On the structure of non-planar strong coupling corrections to correlators of BPS Wilson loops and chiral primary operators

Starting with some known localization (matrix model) representations for correlators involving 1/2 BPS circular Wilson loop $\cal W$ in ${\cal N}=4$ SYM theory we work out their $1/N$ expansions in the limit of large 't Hooft coupling $\lambda$. Motivated by a possibility of eventual matching to higher genus corrections in dual string theory we follow arXiv:2007.08512 and express the result in terms of the string coupling $g_{\rm s} \sim g^2_{\rm YM} \sim \lambda/N$ and string tension $T\sim \sqrt \lambda$. Keeping only the leading in $1/T$ term at each order in $g_{\rm s} $ we observe that while the expansion of $\langle {\cal W} \rangle$ is a series in $g^2_{\rm s} /T$, the correlator of the Wilson loop with chiral primary operators ${\cal O}_J $ has expansion in powers of $g^2_{\rm s}/T^2$. Like in the case of $\langle {\cal W} \rangle$ where these leading terms are known to resum into an exponential of a"one-handle"contribution $\sim g^2_{\rm s} /T$, the leading strong coupling terms in $\langle {\cal W}\, {\cal O}_J \rangle$ sum up to a simple square root function of $g^2_{\rm s}/T^2$. Analogous expansions in powers of $g^2_{\rm s}/T$ are found for correlators of several coincident Wilson loops and they again have a simple resummed form. We also find similar expansions for correlators of coincident 1/2 BPS Wilson loops in the ABJM theory.

λ. Keeping only the leading in 1{T term at each order in g s we observe that while the expansion of xWy is a series in g 2 s {T , the correlator of the Wilson loop with chiral primary operators O J has expansion in powers of g 2 s {T 2 . Like in the case of xWy where these leading terms are known to resum into an exponential of a "one-handle" contribution " g 2 s {T , the leading strong coupling terms in xW O J y sum up to a simple square root function of g 2 s {T 2 . Analogous expansions in powers of g 2 s {T are found for correlators of several coincident Wilson loops and they again have a simple resummed form. We also find similar expansions for correlators of coincident 1/2 BPS Wilson loops in the ABJM theory.

Introduction and summary
An important direction is to extend checks of AdS/CFT correspondence to subleading orders in 1{N expansion on the gauge theory side or higher genus corrections on the dual string theory side. One of the simplest observables to consider is the expectation value xWy of 1 2 -BPS circular Wilson loop for which the exact in N expressions are available in both SU pN q N " 4 SYM [1,2,3,4] and U pN q kˆU pN q´k ABJM [5,6,7] theories. It was recently observed in [8] that the expressions for xWy expanded first in 1{N and then in large 't Hooft coupling λ have a universal form when written in terms of the corresponding string coupling g s and string tension T " R 2 2πα 1 defined as [9,10] SYM : , i.e. we get (1.3) with c 0 " 1 2π , c 1 " 1 24 , etc. The universal structure of (1.3) is a manifestation of the fact that the two gauge theories are expected to be dual to similar superstring theories in AdS 5ˆS 5 and AdS 4ˆC P 3 where xWy should be given by a string path integral over surfaces ending on a circle at the boundary of AdS. e 2π T in (1.3) is the semiclassical factor corresponding to an AdS 2 minimal surface [11,12,13]. The expansion is done first in small string coupling g s (i.e. large N for fixed T " ? λ) and then in 1{T at each order in g s . The power of string coupling is the Euler number χ " 1´2p of a disc with p " 0, 1, 2, ... handles.
A non-trivial feature of (1.3) is that the leading power of the inverse string tension 1{T at each order in g s is precisely´1 2 χ " p´1 2 , i.e. it is correlated with the power of g s . A string theory explanation of this fact was suggested in [8] by showing that dependence of the string partition function on the AdS n radius R (and thus on the string tension) is controlled by the Euler number of the surface.
Another remarkable fact about the SYM result (1.4) is that the leading large T terms in (1.3) exponentiate [2] (according to (1.4), the coefficients c p in (1.3) are given by c p " 1 2πp!`π 12˘p ) xWy " 1 2π ? T g s e 2π T`π 12 Surprisingly, in the ABJM theory the coefficient of the first subleading correction is the same π 12 as in the SYM case [7] xWy "´N 4π λ`π λ 6N`¨¨¨¯e π ?
2λ " 1 ? 2π ? T g s e 2π T " 1`π 12 (1.6) However, the coefficients of higher order terms (that can be found from [14]) turn out to be different than in the SYM case (1.4), i.e. here the exponentiation does not happen. 3 Instead, we will find (see Appendix E) that in the ABJM case the leading strong-coupling terms in (1.3) can be resummed as where a π 2 gs ? T " 2π λ N " 2π k (see (1.2)).
Our aim below will be to extract similar predictions about the structure of small g s , large T string theory corrections and their possible resummation for other closely related observables for which the exact gauge theory results can be found from matrix model representations following from localization (in some cases generalizing partial results in the literature).
Namely, we shall consider correlators of 1 2 -BPS Wilson loop with chiral primary operators (CPO) and also correlators of several coincident Wilson loops (mostly in the SYM theory). Like in the case of xWy in (1.3) we will observe certain universal patterns in their expansion in small g s and large T that should be related to supersymmetry of these observables. This may hopefully aid future investigations on the dual string theory side.
Let us summarize our main results.

Correlators of 1 2 -BPS Wilson loop with chiral primary operators
In section 2 we shall consider the SYM correlator of a circular Wilson loop with a chiral primary operator O J " tr ϕ J . 4 The correlator xW O J y was originally discussed in [11] at the leading order 3 This may be surprising given that such an exponentiation may be expected in the large tension ("thin handle") approximation on the string theory side [2,8] and the fact that the dual string theories in AdS5ˆS 5 and AdS4ˆCP 3 are similar. 4 As is well known, in N " 4 SYM one can construct Maldacena-Wilson loops with various amounts of supersymmetry [15], e.g. the 1 4 -BPS circular loop [16] and 1 8 -BPS loops [17,18,19]. Correlators of these loops and local operators were considered in [20,21,22]. Correlators of 1 8 -BPS circular loop and various chiral primaries have been computed by localization in [23,24,25,26,27]. Correlators involving Wilson loops in higher representations were discussed in [28,29]. In the planar limit at strong coupling the results were successfully compared with AdS/CFT predictions [11,28,29]. Beyond the planar limit and for J ą 3 the definition of the N " 4 SYM BPS operators dual to single-particle string (supergravity) states requires the addition to tr ϕ J of multi-trace terms (see [30] and references therein). We have verified by explicit calculations that this does not change the qualitative structure of the 1{N expansions discussed below.
in strong coupling in connection with the Wilson loop OPE expansion. In the planar limit this correlator was computed exactly in λ in [20]: xW O J y " I J p ? λq (I J is the Bessel function). We have extended the computation to non-planar corrections; expanded in small g s and then in large T as in (  0 "´1 4π pJ 2´1 q as in [20]). Compared to the series in g 2 s T " λ 3{2 N 2 in (1.3) here the natural expansion parameter turns out to be x " g 2 s T 2 " λ N 2 . Remarkably, it is possible to explicitly sum up all leading large T terms in (1.8) as Here F J is a finite polynomial for odd J and b 1`1 4 g 2 s T 2 times a polynomial for even J. For example, in the J " 2 case one finds simply The same expression applies also to the correlator of W with the dimension 4 dilaton operator O dil which is a supersymmetry descendant of O 2 . 6 For any g s and T the expectation value xW O dil y xWy can be found directly from xWy in (1.3),(1.4) by differentiating over λ so that using (1.1) we have (see [8] and refs. there) The small g s , large T expansion of logxWy following from (1.4) is found to be logxWy " 2πT´1 2 log´4π 2 g 2 s T¯´3 16πT`¨¨¨`π 12 and thereforé Here we ignore the R-symmetry factor Y depending on the choice of the CPO and the scalar coupling in W [20] and the factor of dependence on the operator insertion point which is fixed by conformal invariance (see section 2). 6 For higher J, the generalized dilaton operator O dil,J 1 with non-zero R-charge J 1 and dimension ∆ " 4`J 1 is a supersymmetry descendant of OJ with J " 2`J 1 and thus xW O dil,J´2 y{xWy is the same as (1.8 There is still an interesting connection between the resummed expressions for xWy in (1.5) and the correlator xW O 2 y in (1.9),(1.10): both can be given a "D3-brane" interpretation [31,28]. To recall, for a circular Wilson loop in k-symmetric SU pN q representation in the limit of large k, N and λ with κ " k ? λ 4N " k gs 2 T "fixed one expects that xWy should be given by expp´S D3 q where S D3 is the D3-brane action on the corresponding classical solution [31]. For 1 ! k ! N this should apply also to the Wilson loop in the k-fundamental representation described by a minimal surface ending on a multiply wrapped circle; here one finds [31]: Extrapolating this to the k " 1 case corresponds to the resummation of the expansion in (1.3),(1.4) for fixed ? λ N " gs T (i.e. when g s " T formally are both large, cf. [31,32]). Then S D3 "´2πT´π 12 reproducing the exponential factor in (1.5). Similar D3-brane interpretation is possible also in the case of the correlator xW O J y [28]. 7 Indeed, the resummed expression (1.9) is in perfect agreement with the result found in [28] in the fixed κ " k gs 2 T limit (both from the derivative of the D3-brane action over the corresponding graviton source and from the matrix model in the case of k-fundamental representation) after formally interpolating to the k " 1 case.
In section 3 we shall also consider the correlator xW O J 1 O J 2 y with two chiral primary operators. In the two special cases (a) J 1 " J 2 and (b) J 1 " 2, J 2 " 2J it is possible to reduce their computation to correlators in the Gaussian 1-matrix model. The structure of the resulting 1{N strong coupling expansion is found to be similar to (1.8) where b pjq i are polynomials in J 1 , J 2 (see (3.16), (3.22),(3.23)).

Correlators of coincident Wilson loops
Another class of tractable examples that we shall consider in section 4 are the expectation values of coincident circular Wilson loops xW n y in SYM theory. 8 The n " 2 case in the planar limit was discussed, in particular, in [2,44]. Extending calculation to subleading orders in 1{N in large λ limit and rewriting the resulting expansion in terms of g s and T as in (1.3) we have found that We thank S. Giombi for pointing this out to us. 8 Correlators of separated loops were considered in [33,34,35,36]; supersymmetric configurations with oppositely oriented loops were discussed in [37,38]; for various matrix model calculations, see [39,40,41,42,43].
The analogous expression for n " 3 is where Tph, aq is the Owen T-function (see (4.40)). For general n we found similar expansion (see (4.42) and Appendix D) Thus like for xWy in (1.3) here we get again series in ξ " π g 2 s T while for the correlators with chiral primary operators (1.8),(1.14) the expansion was in powers of x " g 2 s T 2 . In section 4.3 we shall derive a similar expansion for the correlator W p1,´1q of coincident Wilson loops in fundamental and anti-fundamental representations. It turns out that in contrast to (1.15),(1.17) W p1,´1q has trivial connected part, i.e. to all orders in ξ (and to leading order in 1{T ) xW p1,´1q y " xWy 2 (1.18) It would be interesting to explain this fact from the string theory point of view.

Comments on correlators in ABJM
Obtaining the above results in the N " 4 SYM theory case is facilitated by a relative simplicity of the associated Gaussian matrix model. In the ABJM theory the computations of similar correlators involving 1 2 -BPS circular Wilson loop [45] are substantially more involved. The structure of the strong-coupling expansion of the correlators with chiral primary operators is expected to be similar to (1.8). 9 This is suggested by the observation [8] that the expansion (1.3) of xWy looks the same in the SYM and ABJM theories and that, in particular, for the dilaton operator the correlators xW O dil y and xWy should be again related as in the last equality in (1.11). Indeed, the dilaton vertex operator has the same structure in both AdS 5ˆS 5 and AdS 4ˆC P 3 string theories and thus the derivative over the zero-momentum dilaton should be related to the string partition function in the same way as in AdS 5ˆS 5 case in [8], i.e. as in (1.11). Below in Appendix F we shall discuss the computation of correlators of coincident Wilson loops xW n y in ABJM theory. In particular, for the n " 2, 3 we will find suggesting the conjecture that, up to subleading 1{T terms at each order in the genus expansion, here the connected part of the correlator xW n y vanishes, i.e. xW n y » xWy n . This is in contrast to the non-trivial relation (1.17) found for xW n y in the SYM case (but is similar to the behaviour of xW p1,´1q y in (1.18)).

Structure of the paper
In section 2 we compute the 1{N expansion (1.8) of the SYM correlator xW O J y of the 1 2 -BPS Wilson loop with a chiral primary operator starting with its matrix model representation implied by localization. In section 3 we repeat the same analysis for the correlator xW O J 1 px 1 qO J 2 px 2 qy assuming a special (supersymmetric) choice of insertion points x 1 and x 2 that allows a matrix model calculation confirming that its strong coupling expansion has the form (1.14).
In section 4 we consider correlators of coincident BPS Wilson loops. We establish the structure of the expansions in (1.15),(1.16) and prove their exact form by exploiting the Toda integrability structure of the underlying Gaussian matrix model. In section 4.3 we consider the correlator of Wilson loops in the fundamental and in the anti-fundamental representation where special features are expected due to supersymmetry. Indeed, in this case one finds (1.8), i.e. there are no leading order corrections to W p1,´1q beyond those in xW 2 y.
In Appendix A we discuss an attempt [2] to explain the negative power of T in the g 2 s {T term in (1.3) by assuming that for large T one can use supergravity approximation as in [11]. As we explain, this argument may work only if there are non-trivial cancellations of the dominant large T terms that should be implied by supersymmetry. Appendix B contains some technical details of the 1{N expansion of xWy. In Appendix C we consider the 1{N expansion of the correlator xW O J y in the string semiclassical limit J " ? λ " 1. In Appendix D we work out the 1{N expansion of xW n y deriving the expansion (1.17).
The Appendices E and F are devoted to the correlators xW n y of 1 2 -BPS circular Wilson loop in the ABJM theory. In Appendix E we comment on the single Wilson loop case case by reviewing the known matrix model results pointing out that here the expansion has again the same structure as in the SYM case in (1.3) and deriving the representation (1.7). Appendix F discusses correlators of n " 2, 3 coincident Wilson loops where we use the topological expansion of the algebraic curve characterizing the ABJM matrix model to first derive the exact expressions valid for all couplings, and then expand at strong coupling demonstrating the validity of (1.19).

Expansion of xW O J y
In this section we will compute the 1{N expansion of the N " 4 SYM correlator of 1 2 -BPS circular Wilson loop with chiral primary operators O J . As was shown in [20], in the leading planar approximation the expression for this correlator is proportional to the Bessel function, xW O J y " I J p ? λq. This result was obtained by summing all planar rainbow Feynman graphs under the assumption that radiative corrections from planar graphs with internal vertices cancel to all orders in perturbation theory. This result was later confirmed in the framework of supersymmetric localization where xW O J y was computed using a suitable hermitian 2-matrix model [23].
Below we shall first obtain the finite N localization result for this correlator using a simplified equivalent 1-matrix model suggested by similar computations in the N " 2 superconformal models [47,48]. We shall then derive the 1{N expansion of this correlator (up to the 1{N 6 order) with the coefficients being J-dependent combinations of Bessel functions of ? λ. Finally, we will extract the leading large λ behaviour of these coefficients.
In general, the 1 2 -BPS Wilson loop depends [49] on a unit 6-vector n i defining the coupling to the SYM scalars n i Φ i . The chiral primary operator may be chosen as O J " tr`u i Φ i pxq˘J where u is a complex null 6-vector u 2 i " 0. The dependence of the correlator xW O J y on n and u factorizes [20], i.e. is contained only in the overall factor Y pn, uq " pn i u i q J . We shall choose the 6-vector n i in W along the 1-direction and the vector u i to be non-zero only in (1,2) directions, so that where C is a circle of radius R (that can be set to 1 as we assume below). Then Y pn, uq " 2´J {2 ; we will not explicitly indicate this factor in xW O J pxqy as it can be absorbed into normalization of O J discussed below. Note also that ϕ " ϕ I T I where T I are U pN q generators. Let us also assume that the unit-radius circular loop in 4-space px 1 , x 2 , x 3 , x 4 q lies in the px 1 , x 2 q plane (with the center at the origin) and define the "transverse distance" Conformal symmetry implies that [11,50,51] xW O J pxqy " In what follows we shall thus assume that xW O J y stands for the x-independent part of (2.4), i.e. its value at x " 0.

Matrix model formulation
We we will use a hermitian 1-matrix model formulation that computes expectation values like (2.4) using a Gaussian hermitian 1-matrix model with the variable a " a I T I as 10 xOy " ż Da Opaq e´t r a 2 , Da " The explicit map from the gauge theory operator to the matrix model one is where the coupling factor comes from the scaling needed to have the simple normalization in the exponent e´t r a 2 in (2.5). Normal ordering in O J (2.6) amounts to subtraction of all selfcontractions. It is required as the correlators involving the standard (flat 4-space) chiral operator do not have self-contractions (there is no ϕϕ propagator) so that on the matrix model side (derived from gauge theory formulated on 4-sphere) these self-contractions should be explicitly removed (see, e.g., [54] The correlator in (2.5) is computed by Wick contractions with the free propagator xa I a J y " δ IJ . In the following, we will need generic multi-trace correlation functions of the form t n 1 ,...,n " xtr a n 1¨¨¨t r a n y. (2.8) They may be computed by repeated application of the U pN q fusion/fission identities leading to the recursion relations [55] t n " 1 2

Differential relations
Using the methods of [48] we can compute the one-point correlation functions (2.4) in presence of the Wilson loop. Remarkably, they can be found directly from the knowledge of xWy since it is possible to show that for all J one has xW O J y " D J pg, B g q xWy, where D J is a linear differential operator of order J´1. This follows from the matrix model representation of xW O J y and is ultimately related to the supersymmetry. For example, the J " 2 CPO correlator is related to the correlator with the dilaton operator and the latter may be found by differentiation over the coupling g or λ " g 2 N as in (1.11). Explicitly, in the J " 2 case one finds xW O 2 y " xW : tr a 2 :y " The J " 4 case is slightly more complicated xW O 4 y " xW : tr a 4 :y "

12)
11 Subtraction of self-contractions is in turn equivalent to the requirement of orthogonality to all lower dimensional operators [55]. Denoting by tΩαu the (single or multi-trace) operators with dimension strictly less than dim O, one has : O :" O´ÿ α,β xO Ωαy pC´1q αβ Ω β , C αβ " xΩαΩ β y.
where we used the explicit form of : tr a 4 : obtained by resolving the mixing with dimension ă 4 operators. From the relations (2.9), we find (doing Wick contractions) (2.14) A completely similar calculation for O 6 and O 8 gives (2.16) The above differential relations (2.11),(2.14),(2.15) written in terms of λ " N g 2 read Similar representations are found for higher even J and also for odd J, e.g.,

1{N and strong coupling expansion
From the large N expansion of xWy in (B.2) we can then compute the corresponding expansion of the ratios xW O J y{xWy. The strong coupling regime we are interested in is defined by first expanding in large N for fixed λ and then expanding the coefficient of each 1{N term at large λ. We find for J " 2, 4, 6 (the expressions for J " 8 and odd J are similar) The leading (planar) terms in the square brackets are 1´J λ`¨¨¨i n agreement with the expansion of I J p ? λq in [20]. 12 To determine the higher order J-dependent terms in the expansion of xW O J y up to some fixed order in 1{N it is convenient to use the representation derived in [53] xW O J y " Expanding at large N gives Using the identity this gives (including all required additional terms in (2.27)) where we notice that the 1{N correction due to the first term in square brackets in (2.27) happens to cancel due to the Bessel function identity 2pJ´1q λqq " 0. Extending this procedure to determine all terms up to order 1{N 6 , we find where A J are expressed in terms of the modified Bessel functions I n " I n p ? λq This planar 1{N 0 part in the square brackets has the full λ dependence given by the Bessel function ratio IJ p ? λq{I1p ? λq up to a power of λ fixed by the choice of normalization of the operator. Notice that we are considering the U pN q gauge theory. Beyond the planar level, results in the SU pN q gauge theory differ in the subleading terms in the 1{N expansion due to an additional factor exp`´λ 8N 2˘, and also due to 1{N modifications in the fusion/fission relations for the SU pN q generators compared to (2.9).
It is then straightforward to expand the coefficients A pnq J pλq at large λ for any J ě 2. 13

String theory interpretation
Let us now rewrite the above expansions in terms of the string coupling and tension in (1.1), setting N " π T 2 {g s and λ " p2πT q 2 . Let us also choose a particular normalization of the chiral primary operator. One possibility could be to impose as in [20] the condition that the two-point function should be unit-normalized. However, this choice does not appear to be natural in the string theory context. 14 Below we shall assume that the operators O J that should correspond to the string vertex operators should be normalized relative to the matrix model operator O J in (2.6) as 15 ? λ we will then have at the leading planar order xW O J y xWy " in agreement with the canonical normalization of the corresponding string vertex operator. Including subleading corrections and using (B.2), we then obtain from (2.27) the following expression for general value of J 4π T`¨¨¨ı (2.33) 13 The expressions (2.23) and (2.28)-(2.30) apply for J ě 2. The case of J " 0 is trivial while for J " 1 we get xW O1y " g 2 ?
2 xWy by contraction of tr a with tr expp g ? 2 aq. The J ě 2 restriction can be understood at the planar level by noting that the recursion relation leading to the IJ term is based on a recursion over the number of scalar propagator endpoints and this has a regular structure only for J ě 2 (cf. section 2.2 in [20]).
14 For example, the string dilaton vertex has a factor of T " ? λ and no gs " 1{N factors (see, e.g., [8]). Its gauge theory counterpart is the SYM Lagrangian 1 g 2 YM tr F 2 mn`. .. and its 2-point function scales as N 2 . 15 Below we shall use the label OJ for the CPO as in (2.6) even though its normalization will be different.
where dots stand for terms subleading at large T and the value of the overall coefficient reflects our choice of normalization of O J in (2.31).

Resummation of leading strong coupling terms
Separating the leading pg s {T q 2n terms in the brackets in (2.33) we get where dots in (2.35) stand for the terms which are subleading in 1{T at each order in g s . Thus, formally, keeping only F J part of (2.35) is the same as keeping only the terms that are non-vanishing The pattern of the leading coefficients in (2.36) suggests the all-order conjecture As was mentioned in section 1.1 in the Introduction, this resummed expression agrees with the semiclassical D3-brane calculation in [28] generalizing the computation of xWy in [31] to the case of correlators with chiral primary operators.
To explain the reason this agreement, let us recall that the semiclassical D3-brane probe description applies to the expectation value of the circular Wilson loop in the k-symmetric representation and in the limit where κ " k ? λ 4N is fixed for large λ and N . Remarkably, at large N and large λ the result for the k-symmetric Wilson loop is the same as for the simpler k-fundamental Wilson loop [53,56,57] for which the dependence on k is obtained from the k " 1 case by simply rescaling λ Ñ k 2 λ. Hence, in the above large N, λ limit with fixed ? λ N " gs T the semiclassical D3-brane description should also reproduce the result for the Wilson loop in the fundamental (k " 1) representation, but this limit is equivalent to the one we considered when we neglected the subleading 1{T terms in the full expansion (2.33). This leading contribution (2.35),(2.37) may be obtained also by directly from the matrix model saddle point at fixed ? λ N [28]. For odd J the function F J pxq in (2.37) reduces to a polynomial in x, while for even J the series expansion in x does not truncate -in this case F J pxq turns out to be a 1`x{4 times a polynomial in x. Indeed, from the definition of the Chebyshev polynomials T n pcos θq " cospn θq, U n pcos θq sin θ " sin`pn`1qθ˘, For even J, the overall factor ?
4T 2 q 1{2 (that has an imaginary branch point) shows that the gs T expansion has a finite radius of convergence. Explicitly, one finds for F J in (2.37) Starting with the resummed expression (2.35),(2.37) we can formally consider the limit when the parameter x " g 2 s T 2 that was fixed in the resummation is now taken to be large. Using that x q J´1`¨¨¨, and (2.34) we then get 16 One can also consider the limit of large J. The result depends on the assumption about growth of J relative to T . To be able to ignore the 1{T corrections in square brackets in (2.33) and thus use the resummed expression in (2.35),(2.37) J 2 should grow slower than T (i.e. J ! λ 1{4 ). Then c J F J pxq " exppJarcsinh ? x 2 q. Another interesting limit corresponds to the semiclassical large charge expansion in the dual string theory when J " T " 1. In this case the 1{T corrections in (2.33) are not negligible and (2.35),(2.37) cannot be used. This limit will be discussed in Appendix C below.

Comparison of expansions of xWy and xW O 2 y{xWy
Let us recall that the chiral primary operator O J " tr ϕ J with dimension ∆ " J belongs to the same short supermultiplet as the R-charge generalization of the dilaton operator O dil,J 1 " trpϕ J 1 F 2 mn q`... of dimension ∆ " 4`J 1 with J 1 " J´2. The standard ∆ " 4 dilaton operator is the supersymmetry descendant of the J " 2 CPO and thus their correlators with the BPS Wilson loop should be directly related. Indeed, like the dilaton correlator, the CPO correlator can be obtained from xWy by the differentiation over the coupling using (2.11),(2.17),(2.31) (cf. (1.11)) 17 According to (2.35),(2.40),(2.34) the result of the resummation of the strong coupling expansion for the J " 2 case is simply (2.44) 16 The limit gs{T " 1 assumed here is of course formal as in the original expansion we assumed that both gs and 1{T are small. 17 For general J, the supersymmetry relation between the Wilson loop correlators with CPO OJ and with the dilaton operator O dil,J´2 imply that xW O dil,J´2 y can also be obtained from xWy by the differential relations like (2.17)- (2.19). or, in gauge theory notation, 1 The leading strong coupling term here agrees with (2.43) since xWy " e ? λ and thus λB λ logxWy " 1 2 ?
λ`¨¨¨. However, the resummed expression (1.5) for xWy does not lead to (2.44) if substituted into (2.43). As already discussed in the Introduction, the reason why the two resummations are not directly related is that subleading in 1{T terms in xWy cannot be in general ignored in logxWy in (2.43) (see (1.12)-(1.13)). In more detail, the structure of the expansion of xWy is where the values of the coefficients a pnq p may be extracted from (1.4) [2]. Then including the subleading terms we have The resummation of xWy leading to the g 2 s T exponent in (1.5) amounts to dropping all subleading a pnq p corrections in (2.45) but they actually contribute to the leading order terms in (2.46) starting with the order`g 2 s T˘2 . Using that a

Expansion of xW
One may also consider a correlation function of a circular Wilson loop with two scalar chiral primary operators at generic positions x 1 , x 2 . Such correlator is fixed by conformal invariance up to a function of N and λ and two scalar combinations u and v of the positions invariant under the conformal transformations preserving the circle [58]. Explicitly, for xW O 1 px 1 q O 2 px 2 qy where O 1 and O 2 are scalar primary operators of dimensions ∆ 1 ,∆ 2 at points x 1 , x 2 P R 4 and W is the circular 1 2 -BPS loop of unit radius the conformal symmetry implies that 18 Fixing particular values of x 1 ,x 2 and thus of u and v one may then study the 1{N expansion of the resulting function. 18 One can conformally map R 4 Ñ AdS2ˆS 2 so that the circle is mapped to the boundary of AdS2. Then xW O1px1q O2px2qy{xWy is invariant under the 6 isometries of AdS2ˆS2 (corresponding to 6 conformal transformations that preserve the circle in R 4 ). It is expressed in terms of two functions (u and v) of the AdS2 and S 2 geodesic distances between the operators (see [58] for details).
It turns out that for special supersymmetric configurations correlators of certain BPS Wilson loops with local operators may be computed to all orders by localization by reducing them to correlators in a multi-matrix model [24]. Examples include special 1 8 -BPS Wilson loop which is a contour on a 2-sphere S 2 Ă R 4 .
In the general 1 8 -BPS case, one considers [24] the operators O J pxq " tr " x n Φ n pxq`iΦ 4 pxq ‰ J (for x 2 n " 1, n " 1, 2, 3) and the Wilson loop for a contour on S 2 Ă R 4 with the scalar coupling being ş nkl Φ n x k dx l (cf. (2.1)). The special 1 2 -BPS case we are interested in here corresponds to placing the operators at the poles of the 2-sphere and the unit-circle Wilson loop at its equator. This results in the following choice of x 1 and x 2 Then the correlator in (3.1) becomes explicitly and the Wilson loop scalar coupling becomes the same as in (2.1) with Φ 1 Ñ Φ 3 (and R " 1).
For general x 1 , x 2 the correlator (3.3) has the structure (3.1) but its value can be computed by localization at specific positions in (3.2).
In detail, it can be computed using a 3-matrix model with the following action depending on the hermitian matrices X 1 , The connected part of the correlator (3.1) is related to a particular matrix model correlator which admits the following 1{N expansion xtr X J 1 1 tr e X 2 tr X J 2 3 y conn " Q J 1 ,J 2 pλ; N q " For the coefficient Q p1q J 1 ,J 2 pλq of the leading planar contribution one finds [24] Q p1q The 3-matrix model representation (3.4),(3.5) can be translated into a Gaussian 1-matrix model one similar to the one considering in the previous section (cf. (2.5),(2.6)). Indeed, after the change of variables the correlator in (3.5) becomes xtrpA`iCq J 1 trpB´iCq J 2 tr e 2πC y, (3.8) 19 We specialize the expression in [24] to the case of (3.2).
computed in the matrix model with the decoupled Gaussian action S " A 2`B2`C2 . Integrating out the A and B matrices amounts to subtracting from trpA`iCq J 1 and trpB´iCq J 2 their self contractions, resulting in the normal ordering discussed in section 2.1. 20 We then end up with the following correlator in the 1-matrix model for C x: tr C J 1 : : tr C J 2 : tr e 2πC y . (3.9) Below we shall consider two examples of the correlators (3.1). The first has J 1 " J 2 " J and the second J 1 " 2 and J 2 " 2J (J is integer). We shall use them to illustrate the general features of the strong coupling limit of the coefficients of the 1{N expansion of (3.1).
In this case the explicit form of the relation between the matrix model correlator and the function of λ, N in (3.1),(3.2) is where O J are the matrix model operators the notation of section 2.1 (cf. (2.6)), i.e. O J ": tr a J : after renaming C Ñ a. We used that xO J y " 0. 21 The operators O J in (3.10) are assumed to be normalized as in (2.31). Let us consider explicitly the J " 2 case when Here we used the relation xW : ptr a 2 q 2 :y " 1 4 pg 2 B 2 g´g B g q xWy " λ 2 B 2 λ xWy that may be proved using the same method as in section 2.2. As a result, we get the following differential relation for the J " 2 case of (3.10) Using the 1{N expansion of xWy in (B.2) we find Similar calculation can be repeated for higher J and leads to Dividing (3.10) over xWy leads to (cf. (3.1)) In this case the 1-matrix model representations for the correlators (3.5) with J " 2, 3 are 22 The exact differential relations for (3.17), (3.18) are found to be As a result, using (B.2) we get Taking the ratio of (3.18) and xWy in (B.2) and expanding at strong coupling gives 23 The sign is i J 1 p´iq J 2 from (3.8). 23 The absence of 1{T corrections at leading planar order in (3.22) is due to cancellation of the planar I1p

Correlators of coincident circular Wilson loops
As was mentioned in the Introduction, we can also study the 1{N expansion for other observables, like correlators of several circular Wilson loops xW n y. Such correlators were previously discussed in particular in the planar approximation in the n " 2 case with two circular loops in parallel planes separated by some distance; at strong coupling one finds a transitional behaviour [33] at certain critical distance when the associated minimal surface reduces to independent surfaces attached to separate loops [34,35,36].
Here we will consider the limiting case when the loops have the same radii and are coincident. In this case the correlator xW n y can be found exactly using the matrix model methods [2,41,39]. Our aim below will be to work out the large N , large λ expansion of such correlators.

xW 2 y for loops in fundamental representation
The coincident Wilson loops may be considered in generic representations (see, e.g., [39,59]). Let us consider the case of two loops in the fundamental representation. 24 The relevant 1{N expansions may be written in terms of matrix model correlators as xWy " xtr exp´b λ 2N a¯y "  24 Let us note that a discussion of similar correlator in planar limit at strong coupling (i.e. using semiclassical string theory) was in section 6 of [38] where the coincident 1 4 -BPS "latitudes" were considered; the present example of 1 2 -BPS circular loops is a special case.
This expression can be checked by directly evaluating xW 2 y at weak coupling and finite N using the Gaussian matrix model, which gives While (4.3) is exact, it is non-trivial to extract the exact λ dependence of its coefficients in the 1{N expansion so some indirect approach may be required. The first non-planar contribution to the 1{N expansion of (4.3) was computed exactly in λ in [60] (and was checked in [44] by the standard weak coupling perturbation theory) Expanding (4.5) at large λ gives (cf. (1.5)) In general, writing the 1{N expansion as the above previously known expressions (4.5) for the p " 0, 1 terms may be written in terms of the 1 F 2 hypergeometric function as xW 2 y 0 " 1 F 2´3 2 ; 2, 3; λ¯, xW 2 y 1 " This agrees with the result in [41] found using the topological recursion. From the point of view of computational efficiency, our procedure based on the hypergeometric representation of the connected part of the xW 2 y correlator has an advantage that it can be easily coded and extended to higher order terms in 1{N expansion in (4.7). Continuing to order p " 6 in (4.7), expanding for large λ and dropping subleading 1{T terms we get the following generalization of (4.6) xW 2 y » W 2 1´1`7 6 ξ`3 7 72 ξ 2`887 6480 ξ 3`28379 1088640 ξ 4`5045 1306368 ξ 5`1210793 2586608640 ξ 6`¨¨¨¨¨¨¯, (4.12) xW 2 y xWy 2 » 1`ξ`ξ This suggests a natural all-order conjecture for the resummed leading-order strong-coupling terms (cf. (1.5),(1.9)) xW 2 y xWy 2 » 1`8 (4.14) We prove (4.14) using the Toda integrability structure of the underlying Gaussian matrix model in the next subsection.
Let us note that one can easily find also the correlation function of W 2 with J " 2 chiral primary operator. Indeed, the insertion of O 2 is equivalent to λB λ in presence of any power of W in the correlator (cf. (2.11),(2.17)). Then from (4.5) one finds which has a similar structure to the one of the previously found correlator in (2.33)

Resummation of the g 2 s {T expansion using Toda integrability structure
In the Gaussian matrix model case, the Toda integrability structure [61,62,63,64] is a useful alternative to the topological recursion. Let us now show how to use it to prove the relation (4.14) to all orders in ξ " πg 2 s {T . From (4.2) it follows that we need to find the exponential generating functions (here x, y are free parameters) e N pxq " xtr e xa y " 8 ÿ n"0 x n n! xtr a n y , e N px, yq " xtr e xa tr e ya y conn " (4.17) The Toda hierarchy analysis of [65] shows that 25 The first recursion is solved by reproducing the expression for xWy in (1.4). 25 Note that our normalization of a is different by ? 2 from the one in [65].

Its general solution is
Gpzq " c z e 1 12z 2`1

18432
? π z e 1 12z 2 erf´1 4z¯, (4.31) where the integration constant c should be set to zero to match the leading terms in (4.12). As a result, we find from (4.25) and (4.31) 26 This proves our conjecture in (4.14).

Case of xW 3 y
Similar approach can be applied also for higher correlators xW n y. For n " 3 we need the generating functions with 3 arguments e N px, y, zq " xtr e xa tr e ya tr e za y conn , The Toda recursion relation here reads Writing it in terms of the functions t in (4.33), w in (4.21) and σ in ( Solving for U pxq and using that gives the analog of (4.14),(4.32) (ξ " πg 2 s {T ) where Tph, aq is the Owen T-function Explicitly, the first few terms in the expansion of (4.39) in powers of ξ are thus (cf. (4.13))

Correlator of loops in fundamental and anti-fundamental representations
Let us consider now a correlator of one Wilson loop in k-fundamental and another in k-antifundamental representation of U pN q. In the matrix model description it is given by (cf. (2.7)) xW pk,´kq y " xW pkq W p´kq y " xtr U k tr U´ky , U " e g ? 2 a . (4.43) We will focus on the k " 1 case as (like in the case of k-fundamentalk-fundamental correlator discussed above) the dependence on k can be recovered by the rescaling g Ñ kg or λ Ñ k 2 λ. Instead of xW 2 y in (4.3) here one finds [41] 28 (4.44) Its weak coupling expansion reads (cf. (4.4)) The first two terms in the 1{N expansion are as in (4.5),(4.7): 28 The peculiar first term in the r.h.s. of (4.44) is due to would-be term in xW pk,k 1 q y proportional to a certain Laguerre L piq j`´p k`k 1 q 2 λ 4N˘c ontribution that happens to be λ independent for k`k 1 " 0.

Acknowledgments
We are grateful to Simone Giombi for a collaboration at an early stage of this project and many useful remarks and suggestions. We also thank Nadav

A On g 2 s {T term in xWy from supergravity approximation
As discussed in the Introduction, the form T 1 2´p of the string tension dependence of the leading strong-coupling terms in the 1{N expansion (1.3) of xW y has a string-theory explanation [8] based on the dependence of the ratio of the string fluctuation determinants (evaluated on a genus p surface) on the AdS radius.
At the same time, since in the large T limit the contributions of massive string modes in the virtual exchanges may be expected to be suppressed, one may hope [2], by analogy with a related discussion in [11], to give an alternative explanation of this dependence based on including only the massless (supergravity) modes in computing string loop corrections to xW y. If such a "supergravity" approach could be shown to work this would allow one to compute, e.g., the leading "one-handle" g 2 s {T correction in (1.3), (1.4) xWy " ? T 2π g s e 2πT ! 1`π 12 including its coefficient. As we will explain below, such a computation does not appear to be straightforward as the specific 1 T dependence of the g 2 s term on the string tension should be a consequence of a subtle supersymmetry-related cancellations of more dominant (for T " 1) terms. Also, specific coefficients will depend on a particular choice of the "string" UV cutoff (Λ " 1 ? α 1 " ? T , see, e.g., [68]). One may represent the contribution of a thin handle attached to a disc by the sum of massless exchanges, each given by the two massless vertex operators V (integrated over the disc) connected by the corresponding target space "massless" propagator. For example, in the flat target space case for the dilaton exchange in the bosonic string theory in D dimensions we would have 30 For large T this should be evaluated near the relevant minimal surface (flat disc for the circular Wilson loop in the flat space case). The relevant exchange contribution will be proportional to where Gpx´x 1 q is the massless Green's function in D dimensions. The coefficient of the massless scalar kinetic term in the tree-level string effective action is T D{2´1 so that the inverse of this factor is to be included into X. As a result, we will get where x m pσq represents the minimal surface. Since the integrals are dominated by the shortdistance region σ " σ 1 where x i " σ i (i " 1, 2) (the D´2 coordinates x r transverse to the disc vanish on the classical solution) we thus find Here Λ " 1 is a UV cutoff that in the string theory context should have the interpretation of a modular integral cutoff set up by the string tension. Then, up to subleading terms in Λ dropped in (A.5), X " g 2 s T universally for any target space dimension D. Since this argument involves just the short-distance region, the result should not be sensitive to the target-space geometry. Indeed, the same expression is found by starting with the D " 10 theory in AdS 5ˆS 5 and compactifying on S 5 , i.e. considering as in [11] the 5d dilaton with dimension ∆ " 4`k where k is KK momentum. In this case δ pDq`x´x pσq˘in (A.2) is replaced by rKpxqs ∆ Y I k pyq where K is the bulk to boundary propagator in AdS 5 and Y I k is S 5 spherical harmonic. G in (A.3) is replaced by the AdS 5 bulk-to-bulk propagator. Taking into account the kdependent normalization factors (see [11]), summing over k and extracting the leading UV divergent part of the resulting analog of (A.4) we end up with same result X " g 2 s T as in (A.5). This is different from the expected g 2 s {T ratio in (A.1). 31 As already mentioned, details of compactification should not actually matter as the highest divergence depends on the power of the UV singularity of the D " 10 massless propagator and is thus universal. In particular, the same result should be found also in the AdS 4ˆC P 3 case.
It is possible that once one adds together similar exchanges of all D " 10 supergravity modes, the leading UV singularity will be reduced by 4 powers of the cutoff Λ due to supersymmetry cancellations. In this case one will end up with the following analog of (A.5) (here D " 10) The computation of the explicit form of the λ dependent coefficients in the 1{N expansion of the circular Wilson loop correlator xWy in the N " 4 SYM theory first appeared in Appendix A of [2] starting with a matrix model ansatz. A convenient algorithm to find these coefficients to any order in 1{N is discussed in [32] and leads to the following compact representation (cf. (2.7) and footnote 1) a y " 2 ?
λ Res Expanding H around x " 0 and taking the residue gives the following explicit expansion in terms of Bessel functions (I n " I n p ? λq) Keeping only the leading term at large λ at each order in 1{N we observe the exponentiation (1.5) originally found in [2] xWy » 31 A potential problem in a similar argument originally suggested in [2] appears to be with the contribution of summation over the KK modes that gives T 5{2 factor rather than T 1{2 assumed there. Indeed, the kinetic term of the KK dilaton has a prefactor B k " r2 k´1 pk`1qpk`2qs´1 that then enters in inverse power in the propagator. Also, the summation over quantum numbers of spherical harmonics with fixed J 2 " kpk`4q gives ř I Y I k Y I k " 2´kpk`2qpk`3q (see [11] for details), so that at the end we get ř k pk`1qpk`2q 2 pk`3q which diverges as ř Λ k k 4 " Λ 5 " T 5{2 as appropriate for a 5-space. We thank S. Giombi for a discussion of this argument.

B.2 On the origin of the N {λ 3{4 prefactor in xWy
Let us explain the origin of the leading strong-coupling prefactor N {λ 3{4 in (B.3) without resorting to the exact Laguerre representation (1.4) of xWy. Let us start with a generic (one-cut) matrix model with potential V and coupling g Z " Let t " N g be the analog of 't Hooft coupling. The planar resolvent for the one-cut distributions on ra, bs is where P pzq is a polynomial chosen so that to reproduce the correct asymptotics of ω 0 pzq at large z.
For an even potential V this gives ω 0 pzq " 1 z`O p 1 z 3 q. The analog of the Wilson loop expectation value is given by where the contour encircles the cut r´a, as " r´?2t, ?
Let us work out directly the large t expansion of the intermediate expression in (B.7). By saddle point analysis´?
Setting x " x˚`δx, expanding in δx and taking the large t limit gives expr ? 2t x`1 2 logp1´x 2 qs " expr ? 2t´1 4 log t`¨¨¨s´p2t`¨¨¨q pδxq 2`¨¨¨a nd thus where t´1 {4 comes from the matrix model "action" evaluated at x˚while an additional t´1 {2 comes from integration over the quadratic fluctuations. Similar analysis can be repeated at subleading order in 1{N . The derivation is less transparent, but the same saddle point argument gives the next term in the form t 3{4 {N , as expected from the exact solution (1.4).
As an aside, it may be of interest to generalize the above discussion to the case of the matrix model with a monomial potential V pxq " 1 2n x 2n . Then the resolvent is given by (B.5) with the following polynomial P pzq " P n pzq (e.g., for n " 2, 3) , a "´8 t 3¯1

{4
; Then in the quartic potential (n " 2) case we find for (B.6) The large a " t 1{4 expansion gives The prefactor of e a thus scales is " t´3 {8 N . In the sextic (n " 3) potential case one finds a similar result with the prefactor " t´1 {4 N . For general n, it is easy to check that the details of the polynomial P n pzq are not important and each of its terms contributes at the same order at large t; as a result (c is a numerical constant) xWy planar 9 N t´3 4n exp`c t The equation with the + sign is solved by the expression coinciding with the D3-brane action evaluated on the corresponding semiclassical solution [31] (see also [56]) Including higher orders in 1{N is straightforward. For instance, the next correction in (B.15) is obtained from in agreement with [57].
On the string theory side, taking the semiclassical limit one finds that the leading correction to the correlator xW O J y is described by a classical string solution [70,4]. One may consider the same limit also directly in the matrix model result for the correlator (2.27), (2.28). This requires the expansion of I J p ? λq in the limit (C.1) which can be found by starting from the Debye expansion of the Bessel J function 32 and analytically continuing to sech α " i J " i J ?
λ . This leads to This generalizes the leading exponential factor e ? λ f pJq found in [70] to subleading terms in 1{ ? λ and 1{N . D 1{N expansion of xW n y Let us consider the correlators xW n y with n ą 2. Expanded in large N , the connected part xW n y conn starts at order N 2´2n , i.e. one has the relations These relations can be easily checked using weak coupling expansions derived from the matrix model; like in (4.4) we get that indeed satisfy (D.1). From those relations we see that starting with the order 1{N 6 expansion of xW n y for n " 1, 2, 3, 4, we can determine the order 1{N 6 corrections in a closed form for all higher n ą 4. For the n " 1 case the 1{N expansion in terms of Bessel functions was given in Appendix B.1. For n " 2 we can use the results obtained in section 4. The 1{N 4 correction in the n " 3 case is easily found by matching the weak coupling expansion and this also fixes the same-order correction in n " 4 case. As a result, we find (I n " I n p ? λq): 33 λp24`131λq 2880 λp23040`56160λ`40920λ 2`6 209λ 3 q 5806080 Applying repeatedly the relations like (D.1) to determine the same expressions for 1 N n xW n y with n ą 4, we obtain the following general result 1 N n xW n y " 35λ 7{2 pn´1qpn´2qp216n 3`1 08n 2´7 38n´2033qI 3 0 λ 2 pn´1qp´2016p19`6nq´42p´4066`557n`600n 2`1 80n 3 qλq I 2 0 I 1 33 One can use recursion relations to bring all Bessel functions to I0 and I1 at the price of introducing polynomials in λ. In some cases, simpler expressions may be obtained in terms of higher index Bessel functions.

E 1{N expansion of xWy for 1 2 -BPS Wilson loop in ABJM
The localization computation in [14] proved that the expectation value of the 1 2 -BPS circular Wilson loop in ABJM theory to all orders in 1{N expansion at fixed level k (i.e. in the M-theory limit) is Let us set k " N λ as in (1.2) and consider the limit of N Ñ 8. 34 Using that the Airy function may be replaced by its asymptotic expansion we find for the resulting exponential factor in (E.1) Keeping only the leading large λ terms (or doing the shift λ Ñ λ`1 24 [7]) this gives the e π ? 2λ factor in (1.6). 35 The pre-exponential part of the ratio of the Airy functions in (E.1) is 1`λ 2 2pλ´1 24 qN 2`¨¨¨. At large λ it is 1 up to subleading contribution " λ{N 2 (instead of the leading λ 2 {N 2 " 1{k 2 " g 2 s {T 34 In the M-theory limit, i.e. expanding (E.1) in 1{N while keeping k fixed we get Note that a similar large N , fixed k expansion of the free energy of the ABJM theory on the 3-sphere considered in [71] contains an additional log N term. 35 Let us note that the expression (E.1) is expected to be valid up to terms which are exponentially suppressed at large N [14]. Such terms may not be a priori negligible in the type IIA string theory limit with fixed λ " N {k. Nevertheless, if one is interested only in the leading large λ corrections it seems reasonable to neglect these exponential corrections.
coming from the expansion of csc`2 π k˘i n (E.1)). This leads to the simple expression (1.7) for the sum of the leading large λ terms in xWy (cf. (1.6)) xWy » 1 2 csc`2 πλ N˘e π ?
2λ "´N 4π λ`π λ 6N`7 The log of (E.4) has the following expansion (2π λ N " a π Compared to the SYM case where the analogous expansion representing the leading-order terms at strong coupling stops at g 2 s {T and thus gives a simple exponentiation in (1.5), this does not happen in the ABJM case. Other differences emerge even at planar level when subleading corrections in large T are considered. While in SYM we have´3 16πT in (1.12), in the ABJM the analogous term in (E.5) is´π 48 T , i.e. the coefficient of the 1{T correction in the planar part of xWy in (1.6),(E.1),(E.4) has an opposite power of π. 36 Let us also note that the structure of (E.5) is essentially similar to the one that appears when one replaces the circular 1 2 -BPS loop by the latitude loops considered in [72]. Also, one can consider the Z r abelian orbifolds of the ABJM theory with reduced amount of supersymmetry where the expectation value of the 1 2 -BPS loop was computed in [73]. The exact expression for xWy differs due to the dependence on the integer r. Nevertheless, in the large N limit at fixed λ " N {k we again obtain a simple prefactor p2rq´1 cscr2π{prkqs (with the ratio of the prefactors in the Airy functions being again 1 to the leading order).

F Correlators of coincident 1 2 -BPS Wilson loops in ABJM
Here we shall study the strong coupling expansion of the expectation value of coincident circular Wilson loops in the ABJM theory.We shall first present perturbative results at weak coupling for finite N , then consider the 1{N expansion with coefficients that are exact functions of the coupling λ and then consider the strong coupling limit.

F.1 Weak coupling expansion
Using the notation of [5] the ABJM matrix model partition function may be written as with the two couplings associated with the two factors in the gauge group U pN qˆU pN q being Setting M " diagpµ 1 , . . . , µ N q and x M " diagpν 1 , . . . , ν N q and using the U pN |N q matrix block notation U " diagpM,´N q, the matrix model counterpart of the 1 2 -BPS Wilson loop reads [45] W " Str U " tr e M`t r e

(F.5)
At leading order we have the expected large N (planar) factorization xW 2 y " xWy 2`O pN 0 q with corrections to it being In general, we can write the 1{N expansion of the connected part of the correlator in the form where at weak coupling Σ p0q pαq " α 2´α We shall now apply the algebraic curve solution of the ABJM matrix model in order to obtain the closed expressions for these two functions valid for all values of the coupling α " 2πi λ and then consider their expansion at strong coupling λ " 1.

F.2 Algebraic curve solution and strong coupling expansion
As discussed in detail in [6,7,14], the ABJM model may be solved after considering it as a restriction of the lens space model Lp2, 1q with generic left and right gauge group ranks and couplings (see also [74,75]). Let us denote by z the large N continuum limit of the eigenvalues µ i and ν i in (F.1). At leading order, the eigenvalues condense on two cuts C 1 " pa´1, aq, C 2 " p´b´1,´bq in the Z " e z plane. Using mirror symmetry, the position of the branch points may be expressed in terms of the coupling λ using the following implicit parametrization [6] apκq " 1 2 " 2`iκ`aκ p4i´κq Integrating the eigenvalue densities along the two cuts, the explicit expression of the 1 2 -BPS loop expectation value is reduced to a residue at infinity (cf. (B.6))

2N
xWy " where ωpZq is the resolvent of the ABJM matrix model. At the planar level, one can use the explicit expression of ωpZq in [76] to obtain the large N part of (F.4) (κ " κpαq, α " 2πi λ) (F.14) The same approach can be applied to the calculation of (F.7) 1 p2N q 2 " xW 2 y´xWy 2 ‰ " where each integration is around both cuts and ω 2 is the connected part of the two-point resolvent.
The large N leading order term ω 2,0 was computed in a generic two-cut hermitian matrix model [77] and provides the first term Σ p0q in (F.7). Evaluating the double residue at infinity gives  where W 1 is the leading-order planar strong coupling part in xWy in (1.6),(E.4),(F.14). In terms of the dual string theory parameters in (1.2)) it reads (F.20) The leading correction 4π 2 λ 2 3N 2 " π 6 g 2 s T in (F. 19) is just twice the correction in xWy in (E.4),(1.6) corresponding to the factorized contribution xWy 2 while the connected contribution is thus subleading (λ 3{2 vs. λ 2 ) at large λ. We conclude that to leading order at strong coupling xW 2 y factorizes (cf. (1.19)) xW 2 y " W 2 1´1`π 6 g 2 s T`¨¨¨¯" xWy 2`. .. , (F. 21) i.e. the connected contribution (F.7) is subleading at large λ " T 2 at order 1{N 2 " g 2 s . It is tempting to conjecture that this factorization continues to be true also at higher orders in 1{N (as that happened in the SYM case for the fundamental -anti-fundamental Wilson loop correlator (4.55)). A test of this conjecture requires a much more involved calculation of Σ p1q term in (F.7) presented in the next subsection. Since W 2 1 " N 2 this requires computing the 1{N 4 " g 4 s term in the brackets in (F.19).
(F. 22) Expanded at weak coupling (F.22) is in agreement with (F.8). At strong coupling (i.e. large κ in (F.10),(F.13)), we obtain for the leading term  We conclude that there is no pg 2 s {T q 2 correction to (F.21), i.e. we have xW 2 y " xWy 2 to this order.

F.2.2 xW 3 y
It is interesting to consider also the first correction to the correlator xW 3 y of the three coincident Wilson loops at the leading order at strong coupling. xW 3 y may be again decomposed into factorized and connected contributions. From the usual scaling arguments, the g 2 s {T correction may come only from xWy 3 and xWyxW 2 y conn while corrections to xW 3 y conn start at order pg 2 s {T q 2 . From the above result (F.19) for xW 2 y (implying that xW 2 y conn is subleading) it follows that the g 2 s {T term in xW 3 y is precisely three times that in xWy, i.e. comes only from xWy 3 . At the next pg 2 s {T q 2 order we may have contribution only from xW 3 y conn , since according to the result of the previous subsection there is no such leading term in xW 2 y conn .