Three-point functions in the SU(2) sector at strong coupling

Extending the methods developed in our previous works (arXiv:1110.3949, arXiv:1205.6060), we compute the three-point functions at strong coupling of the non-BPS states with large quantum numbers corresponding to the composite operators belonging to the so-called SU(2) sector in the $\mathcal{N}=4$ super-Yang-Mills theory in four dimensions. This is achieved by the semi-classical evaluation of the three-point functions in the dual string theory in the $AdS_3 \times S^3$ spacetime, using the general one-cut finite gap solutions as the external states. In spite of the complexity of the contributions from various parts in the intermediate stages, the final answer for the three-point function takes a remarkably simple form, exhibiting the structure reminiscent of the one obtained at weak coupling. In particular, in the Frolov-Tseytlin limit the result is expressed in terms of markedly similar integrals, however with different contours of integration. We discuss a natural mechanism for introducing additional singularities on the worldsheet without affecting the infinite number of conserved charges, which can modify the contours of integration.

recently become available [4][5][6][7]. Detailed comparison of the results in two regimes may allow us to identify the common non-trivial structure beyond kinematics.
As it will be evident, the computation of the three-point functions of non-BPS states in string theory in a curved spacetime is quite non-trivial even at the leading semi-classical level. In the first of such attempts [8], the contribution from the AdS 2 part was evaluated for the string in AdS 2 × S k , where the string is assumed to be rotating only in S k . Since the contribution from the sphere part was not computed in [8], the complete answer for the three-point function was not given. In this context, our present work can be regarded as (the extended version of) the completion of the work initiated by [8].
At about the same time, computation of the three-point functions for different type of heavy external states was attempted by the present authors [9]. We took as the external states the so-called Gubser-Klebanov-Polyakov (GKP) strings [10] spinning within AdS 3 with large spins. In this work, the contribution to the three-point function from the action evaluated on the saddle point configuration was computed by a method similar to the one in [8]. However, unlike the case of [8], the GKP string is not point-like on the boundary, and hence the contributions from the non-trivial vertex operators were 1 Actually the global symmetry of this sector is SO(4) = SU(2) × SU(2), as we will emphasize later.
needed to give the complete answer. Since the precise form of such vertex operators were not known, again the computation had to be left unfinished. This difficulty was later overcome by the development of a new integrability-based method built on the state-operator correspondence and the contribution of the non-trivial wave functions of the external states was obtained [11]. Combined with the contribution from the action evaluated previously, this gave the full answer for the three-point function of the GKP strings in the large spin limit [11].
These works paved the way for the present investigation for more general external states in the product space EAdS 3 × S 3 . However, the applications of our general methods developed in [9,11] to the present case are not quite straightforward. One difficulty is that the external states in the S 3 sector, which are taken to be general one-cut finite gap solutions [12][13][14][15] for the purpose of making comparison with the weak coupling result, are much more structured than the large spin limit of the GKP solutions. In particular, practically. Also, it should be mentioned that this is the first time where we have to combine the contributions from the two different sectors, S 3 and EAdS 3 , which nevertheless are interconnected through the Virasoro constraints. We shall see that when these contributions are put together, considerable simplifications occur, showing the intimate interrelation between them, as expected.
The end result of our rather involved computation is a remarkably simple formula for the three-point function, which exhibits intriguing features. First, one recognizes the expressions to be quite analogous to those that appear in the weak coupling result, even before taking any special limits. A priori it is not obvious why the result in the strong coupling limit should resemble the weak coupling answer so closely. This resemblance becomes more conspicuous upon taking the so-called Frolov-Tseytlin limit, where the angular momentum J for the S 3 rotation is quite large so that the ratio √ λ/J, where λ is the 't Hooft coupling, is small. In this limit the integrands of the integrals expressing the answer become almost identical. However, the integration contours do not quite match. This is not immediately a contradiction since there is no rigorous argument why three-point functions should agree exactly in that limit. Nevertheless, it is of interest to look for a possible mechanism to modify the contours. One important fact to be noted in this regard is that, in addition to the ordinary one-cut solutions we used for our external states, there exist different types of one-cut solutions which can be obtained by taking certain degeneration limits of multi-cut solutions. Since the values of the infinite number of conserved charges do not change in this limiting procedure, these solutions should be considered on equal footing with the corresponding ordinary one-cut solutions.
The important difference, however, is that such a "degenerate solution" has one or more additional singularities on the worldsheet. Since the determination of the contours of integration depends crucially on the analytic structure of the saddle point configuration, this phenomenon provides an example of a natural mechanism by which the contour of integration in the formula for the three-point function can be modified. This issue, however, should be studied further in future investigations. Now as this article has become rather lengthy due to various steps of somewhat involved analyses, it should be helpful to give a brief preview of the basic procedures and exhibit the main result. The next subsection will be devoted to this purpose.

Preview and the main result 1.2.1 The set-up
The three-point function we wish to compute in the semi-classical approximation has the following structure: (1.1) It consists of the contribution of the action and that of the vertex operators, evaluated on the saddle point configuration denoted by X * . The subscript signifies a small cut-off which regulates the divergences contained in S and V i . As we shall show, these divergences cancel against each other and the total three-point function is completely finite. The vertex operator V i [X * ; x i , Q i ] is assumed to carry a large charge Q i of order O( √ λ) and is located at x i on the boundary of the AdS space.
In the case of a string in EAdS 3 × S 3 , the action and the vertex operators are split into the EAdS 3 part and the S 3 part. Their contributions are connected solely through the Virasoro constraint T (z) EAdS 3 + T (z) S 3 = 0 (and its anti-holomorphic counterpart).
In the semi-classical approximation, an external state is characterized by the asymptotic behavior of a classical solution, which should be the saddle point configuration for its two-point function. However, conformally invariant vertex operator which creates such a state is practically impossible to construct at present. Moreover, even if one had the vertex operator, it is of no use since the explicit saddle point solution X * on which to evaluate the vertex operator (and the action) cannot be obtained by existing technology.
Such difficulties, although seemingly insurmountable, can be overcome with the aid of the integrable and analytic structure of the system. For this purpose, it is convenient to formulate the string theory in question as a non-linear sigma model. Since the treatment of the S 3 part and the EAdS 3 part are essentially the same in this regard, we shall focus primarily on the S 3 part in this summary. The basic information is then contained in the right-current j ≡ Y −1 dY and the left-current l ≡ dYY −1 , where Y is the 2 × 2 matrix with unit determinant composed of the embedding coordinates Y I (I = 1, 2, 3, 4) of S 3 in the manner which makes the classical integrability of the system manifest. The information of the infinite number of conserved charges is encoded in the monodromy matrix Ω(x) = P exp(− J (x)), the eigenvalues of which are given by Ω(x) ∼ diag(e ip(x) , e −ip(x) ), where p(x) is the quasimomentum. One can then define the spectral curve Γ by det(y1 − Ω(x)) = 0, which describes a two-sheeted Riemann surface in the variable x with a number of cuts with additional singularities. To each such curve corresponds a classical "finite gap solution" [12][13][14][15], which can be constructed in terms of the solutions of the so-called (right and left) auxiliary linear problems, to be abbreviated as ALP throughout, given by The solutions ψ andψ are expressed in terms of the Riemann theta functions and the exponential functions, which depend on the data of the curve such as the location of the branch points and other singularities. We will be interested in the "one-cut solution", the curve for which has a single square root branch cut of finite size 2 , since the vertex operators producing such solutions should correspond to the composite operators in the SU (2) sector in N = 4 super Yang-Mills theory. Now with this setup, let us sketch how one can compute the three point functions with the above one-cut solutions 3 as external legs.

Evaluation of the contribution of the action
First consider the evaluation of the action part. As described in [9] for the GKP string and will be detailed for the case of our interest, the action integral can be written in the form S ∼ Σ ∧ η, where and η are, respectively a holomorphic 1-form and a closed 1-form defined on the double coverΣ of the worldsheet. By using the Stokes theorem, this can be rewritten as a contour integral S ∼ ∂Σ Πη, where ∂Σ is the boundary ofΣ and the function Π = z is single-valued onΣ. This expression for the action can be further rewritten, using a generalization of the Riemann bilinear identity developed in [9], into a sum of products of certain contour integrals. The important point is that the contours of these integrals interconnect the vertex insertion points z 1 , z 2 , z 3 , thereby correlating the behaviors around these points. Therefore, to compute the integral it is natural to study the behavior of the eigenfunctions of the ALP around z i and more importantly along the paths connecting z i and z j .
Although we do not know the exact saddle point solution for the three-point function, we do know the behavior in the vicinity of each z i since it should be the same as the onecut solution discussed above. This provides the form of the currents needed to analyze the ALP around z i . Clearly there are two independent solutions around each z i and one can compute the local monodromy matrix Ω i belonging to SL(2,C), which mixes these solutions upon going around z i . Then one can take the basis of the solutions of ALP at z i to be the eigenvectors of Ω i , denoted by i ± , belonging to the eigenvalues e ±ip i (x) of Ω i . These eigenvectors are normalized 4 with respect to the SL(2,C) invariant product ψ, χ ≡ det(ψ, χ), to be refereed to as Wronskian throughout this article, as i + , i − = 1.
To gain information about the solution of ALP valid in the entire worldsheet, one can make the "WKB expansion" with ζ = (1−x)/(1+x) as the small parameter corresponding to . One then finds that the same contour integrals with which the action is expressed appear in the WKB expansion of the Wronskians i ± , j ± . Therefore our task is reduced 2 As already mentioned in the introduction, this class can contain solutions which are obtained from m-cut solutions by shrinking m − 1 of them to infinitesimal size. They may play important roles in obtaining all possible three-point functions of this category. 3 When it is not confusing, we use one-cut solution to refer either to the one-cut solution of ALP or the solution of the original equation of motion reconstructed in terms of such solutions. 4 For the normalization of each eigenvector, see section 2.3.
to their computation.
The crucial information about such Wronskians is contained in the global consistency condition of the monodromy matrices given by Since Ω i 's cannot in general be diagonalized simultaneously, this serves as a highly nontrivial constraint. In fact this condition allows one to express certain products of two Wronskians in terms of the local quasi-momenta p i (x)'s, an example of which is given by (1.7) It turns out that the knowledge of the Wronskians, such as 1 + , 2 + , between the eigenfunctions at different insertion points is of utmost importance. All the basic quantities, namely the contour integrals giving the contribution of the action and the wave functions, to be discussed shortly, can be expressed in terms of the Wronskians.
Therefore the crucial task is to separate out, from the relations such as (1.7), the individual Wronskian i ± , j ± . This can be achieved if we know which of the two factors is responsible for each zero and the pole on the spectral curve, produced by the expression on the right hand side. This information dictates the analyticity property of the individual Wronskian in x and by solving the appropriate Riemann-Hilbert problem we can obtain the Wronskians.
As an example, consider the poles produced by the zeros of sin p 1 (x) on the right hand side of (1.7), namely at p 1 (x pole ) = nπ. These are the singular points of the spectral curve where the monodromy matrix Ω 1 (x pole ) takes the form of a Jordan block and the bigger of the two eigenvectors 1 ± diverges. This means that the Wronskian on the left hand side of (1.7) involving such a "big solution" must be responsible for these poles. Now which eigenvector is big and which is small near z i depends on the value of x. In the case of the ordinary one-cut solution its explicit form tells us that it is dictated by the sign of Re q i (x). This means that across the line Re q i (x) = 0, the analytic property of the eigenfunctions i ± changes. We can then extract the regular part of the Wronskian between "small solutions" by using the well-known technique of Wiener-Hopf decomposition, which takes the form of a convolution integral with the contour along the line Req i (x) = 0. Due to the two-sheeted nature of the spectral curve, the kernel of the decomposition formula must be appropriately generalized.
Now the remaining analysis, namely that of the zeros of the right hand side of (1.7), is similar in spirit but is much more complicated because it involves the interplay between the three local quasi-momenta p 1 (x), p 2 (x), p 3 (x) and requires a certain knowledge of the global properties of the solutions of the ALP on the spectral parameter plane. To properly deal with this problem, we will introduce a notion of the "exact WKB curve".
Also, since each p i (x) is double valued, the convolution kernel will be defined on an eightsheeted Riemann surface. Moreover it turns out that the contour of integration must be determined not just by Re q i (x) = 0 for each i but also by certain global "connectivity conditions" expressed in terms of the quantity N i ≡ |Re p i (x)|. Despite such technical complexities, we will be able to compute the desired Wronskians in terms of the quasimomenta p i (x).
With the procedures described above, one obtains the contribution from the action for the S 3 part. Further, in an analogous manner, the corresponding contribution from the EAdS 3 part can be computed.

Evaluation of the contribution of the vertex operators
Let us now turn to the computation of the contribution of the vertex operators. To this end, we extend the powerful method developed in our previous work [11] for the GKP string to more general string. It is based on the state-operator correspondence and the construction of the corresponding wave function in terms of the action-angle variables. If one can construct the action-angle variables (S i , φ i ), the wave function can be constructed simply as where E({S i }) is the worldsheet energy 5 . Although the construction of such variables for a non-linear system is prohibitively hard in general, for integrable systems of the present type there exists a beautiful method [13][14][15], based on the Sklyanin's separation of variables [16], which allows us to construct them from the Baker-Akhiezer eigenvector ψ, which is the solution of ALP satisfying the monodromy equation of the form Ω(x; τ, σ)ψ(x; τ, σ) = e ip(x) ψ(x; τ, σ). More precisely, the dynamical information is encoded in the function n · ψ(x, τ ), where n = (n 1 , n 2 ), to be specified later, is referred to as the "normalization vector". It is known that for an m-cut solution n · ψ(x, τ ) as a function of x has m zeros at certain positions x = {γ 1 , γ 2 , . . . , γ m } and the dynamical variables z(γ i ) and p(γ i ), where z = √ λ (x + x −1 ) /(4π), can be shown to form canonical conjugate pairs. Then by making a suitable canonical transformation, one can construct the action-angle variables (S i , φ i ), where, in particular, the angle variables are given by 5 The sum of such energies of course vanishes for the total system due to the Virasoro constraint.
the generalized Abel map Here, ω i are suitably normalized holomorphic differentials (with certain singularities depending on the specific problem) and x 0 is an arbitrary base point. In the case of the one-cut solution of our interest, we have one angle variable φ R associated with the right ALP shown in (1.4) and one left angle variable φ L associated with the left ALP described in (1.5). Hereafter we will only refer to the "right sector" for brevity of explanation.
Now as we shall describe in section 2.2, we can write down a simple formula which reconstructs the classical string solution from the Baker-Akhiezer vector 6 ψ. Therefore, with a choice of the normalization vector n, one can associate the angle variable φ R (n) to a classical solution, through the (zeros of the) quantity n · ψ.
Let Y denote the form of the three-point saddle solution near the vertex insertion point z i . We will call this part of the solution the ith leg. As we have to normalize the three-point function by the two-point function for each leg, what we wish to compute is the angle variable φ R (n) associated to Y relative to the one φ ref R (n) associated to the "reference two-point solution" Y ref which is created by the same vertex operator 7 at z i . Now the vertex operators of our interest are those which correspond to the gaugeinvariant composite operators in the SU(2) sector of the super Yang-Mills theory. As we discuss in detail in section 4, the basic operators of that category are the chargediagonal operators which are "highest weight" with respect to the global symmetry group SU(2) R × SU(2) L SO(4). Focusing just on the SU(2) R property, one can characterize such an operator by what we call a "polarization spinor", in this case n diag = (1, 0) t , which is annihilated by the raising operator of SU(2) R . More general operator of our interest can then be obtained from such a diagonal operator by an SU(2) R (×SU(2) L ) rotation and is characterized by the polarization spinor n, obtained from n diag by the corresponding SU(2) R rotation. In this way, each vertex operator is associated with such a spinor n.
What is important is that this "polarization spinor" n can be shown to be identical to the "normalization vector" n which determines the angle variable φ R (n) through the quantity n · ψ. As was elaborated in our previous work [11], once the normalization vector n is specified, the relative shift ∆φ R = φ R − φ ref R of the angle variable for the three-point solution Y around z i from that for the two-point reference solution Y ref can be computed from the knowledge of the transformation matrix V ∈ SL(2,C) which connects Y and Y ref in the manner Y = Y ref V in the vicinity of z i . As it will be shown in section 4, the allowed form of V can be deduced from the property that both Y and Y ref are produced from the same vertex operator characterized by the polarization spinor n.
Then, by using the master formula developed in [11], we can express ∆φ R in terms of n, the solutions of the Baker-Akhiezer functions corresponding to Y ref , and the parameters describing V . Applying this procedure to each leg of the three-point function and by making use of a relation between the normalization vector n and the value of ψ at x = ∞, we can express the wave function (for the right sector) in terms of the Wronskians as Here n i is the polarization spinor associated with the vertex operator V i at z i and R i is the absolute value of the SU(2) R charge carried by V i . Note that the kinematical part expressed in terms of n i , n j is clearly separated from the dynamical part, which again is composed of the Wronskians of the solutions of the ALP. The wave function for the EAdS 3 part can be obtained in a similar fashion. In that case, the Wronskians n i , n j can be expressed in terms of the difference of the landing positions of the three legs on the boundary of EAdS 3 and yield the familiar coordinate dependence of the three-point functions.

Final result for the three-point function
We now have all the ingredients for the evaluation of three-point functions. Substituting the explicit expressions of the Wronskians i ± , j ± into the action and the wave function and assembling the contributions from the S 3 part and the EAdS 3 part together, we find that remarkable simplifications take place in the sum. The final result for the general one-cut external states is thus found to be where the prefactor 1/N comes from the string coupling constant g s and the logarithm of the structure constant C 123 is given by z(x) (dp 1 + dp 2 + dp 3 ) 2πi ln sin z(x) (dp 1 + dp 2 − dp 3 ) 2πi ln sin z(x) (dp 1 − dp 2 + dp 3 ) 2πi ln sin z(x) (dp 1 + dp 2 + dp 3 ) 2πi ln sin The notations used in the above expressions are as follows. In the equation (1.11), ∆ i is the conformal dimension of the i-th vertex operator V i and n i andñ i are the polarization spinors for V i with respect to SU(2) R and SU(2) L . In the expression for ln C 123 , p i andp i are the quasi-momenta for the i-th leg for the S 3 part and the EAdS 3 part respectively.
z(x) is the Zhukovsky variable given in (2.36). The symbols M uuu ±±± and Γ u j − denote the contours of integration for the S 3 part andM uuu ±±± andΓ u j − are the contours for the EAdS 3 contribution. The last term Contact stands for some special terms which depend on the detail of the external states. It should be noted that the result above for the threepoint function for the operators corresponding to general one-cut solutions is already reminiscent of the expression in the weak coupling regime. In section 7, we demonstrate that our formula gives the correct result for the case of three BPS operators and that it reduces to the two-point function in the limit when the charge of one of the operators becomes negligibly small. Further, we analyze the Frolov-Tseytlin limit for the case of one non-BPS and two BPS operators and find that the integrals giving the three-point coupling take extremely similar forms, except for different contours of integration. For this issue, we point out the existence of a natural mechanism by which the contours can be modified. Now we briefly indicate the organization of the rest of this article: In section 2, we begin with the description of the string in EAdS 3 × S 3 spacetime and discuss the one-cut solutions we will consider in this work. In section 3, we will study the contribution of the action for the S 3 part to the three-point function and show that the action can be re-expressed in terms of certain contour integrals. In section 4, we describe the evaluation of the wave functions for the S 3 part. Characterizing the vertex operator by a polarization spinor and identifying it with the normalization vector determining the angle variable, we apply the master formula for the shift of the angle variables developed in our previous work to construct the wave functions. Section 5 will be devoted to the explicit evaluation of the Wronskians. The main task is to find the analyticity property of the Wronskian from the improved WKB analysis of the ALP. Using this information, we can project out the individual Wronskian from the expression of the product of Wronskians in terms of the quasi-momenta p i (x) by the use of the Wiener-Hopf decomposition. In section 6, all the results obtained up to this point are put together to produce the final result for the three-point functions of the general one-cut external states. In section 7, in addition to some basic checks of our result, we present the analysis of the Frolov-Tseytlin limit and discuss its outcome. Finally, in section 8 we make some important comments on our present work and indicate possible future directions. Several appendices are provided to supply some additional details.

String in EAdS 3 × S 3 and classical solutions
We begin by setting up the formalism to deal with the strings in EAdS 3 × S 3 in subsection 2.1 and describe the classical solutions we will use as the external states of the three-point functions in subsection 2.2. We then give a brief account on the basic set-up of the threepoint function in subsection 2.3.

Preliminaries
In this article, we will exclusively deal with the string propagating in the product space of the Euclidean AdS 3 subspace of AdS 5 (to be denoted by EAdS 3 ) and the sphere S 3 .
If we describe the AdS 5 in terms of the embedding coordinates by where the superscript M is taken to run as M = −1, 0, 1, 2, 3, 4 and the metric is given by η M N = diag (−1, −1, 1, 1, 1, 1), then the EAdS 3 subspace is defined by setting X 0 = X 3 = 0. Therefore we will parametrize the EAdS 3 × S 3 space in the following way 8 : The Poincaré coordinates (x r , z) = (x 0 , x 1 , x 2 , x 3 , z) of AdS 5 are defined in the usual way as where z = 0 corresponds to the boundary of AdS 5 . When restricted to the EAdS 3 subspace 9 , its boundary is the Euclidean plane parametrized by (x 1 , x 2 ).
The action of a string in this space is given by where Λ andΛ are Lagrange multiplier fields. Upon eliminating them the equations of motion become For physical configurations, we must in addition impose the Virasoro constraints, which require that the sum of the stress-energy tensors for the AdS part and the sphere part must vanish. Namely, For the AdS part, we shall take the external states to be those without the two-dimensional spins. Then near the vertex insertion point the saddle point solution should approach the two-point solution, which is known to be point-like. The forms of T AdS (z) andT AdS (z) for such a two-point solution are uniquely determined by their transformation properties as a (2, 0) and a (0, 2) tensor respectively and are given in terms of the conformal dimension ∆ of the vertex operator as (2.10) Therefore, taking into account the Virasoro condition, near each vertex insertion point z i we must have and similarly for the anti-holomorphic parts. In the case of three-point functions, the information of such asymptotic behaviors suffices to determine the form of the energymomentum tensor exactly everywhere. For the EAdS 3 , the holomorphic part takes the form Here and hereafter, we shall omit the subscript AdS for the stress tensor for the AdS part and simply write T (z) andT (z) for T AdS (z) andT AdS (z).
We now discuss the methods for constructing the solutions of the equations of motion with the use of the classical integrability of the system. There exist two apparently different formalisms. One is the sigma model formulation [12,17] and the other is the so-called Pohlmeyer reduction [18,19]. The former deals with variables which transform covariantly under the global symmetry transformations, whereas the latter employs invariant variables. Because of this feature they have advantages and disadvantages depending on the problem one would like to solve. We shall employ both. It should be remarked however that they are actually connected by a "gauge transformation", as shown in Appendix C.2.

Sigma model formulation
Consider first the sigma model formulation. We will focus on the S 3 part, as the EAdS 3 part can be treated similarly. The embedding coordinates {Y I } are conveniently assembled into a 2 × 2 matrix with unit determinant given by 14) which transforms under the global symmetry group SO(4) = SU(2) L × SU(2) R as The quantities of central importance are the "right" and the "left" currents (or connections) j and l respectively, defined by where x is the complex spectral parameter. The two connections J r = J r z dz + J r z dz and J l = J l z dz + J l z dz are related by the gauge transformation of the form Y(d + J r )Y −1 = d + J l . It is useful to note that the energy-momentum tensors and hence the Virasoro conditions can be expressed in terms of the currents in a concise way. We have, in the cylinder coordinate, Central to the construction and the analysis of the solutions of the equations of motion are the right and the left auxiliary linear problems, to be abbreviated as ALP, which are coupled linear differential equations for vector functions: Compatibility of the system of ALP implies the original equations of motion. Upon developing ψ andψ from a point z 0 along a closed spacelike curve, we obtain the right and the left monodromy matrices Ω(x) and Ω(x) respectively as By virtue of the flatness of the connection, expansion of Ω(x) as a function of x around any point yields an infinite number of conserved charges as coefficients. In particular, expansions around x = ∞ and x = 0 yield, in the leading behavior, the Noether charges for the global SU(2) R and SU(2) L , respectively, defined by Indeed, expanding Ω(x; z 0 ) around x = ∞ and x = 0 and using the definitions above, we By diagonalizing Ω(x; z 0 ), we can obtain a quantity independent of z 0 . Since det Ω(x; z 0 ) = 1, its eigenvalues must be of the structure where m is an integer and the right and the left charges R and L are the (positive) eigenvalues of Q R and Q L respectively.
For the study of the ALP and construction of the finite gap solutions of our interest, the analytic property of the quasi-momentum is of critical importance. Such a structure is encoded in the spectral curve defined by poles at x = ±1 with the magnitude of the residue equaling −2πκ. Since p(x) lives on a two-sheeted surface, we specify its branch by defining the signs at these singularities. We shall employ the definition where the + superscript on 1 + signifies that the point is on the first sheet. Similarly, we shall use − superscript for points on the second sheet. We will give a more detailed discussion of the structure of p(x) for the one-cut solutions of our interest in subsection 2.2.
From the structure of the spectral curve and the quasi-momentum p(x) defined upon it, one can extract important information. For this purpose, we first define the a-and b-cycles in the usual way. For the hyperelliptic curve of our interest, an a-cycle is defined as a cycle which goes around the cut on the same sheet. On the other hand, a b-cycle is defined as the one which starts from a point on the first sheet, goes into the second sheet through the cut and eventually comes back to the same point on th first sheet. Clearly, around an a-cycle, we have a i dp = 0. In contrast, the integral along the b-cycle does not vanish in general and gives b i dp = 2πn i , where n i is an integer called the mode number. Now using the a-type cycles, one can define a set of conserved charges called the filling fractions as is the Zhukovsky variable. In particular, the filling fractions S ∞ and S 0 defined with the contours a ∞ and a 0 , which encircle the point at ∞ and 0 respectively, are of special importance since they are related to the global SU(2) R and SU(2) L charges in the following way, as can be checked using (2.30) and (2.31):

Pohlmeyer reduction for a string in S 3
The sigma model formulation we have sketched above is convenient for analyzing the property of the system under the global symmetry transformations. Hence it will be used as the basis of the construction of the wave function corresponding to the vertex operators in section 4. On the other hand, for the analysis of the contribution of the action, which is invariant under the global transformation, the formalism of the Pohlmeyer reduction will be more convenient.
The essential idea of the Pohlmeyer reduction is to describe the motion of the string in a suitably defined moving frame. This then leads to the Lax equations in terms of the connections which are invariant under the global symmetry transformations. Below we shall only sketch the procedures and then summarize the basic equations we will need later. Further details will be given in Appendix B.
In what follows we shall denote a 4-component field A I simply as A and use the The basic moving frame of 4-component fields, to be called q i , (i = 1, 2, 3, 4), are taken as q 1 ≡ Y, q 2 ≡ a∂Y + b∂Y, q 3 ≡ c∂Y + d∂Y and where N is the unit vector orthogonal to Y, ∂Y and∂Y , and the (field-dependent) coefficients a, b, c, d are chosen so that the simple conditions q 2 · q 3 = −2, q 2 2 = q 2 3 = 0 are satisfied. (Note that since Y 2 = 1, we automatically have q 2 1 = 1, q 1 · q 2 = q 1 · q 3 = 0.) Let us define an SO(4)-invariant field γ by the relation ∂Y ·∂Y = TT cos 2γ . (2.38) Then, the coefficients a, b, c, d can be expressed in terms of T,T and γ, giving q 2 and q 3 of the form Once the moving frame is prepared, one can compute the derivatives of q i and express them in terms of q i again. The result can be assembled into the following equations where W is given by  where ρ andρ are defined by Just as in the case of the sigma model formulation, the integrability of the system allows one to introduce a spectral parameter ζ, related to x by without spoiling the flatness conditions. The Lax equation so obtained is given by (2.48) One can consider the auxiliary linear problem also for the Pohlmeyer connections

One-cut finite gap solutions in S 3
We now describe a particular class of solutions to the equations of motion and the Virasoro constraints, which can be constructed by the so-called finite gap integration method [13][14][15]. These solutions describe the local behaviors of the saddle point solution for the three-point function in the vicinity of the vertex insertion point. The class of our interest is characterized by the associated spectral curve having one square-root branch cut of finte size and will be referred to as a one-cut solution. We will first consider the "basic" one-cut solutions, which are customarily referred to as genus 0 solutions, and study their properties in detail. Then, we describe another class of one-cut solutions which are obtained from multi-cut solutions by certain degeneration procedure. We show that they contain additional singularities on the worldsheet, which may play an important role when we compare the three point functions at strong and weak couplings in section 7.

Basic one-cut solution and "reconstruction" formula
A powerful method for constructing a large class of classical solutions in the sigma model formulation is the so-called finite gap integration method. (For a comprehensive review, see [15].) The method consists of two steps. As the first step, the solutions to the left and the right ALP, called the Baker-Akhiezer functions, are constructed by treating the problems as Riemann-Hilbert problems on a finite genus Riemann surface. Namely, by proving that the function satisfying all the required analytic properties is unique, one constructs such a function in terms of the Riemann theta functions and the exponential functions. Then, as the second step, one develops the "reconstruction" formula 10 , which constructs the solutions to the original equations of motion from the knowledge of the Baker-Akhiezer functions. In this subsection, we will describe the simplest class of solutions corresponding to the case of genus zero Riemann surface, or a two-sheeted surface with one square-root branch cut. Such solutions will be referred to as the basic one-cut solutions.
Consider first the right ALP given in (1.4) and let ψ ± (x, z,z) be the Baker-Akhiezer vector which are at the same time the eigenvectors of the monodromy matrix Ω(x) corresponding to the eigenvalues e ±ip(x) respectively. According to the general theory of finite gap integration, ψ ± corresponding to the one-cut solution are given by simple exponential functions as where c + i are constants,σx denotes the point x on the opposite sheet, and ∞ + (∞ − ) is the point at infinity on the first (resp. second) sheet. The quantity dp is the differential of the quasi-momentum p(x), while dq is the differential of the quasi-energy q(x). Just like p(x), the quasi-energy q(x) is defined by the pole behavior at x = ±1 + of the form The structure and the signs of the residue at x = ±1 for q(x) are determined so that the holomorphicity of the solution (2.51) at x ±1 is as dictated by the ALP. For example at x = 1 the holomorphic part of the ALP is dominating and hence the Baker-Akhiezer vector should be holomorphic. This is in fact realized since p(x) = q(x) near x = 1 and hence the exponent of ψ ± is a function of the combination z = τ + iσ. In the same way, at x = −1 the exponent of ψ ± becomes anti-holomorphic as desired.
Now for the left ALP, the Baker-Akhiezer eigenvectors, denoted byψ ± (x, z,z), are given byψ where the notations are similar and should be self-explanatory.
We will be interested in the case where the branch cut runs between u and its complex conjugateū on the spectral curve. Such a cut is described by a factor of the form We define the branch of y(x) to be such that the sign of y(x) is +1 at x = 1 + . Then p(x) and q(x) satisfying the prescribed analyticity properties are fixed to be Here we fixed p(x) and q(x) such that they vanish at the branch points although the analyticity properties only determine the differential dp and dq. This choice is suitable for the purpose of this paper since the solutions to the ALP in the Pohlmeyer gauge. The forms of p(x) and q(x) depend on whether the cut is placed to the right or to the left of x = 1. Substituting these forms into the formulas for ψ ± andψ ± we get the one-cut solutions for the ALP.
Let us now describe the second step, the (re)construction of the solutions of the equations of motion from the Baker-Akhiezer vectors. Although this has been discussed in the literature [13][14][15], we present below a more transparent formula. Let us form a 2 × 2 matrix Ψ in terms of the two independent Baker-Akhiezer column vectors ψ ± satisfying the right ALP as Ψ = (ψ + ψ − ) and consider the quantitỹ Then, by using the definitions l z = ∂YY −1 and j z = Y −1 ∂Y, we can easily show that If we expressΨ in terms of two column vectorsψ ± asΨ = (ψ +ψ− ), the above equations show thatψ ± are actually two independent solutions to the left ALP. This means that there exist solutions ψ ± andψ ± to the right and the left ALP respectively so that Y can be expressed as  Summarizing, we have two types of simple reconstruction formulas By using the reconstruction formula given above, one can write down the general basic one-cut solution explicitly. It can be written in the form [12,15] where the parameters ν i , m i and θ 0 must satisfy the following conditions expressing the equations of motion and the Virasoro conditions: Applying the reconstruction formula (2.65) with the constant matrixΨ(0) taken to be the identity matrix and using the form of ψ + given in (2.51), we easily find that the parameters m i and ν i can be expressed in terms of p(x) and q(x) as The right and the left Noether charges R and L can be computed directly from the solution (2.67) and are given in terms of the parameters ν i , m i and θ 0 in a universal manner as Explicit expressions of R and L in terms of the position of the cut are given in Appendix A.1. As a result, we find that the charges R and L are positive irrespective of the position of the cut. This means that they should be regarded not as the charges themselves but as their absolute magnitudes. On the other hand, the relative magnitude of R and L depends on the position of the cut as In section 4.3.4, we will see that the difference in the relative magnitude corresponds to the difference of the class of vertex operators for which the solution is the saddle point of the two-point function.

One-cut solutions from multi-cut solutions
We now discuss a more general type of "one-cut" solutions, namely the ones with additional cuts of infinitesimal size besides a cut of finite size. As we shall discuss in section 7.5.2, this type of solutions may play an important role in the comparison of the threepoint functions at strong and weak couplings. Besides such specific reason, as these infinitesimal cuts do not contribute to any of the (infinite number of) conserved charges, they should, on general grounds, be considered on an equal footing with the corresponding solutions carrying the same charges. As a matter of fact, it is much more natural to consider solutions with infinite number of infinitesimal cuts, as they correspond to the infinite number of angle variables which must exist for a string theory even when their conjugate action variables have vanishing values 11 . Now adding an infinitesimal cut to the genus g Riemann surface is equivalent to shrinking a cut in the genus g + 1 surface 12 .
As we shall see, depending on the choice of the parameters we either get back an ordinary genus g finite gap solution or we obtain a new solution with additional singularities.
In contrast to the one-cut solution corresponding to genus zero we have been considering, for a genus g finite gap solution with g ≥ 1 the components of the Baker-Akhiezer 11 As already emphasized in [11], in order to construct a three-point solution in the framework of the finite gap method, which is tailored for construction of two-point solutions, inclusion of infinite number of small infinitesimal cuts is necessary as one has to produce an additional singularity corresponding to the third vertex operator. 12 A similar discussion of this process can be found in [20].
vector are given by the following expressions containing ratios of Riemann theta functions Θ(z) in addition to the exponential part: (2.78) As it is not our purpose here to review the details of the finite gap construction, below we will only explain the minimum of the ingredients and refer the reader to a review article such as [15]. Also, for simplicity and clarity, we will focus on the case of the degeneration from g = 1 to g = 0. This suffices to explain the essence of the construction and the generalization to the case of higher genus is straightforward.
For a g = 1 two-cut solution, the Riemann theta function Θ(z) reduces to the elliptic theta function θ(z) defined by where Π is the period given by the integral of the holomorphic differential w over the b-cycle of the torus As usual, w is normalized by the integral over the a-cycle as a w = 1. A(x) appearing in the argument of the Θ-functions is the Abel map defined by h ± (x) are normalization constants and k and ω are the "momentum" and the "energy" defined by the integrals A quantity of importance is the constant ζ γ ± (0) defined by In this formula, K is the "vector of Riemann constants", which for a torus is simply a number proportional to the period Π as 13 (2.84) 13 For its definition for a general genus g surface, see for example [21].
Finally γ ± (0) are certain points 14 on the Riemann surface, which determine the initial conditions for the solution.
Let us now study what happens when we pinch the a-cycle. In order to keep the normalization condition a w = 1 intact, w must behave near the position of the infinitesimal cut x c as w ∼ Now writing the θ-function as we see that the last factor vanishes as Im Π → ∞, except for m = 0. Therefore in this limit we get θ(z) → 1 and one gets the usual genus 0 solution with only the exponential part.
Now if we identify z = kσ − iωτ in the formulas for ψ i given in (2.77) and (2.78), the arguments of the θ-functions containing z are actually of the form z − a, with a constant shift a given by a = ζ γ ± (0) + · · · . What is important is that ζ γ ± (0) diverges as we pinch the a-cycle. First, obviously Im K diverges as πIm Π. Second, if γ ± (0) is at the position of the shrunk cut x c , Im A(γ ± (0)) diverges just like πIm Π: Since A(γ ± (0)) is finite otherwise, we must distinguish two cases: case (a) Im ζ γ ± (0) ∼ 2πIm Π for γ ±(0) = x c and case (b) Im ζ γ ± (0) ∼ πIm Π for γ ±(0) = x c . Therefore let us write a = lIm πΠ + c, where l = 2 or l = 1 and c is a finite constant. Then the θ-function with this shift can be written as As the θ-functions occur in pairs in the numerator and the denominator in ψ i , their ratio goes to a z-independent finite constant in the degeneration limit and we get back the usual g = 0 one-cut solution. In fact, by repeating this type of process, one can produce a finite gap solution from an infinite gap solution, which must be the generic situation for theories with infinite degrees of freedom, such as string theory.
Next consider the case (b). For l = 1, two terms in the series survive in the limit Im Π → ∞, namely m = 0 and m = 1. Therefore we obtain a non-trivial function of the form where C is a constant. In particular, this function can vanish at certain points, the number of which depend on the magnitude of k. Such a θ-function in the denominator of the expressions for ψ i gives rise to additional simple poles on the worldsheet. In distinction to the singularity due to a vertex operator, these singularities do not carry any charges (including infinitely many higher charges) because the solution is obtained without changing the form of p(x).
Although we will not explicitly make use of the degenerate multi-cut solutions discussed above in the bulk of our investigation, they will be recognized in section 7.5.
In what follows, we will describe some important properties of i ± and related states.
Of crucial importance in the computation of three-point functions will be the SL(2,C) invariant product for i ± and j ± given by In the rest of the paper, we shall refer to this skew-product as Wronskian.
For later convenience, let us fix the normalization of the eigenvectors i ± . We will first impose the usual condition In this formula, (τ (i) , σ (i) ) are the local cylinder coordinates around z i , defined by Here we have chosen the origin of τ (i) to be such that τ (i) = 0 on the small circle |z−z i | = i , which will serve to separate the contributions from the action and the wave function in subsequent sections. Using the results of Appendix A, the eigenvectors for the two-point functionî 2pt ± can be computed aŝ where u i andū i are the positions of the branch points of the quasi-momentum p i (x) for the i-th puncture. The conditions (2.96), (2.98) and (2.99) determine the normalization of i ± completely. The important property of the eigenvectors so normalized is that they transform in the following way when they cross the branch cut 15 : This relation will be used in section 5.5 to determine the normalization of certain Wronskians.

Structures of the action for the S part
Let us now start our study of the three-point functions. In what follows, we will denote their structure as In this section, we focus on the contribution of the action for the S 3 part, namely F action .
First, in subsection 3.1, we rewrite the action as a boundary contour integral using the Stokes theorem and then apply the generalized Riemann bilinear identity derived in [9] to bring it to a more convenient form. Next we turn in subsection 3.2 to the analysis of the WKB expansion of the auxiliary linear problem. We then find that the same contour integrals we used to rewrite the action appear also in the WKB expansion of the Wronskians of the solutions to the ALP. Using this relation, we re-express the action in terms of the Wronskians in subsection 3.3. The resultant expression will be used for the explicit evaluation of the contribution of the action in section 6.

Contour integral representation of the action
For the three-point function of our interest, the (regularized) action for the S 3 part of the string is given by where the symbol Σ\{ i } denotes the worldsheet for the three-point function, which is a two-sphere with a small disk of radius i cut out at each vertex operator insertion point z i . Such a point will often be referred to as a puncture also. In [8] and [22], such worldsheet cut-offs are related to the spacetime cut-off in AdS in order to obtain the spacetime dependence of the correlation functions without introducing the vertex operators. In contrast, as we shall separately take into account the contribution of the vertex operators, i 's can be taken to be arbitrary in our approach, as long as they are sufficiently small and the same for the S 3 part and the EAdS 3 part.
As the action is invariant under the global symmetry transformations, it is natural to express (3.4) in terms of the quantities used in the Pohlmeyer reduction. From (2.38), we can indeed write We further rewrite (3.5) by introducing the following one-forms: The second term on the right hand side of (3.7) is added to make η closed, as one can verify using the relation (2.44). With these one-forms, we can re-express the action (3.5) as a wedge product of the form where an extra prefactor i/2 comes from the definition of the volume form, dz ∧ dz = −2i d 2 z. Then denoting the integral of (z) as the action can be rewritten, using the Stokes theorem, as a contour integral along a boundary ∂Σ of a certain regionΣ (see figure 3.1): To determine the proper region of integrationΣ, we need to know the analytic structure of Π(z), which in turn is dictated by that of T (z). As already explained in section 2, in the case of three-point functions the information of the asymptotic behavior of T (z) at each puncture z i is sufficiently restrictive to determine T (z) exactly to be of the form From this, one can show that Π(z) has three logarithmic branch cuts running from the punctures z i , and one square-root branch cut connecting two zeros of T (z), to be denoted by t 1 and t 2 . Therefore, we should takeΣ to be the double cover (y 2 = T (z)) of the worldsheet Σ with an appropriate boundary ∂Σ, so that Π(z) is single-valued on the whole integration region. In what follows, we will consider the case where the branch cut is located between z 1 and z 3 as depicted in figure 3.2. In such a case, the branch of the square-root of T (z) can be chosen so that it behaves near the punctures on the first sheet as (3.12) Although the discussion to follow is tailored for this particular case, the final result for the three-point function, to be obtained in section 6, will turn out to be completely symmetric under the permutation of the punctures.
At this point, we shall apply the generalized Riemann bilinear identity, derived in [9], to the integral (3.10). As the derivation is lengthy, we refer the reader to [9] for details and just present the result 16 . It can be written as where the definition of each term will be given successively below 17 . The first term, Local, denotes the contribution from the product of contour integrals, each of which is just around the puncture and hence called "local". It is of the form where C i is a contour encircling the puncture z i counterclockwise. Here and hereafter, the symbol ( ↔ η) stands for the contribution obtained by exchanging and η in the preceding term. The second term, Double, denotes the double integrals around the punctures given by The third term, Global, denotes the contribution from the product of contour integrals, one of which is along a contour connecting two different punctures. It is given by More precisely, ij denotes the contour connecting z * i and z * j , where z * i is the point near the puncture z i satisfying z * i − z i = i . The barred indices indicate the points on the second sheet of the double cover y 2 = T (z). For instance, Cī is a contour encircling the point zī, which is on the second sheet right below z i . Finally, the term Extra denotes additional terms which come from the integrals around the zeros of √ T , to be denoted by t k , at which η becomes singular, and is given by Here D k is the contour which encircles t k twice as depicted in figure 3.2.
Among these four terms, Local and Double are expressed solely in terms of the integrals around the punctures and are easy to compute. The explicit results, computed in 16 By decomposing the contours ij 's in (3.13) into d and i 's defined in [9], we arrive at the formula derived in [9]. 17 In [8] and [22], the ordinary Riemann bilinear identity was applied to derive an expression similar to (3.13) but without the terms Local and Double . In their cases, Local and Double vanish and the use of the ordinary Riemann bilinear identity is justified. On the other hand, these two terms do not vanish in our case and we must use the generalized Riemann bilinear identity.
Here Λ i 's are given in terms of γ i and ρ i , defined in (A.21) and (A.22) respectively, as It is important to note that Local and Double are real since κ i and g i are all real.
Therefore they contribute exclusively to the imaginary part of the action (3.10) and hence only yield an overall phase of the three-point functions. We shall neglect such quantities in this paper. Among the remaining two types of terms, Extra can be explicitly evaluated as follows.
Since the worldsheet is assumed to be smooth except at the punctures, the quantity √ TT cos 2γ, which is the integrand of the action integral given in (3.5), should not vanish even at the zeros of T (z). This in turn implies that γ is logarithmically divergent at such points in the manner The one-forms and η flip the sign under the exchange of two sheets. Therefore (3.18) is odd whereas (3.19) is even under such sheet-exchange. In (3.19), κ i for i =2 is set to be equal to κ 2 .
Then, by approximating T (z) as T (z) ∼ c(z − t k ) around t k , we can write down the leading singular behavior of η around t k as Thus the integral along D k can be computed as Since there exist two zeros, Extra is twice this integral and hence is given by For later convenience, we shall derive another expression for the action using a different set of one-forms given bȳ and then consider the average of the two expressions. Using the forms above, the action can be written as As compared to (3.10), the expression (3.27) has an extra minus sign, which is due to the property dz ∧ dz = −dz ∧ dz. Applying the generalized Riemann bilinear identity to where Local, Double and Global are given respectively by (3.14), (3.15) and (3.16) with and η replaced by¯ andη. The integrals of¯ andη around the punctures are given by 19 whereΛ i 's are given in terms of γ i andρ i , defined in Appendix A.2, as Again Local and Double are real and they contribute only to the overall phase. On the other hand, Extra can be evaluated just like Extra and yields +πi/3. Thus, by averaging over the two expressions (3.13) and (3.28) and neglecting terms which contribute exclusively to the overall phase, we arrive at the following more symmetric expression: The quantity (3.32) consists of various integrals along the contours C i and ij . Among them, the ones along C i can be easily computed using (3.18) and (3.29). The integral of along ij can also be computed in principle as we know the explicit form of . Thus the major nontrivial task is the evaluation of ij η and ijη . In the rest of this section, we will see how these integrals are related to the Wronskians of the form i ± , j ± , where i ± are the Baker-Akhiezer eigenvectors at z i of the ALP, corresponding to the eigenvalues e ±ip i (x) .

WKB expansions of the auxiliary linear problem
We now perform the WKB expansion of the auxiliary linear problem and observe that the contour integrals of our interest, ij η and ijη , appear in the expansion of the Wronskians between the eigenvectors of the monodromy matrices.
Let us first consider the WKB expansion of the solutions to the ALP. For this purpose, it is convenient to use the ALP of the Pohlmeyer reduction (2.49). The use of (2.49) has two main virtues. First, as Φ's are given explicitly in terms of T (z) andT (z), it is easier to perform the expansion around ζ = 0 or around ζ = ∞. Second, since the connection (2.47) is expressed solely in terms of the quantities invariant under the global symmetry transformation, we can directly explore the dynamical aspect of the problem setting aside all the kinematical information.
We shall first perform the expansion around ζ = 0. To facilitate this task, it is convenient to perform a further gauge transformation and convert (2.49) to the "diagonal gauge", where the ALP take the form In the above,ψ d in the diagonal gauge is defined bŷ and Φ d 's and A d 's are given by Note that the leading terms in the ALP equations as ζ → 0, namely Φ d z for the first equation and A d z for the second, have been diagonalized. Because of this feature, the leading exponential behavior of the two linearly independent solutions around ζ ∼ 0 can be readily determined aŝ For Re ζ > 0 , For Re ζ < 0 , In these expressions, S i→j stands for the quantity where the one-form α is given in (D.41) in Appendix D.2. A remarkable feature of (3.39) is that the integral of our interest ij η makes its appearance in the exponent S i→j . Now to make use of the averaging procedure described in the previous subsection, we need the other type of integrals ijη which appear in Global. To obtain them, we need to expand the Wronskians this time around ζ = ∞. Since the discussion is similar to the expansion around ζ = 0, we will not elaborate on the details and simply give the results: HereS i→j is defined byS whereα is a one-form given in (D.42) in Appendix D.2. Making use of these two types of expansions, we will be able to rewrite the action in terms of the Wronskians, as described in the next subsection.

The action in terms of the Wronskians
We are now ready to derive an explicit expression of the action in terms of the Wronskians.
As shown in the previous subsection, the integrals we used to rewrite the action, namely ij η and ijη , can be extracted from the Wronskians. For instance, consider the integral 21 η, which appears in 2 − , 1 − . Differentiating ln 2 − , 1 − with respect to ζ using (3.38) and (3.39), we get Therefore we can get the integral 21 η by subtracting the first divergent term and then taking the limit ζ → 0. Similarly 21η can be obtained from 2 − , 1 − in the ζ → ∞ limit. Such procedures can be compactly implemented if we use the variable x instead of ζ, which are related as in (2.46). Then, we can write where the "normal ordering" symbol :A(x): ± is defined by (3.45) This precisely subtracts the divergent term mentioned above. Substituting such expressions to the definitions of Global and Global, we can express them in terms of the Wronskians. Then, using (3.32), we arrive at the following expression for the contribution from the S 3 part of the action F action : The first term in (3.46) expresses the contributions of Extra and Extra. The second term A denotes the contribution of ij and ij¯ in Global and Global and is given by where Λ i andΛ i are as given in (3.20) and (3.31) and Λ i →Λ i , →¯ in the last line denotes the terms obtained by replacing Λ i and in the second line withΛ i and¯ respectively. The third term A η is the contribution of ij η and ijη , which is expressed in terms of the Wronskians in the following way: (3.48) The general formula (3.46) will later be used in section 6 to compute the three-point functions.

Structure of the contribution from the vertex operators
Having found the structure of the contribution of the action part, we shall now study that of the vertex operators.

Basic idea and framework
Before plunging into the details of the analysis, let us describe in this subsection the basic idea and the framework, which includes a brief review of the methods developed in our previous work [11].
As explained in detail in [9], the precise form of the conformally invariant vertex operator corresponding to a string solution in a curved spacetime, such as AdS 3 discussed there or EAdS 3 ×S 3 of our interest in this paper, is in general not known. In particular, for a non-BPS solution with non-trivial σ dependence the corresponding vertex operator would contain infinite number of derivatives and is hard to construct. To overcome this difficulty, we have developed in [11] a powerful method of computing the contribution of the vertex operators by using the state-operator correspondence and the construction of the corresponding wave function in terms of the action-angle variables. Although it was applied in [11] to the case of the GKP string in AdS 3 , the basic idea of the method is applicable to more general situations, including the present one, albeit with appropriate modifications and refinements.
Let us briefly review the essential ingredients of the method. (For details, see section 3 of [11].) The state-operator correspondence, in the semi-classical approximation, is expressed by the following equation: can be expressed simply as where the action variables S i and the worldsheet energy E({S i }) are constant.
In the case of the classical string in R × S 3 , the method for the construction of the action-angle variables was developed in [13][14][15], employing the so-called Sklyanin's separation of variables [16]. This method was adapted to the case of the GKP string in AdS 3 in [11] and, as we shall see, can be applied to the present case of the string in EAdS 3 ×S 3 with appropriate modifications. In this method, the essential dynamical information is contained in the two-component Baker-Akhiezer vectors ψ ± , which satisfy the ALP for the right sector and are the eigenvectors of the monodromy matrix Ω More precisely, the dynamical information is encoded in the normalized Baker-Akhiezer vector h(x; τ ), defined to be proportional to ψ(x; τ, σ = 0) (conventionally taken to be ψ + ) and satisfying the normalization condition where n is called the normalization vector. For a finite gap solution associated to a genus g algebraic curve, h(x; τ ) as a function of x is known to have g + 1 poles at the positions x = {γ 1 , γ 2 , . . . , γ g+1 } and the dynamical variables z(γ i ) and p(γ i ), where z is the Zhukovsky variable defined in (2.36), can be shown to form canonical conjugate pairs.
Then by making a suitable canonical transformation, one can go to the action-angle pairs where, in particular, the angle variable is given by the generalized Abel map In fact, the angle variables can be thought to be determined by the quantity n · ψ, since the poles of the normalized vector h occur at the zeros of n · ψ, as is clear from (4.4).
As we are actually dealing with a quantum system using semi-classical approximation, a classical solution should be thought of as being produced by a quantum vertex operator carrying a large charge. Further, since in our framework the vertex operator is replaced by the corresponding wave function, the angle variables defined through a classical solution should be used to describe the wave function of the corresponding semiclassical state.
Now the serious problem is that we do not know the exact saddle point solution for the three-point function. The only information we know is that in the vicinity of each vertex insertion point z i , the exact three-point solution, to be represented by a 2 × 2 matrix Y given by This means that the angle variables associated to Y, as defined relative to the ones asso- where n andñ are the normalization vectors for the right and the left sector and ψ ± (x) and ψ ref ± (x) are the Baker-Akhiezer eigenvectors corresponding to the solutions Y and Y ref respectively and are related by (4.10) How V and V can be obtained will be described in detail in subsection 4.3.
The remaining problem is to fix the normalization vectors n andñ, relevant for the left and the right sectors. In the case of the string which is entirely in AdS 3 [11], we fixed them by the following argument. Consider for simplicity the wave function corresponding to a conformal primary operator of the gauge theory sitting at the origin of the boundary of AdS 5 . Such an operator is characterized by the invariance under the special conformal transformation. Therefore the corresponding wave function and the angle variables comprising it should also be invariant. Explicitly it requires that n · ψ andñ ·ψ must be preserved under the special conformal transformation and this determined n andñ.
The essence of the argument we shall employ for the case of a string in EAdS 3 ×S 3 studied in the present work is the same. However because the structures of the gauge theory operators and the corresponding string solutions are more complicated, we need to generalize and refine the argument. As a result of this improvement, not only has the determination of the normalization vectors become more systematic but also their physical meaning has been identified more clearly. Moreover, the entire procedure of the constructions of the wave functions for the S 3 part and the EAdS 3 part has become completely parallel and transparent. Below we shall begin the analysis first from the gauge theory side.

Characterization of the gauge theory operators by symmetry properties
As sketched above, in order to construct the wave functions expressing the effect of the insertion of the vertex operators, we must study how to characterize the global symmetry properties of the vertex operators and the classical configurations that they produce in their vicinity.
For this purpose, it is convenient to first look at the symmetry properties of the corresponding gauge theory operators. The three composite operators making up the three-point functions in the so-called "SU (2) Table 1: Table 1. The SU(2) R and SU(2) L charges for the basic scalar fields.
These transformation properties are succinctly represented by the 2 × 2 matrix which gets transformed as U L Φ U R , where U L ∈ SU(2) L , U R ∈ SU(2) R . In spite of this SO(4) symmetry, in the existing literature [27] the operators O i are taken to be com-posed of a special pair of fields 21 indicated in Table 2. For example, O 1 is of the form tr (ZZ · · · XZZX · · · Z). In the spin-chain interpretation, Z and X represent the up and the down spin respectively so that O 1 is a state built upon the all-spin-up vacuum state tr Z l on l sites by flipping some of the up-spins into the down-spins which represent excitations. Therefore at each site there is an SU(2) group acting on a spin, and according to For the vacuum state |Z l = | ↑ l it is obvious. As for the excited states, they can be written as the Bethe states i=1 B(u i )| ↑ l , where B(u i ) is the familiar magnon creation operator carrying the spectral parameter u i . It is well-known [28] that such a state is a highest weight state of the total SU(2) R and hence annihilated by the same S + R , provided that the Bethe state is "on-shell", namely that the spectral parameters satisfy the Bethe ansatz equations. Therefore we have found that kinematically all the operators of type O 1 can be characterized as the highest weight state of the total SU(2) R . Now in order to deal with other operators built upon a "vacuum state" different from tr Z l , let us introduce the general linear combinations of Φ I as P · Φ = 4 I=1 P I Φ I . To discuss the transformation property under SU(2) R × SU(2) L , it is more convenient to deal with the matrix Then, we have the representation (4.14) In this notation, P corresponding to Z,Z, X,X take the form As we argued above, all the on-shell states built upon a common vacuum are annihilated by the same S + R . In other words as long as the global transformation property is concerned, the vacuum state can be considered as the representative of all the states built upon it. Further, since the local spin state is identical at each site for the vacuum state we can characterize the vacuum by the form of the "annihilation operator" s + R acting on a single spin state. As it will be slightly more convenient, instead of the annihilation operator, we will use the "raising operator" K = exp(αs + R ), where α is any constant. The vacuum is then characterized by the form of K that leaves its building block invariant.
Let us explain this idea concretely for the operator Z, which is the building block for the simplest vacuum state tr Z l . In the general notation (4.14), we can express Z as . Now let us look for the raising operators K Z and K Z for SU(2) R and SU(2) L respectively, which leave Z invariant. Since Φ transforms intõ K Z ΦK Z , the invariance condition reads This is equivalent to the condition It is easy to find the solutions 22 , which read where β andβ are arbitrary constants.
Next we consider a general case where the vacuum state is given by tr ( P · Φ) l , with arbitrary P . Since, in general, P · Φ does not carry a definite set of left and right charges defined as in Table 1, this state and the ones built upon it by some spin-chain type excitations are not charge eigenstates. Nevertheless, we can characterize this family of states again by the raising operators K andK which leave P · Φ invariant. Just as in (4.16), this condition is expressed as where P corresponds to P . Since P · Φ can be obtained from Z by an SU(2) L × SU(2) R transformation, P can be obtained from P Z by a corresponding transformation of the form Then combined with (4.18) we readily obtain the relation . Comparing this with (4.16) we can express the raising operators K andK in terms of the ones for the operator Z given in (4.17) in the form (4.20) 22 The most general solutions are of the form α β 0 α −1 and α −1 0 β α . However since we are interested in the raising type operators, it is sufficient to consider the operators of the form (4.17).
Now these raising operators can in turn be characterized by the two-component vectors n andñ, which are left invariant under the following action of K andK respectively 23 : Since the overall factor for these vectors are inessential, we can normalize them to have unit length as n † n =ñ †ñ = 1. We shall refer to them as polarization spinors, as they characterize, so to speak, the "direction of polarization" of the highest weight operator P · Φ. It should be noted that from the knowledge of n andñ, one can reconstruct P which is invariant under the raising operators, as in (4.18). In fact, if we set one can easily check that this P satisfies (4.18), with the use of the defining equations (4.21) and a simple formula Let us illustrate these concepts by computing the polarization spinors for the operators Z andZ respectively. For the operator Z we already computed the right and the left raising operators in (4.17). Then it is easy to see that the corresponding polarization spinors n Z andñ Z satisfying K t Z n Z = n Z andK t Zñ Z =ñ Z are given by As a check, from the formula (4.22), we immediately get P Z = 0 0 0 2 , which is the desired form. As for the operatorZ, repeating the similar analysis, the raising operators leaving PZ = 1 + σ 3 invariant can be readily obtained to be with α andα being arbitrary constants. The corresponding polarization spinors can be taken to be Finally consider the normalization spinors for a general operator P · Φ which is related to Z = P Z · Φ through the relation of the form (4.19). Since the raising operators for such an operator are obtained from those for Z in the manner (4.20), the polarization vectors n andñ are expressed in terms of n Z andñ Z as (4.26) 23 Intentionally we are using the same letters n andñ for the vectors introduced here as those used previously for the normalization vectors. This is because they will be shown to be identical.
As an application of this formula, let us re-derive nZ andñZ from this perspective. Since PZ = 1 + σ 3 and P Z = 1 − σ 3 , it is easy to see that they are related by an SU(2) L × SU(2) R transformation of the form In fact this transformation realizes the mapping (Z, X) → (Z, −X). Substituting the forms of U L and U R into the above formula (4.26), we obtain U t R n Z = (1, 0) t and Summarizing, we can say that, as far as the global symmetry properties are concerned, the operators of type O 1 and O 3 are characterized by the polarization spinors n Z andñ Z , while the operators of type O 2 are associated with nZ andñZ. For more general operators built upon the vacuum tr ( P · Φ) l , the corresponding polarization spinors are obtained from n Z andñ Z by appropriate transformations which connect P with P Z as shown in (4.26).
The importance of the above analysis is that, as we shall describe below, precisely the same characterization scheme must be valid for the vertex operators in string theory which correspond to the gauge theory composite operators like O i . Moreover, it will be shown that the polarization spinors introduced purely from the group theoretic point of view above will be identified with the "normalization vectors" that appeared in  in section 4.2) and its conjugate 24 . In what follows, we shall denote such a solution by Y diag . Then we can associate a pair of polarization spinors n Z andñ Z and the raising operators (4.17) to the vertex operator that produces the solution. For convenience, we display them again with appropriate renaming: All the solutions describing a two-point function of mutually conjugate operators, OO , can be obtained from this basic solution Y diag by an SU(2) R × SU(2) L transformation.
Since a normalized three-point function in the gauge theory can be obtained by dividing an unnormalized one by OO -type two-point functions as This relative difference is the quantity of interest since we need to normalize the threepoint function by the two-point functions. In general V belongs to SL(2, C) R ⊃ SU(2) R , since the three-point interaction is necessarily a tunneling process. In contrast the refer- By the same token we can associate the polarization spinor n and the raising transformation K to the local solution Y. However since Y is produced by the same vertex operator as Y ref , we must have n = n ref .
As for K, just as in (4.34), under the transformation Since this operator must leave n and hence n ref invariant, we must have for some β . Substituting the relation (4.34), we get At this stage we need not know the actual values of a and b in these formulas. b will turn out to be irrelevant and a will be expressed in terms of certain Wronskians.

Construction of the wave function for the right sector
We are now ready for the construction of the wave function for the right sector using the formula for the shift of the angle variable φ R given in (4.8).
First we need to fix the normalization vector n appearing in that formula. As we shall show, the answer is that it coincides precisely with the polarization spinor n introduced from the group theoretical point of view in (4.21) in subsection 4.2. Recall that in the formalism developed in [13][14][15], the zeros of n · ψ(x), where ψ is the Baker-Akhiezer vector and n is the normalization vector, determines the angle variables. When one makes a global SL(2,C) R transformation V R on the string solution Y like Y → YV R , the Baker-Akhiezer vector transforms like ψ → V −1 R ψ. In particular, take V R to be the raising operator K under which the vertex operator producing the solution Y is invariant. Then the wave function corresponding to the vertex operator and hence the angle variables comprising it must also be invariant. This means that the zeros of n · (K −1 ψ) = (K t n) · ψ must coincide with the zeros of n · ψ and hence we must have K t n ∝ n. However since K is similar to K diag , it is clear that the constant of proportionality can only be unity and n must satisfy K t n = n. This, however, is nothing but the definition of the polarization spinor given in (4.21). In other words, the proper choice of the normalization vector for constructing the wave function is precisely the polarization spinor associated to the vertex operator to which the wave function corresponds.  The right hand side can be evaluated using the representation (4.38) as where ψ diag ± (x) is the Baker-Akhiezer vector for Y diag , which is related to ψ ref and see if it has the same transformation property as the corresponding operator in the gauge theory. As it will be checked later in this subsection, it turned out that the case (b) is the correct choice. Therefore we will take Substituting them into (4.40), we obtain the important relations As for the polarization spinor, observe that by inspection the following relation holds:  which will be extremely important.
As for the same quantity appearing in n · ψ + (∞), we can safely set it to zero from the beginning since n · ψ ref + (∞) is non-vanishing. In this way we find that n · ψ ref ± (∞)'s all cancel out and we are left with an extremely simple formula for ∆φ R given by (4.49) Note that the shift depends only on the quantity a, which parametrizes the scale transformation not belonging to SU(2) R , showing the tunneling nature of the effect.
Let us now write the formula (4.47) for the operator at z i with a subscript i as Then, from the definition of the Wronskian we obtain n i , n j = a i a j i − , j − ∞ . Writing out all the relations of this form and forming appropriate ratios, we can easily extract out each a 2 i . The result can be written in a universal form as Then substituting this expression into the formula (4.49) we obtain the shift of the angle variable φ R at the position z i as This formula is remarkable in that it cleanly separates the kinematical part composed of n i , n j and the dynamical part described by i − , j − ∞ .
As the last step of the construction of the wave function, we need to pay attention to the convention of [15] that we are adopting. In that work, the Poisson bracket is defined to be {p, q} = 1 for the usual momentum p and the coordinate q. In this convention the Poisson bracket of the action angle variables was worked out to be given by {φ, S} = 1. In other words the action variable S corresponds to q and the angle variable φ corresponds to p. Therefore upon quantization in the angle variable representation, we must set S = i∂/∂φ. This means that the wave function that carries charge S is given by e −iSφ , not by e iSφ .
Recalling the relation (2.37) between the action variable S ∞ and the right charge R, namely S ∞ = −R, and employing the formula (4.51), the contribution to the wave function from the right sector is obtained as Let us now list the basic results for the left sector, omitting the intermediate details.
Just as for the right sector, the formulas below are valid for any type of operator. Using these formulas, we obtain the contribution to the wave function from the left sector as where we used the gauge invariance of the Wronskians and replaced ĩ + ,j + with i + , j + .
Together with Ψ S 3 R obtained in (4.52) we now have the complete wave function for the S 3 part. It is of the structure (4.60) Let us explain each term (4.60) in order. The first term V kin stands for the kinematical part composed of the Wronskians n i , n j and ñ i ,ñ j , The second term V dyn refers to the dynamical part consisting of the Wronskians i − , j − ∞ and ĩ + ,j + 0 , The last term V energy denotes the contribution involving the worldsheet energy shown in the last term of (4.2). Such a term is necessary for the following reason. As explained below (2.97) and at the beginning of section 3.1, we evaluate our wave function on the circle defined by τ (i) = 0, corresponding to ln |z − z i | = ln i . On the other hand, the wave function introduced through the state operator mapping in (4.1) is defined on the unit circle described by ln |z − z i | = 0. The term V energy is needed to fill this gap. As the energy of the each external state is given 27 by 2 √ λκ 2 i , V energy can be evaluated explicitly as Before ending this subsection, let us make two comments. First, it is not guaranteed at this stage that the wave function thus constructed produces a correctly normalized twopoint function. In addition, as discussed in [11], there may be additional contributions which come from the canonical change of variables, {Y, ∂ τ Y} → {φ i , S i }. However, in section 7.3, it will be checked that our result for the three-point function reproduces the normalized two-point function in an appropriate limit. Therefore we can a posteriori confirm that the wave function is properly normalized and the additional contributions are absent. Second, one recognizes that the power of n i , n j , namely R i + R j − R k , is the familiar combination, made out of conformal weights and spins, for the coordinate differences in the three-point functions of a conformal field theory, except for the overall sign. In the next subsection, we will elaborate on this structure of the power from the point of view of the dual gauge theory. Also in section 6.2, where we construct the wave function for the EAdS 3 part, the above difference in the overall sign will be explained.
Summarizing, the product of (4.52) and (4.59) gives the general form of the wave functions for the three-point function. It is expressed in terms of the two types of Wronskians.
One type is the Wronskians between the solutions of the ALP around vertex insertion points. They will be evaluated in section 5. The others are the Wronskians between the polarization spinors associated with the vertex operators, which are of purely kinematical nature and hence should be common to the string and the gauge theory sides.

Correspondence with the gauge theory side
We shall now examine our formula for the wave function from the point of view of correspondence with the gauge theory side. Now consider the right and the left charges carried by O i . From Table 1, and the compositions of O i , we get, for example, R 1 = 1 2 l 1 − M 1 , L 1 = 1 2 l 1 , etc.. Then, computing the powers of interest we get

Products of Wronskians in terms of quasi-momenta
To obtain the information of the Wronskian i ± , j ± between the eigenvectors of the ALP at different points, we need some condition which governs the global property of such Wronskians. As we shall see, such a condition is provided by the global consistency condition for the product of the local monodromy matrices Ω i associated with the vertex insertion points z i . Since the total monodromy must be trivial upon going around the entire worldsheet, we must have Although this appears to be a rather weak condition, it is sufficiently powerful to determine the forms of certain products of the Wronskians in terms of the quasi-momenta p i (x), as discussed in [8,9]. Let us quickly reproduce those expressions. Take the basis in which Ω 1 is diagonal, namely Since the set of eigenvectors j ± at z j form a complete basis, one can expand the eigenvectors i ± at z i in terms of them in the following way: Making use of this formula, Ω 2 can be expressed in the Ω 1 -diagonal basis as where the matrix M ij , effecting the change of basis, is given by Now owing to the constraint (5.1), Ω 1 and Ω 2 must satisfy the following relation: tr (Ω 1 Ω 2 ) = tr Ω −1 3 = 2 cos p 3 .
In a similar manner, products of certain other Wronskians can also be obtained, which are summarized as the following set of equations 29 : What we need for the computation of the three-point functions, however, are the individual Wronskians and not just the products given in (5.9)-(5.14). Such a knowledge will be extracted based on the analytic properties of the Wronskians regarded as functions of the complex spectral parameter x. We will analyze such properties in the next two subsections.

Analytic properties of the Wronskians I: Poles
An individual Wronskian, viewed as a function of x, is almost uniquely determined 30 by its analytic properties, namely the positions of the poles and the zeros. From the expressions exhibited in (5.9)-(5.14), we know that the products of Wronskians have poles at sin p i = 0 and zeros at sin ((±p 1 ± p 2 ± p 3 )/2) = 0. Therefore the question is which factor of the product is responsible for such a pole and/or a zero. In this subsection, we will describe how to analyze the structure of the poles.
To illustrate the basic idea, we will consider the Wronskians 1 + , 2 + and 1 − , 2 − as examples, for which the product is given by 29 Note that the equations (5.9)-(5.14) appear slightly different in form from those derived in [9]. This is because 2 + in this paper corresponds to 2 − in [9] and 2 − in this paper corresponds to −2 + in [9]. 30 As we will discuss later, the Wronskian also contains essential singularities at x = ±1. In addition, an overall proportionality constant cannot be determined by the positions of zeros and poles. These ambiguities will be fixed in section 5.5.
Let us focus on the pole associated with sin p 1 = 0 and denote the position of the pole by x pole . There are two types of points at which sin p 1 vanishes, the branch points and the "singular points". First consider the case where x pole is a singular point, at which the two eigenvalues of the monodromy matrix Ω 1 degenerate to either +1 or −1. This, however, does not mean that Ω 1 is proportional to the unit matrix for the following reason: If Ω 1 ∝ 1, the monodromy condition Ω 1 Ω 2 Ω 3 = 1 forces p 2 to be equal to +p 3 or −p 3 modulo π. However, since p 1 , p 2 and p 3 can be chosen completely independently, there is no reason for such special relation to hold. Thus, the only remaining possibility is that the monodromy matrix Ω 1 takes the form of a Jordan-block at x = x pole , namely, In this case, the eigenvectors 1 + and 1 − degenerate at x = x pole and we have one eigenvector. To see what happens at x = x pole more explicitly, let us study the asymptotic behavior of 1 ± near z 1 . In the vicinity of each puncture, the saddle point solution for the three-point function can be well-approximated by an appropriate solution for a twopoint function. Consequently, the eigenvectors for the three-point function 1 ± can also be approximated near z 1 by the eigenvectors for the two-point function 1 2pt ± . As shown in (2.96), this structure can be seen most transparently in the Pohlmeyer gauge. Working out the subleading corrections, we obtain the following expansion for the eigenfunctionŝ 1 ± :1 Here τ (1) and σ (1) are the local coordinates near z 1 given in (2.97) and c k andc k are 2 × 2 matrices dependent only on σ (1) and x. The constants a k in the exponents are such that successive terms are becoming smaller by exponential factors as τ → −∞. An important observation is that since1 2pt ± are eigenfunctions corresponding to a two-point function, they are insensitive to the global monodromy constraint (5.1) on the three-point function and hence non-degenerate at x = x pole . An apparent puzzle now is how exponentially small corrections can produce the degeneracy of1 ± .
The answer is the following. Since one of the solutions1 2pt ± is exponentially increasing (i.e. big) and the other is decreasing (i.e. small) as τ → −∞, let us consider the case where1 2pt + is big and1 2pt − is small. Now for1 ± to become degenerate at x = x pole , logically there are three possibilities (a)1 + = α1 − , α = finite , (5.18) First, since1 2pt + is much larger than1 2pt − by assumption, the case (a) cannot occur. Now consider the case where x is slightly different from x pole . Then β is large but finite and the relations (b) or (c) must be realized approximately. But it is obvious that (b) is the only consistent relation since exponentially small solution can appear in the big solution but not the other way around. Therefore we must have the situation It is clear from this expression that which one of theî ± diverges as z → z i is governed by the sign of the real part of the quasi-energy q(x). Since the divergence of the eigenfunction produces a pole on the Wronskian containing it, we can determine which Wronskian of the product is responsible for the pole with the following general rule: At sin p i = 0, the Wronskians behave as Hence, for Re q(x) > 0 the pole occurs in i − , * , while for Re q(x) < 0 it occurs in i + , * .

Analytic properties of the Wronskians II: Zeros
Having determined the pole structure, let us next discuss the zeros of the Wronskians. The determination of the zeros is substantially more difficult since, in contrast to the poles which are local phenomena, the zeros are determined by the global properties on the Riemann surface. As shown in previous works [24][25][26], the notion of the WKB curve [23] is one of the main tools to explore such global properties. However, as its name indicates, the WKB curve is useful only when the leading term in the WKB expansion is sufficiently accurate. For this reason, it is not powerful enough to fully determine the zeros of the Wronskians in the whole region of the spectral parameter space. In this subsection we shall introduce an appropriate generalization of the WKB curve, to be called the exact WKB curve, to overcome this difficulty. 31 Remark: This does not mean of course that there is only one solution at the degeneration point. There must exist another independent solution of new structure, namely the structure which is different from1 ± . However, as long as we stick to this basis, what we see is that one of the solutions diverges and disappears.

WKB approximation and WKB curves
In order to motivate the generalized version, we shall first briefly review the ordinary WKB curves defined in [23].
When the expansion parameter ζ is sufficiently small, the leading term of the WKB expansion for the solutions to ALP (3.33) around z i is given bŷ Along the WKB curve, the magnitude of the leading term in the WKB expansion (5.25) increases or decreases monotonically, until they reach a zero or a pole of T (z). Thus, if two punctures z i and z j are connected by a WKB curve and the spectral parameter ζ is sufficiently small, the small solution s i defined around z i will grow exponentially as it approaches the other puncture z j . In other words, the small solution s i behaves like the big solution around z j . Therefore s i will be linearly independent of s j and hence the Wronskian between these two small solutions s i , s j must be non-vanishing.
With this logic, we conclude that the Wronskians i ± , j ± are non-vanishing if the  figure) and a pole (a red circle in the right figure). There are exactly three WKB curves that emanate from a zero. In contrast, there are infinitely many WKB curves radiating from a pole in all directions.

Exact WKB curves
Evidently, the analysis above is valid only in a restricted region of the spectral parameter plane where the approximation by the leading term of the WKB expansion is reliable.
Actually, even if we improve the approximation by going to the next order approximation, we still cannot cover the entire spectral parameter plane because such an expansion is only an asymptotic series. It is indeed possible that as we change x the small and the big solutions interchange their roles. Such a phenomenon is clearly non-perturbative and cannot be captured by the usual expansion. So to understand the structure of the zeros on the whole spectral parameter plane, it is necessary to generalize the notion of WKB curves in a non-perturbative fashion.
In order to seek such an improvement, we need to look closely at the general structure of the conventional WKB expansion. Let us denote the components of the solutionψ d to the ALP in the diagonal gauge (3.33) aŝ By substituting (5.27) into the ALP (3.33), we obtain the equations for the components ψ (1) and ψ (2) . Then, upon eliminating ψ (2) in favor of ψ (1) , we get a second-order differential equation for ψ (1) . To solve this equation, we expand ψ (1) in powers of ζ in the form (5.28) One can then determine the one-forms W n order by order recursively. This procedure is described in Appendix D.1. As a result of such a computation, we find that the WKB expansions for two linearly independent solutions to the ALP can be expressed in the following form: Here W WKB ≡ W z WKB dz + Wz WKB dz is the one-form defined as a power series in ζ, with the leading term given by √ T dz/(2ζ). On the other hand, the functions f (1) With this structure in mind, we now introduce an improved notion of the WKB curve, to be called the "exact WKB curve", by writing the exact solutions to the ALP in the where f (1) ex and f (2) ex are given by where f ex (1) and f ex (2) are given by 2 .

(5.35)
Using the definition (5.32), let us now discuss the generalization of the WKB curves.
The quantity √ T dz/ζ used to define the original WKB curves is proportional to the leading term in the expansion of W WKB . Therefore the most natural generalization of the WKB curves would be to use W ex , which is a non-perturbative completion of W WKB , to define them as Im (W ex (z; ζ)) = 0 . The precise definition is given as follows: The exact WKB curves associated to the puncture z i are defined as the curves satisfying the equation where W (i) ex is the exponential factor for the solution s i , which is the smaller of the two eigenvectors i + and i − . Explicitly, it is defined through the expression Let us now make several comments. First, it is easy to see that this definition of the exact WKB curves reduces to that of the ordinary WKB curves when ζ is sufficiently small. Second, as in (5.34), with a flip of sign in the exponent, we can obtain another which is big near the puncture z i and satisfies s i , b i = 1. Such a solution b i , however, is not guaranteed to be an eigenvector since the eigenvector distinct from s i is in general given by a linear combination of the form b i + cs i . Now the definition of EWKB (i) given above refers to a specific puncture from which the curves emanate. In order for the notion of the exact WKB curve to be valid for the entire worldsheet, we must guarantee that the definitions of EWKB (i) 's for i = 1, 2, 3 are consistent in the region where they overlap. To check this, let us consider the behavior of the small solution s i as we follow an EWKB (i) . Along such a curve the phase of the exponential factor of s i stays constant, while its magnitude increases monotonically 32 , until it reaches some endpoint. Consider the case in which this endpoint is the puncture at z j . In such a case, we know that s i grows exponentially as it approaches z j and in fact behaves like a big solution b j , up to an admixture of the exponentially small solution s j .
Thus, with sufficient accuracy, s i can be expressed in the small neighborhood of z j as But since the exponent of the small solution s j , which is used to define EWKB (j) , is the same as that of b j except for the sign, we see that by definition the curve we have been following becomes an EWKB (j) curve in the vicinity of z j , when z i and z j are connected by such a curve. Therefore the definitions of EWKB (i) and EWKB (j) are indeed globally consistent.
Let us now make use of the exact WKB curves to determine the analytic properties of the Wronskians. First, by following exactly the same logic as in the case of the ordinary WKB curves, we can immediately conclude that the Wronskian involving two small solutions s i and s j must be nonzero if two punctures z i and z j are connected by some exact WKB curves. Although this is an extremely useful information, the problem seems to be that, unlike the ordinary WKB curves, we do not know the configurations of the EWKB curves since the exact solutions to the ALP are not available.
Nevertheless, we shall show below that by making use of a characteristic quantity defined locally around each puncture for the EWKB curves, it is possible to fully classify the topology (connectivity) of the curves on the entire worldsheet. The quantity in question is the "number density" of the EWKB curves emanating from a puncture at z i . To motivate its definition, consider two such curves which emanate from z i and end at z j and let the constant phase of W (i) ex along the two curves be φ 1 and φ 2 . Evidently the magnitude of the difference |φ 1 − φ 2 | is the same around z i and around z j , that is, it is conserved. If there is no singularity in the region between these lines, we can draw in more EWKB curves connecting z i and z j . Because of the property of the constancy of the phase difference noted above, it is quite natural to draw the curves in such a way that the difference of the phases of the adjacent curves is some fixed unit angle. Going around z i and counting the number of such lines, we can define the number (density) of the EWKB (i) curves as 33 41) 32 Strictly speaking the small eigenvector (5.38) also contains a prefactor in front of the exponential. This prefactor, however, does not play a significant role in our discussion since it drops out if we consider the ratio of two solutions s i /b i . It is in fact sufficient to know the ratio in order to identify the small solution and the big solution. 33 In (5.41), we have chosen a convenient normalization of N i .
where C i is an infinitesimal circle around z i . Although N i is not an integer in general, we will call it "a number of lines". Actually we can express N i in a more explicit way. From the asymptotic behavior of i ± (2.96), we can obtain the form of W (i) ex near z i as Here (τ (i) , σ (i) ) is the local coordinate defined in (2.97), and + or − sign is chosen depending on which of the solutions i ± is small. Substituting (5.42) into the definition (5.41), we obtain a simple expression Since the phase around the puncture is governed by the local monodromy, it is natural that N i can be expressed in terms of p i (x).
Before we make use of the concept of N i in a more global context, let us derive two important properties of the EWKB (i) 's which will be necessary for determining their configurations.
The first property will be termed the non-contractibility. It can be stated as follows: "All the exact WKB curves which start and end at the same puncture are noncontractible." In other words, such curves must go around a different puncture at least once. The proof is simple. Recall that the Wronskians between small solutions should be nonzero if two punctures are connected by an exact WKB curve. If we apply this statement to the same puncture z i connected by an EWKB curve, we would conclude that s i , s i is non-zero, which is clearly false. The only way to be consistent with the general assertion above is that the curve is non-contractible and the solution gets transformed by the non-trivial monodromy Ω as it goes around other punctures. In this case the Wronskian is of the form s i , Ωs i , which need not vanish.
The next property is concerned with the endpoints of the exact WKB curves. It can be stated as follows: " All but finitely many exact WKB curves terminate at punctures. " The proof can be given as follows. As in the case of the ordinary WKB curves, the possible endpoints are the zeros or the poles of W On the other hand, in the second case, which we shall call asymmetric, not all the triangle inequalities are satisfied. For example, one is violated like In this case, one can readily convince oneself that, while all the curves emanating from z 2 and z 3 end at z 1 , there must exist a non-contractible curve connecting z 1 to itself. This is depicted in the right figure of figure 5.2. 34 The big solution b i cannot vanish at such points so as to ensure the normalization condition s i , b i = 1. 35 In the case of the usual WKB curves, W ex ∼ T (z)dz and hence N i is proportional to κ i . Classification by the triangle inequalities for κ i already appeared in [9]. In this way, we can completely classify the configurations of the exact WKB curves from the local information N i = |Re p i (x)|. Note that N i depends on x. In fact it happens that as x changes a symmetric configuration can turn into an asymmetric configuration and vice versa. In an application of the present idea to the classical three-point function in Liouville theory [29], it was checked that such a transition must be taken into account in order to obtain the correct result. Below, we will see explicitly how the patterns of the configurations of the exact WKB curves analyzed above can be used to determine the zeros of the Wronskians.

Determination of the zeros of the Wronskians
As an example, let us focus on the factor sin below applies to all the other cases straightforwardly.) From the relations (5.9)-(5.14), we find that the products of Wronskians that become zero are appear. This is in fact a general feature and holds also for other situations. Now, let us present two theorems, which will be useful in the determination of the zeros. The first theorem is the following assertion, which we have already proved: We can now analyze the zeros of the Wronskians using these theorems. First consider the symmetric case. Since one of the states i ± must be a small solution, either S + or S − must contain two small solutions. For a symmetric configuration, they must be connected by an exact WKB curve. Then by theorem 1 the Wronskian between them must be nonvanishing. Theorem 2 further asserts that all the Wronskians for the members of that set are non-vanishing, while the ones for elements of the other set all vanish.
Next, consider the asymmetric case. For simplicity, let us assume that N 1 > N 2 + N 3 is satisfied 36 . In such a case, there exist exact WKB curves which start from z 1 , go around z 2 (or z 3 ), and return to z 1 . To make use of the existence of such a curve, consider the following Wronskians: 36 Generalization to other cases is straightforward.
To compute them, we first note that 1 ± can be expressed in terms of 2 ± in the following Then, applying Ω 2 to (5.49) and substituting them to (5.48), we can express (5.48) in terms of the ordinary Wronskians as Consider the case where 1 + is the small solution.
Since Ω 2 1 + can be obtained by paralleltransporting 1 + along the exact WKB curve which starts and ends at z 1 , Ω 2 1 + must behave as the big solution around z 1 . Therefore, the Wronskian 1 + , Ω 2 1 + is non-vanishing in this case. Then from (5.50) it follows that 1 + , 2 + must also be non-vanishing. Applying the theorem 2, we conclude that the Wronskians between the members of S + are nonvanishing and those of S − all vanish. In an entirely similar manner, when 1 − is the small eigenvector, we obtain the result where the roles of S + and S − are interchanged.
Performing similar analyses for the other cases, we obtain the general rules summarized below.
Rule 1: Decomposition of the eigenvectors into two groups.

Rule 2: Symmetric case.
When the configuration of the exact WKB curves is symmetric, the Wronskians from the group which contains two or more small solutions are nonzero whereas the Wronskians from the other group are zero.

Rule 3: Asymmetric case.
When the configuration of the exact WKB curves is asymmetric and N i 's satisfy N i > N j + N k , the Wronskians from the group which contains the smaller of the two solutions i ± are nonzero whereas the Wronskians from the other group are zero.
In the next subsection, we will utilize these rules to evaluate the individual Wronskians.

Individual Wronskian from the Wiener-Hopf decomposition
Making use of the data for the analyticity of the Wronskians obtained in the previous subsection, we now set up and solve a Riemann-Hilbert problem to decompose the product of Wronskians and extract the individual Wronskians.The standard method for such a procedure is known as the Wiener-Hopf decomposition, which extracts from a complicated function a part regular on the upper half plane and the part regular on the lower half plane.
The typical set up is as follows. Suppose F (x) is a function which decreases sufficiently fast at infinity and can be written as a sum of two components where F ↑ (x) is regular on the upper half plane while F ↓ (x) is regular on the lower half plane. Then, each component, in the region where it is regular, can be extracted from where Im x < 0 should be expressed as Note that the first term F (x) on the right hand side can be thought of as due to the integral along a small circle around x = x.
To apply this method to the case of our interest, namely to the equations (5.9)-(5.14), we take the logarithm and represent them in a general form as (5.55) Here i denotes a + or − sign. In this process, we have neglected the contributions of the form ln(−1), since they only contribute to the overall phase of the three-point functions.
Our aim will be to express each of the terms on the left hand side of (5.55) in terms of some convolution integrals of the functions on the right hand side. To put it in another way, we wish to decompose each term on the right hand side into contributions coming from each term on the left hand side. Since the quasi-momentum p i (x) is defined on a Riemann surface with branch cuts, we need to generalize the Wiener-Hopf decomposition formula in an appropriate way, as discussed below.

Separation of the poles
Let us first decompose the terms of the form − ln sin p i , which give rise to poles of the Wronskians. As shown in the previous section, which Wronskian develops a pole is determined purely by the sign of the real part of the quasi-momentum q i (x). Therefore, we should be able to decompose the quantity − ln sin p i by using a convolution integral along the curve defined by Re q i = 0. For the ordinary Wiener-Hopf decomposition, the convolution kernel is given simply by 1/(x − x ). In the present case, however, we have a two-sheeted Riemann surface and hence we must make sure that the kernel has the simple pole only when x and x coincide on the same sheet. When they are on top of each other on different sheets, no singularity should occur. The appropriate kernel with this property is given by When x and x get close to each other but on different sheets, the square root factor tends to −1 canceling the +1 term and hence the kernel is indeed regular. Furthermore, in the limit that x tends to ∞, the kernel K i (x ; x) decreases like (x ) −2 , which is sufficiently fast for our purpose.
With such a convolution kernel, we can carry out the Wiener-Hopf decomposition in the usual way. Namely the term − ln sin p i (x) can be decomposed into the contributions of i + , j j (x) and i − , j − j (x) as where the convolution integral is defined as As for the contours of integration, Γ i + is defined by Re q i = 0 and Γ i − stands for −Γ i + .
The direction of the contour Γ i + is defined such that i + , j j (x) does not contain poles can re-express the convolution integrals (5.57) and (5.58) as integrals only on the first (or the upper) sheet.: Here, Γ u i ± denotes the portion of Γ i ± on the upper-sheet of the spectral curve and the kernel K i (x ; x) (without a hat) is defined by . In what follows, such contributions will be referred to as contact terms.

Separation of the zeros
Next we shall discuss the decomposition of the first two terms on the right hand side of (5.55), which are responsible for the zeros of the Wronskians. To perform the decomposition, again we need to determine the appropriate convolution kernel and the integration contour.
Let us first discuss the convolution kernel. As the terms of our focus depend on all the quasi-momenta p i (x)'s, the appropriate convolution kernel must be a function on the Riemann surface which contains all the branch cuts of the p i (x)'s. Such a kernel can be easily written down as a generalization of the expression (5.56) and is given by Since there are two choices of sign for each square root factor on the right hand side of (5.63), K all is properly defined on the eightfold cover of the complex plane. In what follows, we distinguish these eight sheets as { * , * , * }-sheet, where the successive entry * is either "u" denoting upper sheet or "l" denoting lower sheet, referring to the two sheets for p 1 (x), p 2 (x) and p 3 (x) respectively. It is clear that the kernel (5.63) has a pole with a residue +1 at x = x only when two-points are on the same sheet. Therefore it has a desired property for the Wiener-Hopf decomposition.
Let us next turn to the contour of integration. As discussed in the previous section, the zeros of the Wronskians are determined by the following two properties: (i) The connectivity of the exact WKB-curves and (ii) the relative magnitude of the eigenvectors i ± .
Therefore, curves across which these two properties change can be the possible integration contours. Corresponding to the properties (i) and (ii) above, there are two types of integration contours; the curves defined by Re q i (x) = 0 and the curves defined by  Let us now show that the kernel K all used in (5.64) and (5.65) can be effectively replaced by simpler combinations of the form (K i + K j ) /8. To explain the idea, consider the following integral as an example: As the first step, we make a change of integration variable from x toσ 3 x , whereσ i denotes the holomorphic involution with respect to p i , namely the operation that exchanges the two sheets associated with p i . Although this clearly leaves the value of the integral intact, the form of the integral changes. One can easily verify that the following transformation formulas for the integrand and the contours hold: In the second line (5.68), the "sign-flipped kernel" K all is defined by Making such transformations, we can re-express the integral (5.66) as all (x ; x) ln sin Just as before, we neglected the contributions of the form ln(−1) as leading to pure phases. Also, the same remarks made below equations (5.64) and (5.65) on the position of x relative to the contour lines apply to the expressions (5.72) and (5.73) above.
Finally, for later convenience, let us further re-write the above expressions as integrals performed purely on the {u, u, u}-sheet. Each contour M 1 2 3 has parts on the eight different sheets denoted by M u,u,u 1 2 3 , M u,u,l 1 2 3 , etc., where the superscripts indicate the relevant sheet in an obvious way. Consider for example the first integral in (5.72) along the contour M +++ . The form as given is for the portion M uuu +++ . For the portion denoted by M ulu +++ for example, if we wish to express its contribution in terms of an integral on the {u, u, u}-sheet, we need to change the sign of K 2 and p 2 . Then the integral becomes identical to that of the first term in the second line of (5.72), except along M uuu +−+ . In similar fashions we can re-express the contributions from the eight parts of M +++ in terms of the integrals on the {u, u, u}-sheet. After repeating the same procedure for the rest of the three terms in (5.72), one finds that the net effect is that each term of (5.72) is multiplied by a factor of eight, with each contour restricted to the {u, u, u}-sheet. In this way we obtain the representations The results obtained in this subsection and the previous subsection are both expressed in terms of certain convolution integrals on the spectral curve. Thus, in what follows, we will denote their sum by Conv i ± , j ± .
Before ending this subsection, let us make one important remark. Although each convolution integral obtained so far is divergent at x = ±1, the divergence cancels 38 in the sum Conv i ± , j ± . Thus the contribution singular at x = ±1 must be separately taken into account as we will do in the next subsection. 38 One can confirm this by expanding the convolution integrals around x = ±1.

Singular part and constant part of the Wronskians
In addition to the main non-trivial parts determined by the Wiener-Hopf decomposition described above, there are two further contributions to the Wronskians. One is the contribution singular at x = ±1, coming from such structure in the connections used in ALP.
The other is the possibility of adding a constant function on the spectral curve. In this subsection, we will determine these two contributions.
Let us first focus on terms singular at x = 1. To determine such terms, we will need the WKB expansions around x = 1 for all the Wronskians, not just the ones that were discussed in section 3.2, namely i + , j + and i − , j − . This is because of the following reason: Although the formulas we obtained for the contribution of the action and that of the wave function appear to contain Wronskians of the type i + , j + and i − , j − only, we must understand their behavior when they are followed into the second sheet as well in order to know the analyticity property on the entire Riemann surface. As shown in (2.100), when we cross the branch cut associated with p i (x) into the lower sheet, the eigenfunctions i + and i − behave like i − and −i + on the upper sheet, respectively, .
Therefore the behavior of i + , j + on the {u, l, * }-sheet can be obtained from the behavior of i + , j − on the {u, u, * }-sheet, etc.
Now the WKB expansions of the Wronskians of the type i + , j − can be obtained from those of i + , j + by the use of the following Schouten identities: Indeed these identities can be regarded as the equations for the six unknown Wronskians of the form i + , j − . If we consider all the combinations of i, j and k in (5.76), we obtain three independent equations. Combining them with the equations (5.12)-(5.14) for the products of the Wronskians, we can completely determine i + , j − 's in terms of i + , j + in the following form: From these expressions, we can obtain the WKB-expansion for every Wronskian using the results for i + , j + .
The singular term of the Wronskians is given simply by the leading term in the WKB expansion. For instance, the singular terms for i + , j + and i − , j − at x = 1 on the {u, u, u}-sheet is determined from the expansion (3.37) and (3.38) as Then by using (5.77)-(5.82) we can determine the singular terms for i + , j − on the {u, u, u}-sheet as In order to determine the singular terms completely, we also need to understand the singular behavior on other sheets. As already described, this can be done by utilizing the fact that i + and i − transform into i − and −i + respectively as one crosses a branch cut associated to p i (x). For instance, applying this rule we can easily find that the singular term for 1 + , 2 + must behave in the following way on each sheet: T dz (on the {u , u , * }-sheet) , T dz (on the {u , l , * }-sheet) , T dz (on the {l , u , * }-sheet) , T dz (on the {l , l , * }-sheet) .

(5.95)
Combining all these results, it is possible to write down the expression on the entire Riemann surface which gives the correct singular behavior on the respective sheet. It is given by (5.96) Here and hereafter, we will use the notation Sing ± [f (x)] to denote the singular term of In an entirely similar manner, we can determine the terms singular at x = −1 as Singular terms for other Wronskians at x = ±1 can be determined in a similar manner.
The remaining issue is the ambiguity of adding a constant function to the logarithm of the Wronskian. Such an ambiguity can be fixed by once more utilizing the property that i ± that i + (i − ) transforms into i − (−i + ) as it crosses the branch cut of p i . This leads to the following constraint for the Wronskians It turns out that all the results obtained so far satisfy (5.98). Since this property gets lost upon adding a constant to the logarithm of the Wronskian, it shows that our results are already complete and we should not add any constant functions.

Complete three-point functions at strong coupling
Up to the last section, we have developed necessary methods and acquired the knowledge of the various parts that make up the three-point functions of our interest. Now we are ready to put them together and see that they combine in a non-trivial fashion to produce a rather remarkable answer.
First in subsection 6.1, we obtain the complete result for the S 3 part by putting together the contribution of the action and that of the vertex operators. These two contributions combine nicely to produce a simple expression in terms of integrals on the spectral curve. Then, adapting the methods developed for the S 3 part, we evaluate in subsection 6.2 the EAdS 3 part of the three-point function. Our focus will be on the differences between the S 3 and EAdS 3 contributions. Finally in subsection 6.3, we present the full answer by combining the contributions of the S 3 part and the EAdS 3 part. We will see that the structure of the final answer closely resembles that of the weak coupling result. Detailed comparison for certain specific cases will be performed in section 7.

The S 3 part
Before we begin the actual computations, let us summarize the structure of the contributions from the S 3 part to the logarithm of the three-point function, which we denote by F S 3 . As was already indicated in section 2.3, F S 3 consists of the contribution of the action and that of the vertex operators, namely Each contribution can be further split into several different pieces as Among these terms, A , V kin and V energy have already been evaluated respectively in (3.47), (4.61) and (4.63). Thus, our main task will be to compute A η and V dyn . As shown in (3.48) and (4.62), A η is given by the normal ordered derivatives of the Wronskians, :∂ x ln i + , j + : ± , whereas V dyn is given by the Wronskians evaluated at x = 0 and x = ∞, ln i + , j + | ∞ and ln i − , j − | 0 . From the discussion in section 5, we know the Wronskians are comprised of two different parts, the convolution-integral part Conv [ln i * , j * ] and the singular part Sing ± [ln i * , j * ]. They both contribute to A η and V dyn . In what follows, we examine these two parts separately and evaluate their contributions to A η and V dyn .

Contributions from the convolution integrals
We begin with the computation of the convolution integrals. To illustrate the basic idea, let us study Conv Applying similar analyses to other Wronskians and using the formulas (3.48) and (4.62), we can obtain the contributions of the convolution integrals to A η and V dyn , which will be denoted by Conv [A η ] and Conv [V dyn ]. They are given by To simplify the expressions, we have introduced the double bracket notation , to denote sum of three terms with designated combinations of signs, defined as a i 123 = a 1 + a 2 + a 3 , a i 3 12 = a 1 + a 2 − a 3 , etc. , (6.5) Also, we have employed the abbreviated symbols It turns out that the two contributions (6.3) and (6.4) combine to give a remarkably simple expression displayed below. This is due to the crucial relation of the form where z(x) on the right hand side is the Zhukovsky variable, defined in (2.36). Although this equality can be verified by a direct computation using the explicit form of p i (x) for the one-cut solutions given in (2.58), it is important to give a more intuitive and essential understanding. Note that the right hand side of (6.7) is proportional to the integrand of the filling fraction given in (2.35). Therefore when integrated over appropriate a-type cycles, it produces the corresponding conserved charges. In other words, it is characterized by the singularities associated with such charges. Now observe that the left hand side precisely consists of terms which provide such singularities. The first two terms are responsible for the singularities at x = ±1, while the last two terms contain the poles at x = ∞ and x = 0 associated with the charges S i ∞ and S i 0 respectively. Furthermore, it should be emphasized that the formula above unifies the contributions in two sense of the word. First, it unites the contributions from the action, represented by the first two terms, and those from the vertex operators, represented by the last two terms. Only when they are put together one can reproduce all the singularities of the right hand side.
Second, the expression obtained on the right hand side is universal in that all the specific data shown on the left hand side, namely κ i , S i ∞ and S i 0 , are contained in one quantity p i (x). As we shall discuss in section 6.2, this feature allows us to write down the same form of the result (except for an overall sign) given by the right hand side of (6.7) for the contributions from the EAdS 3 part, using the quasi-momentum for that part of the string. Now, applying (6.7) we can rewrite the sum T conv ≡ Conv [A η ] + Conv [V dyn ] into the following compact expression: z(x) (dp 1 + dp 2 + dp 3 ) 2πi ln sin z(x) (dp 1 + dp 2 − dp 3 ) 2πi ln sin z(x) (dp 1 − dp 2 + dp 3 ) 2πi ln sin z(x) (−dp 1 + dp 2 + dp 3 ) 2πi ln sin In the last line, we included the possible contributions from the contact terms, denoted by Contact.

Contributions from the singular part of the Wronskians
We now turn to the computation of the singular part Sing ± [ln i * , j * ]. By substituting the expressions for the singular part of the Wronskians, such as (5.96) and (5.97), into the formulas (3.48) and (4.62), we can evaluate the contributions of the singular part in a straightforward manner. From this calculation, we find that a part of the terms contribute only to the overall phase of the three-point functions. For instance, the first and the third term in (5.96), which are proportional to ±πi(κ 1 + κ 2 − κ 3 ), will only yield an overall phase owing to the factor of πi. Just as before, we will ignore such contributions in this work. Then the contributions of Sing + [ln i * , j * ] to A η and V dyn , denoted by Sing and Sing + [V dyn ], are obtained as and . and Now just as we did for Conv [A η ] + Conv [V dyn ], we can make use of the relation (6.7) to rewrite the sum Sing ± [A η ] + Sing ± [V dyn ] into much simpler forms. The results are Sing + [A η ] + Sing + [V dyn ] =:z(x) dp 1 dx + dp 2 dx − dp 3 dx : + 21 + :z(x) dp 1 dx − dp 2 dx + dp 3 dx and (6.14) The expressions :z(x) dp i /dx: ± in (6.13) and (6.14) above can be evaluated using the explicit form of the quasi-momentum, given in (2.58), as 39 :z(x) dp i dx This provides fairly explicit forms for the expressions Sing ± [A η ] + Sing ± [V dyn ].

Result for the S 3 part
We can now combine the results obtained so far and obtain the net contribution of the S 3 part. Recall that the general structure of the S 3 part of the three-point functions we have computed is of the form Among the various terms shown above, those which can be expressed in terms of the contour integrals of or¯ can be combined and evaluated using the explicit form of :z dp i /dx: ± given in (6.15). The result is Since and¯ behave near the punctures as the expression (6.17) diverges in the following fashion as the regularization parameters i 's tend to zero: Notice, however, that this divergence is precisely canceled by the second term of (6.16).
Therefore, the quantity (6.16) as a whole is finite in the limit i → 0. This is as expected for correctly normalized three-point functions.
Let us summarize the final result for the logarithm of the three-point functions coming from the S 3 part. It can be written in the form where V kin is the kinematical factor depending only on the normalization vectors given in (4.61), T conv is the sum of the contributions from the convolution integrals (6.8), and T sing , which is given in (6.17), represents the sum of A defined in (3.47) and the contributions from the singular parts of the Wronskians.

The EAdS 3 part
We now discuss the contributions from the EAdS 3 part. Since the logic of the evaluation is almost entirely similar, we will not repeat the long analysis we performed for the S 3 part.
In fact it suffices to explain which part of the analysis for the S 3 part can be "copied" and which part has to be modified.

Contribution from the action
Let us begin with the contribution from the action integral. Since EAdS 3 and S 3 are formally quite similar, the computation of the action integral can be performed in exactly the same manner. There is, however, a simple but crucial difference. It is the overall sign of the integral. For EAdS 3 , the counterpart of the matrix Y shown in (2.14) is given by where The right current is then defined aŝ Now compare the expressions of the stress tensors and the action integrals for S 3 and EAdS 3 , expressed in terms of the respective right current. They are given by This shows that while we have the equality tr (ĵ zĵz ) = tr (j z jz) = κ 2 , the signs in front of the action integrals are opposite. Therefore all the results for the action integral are formally the same as those for the S 3 case, but with opposite signs. This will lead to various cancellations with the contributions from the S 3 part, as we shall see shortly.

Contribution from the wave function
As for the evaluation of the contribution from the wave function, the basic logic of the formalism developed in section 4 for the S 3 still applies. However, there are a few important modifications, as we shall explain below.
As discussed in our previous work [11], in the case of a string in EAdS 3 the global symmetry group is SL(2, C) R × SL(2, C) L and hence the the raising operators with respect to which we define the highest weight state are the left and the right special conformal transformations given by where β R and β L are constants. Applying our general argument for the determination of the polarization spinors, we readily find It should be noted that, compared to the S 3 case given in (4.23), n diag here for the right sector is the same asñ Z for the left sector there and similarlyñ for the left sector in the present case is identical to n Z for the right sector for the S 3 case. Now the algebraic manipulations for the construction of the wave functions are the same as for the S 3 case up to the computation of the factor e i∆φ . Therefore, for the right sector, we get the same result for the Z-type operator in the left sector, given in (4.58). For example at z 1 we have e i∆φ R,1 = a −2 1 = 1 + , 2 + 3 + , 1 + 2 + , 3 + ∞ n 2 , n 3 n 1 , n 2 n 3 , n 1 (6.29) This is the inverse of the result for S 3 obtained in (4.51) with i − replaced by i + . The result for the left sector is similar. What this means is that the wave function for the EAdS 3 is obtained from the one for the S 3 case by (i) reversing the sign of the powers and (ii) exchanging i + and i − . Abusing the same notations for the polarization spinors and the eigenvectors as in the S 3 case, we get where R i and L i here are the combinations of the conformal dimension ∆ i and the spin S i given by Then the term z in X −1 − X 4 becomes negligible compared to x 2 /z and X µ approaches a null vector, with large components. Such a vector can be parametrized, up to an overall scale, by the boundary coordinates x = (x 1 , x 2 ) as x 2 = x r η rs x s = x r x r , η rs = (+, +) , r, s = 1, 2 . (6.34) As usual, one can map X µ to the matrix X µΣ µ , withΣ µ = (1, σ 1 , σ 2 , σ 3 ), which transforms from left under SL(2, C) and from right under SL(2, C) * . Then, it is well-known that for a null vector X µ the matrix elements of X µΣ µ can be written as a product of spinors (or twistors) as These spinors can be identified precisely as the polarization spinors characterizing a vertex operator which is placed at x on the boundary for the following reasons. First they transform in the correct way: Under the global transformation X µΣ µ → V L (X µΣ µ )V R , we have (σ 1ñ ) α → (V L σ 1ñ ) α and nα → (nV R )α. This is equivalent toñ → V t Lñ and n → V t R n, which are the right transformation laws. Second, these spinors coincide with the polarization spinors given in (6.27) and (6.28) when we bring the point x to the origin of the boundary by the translation by the vector − x. This is effected by the right and the left translation matrices given by Then we get Therefore n andñ can be identified with the polarization spinors for the vertex operator at x on the boundary. Now let n andñ be similar polarization spinors corresponding to a vertex operator at x on the boundary. Then we immediately get which exhibits the familiar coordinate dependence for the three-point function in such a case.

Total contribution from the EAdS 3 part
As we have seen, the structure of the contribution from the EAdS 3 part is essentially the same as that from the S 3 case, except for the important reversal of signs in the powers in the contributing factor (or the terms contributing to the the logarithm of the three-point coupling.) This change of sign occurred both for the action and for the wave function.
As we compute the basic Wronskians in exactly the same way as before and use them to compute the contributions to the logarithm of the three-point function from the action part and the wave function part, we again obtain the expression of the form of the left hand side of (6.7), with the overall sign reversed. Therefore, we can use the identity (6.7) again to obtain the result −z(x )dp i (x )/dx , wherep i denotes the quasi-momentum for the EAdS 3 part of the string. One can check that in fact this rule of correspondence, namely p i (x) →p i (x) and the reversal of sign for the convolution integrals, applies to all the contributions. Thus, combining all the results for the AdS part, the contribution to the logarithm of the three-point function is given by the following expression: +V energy +T sing +V kin +T conv . (6.44) Here,V energy andT sing are equal to −V energy and −T sing respectively,V kin is the kinematical factor given in (6.43), andT conv is the convolution integrals obtained from the unhatted counterpart for the S 3 case with the substitution rule described above.

Complete expression for the three-point function
We are finally ready to put together the contributions from the S 3 part summarized in (6.20) and those from the EAdS 3 part given in (6.44) and present the full answer for the three-point function. As we have already discussed, the divergent terms cancel with each other for the S 3 part and the EAdS 3 part separately. On the other hand, the constant terms proportional to √ λ/6 cancel between S 3 and EAdS 3 contributions. Thus we are left with the kinematical factors and the contributions from the convolution integrals which are of the same structure except for the overall sign. Therefore, factoring the kinematical structure as the logarithm of the structure constant C 123 is finally given by z(x) (dp 1 + dp 2 + dp 3 ) 2πi ln sin z(x) (dp 1 + dp 2 − dp 3 ) 2πi ln sin z(x) (dp 1 − dp 2 + dp 3 ) 2πi ln sin z(x) (dp 1 + dp 2 + dp 3 ) 2πi ln sin z(x) (dp 1 − dp 2 + dp 3 ) 2πi ln sin other two operators become identical. We confirm that the result reduces to that of the two-point function, as expected. Next, in subsection 7.4, we study three-point functions of one non-BPS and two BPS operators, which were studied on the gauge theory side in [4].
We will focus on certain explicit examples and show that the full three-point functions can be expressed in terms of simple integrals which resemble the semi-classical limit of the results at weak coupling [4][5][6][7]. Then, in subsection 7.5, we discuss the Frolov-Tseytlin limit of such three-point functions. In this limit, the integrands in the final expression approximately agree with the ones in the weak coupling, whereas the integration contours are rather different. Lastly, we discuss the possible origin and the implication of this mismatch.

Basic set-up
Before starting the detailed analysis, let us clarify the basic set-ups to be used in this section.
The three-point functions studied extensively on the gauge theory side are those of the following three types of operators (see also As explained in section 4.3.4, such three-point functions vanish unless the conservation laws 40 for the charges, (4.64), are satisfied. Due to these conservation laws, one cannot in general take the operators to be simple BPS states, such as tr (Z l ) or tr (Z l ), which are the highest-weight vectors of the global SU(2) R ×SU(2) L symmetry. Instead, we need to use descendants of the global symmetry to satisfy the conservation laws when we study threepoint functions involving BPS operators [4,27]. While this can be done without problems on the gauge theory side, it leads to certain difficulty on the string theory side. This is because all the classical solutions of string are known to (or believed to) correspond to some highest-weight states. To circumvent this difficulty, below we will utilize the global transformations to make all three operators to be built on different "vacua". On the string theory side, this corresponds to taking the polarization vectors of the three operators, n i 's andñ i 's, to be all distinct. Then no conservation laws will be imposed and we can safely take the limit where some of the operators become BPS while keeping them to be of highest-weight. Since the correlation functions involving descendants can be obtained from the correlation functions involving the highest-weight states by simple group theoretical manipulations, knowledge of the three-point functions for the highest 40 As we have shown in section 4, such conservation laws can be derived also on the string theory side.
weight states is sufficient. In addition, replacing the highest-weight operator with its descendant only modifies the kinematical factor, V kin , of our result and the dynamical parts of three-point functions, which are main subjects of study in this section, will not be affected. After

Case of three BPS operators
Let us first study the correlation functions of three BPS operators. In order to apply the general formula for the three-point functions of one-cut solutions obtained in the previous section, we need the explicit forms of p(x) and q(x) for the BPS operators, which in particular determine the integration contours. Within the bosonic sector, the characteristic feature of a BPS state is that, as it should correspond to a supergravity mode, it is "point-like", meaning that its two-point function is σ-independent. In the language of the spectral curve, it means the absence of a branch cut, since a branch cut corresponds to a non-trivial string mode with σ-dependence.
Now in fixing the forms of p(x) and q(x), there is a subtle problem with the configuration without a branch cut. In the case of one-cut solutions corresponding to non-BPS operators, the constant parts of p(x) and q(x) are fixed in such a way that they vanish at the branch points. Obviously, for configurations without a branch cut, this prescription cannot be applied. One natural remedy would be to start with a non-BPS solution, apply the usual method above to fix the constants and then shrink the cut to obtain a BPS solution. This idea, however, still does not cure the problem since the resultant p(x) and q(x) depend on the points on the spectral curve at which we shrink the branch cut. The existence of such an ambiguity possibly implies that the semi-classical three-point functions are affected by the presence of infinitesimal branch cuts. Although such an assertion sounds counter-intuitive, it is not totally inconceivable since similar effects were already observed in the study of "heavy-heavy-light" three-point functions 41 in [30].
Below we shall fix the ambiguity by employing a prescription which is quite natural from the viewpoint of the correspondence with the spin chain on the gauge theory side. 41 In [30], such effects were called back reactions.
The prescription is to shrink the branch cuts either at These expressions vanish at x = 0.
As discussed in detail in section 5, the contours for the convolution integrals consist of two types of curves. The first type are those defined by Re q i (x) = 0, across which the relative magnitude of i + and i − changes. They determine the integration contours Γ u i − defined in section 5.4 and are depicted in figure 7.1. Note that in the present case, the contours Γ u 1 − and Γ u 2 − coincide since q 1 (x) = q 2 (x). The second type are the curves defined by N i = N j +N k , across which the connectivity of the exact WKB curves changes. Now for a BPS operator, N i = |Re p i (x)| is given by a common function |Re ((x + 1) −1 + (x − 1) −1 )| times the factor −2πκ i , as shown above. Since κ i 's satisfy the triangular inequalities, this means that N i = N j + N k cannot be satisfied. Hence the second type of curves are absent and the integration contours are determined solely by the first type of curves.
With this knowledge, we can now apply the general rules given at the end of section In this way, we find the contours M uuu ±±± to be given by Let us next consider the effects of the contact terms. As argued in section 6, such contribution must be taken into account when x = 0 (x = ∞) is on the left (right) hand side of the integration contours. The effect is most conveniently done by adding a small circle around x = 0 (x = ∞) to the contour for each integration in (6.8). However, in the case of BPS operators, the integration contours terminate right at x = 0 or x = ∞.
Therefore we need to first regularize them by putting a small branch cut slightly away from x = 0 or x = ∞ and then take the limit where the branch cut shrinks to x = 0 or x = ∞. An example of such a procedure is depicted in figure 7.2. Since the sine-functions in the convolution integrals (6.8) turn out to vanish only on the real axis in the case of BPS operators, we can further deform the contours into those on the unit circle. As a result, we find that the S 3 -part of the three-point function is given by U z (dp 1 + dp 2 + dp 3 ) 2πi ln sin p 1 + p 2 + p 3 2 + U z (dp 1 + dp 2 − dp 3 ) 2πi ln sin p 1 + p 2 − p 3 2 + U z (dp 1 − dp 2 + dp 3 ) 2πi ln sin p 1 − p 2 + p 3 2 + U z (−dp 1 + dp 2 + dp 3 ) 2πi ln sin where U denotes the contour which goes around the unit circle clockwise.
(a) Putting a small branch cut away from x = 0.
(b) Shrinking the cut and deforming the contour.
Then, performing a similar analysis as in the case of S 3 -part, we find that the result is again given by the integrals along the unit circle. As the quasi-momenta p i (x) for the S 3 -part and the onesp i (x) for the EAdS 3 -part coincide in the case of BPS operators, we see from the general formula (6.46) that the contributions form these two parts cancel each other completely. Therefore, the three-point function for three BPS operators is given purely by the kinematical factors as

Limit producing two-point function
Having seen that the BPS three-point functions are correctly reproduced from our general formula, let us next discuss the limit where the three-point functions are expected to reduce to two-point functions. As an example, we take two of the operators O 1 and To understand what happens in such a limit, let us draw the two types of curves, namely Re q i = 0 and N i = N j + N k . The first type of curves are depicted in the first and the second figures of figure 7.3. As for the second type, the only curve we need to consider is the curve given by N 3 = N 1 + N 2 . This is because the inequalities N 1 + N 3 ≥ N 2 and N 2 + N 3 ≥ N 1 are always satisfied since N 1 = N 2 in the present case. When the operator O 3 is sufficiently small, the curve defined by N 3 = N 1 + N 2 almost vanishes and we can practically ignore the effects of such a curve. Thus the integration contours are given purely by Re q 1 = Re q 2 = 0. Applying the rules given in the previous section and taking into account the contact terms, we find that the convolution integrals for the S 3 -part are 43 Although the case considered here appears similar to the one studied in the gauge theory [30] with O 3 taken to be small but nonvanishing, there is a difference: In [30], O 1 and O 2 must have slightly different quasi-momenta in the presence of O 3 , due to the conservation law for the magnons. In the present case, however, as we performed the global transformation, no conservation law is imposed and we can take O 1 and O 2 to have identical quasi-momenta.
given by Γ u 1 − +C∞ z (dp 1 + dp 2 + dp 3 ) 2πi ln sin p 1 + p 2 + p 3 2 + Γ u 1 − +C∞ z (dp 1 + dp 2 − dp 3 ) 2πi ln sin z (dp 1 − dp 2 + dp 3 ) 2πi ln sin z (−dp 1 + dp 2 + dp 3 ) 2πi ln sin where C ∞ is the contour encircling x = ∞ counterclockwise and C 0 is the contour encircling x = 0 clockwise. Setting p 1 = p 2 and p 3 = 0 in this formula, we see that in this limit all the terms in (7.7) completely cancel out with each other. Similar cancellation occurs also for the EAdS 3 -part. Therefore the structure constant C 123 of the three-point function in this limit becomes unity and the result correctly reproduces the correctly normalized two-point function given by Here, ∆, R and L are, respectively, the conformal dimension, the (absolute values of the) right and the left global charges, which are common to O 1 and O 2 .

Case of one non-BPS and two BPS operators
Having checked that our formula correctly reproduces the known results in simple limits, let us now study more nontrivial examples. In this subsection, we take up the three-point functions of one non-BPS and two BPS operators, which were studied on the gauge-theory side in [4]. As in [4], we take O 2 to be non-BPS and O 1 and O 3 to be BPS. In this case, the typical forms of the curves corresponding to Re q i = 0 and N i = N j + N k , are given in figure 7.4.
To perform a more detailed analysis, we need to specify the properties of the operators more explicitly, since the precise form of the integration contours depend on such details.
As we wish to analyze the so-called Frolov-Tseytlin limit and make a comparison with the results in the gauge theory in the next subsection, we will take as a representative example the following set of operators carrying large conformal dimensions:    Now, in distinction to the case of three BPS operators, we must also take into account the possible change of the analyticity of the Wronskians as we cross the lines defined by N i = N j + N k . Thus, we must analyze relevant curves drawn in figure 7.6 (a), where the one in green corresponds to N 2 = N 1 + N 3 and the one in blue represents N 1 = N 2 + N 3 .
Across these lines the configuration changes from symmetric to asymmetric. Accordingly, the rule to find the non-vanishing set of Wronskians changes from Rule 2 to Rule 3. Let us focus on the green curve, which is re-drawn in figure 7.7, with additional information.
It turns out that the configuration is symmetric inside the green circles and asymmetric outside, indicated by the letters S and A respectively. Now in the region outside of the arc of the large green circle bordered by the lines representing Γ u 1− , shown in figure 7.7 by the red straight lines, 1 − , 2 + , 3 − are the small solutions, as indicated in the figure. As this is the asymmetric region we apply the Rules 1 and 3 and conclude that the Wronskians among the states {1 + , 2 + , 3 + } are non-vanishing. As we cross the arc into the shaded region inside of the green circle where the configuration is symmetric, still 1 − , 2 + , 3 − are the small solutions but now we must apply the Rules 1 and 2. Then we learn that the We can determine the positions of the singularities numerically and find that most of the singularities lie on the real axis. Avoiding them, we can deform each contour into a sum of the contour along the unit circle and the one which is far from the unit circle. The  results of this deformation are summarized as where, as before, U denotes the unit circle and the primed contours are as depicted in  Let us make a remark on the separation of the integration contours into the unit circle and the large contours. It is intriguingly reminiscent of the expressions for the one-loop correction to the spectrum of a classical string [31]. In that context, the integration along the unit circle is interpreted as giving the dressing phase and the finite size corrections.
Since our results do not include one-loop corrections, it is not at all clear whether our results can be interpreted in a similar way. However, the apparent structural similarity calls for further study.
7.5 Frolov-Tseytlin limit and comparison with the weak coupling result 7.5.1 Frolov-Tseytlin limit of the three-point function We are now ready to discuss the Frolov-Tseytlin limit of the three-point function and compare it with the weak coupling result. Let us briefly recall how such a limit arises.
As shown in [32], the dynamics of the fluctuations around a fast-rotating string on S 3 can be mapped to the dynamics of the Landau-Lifshitz model, which arises as a coherent state description of the XXX spin chain. In such a situation, the angular momentum J of the S 3 rotation can be taken to be so large that the ratio √ λ/J becomes vanishingly small, even when λ is large. For the spectral problem, it has been demonstrated that such a limit is quite useful in comparing the strong coupling result with the weak coupling counterpart. We would like to see if it applies also to the three point functions. For this purpose, we need to know how such a limit is taken at the level of the quasi-momenta.
Since the SO(4) charges of the external states are proportional to κ i , the appropriate limit is to scale all the κ i to infinity while keeping the mode numbers b i dp finite. As already indicated, we have chosen the example in the previous subsection to be such that we can readily take such a limit.
Upon taking the Frolov-Tseytlin limit, two simplifications occur in our formula. First, since the branch points are far away from the unit circle, we can approximate p 2 (x) on the unit circle by a quasi-momentum for a BPS operator, namely Now recall that the contribution from the EAdS 3 part is such that it precisely canceled the S 3 part in the case of the three BPS operators. Since the EAdS 3 part is unchanged for the present case, again the same exact cancellation takes place as far as the integrals over the unit circles are concerned. Therefore we can drop such integrals and obtain our treatment, due to the inability to construct genuine three-point saddle solutions, we describe the effect of the three vertex operators separately except for imposing the global monodromy condition that reflects the essence of their interaction. However, as already emphasized in our previous work [11], if we wish to deal properly with the three-and higher-point functions using algebraic curve setup for a theory with infinite degrees of freedom, one should actually start from the infinite gap solutions and then consider the limits where the infinite number of cuts on the spectral curve degenerate to zero size.
This process is rather non-trivial and it should be possible to produce some extra singularities on the worldsheet. Although we cannot demonstrate this phenomenon explicitly for the three-point solution, we know that already at the level of two-point solution such a mechanism exists, as discussed in some detail in section 2.2.2. There we saw explicitly that a "one-cut" solution obtained from a multi-cut solution in a certain degeneration limit can produce extra singularities without affecting the infinite number of conserved charges carried by the solution. It is certainly expected that such a mechanism would exist also in the case of higher-point solutions. An interesting question is which of the saddle points, those with extra singularities or those without, describe the correlator of the gauge-theory operators. In any case, further studies are definitely needed to clarify this issue.

Discussions
In this paper, we have succeeded in computing the three-point functions at strong coupling of certain non-BPS states with large charges corresponding to the composite operators in the SU(2) sector of the N = 4 super Yang-Mills theory. As we have already given a summary of the main result in section 1.1, we shall not repeat it here. Instead, below we would like to give some comments and indicate some important issues to be clarified in the future.
One conspicuous feature of our result is that even for rather general external states the integrands of the integrals expressing the structure constant exhibit structures quite similar to the corresponding result at weak coupling. This is quite non-trivial since the weak coupling result in the relevant semi-classical regime is obtained from the determinant formula for the inner product of the Bethe states, which is so different from the method employed for strong coupling. This suggests that we should seek better understanding by reformulating the weak coupling computation in a more "physical" way. As a step toward such a goal, an attempt was made in [36], where the inner product of the Bethe states is re-expressed in terms of an integral over the separated dynamical variables. As the notion of the wave function is clearly visible in this formulation, it may give a hint for the common feature of the strong and the weak coupling regimes, if an efficient method to identify the semi-classical saddle point can be developed.
In contrast to the similarity of the integrands, there is a rather clear difference in the contours of the integrals expressing the three-point coupling in the weak and the strong coupling computations. This is not just a quantitative difference but rather a qualitative one. Reflecting the fact that the determinant formula deals with the Bethe roots, the contour of integration in the weak coupling case is around a cut formed by the condensation of such Bethe roots. Information of such a cut is contained in the quasi-momentum p(x). On the other hand, the principal quantity which determines the integration contour is the real part of the quasi-energy q(x), which is conjugate to the worldsheet time τ . Apparently, this notion is not present in the weak coupling formulation.
Together with the possible extra singularities on the worldsheet discussed in section 7.5.2, the question of the contour requires better understanding.
There are a couple of further interesting questions that one should study concerning our result. One is about the limit of our formula where one of the operators is much smaller than the other two. Such three-point functions were first studied on the stringtheory side in [37][38][39] assuming that the light operator does not change the saddle-point configuration of the other two operators. However, a systematic study on the gaugetheory side [30] reveals that the light operator in some cases modifies the saddle-point substantially. By examining the limit of our formula, it would be possible to understand in detail when and how such a "back-reaction" occurs. Another important problem is to understand the physical meaning of the integration along the unit circle in our formula and clarify if it can be interpreted as the contribution from the dressing phase and the finite size correction as in the case of the one-loop spectrum of a classical string [31].
Finally, let us go back once again to the rather simple structure of the integrand we found, similar to the weak coupling result. The simplicity of such a result suggests that there should perhaps be a better more intrinsic formulation for computing the three-point functions. In the existing literature, including this work, the calculation of the three-point function in the strong coupling regime is divided into the computation of the contribution of the action part and that of the vertex operator (wave function) part. As we have seen in section 6, in the process of putting these separate contributions together there occurs a substantial simplification, besides the usual cancellation of divergences. This strongly indicates that such a separation is not essential and one should rather seek relations which reflect the structure of the entire three-point function based on some dynamical symmetry of the theory including the integrable structure. This is of utmost importance since the true understanding of the AdS/CFT duality lies not just in the comparison of the calculations of various physical quantities in the strong and the weak coupling regimes itself but rather in identifying the common principle behind such computations and agreements.
To make the above remark somewhat more concrete, let us recall that the most important ingredient in the computation performed in this paper at strong coupling is the global consistency relations for the monodromy of the solutions of the auxiliary linear problem around three vertex insertion points. Together with the analyticity property in the spectral parameter, the important quantity i ± , j ± , which relates the behavior of the solutions around different insertion points, is extracted and serves as the building block for the three-point coupling. On the other hand, in the weak coupling computations so far performed, the computation of the three-point coupling is reduced to those of the inner products of the Bethe states and their combinations. Although this is an efficient method, it is based basically on the picture of the two-point function and not on some principle which governs the entire three-point function. Therefore we believe that an extremely important problem is to find some functional equations (or differential equations) satisfied by the three-point function, from which one can determine the coupling constant more or less directly. We hope to discuss this type of formulation elsewhere.

Acknowledgment
We would like to thank Y. Jiang, I. Kostov, D. Serban and P. Vieira for discussions. A Details on the one-cut solutions In this appendix, we will provide some further details on the one-cut solutions.
A.1 Parameters of one-cut solutions in terms of the position of the cut In section 2.2 we have given generic expressions for the parameters which characterize the one-cut solutions in terms of the integrals involving p(x) and q(x). If we now use the explicit forms of p(x) and q(x) given in (2.58) and (2.59), one can evaluate the parameters ν i , m i and θ 0 in terms of the position of the cut specified by u. The results take several different forms depending on the region where the cut is located. It is convenient to express them in universal forms by introducing two additional sign factors η 1 and η 0,1 .
Together with the factor already introduced in (2.60), we give their definitions in the following table: Table 3. Sign factors to distinguish between the positions of the cut.
Re u < −1 −1 < Re u < 0 0 < Re u < 1 1 < Re u + − − + η 1 + + + − η 0,1 + + − + Then, ν 1 and ν 2 are obtained as Finally, let us discuss the signs and the relative magnitudes of the parameters and the charges. The signs and the relative magnitude of ν i depend on u. From the formulas for ν i we can check that |Re u| > 1 : ν 2 < ν 1 < 0 , (A.10) |Re u| < 1 : ν 1 < 0 < ν 2 , (|ν 1 | < ν 2 ) . (A.11) As for the angles, we always have The signs of R and L can be checked to be always positive. ( R for the case |Re u| > 1 and L for the case |Re u| < 1 are somewhat non-trivial to check.) The relative magnitude of R and L can be deduced easily from the difference As the sign of ν 2 has already been obtained in (A.10) and (A.11), we immediately get R < L for |Re u| > 1 , (A.14) R > L for |Re u| < 1 . where we used z = τ + iσ coordinate when we compute these quantities 45 .
Using the results in the previous subsection, we can re-express (A. 16 In the case of three-point functions, we can compute these quantities separately for each puncture as (A.23) They will be used in the computation of three-point functions.

A.3 Computation of various integrals
Using the above results, let us compute various integrals which appear in Local and Double in section 3. Around a puncture, one can approximate the behavior of the world-sheet by that of the two-point functions. Thus, when three string states are semi-classically described 1-cut solutions, we expect the following asymptotic behavior of the one-forms: where w i is the local coordinate w i ≡ τ (i) + iσ (i) around the puncture z i .
Using (A.24), one can evaluate various integrals. First, the contour integrals of λ and ω along C i 's are given by On the other hand, the double contour integral, which appears in Double can be computed as follows:

B Pohlmeyer reduction
In this appendix, we will give some details of the Pohlmeyer reduction for the string on Using the equation of motion, one can also show that γ, ρ andρ satisfy the generalized sin-Gordon equation, which is given in (2.44).
Let us next derive a flat connection associated with the system of equations (B.2)-(B.6). For this purpose, it is convenient to introduce the following orthonormal basis: T sin 2γ e iγ q 4 , (B.12) T sin 2γ e −iγ q 4 , (B.13) T sin 2γ q 3 , (B.14) sin 2γ e −iγ q 4 , (B.16) sin 2γ e iγ q 4 , (B.17) By expressing the basis in a matrix form, Owing to the classical integrability of the string sigma model, we can "deform" the above connection without spoiling the flatness by introducing a spectral parameter ζ = (1 − x)/(1 + x) as Then, similarly to the sigma model case, the embedding coordinates Y can be reconstructed by the formula where Ψ L,R are 2 × 2 matrices with a unit determinant, defined by

C.2 Relation between the connections and the eigenvectors
We now discuss the relation between the connections and the eigenvectors of the the Pohlmeyer reduction and those of the sigma model.
First consider the relation to the right connection of the sigma model. From the formulas for ∂Y and∂Y given in (C.6), we can form the right connection j as Then, comparing (C.7) with (B.26)-(B.28), we find that the following gauge transformation connects the flat connections of the two formulations: The eigevectors ψ ± of the sigma model formulation and those of the Pohlmeyer reduction, denoted byψ ± , are related as Note that the factor of i in (C.10) is needed to reproduce the correct normalization condition ψ + , ψ − = 1. Under the global SU(2) R transformation U R , ψ ± transform as ψ ± → U −1 R ψ ± . From the above formulas (C.10) and (C.11) we see that this corresponds to the transformation Ψ R → Ψ R U R , as remarked previously.
In an exactly similar manner, we can construct the left current l's by where the choice of the sign should be the same as in (D. 16). Continuing in this fashion using (D.5) and (D.6), we can determine∂W −1 ,∂W 0 and∂W 1 to bē where W odd (resp. W even ) denotes terms which (do not) change sign under the sign-flip of ∂ w W −1 . Then, by substituting (D. 19) into (D.11) and extracting the terms odd under the above flip of sign, we can obtain the following simple equation expressing W even in terms of W odd : 20) to the direct expansion described above. In particular, with this method it is much easier to take into account the normalization conditions of the eigenvectors i ± given in (2.96).
Although the method has been described in Appendix B of [22], we will spell out the details of the derivation since several additional considerations are necessary in our case.
To illustrate the basic idea, let us take the Wronskian 2 + , 1 + as an example and discuss its expansion. To compute 2 + , 1 + , we need to parallel-transport the eigenvector 1 + , which is defined originally in the neighborhood of z 1 , to the neighborhood of z 2 using the flat connection and compute the Wronskian with 2 + . In the diagonal gauge, this procedure can be implemented in the following way: In this expression t parametrizes the curve joining z * 1 (at t = 0) and z * 2 (at t = 1) and H 0 and V are defined in terms of the connection in the diagonal gauge, given in (3.35), as As the first step toward this goal, let us determine the expansion of the "initial states", 1 d + (z * 1 ) and2 d + (z * 2 ). As explained in section 2.3, the eigenvectors can be well-approximated near the puncture by those of the corresponding two-point functions. Thus, the expansion of the initial states can be obtained from the explicit form ofî 2pt ± given in (2.98) and (2.99) as1 where |e 1 and |e 2 stand for the unit vectors (D. 33) In the limit ζ → 0, the integral over t 1 in the second term will be exponentially suppressed by the factor exp −2 t 1 0 /ζ , except when the interval is short, i.e. 0 < t 1 < O(ζ 1 ).
Thus, to O(ζ 1 ), one can take in