On topological recursion for Wilson loops in \mathcal N=4 SYM at strong coupling
aa r X i v : . [ h e p - t h ] F e b On topological recursion for Wilson loopsin N “ SYM at strong coupling
M. Beccaria a and A. Hasan a a Università del Salento, Dipartimento di Matematica e Fisica
Ennio De Giorgi ,and I.N.F.N. - sezione di Lecce, Via Arnesano, I-73100 Lecce, Italy
E-mail: [email protected], [email protected]
Abstract
We consider U p N q N “ super Yang-Mills theory and discuss how to extract the strongcoupling limit of non-planar corrections to observables involving the -BPS Wilson loop. Ourapproach is based on a suitable saddle point treatment of the Eynard-Orantin topologicalrecursion in the Gaussian matrix model. Working directly at strong coupling we avoid theusual procedure of first computing observables at finite planar coupling λ , order by order in { N , and then taking the λ " limit. In the proposed approach, matrix model multi-pointresolvents take a simplified form and some structures of the genus expansion, hardly visibleat low order, may be identified and rigorously proved. As a sample application, we considerthe expectation value of multiple coincident circular supersymmetric Wilson loops as well astheir correlator with single trace chiral operators. For these quantities we provide novel resultsabout the structure of their genus expansion at large tension, generalising recent results inarXiv:2011.02885. Keywords: supersymmetric Wilson loop, topological recursion, matrix models. ontents x W n O J y at leading order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3 Subleading corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.4 Relating H n to W n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.5 A few sample calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 A Toda recursion for correlators with chiral primaries 25B The polynomial P k for k “ , . . . ,
26C Some details about topological recursion at large tension 29
C.1 Spectral curve, resolvents, and residues . . . . . . . . . . . . . . . . . . . . . . . . . 29C.2 Topological recursion at subleading order . . . . . . . . . . . . . . . . . . . . . . . 31C.3 Expressions for resolvents at leading and first subleading order of poles . . . . . . . 32
D The correlation function x : tr M : W n y The recent papers [1, 2, 3] focused on certain features of higher genus corrections to BPS Wilsonloops in dual theories related by AdS/CFT. By means of supersymmetric localization, gauge theorypredictions are available as matrix model integrals that depend non-trivially on the number ofcolours 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 modelor by perturbative loop equation methods, like topological recursion [4]. On the string side, thegauge theory parameters N, λ may be replaced by the string coupling g s and tension T . World-sheet genus expansion is a natural perturbation theory controlled by powers of g s accompanied bycorrections in inverse string tension, i.e. σ -model quantum corrections. The two expansions areexpected to match according to AdS/CFT, but practical tests are of course non-trivial. On thegauge side, a rich set of predictions is obtained extracting the dominant strong coupling corrections1rder by order in { N , i.e. well beyond planar level. On string side, this should reproduce thelarge tension limit T " at specific genera, whose independent determination is obviously veryhard beyond leading order. In spite of that, one can still look at manifestations of its expectedstructural properties in the { N gauge theory expansion.The simplest example where this strategy may be concretely illustrated is the expectation value x W y of the -BPS circular Wilson loop in U p N q N “ SYM. The expression for x W y is knownat finite N and λ “ N g exactly [5, 6, 7, 8] and is given by the Hermitian Gaussian one-matrixmodel average x W y “ ż D M tr e ? λ M e ´ N tr M “ e λ N L N ´ ˆ ´ λ N ˙ . (1.1)In this case, the relation among the gauge theory parameters λ, N and g s , T in the dual AdS ˆ S IIB superstring is [9] g s “ λ πN , T “ ? λ π . (1.2)At large tension, (1.1) takes the following form x W y “ π ? Tg s e πT ` π g s T „ ` O p T ´ q T " “ e π T f ˆ g s T ˙ , (1.3) f p x q “ x ´ { exp ˆ π x ˙ . (1.4)The structure of (1.3) is consistent with the dual representation of the Wilson loop expectationvalue as the string path integral over world-sheets ending on a circle at B AdS. A similar large tension analysis is presented in [2] for other quantities related again to the -BPSWilson loop in N “ SYM. In particular, one can consider the normalised ratio of n coincidentWilson loops. This requires consideration of matrix integrals which are generalisations of (1.1),but whose { N expansion is much more difficult to extract. The semiclassical exponential factors „ e πT cancel and the ratio x W n y { x W y n is again organised in powers of g s { T , cf. (1.3), x W n yx W y n T " “ W n ˆ πg s T ˙ , (1.5)where the first three terms of the scaling function W n have been computed in [2] and read W n p x q “ ` n p n ´ q x ` n p n ´ q p n ´ q p n ` q x ` n p n ´ q p n ` n ´ n ´ n ` q x ` ¨ ¨ ¨ . (1.6) The exponential factor exp p πT q comes from the AdS minimal surface [10, 11, 12]. Upon expansion in g s , thepower of the string coupling is minus the Euler number of a disc with p handles ( χ “ ´ p ). The fact that eachpower of g s is accompanied at large tension by a factor {? T is non-trivial and explained in [1]. A similar structureholds for Wilson loops in ABJM theory, dual to string on AdS ˆ CP . See [13] for a recent application of such coincident loops in matrix models associated with JT gravity. Indeed, in this case one does not have a simple result like (1.1), but instead multiple finite sum of „ N terms,see for instance Eq. (4.3) in [2] for n “ . W with a single trace chiral operator O J „ tr Φ J [10, 14] recently reconsidered in [2]. In thiscase, the large tension limit is characterised by a different scaling combination x W O J yx W y T " “ J ˆ π ˙ J { T F J ˆ g s T ˙ , F J p x q “ J ? x sinh ˆ J arcsinh ? x ˙ , (1.7)where we draw attention to the non-trivial dependence of F J p x q on the R-charge J . Beyond proving general structures as in (1.3), (1.5), and (1.7), it is important to developmethods to determine the detailed form of scaling functions like f , W n and F J . A common approachis to compute the { N expansion at finite planar coupling λ in the Hermitian Gaussian one-matrixmodel, and then take the strong coupling limit λ " . For instance, in the case of x W y , one hasthe exact representation at finite λ [16] x W y “ N ? λ Res x “ ” e λ N H p ? λ N x q 8 ÿ n “ I n p? λ q x ´ n ı , H p x q ” ´ coth x ´ x ¯ . (1.8)From (1.8), we get all coefficients of the { N power series in terms of explicit combinations ofmodified Bessel functions ( I n ” I n p? λ q ), see also [6], x W y “ N I ? λ ` λI N ` N ´ λ { I ´ λ I ¯ ` N ´ λ I ´ λ { I ` λ I ¯ ` ¨ ¨ ¨ . (1.9)When each term of this expression is expanded at large λ , the result takes the simple exponentialform (1.3). Of course, the case of x W y 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 x W O J y admits the representation [17] x W O J y “ ˆ π ˙ ´ J { N ? λ e λ N ¿ dz πi z J e ? λ z ´ ` ? λ N z ¯ N ”´ ` ? λ N z ¯ J ´ ı , (1.10)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 increasinglydifficult to extract the genus expansion order by order in { N at finite λ . An example are multiplecoincident Wilson loops x W n y – not to be confused with multiply wound loops – or multi-tracechiral operators [18]. In this case, exact expressions are not available or are too cumbersome to beuseful. Toda recursion relations [19, 20, 21, 22, 23] are a possible method to determine the { N expansion, but work well only for simple observables [2] (and their scope is limited to the Gaussianmatrix 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. In practice,a serious bottleneck in applying this method is the rapid increase of computational complexity athigher genus, see for instance [31]. For these reasons, it seems important to devise a version oftopological recursion suitable for strong coupling directly. Through an analytical continuation it is possible to capture F J by a D3-brane calculation, see [15]. See also [26, 27, 28, 29, 30] for other recent applications of topological recursion to N “ SYM.
3n this paper, we take a first step in this direction. We illustrate a practical approach to workout 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 presentan algorithm for computing the function W n p x q in (1.5) at any desired order with minor effort, andwe illustrate remarkable exponentiation properties of the dominant terms at large n . This resultwill be cross checked by means of an extension to all n of the Toda recursion method used in [2]for n “ , . As a second application, we shall prove that the structure of (1.7) is rather specialand does not extend to the normalized correlators of a chiral primary single trace operator withmultiple coinciding Wilson loops, i.e. ratios x W n O J y { x W n y when n ą . Instead, we prove thatthe relevant scaling variable is g s { T and that the dependence on the R-charge is x W n O J yx W n y T " “ J ˆ π ˙ J { n „ T ` p J ´ q H n ˆ πg s T ˙ , (1.11)where the function H n p x q is independent of J and may be computed in terms of W n by the relation H n p x q “ x π „ ` n p log W n p x qq . (1.12)The derivation of these results is straightforward in the framework of the strong coupling versionof topological recursion, and far from trivial by other methods. A similar approach is expected tobe 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 beinteresting 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 oftopological recursion for N “ SYM and its application to the evaluation of x W y . In Section 3 weshow how to perform a saddle point expansions at strong coupling in the considered problems. Weclarify what are the relevant features of resolvents in that regime. Section 4 presents the strongcoupling version of topological recursion, capturing the reduced resolvents. In Section 5 we applythis formalism to our first application, i.e. the computation of x W n y at large tension. In Section5.1, as a non-trivial check of our approach, the same results are obtained by solving in the strongcoupling limit a suitable Toda recursion for correlators of traced exponentials in the Gaussianmatrix model. Finally, in Section 6 we discuss the correlators x W n O J y between coincident Wilsonloops and a single trace chiral operator. The relation with the scaling function characterising x W n y is proved in Section 6.4. For a Hermitian one-matrix model with potential V , the spectral curve is defined by [25, 34] y ´ V p x q y ` P p x q “ , P p x q “ N B tr V p x q ´ V p M q x ´ M F , (2.1)where x O p M qy “ ş DM e ´ N tr V p M q O p M q and normalization is fixed by x y “ . In the Gaussiancase, V p M q “ M , cf. (1.1), and the curve (2.1) takes the form y ´ xy ` “ , (2.2)4dmitting the rational (complex) parametrization x “ z ` z , y “ z . (2.3)The n -point resolvent is defined as the connected correlator W n p x , . . . , x n q “ B tr x ´ M ¨ ¨ ¨ tr x n ´ M F c , (2.4)and admits the following genus expansion at large NW n p x , . . . , x n q “ ÿ g “ N n ´ ` g W n,g p x , . . . , x k q . (2.5)The functions W n p x , . . . , x n q may be traded by multi-differentials on the algebraic curve (2.2) ω n,g p z , . . . , z n q “ W n,g p x p z q , . . . , x p z n qq dx p z q ¨ ¨ ¨ dx p z n q . (2.6)Multi-trace connected correlators may be computed as contour integrals around the cut Cź i tr O i p M q G c “ ÿ g “ N n ´ ` g p π i q n ¿ ω n,g p z , . . . , z n q ź i O i p x p z i qq . (2.7)Higher genus resolvents obey the topological recursion ω , p z q “ z ˆ ´ z ˙ dz, ω , p z , z q “ dz dz p z ´ z q , (2.8) ω n,g p z , z q “ Res ζ “ , ´ K p z , ζ q „ ω n ` ,g ´ p ζ, ζ ´ , z q ` ÿ h ď g ÿ w Ă z ω | w | ,h ` p ζ, w q ω n ´| w | ,g ´ h p ζ ´ , z z w q ,K p z, w q “ w p w ´ qp z ´ w qp zw ´ q dzdw . where z “ p z , . . . , z n q , w is a subset of z (preserving the order of the variables), | w | is the numberof elements of w , and z z w is the complement of w in z . In the double sum we exclude the twocases p h, w q “ p , Hq and p h, w q “ p g, z q . The recursion (2.8) allows to compute the followingquantities in triangular sequence ( the number under brace is the total weight g ` n ) ω , loomoon Ñ ω , Ñ ω , Ñ ω , looooooooooomooooooooooon Ñ ω , Ñ ω , Ñ ω , Ñ ω , loooooooooooooooomoooooooooooooooon Ñ ¨ ¨ ¨ . (2.9)Apart from the seeds ω , and ω , , all other resolvents have poles in the z i variables only at thespecial points ˘ . The first entries in (2.9) read (omitting the dz ¨ ¨ ¨ dz n differentials) ω , p z q “ z p z ´ q , Connected correlators x X X ¨ ¨ ¨y c are functional derivatives of the logarithm of the generating function ofcorrelators with respect to sources coupled to X i operators. , p z , z , z q “ ´ p z ´ q p z ´ q p z ´ q ` p z ` q p z ` q p z ` q ,ω , p z , z q “ p z ´ q p z ´ q „ z z p ` z q p ´ z ` z q ` z z p ` z q p ´ z ` z q` p z ` z q ` z p z ` z q ` z p z ` z ` z q ` z p z ` z ` z q` z p z ´ z ´ z ` z q ` z p z ´ z ´ z ` z q` z p z ´ z ` z ´ z ` z q ` z p ´ z ` z ` z ´ z ` z q` z p ´ z ` z ` z ´ z ` z q ,ω , p z q “ ´ z p ` z ` z qp´ ` z q , (2.10)and so on. The expression of ω , shows how explicit results become quickly unwieldy. Analysis of the simple loop x W y It is useful illustrate how resolvents are used to computethe genus expansion of the simple loop expectation value x W y . We have x W y “ ż D M tr e ? λ M e ´ N tr M N Ñ8 “ N ÿ g “ N g x W y g , x W y g “ πi ¿ ω ,g p z q e ? λ p z ` { z q . (2.11)The leading term is simply x W y “ ¿ dz πi z ˆ ´ z ˙ e ? λ p z ` { z q “ ? λ I p? λ q , (2.12)in agreement with the well known planar result. The next-to-leading term is x W y “ ¿ dz πi z p z ´ q e ? λ p z ` { z q (2.13)The contour encircles all three singular points, but one can check that there are no residues from z “ ˘ . Thus, integrating by parts two times gives x W y “ λ ¿ dz πi z e ? λ p z ` { z q “ λ I p? λ q , (2.14)which is the well known { N correction. A similar manipulation can be repeated for the nextorder. Integrating by parts five times gives x W y “ ¿ dz πi z p ` z ` z qp z ´ q e ? λ p z ` { z q “ λ { ¿ dz πi ˆ z ` z ˙ e ? λ p z ` { z q “ λ { „ I p? λ q ` I p? λ q “ „ λ { I p? λ q ´ λ I p? λ q , (2.15)in agreement with the { N term in (1.9). We use the generating function e x p z ` { z q “ ř n “´8 I n p x q z n and the identity I p x q ´ I p x q “ x I p x q .
6n the case of x W y , this method may be extended to all orders in the { N expansion, andcan also be generalized to give explicit Bessel function combinations for higher point resolvents atfinite λ , see for instance [31]. Nevertheless, the calculation quickly becomes impractical at higherorders due to the very involved expressions that are generated going recursively through the chainof evaluations (2.9). Also, as we explained in the introduction, we are ultimately interested inextracting the large tension limit and want to bypass the cumbersome procedure of first obtainingexact expressions at finite λ , and then expand them at λ " . For instance, in the above genus-two contribution both Bessel functions give a similar leading asymptotic contribution due to theexpansion I n p? λ q “ ? π λ ´ { ˆ ` n ´
18 1 ? λ ` ¨ ¨ ¨ ˙ e ? λ ` ¨ ¨ ¨ , (2.16)and it would be desirable to pin the total contribution in a more direct way. To this aim, one needsto study (2.8) working at strong coupling from the beginning and making more transparent theorigin of the dominant terms. The next section will be devoted to this problem. In this section, we discuss how to extract dominant terms from integrals like (2.12) by saddle pointevaluation. Although this is a fairly well known topic, we want to emphasize some specific technicalissues 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 I p σ q “ ¿ dz g p z q e ´ σ f p z q . (3.1)Suppose that f p z q has a critical point ¯ z where f p ¯ z q “ . Deforming the contour such that itpasses through ¯ z with constant Im f p z q along the contour locally around ¯ z , we write ( ¯ f “ f p ¯ z q , ¯ f “ f p ¯ z q , z p q “ ¯ z ) I p σ q “ e ´ σ ¯ f ż dt dzdt g p z p t qq e ´ σ ¯ f t `¨¨¨ . (3.2)If g p ¯ z q is finite, we simply extract it from the integral and perform the Gaussian integral. In thefollowing, we shall be interested in the case when g has an odd zero or an even pole around thesaddle point. In the case of a zero with g t Ñ “ A t m ´ ` B t m ` ¨ ¨ ¨ , (3.3)we just include it in the Gaussian integration and get I p σ q “ e ´ σ ¯ f ż dt r At m ´ z p q ` p Bz p q ` Az p qq t m ` ¨ ¨ ¨ s e ´ σ ¯ f t `¨¨¨ “ ? π r B z p q ` A z p qs e ´ σ ¯ f p m ´ q !! p σ ¯ f q ´ m ´ ` ¨ ¨ ¨ . (3.4)In the case of a pole with g t Ñ “ A t ´ m ` ¨ ¨ ¨ , (3.5)7e compute the finite quantity d m dσ m „ e σ ¯ f I p σ q “ ˆ ´ ¯ f ˙ m ż dt dzdt g p z p t qq t m e ´ σ ¯ f t `¨¨¨ “ ˆ ´ ¯ f ˙ m z p q A d πσ ¯ f ` ¨ ¨ ¨ . (3.6)Integrating back in σ gives then I p σ q “ π A z p q e ´ σ ¯ f p´ q m Γ p m ` q ˆ σ ¯ f ˙ m ´ ` ¨ ¨ ¨ . “ ? π A z p q e ´ σ ¯ f p´ q m p σ ¯ f q m ´ p m ´ q !! ` ¨ ¨ ¨ . (3.7) Revisiting x W y at strong coupling These formulas may be applied to contour integrals in-volving Wilson loops and higher order resolvents. Let us illustrate this once again in the case ofthe simple Wilson loop (1.1). The planar contribution in (2.12) has σ “ ? λ , f p z q “ ´ p z ` { z q and g p z q “ z ` ´ z ˘ . The dominant contribution at large λ comes from the saddle point at z “ which is a zero of g p z q of linear order. The parametrization is z p t q “ e it thus ¯ f “ . Expanding g p z q around the zero and taking the first even term gives (3.3) with A “ i and B “ and m “ .Evaluation of (3.4) gives then x W y “ c π λ ´ { e ? λ ` ¨ ¨ ¨ , (3.8)in agreement with (1.3). All the higher genus corrections have even poles at z “ ˘ . Again, theleading contribution comes from z “ and may be computed using (3.7). For instance, at genusone we have g p z q “ πi z p z ´ q , g p z p t qq “ ´ i π t ` ¨ ¨ ¨ Ñ A “ ´ i π , m “ , (3.9)and x W y “ ? π ´ i π i e ? λ p? λ q ´ ` ¨ ¨ ¨ “ λ { ? π e ? λ ` ¨ ¨ ¨ . (3.10)Similarly at genus 2 and higher we can check that this procedure reproduces the expansion (1.3).Higher order corrections in {? λ may also be computed in the same way just by doing Gaussianintegration with more accuracy. For instance, we know that (up to exponentially suppressed terms) x W y “ λ I p? λ q “ e ? λ ˆ λ { ? π ´ λ { ? π ` ¨ ¨ ¨ ˙ (3.11)and we reproduce this expansion by the convenient change of parametrization z ` z “ ´ u . (3.12) This is equivalent to an implicit integration by parts. In both cases we have to be careful about the poles at t “ since a non-zero residue for the pole causes a discontinuity in the contour. In our discussion, this will notmatter because topological recursion ensures that this residue is always zero, when computing expectation values offunctions of the matrix model variable. See last section for examples and Appendix C for general details. z “ e it , this gives u “ t and one gets x W y “ e ? λ π ż du u p ´ u q { e ´ ? λ u “ e ? λ π ż du ˆ u ` u ` ¨ ¨ ¨ ˙ e ´ ? λ u (3.13) “ e ? λ π π „
132 1Γ p ` q ˜ ? λ ¸ ´ { ´ p ` q ˜ ? λ ¸ ´ { ` ¨ ¨ ¨ “ λ { ? π ´ λ { ? π ` ¨ ¨ ¨ , 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.
We now look for a simplification of topological recursion (2.8) based on considering the principalpart of resolvents at z i “ , i.e. the terms that dominate at strong coupling. Let us denote thehighest pole part by ˆ ω n,g . Introducing ∆ i “ z i ´ , the resolvents in (2.10) reduce to the compactexpressions ˆ ω , p ∆ q “
116 ∆ , ˆ ω , p ∆ , ∆ , ∆ q “ ´
12 ∆ ∆ ∆ , ˆ ω , p ∆ , ∆ q “ ` ∆ ` ∆ , ˆ ω , p ∆ q “ ´ . (4.1)The (total) degree of the pole terms is p g ´ q ` n . In general, only even powers of ∆ i appear. Ifsuch an Ansatz is plugged into the topological recursion, one can compute the associated resolventand project onto the maximal pole part. For instance, the last four resolvents in (2.9) become,after projection, ˆ ω , p ∆ , ∆ , ∆ , ∆ q “ ´ p ∆ ∆ ∆ ` ∆ ∆ ∆ ` ∆ ∆ ∆ ` ∆ ∆ ∆ q ∆ ∆ ∆ , ˆ ω , p ∆ , ∆ , ∆ q “ ∆ ∆ „ ∆ ` ∆ ∆ p ∆ ` ∆ q` p ` ∆ ` ∆ ` q ` p ∆ ` ∆ ` ∆ q , ˆ ω , p ∆ , ∆ q “ ´ p ` ∆ ` ∆ ` ∆ ` ∆ ` q ∆ , ˆ ω , p ∆ q “ , (4.2)9hich are very compact expressions, compared with the full resolvents. Being symmetric functions,we can further simplify in terms of elementary symmetric polynomials e k p x , . . . , x n q “ ÿ ď i ă i 㨨¨ă i k ď n x i ¨ ¨ ¨ x i k , (4.3)where x “ . One finds indeed the concise expressions ˆ ω , “ ´ e e , ˆ ω , “ e e ´ e e e ` e , ˆ ω , “ ´ e e ` e e ´ e e . (4.4)Further results are collected in Appendix C.3. Remark:
Of course, the key point of the method is to use ˆ ω projected resolvent in the topologicalrecursion and never using the full ω ’s. As a first application, we consider the large tension limit of x W 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 “ . Later, we shall extend the analysis to a generic number n of coinciding loops. For n “ , the { N expansion of @ W D has been considered in [35, 36, 31, 2] and its first terms read N @ W D “ ˆ ? λ I ˙ ` ? λ N ˆ I I ` I I ˙ ` N „ λ I ´ ? λ p ` λ q I I ` ` λ ` λ I ` ¨ ¨ ¨ , (5.1)where I n ” I n p? λ q . The associated connected correlator is @ W D c “ ÿ g “ N g @ W D c ,g “ ? λ I I ` N „ λ I ´ λ { I I ` λ p ` λ q I ` ¨ ¨ ¨ . (5.2)Expanding at large λ and keeping the leading contribution at each order in { N gives @ W D c , “ π e ? λ ˆ ´
14 1 ? λ ` ¨ ¨ ¨ ˙ , @ W D c , “ λ { π e ? λ ˆ ´ ? λ ` ¨ ¨ ¨ ˙ . (5.3)Let us show how these contributions can be easily recovered from the “maximal poles” topologicalrecursion. We start from the 2-point formula @ W D λ " ,g “ p πi q ¿ ˆ ω ,g p z , z q e ? λ p z ` { z q e ? λ p z ` { z q . (5.4)10he genus 0 contribution is special being related to the universal Bargmann kernel and having nopoles at z , “ . It is @ W D c , “ ¿ dz πi dz πi p z ´ z q e ? λ p z ` { z q e ? λ p z ` { z q “ ¿ dz πi dz πi ÿ n “ n z ´ ´ n w n ´ ÿ p “´8 z p I p p? λ q ÿ q “´8 w q I q p? λ q“ ÿ n “ n I n “ ? λ ÿ n “ p I n ´ ´ I n ` q I n “ ? λ I I , (5.5)where in the last line we used the basic recursion of (modified) Bessel functions and the fact thatthe infinite sum is telescoping. Starting at genus 1 we can apply the formula (3.7) for the factorizedpoles. For instance, the first correction is @ W D λ " , “ ¿ dz πi dz πi „ p z ´ q p z ´ q ` p z ´ q p z ´ q ` p z ´ q p z ´ q e ? λ p z ` { z q e ? λ p z ` { z q p . q Ñ e ? λ „ p h h ` h h q ` h , (5.6)where the numerical constants h m are h m “ p´ q m ? π λ m ´ p m ´ q !! . (5.7)Replacing (5.7) in (5.6) reproduces the leading term in the second expression in (5.3). Extension to x W n y and high order calculation Similarly to (5.6), we can exploit the resol-vents in (4.1) and (4.2) (together with other ones in Appendix C) to evaluate the saddle pointintegrals needed to compute x W n y at high order in the genus expansion. Remarkably, this can bedone for a generic n . To this aim, we introduce the variable ξ “ λ N “ πg s T , (5.8)and the connected correlators @ W D c “ ´ x W y ` @ W D , @ W D c “ x W y ´ x W y @ W D ` @ W D , @ W D c “ ´ x W y ` x W y @ W D ´ @ W D ´ x W y @ W D ` @ W D , etc. . (5.9)Normalizing by suitable powers of the simple Wilson loop, we obtain the following results valid upto order O p ξ q : @ W D c x W y T " “ ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ , @ W D c x W y T " “ ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ , W D c x W y T " “ ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ , @ W D c x W y T " “ ξ ` ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ , @ W D c x W y T " “ ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ , @ W D c x W y T " “ ξ ` ξ ` ξ ` ¨ ¨ ¨ , @ W D c x W y T " “ ξ ` ξ ` ¨ ¨ ¨ , @ W D c x W y T " “ ξ ` ¨ ¨ ¨ . (5.10)From connected correlators we obtain correlators of n coincident Wilson loops using the combina-torial formula x W n yx W y n “ ` ÿ k ă n ÿ π P P p k,n q n ! S p π qp n ´ k ´ | π |q ! ś p P π p p ` q ! ź p P π @ W p ` D c x W y p ` , (5.11)where P p k, n q is the set of integer partitions π of k satisfying k ` | π | ď n where | π | is the numberof elements of π , and S p π q is the symmetry factor of partition π given by products of m ! for eachgroup of m equal elements in π . This expression follows from the fact that x W n y can be writtenas a sum over just the partitions of n . Since we divide by x W y n , all the parts of a given partitionthat are disappear, leaving the partitions of n with every part at least . Such partitions can beseen 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 x W m y c x W y m “ O p ξ m ´ q , to obtain the expansion of x W n yx W y n up to ξ m we canrestrict the sum in (5.11) to k ă m . Furthermore, taking into account that a part p enters theabove expression as x W p ` y c x W y p ` , we need the terms corresponding to all the partitions of m to obtainthe result up to ξ m . Let’s consider a couple of examples. For m “ the only possible partition is and we obtain x W n yx W y n “ ` n ! p n ´ q !2! @ W D c x W y ` O p ξ q “ ` n p n ´ q @ W D c x W y ` O p ξ q . (5.12)For m “ we have three partitions, i.e. and and p , q . The last one has a symmetry factor oftwo. So we obtain x W n yx W y n “ ` n ! p n ´ q !2! @ W D c x W y ` n ! p n ´ q !3! @ W D c x W y ` n ! p n ´ q !2!2!2! ˜ @ W D c x W y ¸ ` O p ξ q (5.13) “ ` n p n ´ q @ W D c x W y ` n p n ´ qp n ´ q @ W D c x W y ` n p n ´ qp n ´ qp n ´ q ˜ @ W D c x W y ¸ ` O p ξ q . In a similar way, to obtain the result up to ξ we will have to add all terms corresponding tothe partitions of three to the above results and so on. We now have all the ingredients needed to12valuate the above expression to ξ . The final result is, cf. (1.5) W n p ξ q “ ` p n ´ q n ξ ` n ` n ´ n ´ n ` ˘ ξ ` n ` n ` n ´ n ´ n ` n ´ ˘ ξ ` n ` n ` n ´ n ´ n ´ n ` n ´ n ` ˘ ξ ` n ` n ` n ` n ´ n ´ n ´ n ` n ` n ` n ´ ˘ ξ ` n ` n ` n ` n ` n ´ n ´ n ` n ` n ´ n ` n ´ n ` ˘ ξ ` n ` n ` n ` n ` n ´ n ´ n ´ n ` n ` n ´ n ` n ´ n ` n ´ ˘ ξ ` n ` n ` n ` n ` n ` n ´ n ´ n ´ n ` n ` n ´ n ` n ´ n ` n ´ n ` ˘ ξ ` ¨ ¨ ¨ , (5.14) where the large tension limit is understood. This is the extension to order ξ of the cubic resultin Eq. (1.17) of [2]. The special cases n “ and n “ are W p ξ q “ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ , W p ξ q “ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ , (5.15)and agree with the exact expressions [2], W p ξ q “ ` e ξ c π ξ erf ´c ξ ¯ , W p ξ q “ ` e ξ c π ξ erf ´c ξ ¯ ` π ? ξ e ξ ” ´ T ´a ξ, ? ¯ı , (5.16)where T p h, a q “ π ż a dx ` x e ´ h p ` x q , (5.17)is the Owen T-function. The coefficient of ξ k is a polynomial in n of degree k . A remarkablesimplification is achieved by writing (5.14) in exponential form log W n p ξ q “ ÿ k “ n p n ´ q P k p n q ξ k , (5.18)since P k turns out to be a polynomial of (approximately half) degree k ´ . Explicitly, one finds P “ , P “ ´ ` n , P “ ´ n ` n , “ ´ ` n ´ n ` n ,P “ ´ n ` n ´ n ` n ,P “ ´ ` n ´ n ` n ´ n ` n ,P “ ´ n ` n ´ n ` n ´ n ` n ,P “ ´ ` n ´ n ` n ´ n ` n ´ n ` n , (5.19)with leading terms at large n following the pattern P k “ k ´ k ? π Γ p k ´ q Γ p k ` q n k ´ ` k ´ k „ ´ p k ´ qp k ´ q Γ p k ´ q? π Γ p k ` q n k ´ ` ¨ ¨ ¨ . (5.20) The genus expansion of (1.3) is efficiently computed by exploiting the Toda integrability of the1-matrix Hermitian Gaussian model [23]. In general, correlators in this model are constrained byintegrable differential equations [20, 21, 22] that in Gaussian case take the Toda form [19]. Noticethat in [23] the matrix model measure is exp p´ tr Ă M q without explicit N factor. This will be theconvention throughout this section. After defining the connected correlators e p x , . . . , x k q “ A tr e x Ă M ¨ ¨ ¨ tr e x k Ă M E c , (5.21)one has e N ` p x q ` e N ´ p x q “ e N p x q ` x N e N p x q , (5.22) e N ` p x, y q ` e N ´ p x, y q “ e N p x, y q ` p x ` y q N e N p x, y q ´ x y N e N p x q e N p y q . (5.23)The general structure is e N ` p x , . . . , x k q ` e N ´ p x , . . . , x k q ´ „ ` N p x ` ¨ ¨ ¨ ` x k q e N p x , . . . , x k q “ g N p x , . . . , x k q , (5.24)where g N may be read from the non-leading terms of the cumulant expansion of x X ¨ ¨ ¨ X k y c andreplacing @ X I ¨ ¨ ¨ X I p D Ñ p x I ` ¨ ¨ ¨ ` x I p q N e N p x I , . . . , x I p q . (5.25)Here we are interested in the specialization to x i “ b λ N . Hence, defining e p k q p λ, N q “ e N ¨˚˚˝c λ N , . . . , c λ N loooooooooomoooooooooon k terms ˛‹‹‚ , (5.26)14e have the equations e p k q ˆ N ` N λ, N ` ˙ ` e p k q ˆ N ´ N λ, N ´ ˙ ´ ˆ ` k λ N ˙ e p k q p λ, N q “ g p k q p λ, N q , (5.27)where g p k q p λ, N q is obtained from the non-leading terms of the cumulant expansion of @ X k D c andreplacing x X p y Ñ p λ N e p p q p λ, N q . (5.28)The explicit coefficients of the cumulant expansion of x X p y c may be expressed in terms of integerpartitions π “ p m m ¨ ¨ ¨ q of p x X p y c “ ÿ π P P p p q p´ q | π |´ p| π | ´ q ! σ p π q ź r x X r y m r , σ p π q “ p ! ś r p r ! q m r m r ! . (5.29)Hence, the equations are e p k q ˆ N ` N λ, N ` ˙ ` e p k q ˆ N ´ N λ, N ´ ˙ ´ ˆ ` k λ N ˙ e p k q p λ, N q“ ÿ π P P p p q π ‰p p q p´ q | π |´ p| π | ´ q ! σ p π q ź r ˆ r λ N e p r q p λ, N q ˙ m r . (5.30)The large tension scaling Ansatz is e p k q p λ, N q “ e k ? λ F k p ξ q , ξ “ λ { N . (5.31)Replacing in the Toda equations gives e k ? λ „ k ξ { p k ξ ´ q F k p ξ q ´ k ξ { F k p ξ q N ´ { ` ¨ ¨ ¨“ ÿ π P P p p q π ‰p p q p´ q | π |´ p| π | ´ q ! σ p π q ź r ´ r ξ { N ´ { e r ? λ F r p ξ q ¯ m r . (5.32)The only partitions that may give a contribution have | π | “ ř m r “ . One case is when k iseven and then the partition is π “ p M , M q , or when k is split into the sum of two different parts π “ p q, p k ´ q qq with q ‰ k { . Denoting by an apex such partitions, we have (using ř r rm r “ k ) k ξ p k ξ ´ q F k p ξ q ´ k F k p ξ q “ ÿ π P P p p q π ‰p p q p´ q | π |´ p| π | ´ q ! σ p π q ź r ` r F r p ξ q ˘ m r . (5.33)Finally, evaluating the r.h.s. for the two relevant kinds of partitions, we obtain the differentialequation k ξ p k ξ ´ q F k p ξ q ´ k F k p ξ q “ ´ k ´ ÿ p “ ˆ kp ˙ p p k ´ p q F p p ξ q F k ´ p p ξ q , (5.34) Notice that distinct partitions are ordered. F k p ξ q ` ξ p ´ k ξ q F k p ξ q “ k k ´ ÿ p “ ˆ kp ˙ p p k ´ p q F p p ξ q F k ´ p p ξ q . (5.35)The first instance k “ gives F p ξ q ` ξ p ´ ξ q F p ξ q “ Ñ F p ξ q “ C ξ ´ { e ξ . (5.36)The constant is fixed by (1.3) and gives F p ξ q “ ? π ξ ´ { e ξ . (5.37)We shall be interested in the ratios R k p ξ q “ F k p ξ q F p ξ q k . (5.38)They obey R k p ξ q ´ k ´ ξ „ ` k p k ` q ξ R k p ξ q “ k k ´ ÿ p “ ˆ kp ˙ p p k ´ p q R p p ξ q R k ´ p p ξ q . (5.39)We also know that R k p ξ q “ O p ξ k ´ q . This gives the integration constant and the explicit recurrencerelation R p ξ q “ ,R k p ξ q “ e k p k ´ q ξ ξ k ´ ż ξ dz e ´ k p k ´ q z z ´ k k k ´ ÿ p “ ˆ kp ˙ p p k ´ p q R p p z q R k ´ p p z q . (5.40)This recursion provides the expressions in (5.10) to be plugged into (5.11) in order to compute thescaling functions W n p ξ q . Just to give an example, using (5.40) one may easily extend the last linein (5.10) and find @ W D c x W y T " “ ξ ` ξ ` ξ ` ξ (5.41) ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ξ ` ¨ ¨ ¨ . This allows to compute the polynomials in (5.19) at higher order. For instance P “ ´ n ` n ´ n ` n ´ n ` n ´ n ` n , “ ´ ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n , (5.42)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 R p ξ q “ c πξ e ξ { erf ˜c ξ ¸ ,R p ξ q “ ´ π ? ξ e ξ „ ´ ` T ˆa ξ, ? ˙ , (5.43)where T is the Owen function, cf. (5.16). The expression for R may be obtained by continuingthe 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 insertionsof local chiral operators, see the discussion in Appendix A. In this case, one has to keep usingtopological recursion, as discussed in the next Section.
In this section, we address the problem of computing the correlator between multiple coincidentWilson loops W n and a single trace chiral operator. In other words, we want to generalize (1.7)and prove (1.11), (1.12). To properly define chiral primaries let us recall that the -BPS Wilsonloop, associated with tr ` e ? λ M ˘ in the Gaussian matrix model, cf. (1.1), stands for the operator W “ tr P exp " g YM ż C dσ r i A µ p x q x µ p σ q ` R Φ p x qs * , (6.1)where C is a circle of radius R (set to unity in the following), and Φ is one of the six real scalars t Φ I u I “ ,..., in N “ SYM. Single trace chiral operators take the general form O J “ tr p u I Φ I p x qq J where u I is a complex null 6-vector obeying u “ [14]. The dependence of the correlator x W O J y on u I and the choice of coupling between the loop and the scalars factorizes and will be absorbed in theoperator normalization [37]. With the same conventions as in [2], the matrix model representativefor the chiral operator O J is O J “ N ˆ π ˙ J { ´ : tr M J : , (6.2)17here normal ordering subtracts self-contractions and is necessary to map matrix model correlatorsto R quantum expectation values [38, 39]. At leading order in large tension, the correlatorbetween a single Wilson loop and the chiral operator O J obeys (1.7) in terms of a scaling functionthat depends on the specific ratio g s { T and has a non-trivial dependence on J . The most naturalscaling dependence is actually on g s { T as in (1.5). Several cancellations occur and are responsiblefor the relevant variable being g s { T . We shall show that this pattern changes in the case of thecorrelator between multiple coincident Wilson loops and one chiral operator. The above mentionedcancellations 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 willconveniently 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 p n, J q “ p , q in Appendix A. As a preliminary step we first address the issue of the effects of normal ordering in (6.2) and therole of multi-trace operators. It is instructive to look at the first cases at low J . A straightforwardexplicit calculation gives (we restrict to even J for the purpose of illustration) : tr M : “ tr M ´ N , : tr M : “ tr M ´ tr M ´ N p tr M q ` N ` N , : tr M : “ tr M ´ tr M ` ˆ ` N ˙ tr M ` N ˆ ´ p tr M q ´ tr M tr M ` p tr M q ˙ ´ N ´ N , (6.3)and so on. In general, terms involving products of k traces are „ { N k ´ at large N . We will write : tr M J : “ ÿ k ě N k ´ “ : tr M J : ‰ k . (6.4)where the operators in “ : tr M J : ‰ k have coefficients O p q at large N . Now, let us consider thegenus g contribution to the connected correlator @ W n O p k q D c where O p k q is any arbitrary k -traceoperator. We can write, cf. (2.5), A W n O p k q E c ˇˇˇ genus g (6.5) “ N g ` k ` n ´ p πi q k ` n ¿ ω n,g p z , . . . , z n ` k q O p k q p x p z q , . . . , x p z k qq exp ˜ λ n ÿ l “ x p z k ` l q ¸ . In the case of O p k q “ r : tr M J : s k , taking into account the extra factor N k ´ in (6.4), we find that @ W n r : tr M j : s k D c | genus g scales as N ´p g ` k ` n ´ q . Finally, let us pin the dependence on λ " . The choice of normalization in (6.2), and in particular the overall power of N , is dictated by string theory andmakes direct contact with the associated natural vertex operators [1]. Another standard choice is to require a fixednormalization of the chiral operators 2-point functions as in [14]. The k -trace part may have an explicit N dependence as in “ : tr M : ‰ which has a piece p ` { N q tr M whose N Ñ 8 limit is finite. : tr M J : does not depend on λ explicitly, the strong coupling limit of theexpectation value of Wilson loop with chiral operators corresponds to maximizing the order ofpoles of variables corresponding to Wilson loop or conversely minimizing the order of the polesof the variables that correspond to the : tr M J : operator. The total order of the poles of ω n,g is p g ´ q ` n , as discussed in section 4. According to the saddle point analysis this implies thefinal scaling behaviour @ W n r : tr M J : s k D c ˇˇ genus g „ λ p g ` n ` k ´ q N g ` k ` n ´ . (6.6)This gives the leading power at large λ for all genera. In particular, a term with an overall N P factor will be accompanied by the following powers of λλ ´ k ˆ λ N ˙ P , (6.7)showing that multiple trace contributions are suppressed. Besides, since the saddle point expansionhas relative corrections in powers λ ´ { „ { T , double trace corrections to normal ordering cannotbe 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 doubletrace operators . At the planar level, as is well known, the double trace part doesn’t contribute.At { N level there are two relevant cumulants corresponding to p k, g q “ p , q and p , q . Their λ dependence can be obtained using the explicit strong coupling resolvents given (4.2) as respectively, N ¿ z exp ˜ ? λ x p z q ¸ ˆ ω , p z , z q „ λ N , N ¿ z exp ˜ ? λ x p z q ¸ ˆ ω , p z , z , z q „ λ N . (6.8)The first contribution is dominant in the large tension limit and in fact we would have to expandit to three orders in λ before the the second one becomes effective. Since, the rate of growth of theexponent of λ is for g but only for k , as g and k increase or as cumulants are multiplied, thegap between the contributions of single and higher trace operators only increases. x W n O J y at leading order As a result of the above discussion, we can restrict ourselves to the single trace part of normalordering, i.e. the planar approximation. According to [40, 41], it may be written in terms ofChebyshev polynomials and, in the z variable, it reads : tr M J : Ñ z J ` z J ` δ j, ` . . . . (6.9)The relevant connected correlators @ W n : tr M J : D c,g are x W n O y c,g “ p πi q n ` ¿ ω n,g p z , . . . , z n ` q z ´ J exp « ? λ p x p z q ` ¨ ¨ ¨ ` x p z n ` qq ff . (6.10) This is of course contingent on these first two contributions from single trace operators not vanishing once wetake residue integrals corresponding to the chiral operator. As we now show this is indeed the case. This remarkwill be important later. We deform the z integration shrinking it around z “ . This is possible because residues vanish at z i “ ˘ and, in particular, we can drop the parts that are not singular at z “ .
19n the strong coupling limit we use the strong coupling resolvent ˆ ω n,g p z q and keep in it only thoseterms that minimize the order of the poles of z at . This can be done by going through one stepof topological recursion. This corresponds to starting with ω g,n ´ p z , . . . z n q and using: n ` ź i “ p z i ´ q k i Ñ n ÿ j “ k j ` p z ´ q p z j ´ q n ` ź i “ p z i ´ q k i ` ¨ ¨ ¨ . (6.11)We can now integrate over z , ¨ ¨ ¨ , z n in the saddle point approximation. The above factor of k j ` ensures that the result has a very simple relation to x W n y c,g in the strong coupling limit, i.e. @ W n : tr M J : D λ " c,g “ nλ x W n y λ " c,g Res z “ dz p z ´ q z J ` ¨ ¨ ¨ “ J n ? λ x W n y λ " 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 x W n y λ " “ J n ? λ ` ¨ ¨ ¨ . (6.13)This is of course expected from the known results for n “ and n “ , see [2]. To go beyond leading order we need to carry out topological recursion with poles of one subleadingorder included. It is convenient to change variables from z to u , cf. (3.12), and write ω n,g p u , . . . , u n q “ ˆ ω n,g ` δω n,g ` . . . . (6.14)Where δω n,g includes the poles of total degree g ` n ´ , see Appendix C for full details of theprocedure. Using (6.14) we can compute the one-variable resolvents obtained after integration ofall but one variable, ¯ ω n,g p z q “ p πi q n ¿ ω n ` ,g p u p z q , u , . . . u n q . (6.15)Due to our previous discussion, cf. (6.6), the first two orders in the {? λ expansion at large λ canbe computed by ignoring mixing with multi-trace operators and using the simple correspondencein (6.9). Thus, we simply obtain @ W n : tr M J : D c,g “ Res z Ñ ¯ ω n,g p z q z J . (6.16)This computes the connected part of the correlator but we can also define a function that similarlycomputes the full correlator, i.e. ¯Ω n,g p z q “ n ÿ k “ g ÿ h “ ˆ nk ˙ A W n ´ k E g ´ h ¯ ω k,h p z q Ñ @ W n : tr M J : D “ Res z Ñ ÿ g “ N g ´ ¯Ω n,g p z q z J . (6.17) Let us remind that the presence of dz in the residue is formal at this level and could be omitted. Nevertheless,it is convenient to keep it to emphasize transformation properties under change of variables. We change back to z -coordinates for the free variable because this is convenient to compute expectation valueswith a chiral operator.
20o compute @ W n : tr M J : D { x W n y it is convenient to expand ¯Ω n,g p z q as: ¯Ω n,g p z q “ U n, p z q x W n y c,g ` U n, p z q x W n y c,g ´ ` ¨ ¨ ¨ ` U n,g p z q x W n y c, ` . . . , (6.18)where each U n,g p z q is determined recursively genus by genus and final dots stand for a correctionof order O pp {? λ q g ` q relative to the leading order. Then it can be seen that, @ : tr M J : W n D x W n y “ Res z Ñ ÿ g “ N g ´ U n,g p z q z J . (6.19)The functions U n,g p z q depend also on λ . To the leading order in λ , U n, p z q can be read from(6.12). To get a non-vanishing result for all other U n,g we need to go beyond ˆ ω 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: U n, p z q “ dz ˜ n ? λ p z ´ q ` f n, p z qp z ´ q ¸ ` . . . ,U n,g p z q “ dzf n,g p z q λ g p z ´ q ` . . . . (6.20)Where f n,g p z q are polynomials of degree at most and independent of λ . Two out of the 4 freecoefficients are determined by the requirement from topological recursion that U n,g ` z ˘ “ ´ U n,g p z q .Another one can be fixed by requiring that x : tr M : W n y c,g vanishes for g ą . . Combiningthese two requirements we obtain U n,g “ dz c n,g λ g z p z ´ q . (6.21) Explicit results
After having clarified the general structure of topological recursion for thequantities we need, let us present explicit results. For the ’critical’ case n “ we find U , p z q “ dz ˜ ? λ p z ´ q ` z p z ´ q ¸ ` . . . ,U , p z q “ dz λ { z p z ´ q ` . . . , (6.22)while the higher U ,g p z q vanish i.e c ,g “ for g ą . This can be seen as consistency check and isa result of the cancellations required to reorganize the series for x W : O J : y as in (1.7). To calculatenon-vanishing terms in U ,g p z q for g ą we will need to keep more than leading terms in ω n,g .These peculiar cancellations do not occur for n ą and make the calculation of subleadingcorrections possible with our level of accuracy. We find ÿ g N g ´ U ,g p z q “ N ˆ ? λ p z ´ q ´ z p z ´ q ` z p z ´ q ” λ { N ´ λ N ` λ { N , This may be shown by explicit splitting of U p N q into U p q ˆ SU p N q , see Appendix D. Recall that we expect a major change of features when moving from n “ to n ą . λ N ` λ { N ´ λ N ı˙ ` . . . ÿ g N g ´ U ,g p z q “ N ˆ ? λ p z ´ q ´ z p z ´ q ` z p z ´ q ” λ { N ` λ N , ´ λ { N ` λ N ´ λ { N ı˙ ` . . . , ÿ g N g ´ U ,g p z q “ N ˆ ? λ p z ´ q ´ z p z ´ q ` z p z ´ q ” λ { N ` λ N ´ λ { N ´ λ N ı˙ ` . . . , ÿ g N g ´ U ,g p z q “ N ˆ ? λ p z ´ q ´ z p z ´ q ` z p z ´ q ” λ { N ` λ N ` λ { N ı˙ ` . . . . (6.23)As a result of this, the dependence on J in x W n : tr M j : y x W n y is much simpler for n ą than in the n “ case, cf. (1.7). Indeed, from the above, it has to be proportional to Res z “ dz p z ´ q z J ´ “ J ` J ´ ˘ . (6.24)This means in that the structure of large tension limit of x W n O J yx W n y is given by (1.11). The first fewterms of H n p x q can be calculated from (6.23) and read H p x q “ x π ´ x π ` x π ´ x π ` x π ´ x π ` . . . , H p x q “ x π ` x π ´ x π ` x π ´ x π ` . . . , H p x q “ x π ` x π ´ x π ´ x π ` . . . , H p x q “ x π ` x π ` x π ` . . . . (6.25) 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 . Tothis aim we take J “ in the topological recursion result (1.11) and write x W n O yx W n y T " “ π n p T ` H n q . (6.26)The l.h.s. may be traded for a logarithmic derivative of x W n y due to the matrix model identity x W n O y “ λ ddλ x W n y . (6.27)Hence we have λ ddλ log x W n y T " “ π n p T ` H n q , (6.28)22nd a short calculation gives the relation H n p x q “ x π „ ` n ddx log W n p x q . (6.29)Replacing W n by its evaluation by means of (5.40) and using the series expansion (5.14), we get H n p x q “ ´ ` n π x ` p´ ` n qp´ ` n q π x ` p´ ` n qp ´ n ` n q π x ` p´ ` n qp´ ` n ´ n ` n q π x ` ¨ ¨ ¨ , (6.30)in agreement with (6.25). Of course, the exact determination of W n by Toda recursion means thatwe can provide easily all order expansion of the H n function by means of (6.29). Let us give some examples of (1.12) by explicit computations. For n “ we need the explicit exactexpansion N @ W D “ ” ? λ I ı ` ? λ N ” I I ` I I ı ` N ” λ I ´ ? λ p ` λ q I I ` ` λ ` λ I ı ` N ” ´ λ p ` λ q I ` ? λ p ` λ ` λ ` λ q I I ´ ` λ ` λ ` λ I ı ` O ´ N ¯ . (6.31)Using (6.27) we work out the case p n, J q “ p , q @ W O D x W y “ ? λ `¨ ¨ ¨` N „ λ { `¨ ¨ ¨ ` N „ ´ λ `¨ ¨ ¨ ` N „ λ { `¨ ¨ ¨ `¨ ¨ ¨ . (6.32)Comparing with (1.11) gives the first terms H p x q “ π ˆ x ´ x ` x ` ¨ ¨ ¨ ˙ , (6.33)in agreement with (6.30). In this case we can give the exact expression in a reasonable compactform using the first equation in (5.16) H p x q “ ` x ´ ` e x { ? πx erf `a x ˘ . (6.34)A similar calculation can be repeated for n “ . In this case we have N @ W D “ ” ? λ I ı ` N ´ I I ´ ? λ I ¯ ` N ” λ { I I ´ ` λ I I ` ` λ ` λ ? λ I ı N ” λ I ´ λ { p ` λ q I I ` ` λ ` λ ` λ I I ´ ` λ ` λ ` λ ? λ I ı ` ¨ ¨ ¨ . (6.35)This gives @ W O D x W y “ ? λ `¨ ¨ ¨` N „ λ { `¨ ¨ ¨ ` N „ λ `¨ ¨ ¨ ` N „ ´ λ { `¨ ¨ ¨ `¨ ¨ ¨ . (6.36)Comparing with (1.11) we obtain H p x q “ π ˆ x ` x ´ x ` ¨ ¨ ¨ ˙ , (6.37)in agreement with (1.12). As in (6.34), one can give a closed formula for this function in terms of thespecial error and Owen-T functions. As a final check, probing the peculiar simple J dependence in(1.11), we consider the case p n, J q “ p , q . To analyze this case by expansion of exact expressionsat finite λ we need the Bessel function expansion of @ W O D where O “ N a π : tr M : and : tr M : “ tr M ´ tr M . By matching a large number of weak coupling perturbative coefficients,we find @ W O D “ N c π " ´ I I λ ` p ` λ q I λ { ` N „ ´ ? λI ` p ` λ q I I ´ I ? λ ` N „ ? λ p ` λ ` λ q I ` p´ ´ λ ` λ q I I ` p ` λ ´ λ ` λ q I ? λ ` N „ ? λ p´ ´ λ ´ λ ` λ q I ` p ` λ ` λ ´ λ ` λ q I I ` p´ ´ λ ´ λ ` λ ` λ q I ? λ ` ¨ ¨ ¨ * . (6.38) This gives @ W O D x W y “ c π " ? λ `¨ ¨ ¨` N „ λ { `¨ ¨ ¨ ` N „ ´ λ `¨ ¨ ¨ ` N „ λ { `¨ ¨ ¨ `¨ ¨ ¨ * . (6.39)This expansion should be compared with the n “ J “ case of (1.12), i.e. ˆ π ˙ { p T ` H p x qq “ c π „ ˆ πT ` x ´ x ` x ` ¨ ¨ ¨ , (6.40)and indeed we find that this is equivalent to the previous expansion (6.33). Acknowledgments
MB and AH are supported by the INFN grant GSS (Gauge Theories, Strings and Supergravity).24
Toda recursion for correlators with chiral primaries
The genus expansion of the ratio x W O y { x W y may be computed by (6.27) in terms of x W y .Alternatively, it is equivalent to use the integral representation (1.10) derived in [17]. Here, wewant to show how such correlators may be treated by Toda recursion, as an illustration, generalizingthe treatment in App. B.3 of [2]. From e N p x, y q “ B tr e x b Nλ M tr e y b Nλ M F ´ B tr e x b Nλ M F B tr e y b Nλ M F , (A.1)we have B x e N ´ x, b λ N ¯ˇˇˇ x “ e N ´b λ N ¯ “ Nλ @ W : tr M : D x W y “ @ W : tr a : D x W y , (A.2)where M “ c λ N a „ tr e M e ´ Nλ tr M “ tr e b λ N a e ´ tr a , (A.3)to make contact with the expressions in [2]. The relevant Toda equation is (5.23). Taking twoderivatives involves the auxiliary quantity B x e N ´ x, b λ N ¯ˇˇˇ x “ e N ´b λ N ¯ “ c Nλ x W tr M yx W y “ ? x W tr a yx W y . (A.4)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 x W y “ N J ´ ? λ C J p ζ q , ζ “ g s T “ λ N . (A.5)The Toda recursion takes the form a N p N ´ q a ζ C ˆ NN ´ ζ ˙ e N ´ p? N ζ q e N p? N ζ q ` a N p N ` q a ζ C ˆ NN ` ζ ˙ e N ` p? N ζ q e N p? N ζ q´ N a ζ p ` ζ q C p ζ q ´ ? ζC p ζ q ` ζ “ (A.6)The expansion at large N with fixed ζ require to study the asymptotic behaviour of e N p? N µ q atfixed µ . Recall that e N p x q “ e x L N ´ p´ x q , e N p x q ` x e N p x q ´ p N ` x q e N p x q “ . (A.7)Setting x “ ? N µ and expanding the differential equation gives e N p? N µ q “ N ´ { exp „ N f p µ q ` f p µ q ` N f p µ q ` ¨ ¨ ¨ , (A.8)with f p µ q “ µ a µ ` ` arcsinh µ , f p µ q “ ´
32 log µ ´
14 log p µ ` q . (A.9)25his is enough to derive the relevant terms in the expansion e N ˘ p? N ζ q e N p? N ζ q “ e ˘ arcsinh ? ζ „ ` N ˘p ` ζ q ´ ? ζ ? ` ζ p ` ζ q ` O ˆ N ˙ ¨ ¨ ¨ . (A.10)Using this in the expansion of (A.6) gives C p ζ q ´ ζ C p ζ q ` ´ ` ? C p ζ q ζ ? ` ζ “ . (A.11)It is easy to check that C p ζ q ” ? , so that C p ζ q “ a ` ζ ` k a ζ, (A.12)where k is a constant that we set to zero by analyticity. The result agrees with [2], see Eqs.(2.34,2.35, 2.40) there. B The polynomial P k for k “ , . . . , The polynomials P k p n q have been defined in (5.18) and their expression for k up to 10 have beengiven in (5.19) and (5.42). The expressions for k “ , . . . , are given below. P “ n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` ,P “ n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ ,P “ n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` ,P “ n ´ n ` n ´ n ` n n ` n ´ n ` n ´ n ` n ´ n ` n ´ ,P “ n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ` n ´ n ` n ´ ,P “ n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ` n ´ n ` n ´ n ` n ´ n ` ,P “ n ´ n ` n ´ n ` n ´ n ` n ´ n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ , “ n ´ n ` n ´ n ` n ´ n ` n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` ,P “ n ´ n ` n ´ n ` n ´ n ` n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ ,P “ n ´ n ` n ´ n n ´ n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` n ´ n ` (B.1) C Some details about topological recursion at large tension
Here we summarize some details about topological recursion that are relevant to the strong couplinglimit of correlation functions studied in the main text. Our presentation will be for the Gaussianmatrix model although most of the statements have straightforward generalizations to a generalgenus 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. The two sheets are glued along the cut on which the eigenvalues condense in the large N limit. Thecoordinate z defined in (2.3) maps these two sheets to the Riemann sphere as shown in Fig. 1. Ageneric value of x has two preimages since x p z q “ x ` z ˘ , if | z | ‰ then for one of these preimages | z | ą and for other | z | ă . These are the two sheets which have been mapped to the exterior andinterior respectively of unit circle on the z -plane. Let’s now focus on the unit circle itself on whichwe write z “ exp p it q . Then x p z p t qq “ t . So as z goes from to π , x p z p t qq goes from to ´ .This is one copy of the cut while the other copy is corresponds to t going from π Ñ π „ . Thetwo copies of the cut are joined at z “ and z “ ´ which correspond to x “ and x “ ´ i.ethe end points of the cut. These are the only two values of x which have a single preimage. These This holds more generally in 1-cut cases, not necessarily Gaussian. The spectral curve has genus s when thelarge N limit is associated with s ` disconnected cuts. dx . Lastly, notice that although y is not a single valued functionof x , it is a single valued function of z . Note that the unit circle is also the contour for the saddlepoint approximation, the saddle point integral is actually done over a double copy of the cut. BA x -plane (two sheets) BA z -plane Figure 1: Illustration of the spectral curve.
Left: the two x -sheets connected by the red cut. Thepoints A and B are in different sheets. Right:
In the z -plane the circle is formed from two copiesof 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.The resolvents ω n,g p z , . . . , z n q are all meromorphic multi-differentials on the z -plane whichpoles only at the branch point z “ ˘ . One of results of the topological recursion is the antisym-metry property: ω n,g ˆ z , z , . . . , z n ˙ “ ´ ω n,g p z , . . . , z n q . (C.1)As a consequence, correlation functions of the polynomials formed from the trace of the matrix M don’t receive any contribution from the poles of the resolvents. This follows from the fact thatthese matrix observables map to polynomials f p x p z q , . . . , x p z n qq . And since x p z q “ x ` z ˘ , thesame is true of f too. Since z Ñ z leaves ˘ fixed, changing variables to z , we obtain Res z “˘ ω n,g p z , . . . , z n q f p x p z q , . . . , x p z n qq “ Res z “˘ ω n,g ˆ z , z , . . . , z n ˙ f ˆ x ˆ z ˙ , . . . , x p z n q ˙ “ ´ Res z “˘ ω n,g p z , . . . , z n q f p x p z q , . . . , x p z n qq “ . (C.2)As a result the sole contribution to the correlation function of f comes from its own poles, whichfor a polynomial of x i are at z p x i q “ (inside the contour) and z p x 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 p z , . . . , z n q f p x p z q , . . . , x p z n qq “ Res z i “ ω n,g p z , . . . , z n q f p x p z q , . . . , x p z n qq . (C.3)The same logic works for any holomorphic function of x among them the Wilson loop. As wehave seen in practice, the contour integral is more convenient for the strong coupling expansion ofWilson loops while the residue at is simpler for chiral operators. Nevertheless this vanishing ofresidues at ˘ ensures that there is no ambiguity in the saddle point prescription, since we cansmoothly deform the contour past the branch points, as illustrated in Fig. 2.30 -plane Figure 2: The contour integral for comput-ing the matrix correlator f p x q in z -plane (thebig circle) gets contribution only from thepole of f p x p z qq at z “ . The residue atpoles 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 approximationto subleading orders can be seen as a reparameterization of spectral curve as z p u q “ exp ´ i arcsin ´ u ¯¯ . (C.4)This change of variables maps the z -plane to a cylinder u “ φ ` ir where φ parameterizes a circle ofradius extending from ´ to while r a real line. In this manner u is the local complex coordinateon an infinite cylinder. This cylinder is compactified to a sphere by identifying the circle at u “ i with one point and the circle at u “ ´ i with another point. In the u -coordinate the branchpoint are mapped to and ´ „ . In the strong coupling limit the dominant contribution to theexpectation value of Wilson loops comes from u “ while the contour for saddle point integral isthe circle r “ .These coordinates also turns out to be somewhat simpler for carrying out topological recursion.Changing variables, the topological recursion formula (2.8) becomes ω n,g p u , u q “ Res v Ñ , ˘ K p u , v q ” ω g ´ ,n ` p v, ´ v, u q ` ÿ h ď g ÿ r Ă u ω h, | r | p v, r q ω g ´ h,n ´| r | p´ v, u { r q ı . (C.5)In terms of these variables the recursion Kernel K is K p u, v q “ ´ i du v ? ´ u p u ´ v q dv . (C.6)Apart from the factor ? u ´ which is independent of v and as a result gives an overall mul-tiplicative factor, the kernel is homogeneous in these coordinates if we only keep the residue at u “ . This makes it easier to separate out the contribution of different orders. Indeed defining ˆ K p u, v q “ ´ i ? ´ u K p u, v q , we see that Res v Ñ ˆ K p u, v q duv k “ ´ du u k ` , Res v Ñ ˆ K p u, v q dv p v ´ w q v k “ ´ du k ÿ i “ i ` w i ` u k ´ i ` . (C.7)31o ˆ K p u, v q uniformly increases the degree of the poles of differential it acts on by . This simplifi-cation in the recursion kernel is a trade off due to the fact that the starting point of the recursion ω , p u, v q is now more complicated being given by w , p u, v q “ du dv ? ´ u ? ´ v ` u ` v ´ u v ´ uv ? ´ u ? ´ v ˘ , (C.8)and, for the purposes of carrying out topological recursion, it will be expanded into a double powerseries easily. Another simplification is that in these coordinates the antisymmetry property (C.1)reads ω n,g p´ u , . . . , u n q “ ω n,g p u , . . . , u n q . (C.9)This means in particular that ω n,g p u , . . . , u n q “ i n du . . . du n f n,g ˆ u , . . . , u n ˙ , (C.10)for some symmetric polynomials f n,g . As a result the poles encountered in the saddle point integralsare always of even order. Finally, we observe that all ω g,n computed through the topologicalrecursion have poles of order at least at u “ and as a result for the first two orders of polesthat we need we can ignore the residues at ˘ in (C.5). C.3 Expressions for resolvents at leading and first subleading order of poles
Now we present some of the resolvents needed to compute various explicit expansions presented inthe main text (5.10, 6.22, 6.23). We do this by presenting f n,g ´ u , . . . , u n ¯ as defined above in(C.10). These are symmetric polynomials of their arguments u i and to keep the expression relativelycompact we present them in terms of elementary symmetric polynomials, cf. (4.3). Similarly to ourdecomposition of ω n,g “ ˆ ω n,g ` δω n,g we divide f n,g into leading ˆ f n,g and subleading δf n,g pieces. Leading order ˆ ω g,n n “ f , “ e , ˆ f , “ ´ e , ˆ f , “ e , ˆ f , “ ´ e , ˆ f , “ e , ˆ f , “ ´ e , f , “ e , ˆ f , “ ´ e . (C.11) n “ f , “ e e ´ e , ˆ f , “ ´ e e ` e e ´ e e , ˆ f , “ e e ´ e e ` e e ´ e e ` e , ˆ f , “ ´ e e ` e e ´ e e ` e e ´ e e ` e e , ˆ f , “ e e ´ e e ` e e ´ e e ` e e ´ e e ` e e ´ e , ˆ f , “ ´ e e ` e e ´ e e ` e e ´ e e ` e e ´ e e ` e e ´ e e , ˆ f , “ e e ´ e e ` e e ´ e e ` e e ´ e e ` e e ´ e e ` e e ´ e e ` e . (C.12) n “ f , “ ´ e , ˆ f , “ e e ´ e e e ` e , ˆ f , “ ´ e e ` e e e ´ e e ´ e e e ` e e e ` e e ´ e , ˆ f , “ e e ´ e e e ` e e ` e e e ´ e e e e e ´ e e e ` e e e ` e e e ´ e e e ` e ´ e e , ˆ f , “ ´ e e ` e e e ´ e e ´ e e e ` e e e ` e e e ´ e e ´ e e e ` e e e ´ e e e ` e e e ´ e e ` e e e ´ e e e ` e e e ´ e e e ` e e ´ e e ´ e , ˆ f , “ e e ´ e e e ` e e ´ e e e ` e e ´ e e e ` e e e ` e e e ´ e e e ` e e ´ e e e ` e e e ´ e e e ` e e e ´ e e e ` e e ` e e e ´ e e e ` e e e ´ e e e ` e e e ´ e e e ´ e e e ` e e ` e e e ` e ´ e e . (C.13) n “ f , “ ´ e e , ˆ f , “ e e ´ e e e ` e e e ` e e ´ e , ˆ f , “ ´ e e ` e e e ´ e e e ` e e ´ e e e ` e e e e ´ e e e ` e e e ´ e e e ` e e ´ e e e , ˆ f , “ e e ´ e e e ` e e e ` e e e ´ e e e e ` e e e ` e e e ´ e e e e e e e ´ e e e ` e e ` e e e ´ e e e e ` e e e e ` e e e ´ e e e e ` e e e ´ e e ´ e e ´ e e ´ e e ´ e e e ` e e . (C.14) n “ , , , , f , “ e e ´ e e , ˆ f , “ e e ´ e e e ` e e e ` e e e ´ e e e ` e ´ e e e , ˆ f , “ ´ e e ` e e e ´ e e , ˆ f , “ e e ´ e e e ` e e e ` e e e ´ e e e ´ e e e e ` e e e ´ e e ` e e ` e e e ´ e , ˆ f , “ ´ e e ` e e e ´ e e e ´ e e ` e e ˆ f , “ e e ´ e e e ` e e e ` e e e ´ e e e ´ e e e e ` e e e ´ e e e ` e e e ` e e e e ´ e e e ` e ` e e e ´ e e e ´ e e e , ˆ f , “ ´ e e ` e e e ´ e e e ´ e e e ` e e e ` e e e ´ e e , ˆ f , “ e e ´ e e e ` e e e ` e e e ´ e e e ´ e e e e ` e e e ´ e e e ` e e e ` e e e e ´ e e e ` e e e e ´ e e e e ´ e e e e ` e e e ` e e ´ e e e ` e e ´ e e e ` e e e ` e e e ´ e , ˆ f , “ ´ e e ` e e e ´ e e e ´ e e e ` e e e ` e e e e ´ e e e ` e e ´ e e ´ e e e ` e e . (C.15) First subleading order δω n,g “ δf , “ e ,δf , “ ´ e ,δf , “ e ,δf , “ ´ e ,δf , “ e . (C.16) n “ δf , “ e ` e e ,δf , “ ´ e ` e e ` e e ´ e ,δf , “ e ´ e e ´ e e ` e e ´ e e ,δf , “ ´ e ` e e ` e e ´ e e ` e e ´ e e ` e . (C.17) n “ δf , “ ´ e ,δf , “ e e ` e e ´ e e ` e e ,δf , “ ´ e e ´ e e ` e e ` e e e ´ e e ´ e e ` e e e ` e ` e e ,δf , “ e e ` e e ´ e e ´ e e e ` e e ` e e ` e e e ´ e e ´ e e e ` e e ´ e e e ` e e ` e e e ´ e e ´ e e . (C.18) n “ , δf , “ e ´ e e ,δf , “ e e ´ e e e ` e e ` e e e ` e e ´ e e , f , “ ´ e e ´ e e ` e e e ` e e e ´ e e ´ e e e , ´ e e e ` e e e ` e e ´ e e e ` e e e , ` e e e e ´ e e ` e e ` e e ´ e e ´ e e ,δf , “ ´ e e ` e e ` e e ,δf , “ e e ´ e e ´ e e e ` e e e ` e e e ` e e e ´ e e ´ e e e ´ e e e ´ e e ` e e . (C.19) D The correlation function x : tr M : 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 genuscorrections beyond the leading order. This can be easily proved by starting from the followingsplitting of M in the U p N q theory M “ ˜ M ` mN , ˜ M “ M ´ N tr M , m “ tr M , (D.1)where ˜ M is the traceless part. The matrix model partition function becomes Z “ ż dm ż d ˜ M δ p tr ˜ M q exp ˆ ´ N tr ˜ M ´ m ˙ . (D.2)For the Wilson loop operator, the splitting (D.1) impliestr exp ˜ ? λ M ¸ “ exp ˜ ? λ N m ¸ tr exp ˜ ? λ M ¸ . (D.3)As a result the expectation value of n coincident Wilson loops takes the form x W n y “ x W n y traceless ż dm exp ˆ ´ m ` nλ N m ˙ , x W n y traceless “ ż d ˜ M δ p tr ˜ M q „ tr exp ˆ λ M ˙ n exp ˆ ´ N tr ˜ M ˙ . (D.4)In the case of x m W n y , we obtain the same integral for ˜ M with an extra insertion of m in the m -integral. As a result, the “traceless” part x W n y traceless cancels and we obtain x m W n yx W n y “ ş dm m exp ´ ´ m ` n ? λ N m ¯ş dm exp ´ ´ m ` n ? λ N m ¯ “ n ? λ N . (D.5)This is just the leading order result obtained in (6.13) and specialized to J “ . The abovediscussion shows that it is in fact exact. 37 eferences [1] S. Giombi and A. A. 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