Half-BPS Wilson loop and AdS$_2$/CFT$_1$

We study correlation functions of local operator insertions on the 1/2-BPS Wilson line in ${\cal N}=4$ super Yang-Mills theory. These correlation functions are constrained by the 1d superconformal symmetry preserved by the 1/2-BPS Wilson line and define a defect CFT$_1$ living on the line. At strong coupling, a set of elementary operator insertions with protected scaling dimensions correspond to fluctuations of the dual fundamental string in AdS$_5 \times S^5$ ending on the line at the boundary and can be thought of as light fields propagating on the AdS$_2$ worldsheet. We use AdS/CFT techniques to compute the tree-level AdS$_2$ Witten diagrams describing the strong coupling limit of the four-point functions of the dual operator insertions. Using the OPE, we also extract the leading strong coupling corrections to the anomalous dimensions of the"two-particle"operators built out of elementary excitations. In the case of the circular Wilson loop, we match our results for the 4-point functions of a special type of scalar insertions to the prediction of localization to 2d Yang-Mills theory.


Introduction
In the N = 4 supersymmetric Yang-Mills theory, it is natural to consider Wilson loop operators that include couplings to the six scalars I in the theory [1,2] W = trP e dt iẋ μ A μ +|ẋ|θ I I , (1.1) where x μ (t) is a closed loop, and θ I (t) is a unit 6-vector. For generic contour and scalar couplings, these operators are only locally supersymmetric, but special choices of x μ and θ I lead to Wilson loops preserving various fractions of the superconformal symmetry of the N = 4 SYM theory [3,4]. The most supersymmetric operator is obtained by taking the contour to be an infinite straight line (or circle), and θ I a constant 6-vector, corresponding to a fixed direction in the scalar space: in this case the Wilson loop is 1/2-BPS, i.e. it preserves 16 of the 32 supercharges of the superconformal group P SU (2, 2|4). Making the choice θ I I = 6 , this 1/2-BPS straight line operator is given by where we have identified the Euclidean time x 0 = t ∈ (−∞, ∞) to be the line that defines the operator.
In this paper we will be interested in the computation of correlation functions of local operators inserted along the straight Wilson line, defined as follows. Given some local operators O i (t i ) transforming in the adjoint representation of the gauge group, one can define the gauge invariant correlator [5] (1. 3) The SU (N) indices are contracted with the Wilson lines joining the various points, making this a gauge invariant observable. Since the expectation value of the straight Wilson line is trivial, this definition satisfies 1 = W = 1. More generally, one should normalize the correlator on the right-hand side by the expectation value of the Wilson loop without insertions (this is relevant in the case of the 1/2-BPS circular loop, which has a non-trivial expectation value [6][7][8]). Note that, since operator insertions are equivalent to deformations of the Wilson line [5,9], the complete knowledge of the correlators (1.3) would, at least in principle, allow to compute the expectation value of general Wilson loops which are deformations of the line or circle.
To understand the structure of the correlators (1.3), it is useful to recall the symmetries preserved by the 1/2-BPS Wilson line. First, it is clear that it preserves an SO (5) subgroup of the SO(6) R R-symmetry that rotates the 5 scalars a , a = 1, . . . , 5 that do not couple to the Wilson loop. In addition, it preserves an SO(2, 1) × SO (3) subgroup of the 4d conformal group SO (2,4), where the SO(3) corresponds to rotations around the line, and the generators of SO (2,1) correspond to dilatations, translation and special conformal transformation along the line. This SO(2, 1) is the d = 1 conformal group. Together with the 16 supercharges preserved by the loop, the symmetries of 1/2-BPS Wilson lines form the d = 1, N = 8 superconformal group OSp(4 * |4).
It follows that operator insertions along the Wilson line are classified by their representations under the OSp(4 * |4) symmetry. In particular, they are labeled by their scaling dimension , corresponding to a representation of SO (2,1), and by a representation of the "internal" (from the point of view of the line) symmetry group SO(3) × SO (5). The set of correlators (1.3) are then constrained by the d = 1 conformal symmetry in a way analogous to higher dimensional CFTs. They can be interpreted as characterizing a defect CFT 1 living on the Wilson line [5,10,9]. This CFT 1 should then be fully determined by its spectrum of scaling dimensions and OPE coefficients. Because the "double-bracket" correlators (1.3) satisfy all the usual properties of CFT correlation functions, we may often talk about the O i (t i ) as operators in a 1d CFT, without referring to their (non-local) origin in SYM theory.
Among the possible operator insertions, a special role is played by a set of "elementary excitations" that fall into a short representation of the OSp(4 * |4) symmetry with 8 bosonic plus 8 fermionic operators, and have protected scaling dimensions. The bosonic operators are the 5 scalars a (with dimension = 1) that do not couple to the Wilson line, which have = 1, and the components of the field strength F ti ≡ iF ti + D i 6 (with dimension = 2) along the directions i = 1, 2, 3 transverse to the line. F ti is also known as the displacement operator, which measures the change of the Wilson loop under deformations orthogonal to the contour (this can be defined for any defect in a CFT). 2 The fact that these operators have protected scaling dimensions implies that their 2-point functions (in the sense of (1.3)) computed in planar SYM theory take the exact form , F ti (t 1 )F tj (t 2 ) = δ ij C F (λ) t 4 12 , (1.4) where the 't Hooft coupling λ dependence appears only in the normalization factors. These are proportional to the so-called Bremsstrahlung function B(λ) defined in [14] C (λ) = 2B(λ) , The three-point functions of these elementary bosonic excitations vanish by the SO(3) × SO (5) symmetry. The four-point functions are expected to be non-trivial functions of the positions t r (constrained by the 1d conformal symmetry as reviewed in Section 3 below) and of the coupling constant λ. Little is known about their structure apart from the leading perturbative term in the four-point of F ti computed in [9]. In this paper, we will compute these four-point functions at strong coupling using the string theory in AdS 5 × S 5 dual to planar N = 4 SYM. At strong coupling, Wilson loops are related by duality to open string minimal surfaces in AdS 5 ending on the contour defining the loop operator at the boundary. In the case of the 1/2-BPS Wilson line (or circle), the relevant minimal surface is an AdS 2 embedded in AdS 5 (and sitting at a point on the S 5 ). The fundamental open string stretched in AdS preserves the same OSp(4 * |4) as the 1/2-BPS Wilson line (see e.g. [15]). In particular, the 1d conformal group SO(2, 1) is realized as the isometry of AdS 2 .
As we will review in Section 2, expanding the string action in static gauge around the minimal surface solution, one finds [16] that the AdS 2 multiplet of fluctuations transverse to the string includes 5 massless scalars y a corresponding to the S 5 directions, three massive scalars x i with m 2 = 2 corresponding to AdS 5 fluctuations, and 8 fermionic modes with m 2 = 1. It is then natural to identify these 8 + 8 excitations, which may be thought as fields living in AdS 2 , with the elementary CFT 1 insertions described above [10,17,18]. Indeed, the standard relation m 2 = ( − d) between AdS d+1 scalar masses and the corresponding CFT d operator dimensions in the present case implies that the massless y a fields should be dual to = 1 operators in CFT 1 , 2 The fact that the displacement operator has protected dimension = 2 for a line defect in a 4d CFT is a general result, and follows from a Ward identity for the breaking of translations in the directions orthogonal to the defect, see e.g. [11][12][13]. namely the scalars a , while the three AdS 5 fluctuations x i with m 2 = 2 should be dual to the field strength operators F ti with = 2. 3 In general, in AdS/CFT the closed superstring vertex operators are mapped to single-trace gauge invariant local operators in the SYM theory. Including the open-string sector (with open strings ending at the boundary) one should be able to describe the gauge-invariant operators (1.3) that correspond to insertions of general local operators along the Wilson loop. In this paper we will limit our considerations only to insertions corresponding to the operators with protected scaling dimensions, that should be dual to "light" fields on the AdS 2 string world-sheet as described above. It would be of course interesting to work out the strong coupling description of insertions that develop large anomalous dimensions at strong coupling, such as, for instance, the insertion of 6 [21]. It is natural to expect that the dual of this type of "heavy" insertions have m 2 ∼ 1/α ∼ √ λ, corresponding to massive states of the open string. In terms of CFT 1 scaling dimensions, this implies that at strong coupling the spectrum of operators on the Wilson line has a large gap gap ∼ λ 1/4 , similarly to what happens for the closed string states.
The expansion of the Nambu action around the classical solution yields the interaction vertices between the light fields. We will use these vertices to compute the corresponding tree-level Witten diagrams in AdS 2 and extract the strong coupling prediction for the four-point functions of the protected insertions on the Wilson line. This is depicted schematically in Fig. 1. The calculation is similar to those in [22,23], however, we emphasize that the interpretation is different. In the supergravity calculations of [22,23], one computes correlation functions of single-trace local operators, dual to closed string states, and the expansion parameter is 1/N 2 . In our case, we compute the correlators (1.3) of insertions on the Wilson line, and the expansion parameter for the AdS 2 Witten diagrams is the inverse string tension, or 1 √ λ . Note that the 2d theory defined by the fundamental string action is expected to be UV finite, and thus the duality with the 1d CFT at the boundary should hold for any value of the coupling. In particular, the calculation of AdS 2 Witten diagrams involving loops should be well defined here. 4 Note also that the AdS 2 worldsheet is not decoupled from the rest of the AdS 5 × S 5 bulk. For instance, one can consider processes where the worldsheet interacts with closed string modes 3 The spectrum of quadratic superstring fluctuations is the same as in the case of "non-relativistic limit" of AdS 5 × S 5 superstring [19] and was suggested [10] to be related via AdS 2 /CFT 1 to the OSp(4 * |4) invariant N = 8 superconformal quantum mechanics of [20]. 4 For instance, by computing loop corrections to the boundary-to-boundary propagator one can verify that the elementary excitations are protected, as well as check the strong coupling expansion of the function B(λ).
propagating from the worldsheet to a point on the boundary away from the line (note, however, that these processes are suppressed in the large N limit). This corresponds to correlation functions such as W trZ J [24][25][26] (and more generally one may consider mixed correlators of defect operators and operators inserted away from the defect). The picture is similar to the one discussed in [27][28][29] (see also, e.g., [11][12][13]30,31] for recent related work) where one considers an AdS d brane inside AdS d+1 , and there is a defect CFT living at the boundary of AdS d . In our case, we have an AdS 2 worldsheet inside AdS 5 , and a codimension 3 defect in CFT 4 (the Wilson line) at the boundary of AdS 2 . 5 Using the OPE expansion, we can also extract from the tree-level four-point functions the leading strong coupling corrections to the scaling dimensions of the "two-particle" operators built of products of two of the protected insertions (with an arbitrary number of t-derivatives in between). For instance, we find that the SO ( with O(t) = a a (t) in the present case. In principle, the spectrum of dimensions of operators inserted on the Wilson line should be accessible from the TBA approach of [35,36], and it would be interesting to reproduce our results in this integrability-based framework. 6 More broadly, it would be important to see how integrability is reflected in the structure of the Witten diagrams one computes in the AdS 2 worldsheet theory, perhaps uncovering an analog of the factorization of the S-matrix in integrable theories in flat space. While we focus on the straight line for most of the paper, our results can be also mapped to the circle by a (large) conformal transformation, as explained in Section 6. For a particular class of S 5 insertions on the circular loop that are expected to be captured by localization [37][38][39][40], we show in Section 6 that the result of the Witten diagram calculation in AdS 2 precisely matches the exact prediction derived from localization to 2d YM theory.
As was appreciated in recent discussions of AdS 2 /CFT 1 in the context of dilaton-gravity models [41][42][43][44][45] one can think of a system in AdS 2 as having asymptotic 1d reparametrization symmetry that is spontaneously broken down to SO(2, 1), which is the isometry of AdS 2 metric. In our present case the original definition of the Wilson loop (1.1) has a reparametrization invariance which is fixed by the identification x 0 = t in (1.2), and the remaining conformal symmetry is the SO(2, 1) subgroup of the 4d conformal group that preserves the line. It is important to 5 One may also consider the D3 and D5 branes dual to 1/2-BPS Wilson loops in higher-rank symmetric and antisymmetric representations [32,33,15,34]: these branes have AdS 2 × S 2 or AdS 2 × S 4 worldvolumes, and preserve the same OSp(4 * |4) symmetry as the fundamental string. Computing AdS 2 Witten diagrams in this case (after KK reduction on the sphere factors) should yield the correlators (1.3) where the trace is taken to be in the symmetric or antisymmetric representations of rank k ∼ N . 6 It would also be interesting to use integrability to reproduce a weak coupling Feynman graph approach to dimensions of operator insertions on the Wilson line.
stress that compared to the gravitational AdS 2 models in [41][42][43][44][45] our bulk action (2.4) is defined in fixed AdS 2 background, i.e. does not contain gravity: before fixing the static gauge 7 the string action (2.1) is reparametrization invariant, but gravity never becomes dynamical in critical superstring theory. In line with this, the boundary theory has no analog of the pseudo-Goldstone mode [42] related to the (spontaneously broken) reparametrizations.

AdS 5 × S 5 string action in static gauge as AdS 2 bulk theory action
The bosonic part of the superstring action in AdS 5 × S 5 has the standard form where σ μ = (t, s) are Euclidean world-sheet coordinates, r = (0, i) = (0, 1, 2, 3) label coordinates of the Euclidean 4-boundary and a = 1, ..., 5 are S 5 labels. The minimal surface corresponding to the straight Wilson line at the boundary is described by The corresponding induced metric is that of AdS 2 , i.e. g μν dσ μ dσ ν = 1 s 2 (dt 2 + ds 2 ). We will study correlators of small fluctuations of "transverse" string coordinates (x i , y a ) near this minimal surface that will thus propagate in the induced AdS 2 metric. The resulting global symmetry of the bosonic action will thus be SO(2, 1) × [SO(3) × SO (6)]. To make the SO(2, 1) symmetry (which will be the conformal symmetry at the corresponding 1d boundary theory) manifest it is useful to choose the AdS 2 adapted coordinates and fix the static gauge in which z and x 0 do not fluctuate. The relevant embedding of AdS 2 into AdS 5 is described by ( Starting with the Nambu action and fixing the static gauge by the conditions on x 0 and z as in (2.2) we get where g μν = 1 s 2 δ μν is the background AdS 2 metric. This action can be interpreted as that of a straight fundamental string in AdS 5 × S 5 stretched along z, i.e. from the boundary towards the center of AdS 5 . It may be also viewed as a 2d field theory of 3 + 5 scalars in AdS 2 geometry with manifest symmetry SO(2, 1) × [SO(3) × SO (6)]. Interpreted as a 2d bulk AdS 2 theory, it 7 Defining the Wilson loop expectation value in string theory in conformal gauge where one has two more (compared to physical static gauge) dynamical coordinates and ghosts one would end effectively with an integral over boundary reparametrizations (see [46][47][48]). In this case the identification between the operators on the Wilson line on the gauge theory side and the string excitations appears to become more intricate. This question deserves further study.
should thus have a CFT 1 dual living at the z = s = 0 boundary. As explained in the Introduction, this CFT 1 can be viewed as the defect CFT defined by operator insertions on the straight Wilson line.
Expanding this action in powers of x i and y a we get Thus x i are 3 massive (m 2 = 2) and y a are 5 massless scalars propagating in AdS 2 .
One may also include the fermionic terms coming from the corresponding AdS 5 × S 5 superstring action as in [16] (there will also be eight 2d fermions with mass 1). The resulting 2d theory should be UV finite and thus should be dual to a quantum 1d CFT at the boundary for any value of the coupling T = where ... AdS 2 is the expectation value in the 2d theory (2.4) corresponding to Witten diagrams with bulk-to-boundary propagators attached to the points t 1 , ..., t n at the boundary. As discussed in the Introduction, the X ∼ y a in (2.4) will correspond to the scalar operators O ∼ a (a = 1, ..., 5) of dimension = 1 while X ∼ x i will correspond to the generalized field strength The relation (2.10) can be understood as follows. The correlators 3) can be found by first computing a wavy-line Wilson loop expectation value W (C) , taking functional derivatives over the contour function C(t) and then setting it to be a straight line. At weak coupling this procedure was followed in [9]. At strong coupling W (C) is assumed to be given by the AdS 5 × S 5 open string path integral with Dirichlet boundary conditions (implying that disc-like or half-plane like world-surface ends on a contour at the boundary of AdS 5 ×S 5 ). To leading order in large √ λ expansion that means computing the minimal area of the corresponding surface, i.e. the value of the (Euclidean) string action on the classical solution of the Dirichlet problem. In the present case of the string action in the static gauge (2.4) interpreted as a 2d field theory in AdS 2 this is equivalent to the standard AdS/CFT procedure of computing the generating functional for the corresponding CFT 1 boundary correlators or X(t 1 )X(t 2 )....X(t n ) AdS 2 . Expanding the resulting on-shell value of the string action in powers of small deviations of the boundary curve from the straight line will then give the correlators that can be equivalently found by computing the bulk Green's functions connected to the boundary points by the bulkto-boundary propagators. One can check the relation (2.10) explicitly at the 2-point level using the wavy-line solution of [49] (see also [50]), reproducing the string tree-level [49,51] and the 1-loop [52] corrections in the strong-coupling expansion of the B(λ) function in (1.4), (1.5).
The Lagrangian (2.5) has no cubic terms, so the contribution to the simplest 4-point tree-level correlation functions of x i and y a will be given just by the contact 4-point vertices in (2.7)-(2.9). Below we will compute the corresponding Witten diagrams in AdS 2 connecting the 4-vertices to the boundary points by bulk-to-boundary propagators as in, e.g., [22,23]. As we will be interested only in leading large λ (tree-level) bosonic field correlators we will ignore the fermions.
Note that while we have made a particular choice of AdS 5 coordinates in (2.3) the result for the on-shell AdS 2 amplitudes (i.e. boundary operator correlation functions) should be invariant under local field redefinitions (at least in the case of separated boundary points controlled by conformal invariance).
After including fermions and fixing kappa-symmetry gauge the superstring action (generalizing (2.1), (2.4)) expanded near the 1/2 BPS straight line minimal surface should be describing a globally supersymmetric field theory [16] for the OSp(4 * |4) multiplet of 8 + 8 bosons and fermions in AdS 2 . Same symmetry appears on the dual gauge theory side. While in this paper we will discuss only 4-point correlators of bosonic coordinates, this supersymmetry should allow also to determine the correlators involving fermionic excitations.

Four-point functions and conformal blocks in CFT 1
Before proceeding to computation of correlators of 2d fields in the AdS 2 theory (2.4), (2.5) let us make some general remarks about the structure of four-point functions in CFT 1 Note that the usual cross ratios u, v are not independent in d = 1, i.e.
This is because the SO(2, 1) symmetry allows one to fix three points on the line, leaving a single free real parameter as the position of the fourth point. For example, if we set t 1 = 0, t 3 = 1, t 4 = ∞, then χ corresponds to the position t 2 of the second operator.

OPE expansion
As in the case of higher dimensional CFT, the function G(χ) in (3.1) has an OPE expansion

4)
where h is the scaling dimension of the exchanged operator, c , We will also need the case of correlator of operators with pairwise equal dimensions (3.5) Here the conformal block expansion reads [53] Note that in (3.5) we have written the result by choosing the 12 → 34 channel (corresponding to χ → 0), which will be more convenient below. Of course, one may also write the 4-point function in the form

Generalized free field OPE coefficients
It will be useful for what follows to collect some results for the OPE coefficients of generalized free fields (see, e.g., [54][55][56][57]12]). In the case of the 4-point function of identical operators of dimension , the generalized free field 4-point function has G(u, v) where we assumed unit normalization of the 2-point function. The operators exchanged in the OPE are just the identity and the tower of "two-particle" operators of dimension 2 + 2n, n = 0, 1, . . . . The corresponding OPE coefficients are given explicitly by as one can verify from the identity ∞ n=0 c , ;2 +2n χ 2 +2n While operators with odd number of derivatives do not appear in the OPE of identical O 's, it will be useful for the case of operators carrying a flavor index (where odd n can appear in the antisymmetric channel) to note the following result for the sum over odd n: ∞ n=0 c , ;2 +2n+1 χ 2 +2n+1 2 F 1 (2 + 2n + 1, 2 + 2n + 1, 4 + 4n + 2, χ) (3.12) In the case of pairwise identical operators, we have (cf. (3.5)) Here the operators exchanged in the small χ expansion are for all integer n (both even and odd). The corresponding OPE coefficients are found to be (3.14) Indeed, one may verify that this agrees with the OPE expansion in (3.6) by checking that ∞ n=0 c 1 , 2 ; 1 + 2 +n χ 1 + 2 +n 2 F 1 ( 1 + n, 2 + n, 2 1 + 2 2 + 2n, χ) = χ 1 + 2 . (3.15)

Four-point function of S 5 fluctuations
In this section we compute the tree-level 4-point Witten diagram of the S 5 fluctuations y a in the AdS 2 action in (2.5). As reviewed above, these are dual to the 5 SYM scalars a , a = 1, . . . , 5 (that do not appear in the exponent of the half-BPS Wilson line operator) inserted along the line. The strong-coupling limit of the SYM correlator 1.3 should be given by the tree-level string coordinate amplitude as in (2.10).
where χ is the conformally invariant cross ratio (3.2). In writing (4.1), we used the fact that the operators a have protected dimension = 1, i.e. that their exact two-point function is 8 In (4.1) we factored out C (λ) 2 so that in the OPE limit χ → 0 we have G a 1 a 2 a 3 a 4 (χ) = δ a 1 a 2 δ a 3 a 4 + O(χ). The two-point normalization factor C (λ) is related to the Bremsstrahlung function defined in [14,36]. We can always absorb this factor in the normalization of the operators, and we will do so in the following by choosing a canonical form of the bulk-to-boundary propagators. The function G a 1 a 2 a 3 a 4 (χ) (which is also a non-trivial function of the coupling λ) can be decomposed into its SO(5) singlet, symmetric traceless and antisymmetric parts, a 1 a 2 δ a 3 a 4 + G T (χ) δ a 1 a 3 δ a 2 a 4 + δ a 2 a 3 δ a 1 a 4  At strong coupling, these functions are expected to have the expansion (working in perturbation theory) The leading terms here correspond to the disconnected contribution to the 4-point function, namely diagrams with two "boundary-to-boundary" propagators, (see Fig. 2), and are given by the generalized free field expression (cf. (3.8)) 9 δ a 1 a 2 δ a 3 a 4 + χ 2 δ a 1 a 3 δ a 2 a 4 + χ 2 (1 − χ) 2 δ a 1 a 4 δ a 2 a 3 (4.5) which yields The functions appearing at order 1 √ λ in (4.4) correspond to the leading contribution to the connected 4-point function at strong coupling, which comes from tree-level connected Witten diagrams. These are given by the 4-vertices in (2.9) with four bulk-to-boundary propagators attached. 9 Note that the separation between a connected and a disconnected contribution defined in (4.5) is natural from the point of view of the AdS 2 worldsheet perturbation theory, to all orders in 1/ √ λ: in general, the disconnected contribution is given by a pair of loop-corrected boundary-to-boundary propagators. In the weak coupling limit, on the other hand, it is straightforward to see that the leading contribution in the planar limit is δ a 1 a 2 δ a 3 a 4 + χ 2 (1−χ) 2 δ a 1 a 4 δ a 2 a 3 , which is not exactly of the form (4.5), indicating that the connected contribution, defined from the point of view of the AdS 2 perturbation theory, should, in fact, contribute at leading order at small λ.
We will adopt the following normalization of the bulk-to-boundary propagator (in general dimension d) (4.7) In this normalization [58,59], the tree level two-point function of the dual boundary operator is . 10 In the present case of d = 1 and = 1, we then have (t ≡ x 0 ) When one has only quartic contact diagrams (as in our present case), all tree-level 4-point functions can be written in terms of the D-functions [22,61,62] defined in the general case of AdS d+1 as Note that derivatives in the vertices can be dealt with by using the identity (here ∂ μ = (∂ z , ∂ r ), r = 0, 1, 2, ..., d − 1 and g μν = z 2 δ μν ) g μν ∂ μK 1 (z, x; x 1 ) ∂ νK 2 (z, x; x 2 ) (4.10)

Connected part of the four-point function
Returning to our case of d = 1, let us write the tree-level connected 4-point function (4.1) of y a coordinates in (2.5) as where Q 4y is obtained from the vertex L 4y in (2.9). Explicitly, we find 10 Note that this differs from the normalization adopted in [60], where C =  To write the result in a manifestly conformally invariant form, it is convenient to introduce the "reduced" D -functions that are functions of cross-ratios only. In general d, they are defined in terms of (4.9) as [62]

D 1 2 3 4 can be written explicitly as the following Feynman parameter integral
(4.14) In d = 1 where u = χ 2 , v = (1 − χ) 2 we get D 1 2 3 4 as a function of a single variable χ . When the indices i are integers, the integral (4.14) can be evaluated explicitly. The basic example appearing in our calculations is One can check that this agrees with the d = 1 limit of the well-known result in general d after we set z =z = χ (cf. (3.3)). The D -functions with higher integer indices can be either evaluated directly using (4.14), or expressed in terms of derivatives of D 1111 using the identities listed in [62]. Evaluating all the relevant integrals, the final result for (4.11) takes the form where we factored out (C =1 ) 2 so that G a 1 a 2 a 3 a 4 (χ) corresponds to a canonical unit normalization. Separating out the singlet (S), symmetric traceless (T ) and antisymmetric (A) channels as in (4.3), (4.4) G a 1 a 2 a 3 a 4 (1) T (χ) δ a 1 a 3 δ a 2 a 4 + δ a 2 a 3 δ a 1 a 4 − 2 5 δ a 1 a 2 δ a 3 a 4 a 1 a 3 δ a 2 a 4 − δ a 2 a 3 δ a 1 a 4 , Here and in what follows log |χ| ≡ 1 2 log(χ 2 ) and log |1 − χ| ≡ 1 2 log (1 − χ) 2 ] where χ ∈ (−∞, ∞). Alternatively, we may assume that χ ∈ (0, 1) (which, in particular, is sufficient for considerations of the OPE below) and thus omit the absolute values, log |χ| → log χ and log |1 − χ| → log(1 − χ). This is sufficient for obtaining the expressions on the entire real line using analytic continuation in χ (and crossing symmetry).
We can expand the above functions in the OPE limit χ → 0 as Since the term of order χ 2 log |χ| is absent from G T (χ), this result implies that the symmetric traceless "two-particle" operators (a b) do not have an anomalous dimension. This is as expected since these operators, such as Z J with Z = 1 + i 2 (inserted into the Wilson line) are BPS and hence protected [5,36].
On the other hand, the singlet a a acquires an anomalous dimension due to the presence of the χ 2 log |χ| term in (4.20). The same is true for other two-particle operators encoded in the higher powers χ 2+n log |χ|. We will extract their scaling dimensions systematically in the next subsection. Note that in the antisymmetric channel there is no operator of dimension 2 While one can consider the insertion of the operator [ a , b ] on the Wilson line, this operator sits in the same supersymmetry multiplet as 6 [9] and thus is expected to acquire a large anomalous dimension at strong coupling (presumably of order λ 1/4 [21] at large λ). Thus, it is expected to decouple from our perturbative calculations.

Scaling dimensions of two-particle operators from OPE
Each of the functions G S (χ), G T (χ) and G A (χ) in (4.18) is expected to have an OPE expansion of the form (3.4). To the leading order, where the 4-point function is given by the generalized free field expressions, the exchanged operators entering this expansion are the "two-particle" operators of the form ∼ a ∂ n t b (as always understood as attached to the Wilson line), as reviewed in Section 3.2. These can be decomposed in the irreducible representations of SO (5) (4.23) The connected 4-point functions computed in the previous subsection encode the 1 √ λ corrections to the scaling dimension of these operators, as well as the correction to the corresponding OPE coefficients. However, a difficulty arises in directly extracting this CFT data from the 4-point functions because of operator mixing. 11 Due to degeneracies in the leading order two-particle spectrum, at the interacting level some of the operators in (4.23) can mix with twoparticle operators with the appropriate quantum numbers built out of generalized gauge field strength F ti or fermions (recall that the 8 fermions transform in the (2,4)
Let us start our analysis with the symmetric traceless channel. In this case, we expect that the corresponding operators [ ] T 2n should not be affected by mixing because there are no other two-particle operators with the same quantum numbers. We can write (4.25) where the sum is over the primaries [ ] T 2n appearing in the OPE. At large λ, we can write their dimension and the OPE coefficient as Plugging the expansion (4.26) into (4.24), we get (see (4.6)) (4.27) Comparing this with the generalized free field result in (3.10), the leading OPE coefficients are found to be  (1) log |χ| + . . . , (4.29) we see that the anomalous dimensions are determined by the log |χ| terms in G (1) where G (1) T (χ) log |χ| is the function multiplying log |χ| in (4.19). This equation can be solved for any n with the help of the orthogonality relation [54] where the integral is over a contour around the origin in the complex plane. This result is valid for any , and can be verified, for instance, by using the series expansion of the hypergeometric function in (4.25). Using (4.31), we get from (4.30) Evaluating the residue, we find that the result takes the remarkably simple form

3) is given by
The vanishing of the anomalous dimension for n = 0 reflects the fact that the operator (a b) is protected. The operators with n > 0 are unprotected and belong to a long superconformal multiplet. Note that the anomalous dimension is negative for all n > 0, indicating an effective attractive interaction between single-particle states. Plugging the expansion (4.26) into (4.24) results also in the following equation which determines the leading strong-coupling correction to the OPE coefficients ∞ n=0 χ 2+2n c (1) [ where G (1) T (χ) no−log |χ| denotes the part of (4.19) which does not involve the log |χ| term. It is straightforward to use this to extract c (1) [ ] T 2n for any given n. For example, the results for n = 0, 1, 2 including both order zero (4.28) and 1 √ λ correction are (4.37) A relation of this type was found empirically in [54] and proved in [55] (see also [63]). Explicitly, we get where H n = n k=1 1 k is the harmonic number. In the singlet and antisymmetric channels, the leading order OPE coefficients are determined by (4.39) and using the results in Section 3.2 are found to be (4.40) In view of the orthogonality relation (4.31), we can extract the anomalous dimensions as A (χ) log |χ| . (4.41) Evaluating the residues we find, as in (4.33), simple quadratic polynomials in n γ (1) However, due to the mixing issues described above, these expressions should be viewed as "averages" of the anomalous dimensions over the operators appearing in the mixing (weighted by the corresponding OPE coefficients). 13 The exception is the singlet operator a a with n = 0, which cannot mix with any other operator. In this case, (4.42) yields Following similar approach as used above in the [ ] T 2n case, we can also extract the corresponding OPE coefficient It is interesting to notice that in all of the above expressions (4.34), (4.42) the large n limit of the scaling dimensions has the same asymptotic form We will find below that (4.45) is true also for the scaling dimensions extracted from the mixed x 2 y 2 -correlators and x 4 -correlators. Note that this implies that the perturbative result should not be trusted when n becomes of order √ λ, because then the leading term is comparable to the first perturbative correction (also, in such regime, contributions of massive string states should be already important, presumably corresponding to non-perturbative corrections to the 4-point function). Nevertheless, the form (4.45) is suggestive of a semiclassical limit with n, √ λ 1 and ν ≡ n √ λ fixed. Our results then suggest that in this limit the dimensions of such "two-particle" operators have a universal strong-coupling form This behavior may be captured by a semiclassical string calculation, analogous to the one in [5,65,66] where the insertions carried large R-charge, while here we just need large SO(2, 1) quantum number and no R-charge.

Four-point functions with AdS 5 fluctuations
Starting with the AdS 2 Lagrangian (2.5)-(2.9) we may also compute other four-point correlators involving AdS 5 coordinates x i which are dual to the dimension = 2 operator F it inserted on the Wilson line. Explicitly, below we will compute (cf. (2.10)) .

(5.2)
Since the Wilson line is 1/2-BPS these two correlators should be related to the correlation function of four S 5 fluctuations by supersymmetry transformations.

Two AdS 5 and two S 5 fluctuations
The leading-order contribution to the connected part of the correlator (5.1) may be written as (cf. (4.7), (4.8)) where Similarly to the discussion in Section 4.2, we may also extract the scaling dimensions of two-particle operators appearing in the OPE. In this case the relevant operators are ) and correspond to mixed xy two-particle states. Let us first rewrite the 4-point function (5.1), (5.3) by relabeling t 2 ↔ t 3 where from (5.5) we get The corresponding disconnected contribution appearing at leading order is (see (3.13)) Using (3.14), this determines the leading order OPE coefficients appearing in the expansion (3.6) (n + 2) (n + 4) (n + 5) 6 (n + 1) (2n + 5) . (5.10) To extract the anomalous dimensions, we can use the following generalization of the orthogonality relation (4.31) dz 2πi , h − a, 2h, z) .
In our case, we need = 3 and a = 1, see (3.6). Then the anomalous dimensions are given by where we have used that G (1) xy (χ) log |χ| = 2χ 3 (2χ − 1). Evaluating the residue, we find γ (1) [ F] n = − Let us separate the cases of even and odd n. For even n we expect that the operators [ F] 2n can mix with two-fermion states in the same representation. 14 For odd n, on the other hand, we do not expect mixing with two fermion states, and from (5.13) we get For large n, we recover the universal form (4.45), (4.46) found from the analysis of the y-correlators. Note that the dimension [ F] 2n+1 in (5.14) is the same as [ ] T 2n in (4.34) for n = n + 1. This is consistent with the fact that these operators should belong to the same long supermultiplet.

Four AdS 5 fluctuations
Finally, let us compute the four-point function of the three AdS fluctuations x i (5.2) using similar normalization for the connected part as in (5.3) (χ) , (5.15) with (cf. (4.18)) G (1) The two-particle states encoded in the OPE of the 4-point function of x fluctuations are The calculation of their anomalous dimensions follows the same steps as outlined in the previous sections. The disconnected contributions to the 4-point function are (cf. (4.6)) (5.23) As explained in Section 4.2, the singlet operators [FF] S 2n can mix with and two-fermion operators, and also [FF] A 2n+1 can mix with two-fermion states in the same representation. Therefore, the corresponding anomalous dimensions above should be viewed as averages and more work would be needed to disentangle the mixing. The symmetric traceless operators [FF] T 2n are not expected to mix, and from (5.23) we hence get their dimensions to be is the same as [ F] 2n+1 in (5.14) and also [ ] T 2n+2 in (4.34), indicating that these operators belong to the same supermultiplet.

Circular Wilson loop: comparison to localization
In the above calculations we assumed the straight Wilson line at the boundary. However, one can map the straight line to the circle by a conformal transformation, which allows then to translate correlators of operator insertions on the line to those on the circle. Explicitly, we can perform the transformation t → tan(τ/2), where −π < τ < π is the coordinate along the circle. Under this transformation, the two-point function of an operator O inserted in the Wilson loop changes as Note that the expectation value of the circular half-BPS Wilson loop is not trivial and given at large N by the well-known expression [6][7][8] and hence the double-bracket correlator in (6.1) requires a normalization factor given by this expectation value. On the string theory side, the transformation from boundary line to circle simply amounts to changing coordinates on the Euclidean AdS 2 worldsheet from the Poincare metric we have been assuming above to the hyperbolic disk metric In the case of the four-point function of S 5 fluctuations, it appears to be possible to compare our results to some localization prediction. In a series of papers [4,[37][38][39]67] it was proposed that correlation functions in a subsector of supersymmetric Wilson loops and local operators in N = 4 SYM can be computed via localization in terms of 2d YM theory. The relevant Wilson loops, first introduced in [68], are defined on generic contours on an S 2 subspace of R 4 (or S 4 ), and couple to three of the scalar fields in the SYM theory, say 1 , 2 , 3 , in a way prescribed by supersymmetry: where x i parametrize a unit two-sphere x 2 1 + x 2 2 + x 2 3 = 1. With such couplings to scalars i , the Wilson loops (6.6) are 1/8-BPS for generic contour C. These operators are mapped under localization 15 to usual Wilson loops in 2d YM on S 2 . The 1/2-BPS circular Wilson loop is a special case obtained by choosing the contour to be a great circle on S 2 . For instance, taking the equator x 1 = cos(τ ), x 2 = sin(τ ) gives the 1/2-BPS operator which couples to 3 only. In this section, we will use this convention for the scalar that couples to the 1/2-BPS Wilson loop, to adhere with the definition (6.6) used in the original papers.
The relevant local operators appearing in the localization setup are chiral primaries with specific position-dependent combination of scalars, which were first studied in [69]. Recall that a convenient way to write a chiral primary is in terms of an auxiliary null 6-vector The local operators that are captured by localization are inserted on the S 2 and have the form where x 1 , x 2 , x 3 is the point on S 2 where the operator is inserted. This means that the null 6-vector is position-dependent and given by (x) = (x 1 , x 2 , x 3 , i, 0, 0). These operators are mapped by localization [39,38] to powers of the Hodge dual (i * F ) J of the 2d YM field strength, and one can then compute general mixed correlation functions of Wilson loops and local operators using the 2d YM theory. A crucial property of the operators (6.8) is that their correlation functions are position independent, at any coupling [69]. From the point of view of localization to 2d YM theory, this can be understood as the fact that correlation functions of the field strength dual * F are position independent. 16 In addition to considering correlation functions of Wilson loops with local operators inserted away from the loop, as in [25], one can also insert the local operators (6.8) along the Wilson loop, which is our main interest here. A calculation of this type was carried out in [70], where the calculation in the 2d theory (to the leading order considered there) was found to be in agreement with the integrability-based results of [71].
To make contact with the calculation in Section 4, we should consider the 4-point function of the operators (6.8) inserted along the circular loop (τ 1 ) · (τ 1 ) (τ 2 ) · (τ 2 ) (τ 3 ) · (τ 3 ) (τ 4 ) · (τ 4 ) circle , (6.9) where, since the operators are inserted on the great circle in the (12)-plane, the null 6-vectors are given by 17 (τ k ) = cos τ k , sin τ k , 0, i, 0, 0 ≡ k , k= 1, . . . , 4 . (6.10) Let us first check that the two-point function of such operators along the circle is indeed position independent. We have (τ 1 ) · (τ 1 ) (τ 2 ) · (τ 2 ) circle = C (λ) (τ 1 ) · (τ 2 ) 2 sin τ 12 2 2 = − 1 2 C (λ) . (6.11) As expected, the factor in the numerator coming from the τ -dependent null vector cancels the position dependence of the denominator. To find (6.9), we just have to contract the SO (5) index structures in the result in Section 4 with the vectors k . Note that 1 · 2 3 · 4 = 2 sin τ 1 −τ 2 2 2 sin τ 3 −τ 4 2 2 , 1 · 2 3 · 4 1 · 3 2 · 4 = χ 2 , 1 · 2 3 · 4 1 · 4 2 · 3 = χ 2 (1 − χ) 2 . (6.12) Using these relations and the decomposition in (4.18), we find for the unit-normalized connected part of the 4-point function A (χ) . (6.13) Plugging in the explicit functions of cross-ratio from (4.19), one can verify that the position dependence completely cancels out and we end up with (6.14) Let us now compare this result with the prediction of localization. One should compute the 4-point function F (τ 1 )F (τ 2 )F (τ 3 )F (τ 4 ) YM 2 in 2d YM, where we introduced for convenience the shorthand F ≡ i * F for the dual of the field strength, which is inserted four times along the circular Wilson loop. A shortcut to this calculation may be obtained by starting from a more 17 Recall that in this section we are assuming that 3 is the scalar that couples to the Wilson loop, so the S 5 fluctuations y a are dual to 1 , 2 , 4 , 5 , 6 . general contour C and noticing that insertions of i * F are equivalent to taking derivatives of the Wilson loop expectation value with respect to the area. 18 For a general contour C singling out areas A 1 , A 2 on S 2 , with A 1 + A 2 = 4π (we take unit radius), the invariance under area preserving diffeomorphisms of 2d YM implies that the expectation value is given by the same expression (6.2) up to an area-dependent rescaling of the coupling This is the expectation value of the general 1/8-BPS operator (6.6). The 1/2-BPS circle corresponds to the special case A 1 = 2π when λ = λ. Then, taking derivatives of log W A 1 with respect to A 1 and setting A 1 = 2π after that yields the connected correlators of i * F inserted along the circle. For instance, the two-point function is given by . (6.16) This implies that in (6.11) C (λ) = √ λI 2 ( √ λ) , in agreement with the Bremsstrahlung function of [14].