 4 below.

Example: tr M
To illustrate this Padé-conformal extrapolation technique, we first consider the expansions of tr M and tr M 2 , for which we can compare with analytic results found in section 4. But we stress that the power of this method is in cases when such analytic comparisons are not available, and one is only presented with a truncated series, and possibly some physical intuition about the singularity structure. For tr M we have the exact expansion (cf. (4.12)) The ratio c k`1 {c k is plotted in the left panel of Figure 1 based on the first 150 terms, indicating an alternating series with radius of convergence 1. The fact that the leading singularity is logarithmic is shown by the fact that c k " p´1q k kˆc onstant as k Ñ 8. See the right panel in Figure 1. The fact that the leading singularity is a branch point is also indicated by the Padé poles, which are shown in the left panel of Figure 2, accumulating along the negative real axis to the branch point atλ "´1.
After the conformal map (6.4), followed by re-expansion to 150 terms in z, the poles of the resulting diagonal Padé approximant are shown in the right panel of the same figure. This Figure indicates the existence of branch point singularities at the z plane images ofλ "´1,´4,´9,´16. The data becomes noisy at the conformal image of´25, with unphysical poles appearing inside the unit disk. These can be resolved by taking more terms in the original expansion.
We now map this Padé approximant back to the physicalλ plane using the inverse conformal map in (6.4), and plot to large λ. Figure 3 compares the diagonal Padé extrapolation (orange curve), divided byλ, with the analytic largeλ limit of 1 8 (blue curve) and the Padé-conformal extrapolation (green curve). The first plot extends out toλ " 10 4 , while the second plot extends out toλ " 10 6 . Note that the Padé approximant eventually breaks down atλ « 1.5¨10 4 , while the Padé-conformal approximant extends much further to very largeλ. We stress that exactly the same input coefficient data was used in producing these two extrapolations, illustrating the dramatic effect of the conformal map.
A similar analysis can be applied to tr M 2 where we do not have a simple closed form expression for the expansion coefficients, but there is a systematic way to expand to very high order (multiple hundreds of terms, see (4.21)). The resulting structure is very similar to that for the tr M case discussed above, so we do not repeat the analogous plots. These poles lie on the negative real axis and accumulate toλ "´1. On the right we show the poles in z-plane after application of the conformal transformation (6.4) followed by Padé. In this case collinear singularities on the lineλ P p´8,´1s are separated and made visible as arcs converging to points on the unit circle in the z plane. These agree with a similar analysis for the whole ∆F , see (6.5).

∆F
Let us now consider the large λ extrapolation of the full ∆F . We begin with the small λ expansion discussed in section 3.1. We generated 150 terms of this expansion, with 450 digit precision for the coefficients. The coefficients are sums of products of odd ζ 2k`1 -values, but it is faster to work with finite but high precision coefficients. The ratio test and Padé analysis again indicate a leading singularity at λ "´π 2 , so we make the same conformal map (6.4) and subsequent Padé approximant and inverse map back to the physical λ plane. Figure 4 shows the result, and we again see that the Padé-conformal extrapolation extends to a much larger value of λ. This extrapolation shows that the functional form of the large λ behavior is (left panel of the figure) ∆F pλq " This functional form matches the result of resumming the leading large λ terms of tr M n in (5.9), and the coefficients approximately agree. We stress that the only input information used for this extrapolation from small λ to large λ was the list of 150 perturbative coefficients. To get a better estimate of the result requires fitting the ratio ∆F { ? λ and it is hard to support a specific functional form. The slow convergence shown in the right panel of Figure 4 should be due to the expected logarithmic corrections in (1.14) if they do not happen to cancel in ∆F .  ? λ at large λ. The convergence of the coefficient to the asymptotic value is steady but slow, as illustrated in the right panel on a logarithmic scale. See Section 6.2 for a more refined estimate of the overall coefficient.

Evaluation of ∆F at large λ using truncation method
In this subsection we use a complementary numerical method in order to extract the precise large λ behaviour of ∆F . Starting from the expansion (5.3), let us denote by M K the KˆK matrix which is the linear in λ part of M , truncated to the first K rows and columns. Then To determine the large λ behaviour of ∆F pλq, we need to take first K Ñ 8, and then λ Ñ 8. λ where F K is defined in (6.8), and K " 20, 40, . . . , 260 from bottom to top. For each K, there is a maximal value µ K . In the right panel we plot µ K vs. K (blue dots) and compute its best fit (orange dashed line) with a constant plus a leading " K´1 {2 and subleading " K´1 terms. The best fit parameters are in (6.10).
To bypass this double limit procedure, we will fix K, increase λ until the ratio ∆F K pλq ? λ reaches a maximum and, finally, extrapolate µ K to K Ñ 8. According to (5.10), the expected value is c 1 " 1 2π . The explicit numerical results are collected in Figure 5. In the left panel we show the curves ∆F K pλq ? λ for K " 20, 40, 60, . . . 260. For each K a maximum in (6.9) is reached at a value of λ that increases with K. The maximum value µ K is shown in the right panel of the figure and fitted by the dashed curve µ fit K " 0.158´0 . We use the label "N " 2" to separate the genuine correction to the N " 4 result. The explicit calculation starting with the matrix model expression (2.8), (3.3) gives 32 p8 π 2 q 3`1 05ζ 7 2 p8 π 2 q 4¯λ 5`´´5ζ5 256 p8π 2 q 3`1 05ζ 7 16 p8π 2 q 4´2 205ζ 9 4 p8π 2 q 5¯λ 6´´5 ζ 5 12288 p8π 2 q 3`7 5ζ 2 5 2 p8π 2 q 6`3 5ζ 7 128 p8π 2 q 4´2 205ζ 9 32 p8π 2 q 5`1 0395ζ 11 2 p8π 2 q 6¯λ 7`O pλ 8 q. (A. 3) The function ∆qpλq in (1.2),(1.10) is obtained dividing by xWy 0 and this gives precisely (3.5), i.e. the result consistent with (2.13). We checked the relation (2.13) to order Opλ 20 q by an independent computation of both ∆F and ∆q. Let us note that each of the monomials in the ζ n -values appears in the weak-coupling expansion with a simple single-power dependence on λ. The corresponding leading 1{N 2 corrections in xWy orient happen to have its non-trivial dependence on λ via the Bessel function factor " λ´1 {2 I 1 p ?
λq. This property can be proved for specific monomials in ζ n -values by the methods described in [19]. It may be made explicit by collecting terms in (A.3) as xWy N"2 1 "´1 5ζ 5 4 p8π 2 q 3 λ 4´1`λ 8`λ Here, one can see that each monomial in the ζ n -values is multiplied by the expansion of the factor xWy 0 " 2 ? λ I 1 p ? λq. As discussed in [19], this property is important for the relation (2.13) to hold.

B Weak coupling expansion of tr M 2
Here we shall provide the proof of the result (4.21) for the weak-coupling expansion of tr M 2 . We begin with the expansion of the product of two Bessel functions as a series of Jacobi polynomials The explicit values of the coefficients b n can be easily computed from (C.2) tb n u n"1,2,... " 1 4 , Thus, leading and subleading large-order behaviour terms determine the Taylor expansion of the analytic function φptq which multiplies the branch-cut factor in (D.1). The function ψptq can be extracted similarly by multiplying f ptq through by´1´t t 0¯g , and applying the same procedure. If the singularity is logarithmic, where φptq and ψptq are analytic near t 0 , then the Taylor expansion coefficients of f ptq near the origin have large-order (k Ñ 8) growth . .
This logarithmic behaviour is found for the expansion coefficients of ∆F pλq, as shown in the right hand panel of Figure 1.