Exact correlators on the Wilson loop in \( \mathcal{N}=4 \) SYM: localization, defect CFT, and integrability

Journal of High Energy Physics, May 2018

Abstract We compute a set of correlation functions of operator insertions on the 1/8 BPS Wilson loop in \( \mathcal{N}=4 \) SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the ’t Hooft coupling and the rank of the gauge group. When applied to the 1/2 BPS circular (or straight) Wilson loop, our results provide an infinite family of exact defect CFT data, including the structure constants of protected defect primaries of arbitrary length inserted on the loop. At strong coupling, we show precise agreement with a direct calculation using perturbation theory around the AdS2 string worldsheet. We also explain the connection of our results to the “generalized Bremsstrahlung functions” previously computed from integrability techniques, reproducing the known results in the planar limit as well as obtaining their finite N generalization. Furthermore, we show that the correlators at large N can be recast as simple integrals of products of polynomials (known as Q-functions) that appear in the Quantum Spectral Curve approach. This suggests an interesting interplay between localization, defect CFT and integrability.

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Exact correlators on the Wilson loop in \( \mathcal{N}=4 \) SYM: localization, defect CFT, and integrability

Accepted: May Exact correlators on the Wilson loop in Washington Road 0 Princeton 0 U.S.A. 0 0 1 Einstein Dr , Princeton, NJ 08544 , U.S.A 1 School of Natural Sciences, Institute for Advanced Study 2 Department of Physics, Princeton University We compute a set of correlation functions of operator insertions on the 1=8 BPS Wilson loop in N = 4 SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the 't Hooft coupling and the rank of the gauge group. When applied to the 1=2 BPS circular (or straight) Wilson loop, our results provide an in nite family of exact defect CFT data, including the structure constants of protected defect primaries of arbitrary length inserted on the loop. At strong coupling, we show precise agreement with a direct calculation using perturbation theory around the AdS2 string worldsheet. We also explain the connection of our results to the \generalized Bremsstrahlung functions" previously computed from integrability techniques, reproducing the known results in the planar limit as well as obtaining their nite N generalization. Furthermore, we show that the correlators at large N can be recast as simple integrals of products of polynomials (known as Q-functions) that appear in the Quantum Spectral Curve approach. This suggests an interesting interplay between localization, defect CFT and integrability. AdS-CFT Correspondence; Conformal Field Theory; Supersymmetric Gauge - 4 5 7 8 2.1 2.2 3.1 3.2 3.3 3.4 4.1 4.2 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 7.1 7.2 6 Weak- and strong-coupling expansions Weak coupling expansion Strong coupling expansion Comparison to string theory Emergent matrix model at large N DL and QL(x) as a matrix model Classical limit of the matrix model Conclusion Correlators on the 1=2 BPS Wilson loop from OPE Construction of operators from the Gram-Schmidt orthogonalization A remark on the 1=8 BPS Wilson loop Results for topological correlators Generalized Bremsstrahlung functions Cusp anomalous dimension and Bremsstrahlung function Relation to the two-point function and the result at nite N Large N limit Integral expression for topological correlators Properties of QL(x) Comparison with the results from integrability Variations of the measure Nonplanar corrections to the measure 1 Introduction 2 Topological correlators on the 1=8 BPS Wilson loop 1=8 BPS Wilson loop Correlators on the 1=8 BPS loop 2.3 1=2-BPS Wilson loop and defect CFT 3 Computation of the correlators A Explicit results for operators with L 3 A.1 Generalized Bremsstrahlung { i { Introduction The exact solution to an interacting quantum eld theory in four dimensions would mark a breakthrough in theoretical physics, although it still seems out of reach at present time. In supersymmetric theories, one can make some progress since there are observables that preserve a fraction of the supersymmetries and are therefore often amenable to exact analytic methods, most notably supersymmetric localization [1]. Another powerful method, which is currently the subject of active exploration, is the conformal bootstrap, see e.g. [2] for a recent review. This approach uses conformal symmetry instead of supersymmetry, and has been remarkably successful in deriving bounds on a landscape of theories from a minimal set of assumptions [3, 4]. The third way towards this goal is integrability [5]. Although the applicability of integrability is much smaller than the other two since it applies only to speci c theories, the advantage is that it works not only for supersymmetric observables but for nonsupersymmetric ones as well. It also allows one to compute them exactly as a function of coupling constants, rather than giving general bounds. In this paper, we consider quantities which may stand at the \crossroads" of all these three methods. More speci cally, we study the correlation functions of local operator insertions on the 1=8-BPS Wilson loop in N = 4 supersymmetric Yang-Mills theory (SYM). The supersymmetric Wilson loop in N = 4 SYM has been an active subject of study since the early days of AdS/CFT correspondence [6, 7]. The 1=2-BPS circular Wilson loop, which preserves a maximal amount of supersymmetry, was computed rst by summing up a class of Feynman diagrams [8, 9], and the exact result for its expectation value was later derived rigorously from supersymmetric localization [1], which reduces the problem to a simple Gaussian matrix model. The result is a nontrivial function of the coupling constant, which nevertheless matches beautifully with the regularized area of the string in AdS at strong coupling, providing key evidence for the holographic duality. The computation was subsequently generalized to less supersymmetric Wilson loops, such as the 1=4 BPS circular loop [10], and a more general in nite family of 1=8-BPS Wilson loops de ned on curves of arbitrary shape on a two-sphere [11, 12]. For such loops, an exact localization to 2d Yang-Mills theory was conjectured in [11, 12], and later supported by a localization calculation in [13]. Because of the invariance under areapreserving di eomorphisms of 2d YM theory, one nds that the result for the expectation value of the 1=8 BPS Wilson loop depends only on the area of the region surrounded by the loop. The localization relation to the 2d theory was checked in a number of nontrivial calculations, see e.g. [14{21]. It was also used in [22] to compute various important quantities de ned on the Wilson loop, such as the two-point function of the displacement operator and the related \Bremsstrahlung function". It was based on the observation that one can insert a displacement operator by di erentiating the expectation value of the Wilson loop with respect to its area A; D The purpose of this paper is to show that there are in nitely many other observables that can be computed using the results from localization. They are the correlation functions { 1 { of special scalar insertions ~ L inside the Wilson loop trace, where the scalar ~ is chosen so that the correlators are independent of the positions of the insertions.1 Similarly to the displacement operator, one can relate the insertion of ~ 's to the area-derivative of the Wilson loop, essentially because ~ turns out to correspond via localization to insertions of the Hodge dual of the 2d YM eld strength. However, one key di erence from the analysis in [22] is that after taking the multiple area derivatives, to de ne the properly normalordered operators one has to perform the so-called Gram-Schmidt orthogonalization to make ~ k's for di erent k orthogonal to each other (and, in particular, also orthogonal to the identity, i.e. their one-point functions vanish). After doing so, the result for the two-point function takes a particularly compact form and exhibits a simple determinant For higher-point functions, the result can be written succinctly in terms of certain polynomials FL(X), which by themselves are expressed in terms of determinants: h A0=A See section 5 for further details including the de nition of FL. As a special application of our analysis, we explain the relation of our correlators on the Wilson loop to the \generalized Bremsstrahlung function" BL( ) (whose de nition is reviewed in more detail in section 4 below), which was computed previously in the planar large N limit from integrability [25, 26]. In particular we nd that HL( ) 1 2 DL+1 DL A=2 2 : At large N , we show that this agrees with the integrability result. Moreover, since the Wilson loop expectation value appearing in the calculation of the determinants DL is known for any N; via localization, this provides the nite N form of the generalized Bremsstrahlung function BL( ). Our results are valid for the general 1=8-BPS Wilson loop de ned on an arbitrary contour on S2, but perhaps the most interesting case is the 1=2-BPS loop. Since the 1=2-BPS loop is circular (or, by a conformal transformation, a straight line) it preserves a SL(2; R) conformal subgroup, and therefore can be viewed as a conformal defect of the 4d theory. The correlation functions of operator insertions on the 1=2-BPS loop are then constrained by the SL(2; R) d = 1 conformal symmetry, or more precisely by the OSp(4 j4) SL(2; R) SO(3) SO(5) 1d superconformal symmetry [27]. Some of the properties of this defect CFT were recently studied at weak [28{31] and strong coupling [31, 32]. 1The correlators of Wilson loops and local operators of similar kind, but inserted away from the Wilson loop, was studied in earlier literature, e.g. [18{20]. 2A symmetric matrix of the type appearing here, which satis es Mij = Mi+j, is sometimes called )L, where Y is a null polarization 5-vector. Such operators transform in the rank-L symmetric traceless representation of SO(5), and they belong to short representations of the 1d superconformal group, with protected scaling dimension = L. Because their 2-point and 3-point functions are fully xed by the SL(2; R) symmetry, the restriction to the topological choice of polarization vectors still allows one to extract exact results for the 2-point normalization and 3-point structure constants of the general defect primaries (Y )L. Unlike the analogous case of single trace chiral primaries of the 4d theory, which are dual to protected closed string states, the structure constants in the present case are found to have a highly non-trivial dependence on the coupling constant. Our construction provides exact results for such structure constants of all operators in this protected subsector, which should provide valuable input for a conformal bootstrap approach to the Wilson-loop defect CFT (see e.g. [33]). The connection to integrability techniques emerges in the planar limit. At large N , we found that the results can be rewritten as a simple integral h with the measure d given in (5.5) (see also other forms of the measure (5.45) and (5.46)). This is by itself an interesting result, but what is more intriguing is that the function QL(x) that appears in the formula is directly related to the Quantum Spectral Curve [34], which is the most advanced method to compute the spectrum of the local operators in N = 4 SYM. The appearance of such functions in our setup hints at a potential applicability of the Quantum Spectral Curve to the problem of computing correlation functions. The rest of this paper is organized as follows: in section 2, we review the construction of the 1=8 BPS Wilson loop and explain the de nitions of the correlators that we study in this paper. We then relate them to the area derivative of the Wilson loop in section 3 by using the OPE and the Gram-Schmidt orthogonalization. The nal result for the correlator at nite N is given in subsection 3.4. In section 4, we apply our method to compute the nite-N generalization of the generalized Bremsstrahlung function. We then take the large N limit of our results rewriting the correlators as an integral in section 5. The results at large N are expanded at weak and strong coupling in section 6 and compared against the direct perturbative computations on the gauge theory side and on the string theory side respectively. We also provide a matrix-model-like reformulation of the large N results in section 7. Finally, section 8 contains our conclusion and comments on future directions. 2 Topological correlators on the 1=8 BPS Wilson loop In this section, we explain the de nitions of the topological correlators that we study in this paper and discuss their relation to the defect-CFT data. { 3 { BPS Wilson loop lives on S2 and couples to a scalar as prescribed in (2.1). The expectation value of such a loop depends only on the area A of the region inside the loop (the red-shaded region in the gure). Note that, although \the region inside/outside the loop" is not a well-de ned notion, such ambiguity does not a ect the expectation value since it is invariant under A ! 4 A, which exchanges the regions inside and outside the loop. 2.1 1=8 BPS Wilson loop The 1=8 BPS Wilson loops is a generalization of the standard Wilson loop and it couples to a certain combination of the N = 4 SYM scalars, as well as the gauge eld [8{12]. In order to preserve 1=8 of the superconformal symmetry, the contour C must lie on a S2 subspace of R4, which we may take to be parametrized by x21 + x22 + x23 = 1, and the coupling to the scalars is prescribed to be W 1 N tr P heHC(iAj+ kjlxk l)dxj i where i; j; k = 1; 2; 3, and we pick three out of the six scalar elds to be coupled to the loop. In what follows we will focus for simplicity on the fundamental Wilson loop, namely the trace in (2.1) is over the fundamental representation of the gauge group U(N ). However our construction below can be easily extended to arbitrary gauge group and arbitrary representations. The expectation value of this Wilson loop can be computed by supersymmetric localization [1, 13]. The result only depends on the rank of the gauge group N , the coupling constant gYM and the area of the subregion inside the contour C, see gure 1, which we denote by A [10{13]: (2.1) (2.2) (2.3) (2.4) To emphasize its dependence on the area, we sometimes denote hWi as hW(A)i. This matrix model integral can be evaluated explicitly [9] as hWi = 1 N operator (2.1) couples to a single scalar eld. This special case corresponds to the 1=2-BPS Wilson loop, see section 2.3 below. In the large N limit, the result simpli es and can be expressed in terms of the Bessel function [8]: This can also be rewritten in terms of the -deformed Bessel functions introduced in [26], hWijlarge N = p 0 2 + I1( p 0) : n # 2 ; In = 1 2 In p " + n 2 ( 1)n 1 2 2 ; where ~ is a position-dependent scalar,4 The BPS correlators we study in this paper are given by the following choice of the 1 N h tr P O1( 1) On( n)eHC(iAj+ kjlxk l)dxj i N = 4 SYM : (2.8) 2 [0; 2 ] and i's are the positions of the operator as hWijlarge N = p Ia=2 : 1 2 In (2.7), all the area dependence is encoded in the function I1a=2. This property turns out to be very useful when we later derive the integral expression for the topological correlators at large N . 2.2 Correlators on the 1=8 BPS loop The correlation function of the local operators on the Wilson loop is de ned by3 Oi( i) ~ Li ( i) ; ~ ( ) = x1( ) 1 + x2( ) 2 + x3( ) 3 + i 4 : An important property of such correlators is that they do not depend on the positions of the insertions i's. This follows5 from the fact that, after localization, these operators are 4Throughout this article, we use the convention in which the scalar propagator reads 3Note that here we do not divide the correlator by the expectation value of the Wilson loop hWi. ( I (x1))a b ( J (x2))c d = gYM ad cb IJ ; 2 8 2jx1 x2j2 where a-d are the U(N ) matrix indices, and I; J = 1; : : : ; 6. 5Alternatively, one should be able to show that the (twisted) translation generator which moves the positions of the insertions is Q-exact with Q being one of the supercharges preserved by the con guration. { 5 { Therefore, one can view the correlators on the 1=2-BPS Wilson loop as correlators of a defect CFT. To make this point precise, one needs to consider the normalized correlator, which is obtained by dividing the bare correlator (2.8) by the expectation value of the Wilson loop: hhO1( 1)O2( 2) On( n)ii After the normalization, the expectation value of the identity operator becomes unity and the correlators obey the standard properties of the defect CFT correlators. Using these normalized correlators, one can extract the defect CFT data from the topological correlators. To see this, let us consider general two- and three-point functions of BPS operators on the 1=2-BPS loop: GL1;L2 = hh(Y1 ~ )L1 ( 1) (Y2 ~ )L2 ( 2)iicircle ; GL1;L2;L3 = hh(Y1 ~ )L1 ( 1) (Y2 ~ )L2 ( 2) (Y3 ~ )L3 ( 3)iicircle : In (2.14), ~ ( 1; 2 ; 4 ; 5 ; 6) and Yi's are ve-dimensional complex vectors satisfying Yi Yi = 0. Unlike the topological correlators (2.8), the correlators (2.14) depend on the positions of the insertions, and the vectors Yi. However, because of the conformal symmetry and the SO(5) R-symmetry, their dependence is completely xed to be:6 GL1;L2 = nL1 ( ; N ) GL1;L2;L3 = cL1;L2;L3 ( ; N ) L1;L2 (Y1 Y2)L1 (2 sin 212 )2L1 ; (Y1 Y2)L12j3 (Y2 Y3)L23j1 (Y3 Y1)L31j2 2 sin 212 2L12j3 2 sin 223 2L23j1 2 sin 231 2L31j2 ; mapped to eld-strength insertions in two-dimensional Yang-Mills theory (see section 3.1 for further explanation), which enjoys invariance under area-preserving di eomorphisms, making it almost topological. Because of their position independence, we will call them \topological correlators" in the rest of this paper. 1=2-BPS Wilson loop and defect CFT For special contours, the Wilson loops preserve higher amount of supersymmetry. Particularly interesting among them is the 1=2-BPS Wilson loop, whose contour is a circle along the equator and which couples to a single scalar 3 [8, 9]: HJEP05(218)9 with ij i j and Lijjk Lk)=2. Here nL1 is the normalization of the two-point function while cL1;L2;L3 is the structure constant. As shown above, both of 6Of course, one may also write the analogous result for the straight line geometry, which is related to the circle by a conformal transformation. { 6 { (2.12) (2.13) (2.14) (2.15) these quantities are nontrivial functions of and N . Note that, although we often set the normalization of the two-point function to be unity in conformal eld theories, for special operators the normalization itself can have physical meaning.7 For instance, the length-1 operator (Y ~ ) is related to the displacement operator and has a canonical normalization which is related to the Bremsstrahlung function [22]. Now, if we go to the topological con guration by setting the vectors Yi to be Yi = (cos i; sin i; 0; i; 0; 0) ; GL1;L2 jtopological = GL1;L2;L3 jtopological = 1 2 1 2 L1 This shows that the topological correlators compute the normalization and the structure constant in the defect CFT up to trivial overall factors. Alternatively, one can consider GL1;L2;L3 (GL1;L1 GL2;L2 GL3;L3 )1=2 topological = cL1;L2;L3 (nL1 nL2 nL3 )1=2 ; (2.16) (2.17) (2.18) HJEP05(218)9 and get rid of the overall factors. The quantity which appears on the right hand side of (2.18) is a structure constant in the standard CFT normalization; namely the normalization in which the two-point function becomes unity. Note that, for higher-point functions, there is no such a direct relation between the general correlators and the topological correlators: the general higher-point correlators are nontrivial functions of the cross ratios while the topological correlators do not depend at all on the positions. Thus for higher-point functions, one cannot reconstruct the general correlators just from the topological correlators. 3 Computation of the correlators We now compute the correlators on the 1=8 BPS Wilson loop h ~ L1 ( 1) ~ L2 ( 2) ~ Ln ( n)i ; (3.1) using the results from localization. We rst discuss the correlators on the 1=2 BPS Wilson loop from the OPE perspective and then present a general method that applies also to the 1=8 BPS Wilson loop. 3.1 Correlators on the 1=2 BPS Wilson loop from OPE When all the operators are length-1 and the Wilson loop is circular (or equivalently 1=2 BPS), the correlators (3.1) were already computed in [32]. Let us st brie y review their 7Other examples are the stress-energy tensor and the conserved currents, whose two-point functions are related to CT and CJ . { 7 { computation: by performing localization, one can reduce the computation of the 1=8 BPS Wilson loop in N = 4 SYM to the computation of the Wilson loop in two-dimensional Yang-Mills theory in the zero instanton sector [13]. Under this reduction, the insertion of the position-dependent scalar ~ is mapped to the insertion of the dual eld strength F2d: Using this correspondence,8 one can insert ~ 's on the circular Wilson loop by di erentiating its expectation value with respect to the area A: ~ , i F2d : h ~ L ~ ijcircle = A=2 : Using the expression (3.3), one can compute arbitrary correlation functions of single-letter insertions ~ . To study more general BPS correlators, we also need to know how to insert operators of longer length, ~ L with L > 1. The rst guess might be to relate it simply to the L-th derivative of the Wilson loop, ~ L This guess, however, turns out to be incorrect. To see why it is so, let us consider ~ 2 as an example. We know that the second derivative of hWi corresponds to the insertion of two ~ 's on the Wilson loop. Since the correlator we are studying is topological, one can bring the two ~ 's close to each other without a ecting the expectation value and rewrite them using the operator product expansion. This procedure does produce the length-2 operator ~ 2 as we wanted, but the problem is that it also produces other operators:9 OPE ~ ~ = ~ 2 + c1 ~ + c0 1 : : ~ 2 : = ~ ~ c1 ~ c01 : (3.2) (3.3) Here ci's are some numerical coe cients and 1 is the identity operator. Thus, to really get the length-2 operator, one has to subtract these unnecessary OPE terms from ~ ~ : Here ~ ~ on the right hand side denotes two single-letter insertions at separate points while : ~ 2 : is a length-2 operator inserted at a single point. Since this subtraction procedure is conceptually similar to the normal ordering, we hereafter put the normal-ordering symbol : : to the operator obtained in this way. The coe cients ci's are nothing but the OPE coe cients of the topological OPE (3.5). They are thus related to the following three-point functions: c1 / h ~ ~ ~ ijcircle ; c0 / h ~ ~ 1ijcircle : sides in [21]. with k > 2. 8At weak coupling, this correspondence was checked by the direct perturbative computation on both 9Owing to the representation theory of SO(5), the OPE does not produce higher-charge operators, ~ k { 8 { If we were using the operators, ~ =h ~ ~ i1=2 whose two-point function is unit-normalized, the constants of proportionality in (3.7) would have been unity. However the operators we are using here are not unit-normalized and one has to take into account that e ect. This leads to the following expressions for the coe cients c1 and c0: hWi A=2 ; We can repeat this procedure to express operators of arbitrary length in terms of single-letter insertions and compute their correlation functions. Although these procedures can be easily automated using computer programs, they do not give much insight into the underlying structure. In the next section, we discuss a simpler way to reorganize these procedures which also leads to a simple closed-form expression. 3.2 Construction of operators from the Gram-Schmidt orthogonalization As a direct consequence of the subtraction procedures (3.6), the operators constructed above satisfy the following important properties: The operator basis with such properties turns out to be unique and can be constructed systematically by using the so-called Gram-Schmidt orthogonalization. As we see below, it also allows us to write down a closed-form expression for the operators : ~ L : . The Gram-Schmidt orthogonalization is an algorithmic way of getting the orthogonal basis from a given set of vectors. It was recently applied in the computation of Coulomb branch operators in N = 2 superconformal theories in [37]. Its large N limit was discussed in [38] while the case for N = 4 SYM was analyzed further in [39, 40]. What we describe below is a new application of the method to the correlators on the Wilson loop. To get a glimpse of how it works, let us orthogonalize two arbitrary vectors fv1 ; v2g. A simple way of doing so is to de ne new vectors as u1 = v1 ; u2 = v2 hv1; v2i v1 ; where h ; i denotes the inner product between two vectors. This is of course just an elementary manipulation, but the key point is that one can re-express (3.9) as u1 = v1 ; u2 = 1 { 9 { (3.9) (3.10) where j j denotes a determinant of a matrix. This expression can be readily generalized to the case with more vectors. The result reads uk = 1 dk 1 dk = For details of the derivation, see standard textbooks on linear algebra. The new vectors de ned above are orthogonal but not normalized. Their norms can be computed using the de nitions above and we get huk ; uli = dk dk 1 kl : We now apply the Gram-Schmidt orthogonalization to the set of single-letter insertions f1 ; ~ ; ~ ~ ; : : :g. The norms between these vectors are given by the two-point functions, which can be computed by taking derivatives of hWi, h ~ ~ ~ L M hWi We then get the expression for the operator : ~ L : , : ~ L : = 1 DL DL = hWi (1) hWi hWi (L 1) . hWi hWi hWi hWi (1) hWi . . . (L 1) hWi hWi hWi . . . hWi (1) (2) (L) . . . . . . hWi hWi hWi hWi hWi hWi ~ L (L 1) (L) (@A)k hWi. Let us emphasize that this method applies to general 1=8 BPS Wilson loops. To get the result for the 1=2 BPS loop, one just needs to set A = 2 at the end of the computation. For small values of n, one can check explicitly that this expression coincides with the operators obtained by the recursive procedure outlined in the previous subsection. One can also check that the basis obtained in this way satis es the aforementioned three properties. Owing to the property (3.12), the two-point function of the operators ~ L is given by a ratio of determinants: For the 1=2 BPS loop, this provides an exact result for the normalization of the two-point function in the defect CFT (see the discussions in section 2.3), h : ~ L : : ~ M : i = DL+1 DL LM : nL = ( 2)L DL+1 DL A=2 As it is well-known, the result for L = 1 is related to the normalization of the displacement operators while the results for L > 0 provide new defect-CFT observables.10 We will later see in section 5.3 that the large-N limit of these determinants is related to the determinant representation of the generalized Bremsstrahlung function derived previously in [25, 26]. 3.3 A remark on the 1=8 BPS Wilson loop As mentioned above, the Gram-Schmidt process can be applied to the general 1=8 BPS Wilson loops. At the level of formulas, one just needs to keep the area A general in (3.14) and (3.19). However, there is one important qualitative di erence which we explain below. Unlike the 1=2 BPS Wilson loop, the rst-order derivative hWi the general 1=8 BPS Wilson loop. This means that the single-letter insertion ~ has a nonvanishing one-point function; in other words, the two-point function of ~ and the identity operator 1 is nonzero. Therefore, to de ne an orthogonal set of operators, one has to perform the subtraction even for ~ . In fact, by applying the Gram-Schmidt orthogonalization, (1) does not vanish for hWi hWi We thus need to distinguish : ~ : from ~ . This was one of the reasons why we preferred to put the normal-ordering symbol when de ning the operator : ~ J : . 3.4 Results for topological correlators Using the closed-form expression (3.14), one can compute higher-point functions of : ~ N : . To express the result, it is convenient to introduce a polynomial (3.15) (3.16) (3.17) hWi (1) hWi hWi (L 1) . . . 1 hWi hWi . . . hWi X (1) (2) (L) . . . hWi hWi . . . hWi XL (L) (L+1) 10Although the normalization of the operators is usually not meaningful, for this class of operators, there is a canonical normalization induced by the facts that ~ is related to the displacement operator and : ~ L : is essentially a product of L ~ 's. By replacing Xk by ~ ~ , one recovers : ~ L : . In terms of these polynomials, the higherpoint function reads k m k=1 Let us make two remarks regarding this formula: rst, the derivatives @A0 's on the right hand side act only on the last term hW(A0)i (not on the coe cients of the polynomials FLk ). Second, the polynomial FL is not just a technical tool for writing down higherpoint correlators, but it gives an explicit map between the OPE and the multiplication of polynomials. To see this, consider a product of two such polynomials. Since the product is also a polynomial, one can express it as a sum of FL's, HJEP05(218)9 FL1 (X)FL2 (X) = cL1;L2;M FM (X) ; where cL1;L2;M is a \structure constant" for the multiplication of polynomials. This expansion can be performed also on the right hand side of (3.19). On the other hand, we can perform a similar expansion on the left hand side of (3.19) using the OPE, (3.20) (3.21) (3.22) : ~ L1 : : ~ L2 : = cL1;L2;M : ~ M : : Equating the two expressions, we conclude that these two structure constants must coincide, namely cL1;L2;M = cL1;L2;M . This provides an interesting correspondence between the multiplication of polynomials and the OPE. We can also express the results more explicitly in terms of determinants. For this purpose, we rst perform the Laplace expansion of the polynomial FL(X): L 1 DL n=0 Here DL(i;j) is a minor of DL obtained by deleting the i-th row and j-th column. We then substitute this expression into (3.19) to get L1+L2 X M=0 L1+L2 X M=1 We can also perform one of the sums explicitly to reconstruct a determinant: the result with ntot reads h at the origin, and the insertions ZL. The scalar coupling of each semi-in nite line is given by the vector ~n1;2, and the relative angle between the two vectors is . The divergence from this Wilson line is controlled by the generalized Bremsstrahlung function. . . . hWi hWi (L 1) (n) hWi hWi (L) (n+1) hWi (L) (L+1) hWi . . . hWi hWi for \extremal" correlators which satisfy L1 = Pm that survives and we get a simpler formula Importantly, D~ L;n vanishes unless n L since otherwise the last row coincides with one of the rows above. This allows us to restrict the sum in (3.24) to n0tot L1. In particular, k=2 Lk, there is only one term in the sum h For general correlators, the expression (3.24) is not very concise as it involves several terms. The results for two- and three-point functions of operators with L 3 are given explicitly in appendix A. We will later see in sections 5 and 7 that in the large N limit there is an elegant reformulation in terms of integrals and a matrix model. 4 Generalized Bremsstrahlung functions Z 0 1 As an application of our method, in this section we compute the so-called \generalized Bremsstrahlung function". The result provides nite-N generalization of the planar results computed previously in [25, 26] using integrability [41, 42]. 4.1 Cusp anomalous dimension and Bremsstrahlung function Let us rst recall the de nition of the generalized Bremsstrahlung function. Consider the following cusped Wilson line with insertions (see also gure 2): WL( ; ) P exp d hiA x_ 1 + ~ ~n1jx_ 1ji ZL P exp d hiA x_ 2 + ~ ~n2jx_ 2ji : (4.1) 0 Here Z = 3 + i 4 and the x1;2(t) and ~n1;2 are given by As shown above, WL is parametrized by the two angles and . When = BPS and the expectation value hWLi is controlled by the cusp anomalous dimension nite. However, if 6 = , it has the divergence hWL( ; ) i L: UV rIR L( ; ) L( ; ) = ( )HL( ) + O(( )2) ; Here UV and rIR are the UV and IR (length) cuto s respectively. The cusp anomalous dimension can be expanded near and the leading term in the expansion reads BL( ): The function HL is related to the quantity called the generalized Bremsstrahlung function For L = 0, BL( ) is related to the energy emitted by a moving quark [22] and this is why it is called the generalized Bremsstrahlung function. 4.2 Relation to the two-point function and the result at nite N To compute BL from our results, one has to relate it to the topological correlators. For L = 0 this has already been explained in [22]. As we see below, essentially the same argument applies also to L 6= 0 (see also [21]). The rst step is to consider a small deformation away from the BPS cusp by changing the value of . Then, the change of the expectation value can be written as where 0 is hWLi = hWLi d hh 0( )iicusp 0 = sin and hh iicusp is the normalized correlator of the scalar insertion on the cusped BPS Wilson loop WL( ; = ). Using the invariance of WL under the dilatation around the origin, the -dependence of hh 0iicusp can be xed to be 1 hh 0( )iicusp = hh 0( = 1)iicusp : We can then compare (4.3) with (4.6) (introducing the UV and IR cuto s to evaluate the integral), to get L = ( )hh 0( = 1)iicusp + O(( )2) : (4.2) , WL is (4.3) cusped Wilson line to a con guration depicted above. The red and black semi-circles correspond to the two semi-in nite lines in gure 2 of the same color. The angle between the two semi-circles is . The loop divides the S2 into two regions with areas 2 2 . (Note that we already set = in this gure.) S2 by the conformal transformation, The second step is to map the BPS cusp WL( ; = ) to the 1=8 BPS Wilson loop on x1 = 2X2 1 + X12 + X22 ; x2 = 2X1 1 + X12 + X22 ; x3 = 1 X12 1 + X12 + X22 X22 : Here xi's are the (embedding) coordinates of S2 while Xi's are the coordinates on R2 where the cusped Wilson loop (4.2) lives. After the transformation, and changing variables by = cot(t=2), the two semi-in nite lines of the cusped Wilson loop are mapped to the two arcs on S2 (see also gure 3), (x1; x2; x3) = ((0; sin t; cos t) ( sin sin t; cos sin t; cos t) 0 < t < t 2 ; (4.10) (4.11) where the rst arc (0 < t ) and the second arc ( < t 2 ) correspond to the black and the red lines in gure 2 respectively. The rst arc couples to 1, and the second one to cos 1 + sin 2, in accordance with our conventions (2.1) for the 1/8-BPS loop. As shown in gure 3, the resulting Wilson loop has cusps at the north and the south poles (tN and tS) with insertions ZL and ZL respectively. The insertion 0( = 1) is mapped to the insertion at a point11 te where the red arc intersects the equator of S2. We then arrive at 11In terms of the parametrization given in (4.11), tS = 0, tN = and te = 3 =2. the relation between the expectation values, where h i denotes a (un-normalized) correlator on the Wilson loop on S2. Now, a crucial observation is that one can complete the insertion 0(te) to the position dependent scalar 0 i 4 = sin 4 vanishes owing to the charge conservation. Furthermore, ZL(tN ) and ZL(tS) can be identi ed with : ~ L : . We thus arrive at the following relation,12 Note that ~ in the middle is not normal-ordered since it comes directly from the deformation of the loop. Since the area surrounded by this loop is given by A = 2 2 , one can express From (4.9) and (4.15), we can compute the generalized Bremsstrahlung function as (4.12) (4.13) (4.15) (4.16) (4.17) (4.18) DL+1 : DL As given in (3.14), DL is the following simple determinant, with A = 2 2 . In the limit ! 0, the formula takes a particularly simple form, hWi (1 i; j L) ; BL(0) = 4 DL+1 DL =0 = DL+1 DL A=2 : This is the main result of this section. In the next section, we will see that the formula (4.16) reproduces the results in [25, 26] in the large N limit. Note, however, that our results (4.16) is also valid at nite N : all one has to do is to plug in (4.17) the nite N form of the Wilson loop expectation value, which is given in (2.3). The rst few explicit results for = 0 are given in the appendix. 5 Large N limit In this section, we study in detail the topological correlators in the large N limit. In particular, we derive a simple integral expression. 12Precisely speaking, the area derivative can also act on the operator : ~ L : (in addition to inserting an extra single-letter insertion) since it is given by a sum of single-letter insertions with the area-dependent coe cients: : ~ L : = ~ L + c1(A) ~ L 1 + : (4.14) However, since the leading coe cient is 1, @A : ~ L : only starts with ~ L 1. Therefore, one can always express @A: ~ L : as a sum of : ~ k : with k < L. We thus conclude that such contributions vanish because of the orthogonality, h: ~ k : : ~ L : i = 0 for k < L, and do not a ect (4.15). 5.1 As mentioned before, an important simpli cation in the large N limit is that hWi can be expressed in terms of the deformed Bessel function (2.7). A nice feature of the deformed Bessel function is that it admits an integral expression [43], In = I dx 2 ixn+1 sinh(2 g(x + 1=x))e2g (x 1=x) ; where here and below we use the notation Applying this to (2.7), we can express hWi and its derivatives simply as where the measure d is de ned by g hWi = p 4 I I : d ; d 2 g hWi ; d = dx sinh(2 g(x + 1=x))ega(x 1=x) Second, they are normalized as Third, since QL(x) is a polynomial of X = g(x x 1), it follows that I d (x) QL(x)QM (x) = DL+1 DL (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) 2 and the 1=2-BPS Wilson loop corresponds to a = 0. Combining this integral expression with the formula (3.19), we obtain a simple integral expression for the multi-point correlators, h the rest of this section, it is related to the Quantum Spectral Curve [34]. The functions QL(x) have several important properties. First, owing to the orthogonality of the two-point functions, they satisfy the following orthogonality relation: Furthermore, they satisfy the following equalities: The rst equality follows from I I d (x) xQL(x)QM (x) = 0 ; d (x) x2QL(x)QM (x) = DL+1 DL where in the second equality we used the property of the measure,13 R d ( 1=x) = R x2d (x). In a similar manner, the second equality can be proven: d (x) x2QL(x)QM (x) = d ( 1=x) QL( 1=x)QM ( 1=x) = d (x) QL(x)QM (x) : The equalities (5.8), (5.11) and (5.12) imply that QL(x) and xQL(x) for L 2 N 0 together form a set of orthogonal functions under the measure d (x). They are in fact the Gram-Schmidt basis obtained by applying the orthogonalization to the set of functions f1; x; x 1; x2; x 2; : : :g. As we see below, this characterization of the functions QL(x) plays a key role in identifying them with the functions introduced in the integrability-based approaches [26, 43]. 5.3 Comparison with the results from integrability We now prove the equivalence between our results and the results obtained previously from integrability [25, 26]. For this purpose, we rst show that QL coincides with the function PL, which was introduced in [25, 26] and later shown to be directly related to the so-called \Q-functions" in the Quantum Spectral Curve [44]. The equivalence of other quantities, including the generalized Bremsstrahlung functions, follow from it. As the rst step, let us recall the polynomials PL de ned in [26]: I I I 1 2 . . . I I 0 1 . . . I2L I2L 1 x L x1 L . . . I2 2L I1 2L I3 2L I2 2L . . . I 1 xL 1 . . . I 0 xL ; L 0 ; (5.15) 13Note that an extra minus sign comes from a change of the direction of the contour. (5.11) (5.12) (5.13) The expression (5.50) contains an extra dependence on the area, (2 a)2. However, this can be absorbed into the integral by using 0 48N 2 I2( p 0) = = Applying this to the rst nonplanar correction, we get with with and performing the integration by parts. As a result, we get 0 48N 2 I2( p 0) = dx e2 g(x+ x1 )ega(x x1 ) f (2 g(x + 1=x)) ; f (z) = (2 g)4 z2 N 2 3z + 3 3z4 : Thus, using the exponential measure for the planar part, one can write down the corrected measure d 1=N as (5.50) (5.51) (5.52) (5.53) (5.54) (5.55) (5.56) (5.57) + (2 g)4 z2 N 2 3z + 3 3z4 + O(1=N 4) : Since the area dependence only appears in the exponent ega(x 1=x), the expectation value of the Wilson loop and its derivatives retain the following simple expressions: hWi (n) = I d 1=N g(x From this, it follows that the integral expression for the topological correlators (5.6) and the orthogonality condition for QL(x) (5.8) still hold if we replace d with d 1=N . Repeating the same analysis at higher orders, we can determine the corrections to the measure order by order. After working out rst several orders, we found17 the following relation between the terms that appear in the expansion of hWi and the corrections to F (z): ( 0) n2 In( p 0) Integration by parts ! (4 g)2 n r 2 z e zp zKn+ 12 ( z) 17We only checked the relation by Mathematica and did not work out a proof. It would be nice to prove and establish the relation. Note that although the right hand side involves the modi ed Bessel function Kn+ 12 , it actually reduces to a rational function of z. Now, applying this relation to the expansion of hWi given by Drukker and Gross [9], 2 hWi = p 0 I1( p 0) + X 1 k=1 1 N 2k k 1 X Xks s=0 0 4 we obtain the following expansion of F (z): z In (5.58) and (5.59), Xks is a numerical coe cient de ned by the following recursion: 4Xks = X10 = 3k 1 12 3k ; s 2 s Xks 11 + 3k 1 s Xks 1 ; Xkk = 0 : (5.60) (6.1) (6.2) It would be interesting to try to resum the series (5.59) and to consider the nonperturbative corrections. 6 Weak- and strong-coupling expansions We now discuss the weak- and the strong-coupling expansions of topological correlators on the 1=2-BPS Wilson loop at large N , and compare them with the direct perturbative results. In particular, we focus on the three-point functions since the topological correlators are closed under the OPE and the match of the three-point functions (or equivalently the OPE coe cients) automatically guarantee the match of higher-point functions. In both cases, we rst compute the expansion of the polynomials QL(x): QL(x) = (Q0L(x) + g2Q1L(x) + Q0L(x) + g1 Q1L(x) + g g 1 1 : To determine the expansion, it is convenient to use the symmetrized measure d sym (5.45) and perform the change of variables from x to y i(x can be rewritten as the following integral of y: x 1)=2. Then, the integral over x I d sym ( ) = Note that we set a = 0 since we consider the 1=2-BPS loops. In what follows, we use this representation for the measure to compute the weak- and the strong-coupling expansions. 6.1 Let us expand the measure (6.2) at weak coupling, At the leading order, it is given by : This coincides with the measure for the Chebyshev polynomials of the second kind. Thus, taking into account the di erence of the normalization, we conclude that QL at the leading order at weak coupling is given by where UL(y) is the Chebyshev polynomial of the second kind determined by the following recursion relation: Q0L(x) = ( ig)LUL(y) ; U0(y) = 1 ; U1(y) = 2y ; UL+1(y) = 2yUL(y) UL 1(y) : Having identi ed QN with the Chebyshev polynomial, one can now compute the twoand the three-point functions by using the identities, 2 Z dyp1 y2UL(y)UM (y) = LM ; UL(y)UM (y) = (L M ) : M k=0 X UL M+2k(y) (6.3) (6.4) (6.5) (6.6) (6.7) (6.8) (6.9) Using these identities to evaluate the integral expressions for the correlators (5.6), we get where Ltot is given by and the symbol dL1;L2;L3 denotes h : ~ L1 : : ~ L2 : i O(g0) h : ~ L1 : : ~ L2 : : ~ L3 : i O(g0) = ( g2)L1 L1;L2 ; 2 Ltot = ( g ) 2 dL1;L2;L3 ; Ltot L1 + L2 + L3 ; dL1;L2;L3 = <1 8 :0 As shown in (6.10) the three-point function is nonzero only when the triangle inequalities are satis ed and the sum of the lengths of the operators is even. These results precisely match the tree-level planar Wick contractions. Note that, at this order, the expectation value of the Wilson loop is 1 and there is no distinction between the un-normalized and the normalized correlators hh ii. Let us now discuss the one-loop correction. At one loop, the measure receives an additional contribution, This change of the measure induces the change of the orthogonal polynomials Q1L since they need to satisfy the modi ed orthogonality condition Z Z Furthermore, in order to keep the normalization condition (5.9), the correction Q1L must be a polynomial of y with the order < L. One can solve these conditions using the equality HJEP05(218)9 and the result reads 1 4 (1 y2)UL(y) = (2UL(y) UL+2(y) UL 2(y)) ; Q1L(y) = ( ig)L 2 2 3 UL 2(y) : We can then compute the correction to the two- and the three-point functions using the integral representation for the correlators (5.6) as follows: (6.11) (6.12) (6.13) (6.14) (6.15) (6.17) (6.18) (6.16) h : ~ L1 : : ~ L2 : i O(g2) : ~ L1 : : ~ L2 : : ~ L3 : i O(g2) (2 g)2( g2)L1 where da;b;c is 1 only when a + b + c is even and they satisfy the triangular inequality, (otherwise zero). Using the identities18 dL1;L2 2;L3 = dL1;L2;L3 dL1;L2;L3 2 = dL1;L2;L3 L2+L3;L1 L3+L1;L2 L1+L2;L3 + L3+L1;L2 2 ; L2+L3;L1 + L1+L2;L3 2 ; we can rewrite the three-point function also as h : ~ L1 : : ~ L2 : : ~ L3 : i O(g2) = (3dL1;L2;L3 L1+L2;L3 L2+L3;L1 L3+L1;L2 ) : 18These identities can be derived by expressing da;b;c as a product of step functions and using the fact that (x + 1) = (x) + x; 1 and (x 1) = (x) x;0. O(g2), we get the following results for the normalized correlators: By dividing the correlators by the expectation value of the Wilson loop hWi = 1 + 2 2g2 + hh: ~ L1 : : ~ L2 : ii O(g2) = hh: ~ L1 : : ~ L2 : : ~ L3 : ii O(g2) = 2( g)2( g2)L1 They are in perfect agreement with the direct one-loop computation performed in [29]. For completeness, let us also present the structure constant in the standard CFT normalization; namely the normalization in which the two-point functions become unity. The result up to O(g2) reads t = p with given by y2) y = 0 : Expanding the measure around this saddle point and performing the change of variables 2 gy, we obtain the following expression for the measure at strong coupling: 1 dy 2 Z 1 y2) = 1 e4 g (2 )5=2g3=2 e t2 dt ; 1 g t 8 4 ; d 0(t) + d 1(t) + O(g2) + O(e g) : 2 3 1 dy y=y (6.19) (6.21) (6.22) (6.23) (6.24) (6.25) Strong coupling expansion Let us consider the expansion at strong coupling. Here we send g ! 1 while keeping the lengths of the operators Li's nite. In this limit, the integral : ~ L1 : : ~ L2 : : ~ Ln : i = 2 Z 1 can be approximated by its saddle point, At the leading order, the measure d 0 is simply a gaussian. As is well-known, this is nothing but the measure for the Hermite polynomials. Thus, QL(x) at strong coupling is Q0L(x) = ( i)L g 2 L=2 HL(t) : Here the factor ( i)L(g=2 )L=2 comes from the normalization of QL (5.9), and HL(t) is the Hermite polynomial de ned by H0(t) = 1 ; H1(t) = 2t ; HL(t) = 2tHL 1(t) 2(L 1)HL 2(t) : We can then compute the two- and the three-point functions using the properties of the Hermite polynomials, 1 dt e t2 HL(t)HM (t) = 2LL!p LM ; HL(t)HM (t) = M X k=0 2M kL!M ! (L M + k)!(M The results are given by HL M+2k(t) (L M ) : HJEP05(218)9 : ~ L1 : : ~ L2 : i h : ~ L1 : : ~ L2 : : ~ L3 : i = e4 g e4 g L1! L1L2 ; Lt2ot L1!L2!L3! dL1;L2;L3 ; Lk)=2. Note that the overall coe cient e4 g=(2(2 g)3=2) is precisely the expectation value of the circular Wilson loop at strong coupling. Therefore, the normalized correlators take the following simple form: hh: ~ L1 : : ~ L2 : ii g!1 hh: ~ L1 : : ~ L2 : : ~ L3 : ii g!1 = = g L1 g L1! L1L2 ; Lt2ot L1!L2!L3! dL1;L2;L3 : These results reproduce the strong-coupling answer, which is given by the generalized free elds in AdS2. Let us now compute the correction to this strong coupling answer. At the next order, the measure receives a correction d 1, given by (6.24). As in the weak-coupling analysis, the change of the measure induces the correction to QL since they have to satisfy the modi ed orthogonality condition: To solve this condition, we use the following property of the Hermite polynomial: 1 t4HL(t) = 1 16 H4(t) + 12H2(t) + 12 1 HL(t) d 1Q0LQ0M + = HL+4 + + (2L 1) 2L + 3 4 L! (L 2)! HL+2 + HL 2 + 3(2L2 + 2L + 1) (L L! 4 4)! HL 4 : HL (6.26) (6.27) (6.28) (6.29) (6.30) (6.31) Q1L(x) = ( i)L 8 : ~ L1 : : ~ L2 : i O(1=g) : ~ L1 : : ~ L2 : : ~ L3 : i O(1=g) Using this result, we can compute the correction to the two-point function as L! (L 2)! L! (L 4)! (2L 1) HL 2(t) + HL 4(t) : (6.32) e4 g e4 g Ltot 2 L1! L1L2 32 g 3 3 64 g (2L12 + 2L1 + 1) ; (Lt2ot + 2Ltot + 2) (6.33) (6.34) (6.35) (6.37) Since the expectation value of the Wilson loop can be expanded at strong coupling as hWi e4 g 1 3 32 g + O(1=g2) ; the normalized correlators are given by hh: ~ L1 : : ~ L2 : ii O(1=g) hh: ~ L1 : : ~ L2 : : ~ L3 : ii O(1=g) g L1 Ltot 2 L1! L1L2 32 g 3 (2L12 +2L1) ; 3 64 g (Lt2ot +2Ltot) L1!L2!L3! dL1;L2;L3 : 1=2 64 g 1 2 As we will see in the next subsection, these results are in perfect agreement with the direct strong-coupling computation. normalization at strong coupling: Using these results, we can also compute the structure constant in the standard CFT hh: ~ L1 : : ~ L2 : : ~ L3 : ii hh: ~ L1 : : ~ L1 : iihh: ~ L2 : : ~ L2 : iihh: ~ L3 : : ~ L3 : ii p L12j3!L23j1!L31j2! L1!L2!L3! dL1;L2;L3 1+ Comparison to string theory 3(L21 +L22 +L23 2(L1L2 +L2L3 +L3L1)) 1 g2 +O : (6.36) SB = p 2 Z 1 2 In this section we show that the strong coupling expansion of the localization results derived above precisely matches the direct perturbative calculation using the AdS5 S5 string sigma model. As is well-known, on the string theory side the 1/2-BPS (circular or straight) Wilson loop is dual to a minimal surface with the geometry of an AdS2 embedded in AdS5 (and pointlike in the S5 directions). The dynamics of the string worldsheet uctuations is most conveniently described using the Nambu-Goto action in static gauge. The bosonic part of the string action up to the quartic order was written down explicitly in [32] and it reads d Here g is the AdS2 worldsheet metric, ya; a = 1; : : : ; 5 are the massless uctuations in the S5 directions, which are dual to the scalar insertions a on the gauge theory side, and xi; i = 1; 2; 3 are the m2 = 2 uctuations in AdS5 dual to insertions of the displacement operator [46, 47]. For the explicit form of the quartic vertices, see [32]. Note that there are no cubic vertices between the elementary bosonic uctuations. Let us rst review the result for the tree-level connected four-point function of the ya uctuations computed in [32], and its agreement with the localization prediction. Taking the circular geometry at the boundary, it takes the form hY1 y( 1)Y2 y( 2)Y3 y( 3)Y4 y( 4)icAodnSn2: = GT ( )+ (GT ( )+GA( ))+ (GT ( ) GA( )) : p 2 2 2 Y1 Y2 Y3 Y4 1 (4 sin 212 sin 234 )2 p where is the cross-ratio and ; are SO(5) cross-ratios 2 5 = sin 212 sin 234 sin 213 sin 224 = Y1 Y3 Y2 Y4 Y1 Y2 Y3 Y4 Y1 Y4 Y2 Y3 Y1 Y2 Y3 Y4 with Yi null polarization 5-vectors. The functions of cross-ratio GS;T;A( ) appearing in the 4-point function above correspond to singlet, symmetric traceless and antisymmetric channels, and their explicit form can be found in [32]. In writing (6.45) we have taken the normalization of the y uctuations such that the leading order 2-point function computed from the string action reads19 hY1 y( 1)Y2 y( 2)iAdS2 = Y1 Y2 2 2 (2 sin 212 )2 This normalization agrees in the strong coupling limit with the normalization we adopted on the gauge theory side, which gives hhY1 ( 1)Y2 ( 2)ii = I2( p ) 2 2 I1( Y1 Y2 p ) (2 sin 212 )2 = 2 2 1 + : : : Y1 Y2 (2 sin 212 )2 : : 2 3 p (6.38) (6.39) (6.40) (6.41) (6.42) (6.43) (6.44) We now specialize to the topological boundary operators, by choosing the polarizations Yi = (cos i; sin i; 0; i; 0; 0) : By analogy with the notation introduced earlier on the CFT side, let us de ne y~( ) cos( )y1( ) + sin( )y2( ) + iy4( ) ; 19In [32] instead a canonical normalization of the kinetic term was used, so that the leading 2-point function was the overall independent. The normalization in (6.41) is actually the one which is naturally induced by dependence in the string action, upon adopting the standard AdS/CFT dictionary to compute the tree-level 2-point function, see [45]. F F F F F F F to next-to-leading order at strong coupling. The grey blob in the middle gure denote the one-loop correction to the \boundary-to-boundary" y propagator. HJEP05(218)9 which is dual to the insertion of ~ and has the constant 2-point function given at leading order by hy~( 1)y~( 2)iAdS2 = 4 2 . Then, using the explicit form of GS;T;A( ), one nds the position independent result for the connected 4-point function hy~( 1)y~( 2)y~( 3)y~( 4)icAodnSn2: = 3 16 4 : The full 4-point function to the rst subleading order also receives contribution from disconnected diagrams, as shown in gure 4. In addition to the leading tree-level generalized free- eld Wick contractions, there are corrections of the same order as (6.45) coming from disconnected diagrams where one leg is one-loop corrected, see the gure. While these corrections have not been computed explicitly yet from string theory, we will assume below that they reproduce the strong coupling expansion of (6.42).20 Then, the 4-point function of single-letter insertions computed from the AdS2 string theory side reads to this order hy~( 1)y~( 2)y~( 3)y~( 4)iAdS2 = 4 2 !2 " 1 3 2 2 3 +: : : = 3 p where the rst term in the bracket is the contribution of disconnected diagrams, and the second term the one of the tree-level connected diagram. This precisely matches the strong coupling expansion of the localization result hh ~ ~ ~ ~ 16 4 + 3 2 4 3 p 4 4I1( I0( p ) p ) : Having reviewed the matching of the ~ 4-point function, let us now move to the computation of the two-point and three-point functions of arbitrary length insertions : ~ L : . The Witten diagrams contributing to the 2-point function to the rst two orders in the strong coupling expansion are given in gure 5 (as in gure 4 above, there are one-loop corrections to the diagrams involving free- eld Wick contractions, that for brevity we do not depict in the gure). The contribution of the diagram involving the 4-point vertex can be obtained from the 4-point result (6.38) by taking Y2 ! Y1; Y3;4 ! Y2, and taking the limit 20Alternatively, one may consider normalized correlators as in (6.35), where such corrections drop out in the ratio. (6.45) (6.46) (6.47) ture the case L = 4 is shown). The diagrams on the left, corresponding to generalized free- eld contractions, also receive a subleading correction where a y-propagator is one-loop corrected. HJEP05(218)9 2 ! 1; 4 ! 3 at small , and so we get 2. From [32], we have GT ( ) = 3 2 + : : : and GA( ) = O( 3 log( )) 2 h(Y1 y( 1))2(Y2 y( 2))2iAdS2 = 3 p (Y1 Y2)2 64 4 sin4 12 2 = 3 16 4 ; (6.48) where the rst equality is valid for any choice of the null polarization vectors, and in the second equality we have specialized to the topological con guration. We can now use this result and some elementary combinatorics to compute the 2-point functions for arbitrary length. We nd y~Ly~LiAdS2 = 4 2 !L 2 4L! 1 3L 2 L 2 !2 3 (L 2)! + : : :5 (6.49) L 2 ! The rst term in brackets corresponds to the generalized free eld Wick contractions: there are clearly L! such contractions, and the factor (1 + : : :)L = 1 2 3L + : : : accounts p for the one-loop correction of the boundary-to-boundary legs, as discussed above. The second term in brackets corresponds to the diagrams involving the 4-point vertex shown in gure 5: there are ways of picking two y's on each operator, and (L 2)! free- eld contractions among the remaining y's. This result can be simpli ed to y~Ly~LiAdS2 = p 4 2 !L L! 1 p L(L + 1) + : : : ; (6.50) 3 2 3 p 4 3 which indeed precisely agrees with the localization result given in (6.29) and (6.35). Similarly, the diagrams contributing to the 3-point function hh: ~ L1 : : ~ L2 : : ~ L3 : ii are shown in gure 6. The leading contribution is given again by free- eld Wick contractions. Let us de ne the number of such contractions to be nL1;L2;L3 L1!L2!L3! dL1;L2;L3 with dL1;L2;L3 given in (6.10). At the subleading order, there are two topologies which involve the 4-point vertex: one where the vertex connects two y's belonging to two di erent FL3 FL3 FL3 operators (in the picture the case L1 = L2 = 4; L3 = 2 is shown). In addition, there are one-loop corrections to the generalized free- eld diagrams shown on the left. operators, and one where it connects two y's from one operator and two y's from two separate operators, see the gure.21 The rst type of diagram can be computed using (6.48). For the second type of diagram, again taking the limit of the 4-point result (6.45) by setting Y2 ! Y1; Y3 Y2; Y4 Y3 and similarly for the i points, one nds h(Y1 y( 1))2 Y2 y( 2) Y3 y( 3)iAdS2 = 3 p (Y1 Y2)(Y1 Y3) 64 4 sin2 212 sin2 13 2 = 3 16 4 where we have specialized to the topological con guration in the second step, but the rst equality holds in general. Then, working out the relevant combinatorics and putting all the contributions together, we nd for general lengths hy~L1 y~L2 y~L3 iAdS2 = 3 3 nL1 2;L2 2;L3 nL1 2;L2 1;L3 1 p ! Lt2ot " 4 2 L1 2 2 nL1;L2;L3 1 2 3(L1 +L2 +L3) +: : : 2 L2 +nL1 2;L2;L3 2 L1 2 2 L3 +nL1;L2 2;L3 2 L2 2 L3 2 L1 L2L3 +nL1 1;L2 2;L3 1 L2 L1L3 +nL1 1;L2 1;L3 2 2 2 L3 L1L2 ; (6.52) (6.53) with Ltot = L1 + L2 + L3. This simpli es to y~L1 y~L2 y~L3 iAdS2 = 4 2 ! Lt2ot nL1;L2;L3 1 3Ltot(Ltot + 2) 16p + : : : ; (6.54) again in complete agreement with the localization prediction (6.29) and (6.35). In a similar way, one can compute higher-point correlation functions of : ~ L : insertions to next-to-leading order at strong coupling. While for the topological operators the agreement of these should follow from the agreement of 2-point and 3-point functions shown above, to dispel any doubt we have explicitly veri ed in various higher-point examples that the localization results are indeed correctly reproduced by string perturbation theory around the AdS2 minimal surface. 21Note that there is no diagram where the 4-vertex connects three y's on the same operator, as this vanishes by SO(5) symmetry: in terms of the null polarization vectors, it necessarily involves a factor Yi Yi = 0. with Xi xi 1). Here we used the exponential measure (5.46) for later convenience, but the results in this subsection are equally valid if we substitute it with d or d sym. The determinant in (7.1) has the structure of the Vandermonde determinant and it can be rewritten as DL = d exp(xk) ; (7.1) YL I k=1 X2 1 . . . X1 X22 . .. . . . XLL 1 XLL X1L 1 X2L . . . XL2L 2 ! 1 X2 . . . X1 X22 . .. . . . XLL 1 XLL X1L 1 X2L . . . XL2L 2 = Y Xk 1 Y(Xj k i<j Xi) : Since the measure factors in (7.1) are symmetric under the permutation of the indices, we can replace the right hand side of (7.2) with its symmetrized version, Y Xk 1 Y(Xj k i<j Xi) ! L! i<j Xi) X ( 1)j j Y Xkk 1 : 2SL We then realize that the sum over the permutation is precisely the de nition of the Vandermonde determinant. We can thus replace the determinant part by In this section, we reformulate our results in the planar limit as a matrix model. We follow closely the approach in the integrability literature [25, 43], but the resulting matrix model is slightly di erent. This reformulation would be useful for studying the semi-classical limit where Li and g are both send to in nity while their ratios are kept nite. We present preliminary results for the semi-classical limit leaving more detailed analysis for future investigation. DL and QL(x) as a matrix model Using the integral representations (5.3) and (5.4), the equation (3.14) can be re-expressed as X1 X22 . .. . . . XLL 1 XLL X1L 1 X2L . . . XL2L 2 Qi<j (Xi Xj )2 L! Therefore, we obtain the multi-integral expression, DL = gL(L 1) L! YL I k=1 d exp(xk) ! Y(xi i<j xj )2 1 + 1 xixj 2 ; (7.2) (7.3) (7.4) (7.5) Note that this matrix-model-like expression is similar but di erent from the matrix model for m2L, derived in [43]. One notable di erence is that the integral in [43] contains 2L integration variables while the integral derived here contains only L integration variables. As proven in section 5.3, the two determinants are related by (5.42). One can also express the polynomial FL as a multiple integral. Applying the integral expression (5.4) to (3.18), we get FL[X] = 1 DL YL I k=1 d exp(xk) XLL 1 XLL X2 1 . . . 1 X1 X22 X . .. . . . X1L X2L+1 . . . XL2L 1 XL : (7.6) xk xk 1. Now the determinant part in the integrand can be evaluated as XLL 1 XLL X2 1 . . . 1 X1 X22 X . .. . . . X1L X2L+1 . . . XL2L 1 XL = Y Xk 1(X k Xk) Y(Xj i<j Xi) : Thus, after symmetrization, we get FL(X) = gL(L 1) "YL I L!DL k=1 d exp(xk) X g(xk xk 1) Y(xi xj )2 1+ i<j 1 xixj As can be seen from this expression, FL(X) is the analogue of the characteristic polynomial of the matrix model, which is obtained by inserting det(X M ) in the integral of the matrix M . After the change of the variables X = g(x x 1), it can be rewritten as QL(x) = FL(g(x x 1)) L!DL k=1 d exp(xk) (x xk) 1+ 1 xxk Y(xi xj )2 1+ i<j 1 xixj 7.2 Classical limit of the matrix model Let us now consider the limit where g and Li's are sent to in nity while their ratios remain nite. This limit corresponds to a classical string con guration in AdS and therefore is called the (semi-)classical limit. The integral expression (7.5) can be rewritten as DL = gL(L 1) I (4 g)LL! YL dxk(1 + xk 2) eSL(x1;:::;xk) ; where the action is given by SL = L k=1 X 2 g xk + 1 xk + ag xk + 2 X log (xi xj ) 1 + k=1 1 xk 2 i L i<j 2 : 2 : (7.7) (7.8) (7.9) 1 xixj (7.10) : (7.11) two branch cuts; the one coming from the condensation of xk and the other coming from the condensation of 1=xk. In the classical limit, the integral can be approximated by the saddle point @SL=@xk = 0. To compare with the result from integrability, it is convenient to introduce the rapidity variables uk g xk + : x2 4 2 j6=k (xk 1 5 = As in the usual large N matrix models, we expect that xk's condense into a branch cut in the classical limit as shown in gure 7. To describe the limit, it is convenient to introduce a function pL(x) de ned by pL(x) 2 1 x2 4 2g k=1 (x 1 xk) 1 + xx1k 3 5 : Then, the saddle-point equation (7.13) can be rewritten as where xk denote the two di erent sides of the branch cut. Since p(x) has the symmetry 1 2 [pL(xk + ) + pL(xk )] = ; pL(x) = pL( 1=x) ; (7.12) (7.13) (7.14) (7.15) (7.16) owing to (7.16). 1 2 pL x 1 + + pL x 1 It turns out that the function pL(x) coincides with the quasi-momentum computed = a=2). To see this, let us rewrite (7.14) using the identity, where we de ned xk with k > L as pL(x) = a x2 + 1 2 x2 1 y 2L X xk and it agrees with (3.15) in [43] (after appropriate reordering of xk's). These two agreements guarantee that our p(x) has the same analytic properties as the quasi-momentum in [43], which uniquely specify the function. We thus conclude that the two functions must be the same. Using the quasi-momentum pL(x), we can also express the semi-classical limit of QL(x). By taking the saddle-point of the integral expression (7.9), we obtain where xk's are the saddle-point values of the integration variables, which satisfy (7.13). Using the de nition of pL(x), we can also rewrite (7.22) as Z u(x) QL(x) gL exp du0 pL(u0) + a u0 2g p(u0)2 4g2 where we introduced the rapidity variable u de ned by u g(x + 1=x). 22Precisely speaking, there is a small di erence from (3.13) in [43]: in their case, the number of roots is 2L + 1 whereas it is 2L in our case, even after we doubled the number of roots by (7.20). However, this di erence does not a ect the leading semiclassical answer. xk L (k > L) : 1 2 [pL(xk + ) + pL(xk )] = 1 L + 1 L 2L the branch cut of xk's is accompanied by another branch cut that is formed by Around this other branch cut, pL(x) satis es (7.18) (7.17) (7.19) (7.20) (7.21) (7.22) (7.23) the transformation x ! be re-expressed as The expression (7.19) coincides22 with the de nition of the quasi-momentum (3.13) in [43] if we take into account the fact that the distribution of xk's in [43] are symmetric under 1=x. Furthermore, using (7.20), the saddle-point equation can 1 x x 2 1 g L Y(x k=1 xk) 1 + 1 xxk ; Given the match of the quasi-momentum, we can follow the argument of [43] and show that the semi-classical limit of our matrix model correctly reproduces the Bremsstrahlung function computed from classical string. More interesting and challenging would be to compute the semi-classical limit of the structure constants using the integral representation (5.6) and the asymptotic formula for QL (7.23). We leave this for future investigation. Before ending this section, let us also point out that one can study the nonplanar corrections to DL and QL by replacing the measure d exp in (7.5) and (7.9) with the nonplanar measure d 1=N given in section 5.5. It would be interesting to analyze the classical limit of the nonplanar corrections using our matrix model and match it with a classical string con guration. 8 In this paper, we computed a class of correlation functions on the 1=8 BPS Wilson loop by relating them to the area derivatives of the expectation value of the Wilson loop. When restricted to the 1=2 BPS loop, the results provide in nitely many defectCFT data. As a byproduct, we also obtained nite-N generalization of the generalized Bremsstrahlung function. Let us end this paper by mentioning several future directions worth exploring: rstly, it would be interesting to generalize our analysis to include operators outside the Wilson loop. In the absence of insertions on the loop, such correlators were computed in [18, 19] using the relation to 2d YM. Combining their results with our method, it should be possible to compute the correlators involving both types of operators. Work in that direction is in progress [48]. Once such correlators are obtained, one can try to numerically solve the defect CFT bootstrap equation [33] using these topological correlators as inputs. Another interesting direction is to apply our method to other theories, in particular to N = 2 superconformal theories in four dimensions, for which the Bremsstrahlung function was recently studied in [49]. Having exact correlators for these theories would help us understand their holographic duals, including the dual of the Veneziano limit of N = 2 superconformal QCD [50]. At large N , we have shown that the correlators are expressed in terms of simple integrals. A challenge for the integrability community is to reproduce them from integrability. In the hexagon approach to the structure constants [51, 52], the results are given by a sum over the number of particles. At rst few orders at weak coupling where the sum truncates, it is not so hard to reproduce our results [ 29, 53 ]. A question is whether one can resum the series and get the full results. In many respects, the topological correlators on the Wilson loop would provide an ideal playground for the hexagon approach; one can try to develop resummation techniques, x potential subtleties (if any), and compute nonplanar corrections [54, 55]. Lastly, the appearance of the Q-functions in our large-N results suggests deep relation between localization and the Quantum Spectral Curve. It is particularly intriguing that there is a one-to-one correspondence between the multiplication of the Q-functions and the operator product expansion of the topological correlators. A similar observation was recently made in [56] in a slightly di erent context: they found that the correlators on the Wilson loop in the so-called ladders limit [57], which can be computed by resumming the ladder diagrams [30], simplify greatly when expressed in terms of the Q-functions of the quantum spectral curve. Exploring such a connection might give us insights into the gauge-theory origin of the Quantum Spectral Curve. Acknowledgments We thank N. Gromov, P. Liendo, C. Meneghelli and J.H.H. Perk for useful discussions and comments. SK would like to thank N. Kiryu for discussions on related topics. The work of SG is supported in part by the US NSF under Grant No. PHY-1620542. The work of SK is supported by DOE grant number DE-SC0009988. A Explicit results for operators with L 3 In this appendix we collect some explicit results for 2-point and 3-point functions of operators with L 3. We restrict for simplicity to the case of the 1/2-BPS loop. In terms of the area-derivatives of the Wilson loop expectation value, one gets for the 2-point functions hh: ~ : : ~ : ii = W(2) hh: ~ 2 : : ~ 2 : ii = W W(4) hh: ~ 3 : : ~ 3 : ii = W(2) W (W)2 W(6) W W(2) (W(2))2 (W(4))2 and for the 3-point functions hh: ~ 2 : : ~ : : ~ : ii = hh: ~ 2 : : ~ 2 : ii hh: ~ 2 : : ~ 2 : : ~ 2 : ii = hh: ~ 3 : : ~ 2 : : ~ 1 : ii = hh: ~ 3 : : ~ 3 : ii hh: ~ 3 : : ~ 3 : : ~ 2 : ii = (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) 2(W(2))3 3W(2)W(4)W +W(6)(W)2 (W)3 (W(2))3W(6) +(W(2))2( W(4))2 +W(8)W 2W(2)W(4)W(6)W +(W(4))3W : (W(2))2(W)2 Here W < W > jA=2 and W(k) @Ak < W > jA=2 (similar expressions hold for the general 1/8-BPS loop, but they also involve derivatives of odd order). Using the Wilson loop expectation value (2.3), one can obtain in a straightforward way the explicit nite N results in terms of Laguerre polynomials, but the expressions are rather lengthy and we do not report them here. In the planar large N limit, the above correlators can be expressed hh: ~ : : ~ : ii = hh: ~ 2 : : ~ 2 : ii = hh: ~ 3 : : ~ 3 : ii = I2 p 4 2I1 p p 3 16 4 I0 16 4I1 p p I0 p 2 4I1 p p and for the 3-point functions: hh: ~ 2 : : ~ : : ~ : ii = hh: ~ 2 : : ~ 2 : ii 3 p (5 +72)I0 64 6I1 p I2 p p 3(13 +144)I0 32 6 I0 p 3( (32 3 )+288)I1 64 6p I0 p p 2I1(p ) p 3=2I0 32 6I1 p p 3 3 32 6 8 6I1 p p 3 (5 +72)I0 256 8I1 p 2I2 p 3( (2 +579)+6192)I0 64 8I2 3( ( (9 112)+4960)+34176)I1 256 8 I2 p 3 p (127 +1920)I0 128 8I1 p I2 2 : 3 p ( +40)I0 p 32 6I1 p p p 3 4 6 3( (5 757) 6336)I1 32 8p I0 p p I0 p p 2I1(p ) 2 4 4 p 2I1(p ) p (A.11) (A.12) (A.13) (A.14) : (A.15) (A.16) (A.17) A.1 Let us also list the rst few results for the generalized Bremsstrahlung function, focusing on the case = 0 given by eq. (4.18). Using the same notation as above, the L 2 results in terms of area-derivatives of the Wilson loop expectation value read W 2W(2) W W(4) 2(W(2))2W(4)W W(2)W(6)(W)2 + 2(W(4))2(W) (W(2))3W W(2)W(4)(W)2 Plugging in (2.3), one can nd the explicit nite N results. For instance, we obtain 16 2N 16 2N 1 + 2L2N 2 L1 N 1 4L2N 2 L1 N 1 4N 4N 4N 4N 6 2L3N 3 2L2N 2 4N 4N + L2N 2 + L1 4N 4N (A.18) : (A.19) Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE]. [2] D. 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Simone Giombi, Shota Komatsu. Exact correlators on the Wilson loop in \( \mathcal{N}=4 \) SYM: localization, defect CFT, and integrability, Journal of High Energy Physics, 2018, 109, DOI: 10.1007/JHEP05(2018)109