Large Charges on the Wilson Loop in $\mathcal{N}=4$ SYM: Matrix Model and Classical String

We study the large charge sector of the defect CFT defined by the half-BPS Wilson loop in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. Specifically, we consider correlation functions of two large charge insertions and several light insertions in the double-scaling limit where the 't Hooft coupling $\lambda$ and the large charge $J$ are sent to infinity, with the ratio $J/\sqrt{\lambda}$ held fixed. They are holographically dual to the expectation values of light vertex operators on a classical string solution with large angular momentum, which we evaluate in the leading large $J$ limit. We also compute the two-point function of large charge insertions by evaluating the on-shell string action, supplemented by the boundary terms that generalize the one introduced by Drukker, Gross and Ooguri for the Wilson loop without insertions. For a special class of correlation functions, we reproduce the string results from field theory by using supersymmetric localization. The results are given by correlation functions in an"emergent"matrix model whose matrix size is proportional to $J$ and whose spectral curve coincides with that of the classical string. Similar matrix models appeared in the study of extremal correlators in rank-1 $\mathcal{N}=2$ superconformal field theories, but our results hold also for non-extremal cases.


Introduction and summary
The study of operators with large quantum numbers has played a central role in the development of the gauge/string duality and AdS/CFT correspondence [1,2]. Under the duality, operators with large quantum numbers are dual to macroscopic string states, which can be quantized by semiclassical methods. For instance, the scaling dimensions of single-trace operators with large charges in the gauge theory are dual to the energies of semiclassical closed string states in AdS. The investigation of such large charge sectors of AdS/CFT also paved the way to the discovery of integrable structures on the gauge and string sides, leading to rather non-trivial dynamical tests of the correspondence (see for instance [3] for a review of this extensive topic). More recently, general properties of the large charge expansion in CFTs with global symmetries have been extensively studied using effective field theory methods, see for instance [4][5][6][7] and [8] for a recent review.
In this work, our focus is on the large charge sector of the defect CFT associated with the half-BPS Wilson loop in the N = 4 supersymmetric Yang-Mills (SYM) theory. This is a well-known generalization of the usual Wilson loop operator that couples to one of the scalar fields of the theory, and is supported on a circular (or infinite straight line) contour. It preserves a one-dimensional superconformal group OSp(4 * |4) that includes 16 supercharges (half of the superconformal symmetry of the N = 4 SYM theory) and the bosonic subgroup SL(2, R) × SO(3) × SO (5). Here SL(2, R) is the 1d conformal symmetry preserved by a circle (or line), SO(3) is the symmetry under spacetime rotations transverse to the defect, and SO(5) is the residual R-symmetry that rotates the five scalars that do not couple to the Wilson loop. The correlation functions of operator insertions on the Wilson loop (together with correlation functions involving "bulk" single-trace operators) define a rather rich defect CFT that is amenable to studies via holography [9][10][11], localization [11][12][13], conformal bootstrap [14][15][16] and integrability [9,[17][18][19][20][21] [10]. Similarly, one can also compute correlation functions of "composites" of those excitations (see for instance [12] for some explicit examples). This perturbative approach via AdS 2 Witten diagrams is appropriate as long as the quantum numbers of the inserted operators are small compared to the string tension, or √ λ (where λ = g 2 YM N is the gauge theory 't Hooft coupling, and in this paper we focus on the planar limit throughout). When the quantum numbers become of order √ λ, the boundary insertions are of the same order as the string action and one then expects a new open string solution to dominate the path-integral.
A special class of operator insertions that are ideal for exploring this regime are the chiral primaries corresponding to products of scalar fields transforming in the symmetric traceless representations of the SO(5) R-symmetry. These operators have protected scaling dimensions, but their two-point and three-point functions are non-trivial functions of the coupling constant that can nevertheless be computed exactly via localization [12,13]. More generally, localization allows to compute the n-point functions in a subsector of "topological" chiral primaries (to be reviewed below), which have position independent correlation functions.
In the large charge regime, the simplest object to consider in the defect CFT is the insertion of two chiral primaries with charge J such that On the string theory side, the worldsheet corresponding to this configuration was identified first in [9] in the limit J √ λ → ∞, and later generalized to finite J √ λ in [28]. 2 We review the solution in detail in Section 4. Starting from this string worldsheet and inserting additional operators with small quantum numbers, one can then obtain new results on correlation functions of two "heavy" (large charge) operators and any number of "light" operators, normalized by the heavy-heavy two-point function. See Figure 1. These correlators include in particular the defect CFT OPE data in the heavy-heavy-light configuration. In this paper we focus on the leading order in the large charge limit (1.1), corresponding to the classical analysis on the string side, while subleading corrections corresponding to quantum fluctuations will be discussed in the companion paper [29].
A more basic quantity that one can also study is the correlation function of the two heavy operators alone. Since the operators are protected with dimension ∆ = J, their two-point function on the Wilson loop has the form where d is the distance between the operators, and n J (λ) is a non-trivial function of the coupling constant and charge J. This normalization constant has physical meaning because the operators are protected (for instance, for J = 1 it is related to the Bremsstrahlung function [18]). In the large charge limit, Eq. (1.1), it is expected to exponentiate as n J (λ) ∼ e √ λf (J/ √ λ) , where the non-trivial function in the exponent should correspond to the classical string action. In Section 4 we discuss this holographic two-point calculation in detail (refining and extending earlier attempts in [30]). We will propose, in particular, that a suitable modification of the boundary term prescription of [31] is needed in this case to properly account for the presence of the boundary vertex operators for the large charge insertions.
A non-trivial test of the string theory predictions can be obtained by comparing to the localization approach, which was developed in [12,13]  In this paper, we take two of them (Z J andZ J ) to have large charges (J ∼ √ λ 1) and the others (O j (ϕ j )'s) to be light. b. At strong coupling, the same correlation functions are described by a non-trivial classical string solution which ends at the Wilson loop on the boundary. In the absence of the insertions, the string worldsheet lies on the Euclidean AdS 2 subspace inside AdS 5 × S 5 . The large charge insertions deform this solution and the string worldsheet has a non-trivial profile around the insertion points. As an illustration of the string solution, the figure above shows a density plot of the worldsheet curvature. The leading large J answer can be computed by evaluating the light vertex operators on this deformed solution.
J ∼ √ λ are inserted, the localization results can be recast in terms of the "planar" limit of a matrix model of a J × J Hermitean matrix. 3 Using this matrix model description, in Section 3 we obtain the localization prediction for the two-point function of the heavy operators as well as the topological n-point functions of the two heavy operators and any number of light primaries, working to leading order in the large charge limit. The results are found to be in complete agreement with the string theory predictions we obtain in Section 4.
On the string theory side, in addition to the modified boundary term mentioned above, an additional ingredient that is crucial to obtain the correct result for the correlation functions is the averaging over a classical modulus of the string solution-namely, a constant angle φ 0 corresponding to the circle on S 5 that is dual to the charge J. This averaging is similar to the one recently advocated in [32], and previously in [33] (see also [5] for the analogous effect in the effective field theory description of the large charge expansion of CFTs).
It is interesting to note that our matrix model shares some features with the matrix model derived in [34], which describes the extremal correlation functions in N = 2 SCFTs.
In particular, the size of the matrix in both cases is given by the R-charge of the operators.
However, there are also notable differences. Firstly, while the matrix model in [34] describes a subsector of the rank-1 SCFTs 4 e.g. SU (2) N = 4 SYM, our matrix model describes a subsector of U (N ) N = 4 SYM at large N , which is in a completely different parameter regime. Secondly, in the setup discussed in [34], all the physical information is in the two-point function, 5 and correspondingly, the matrix model only computes such two-point functions. By contrast, in our setup, there are non-extremal higher-point functions which encode non-trivial physical information, and we can study them using our matrix model. Finally and perhaps most importantly, the classical spectral curve of our matrix model, which controls the leading large charge answer, coincides with the spectral curve of the classical string solution describing the holographic dual of the large charge operators on the Wilson loop. This provides a clear physical interpretation of the spectral curve of the large charge matrix model [28]. In addition, it suggests that the structure observed in the topological subsector may extend to more general non-SUSY sectors. This optimism comes from the fact that the spectral curve of the classical string is known to control the full non-protected spectrum of semiclassical fluctuations on the worldsheet [36,37].
There are several future directions worth exploring. First, it would be interesting to consider large charge limits of the correlation functions which involve "bulk operators", namely operators outside the defect (previous work on some of these observables include [38][39][40]). Studying such correlation functions would allow us to analyze the interplay between the large charge limit and the defect crossing equation. Second, the generalization of our results to 1/8-BPS Wilson loops [41][42][43] should be possible. In this case, the relation to the defect CFT will be lost since general 1/8-BPS Wilson loops do not preserve the conformal symmetry. Nevertheless, we can analyze the correlation functions on these Wilson loops both from the localization and the (semi-)classical string. Third, the localization analysis can be applied away from the large N limit, in particular to SU (2) N = 4 SYM [12]. The large charge limit in this case would be more directly related to large charge limits of rank-1 SCFTs studied in the literature [34,44], and it would be interesting to explore their relation.
The remainder of this paper is organized as follows. In Section 2 we review some basic facts about the defect CFT on the half-BPS Wilson loop and introduce the large charge operators of interest in this paper. In Section 3 we review the localization results for the correlation functions on the Wilson loop and the matrix model describing the large charge sector, and then use it to compute the two-point function of the heavy operators and the topological n-point functions of two heavy and any number of light operators. In Section 4 we introduce the string solution describing the insertion of two large charges on the Wilson loop, and then discuss the computation of the two-point function of the heavy operators and the correlation functions of the two heavy and any number of light operators. The appendices include some technical details on the "quasi-momentum" associated to the matrix model and on the supersymmetries preserved by the string solution.

Large charges in the Wilson loop defect CFT
This section introduces the defect CFT observables we will analyze in this paper. First, our conventions for N = 4 SYM: we work in Euclidean signature and use the standard Cartesian coordinates x µ , µ = 1, . . . , 4, on R 4 . We take the gauge group to be U (N ), denote the Yang-Mills coupling constant g YM , and the 't Hooft coupling λ ≡ g 2 YM N . Below we will also often adopt the notation g ≡ SYM whose contour is a circle and which couples to a single scalar. The Wilson loop is whereẋ µ = dx µ dϕ and ϕ is a coordinate parameterizing the loop. Below we will mostly work with the explicit parameterization x µ (ϕ) = (cos ϕ, sin ϕ, 0, 0) with ϕ ∈ [−π, π) (in which case |ẋ| = 1). The gauge field A µ and the scalars Φ I , I = 1, . . . , 6, transform in the adjoint of U (N ) and the trace is taken in the fundamental. The expectation value of W, as well as its planar limit (N → ∞, g fixed) and supergravity limit (N → ∞, g 1), is well known [45][46][47]: The Wilson loop preserves an SL(2, R) subgroup of the SO(5, 1) conformal group of N = 4 SYM and defines a one dimensional defect CFT in which local adjoint operators O i (x), i = 1, . . . , n are inserted along the contour [9-15, 19-21, 23, 24, 48, 49]. We will use the "single bracket" notation for the unnormalized defect correlators given by ) and the path ordering in the N = 4 SYM correlator acts both on the local operators and on the Wilson loop. It is also convenient to define the "double bracket" defect correlators normalized by the Wilson loop expectation value which satisfy 1 = 1. The defect correlators obey the axioms outlined in Appendix A of [50] for correlators in a one dimensional CFT. Since the defect is a circle, the conformal correlators are composed of ratios of the chordal distances, which we will denote Although our discussion is framed in terms of the circular Wilson loop, the analysis of the where t ij ≡ t i − t j . Thus, we may readily switch between correlators on the circle and correlators on the line by exchanging chordal distances with Euclidean distances. 6 One way to explicitly parametrize the half-BPS line is to pick the contour x µ (t) = (t, 0, 0, 0) and let Chiral primaries. We are interested in the large charge sector of defect correlators of chiral primaries of the form . (2.10) Here, J ij|k ≡ (J i + J j − J k )/2. The 3-pt function is zero unless J 1 , J 2 and J 3 satisfy the triangle inequality (J 1 + J 2 ≥ J 3 , plus permutations) and sum to an even integer.
Furthermore, the chiral primaries have a topological sector, which is accessible for finite N and g using localization [12], in which the correlators are position independent and the OPE is closed. We will discuss the topological sector in greater detail in Section 3, but we presently define the topological operators to be chiral primaries whose polarization vectors are correlated with their positions in the following way: 7 We will usually drop the explicit dependence on ϕ. Since (ϕ i )· (ϕ j ) = − 1 2 d 2 ij , the topological two and three-point correlators are manifestly constant: In accordance with localization, higher-point correlators are also constant. Given Eqs. (2.9)-(2.10), the topological two and three-point functions fully determine the general two and three-point functions, but the same is not true of the higher-point functions because their form is not fixed up to an overall constant by conformal symmetry.
Correlators in the large charge sector. In this work, we study the correlators of 2 + n chiral primaries, two of which have charge J and n of which have charges 1 , . . . , n . We analyze the correlators in the large charge regime, in which we first take N → ∞ with g, J, and i held fixed, and then take g → ∞ with i and J ≡ J g , (2.14) held fixed. Thus, in the planar limit and strongly coupled regime, the two operators with charge J are the same size as the coupling and the operators with charges i are parametrically smaller. We call the former "large charges" or "heavy operators" and the latter "finite charges" or "light operators." In the present work we focus mostly on the leading behavior, and postpone the analysis of subleading corrections in 1/g to [29].
In the first half of our analysis, in Section 3, we start from the planar-exact integral representation of the topological correlators that was derived in [12], and use it to derive a matrix model that describes the large charge sector. This lets us determine, to leading order, the normalized higher-point correlators as well as the two-point function In the second half of our analysis, in Section 4, we analyze the string dual to the large charge defect correlator where we have chosen the specific chiral primaries is the chordal distance between the large charges. In Eq. (2.17), we have used the rotational symmetry to put the two heavy operators at ϕ = ±ϕ L and assume for simplicity that ϕ L ∈ (0, π/2], but otherwise let the locations of the insertions be general. 8 By evaluating the action and vertex operators of the classical string, we can determine the leading behavior of the two-point function, Eq. (2.17), and of higher correlators with additional insertions of powers of Z andZ. Finally, we will also be able to use the classical string to reproduce the topological correlators computed by the matrix model. This is because Z J andZ J become topological in the antipodal configuration ϕ L = π/2, since and because the light topological operators,Φ , truncate to linear combinations of powers of Z andZ at leading order.

Large charge correlators from localization
In this section, we compute correlation functions with large charge insertions in the topological sector. We achieve this by deriving a matrix model that describes the large charge topological sector and applying the matrix model techniques. We will later see in Section 4 that the results obtained from localization are in perfect agreement with the results from the classical string.

Matrix model for large charge correlators
Integral representation for the topological sector. Let us first review the integral representation of the correlation functions of insertions on the half-BPS Wilson loop in N = 4 SYM that was derived in [12,13].
The correlation functions in the topological sector on the half-BPS Wilson loop at large N are given by the following integral, whereΦ L is the charge L protected topological insertion defined in Eq. (2.11), dµ is the measure dµ = 1 4πg 2. They are polynomials in X ≡ g(x − x −1 ) with a unit leading coefficient: These two properties uniquely determine the functions Q J , and they can be computed systematically by performing the Gram-Schmidt orthogonalization on the set of monomials {1, X, X 2 , · · · }. Applying the Gram-Schmidt procedure, we obtain the simple expression: One important identity that follows from the orthogonalization is the expression for the two-point function The relations, Eq. (3.5) and Eq. (3.7), hold for any choice of the measure dµ as long as the Q's satisfy the aforementioned two properties. This will be important in the analysis of higher-point functions, as we will see shortly.
Matrix model. Another identity that we use in this section is the matrix model expression for the determinant D J . This can be derived by expanding the determinant and expressing each component as an integral (see Section 7 of [12]). The result reads (3.8) Alternatively we can write it in terms of where the integration contours are around the branch cut of X 2 k + 4g 2 . Setting aside the unusual choice of contour, this can be regarded as the eigenvalue integral for the partition function of a J × J Hermitian matrix model with potential e 2π √ M 2 +4g 2 /(8g 2 π 2 i): We can also express the two-point function, Eq. (3.7), directly in terms of this matrix model. For this purpose, we first rewrite D J+1 as If one instead performs a change of variables x i = ie iθi , the interaction term in Eq. (3.8) becomes which coincides with the Vandermonde factor for the matrix model of the orthogonal group SO(2N ).
The terms in the square bracket can be identified with the expectation value of the square of the determinant operator in the matrix model, det(Y − M ) 2 . Dividing this by D J , we where Y ≡ g(y − y −1 ) and • J denotes the expectation value in the matrix model: Let us make several comments about this matrix model. First, Eq. (3.8) was derived initially from integrability in [28] in the computation of the generalized cusp anomalous dimension. It was later re-derived in [12] from localization and was shown to control the two-point function in the topological sector as well.
Second, it has several similarities with the large-charge matrix model in [34], which computes the extremal correlation functions in rank-1 N = 2 SCFTs. In both cases, the matrix model was derived by applying the Gram-Schmidt orthogonalization to the localization results and the sizes of the matrices are related to the charges of the operators. Note however that the parameter regimes described by the two models are quite different. The matrix model in [34] is for the rank-1 SCFTs, a canonical example being N = 4 SYM with the SU (2) gauge group. By contrast, our matrix model is for the large N limit of N = 4 SYM. Another difference is that our model can be applied to the non-extremal correlation functions, as we will see below.
Third, since the measure is given by dµ ∼ e 2πg(x+x −1 ) the standard large N limit in the matrix model corresponds to the limit in which J is large but J/g is kept fixed. In terms of the 't Hooft coupling λ, this corresponds to the limit which is precisely the strong coupling limit in which the theory is described by classical strings. In other words, the large J expansion of the matrix model gives the semi-classical quantization of the classical string.
Higher-point functions. Let us now use the formalism reviewed above to derive a matrix model representation for the correlation functions of two large-charge insertions (Φ J ) and To compute this correlation function, we first deform the measure by exponentiating the light operators: Next, we construct the orthogonal polynomials of X under this measure, which we denote bỹ Using the relations reviewed in the previous section, we can express the two-point function of theQ J 's as a ratio of (deformed) determinants, Then we differentiate the left hand side of Eq. (3.17) with respect to t and set t to zero.
Since both the measure andQ depend on t, the differentiation produces three different terms, However, the latter two terms identically vanish for the following reason: Since the leading coefficient of the polynomialQ J is 1, the derivative dQ J /dt is always a polynomial of lower degree. Thus one can express dQ J /dt| t=0 as a linear combination of Q M with M < L. We can then use the orthogonality of the Q-functions, Eq. (3.7), to conclude that the latter two integrals vanish.
In summary, we arrive at the equality (3.20) If we divide this correlation function by the two-point function of the heavy insertions, we arrive at the simpler formula Now the final step is to translate this into the matrix model. As with D J , the deformed determinant admits a matrix model representation Eq. (3.8) with dµ replaced with dμ. By differentiating it with respect to t and setting it to zero, we get In the matrix model language, after dividing by D J , this corresponds to the expectation value of the "single-trace operator" Tr n j=1 Q j (M ) . Therefore, we arrive at the relation Note that this formula naturally reduces to the original integral representation Eq. (3.1) when J = 0 because • J=1 is a single-integral with the measure dµ, and • J=0 = 0.

Large J analysis
Higher-point functions. To evaluate Eq. (3.23) in the large J limit, it is convenient to use the quasi-momentum discussed in [12,28]: To motivate this definition, let us consider the one form p(x)du, where u is the Zhukovsky variable often used in the integrability literature: An important property of this one form is that it has poles with residue −1 at x = x k and x = −x −1 k and poles with residue J at x = 0 and x = ∞. In addition, it is related to the resolvent in the matrix model that we introduced earlier. Using the standard definition, the resolvent of the matrix model is where we recall that the X k 's are the eigenvalues of the matrix M . Now, by explicit computation, we can check Using Eq. (3.26), we can express the expectation value of the "single-trace" operators as where C is a contour which encircles all the eigenvalues X k 's counterclockwise. To proceed, it is convenient to rewrite the above expression in terms of the x, rather than the X, variables.
To do so, we need to remember that the x-plane is a double cover of the X-plane and each eigenvalue X k has two images x k and −x −1 k in the x-plane. Taking this into account, we obtain the expression where the contours C + and C − encircle x k 's and −x −1 k 's counterclockwise respectively. Note that we used the symmetry of Q j (x) If we take the large J limit, we can approximate p(x) with its classical value p cl (x), which Figure 2: Analytic structure of p cl (x). It has two branch cuts C ± , around which the integral p cl (x)du gives −J, as well as two poles at x = 0 and x = ∞, around which p cl (x)dx gives J. In addition, p cl = π at the right branch points, e ±iθ , and p cl = −π at the left branch points, −e ±iθ .
was computed 10 in [28]. We then get The classical limit of the quasi-momentum p cl is given in terms of elliptic functions [28], but we do not need its explicit form in this paper. Instead we use its following properties (see also Figure 2): • In the x-plane, p cl (x) has two (square-root) branch cuts along a unit circle; one along the arc C + = [e −iθ 0 , e iθ 0 ] and the other along the arc • The parameter θ 0 determining the size of the branch cuts is related to the charge J of the heavy operators by Here K and E are the complete elliptic integrals of the first and second kind. J is a monotonic function of θ 0 satisfying J = 0 when θ 0 = 0 and J → ∞ when θ 0 → π/2. As will be explained in Section 4, the angle θ 0 also controls the extension of the classical string solution in one of the S 5 directions.
• On the branch cuts, p cl satisfies In particular, at the branch points, p cl (x) = ±π. The integrals around the branch cuts and at x = 0, ∞ are given by • The derivative ∂ J p cl (x)du admits a simple expression (see Appendix A for the derivation): Let us also note that, as shown in [28], p cl (x) coincides with the quasi-momentum of the classical string solution 11 to be discussed in the next section, which is a holographic dual of the large charge operator on the Wilson loop. This relation is intriguing for several reasons: solutions, one can also describe the spectrum of general semi-classical fluctuations using the quasi-momentum as was demonstrated in [36,37]. We plan to revisit this (in the context of correlation functions) in the upcoming paper [29].
To evaluate Eq. (3.31), we deform the contour and bring it around x = 0 and x = ∞.
Using the invariance of the integrand under x → −x −1 , we then arrive at where the contour encircles x = 0 counterclockwise and the factor of 2 comes from the contribution at infinity. Plugging this expression into the large J limit of the formula for the 11 For the definition of the quasi-momentum in the classical string, see e.g. [51,52].
The final step is to substitute Q j with its strong coupling limit determined in [12], where H is the Hermite polynomial. As can be seen from this expression, the leading contribution at large g comes from the highest power in the Hermite polynomial. Therefore, for the purpose of computing the leading answer, we can simplify Eq. (3.38) to Performing the integral in Eq. (3.37) analytically, we find that the result is nonzero only when the total length of light operators is even, and it is given by We will later see that this agrees precisely with the result from the classical string.
Two-point function. We can also compute the two-point function, Φ JΦJ , using the matrix model techniques.
For this purpose, we start with the "planar" approximation of the expression Eq. (3.12), The exponent can be computed using the matrix model techniques. Namely, we have To evaluate this integral, we rewrite the logarithmic term as 12 We then get Here we used Eq. (3.34) to evaluate p cl (x)du. Next we deform the contour of x and pick up the residues of poles at x = y , −1/y , 0, ∞. As a result we obtain This condition is satisfied at the right branch points of p cl (i.e., y = e ±iθ 0 ). We thus obtain the following expression for the two-point function in the large J limit: Here we took y = e iθ 0 ; the other choice, e −iθ 0 , will give the same answer.
Before proceeding, let us make a few comments. The last term in Eq.
On the other hand, the left hand side of Eq. (3.48) is given by a ratio of matrix integrals Note that J is also given by an integral of p cl (x)du albeit on a different contour C ± . Therefore Eq. (3.48) can be regarded as a relation between period integrals along two "dual" cycles 13 : See also Figure 3. This is an analog of the relations known in conventional large N matrix models and topological strings, which relate the genus 0 free energy with the prepotential of period integrals [53,54]. Now, to evaluate Eq. (3.48), we first differentiate the exponent with respect to J. This gives two contributions: one is proportional to the deformation of the branch point ∂ J u(y * ) while the other comes from the explicit J-dependence in p cl and J log(gΛ). The first contribution is proportional to π − p cl , which vanishes at the saddle point. Therefore, we only 13 The extra terms in J D can be viewed as a regularization which makes the period integral finite.
need the second contribution, which gives This integral can be evaluated analytically, and after sending the cut-off Λ to infinity, we get It is useful to pause at this point and switch to the compact notation Returning to the computation of ΦJΦJ , it follows from Eq. (3.55) that This lets us integrate the c 2 -dependent piece of the RHS of Eq. (3.53) with respect to J: Meanwhile, the integral of the LHS of Eq. (3.53) yields log Φ JΦJ − log 1 . Putting every-thing together, we finally obtain the two-point correlator, (3.58) We will be able to rederive Eq. (3.58) from the classical action of the dual string in Section 4.

Large charge correlators from the Bremsstrahlung function
The large charge behavior of the topological two-point correlator can alternatively be determined from the ratio of incrementally shifted two-point functions, which can in turn be related to the Bremsstrahlung function, B J . 14 This lets us take advantage of the leading large charge expression for B J determined in [28] and the first subleading correction determined in [55]. We will derive the relation between R J and B J using the localization framework developed in [12], which we therefore first briefly summarize.
Firstly, the half-BPS Wilson loop defined in Eq. (2.1) can be generalized to a 1/8−BPS Wilson loop [42,43], W 1/8 , whose contour x µ (ϕ) is any closed curve on the two-sphere x 2 1 + x 2 2 + x 2 3 = 1 and which couples to the scalars as 15 The expectation value of W 1/8 is found by replacing g → g A(4π−A)  14 The Bremsstrahlung function B 0 was introduced in [18] as a quantity related to the cusp anomalous dimension of the Wilson loop and the radiation emitted by an accelerating quark, and was computed using localization. The generalized Bremsstrahlung function, B J , was defined and computed using integrability in [28] and computed using localization in [12]. 15 We pick this combination of scalar fields to couple to the 1/8-BPS Wilson loop in order to match our conventions for the half-BPS Wilson loop and topological operators in Section 2. 16 More generally, we may equivalently define the topological operator to be x(ϕ) · Φ + in · Φ ⊥ , where Φ ⊥ ≡ (Φ 1 , Φ 2 , Φ 5 ) andn is any unit 3-vector. We use the freedom to pickn = (0, 0, 1) so that this topological operator reduces to the topological operator defined in Eq. along the loop: (3.60) For this subsection, we will use . . . In this way the higher charge topological operators define an orthogonal basis that can be constructed from non-coincident insertions of the unit topological operators using the Gram-Schmidt procedure.
Finally, the topological correlators are closed under the OPE. As a special case, the OPE between a charge J insertion and a unit charge is simplỹ The prefactors on the RHS of Eq. (3.63) can be determined by taking the expectation value after multiplying both sides above byΦ J+1 orΦ J−1 , and also noting that This is a special case of the statement that the extremal topological correlators, for which the largest charge is equal to the sum of all the smaller charges, reduce to two-point functions. Namely, as was shown in [12].
We have now laid all of the groundwork needed to relate the topological two-point function to the Bremsstrahlung function. We begin with the expression derived in [12] (see also [18]) for the Bremsstrahlung function as the second area derivative of the two-point function: To get to the second equality, we used Eq.
which is the desired relation for R J in terms of B J . It holds for any g and J.
We are in particular interested in Eq. Eq. (3.71) and matching the two sides order by order yields the following pair of differential equations: and noting from Eq. (3.56) that b 1 (J/g) = g dc 2 dJ , we find dr 2 dc 2 = −g 2 . Finally, imposing the initial conditions r 2 (0) = r 1 (0) = 0, 18 we arrive at the following particularly simple form for 17 To get this result, we set r = 1 and sin 2 ψ = c 2 in Eqs. (5.11) and (5.12) of [55]. 18 These initial conditions can be justified as follows. As we'll discuss in Section 4.4, the small J expansion of the large charge correlators matches the large J expansion of the strongly coupled finite charge correlators. Thus, we can partly determine the behavior of R J as J → 0 using the expression for the finite charge R J : (3.77) Eq. (3.77) lets us determine the large charge limit of the two-point correlator. Keeping only the leading order term in R J , we find log c 2 (j) . Returning to Eq. (3.70), Eq. (3.77) also lets us determine the four-point correlator: We will see in Section 4 that, in the dual string calculation, 2R J is the classical contribution to the four point function. The other term, −B J , corresponds to the contribution from semiclassical fluctuations, which we will discuss in [29].
Finally, Eq. (3.77) also lets us determine the normalized extremal OPE coefficient, to next-to-leading order. We first note from Eq. (3.56) how c changes when J changes by a finite amount: correlator determined in [12], which is given in Eq. (3.83). From we explicitly see that r 2 → 0 and r 1 → 0 as J → 0. Therefore, Noting the strong coupling expansion of n for finite [12], the normalized extremal OPE coefficient becomes (3.84)

Large charge correlators from the classical string
Now we turn to the analysis of the large charge correlators using the classical string solution in AdS 5 × S 5 that is holographically dual to the Wilson operator with Z J andZ J inserted.
The classical string was discussed previously in a few closely related contexts. The string with J → ∞ was first identified in [9], and its spectrum of fluctuations in the BMN limit was matched to the anomalous dimensions, computed in weakly coupled gauge theory, of "words" inserted on the Wilson loop that are composed of many copies of Z interspersed with a few copies of an orthogonal scalar. Then [30] used the same classical string to explore the J → ∞ limit of the two-point function in Eq. (2.17). Finally, the string with general J is a special case of a more general string solution identified in Appendix E of [28], which was used to determine

Identifying the dual string solution
In this section, generalizing Section 4.1 in [9], we identify the classical string dual to the Wilson operator with Z J andZ J inserted and J finite. We begin by writing the metric in AdS 5 × S 5 using global coordinates on AdS 5 and embedding coordinates on S 5 : where ds 2 AdS 5 = dρ 2 − cosh 2 ρdt 2 + sinh 2 ρ dψ 2 1 + sin 2 ψ 1 dψ 2 2 + sin 2 ψ 2 dψ 2 ds 2 S 5 = dΘ I dΘ I . We let the contour of the half-BPS Wilson operator on the boundary consist of the two antiparallel, antipodal lines that lie at ρ → ∞, ψ 1 = ψ 2 = π 2 and ψ = 0, π, such that the line at ψ = 0 runs in the positive t direction and the line at ψ = π runs in the negative t direction. Furthermore, we put Z J in the infinite past, t → −∞, andZ J in the infinite future, t → ∞. Given this symmetric choice of the Wilson contours in global coordinates, we may restrict our attention to the AdS 3 subspace of AdS 5 with ψ 1 = ψ 2 = π 2 . We could also fix ψ = 0, π and restrict to an AdS 2 subspace but it is convenient to keep track of ψ for when we ultimately change to Poincaré coordinates and map this configuration to the circular Wilson loop with two insertions at ϕ = ±ϕ L , as in Eq. (2.17).
The string moving in the background in Eq. (4.4) is governed by the Nambu-Goto action (recall that we use the notation g ≡ √ λ 4π ), where Ψ is shorthand for the all the coordinates of the string, 2g is the string tension, σ α , α = 0, 1, are the worldsheet coordinates, and h denotes the determinant of h αβ , the metric induced on the worldsheet. Following [9], we identify the classical string dual to the Wilson operator with Z J andZ J in the large charge limit as the solution extremizing Eq. (4.6) with the following properties: 1. The string is incident on the antiparallel lines at ψ = 0, π on the boundary of AdS 3 and is at θ = 0, the north pole of S 2 , when it reaches the boundary.
2. As the string moves away from the antiparallel lines on the boundary, it simultaneously moves away from the north pole in S 2 and reaches a maximum angle, θ 0 ∈ [0, π/2), when it reaches the middle of AdS 3 .
3. The string rotates in S 2 in the φ direction with angular momentum J.
A family of string solutions satisfying the equations of motion and having the first two properties is given by: Here, the worldsheet coordinates, σ 1 ≡ ρ and σ 0 ≡ t, span R, the parameter c ∈ [0, 1) is related to the maximum value of θ by and φ 0 ∈ [0, 2π) is a modulus whose significance will become clear in the next section. The string is depicted in Figure 4. While we have presented the string solution using the Nambu-Goto description, one may also use the conformal gauge and the Polyakov action, as in [9,28].
The solution in the conformal gauge is related to Eq.
which matches Eq. (3.55) derived from the matrix model side. This tells us that the maximum polar angle of the classical string is the same as the angle parametrizing the branch cuts of p cl (see Figure 2), which justifies our use of the same symbol, θ 0 , for both. Therefore, we may also identify the parameter c in the string solution in Eq. (4.7) with the parameter introduced in Eq. (3.54). In fact, as shown in [36], the classical quasi-momentum of the matrix model coincides with that obtained from the string solution.
To connect our discussion above with previous work, we note that the string solution found in [9] is given by Eq. (4.7) when J → ∞ or c = 1. The classical string we consider is a simple generalization of the c = 1 string that preserves the same supersymmetries, as we check explicitly in Appendix B. It is in turn a special case of the string considered in [28] when the Wilson loop on the boundary does not have cusps in either the ambient contour or the internal contour at the locations of the two insertions. 19 To close this section, we transform the solution in Eq.  The solution in these coordinates takes the explicit form When c = 1, this solution is essentially equivalent to the one in [30].
The string in Eq. (4.12) is incident on Z J at ϕ = −ϕ L as τ → −∞, onZ J at ϕ = ϕ L as τ → ∞, and on the unit circle as ρ → ±∞. To approach the specific point ϕ on the circle from the worldsheet, we fix τ and send ρ → η∞, where η = +1 for ϕ ∈ [−ϕ L , ϕ L ] and η = −1 otherwise and τ is determined by the pair of implicit equations cosh τ ± sinh τ cosh τ + η cos ϕ L = 2 sin 2 ϕ ± ϕ L 2 / sin 2 ϕ L . This follows from Eq. (4.12) and we have expressed it in terms of the chordal distances from ϕ to the insertions.
A note about notation: Going forward, we will continue to write d 2 σ ≡ dρdτ for the measure on the worldsheet and use α and β as free or summed worldsheet indices, but we will label particular components of worldsheet tensors using ρ, τ instead of 1, 2.

Two-point function
The two-point correlator in Eq. (2.17) can be computed using a semiclassical evaluation of the string path integral of the appropriate vertex operators weighted by the action: (4.14) Here DΨ is the path integral over all the fields in the superstring sigma model, which are collectively denoted by Ψ. Furthermore, v andv are the vertex operators dual to Z andZ, which are determined by sending the insertions of Θ 4 ± iΘ 5 = sin θe ±iφ to the boundary of the world sheet: The limits are taken in accordance with Eq. (4.13). We have defined the vertex operators with the usual factor of 1/z (recall that the unit charge chiral primaries have unit conformal dimension) that appears in the AdS/CFT dictionary when extrapolating bulk points to the boundary. The vertex operators also include the natural string tension factor 2g = √ λ 2π . And, finally, S[Ψ] in Eq. (4.14) is the complete action of the string, which is the sum of the Nambu-Goto action and a boundary action: for the classical action of the string, and v cl (ϕ) ≡ v(ϕ; Ψ cl (c, φ 0 , ϕ L )),v cl (ϕ) ≡v(ϕ; Ψ cl (c, φ 0 , ϕ L )) (4.18) for the vertex operators evaluated on the classical string solution. The action S cl (c) does not depend on φ 0 or ϕ L , since changing φ 0 is implemented by an isometry on S 2 and changing ϕ L is implemented by an isometry on EAdS 3 . By contrast, the vertex operators, v cl andv cl , do depend on c, φ 0 and ϕ L as well as on ϕ, but we drop the explicit dependence for ease of notation. The two-point function in the large charge limit therefore becomes The term S cl (0) in the exponent comes from the denominator in Eq. (4.14). To understand and apply Eq.  [33] and also [32] for discussions. In our case, the semiclassical string with a given φ 0 is dual to a coherent state of average charge J, and integrating over φ 0 in Eq. (4.19) ensures that the resulting expectation value on the LHS is for an eigenstate of charge J.
Next, we note the explicit form of the vertex operators on the classical string: where ∆ was defined in Eq. (4.12). Above we used Eq. (4.13) to evaluate the limit in terms of ϕ instead of τ and wrote the result in terms of the chordal distances introduced in Eq. (2.5).

Thus, the contribution of the large charge vertex operators to the RHS of Eq. (4.19) is
We also note the explicit form of the metric induced on the worldsheet, Computing the action. As is familiar from the study of classical strings dual to Wilson loops without insertions [31,45,56,57], the string action is not simply the Nambu-Goto or Polyakov action since the area of the minimal surface diverges as it approaches the Wilson contour on the boundary. One possible remedy in that context is to regularize the area by shifting the contour from z = 0 to z = , remove the divergent piece from the area that is proportional to the length of the contour as → 0 + , and interpret it as a renormalization of the mass of the particle moving around the loop [56,57]. An alternative approach is 20 One may also define the large charge vertex operators to be v J ≡ lim τ →−∞ (2g sin θe iφ /z) J andv J ≡ lim τ →∞ (2g sin θe −iφ /z) J , so that we send τ → ∓∞ without sending ρ → ±∞ first. The order of limits does not matter and the resulting classical vertex operators reproduce v J cl (−ϕ L ) andv J cl (ϕ L ), as given by Eq. (4.20).
to take the Legendre transform of the Nambu-Goto action with respect to the bulk radial coordinate [31]. This adds a boundary term to the string action that does not change the equations of motion or the extremal surface but does change the value of the action and makes it well-defined (and, when applicable, in agreement with gauge theory). We adopt this second approach for the computation of the action of the classical string dual to the Wilson loop with insertions, but must modify the precise prescription in [31] to reflect the modified boundary conditions of the string arising from the insertions.
Because of its important role in what follows 21 let us briefly review the boundary term proposed in [31] for the classical string dual to a smooth Wilson loop without insertions.
We call this the DGO boundary term, in reference to the authors' initials. We begin by combining the inverse bulk coordinate u ≡ 1/z with the embedding coordinates Θ I for S 5 into u I ≡ uΘ I . This satisfies u I u I = u 2 . In these coordinates, the metric on EAdS 5 × S 5 can be written Then the induced metric on the worldsheet is h αβ = u 2 ∂ α x µ ∂ β x µ + u −2 ∂ α u I ∂ β u I , and we define the canonical momenta conjugate to u I and u to be In this language, the DGO prescription is to take the Legendre transform with respect to u or, equivalently, with respect to the u I by adding to the Nambu-Goto action the following boundary action: The variation of the Nambu-Goto action plus the DGO action under u I → u I + δu I about a classical solution is then For the variation to be zero, we must send δΠ α I → 0 at the boundary of the worldsheet, which means the complete action is to be viewed as a functional of the momentum Π I instead of the coordinates u I at the boundary. The DGO boundary term can therefore be thought of as imposing Neumann, rather than Dirichlet, boundary conditions on u I . 22 As a simple check of Eq. (4.26), we compute the action of the string with c = 0, which is dual to the half-BPS Wilson loop without insertions. In this case √ h = cosh ρ, ∂ ρ (uΠ ρ u ) = 2g cosh ρ, and ∂ τ (uΠ τ u ) = 2g sech 2 τ sech ρ, which reproduces the well known result This is just the string tension, 2g, times the regularized area of EAdS 2 , −2π, and matches the leading behavior of the gauge theory result in the supergravity regime given in Eq. (2.2).
Our characterization of the boundary term in Eq. (4.26) closely follows the one in [31] except that the original discussion was framed in terms of z and Y I ≡ zΘ I , and the momenta These limits are taken with ρ fixed but general. Finally, the limiting behavior of the momenta By contrast, u 4 + iu 5 and u 4 − iu 5 diverge as τ → ∞ and τ → −∞, respectively and Π 4−i5 and Π 4+i5 approach non-trivial functions of ρ as τ → ∞ and τ → −∞, respectively. 23 Eqs. (4.29) and (4.30) indicate that, as compared to the general prescription in [31], inserting Z J andZ J on the Wilson loop changes the boundary condition of u 4 + iu 5 at τ → −∞ and of u 4 − iu 5 at τ → ∞ from Neumann to Dirichlet. We therefore propose that the correct boundary action is the DGO action in Eq. (4.26), except with the Legendre transformation on u 4 ± iu 5 at τ = ∓∞ removed: Like the Nambu-Goto action, this boundary action is not finite by itself. The bulk and boundary actions should be added as in Eq. (4.16) with IR cutoffs imposed on both ρ and τ , and the sum then approaches a finite value as the cutoffs are removed.
The evaluation of the complete action is now simple, especially using the coordinates in Eq. (4.22). Firstly, we note where Π φ is the angular momentum density introduced in Eq. (4.9). The second two terms in Eq. (4.31) therefore evaluate to Furthermore, one can check that the ρ term in the DGO Lagrangian satisfies ∂ ρ (uΠ ρ u ) = 2g √ h and therefore cancels with the Nambu-Goto Lagrangian, while the τ term simplifies to ∂ τ (uΠ τ u ) = 2g sech 2 τ / cosh 2 ρ − c 2 . Therefore, Then, from Eq. (4.19) and including the contribution of the vertex operators Eq. (4.21), we obtain the two-point function Note that if we put Z J andZ J in the topological configuration by sending ϕ L → π/2, as in Eq. (2.20), we find This is in precise agreement with the localization result in Eq. (3.58).
We should emphasize that although we have for simplicity written the boundary action Eq. (4.31) using the global coordinates ρ and τ , the complete action is independent of the choice of worldsheet coordinates and IR regularization. 24 Before proceeding to the computation of the higher-point functions, we close this section with some comments about the evaluation of the string action. We begin by noting, since Here, we treat c and J as independent variables when taking the partial derivative, and c(J) is the particular value of c satisfying Eq. (4.9). If we view the vertex operators in Eq. (4.14) as additional boundary terms in a "total" action,

40)
24 This can be seen as follows. Firstly, the sum of the Nambu-Goto and DGO actions is manifestly worldsheet reparametrization invariant and one can also readily check that it is finite without need for regularization. Secondly, the first boundary term in Eq. (4.31) can be written in a coordinate independent way as follows (the treatment of the second boundary term is analogous): Here, η Λ is a family of curves that approach Z J as Λ → ∞, λ is a coordinate along η Λ that we may for concreteness take to run from −1 to 1, γ αβ ≡ dη α dλ dη β dλ h αβ is the induced metric on η Λ , n α is a unit normal vector (i.e., dη α dλ n α = 0 and n α n β h αβ = 1), and finally P α ≡ (u 4 + iu 5 )Π α 4+i5 / √ h. The point, then, is that the limit on the RHS of Eq. (4.38) is independent of the particular choice of η Λ (λ) ≡ (ρ Λ (λ), τ Λ (λ)) as long as ρ Λ (+1) → ∞, ρ Λ (−1) → −∞ and max λ∈[−1,1] τ Λ (λ) → −∞ as Λ → ∞, which is what we mean when we say that η Λ "approaches Z J " as Λ → ∞. Next, it is instructive to compare the behavior of Eq. (4.36) in the limit J → ∞ to the results in [30]. Noting that 2J log c → 0 and E(c 2 ) → 1 as J → ∞, we find that in the large Eq. (18) in [30] differs from our result above by an extra 2J in the exponent. 26 This is because the analysis in [30] used the DGO boundary action in Eq.  [38,40,59]. The classical string solution in that case is topologically a cone whose boundary is the disjoint union of a point incident on the local insertion and a circle incident on the Wilson loop. Our analysis is most directly similar to the one in [59], where the action is also computed with a DGO boundary term added to the Nambu-Goto action. 27 In that paper, the DGO term is written not as a bulk integral of a total derivative like in Eq. (4.26), which would give contributions from both boundaries, but rather as a boundary integral over only the boundary incident on the Wilson loop. Given our discussion above, we interpret this to mean that the u I coordinates satisfy Neumann conditions on the boundary incident on the Wilson loop, as in the DGO prescription, but Dirichlet conditions on the boundary incident on the local insertion. 28 Moreover, if we instead write the DGO term in [59] as a bulk integral of a total derivative or equivalently add the DGO term also for the boundary incident on the local operator, then we find the action changes by J, just like 26 Specifically, Eq. (18) in [30] gives the following value for the total action (including the contributions from the vertex operators): Identifying 2 → d as the distance between the insertions, R 2 2πα → 2g as the string tension and as an IR regulator as z → 0 that [30] imposed, we find that the above expression differs from the negative log of Eq. (4.42) by 2J + 4πg + 2J log 2g . The 4πg and 2J log 2g terms in the difference arise because, unlike [30], we normalize the Wilson loop with insertions by the Wilson loop without insertions (see Eqs. (2.4) and (4.14)) and we include a factor of 2g/z in the vertex operators as we approach the boundary (see Eq. (4.15)). Accounting for these differences in convention, the substantive difference between the two results is therefore simply 2J.
27 [38,40] used different, equivalent methods to regulate the semiclassical action. 28 More precisely, we would say that the combination of the u I coordinates dual to Z satisfies the Dirichlet condition on the insertion. The other linear combinations should still satisfy Neumann conditions but the boundary terms implementing their Legendre transforms evaluate to zero and can effectively be dropped.
in Eq. (4.33). Thus, the semiclassical string analysis of Z J andZ J inserted on the Wilson loop is very similar to the analysis of trZ J inserted away from the Wilson loop, except that splitting the boundary of the string worldsheet into a part incident on the local insertions and a part incident on the Wilson loop proper is more subtle in the former case.

Higher-point functions
The procedure we used to compute the large charge limit of the two-point function can be straightforwardly extended to higher-point functions. For every extra insertion of Z (ϕ) in the Wilson loop correlator, we simply add an extra factor of v cl (ϕ) in the integral over φ 0 in Eq. (4.19), and likewise forZ (ϕ) andv cl (ϕ). Thus, in the large charge limit, Here, we let tot ≡ n i=1 i and¯ tot ≡ m j=1¯ j . We could also be interested in higher-point functions in which the light operators are not just finite powers of Z andZ. However, the orthogonal scalars Φ 1 , Φ 2 and Φ 3 are effectively "invisible" in the large charge limit, as is clear from the perspective of the dual string: since Θ i = 0 for i = 1, 2, 3 on the classical string solution, the corresponding classical vertex operators, lim bulk→bdy Θ i /z, are trivially zero. This means that correlators involving light operators of the form ( · Φ) , where · Φ = 1 Φ 1 + 2 Φ 2 + 3 Φ 3 + Z Z + ZZ , reduce to linear combinations of the correlators in Eq. (4.44) after the truncation · Φ → Z Z + ZZ .
Alternatively, we can compute correlators of the more general light operators directly using the classical string by replacing · Φ in the Wilson loop correlator with Z v cl + Zv cl in the integral over φ 0 . (The correlators resulting from such truncation may of course be zero, like when Z = Z = 0, in which case we need to go beyond the classical analysis to determine the leading non-trivial contribution. For instance, to determine the four-point correlator Z JZ J Φ 1 Φ 1 , we need to study fluctuations of the semiclassical string, which will be discussed in [29].) As a special case, we will compute the higher-point topological correlator in Eq. (2.15).
Firstly, we again set ϕ L = π/2 to put the large charges in the topological configuration.
Thus, the integral counts the number of ways of grouping tot objects into two equal halves, and the correlator simplifies to This matches the localization result in Eq. (3.41).
We can also extend the analysis to correlators involving two large operators with unequal charges J and J + , where is held fixed in the large charge limit. The string path integral is dominated by the same saddle point, Ψ → Ψ cl (c(J)), and therefore the classical action and the vertex operators are the same as before. 29 The only difference is that the classical vertex operator for the J + charge has extra copies of v cl (−ϕ L ). For instance, the generalization of Eq. (4.44) is Furthermore, to determine the corresponding topological correlators, we again set ϕ L = π/2, replaceΦ J+ → v cl (− π 2 ) J+ = (gc) J+ e i(J+ )φ 0 andΦ J → (−1) Jv ( π 2 ) J = (−gc) J e −iJφ 0 for the 29 We could also alternatively localize the path integral to Ψ → Ψ cl (c(J )) for any J = J +O(g 0 ), including J = J + . The resulting string correlators, now with vertex operators v cl ,v cl ∝ c(J ) and action S cl (c(J )), have the same leading behavior in the large charge limit. This follows from Eq. (4.41). The analogous statement for the one-dimensional Laplace integral e −N f (x) g(x)dx, with N large and f minimized at x 0 , is that the leading term e −N f (x0) g(x 0 ) is invariant under x 0 → x 0 + a/N because f (x 0 ) = 0. See also the next footnote. heavy operators and use Eq. (4.45) for the light operators. This yields CFT data from a four-point function. The final correlator we consider is the nontopological correlator with two equal large charges and two equal light charges. This is a special case of Eq. (4.44) and is given by (4.50) This four point function contains some simple large charge CFT data. We choose to study the conformal block expansion in the Z Z J →Z Z J channel, so that the smallest exchanged operators also have large charges. (By contrast, the smallest operators exchanged in the Z Z → Z JZ J channel are light).
The operators O n in the OPE of Z and Z J have dimensions of order J or higher. Therefore, two of the three parameters of the hypergeometric function are large, in which case it takes a simplified form [61]: This matches the leading term in the small J expansion of Eq. (4.44), which is given by replacing c 2 → J /π. One consequence of this comparison is that we may view the light vertex operators on the J 1 semiclassical string as arising from the boundary-to-boundary con- It is perhaps interesting to compare Eqs. (4.59) and (4.67) with the general charge dependence of the heavy-heavy-light OPE coefficient that one would expect based on an effective field theory analysis expanding in inverse powers of the large charge. As in [5,7], the OPE coefficient typically grows as a power of the large charge. Though the extremal OPE coefficient indeed has this behavior at small J (see Eq. (4.56)), we see from Eq. (4.67) that it instead saturates to a constant at large J .
Finally, as noted in [38], if J were the only parameter governing the correlators when J is large, then the 1 g J regime probed by Eqs. (4.42) and (4.67) should match the g 1 ∼ J regime accessible to weakly coupled gauge theory, thus creating a bridge between two regimes that a priori seem very different. This agreement was verified in [38], at least schematically, for the correlator of the half-BPS Wilson loop and a single large charge insertion off the Wilson loop. Likewise, in [9] the anomalous dimensions of "words" composed of many copies of Z interspersed with orthogonal scalars inserted on the half-BPS Wilson loop were found to match in both the small g, large J regime and the large g, large J regime. However, we observe that this agreement between the two regimes does not extend to the correlators we have been studying. From either localization [12] or gauge theory [19], one finds that that the two and three-point functions in the weakly coupled limit are Φ JΦJ = (−g 2 ) J 1 − 2π 2 3 g 2 + O(g 4 ) ,

Acknowledgments
We are grateful to Nadav Drukker for useful correspondence. The work of SG and BO is supported in part by the US NSF under Grant No. PHY-1914860.

A Derivative of the quasi-momentum
In this appendix, we derive the expression for the derivative of the quasi-momentum, Eq. (3.35).
For this purpose, we use the fact that the one form p cl du has poles with residue J at x = 0 and x = ∞, while its integral along two branch cuts gives −J. From these properties, we immediately conclude that its derivative ∂ J p cl du satisfies 1. C ± ∂ J p cl du = −1.
In addition, one can check that p cl du flips a sign upon x → −x −1 . This follows from the definition of the (quantum) quasi-momentum, Eq. (3.24).
These conditions are sufficient for fixing the one form up to a constant α: To determine α, we expand the quantum quasi-momentum Eq. (3.24) around x = 0: In the classical limit, x k 's are distributed symmetrically with respect to the real axis (see

B Supersymmetries of the classical string
This appendix demonstrates that the string solution identified in Eq. (4.7) has the same supersymmetries as the string solution analyzed in [9], which corresponds to c = 1, and therefore has the same supersymmetries as the Wilson operator with Z J andZ J inserted.
More precisely, we will impose the supersymmetry conditions identified in Sec. 4.2 of [9] on a string ansatz and will show that Eq. (4.7) give the general solution to the resulting pair of differential equations. We closely follow the approach of [9]. To avoid confusion, it should be noted that some of the notation in this appendix differs from that in the rest of the text.