On topological recursion for Wilson loops in N “ 4 SYM at strong coupling

We consider UpNq N “ 4 super Yang-Mills theory and discuss how to extract the strong coupling limit of non-planar corrections to observables involving the 1 2 -BPS Wilson loop. Our approach is based on a suitable saddle point treatment of the Eynard-Orantin topological recursion in the Gaussian matrix model. Working directly at strong coupling we avoid the usual procedure of first computing observables at finite planar coupling λ, order by order in 1{N , and then taking the λ " 1 limit. In the proposed approach, matrix model multi-point resolvents take a simplified form and some structures of the genus expansion, hardly visible at low order, may be identified and rigorously proved. As a sample application, we consider the expectation value of multiple coincident circular supersymmetric Wilson loops as well as their correlator with single trace chiral operators. For these quantities we provide novel results about the structure of their genus expansion at large tension, generalising recent results in arXiv:2011.02885.


Introduction and results
The recent papers [1,2,3] focused on certain features of higher genus corrections to BPS Wilson loops in dual theories related by AdS/CFT. By means of supersymmetric localization, gauge theory predictions are available as matrix model integrals that depend non-trivially on the number of colours N and 't Hooft planar coupling λ (mass deformations will not be relevant here). The large N expansion may be computed at high order starting from exact expressions in the matrix model or by perturbative loop equation methods, like topological recursion [4]. On the string side, the gauge theory parameters N, λ may be replaced by the string coupling g s and tension T . Worldsheet genus expansion is a natural perturbation theory controlled by powers of g s accompanied by corrections in inverse string tension, i.e. σ-model quantum corrections. The two expansions are expected to match according to AdS/CFT, but practical tests are of course non-trivial. On the gauge side, a rich set of predictions is obtained extracting the dominant strong coupling corrections order by order in 1{N , i.e. well beyond planar level. On string side, this should reproduce the large tension limit T " 1 at specific genera, whose independent determination is obviously very hard beyond leading order. In spite of that, one can still look at manifestations of its expected structural properties in the 1{N gauge theory expansion.
The simplest example where this strategy may be concretely illustrated is the expectation value xWy of the 1 2 -BPS circular Wilson loop in U pN q N " 4 SYM. The expression for xWy is known at finite N and λ " N g 2 YM exactly [5,6,7,8] and is given by the Hermitian Gaussian one-matrix model average In this case, the relation among the gauge theory parameters λ, N and g s , T in the dual AdS 5ˆS 5 IIB superstring is [9] g s " λ 4πN , T " ? λ 2π . (1.2) At large tension, (1.1) takes the following form xWy " 1 2π ? T g s e 2πT`π 12 The structure of (1.3) is consistent with the dual representation of the Wilson loop expectation value as the string path integral over world-sheets ending on a circle at BAdS. 1 A similar large tension analysis is presented in [2] for other quantities related again to the 1 2 -BPS Wilson loop in N " 4 SYM. In particular, one can consider the normalised ratio of n coincident Wilson loops. 2 This requires consideration of matrix integrals which are generalisations of (1.1), but whose 1{N expansion is much more difficult to extract. 3 The semiclassical exponential factors " e 2πT cancel and the ratio xW n y { xWy n is again organised in powers of g 2 s {T , cf. (1.3), xW n y xWy n T "1 " W nˆπ g 2 s T˙, (1.5) where the first three terms of the scaling function W n have been computed in [2] and read W n pxq " 1`n pn´1q 2 x`n pn´1q p3n´5q pn`2q 24 x 2 n pn´1q p15n 4`3 0n 3´7 5n 2´6 10n`1064q 720 (1.6) A third example of scaling functions emerging in the large tension limit are normalised correlators of W with a single trace chiral operator O J " tr Φ J [10,14] recently reconsidered in [2]. In this case, the large tension limit is characterised by a different scaling combination xW O J y xWy where we draw attention to the non-trivial dependence of F J pxq on the R-charge J. 4 Beyond proving general structures as in (1.3), (1.5), and (1.7), it is important to develop methods to determine the detailed form of scaling functions like f, W n and F J . A common approach is to compute the 1{N expansion at finite planar coupling λ in the Hermitian Gaussian one-matrix model, and then take the strong coupling limit λ " 1. For instance, in the case of xWy, one has the exact representation at finite λ [16] xWy " 2N ?
λ Res When each term of this expression is expanded at large λ, the result takes the simple exponential form (1.3). Of course, the case of xWy is particularly simple because of the compact closed formula (1.1) leading to (1.8). Somehow, a similar situation occurs in the case of the scaling function F J in (1.7). Indeed, the correlator xW O J y admits the representation [17] xW O J y "ˆ2 π˙1´J and one can prove (1.7) from this formula, which is exact at finite N and λ [15,2]. However, as soon as the observables under study become more complicated, it is increasingly difficult to extract the genus expansion order by order in 1{N at finite λ. An example are multiple coincident Wilson loops xW n y -not to be confused with multiply wound loops -or multi-trace chiral operators [18]. In this case, exact expressions are not available or are too cumbersome to be useful. Toda recursion relations [19,20,21,22,23] are a possible method to determine the 1{N expansion, but work well only for simple observables [2] (and their scope is limited to the Gaussian matrix model). A more general approach is to take advantage of topological recursion [24,25] which is an efficient way to organise the hierarchy the matrix model loop equations. 5 In practice, a serious bottleneck in applying this method is the rapid increase of computational complexity at higher genus, see for instance [31]. For these reasons, it seems important to devise a version of topological recursion suitable for strong coupling directly.
In this paper, we take a first step in this direction. We illustrate a practical approach to work out topological recursion at strong coupling by isolating dominant contributions at large tension. Despite its simplicity, the method turns out to be rather effective. As an illustration, we present an algorithm for computing the function W n pxq in (1.5) at any desired order with minor effort, and we illustrate remarkable exponentiation properties of the dominant terms at large n. This result will be cross checked by means of an extension to all n of the Toda recursion method used in [2] for n " 2, 3. As a second application, we shall prove that the structure of (1.7) is rather special and does not extend to the normalized correlators of a chiral primary single trace operator with multiple coinciding Wilson loops, i.e. ratios xW n O J y { xW n y when n ą 1. Instead, we prove that the relevant scaling variable is g 2 s {T and that the dependence on the R-charge is where the function H n pxq is independent of J and may be computed in terms of W n by the relation The derivation of these results is straightforward in the framework of the strong coupling version of topological recursion, and far from trivial by other methods. A similar approach is expected to be useful and apply in harder cases with separated Wilson loops or more local operator insertions. Some of these problems can be mapped to multi-matrix models calculations [32] that would be interesting to study by a suitable strong coupling limit of more general topological recursions [33].
The detailed plan of the paper is as follows. In Section 2 we briefly recall the structure of topological recursion for N " 4 SYM and its application to the evaluation of xWy. In Section 3 we show how to perform a saddle point expansions at strong coupling in the considered problems. We clarify what are the relevant features of resolvents in that regime. Section 4 presents the strong coupling version of topological recursion, capturing the reduced resolvents. In Section 5 we apply this formalism to our first application, i.e. the computation of xW n y at large tension. In Section 5.1, as a non-trivial check of our approach, the same results are obtained by solving in the strong coupling limit a suitable Toda recursion for correlators of traced exponentials in the Gaussian matrix model. Finally, in Section 6 we discuss the correlators xW n O J y between coincident Wilson loops and a single trace chiral operator. The relation with the scaling function characterising xW n y is proved in Section 6.4.

Topological recursion for the Gaussian Matrix Model
For a Hermitian one-matrix model with potential V , the spectral curve is defined by [25,34] where xOpM qy " ş DM e´N tr V pM q OpM q and normalization is fixed by x1y " 1. In the Gaussian case, V pM q " 1 2 M 2 , cf. (1.1), and the curve (2.1) takes the form admitting the rational (complex) parametrization The n-point resolvent is defined as the connected correlator 6 W n px 1 , . . . , x n q " and admits the following genus expansion at large N W n px 1 , . . . , x n q " (2.5) The functions W n px 1 , . . . , x n q may be traded by multi-differentials on the algebraic curve (2.2) ω n,g pz 1 , . . . , z n q " W n,g pxpz 1 q, . . . , xpz n qq dxpz 1 q¨¨¨dxpz n q . (2.6) Multi-trace connected correlators may be computed as contour integrals around the cut Higher genus resolvents obey the topological recursion where z " pz 2 , . . . , z n q, w is a subset of z (preserving the order of the variables), |w| is the number of elements of w, and zzw is the complement of w in z. In the double sum we exclude the two cases ph, wq " p0, Hq and ph, wq " pg, zq. The recursion (2.8) allows to compute the following quantities in triangular sequence ( the number under brace is the total weight g`n) 1 N 2g xWy g , xWy g " 1 2πi ¿ ω 1,g pzq e ? λ 2 pz`1{zq . (2.11) The leading term is simply 7 12) in agreement with the well known planar result. The next-to-leading term is The contour encircles all three singular points, but one can check that there are no residues from z "˘1. Thus, integrating by parts two times gives which is the well known 1{N 2 correction. A similar manipulation can be repeated for the next order. Integrating by parts five times gives 15) in agreement with the 1{N 3 term in (1.9). 7 We use the generating function e x 2 pz`1{zq " ř 8 n"´8 Inpxq z n and the identity I0pxq´I2pxq " 2 x I1pxq.
In the case of xWy, this method may be extended to all orders in the 1{N expansion, and can also be generalized to give explicit Bessel function combinations for higher point resolvents at finite λ, see for instance [31]. Nevertheless, the calculation quickly becomes impractical at higher orders due to the very involved expressions that are generated going recursively through the chain of evaluations (2.9). Also, as we explained in the introduction, we are ultimately interested in extracting the large tension limit and want to bypass the cumbersome procedure of first obtaining exact expressions at finite λ, and then expand them at λ " 1. For instance, in the above genustwo contribution both Bessel functions give a similar leading asymptotic contribution due to the expansion and it would be desirable to pin the total contribution in a more direct way. To this aim, one needs to study (2.8) working at strong coupling from the beginning and making more transparent the origin of the dominant terms. The next section will be devoted to this problem.

Saddle point methods for Wilson loops
In this section, we discuss how to extract dominant terms from integrals like (2.12) by saddle point evaluation. Although this is a fairly well known topic, we want to emphasize some specific technical issues that are relevant in the calculations we are interested in. To this aim, we consider the large σ Ñ`8 expansion of a contour integral of the form Suppose that f pzq has a critical pointz where f 1 pzq " 0. Deforming the contour such that it passes throughz with constant Imf pzq along the contour locally aroundz, we write (f " f pzq, f 2 " f 2 pzq, zp0q "z) If gpzq is finite, we simply extract it from the integral and perform the Gaussian integral. In the following, we shall be interested in the case when g has an odd zero or an even pole around the saddle point. In the case of a zero with we just include it in the Gaussian integration and get In the case of a pole with g tÑ0 " A t´2 m`¨¨¨, (3.5) we compute the finite quantity 8 Integrating back in σ gives then Revisiting xWy at strong coupling These formulas may be applied to contour integrals involving Wilson loops and higher order resolvents. Let us illustrate this once again in the case of the simple Wilson loop (1.1). The planar contribution in (2.12) has σ " ? λ, f pzq "´1 2 pz`1{zq and gpzq " 1 z`1´1 z 2˘. The dominant contribution at large λ comes from the saddle point at z " 1 which is a zero of gpzq of linear order. The parametrization is zptq " e it thusf 2 " 1. Expanding gpzq around the zero and taking the first even term gives (3.3) with A " 2i and B " 4 and m " 1. Evaluation of (3.4) gives then λ`¨¨¨, (3.8) in agreement with (1.3). All the higher genus corrections have even poles at z "˘1. Again, the leading contribution comes from z " 1 and may be computed using (3.7). For instance, at genus one we have gpzq " 1 2πi (3.10) Similarly at genus 2 and higher we can check that this procedure reproduces the expansion (1.3). Higher order corrections in 1{ ? λ may also be computed in the same way just by doing Gaussian integration with more accuracy. For instance, we know that (up to exponentially suppressed terms) and we reproduce this expansion by the convenient change of parametrization Using again z " e it , this gives u " 2 sin t 2 and one gets in agreement with (3.11).
Remark: The integrals in (3.13) are apparently divergent, even in Cauchy prescription. Actually, they are evaluated by formulas as (3.7) that hide their original definition as finite contour integrals.

Topological recursion for dominant strong coupling poles
We now look for a simplification of topological recursion (2.8) based on considering the principal part of resolvents at z i " 1, i.e. the terms that dominate at strong coupling. Let us denote the highest pole part byω n,g . Introducing ∆ i " z i´1 , the resolvents in (2.10) reduce to the compact expressionsω 1,1 p∆q " , ω 1,2 p∆q "´1 05 1024 ∆ 10 . (4.1) The (total) degree of the pole terms is 6pg´1q`4n. In general, only even powers of ∆ i appear. If such an Ansatz is plugged into the topological recursion, one can compute the associated resolvent and project onto the maximal pole part. For instance, the last four resolvents in (2.9) become, after projection, which are very compact expressions, compared with the full resolvents. Being symmetric functions, we can further simplify in terms of elementary symmetric polynomials e k px 1 , . . . , x n q " ÿ where x " 1 ∆ 2 . One finds indeed the concise expressionŝ Further results are collected in Appendix C.3.
Remark: Of course, the key point of the method is to useω projected resolvent in the topological recursion and never using the full ω's.

Large tension analysis of coincident Wilson loops
As a first application, we consider the large tension limit of xW n y and, in particular, the ratio (1.5).
As an illustration of the our strategy, we will begin with the doubly coincident Wilson loop, i.e. the case n " 2. Later, we shall extend the analysis to a generic number n of coinciding loops. For n " 2, the 1{N expansion of @ W 2 D has been considered in [35,36,31,2] and its first terms read where I n " I n p ? λq. The associated connected correlator is Expanding at large λ and keeping the leading contribution at each order in 1{N gives Let us show how these contributions can be easily recovered from the "maximal poles" topological recursion. We start from the 2-point formula The genus 0 contribution is special being related to the universal Bargmann kernel and having no poles at z 1,2 " 1. It is where in the last line we used the basic recursion of (modified) Bessel functions and the fact that the infinite sum is telescoping. Starting at genus 1 we can apply the formula (3.7) for the factorized poles. For instance, the first correction is where the numerical constants h m are Replacing (5.7) in (5.6) reproduces the leading term in the second expression in (5.3).
Extension to xW n y and high order calculation Similarly to (5.6), we can exploit the resolvents in (4.1) and (4.2) (together with other ones in Appendix C) to evaluate the saddle point integrals needed to compute xW n y at high order in the genus expansion. Remarkably, this can be done for a generic n. To this aim, we introduce the variable and the connected correlators Normalizing by suitable powers of the simple Wilson loop, we obtain the following results valid up to order Opξ 8 q: From connected correlators we obtain correlators of n coincident Wilson loops using the combinatorial formula where P pk, nq is the set of integer partitions π of k satisfying k`|π| ď n where |π| is the number of elements of π, and Spπq is the symmetry factor of partition π given by products of m! for each group of m equal elements in π. This expression follows from the fact that xW n y can be written as a sum over just the partitions of n. Since we divide by xWy n , all the parts of a given partition that are 1 disappear, leaving the partitions of n with every part at least 2. Such partitions can be seen to be in one to one correspondence with the partitions of integers k ď n such that k`|π| ď n, giving the expression above.
Since on general grounds xW m y c xWy m " Opξ m´1 q, to obtain the expansion of xW n y xWy n up to ξ m we can restrict the sum in (5.11) to k ă m. Furthermore, taking into account that a part p enters the above expression as xW p`1 y c xWy p`1 , we need the terms corresponding to all the partitions of m to obtain the result up to ξ m . Let's consider a couple of examples. For m " 1 the only possible partition is 1 and we obtain xW n y xWy n " 1`n For m " 2 we have three partitions, i.e. 1 and 2 and p1, 1q. The last one has a symmetry factor of two. So we obtain In a similar way, to obtain the result up to ξ 3 we will have to add all terms corresponding to the partitions of three to the above results and so on. We now have all the ingredients needed to evaluate the above expression to ξ 8 . The final result is, cf. (1.5) where the large tension limit is understood. This is the extension to order ξ 8 of the cubic result in Eq. (1.17) of [2]. The special cases n " 2 and n " 3 are and agree with the exact expressions [2], is the Owen T-function. The coefficient of ξ k is a polynomial in n of degree 2k. A remarkable simplification is achieved by writing (5.14) in exponential form since P k turns out to be a polynomial of (approximately half) degree k´1. Explicitly, one finds with leading terms at large n following the pattern

Solution by Toda recursion
The genus expansion of (1.3) is efficiently computed by exploiting the Toda integrability of the 1-matrix Hermitian Gaussian model [23]. In general, correlators in this model are constrained by integrable differential equations [20,21,22] that in Gaussian case take the Toda form [19]. Notice that in [23] the matrix model measure is expp´1 2 tr Ă M 2 q without explicit N factor. This will be the convention throughout this section. After defining the connected correlators one has e N`1 pxq`e N´1 pxq " 2 e N pxq`x 2 N e N pxq, (5.22) e N`1 px, yq`e N´1 px, yq " 2 e N px, yq`p x`yq 2 N e N px, yq´x 2 y 2 N 2 e N pxqe N pyq.
The general structure is e N`1 px 1 , . . . , x k q`e N´1 px 1 , . . . , x k q´"2`1 N px 1`¨¨¨`xk q 2  e N px 1 , . . . , x k q " g N px 1 , . . . , x k q, (5.24) where g N may be read from the non-leading terms of the cumulant expansion of xX 1¨¨¨Xk y c and replacing @ where g pkq pλ, N q is obtained from the non-leading terms of the cumulant expansion of @ X k D c and replacing The explicit coefficients of the cumulant expansion of xX p y c may be expressed in terms of integer partitions π " p1 m 1 2 m 2¨¨¨q of p Hence, the equations are The large tension scaling Ansatz is Replacing in the Toda equations gives The only partitions that may give a contribution have |π| " ř m r " 2. One case is when k is even and then the partition is π " p M 2 , M 2 q, or when k is split into the sum of two different parts π " pq, pk´qqq with q ‰ k{2. 9 Denoting by an apex such partitions, we have (using ř r rm r " k) Finally, evaluating the r.h.s. for the two relevant kinds of partitions, we obtain the differential equation that we rearrange in the form The first instance k " 1 gives The constant is fixed by (1.3) and gives We shall be interested in the ratios They obey We also know that R k pξq " Opξ k´1 q. This gives the integration constant and the explicit recurrence relation This recursion provides the expressions in (5.10) to be plugged into (5.11) in order to compute the scaling functions W n pξq. Just to give an example, using (5.40) one may easily extend the last line in (5.10) and find and so on. Further expressions of P k for k up to 20 are collected in Appendix B.
Remark: Of course, one can also use (5.40) without expanding. This gives exact expressions for R k as iterated integrals. The first two cases are where T is the Owen function, cf. (5.16). The expression for R 4 may be obtained by continuing the iteration but will involve integrals of the T function. A simple general feature of the functions R k is that they are all entire in ξ. Hence, the radius of convergence of (5.14) is infinite for all n.
Remark: One has to keep in mind that Toda recursion methods are not suitable to treat insertions of local chiral operators, see the discussion in Appendix A. In this case, one has to keep using topological recursion, as discussed in the next Section.

Correlator of coincident Wilson loops and a chiral operator
In this section, we address the problem of computing the correlator between multiple coincident Wilson loops W n and a single trace chiral operator. In other words, we want to generalize (1. where C is a circle of radius R (set to unity in the following), and Φ 1 is one of the six real scalars tΦ I u I"1,...,6 in N " 4 SYM. Single trace chiral operators take the general form O J " trpu I Φ I pxqq J where u I is a complex null 6-vector obeying u 2 " 0 [14]. The dependence of the correlator xW O J y on u I and the choice of coupling between the loop and the scalars factorizes and will be absorbed in the operator normalization [37]. With the same conventions as in [2], the matrix model representative for the chiral operator O J is where normal ordering subtracts self-contractions and is necessary to map matrix model correlators to R 4 quantum expectation values [38,39]. 10 At leading order in large tension, the correlator between a single Wilson loop and the chiral operator O J obeys (1.7) in terms of a scaling function that depends on the specific ratio g 2 s {T 2 and has a non-trivial dependence on J. The most natural scaling dependence is actually on g 2 s {T as in (1.5). Several cancellations occur and are responsible for the relevant variable being g 2 s {T 2 . We shall show that this pattern changes in the case of the correlator between multiple coincident Wilson loops and one chiral operator. The above mentioned cancellations do not occur anymore and one has instead the structure (1.11). Besides, the function H n can be computed explicitly in terms of W n , cf. (1.12). To derive such a result, we will conveniently use the strong coupling version of topological recursion. As we remarked previously, Toda recursion is rather cumbersome for these purposes, as illustrated in the example pn, Jq " p1, 2q in Appendix A.

Contribution from multi-trace operators in normal ordering
As a preliminary step we first address the issue of the effects of normal ordering in (6.2) and the role of multi-trace operators. It is instructive to look at the first cases at low J. A straightforward explicit calculation gives (we restrict to even J for the purpose of illustration) and so on. In general, terms involving products of k traces are " 1{N k´1 at large N . We will write : tr M J : where the operators in " : tr M J : ‰ k have coefficients Op1q at large N . 11 Now, let us consider the genus g contribution to the connected correlator @ W n O pkq D c where O pkq is any arbitrary k-trace operator. We can write, cf. (2.5), ω n,g pz 1 , . . . , z n`k q O pkq pxpz 1 q, . . . , xpz k qq exp˜λ 2 n ÿ l"1 x pz k`l q¸.
In the case of O pkq " r: tr M J :s k , taking into account the extra factor 1 N k´1 in (6.4), we find that @ W n r: tr M j :s k D c | genus g scales as N´p 2g`2k`n´3q . Finally, let us pin the dependence on λ " 1. 10 The choice of normalization in (6.2), and in particular the overall power of N , is dictated by string theory and makes direct contact with the associated natural vertex operators [1]. Another standard choice is to require a fixed normalization of the chiral operators 2-point functions as in [14]. 11 The k-trace part may have an explicit N dependence as in " : tr M 6 : ‰ 1 which has a piece 15 4 p1`1{N 2 q tr M 2 whose N Ñ 8 limit is finite.
Since the operator : tr M J : does not depend on λ explicitly, the strong coupling limit of the expectation value of Wilson loop with chiral operators corresponds to maximizing the order of poles of variables corresponding to Wilson loop or conversely minimizing the order of the poles of the variables that correspond to the : tr M J : operator. The total order of the poles of ω n,g is 6pg´1q`4n, as discussed in section 4. According to the saddle point analysis this implies the final scaling behaviour @ W n r: tr M J :s k D cˇgenus g " This gives the leading power at large λ for all genera. In particular, a term with an overall 1 N P factor will be accompanied by the following powers of λ showing that multiple trace contributions are suppressed. Besides, since the saddle point expansion has relative corrections in powers λ´1 {2 " 1{T , double trace corrections to normal ordering cannot be seen even at first subleading order in large λ.
Let us see this explicitly in the simplest case of a single Wilson loop keeping only up to double trace operators . At the planar level, as is well known, the double trace part doesn't contribute. At 1{N 2 level there are two relevant cumulants corresponding to pk, gq " p1, 1q and p2, 0q. Their λ dependence can be obtained using the explicit strong coupling resolvents given (4.2) as respectively, The first contribution is dominant in the large tension limit and in fact we would have to expand it to three orders in λ before the the second one becomes effective. Since, the rate of growth of the exponent of λ is 6 for g but only 2 for k, as g and k increase or as cumulants are multiplied, the gap between the contributions of single and higher trace operators only increases. 12 6.2 xW n O J y at leading order As a result of the above discussion, we can restrict ourselves to the single trace part of normal ordering, i.e. the planar approximation. According to [40,41], it may be written in terms of Chebyshev polynomials and, in the z variable, it reads The relevant connected correlators @ W n : tr M J : D c,g are 13 xW n Oy c,g " 1 p2πiq n`1 ¿ ω n,g pz 1 , . . . , z n`1 q z´J 1 exp « ? λ 2 pxpz 2 q`¨¨¨`xpz n`1 qq ff . (6.10) In the strong coupling limit we use the strong coupling resolventω n,g pzq and keep in it only those terms that minimize the order of the poles of z 1 at 1. This can be done by going through one step of topological recursion. This corresponds to starting with ω g,n´1 pz 1 , . . . z n q and using: We can now integrate over z 2 ,¨¨¨, z n in the saddle point approximation. The above factor of 2k j`1 ensures that the result has a very simple relation to xW n y c,g in the strong coupling limit, i.e. 14 @ W n : tr M J : D λ"1 c,g " nλ 2 xW n y λ"1 c,g Res xW n y λ"1 c,g`¨¨¨. (6.12) The same is true for the full correlator, after expanding into connected correlators, i.e. @ W n : tr M J : D xW n y λ"1 " J n ? λ 2`¨¨¨. (6.13) This is of course expected from the known results for n " 1 and n " 2, see [2].

Subleading corrections
To go beyond leading order we need to carry out topological recursion with poles of one subleading order included. It is convenient to change variables from z to u, cf. (3.12), and write ω n,g pu 1 , . . . , u n q "ω n,g`δ ω n,g`. . . . (6.14) Where δω n,g includes the poles of total degree 6g`4n´8, see Appendix C for full details of the procedure. Using (6.14) we can compute the one-variable resolvents obtained after integration of all but one variable, 15ω n,g pzq " 1 p2πiq n ¿ ω n`1,g pupzq, u 1 , . . . u n q . (6.15) Due to our previous discussion, cf. (6.6), the first two orders in the 1{ ? λ expansion at large λ can be computed by ignoring mixing with multi-trace operators and using the simple correspondence in (6.9). Thus, we simply obtain @ W n : tr M J : D c,g " Res zÑ0ω n,g pzq z J . (6.16) This computes the connected part of the correlator but we can also define a function that similarly computes the full correlator, i.e. Ω n,g pzq " To compute @ W n : tr M J : D { xW n y it is convenient to expandΩ n,g pzq as: Ω n,g pzq " U n,0 pzq xW n y c,g`U n,1 pzq xW n y c,g´1`¨¨¨`U n,g pzq xW n y c,0`. . . , (6.18) where each U n,g pzq is determined recursively genus by genus and final dots stand for a correction of order Opp1{ ? λq g`2 q relative to the leading order. Then it can be seen that, @ : tr M J : W n D xW n y " Res The functions U n,g pzq depend also on λ. To the leading order in λ, U n,0 pzq can be read from (6.12). To get a non-vanishing result for all other U n,g we need to go beyondω n,g and include δω n,g . Restricting ourselves to two leading term in λ, the most general structure possible for U n,g is: Where f n,g pzq are polynomials of degree at most 3 and independent of λ. Two out of the 4 free coefficients are determined by the requirement from topological recursion that U n,g`1 z˘"´U n,g pzq. Another one can be fixed by requiring that x: tr M : W n y c,g vanishes for g ą 0. 16 . Combining these two requirements we obtain U n,g " dz c n,g λ 3g 2 z pz´1q 4 . (6.21) Explicit results After having clarified the general structure of topological recursion for the quantities we need, let us present explicit results. For the 'critical' case n " 1 we find 17 U 1,0 pzq " dz˜? λ 2pz´1q 2`3 z 2pz´1q 4¸`. . . , U 1,1 pzq " dz λ 3{2 z 32pz´1q 4`. . . , (6.22) while the higher U 1,g pzq vanish i.e c 1,g " 0 for g ą 1. This can be seen as consistency check and is a result of the cancellations required to reorganize the series for xW : O J :y as in (1.7). To calculate non-vanishing terms in U 1,g pzq for g ą 1 we will need to keep more than 2 leading terms in ω n,g . These peculiar cancellations do not occur for n ą 1 and make the calculation of subleading corrections possible with our level of accuracy. We find . . , . . , . . .

(6.23)
As a result of this, the dependence on J in xW n :tr M j :y xW n y is much simpler for n ą 1 than in the n " 1 case, cf. (1.7). Indeed, from the above, it has to be proportional to This means in that the structure of large tension limit of xW n O J y xW n y is given by (1.11). The first few terms of H n pxq can be calculated from (6.23) and read . . , . . , . . , . . . (6.25)

Relating H n to W n
The discussion in previous section has led to the expansion (6.25) for the scaling functions H n . Most importantly, we could prove the general structure (1.11), with its peculiar dependence on the J parameter. In this section we show how this can be exploited to express H n in terms of W n . To this aim we take J " 2 in the topological recursion result (1.11) and write xW n O 2 y xW n y T "1 " π n pT`3 H n q. " π n pT`3 H n q, (6.28) and a short calculation gives the relation H n pxq " x 2π Replacing W n by its evaluation by means of (5.40) and using the series expansion (5.14), we get H n pxq "´5`6 n 24π x`p´1`n qp´5`2nq 12π x 2`p´1`n qp133´95n`15n 2 q 60π x 3 p´1`nqp´24159`23611n´7035n 2`6 30n 3 q 1260π x 4`¨¨¨, (6.30) in agreement with (6.25). Of course, the exact determination of W n by Toda recursion means that we can provide easily all order expansion of the H n function by means of (6.29).

A few sample calculations
Let us give some examples of (1.12) by explicit computations. For n " 2 we need the explicit exact expansion Using (6.27) we work out the case pn, Jq " p2, 2q λ 9{2`¨¨¨`¨¨¨. (6.32) Comparing with (1.11) gives the first terms x 3`¨¨¨˙, (6.33) in agreement with (6.30). In this case we can give the exact expression in a reasonable compact form using the first equation in (5.16) 2˘.

(6.34)
A similar calculation can be repeated for n " 3. In this case we have Comparing with (1.11) we obtain H 3 pxq " 1 9πˆ3 in agreement with (1.12). As in (6.34), one can give a closed formula for this function in terms of the special error and Owen-T functions. As a final check, probing the peculiar simple J dependence in (1.11), we consider the case pn, Jq " p2, 3q. To analyze this case by expansion of exact expressions at finite λ we need the Bessel function expansion of @ a π 2 : tr M 3 : and : tr M 3 :" tr M 3´3 tr M . By matching a large number of weak coupling perturbative coefficients, we find " ? λp´5760´1440λ´528λ 2`1 939λ 3 qI 2 0 483840`p 184320`69120λ`8544λ 2´3 9902λ 3`6 209λ 4 qI 0 I 1 3870720 p´368640´184320λ´6144λ 2`4 1128λ 3`2 4815λ 4 qI 2 1

3ˆπ 2˙3
and indeed we find that this is equivalent to the previous expansion (6.33).

A Toda recursion for correlators with chiral primaries
The genus expansion of the ratio xW O 2 y { xWy may be computed by (6.27) in terms of xWy. Alternatively, it is equivalent to use the integral representation (1.10) derived in [17]. Here, we want to show how such correlators may be treated by Toda recursion, as an illustration, generalizing the treatment in App. B.3 of [2]. From to make contact with the expressions in [2]. The relevant Toda equation is (5.23). Taking two derivatives involves the auxiliary quantity To continue, we need the correct Ansatz for the r.h.s. of (A.2) and (A.4) at large tension. This is @ W : tr a J : D (A.5) The Toda recursion takes the form The expansion at large N with fixed ζ require to study the asymptotic behaviour of e N p ? N µq at fixed µ. Recall that Setting x " ? N µ and expanding the differential equation gives This is enough to derive the relevant terms in the expansion (A.10) Using this in the expansion of (A.6) gives It is easy to check that C 1 pζq " 1 2 ?
2 , so that where k is a constant that we set to zero by analyticity. B The polynomial P k for k " 11, . . . , 20 The polynomials P k pnq have been defined in (5.18) and their expression for k up to 10 have been given in (5.19) and (5.42). The expressions for k " 11, . . . , 20 are given below. C Some details about topological recursion at large tension Here we summarize some details about topological recursion that are relevant to the strong coupling limit of correlation functions studied in the main text. Our presentation will be for the Gaussian matrix model although most of the statements have straightforward generalizations to a general genus 0 spectral curve. See [25,34] for pedagogical details and general treatment.

C.1 Spectral curve, resolvents, and residues
For the Gaussian matrix model the spectral curve is a two-sheeted cover of the complex plane. 18 The two sheets are glued along the cut on which the eigenvalues condense in the large N limit. The coordinate z defined in (2.3) maps these two sheets to the Riemann sphere as shown in Fig. 1. A generic value of x has two preimages since xpzq " x`1 z˘, if |z| ‰ 1 then for one of these preimages |z| ą 1 and for other |z| ă 1. These are the two sheets which have been mapped to the exterior and interior respectively of unit circle on the z-plane. Let's now focus on the unit circle itself on which we write z " exppitq. Then xpzptqq " 2 cos t. So as z goes from 0 to π, xpzptqq goes from 2 to´2. This is one copy of the cut while the other copy is corresponds to t going from π Ñ 2π " 0. The two copies of the cut are joined at z " 1 and z "´1 which correspond to x " 2 and x "´2 i.e the end points of the cut. These are the only two values of x which have a single preimage. These are the zeroes of the differential dx. Lastly, notice that although y is not a single valued function of x, it is a single valued function of z. Note that the unit circle is also the contour for the saddle point approximation, the saddle point integral is actually done over a double copy of the cut. In the z-plane the circle is formed from two copies of the cut and separates the two sheets whose images are the outer/inner parts. In particular, the (image of the ) point A is inside the circle.

(C.2)
As a result the sole contribution to the correlation function of f comes from its own poles, which for a polynomial of x i are at zpx i q " 0 (inside the contour) and zpx i q Ñ 8 (outside the contour). These correspond to x Ñ 8 in the two sheets. Hence, we can write ¿ z i ω n,g pz 1 , . . . , z n qf pxpz 1 q, . . . , xpz n qq " Res z i "0 ω n,g pz 1 , . . . , z n qf pxpz 1 q, . . . , xpz n qq.
The same logic works for any holomorphic function of x among them the Wilson loop. As we have seen in practice, the contour integral is more convenient for the strong coupling expansion of Wilson loops while the residue at 0 is simpler for chiral operators. Nevertheless this vanishing of residues at˘1 ensures that there is no ambiguity in the saddle point prescription, since we can smoothly deform the contour past the branch points, as illustrated in Fig. 2.
z-plane Figure 2: The contour integral for computing the matrix correlator f pxq in z-plane (the big circle) gets contribution only from the pole of f pxpzqq at z " 0. The residue at poles of resolvent (dashed circles) vanish.

C.2 Topological recursion at subleading order
The coordinate u defined in (3.13) which is convenient for extending the saddle point approximation to subleading orders can be seen as a reparameterization of spectral curve as zpuq " exp´2i arcsin´u 2¯¯.

(C.4)
This change of variables maps the z-plane to a cylinder u " φ`ir where φ parameterizes a circle of radius 4 extending from´2 to 2 while r a real line. In this manner u is the local complex coordinate on an infinite cylinder. This cylinder is compactified to a sphere by identifying the circle at u " i8 with one point and the circle at u "´i8 with another point. In the u-coordinate the branch point are mapped to 0 and´2 " 2. In the strong coupling limit the dominant contribution to the expectation value of Wilson loops comes from u " 0 while the contour for saddle point integral is the circle r " 0.

(C.5)
In terms of these variables the recursion Kernel K is Kpu, vq "´i du 2v ? 4´u 2 pu 2´v2 q dv . (C.6) Apart from the factor ? u 2´4 which is independent of v and as a result gives an overall multiplicative factor, the kernel is homogeneous in these coordinates if we only keep the residue at u " 0. This makes it easier to separate out the contribution of different orders. Indeed defininĝ Kpu, vq "´i ? 4´u 2 Kpu, vq, we see that (C.7) SoKpu, vq uniformly increases the degree of the poles of differential it acts on by 2. This simplification in the recursion kernel is a trade off due to the fact that the starting point of the recursion ω 2,0 pu, vq is now more complicated being given by and, for the purposes of carrying out topological recursion, it will be expanded into a double power series easily. Another simplification is that in these coordinates the antisymmetry property (C.1) reads ω n,g p´u 1 , . . . , u n q " ω n,g pu 1 , . . . , u n q . (C.9) This means in particular that ω n,g pu 1 , . . . , u n q " i n du 1 . . . du n f n,gˆ1 u 2 1 , . . . , 1 u 2 n˙, (C.10) for some symmetric polynomials f n,g . As a result the poles encountered in the saddle point integrals are always of even order. Finally, we observe that all ω g,n computed through the topological recursion have poles of order at least 4 at u " 0 and as a result for the first two orders of poles that we need we can ignore the residues at˘2 in (C.5).
n " 1 δf 1,1 " 5e 1 128 , δf 1,2 "´4 83e 4 D The correlation function x: trM : W n y In the main text, to prove (6.21), we exploited the fact that x: tr M : W n y has no higher genus corrections beyond the leading order. This can be easily proved by starting from the following splitting of M in the U pN q theory In the case of xmW n y, we obtain the same integral forM with an extra insertion of m in the m-integral. As a result, the "traceless" part xW n y traceless cancels and we obtain xmW n y xW n y " This is just the leading order result obtained in (6.13) and specialized to J " 1. The above discussion shows that it is in fact exact.