Classical integrability for three-point functions: cognate structure at weak and strong couplings

In this paper, we develop a new method of computing three-point functions in the SU(2) sector of the N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathcal{N}=4 $$ \end{document} super Yang-Mills theory in the semi-classical regime at weak coupling, which closely parallels the strong coupling analysis. The structure threading two disparate regimes is the so-called monodromy relation, an identity connecting the three-point functions with and without the insertion of the monodromy matrix. We shall show that this relation can be put to use directly for the semi-classical regime, where the dynamics is governed by the classical Landau-Lifshitz sigma model. Specifically, it reduces the problem to a set of functional equations, which can be solved once the analyticity in the spectral parameter space is specified. To determine the analyticity, we develop a new universal logic applicable at both weak and strong couplings. As a result, compact semi-classical formulas are obtained for a general class of three-point functions at weak coupling including the ones whose semi-classical behaviors were not known before. In addition, the new analyticity argument applied to the strong coupling analysis leads to a modification of the integration contour, producing the results consistent with the recent hexagon bootstrap approach. This modification also makes the Frolov-Tseytlin limit perfectly agree with the weak coupling form.


Introduction
In the preceding few years, a number of substantial advancement have been made concerning the evaluation of the two-and three-point correlation functions in the N = 4 super Yang-Mills theory in the large N limit, not only in the weak and the strong coupling regimes but also at finite couplings, with clever ideas and some assumptions.
As for the two-point functions, one can now compute them quite accurately at an arbitrary coupling using the so-called quantum spectral curve [1], which is a vastly evolved form of its precursor, the thermodynamic Bethe ansatz formalism [2][3][4].
Equally important have been the developments on the computation of the three-point functions involving non-BPS operators, which are imperative in understanding the dynamical aspects of the AdS/CFT correspondence [5][6][7].
At weak coupling, one can systematically study these objects by mapping them to scalar products of the integrable spin chain as demonstrated in the pioneering works [8][9][10][11]. As for the strong coupling, due to the lack of the method of quantization for a string in a relevant curved spacetime, only the semiclassical saddle-point computation appeared to be feasible. However, the initial attempts for some fully non-BPS three-point functions revealed that such a method is already rather challenging and only some partial results were obtained [12,13]. It was only after some non-trivial efforts that these difficulties were overcome and finally rather general class of three-point functions in the SU(2) sector were evaluated [14,15].
Very recently, in a different vein, a nonperturbative framework capable of studying these objects at finite coupling was put forward in [16]. The basic idea of this approach is to decompose the three-point functions into more fundamental building blocks called the hexagon form factors and determine them using assumed all-loop integrability. 1 Although quite powerful, as this method refers only to the magnon and its mirror excitations without referring to their specific origins, it is difficult to see how the gauge theory and the string theory are related. In this sense, our present work connecting the weak and the strong coupling representations based on the known integrability properties should be considered as complementary to such a universal approach. Now concerning both the weak-coupling and the hexagon form factor methods, the three-point functions are expressed in terms of the sums over partitions of rapidities, 2 which become increasingly more complicated as the number of magnons increases. However, it turned out that, in the semiclassical limit, where both the number of magnons and the length of the spin chain become very large, the result at weak coupling can be written in a surprisingly concise form, namely a simple integral on the spectral curve, whose integrand is expressed solely in terms of the so-called pseudo-momenta [19][20][21]. Now it is important to recall at this point that also at strong coupling in the semiclassical approximation the form of the three-point functions exhibits the same simple structure. A natural question then is whether there is an underlying physical mechanism by which one can produce such a simple expression more directly.

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In the case of the two-point functions, a similar question was addressed in [22,23]. In the semiclassical limit, the collective dynamics of magnons is described by the so-called Landau-Lifshitz model, a classically integrable nonrelativistic sigma model which can be obtained as a continuum limit of the Heisenberg spin chain. This formulation allows one to compute the semi-classical two-point function directly using the classical integrability and moreover makes it possible to describe the weak and strong coupling computations in a similar manner.
So the main purpose of the present work is to develop a formulation for the computation of the three-point functions at weak coupling, which in the semiclassical limit produces in a direct way the compact integral expressions similar to those in the strong coupling and to understand its basic mechanism. This will not only be quite useful from the point of view of the computation of the semiclassical limit, which is often physically most interesting, but the understanding of its mechanism would also reveal an aspect of integrability common to apparently disparate regimes. We will indeed see that a formulation extremely similar to that of the strong coupling analysis performed in [14] is possible and it will not only reproduce existing results in the literature but also make predictions for a class of threepoint functions whose semi-classical limit have not yet been computed.
Let us now describe the idea and the structure of our formulation more explicitly. The basic starting point is the result of our previous paper [24] where the tree-level threepoint function in the SU(2) sector can be expressed as the overlap between the singlet state and the three spin-chain states. By preparing a coherent state basis, we can then express such an overlap as a product of integrals over the coherent state variables. Now for the semiclassical situation of our interest, each spin chain reduces to a Landau-Lifshitz string and, more importantly, the overlap can be evaluated by the saddle point method. The situation is quite similar to the one at strong coupling, and just as in that case the determination of the saddle point configuration is quite difficult. However, the similarity to the strong coupling case goes further in the semiclassical situation. We also have the monodromy relation identical in form, derived in [24,25] for the weak coupling, which was one of the crucial ingredients in the strong coupling case in determining the three-point function without the knowledge of the saddle point configuration. This relation is natural and powerful as it is a direct consequence of the classical integrability of the string sigma model and encodes infinitely many conservation laws. Now, with such a monodromy relation at hand, most of the crucial ingredients for the strong coupling computation can be transplanted, with some modifications, to the present weak coupling case. More precisely, what this means is the following:

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It should be noted, however, that there is an important difference from the strong coupling case, concerning the determination of the analyticity property of the Wronskians. For the strong coupling case, the analyticity was determined by assuming the smoothness of the worldsheet for the saddle-point configuration connecting the three strings. In the present case, however, the three spin chains are glued together directly by the singlet projector, which is nothing but a convenient way of performing the Wick contractions dictated by the super Yang-Mills dynamics. There is no concept of worldsheet and hence the smoothness argument above does not apply. Therefore, in this paper we developed a new different argument, which is more powerful and universal. The basic idea is to study the response of certain fundamental quantities to an addition of a small number of Bethe roots. In the semiclassical context, such an addition corresponds to the continuous variation of the filling fraction of the Bethe roots and when applied to the (log of) the structure constant ln C 123 , it reveals that ln C 123 plays the role of the generating function of the angle variables and provides the key equation for obtaining C 123 . On the other hand, as it will be elaborated fully in section 4, we can also apply such a variation to the norm of the exact spin-chain eigenstate. When the original and the deformed states are both on-shell Bethe eigenstates, they must be orthogonal and we demand that this exact quantum property must be smoothly connected to the semiclassical structure for consistency. This requirement will turn out to be powerful enough to determine the configuration of the zeros and the poles on the spectral curve. The Wronskians determined through this logic not only leads to the known semiclassical results for the three-point functions in the literature but also allow us to compute more general SU(2) correlators, which have not been computed before.
It is then extremely interesting to apply this new orthogonality argument to the strong coupling case and see what happens. It turned out that this more universal argument lead to the modification of the integration contours obtained in the previous investigation, and the results with the modified contours are consistent with the hexagon form factor approach of [16] and exactly match the Frolov-Tseytlin limit [26] in the weak coupling regime. This indicates that, as already suspected and discussed in [14], the apparently natural requirement of smoothness of the saddle-point worldsheet configuration in the strong coupling case is not quite correct and our new logic for determining the analyticity in the semiclassical spectral curve is more reliable.
The rest of the paper is structured as follows. In section 2, after reviewing the formulation of the tree-level structure constant in terms of the overlap with the singlet projector, we derive a path-integral representation for such an overlap using the coherent state basis, which is subsequently evaluated by its saddle point in the semi-classical limit. We then show that the variation of the semi-classical structure constant with respect to a conserved charge of the spin chain states produces the angle variable which is canonically conjugate to that charge. In section 3, we construct the angle variables for the Landau-Lifshitz model using its classical integrability. Based on the results in the previous sections, we express, in section 4, the semi-classical structure constant in terms of the Wronskians of the eigenvectors of the monodromy matrices. In section 5, we evaluate such Wronskians, making use of the monodromy relation and the orthogonality of two on-shell states. Putting together JHEP10(2016)042 all the results in the preceding sections, we finally derive the explicit expression for semiclassical structure constants at weak coupling in section 6. In section 7, we describe how the argument developed in the present paper applied to the strong coupling computation modifies the results obtained previously. We conclude in section 8 and indicate several future directions. A few appendices are provided for technical details.
2 Semi-classical structure constant and the monodromy relation

Wick contraction represented as the singlet projection
We begin with a brief review of the two devices introduced in our previous work [28], namely the double spin-chain representation for the SU(2) sector and the interpretation of the Wick contraction as the group singlet projection, which greatly facilitate the construction of the correlation functions.
The four scalar fields φ i (i = 1, 2, 3, 4) forming the so-called the SU(2) sector of the super Yang-Mills theory can be assembled into a 2 × 2 matrix Φã a given by where Z ≡ φ 1 + iφ 2 , Y ≡ φ 3 + iφ 4 andZ andȲ are their hermitian conjugates respectively. Evidently, the symmetry of the SU(2) sector is actually SO(4)=SU(2) L × SU(2) R and the matrix Φ transforms under these two SU(2) factors as Φ → U L ΦU R , where U L (U R ) belongs to SU(2) L (SU(2) R ). This suggests that it is natural to consider the spin-chain consisting of these basic fields as forming a tensor product of two spin-chains, which we called the double spin-chain. Consider first the individual spin states | ↑ and | ↓ and denote them by | ↑ = |1 and | ↓ = |2 for convenience. Then, from the transformation property above, the basic fields correspond to the tensor product states as Z → | ↑ L ⊗ | ↑ R = |1 ⊗ |1 , etc. It is easy to see that this mapping is succinctly summarized as Φã a → |ã ⊗ |a ,ã, a = 1, 2 . (2.2) To construct the correlation functions at the tree level, we need to Wick contract these fields. For the Wick contraction of Φã a Φb b , the only non-vanishing ones are (suppressing the coordinate dependence) ZZ = 1 and XX = 1. This gives the simple formula Φã a Φb b = ãb ab .
(2. 3) In terms of the corresponding spin states, this rule is equivalent to |ã |b = ãb , |a |b = ab .  From the above rules, the Wick contraction between F 1 and F 2 , where F i = |f i ⊗ |f i , can be easily computed as . This form shows that one can perform the Wick contraction by taking the inner product with the singlet projection operator 1| = ab a| ⊗ b|, for both SU(2) L and SU(2) R , (2.7) namely This representation allows us to perform the Wick contractions for any complicated operators easily and systematically. 3 Now let us apply this scheme to the single-trace operators. The contractions which survive in the large N limit are the ones which connect the (L + 1 − i)-th field in the operator O 1 with the i-th field in the operator O 2 , where L is the length common to both operators. Explicitly, an example of this structure looks like This structure motivates us to consider the following tensor product of singlet states, denotes the single-spin state living on the i-the site of the spin chain corresponding to the operator O k . Then, the contractions between the operators can be reproduced by taking the inner product (2.11) Here |Õ k ⊗ |O k and |Õ k ⊗ |O k are the spin-chain states corresponding to the operators O k . For a pictorial explanation, see figure 1-(b).

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Since the tree-level three-point function is essentially given by a product of Wick contractions, one can also map the computation of the three point function to that in the spin-chain Hilbert space: (2.12) As in the previous case, the structure of the singlet state 1 123 | is determined by the structure of the Wick contraction, which is depicted in figure 1-(c). Explicitly, it is given by (2.13) Here L k is the length of the operator O k and L ij = (L i + L j − L k )/2 is the number of Wick contractions connecting O i and O j . Now taking into account the normalization factors correctly, we arrive at the following basic formula for the structure constant: 4 (2.14) In the above, N c denotes the rank of the gauge group. An important consequence of this formalism is the so-called monodromy relation, which is an identity connecting the structure constant with and without the insertion of the monodromy matrix. It was derived in [24,25] and, for the SU(2) R part, it reads 5 (2.15) Here Ω(u) is the monodromy matrix constructed from the Lax operator 16) and the superscripts Ω ±,+|− (u) indicates the shift of the argument by ±i/2 (for a precise definition, see figure 2). The constant factor f 123 (u) is given by The identity (2.15) encodes infinitely many conservation laws for the structure constant. As we will see in section 2.3, the semi-classical limit of (2.15) takes a form identical to the one at strong coupling and will play a key role in the subsequent analysis.
4 See [11] for the origin of the prefactor in (2.14). 5 Here we adopt the normalization of L k (u) to be such that L k (∞) = 1, which is slightly different from the one used in [24,25]. The monodromy matrix in the present normalization can be naturally identified with the monodromy matrix in the Landau-Lifshiz sigma model in the semi-classical limit.

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The monodromy relation given in (2.15). The three-point function with and without the monodromy matrix are equal up to the prefactor, which we omit writing here. The red lines denote parts of the monodromy matrix with a −i/2 shift of the spectral parameter whereas the blue lines denote parts of the monodromy matrix with a +i/2 shift of the spectral parameter.

On-shell Bethe states and polarization vectors
Before discussing the semi-classical limit, let us briefly explain how to characterize the general SU(2) state in the double spin-chain representation.
In the Bethe-ansatz approach, we construct the general eigenstates of the Hamiltonian by first considering the vacuum, which is typically taken to be tr Z L , and then introducing the magnons (X orX) with a set of rapidities satisfying the Bethe ansatz equation. An important property of such states, to be called the on-shell Bethe states, is that they are the highest weight state [27] if the rapidities are all finite. In the double spin-chain representation, this translates toS whereS + (S + ) is the raising operator for the total spin in SU(2) L (SU(2) R ).
In the case of three-point functions, we cannot take all the states to be the ones constructed upon tr Z L since such three-point functions vanish owing to the charge conservation. To study nonvanishing three-point functions, we have to consider the states constructed upon more general vacua, which can be obtained from tr Z L by the SU(2) L × SU(2) R rotations. As shown in [24], such vacua can be characterized in terms of the polarization vectors 6 n andñ, in the following way: 7 tr (ñãn a Φã a ) L (ã, a = 1, 2) . (2.20) The highest weight condition satisifed by the on-shell Bethe states constructed upon this rotated vacuum readsS

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where S + andS + are rotated generators given by The highest weight condition (2.21) will play an important role when deriving the expression for the semi-classical structure constant in section 4.3.

Coherent-state representation and the semi-classical limit of C 123
We will now study the semi-classical limit of the expression (2.14) for C 123 . Unlike the previous methods [11,18,20], where one first evaluates this quantity exactly and then take the semi-classical limit, we shall take the semi-classical limit at the outset by deriving a path integral representation of the structure constant and applying the saddle-point method. This scheme will be seen to be valuable as a novel computational method universally applicable for a large class of SU(2) three-point functions, including the cases which previously could not be treated easily. Actually the more important aspect of this method is that it reveals a cognate structure between the weak coupling computation under consideration and the strong coupling counterpart performed in [14], as we shall see. The semi-classical limit of our interest is a sort of the continuum limit of the Heisenberg spin chain. More precisely, it is the following scaling limit, L → ∞ , M → ∞ , Lp, L/M : fixed . (2.24) Here L, M and p are, respectively, the length of the spin chain, the number of magnons and the momentum of each magnon. As such it is efficiently described by some continuous field along the chain, which should provide a representation of SU (2). The so-called coherent state representation is ideal for such a purpose. It is a representation realized on the coset space SU(2)/U(1), which is isomorphic to a unit sphere S 2 . As briefly reviewed in appendix A, a coherent state representation for a single spin 1/2 state can be taken to be where n 0 = (0, 0, 1) is a unit vector in the z direction and n = (sin θ cos φ, sin θ sin φ, cos θ) is a unit vector in a general direction. To express C 123 in this basis, we just need to insert the completeness relation

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to each inner product in (2.14). As a result, we obtain the following path-integral expression: (2.27) Here ñ and n denote a chain of coherent states m, l] is the overlap between the singlet and the coherent states while the wave functionsΨ and Ψ are defined bỹ Now in the semi-classical limit, this expression can be well-approximated by the saddlepoint of the integrand, which gives 31) where n * k ( ñ * k ) represents the saddle point of the D n * k (D ñ * k ) integral. Evidently, the result (2.31) factorizes into the SU(2) L part and the SU(2) R part. In the discussions in the following sections, we mainly focus on the SU(2) R part since the computation in the SU(2) L part is similar.
Let us now study the semi-classical limit of the monodromy relation. Since the monodromy matrix is an O(1) quantity, the insertion of the monodromy matrix does not affect the saddle point. Thus, in the semi-classical limit, we can replace the monodromy matrix, which is originally the quantum operator acting on the spin chain, with the classical value evaluated on the saddle point given in (2.31). Furthermore, since we scale the spectral parameter as u ∼ L in the semi-classical limit, the shifts of the arguments in Ω ± etc. become negligible and the factor f 123 can be approximated by unity. Therefore we arrive at the relation (2.32) Importantly, (2.32) has exactly the same form as the monodromy relation in the string sigma model. This allows us to transplant most of the crucial ingredients for the strong coupling computation as we shall see in the next section.

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2.4 ln C 123 as the "generating function" of the angle variable Once we choose the operators O i of definite conformal dimensions for which to compute the three-point function, each part of the expression in (2.31) can be explicitly computed in principle with the judicious use of the integrability. This is indeed the approach taken in the previous study at strong coupling [14]. However, in this brute-force method, we shall encoutner extremely complicated intermediate expressions, most of which should cancel in the final result. Therefore, below we shall devise an entirely different method, which at the same time reveals the important meaning of ln C 123 as a whole. This approach also enables us to study the semi-classical states with arbitrary number of cuts in the spectral curve, unlike the method in [14], which was restricted to the so-called one-cut solutions. The basic idea is to see how ln C 123 changes as we introduce a small number of additional Bethe roots. In the semi-classical context, this means to study the variation of the structure constant with respect to the variation of the filling fraction 8 S (m) i given by where the subscript i labels the filling fraction for the different cut belonging to the same operator, while the superscript (m) labels the three different operators. By "integrating" this quantity, one can determine the ratio between the structure constant involving non-BPS operators and the one for three BPS operators, for which all the filling fractions vanish. Specifically, the change of the filling fraction produces the following two effects: (i) a slight change of the saddle point configuration n * and (ii) the direct small change of the wave functions Ψ[S i , n * ] due to δS i . Actually, the contribution from (i) takes the form, and hence it vanishes owing to the saddle-point equation δ C 123 | saddle /δ n * m = 0. Now from the general theory, the wave function in the semi-classical limit is given by the following WKB form where in the present case Q k 's correspond to the coherent-state variables, n, and P k 's to their canonical conjugates. The right hand side of (2.35) can be regarded as the generating function of the canonical transformation. Therefore, by differentiating with respect to the filling fraction, which is known to be the conserved action variables, we obtain i . Putting altogether, we find that ln C 123 plays the role of the "generating function" giving the angle variable under the variation of the filling fraction and we get the simple formula (2.37) Concerning this formula, two comments are in order. First, as we have already indicated by the use of quotation marks, the quantity ln C 123 is not the generating function of the action variables in the usual sense. The precise meaning is that at the saddle point it behavs as if it were a generating function of the value of the angle variable under the variation of S (m) i . The second comment concerns the normalization of the structure constant C 123 or rather the normalization of the operators making the three-point function. As it will be discussed in the next section, in the general integral expressing the angle variable φ i in (3.21), we will not specify the initial point of integration. Therefore the expression (2.37) is actually ambiguous as it stands. To fix this ambiguity, we require that the operators we use produce the normalized two-point functions correctly. This can be achieved in practice by replacing the right hand side of (2.37) by the difference between the angle variable for the three-point function and the one for the two-point function in the following way: Unlike (2.37), the expression (2.38) is entirely unambiguous and we will adopt thhis form in the rest of this article.

Classical integrability of the Landau-Lifshitz model
We shall now apply the general formalism developed in the previous section more explicitly to the semi-classical limit of the Heisenberg spin chain. It is well-known that such a limit gives rise to so-called the Landau-Lifshitz model, a classically integrable field theory in 1 + 1 dimensions.

Landau-Lifshitz model, its Lax pair and the monodromy matrix
Let us briefly summarize the basic properties of the Landau-Lifshitz model and its integrable nature. In the semi-classical limit, the coherent state variable n(m, τ ), where m is an integer specifying the position along the spin chain, becomes a continuous field n(σ, τ ). It is convenient to take the range of σ to be 0 ≤ σ ≤ L, where L is the length of the spin chain. The action is given by 9

1)
9 A review of the derivation is provided in appendix A.

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where g = √ λ/4π is the 't Hooft coupling constant. The first term in (3.1) is the Wess-Zumino term and the s-dependence of n is defined such that n(s = 1) = (0, 0, 1) and n(s = 0) = n. The equation of motion obtained by varying the above action reads One of the important features of this model is its classically integrability, whose clearest manifestation is the existence of the following Lax pair structure where u is the spectral parameter. From the above Lax pair, one can construct the monodromy matrix in the usual way: 10 As in the case of the integrable string sigma model, one defines the quasi-momentum p(x) as the logarithm of the eigenvalue of the monodromy matrix: (3.5) The asymptotic properties of the quasi-momentum at u = 0 and u = ∞ can be easily obtained from the above definitions and contain useful information: its residue at u = 0 is related to the length of the spin chain [23] as while the leading behavior at infinity provides the information of the number M of magnon excitations of the system: The spectral curve is defined from the monodromy matrix as Owing to the singular behavior of the quasi-momentum (3.6), the spectral curve contains an infinite number of points satisfying e 2ip(u * ) = 1. Such points are called the singular points and can be regarded as the infinitesimal branch cut [29,30] (see also figure 3). Figure 3. The structure of the spectral curve at weak coupling. In general, it has several branch cuts and infinitely many singular points, denoted by black dots, which accumulate to u = 0. The singular points can be regarded as degenerate branch points. Now, it is well-known that the information contained in the non-linear Lax equation can be recovered by the simultaneous solution of the auxiliary linear problem given by

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In particular, for the i-th spin chain, the solutions which are at the same time the eigenfunctions of the monodromy matrix Ω i with eigenvalues e ±ip i (u) will be denoted by i ± , i.e.
and they will be of great importance in what follows.

Action-angle variables
As was already indicated in section 2.4, the concept of action-angle variables plays an essential role in the computation of the structure constant. Therefore in this subsection, with the use of the method of Sklyanin [31] we shall construct the action-angle variables for the Landau-Lifshitz model. First, we must compute the Poisson bracket between the elements of the monodromy matrix, which is characterized by the classical r-matrix in the form (3.11) In the case of the Landau-Lifshitz sigma model, since the quantum R-matrix R(x) for the XXX spin chain is well-known, a quick way 11 to obtain the classical r-matrix is to take the classical limit of R(x). Explicitly, we obtain

12)
11 For the first-principle derivation of the classical r-matrix, see appendix B.

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where I is the identity operator and P is the permutation operator. Using the form (3.12) in (3.11), one can obtain the explicit form of the Poisson bracket between the individual components of the monodromy matrix, which are denoted as usual in the form The resulting Poisson brackets obtained in this way are displayed in appendix C. We now describe the Sklyanin's approach [31] for the construction of the action-angle variables. Consider the eigenfunctions ψ ± (u) of the monodromy matrix Ω(u), with eigenvalues e ±ip(u) , i.e. defined by Ω(u)ψ ± (u; τ ) = e ±ip(u) ψ ± (u; τ ) . (3.14) Such eigenstates are called Baker-Akhiezer vectors. Now an important information is encoded in the normalized Baker-Akhiezer vector h(u; τ ) defined to be proportional to ψ + (u; τ ) and satisfying the normalization condition Here n = (n 1 , n 2 ) t is a constant vector with unit norm. In the original formalism by Sklyanin, n can be arbitrary as long as it is independent of the spectral parameter. However, in the present context, we must choose it to be equal to the polarization vector diuscussed in section 2.2 in order to guarantee the highest weight property of the semi-classical wave function (see appendix C for a detailed explanation). For general solutions, there are infinitely many poles in h(u, τ ), the position of which are denoted by γ i , i = 1, 2, . . .. Sklyanin observed that to each such pole γ i , which becomes a dynamical variable through the relations (3.14) and (3.15), a canonical pair of variables are associated. Relegating the details of the derivation to appendix D, the result is the following set of commutation relations (3. 16) where p(γ i ) is the quasi-momentum p(u) with the substitution u = γ i . This shows that (γ i , −ip(γ i ))'s are canonical pairs of variables.
Once the canonical pairs are obtained, one can easily construct the action variables, which should be identified with the conserved filling fractions S i , as Here C i denotes the i-th branch cut. Now to construct the angle variables φ i conjugate to S i , we need to to find the generating function of the type F (γ i , S i ), which effects the canonical transformation from (γ i , −ip(γ i )) to the action-angle variables. Such a function is defined by the properties

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In the present context, the first equation should be viewed as defining the function F , while the second equation should be regarded as the definition of φ i . Therefore, to determine F , we need to integrate the first equation with S i fixed. As the filling fractions are given by the integral of p(u) on the spectral curve, fixing all S i 's is equivalent to fixing the functional form of p(u). Therefore, F can be determined as Next we compute φ i = ∂F/∂S i . This requires changing S i with all the other filling fractions fixed. In the Heisenberg spin chain we started with, this corresponds to adding a small number of Bethe roots to the branch cut C i . As is clear from (3.5), this addition of magnon inevitably changes the asymptotic behavior of p(u) at u = ∞. Therefore, changing S i is precisely equivalent to adding to p(u)du a one-form whose period integral is non-vanishing only for the cycle around C i and the cycle at infinity. Such a one-form should be proportional to a holomorphic differential ω i satisfying the following properties: (3.20) Here ∞ + (∞ − ) denotes the infinity on the first (second) sheet. Using such ω i , the partial derivative ∂F/∂S i is expressed as 12

Angle variables and the Wronskians
In this section, we shall show that the angle variables constructed in the previous section can be expressed in terms of the skew-symmetric product, to be called the Wronskians, of the solutions of the auxiliary linear problem corresponding to the Lax pair and of the polarization vectors. In what follows the Wronskian of any two-component vectors χ a and φ a is defined as

Normalization of the solutions to the auxiliary linear problems
By using the Wronskian, we shall conveniently normalize the solutions k ± of the auxiliary linear problem for the k-th spin chain as In addition to this condition, it is consistent to require that the two solutions k ± are related across the cut by

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Then from (4.2) and (4.3), one can show that k ± develop the following singularity at the branch points of the spectral curve Let us briefly explain how this comes about. As the eigenvectors of the monodromy matrix are determined only up to an overall factor, we may first choose an eigenvector k 0 + which remains non-singular even at the position of the branch points. Then the other eigenvector k 0 − can be obtained by the smooth analytic continuation of k 0 + to the second sheet, since upon this operation the quasimomentum p k (u) flips sign and hence the eigenvalue changes from e ip k (u) to e −ip k (u) . By this definition, k 0 ± clearly satisfy the following relations: Now let us normalize these two eigenvectors so that they satisfy the normalization condition (4.2). This can be achieved by the rescaling Since two eigenvectors k 0 ± become degenerate at the branch points, k 0 + , k 0 − has a simple zero at such points. This yields the singularity structure given in (4.4).
Note that the aforementioned conditions do not completely fix the normalization, since we can always "renormalize" the eigenvectors as In section 4.3, we will utilize this freedom to express the angle variable in terms of the Wronskians.

Separated variables for two-point functions and orthogonality
In order to obtain the formula for the difference of the angle variables appearing in (2.38) in terms of appropriate Wronskians, we must first clarify the structure of the separated variables on the two-sheeted spectral curve. Similar information was crucial also in the case of the strong coupling, treated by the string theory representation. In that case, certain assumptions on the analyticity as a function on the string worldsheet helped determine some important structure. However, in the present case there is no worldsheet and we must devise a different logic to get a handle on the structure of the separated variables. Before delving into the discussion of the case of the three-point function, it is necessary to understand in detail the separated variables for the two-point functions. It will turn out JHEP10(2016)042 that the logic that we shall employ is of such a general validity that it can also be applied for strong coupling, as well as for the weak coupling that we are analyzing.
Let us consider the norm of a physical spin-chain state Ψ|Ψ (or equivalently a twopoint function) and perturb one of the states by adding a small number of Bethe roots to produce the inner product Ψ|Ψ + δΨ . Clearly Ψ|Ψ + δΨ should be non-vanishing for a general perturbation. However, when the perturbed state is such that it becomes on-shell again, Ψ|Ψ + δΨ must vanish because of the orthogonality of different eigenstates of the spin Hamiltonian. Therefore we have It should be emphasized that this is an exact quantum statement. Now we perform the same type of perturbation in the semi-classical regime. Specifically, consider the norm Ψ|Ψ of a semiclassical on-shell state and perturb only the ket state |Ψ by adding a small cut at the position of one of the singular points, 13 u * , which corresponds to adding a small number of Bethe roots. When the added cut is small enough, the log of this quantity (normalized by the original norm) can be expressed as where the derivative with respect to S u * acts only on |Ψ . We have denoted the action variable associated with the degenerate cut at the singular point u * by S u * and δS u * denotes the filling fraction corresponding to the small cut added. Since the state |Ψ is semiclassical, we can evaluate the quantity ∂ Ψ|Ψ /∂S u * using the saddle point approximation. This operation is exactly the same as the one performed on ln C 123 previously, and taking into account the saddle point equation itself the contribution that remains is where φ u * is the angle variable evaluated on the unperturbed state. As the small cut added in this regime is actually made of some number m of on-shell Bethe roots, with the positive integer m being of O(1), we can identify δS u * as m and hence (4.10) together with (4.11) can be written as 14 This means that when the perturbed state |Ψ + δΨ is again on-shell, according to the exact quantum property (4.9), which must hold in the semi-classical regime as well, we must have Ψ|Ψ + δΨ Ψ|Ψ e imφu * → 0 . (4.13) 13 As discussed in [32,33], the on-shell perturbation of the classical solution corresponds to the insertion of an infinitesimal cut at singular points. 14 Note that in the semi-classical limit, anything which does not scale as the length of the chain L can be regarded as small numbers.

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To examine this, let us compute φ u * using the formula (3.21) applied to this case. We have where γ j are the separated variables and ω u * is the holomorphic differential which satisfies the following properties on the first and the second sheet.
1st sheet: 2nd sheet: This means that when one of the γ j 's is at u = u * on the first sheet, φ u * behaves like while if such a situation occurs on the second sheet, we have Thus in order for e imφu * to vanish as dictated by (4.13), there must be a separated variable at each singular point on the first sheet. This information will be of prime importance in section 4.3 and 5.2: in section 4.3, it will provide the information of the zeros and poles of the important quantity n, ψ 3pt + . Such analyticity properties will in turn be imperative in determining those of the Wronskians, in terms of which the correlation functions will be expressed.
We once again stress that the preceding argument only uses the exact quantum property and its validity for the semi-classical regime as a special case, it is applicable regardless of the strength of the coupling constant.

Angle variables expressed in terms of the Wronskians
Using the properties discussed above, we now rewrite the angle variables in terms of the Wronskians.
As described in section 3.2, to construct the angle variables, we need to know the separated variables, which are associated to the poles of the normalized eigenvector of the monodromy matrix given in (3.15). Clearly some of the zeros of n , ψ + , where ψ + is the unnormalized eigenvector, correspond to such poles. However, n , ψ + may contain additional zeros, which do not appear in the normalized eigenvector ψ + / n , ψ + since ψ + itself becomes a zero-vector at such points and the ratio becomes finite. In addition to zeros, n , ψ + in general has poles where ψ + itself diverges. Likewise, these poles do not appear in JHEP10(2016)042 the normalized eigenvector as they cancel between the numerator and the denominator. In what follows, we call these zeros and poles spurious zeros and poles. It is important to note that the positions of the spurious zeros and poles may move if we change the normalization of the eigenvector as (4.7) whereas those of the separated variables do not.
With these properties in mind, let us study the analytic structure of the factor n , ψ 2pt + for the two-point function. When the spectral curve contains m-cuts, there are m "dynamical" separated variables, which evolve in the worldsheet time [29]. In addition to those, there are infinitely many non-dynamical separated variables which are trapped at the singular points on the first sheet of the spectral curve as discussed in the previous subsection. Both of them must correspond to zeros of n , ψ 2pt + . However, if we construct the solution explicitly using the finite gap method [29], we do not find infinitely many zeros corresponding to the nondynamical separated variables. This is because n , ψ 2pt + has spurious poles, which cancel the zeros associated with those separated variables. Furthermore, it has a square-root singularity at the branch points as shown in (4.4). Thus the divisor of n , ψ 2pt + is given by: 15 Here γ 2pt k 's correspond to the separated variables (both dynamical and nondynamical), s l 's denote the singular points on the first sheet, and b m 's denote the branch points.
We now turn to the corresponding quantity for the three-point function n , ψ 3pt + . To compute the normalized three-point functions, it is convenient to use the same normalization for the eigenvectors ψ 2pt + and ψ 3pt + . More precisely, we require ψ 3pt + (and therefore n , ψ 3pt + ) to have the same spurious zeros and poles as ψ 2pt + . This can always be achieved by multiplying by an appropriate function of the spectral parameter as (4.7). However, in that process, we often need to introduce extra spurious zeros and poles to n , ψ 3pt + in order to make c(u) in (4.7) to be single-valued on the spectral curve. Therefore, the general structure of the divisor takes the following form: Here γ 3pt k are the separated variables for the three-point functions whereas the last term (η n and δ n ) correspond to the extra spurious zeros and poles alluded to above.
Let us make two important remarks regarding (4.19). First, owing to the normalization condition ψ 3pt + , ψ 3pt − = 1, ψ 3pt − has zeros at δ n and poles at η n . Since ψ 3pt ± are related to each other by (4.3), η n and δ n must satisfy η n =σδ n , (4.20) 15 The solution for the two-point function has moduli, and for generic values of the moduli n , ψ 2pt + can have other spurious zeros and poles. However, on physical grounds, we expect that it is possible to choose a solution for which such poles and zeros are absent (although the argument is not completely rigorous). For a more detailed discussion on this point, see appendix E.

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whereσ is the holomorphic involution, which exchanges two sheets of the Riemann surface. Second, as (4.19) shows, ψ 3pt + becomes singular at the singular points on the first sheet. This property plays a key role in the determination of the analyticity structure in section 5.2.
Taking into account the analyticity structure discussed above, we now compute the right hand side of (2.38), which is the shift of the angle variables for the three-point function relative to those of the two-point function. (In what follows, we suppress the indices distinguishing operators until the very end when we write down the final expression (4.35).) For this purpose, it is useful to introduce a one-form df defined by the divisor of which is given by Now, using the formula (3.21), we can express the shift ϕ k as where ω k satisfies This expression can be simplified further using the generalized Riemann bilinear identity, 16 which reads Hereω P Q;k andω RS;k are normalized Abelian differentials satisfying 17 Pω Since ω k can be identified with −ω ∞ + ∞ − ;k , we can use the Riemann bilinear identity to rewrite ϕ k as

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where the integration contour starts from ∞ − , goes through the cut C k and ends at ∞ + , and e k is the one-form uniquely characterized by the following conditions: Here the second property follows from (4.20). Finally, substituting the definition of df into (4.27), we obtain 18 This expression can be further rewritten in terms of the Wronskians by judicious use of the highest weight condition. To see this, recall that the on-shell states constructed upon the rotated vacuum with the polarization vector n is annihilated by the operator S + , given in (2.22). Such global charges can be read off from the asymptotic behavior of the monodromy matrix as where the second equality follows from (2.22). This leads to the following asymptotic form of the monodromy matrix in the semi-classical limit: Note that this is true both for two-point functions and multi-point functions. From (4.31) and the asymptotic form of the quasi-momentum (3.7), we can determine the asymptotic behavior of the eigenvectors ψ ± to be of the form where a, b are arbitrary and we have imposed the normalization condition ψ + , ψ − = 1.
In the special case of two-point functions, by the explicit construction based on the finitegap method, we can check 19 that a 2pt = 1. Thus the ratio appearing in (4.29) can be evaluated as For three-point functions, the same quantity can be extracted from different combinations of the Wronskians. For instance, it is easy to verify, using (4.32), that the following combination gives the a −2 3pt for the operator O i :

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Thus, restoring the indices for the operators, we arrive at the final expression, Here e (i) k i is a one form defined on the spectral curve of the i-th operator O i , which satisfies s denote the holomorphic involution and the branch cuts respectively, for the spectral curve of O i .
Let us make a remark on the nature of the angle variables for the general multi-cut solutions that we are capable of dealing with in this paper, in comparison to the previous work [14], where only the one-cut solution could be studied. In that work, the only allowed perturbation is to vary the filling fraction associated with the unique cut and at the same time the one at infinity, i.e. S ∞ , in the opposite direction for consistency. This is why the angle variable conjugate to the global charge S ∞ showed up in the previous work. However, in the more general case of multi-cut solutions, we have to specify the cut to be perturbed among many and the corresponding angle variable is not necessarily the one conjugate to the global charge but the one associated with the more general filling fraction.

Evaluation of the Wronskians
Now that we have expressed the structure constant in terms of Wronskians, our final task is to evaluate these Wronskians.

Products of Wronskians from monodromy relation
Let us recall that at strong coupling, the monodromy relation was of crucial importance and it allowed us to express certain products of Wronskians in terms of quasi-momenta [12][13][14]. Since the relation derived in (2.32) is identical in form to the one in that analysis, one can apply the same argument also to the present case.
First, take the basis in which Ω 1 is diagonal of the form diag(e ip 1 , e −ip 1 ). Since the eigenvectors of different monodromy matrices are related with each other as Ω 2 in this basis is given by where M ij is the basis-transformation matrix defined by Now, using the relation Ω 1 Ω 2 Ω 3 = 1 (2.32), we can express the trace of the monodromy matrix Ω 3 as Substituting the explicit expressions of Ω 1 and Ω 2 above to the right hand side of (5.4), we get Combining this equation with the Schouten identity, we can determine the products 1 + , 2 + 1 − , 2 − and 1 + , 2 − 1 − , 2 + . Products of other Wronskians can be computed in a similar manner and the results can be summarized as where the cyclic permutation of (1, 2, 3) should be applied to (i, j, k).

Analytic properties of the Wronskians
Now to compute three-point functions, we need to know the individual Wronskians, not just their products (5.7). For this purpose, below we need to determine the analytic properties of the Wronskians as a function of the spectral parameter. This knowledge will later be indispensable in decomposing the products into individual Wronskians. Before proceeding, let us make one general remark: since each set of eigenvectors i ± live on a two-sheeted Riemann surface, Wronskians generally live on a 2 3 -sheeted Riemann surface. Each of these eight-fold sheets will be denoted as [ * , * , * ]-sheet, where the n-th entry * is written as "u" for the upper sheet of p n (u) and "l" when it refers to the lower sheet of p n (u). The determination of the analyticity on this eight-sheeted Riemann surface may at first sight seem a formidable task. However, as the eigenvectors on different sheets are related with each other by (4.3), once we know the analyticity of all the Wronskians on the [u, u, u]-sheet, the analyticity on the other sheets can be automatically deduced. For instance, the analyticity of 1 + , 2 + on the [l, u, u]-sheet are the same as the analyticity properties of 1 − , 2 + on the [u, u, u]-sheet. Thus, in what follows, it suffices to determine the analyticity on the [u, u, u]-sheet.
BPS correlators. We first study the simplest possible case, namely the three-point function of BPS operators. A distinct feature of such correlators is that the quasi-momenta do not contain any branch cuts. This simplifies the determination of the analyticity of Wronskians drastically, as we see below. Figure 4. The analytic structure of i + , j + on the [u, u, u]-sheet when all the operators are BPS. Left: in the limit p i → p j and p k → 0, the poles (denoted by black dots) and the zeros (denoted by red dots) are on top of each other, and the Wronskians are free of singularities. Right: since there are no branch cuts, even after p k becomes nonzero and p i starts to differ from p j , the zeros and the poles cannot move away from the [u, u, u]-sheet.
Let us first consider the Wronskians with the same signs, i + , j + and i − , j − . As (5.7) shows, their products contain poles at sin p i = 0 and sin p j = 0, which are at the singular points of the spectral curve. In our normalization of the eigenvectors, the plus solutions i + and j + become singular at the singular points on the first sheet of the spectral curve (see the discussion below (4.19)). This means that the poles on the [u, u, u]-sheet belong to i + , j + (and not to i − , j − ). In addition to poles, the right hand side of (5.7) has zeros at sin(p i + p j + p k )/2 = 0 and sin(p i + p j − p k )/2 = 0. To determine which Wronskian contains these zeros, we first consider the limit where p k → 0 and p i → p j . In this limit, the classical solution for the three-point function approaches to the one for the two-point function. As mentioned in section 4.3, for the two-point function, the eigenvectors are nonsingular even at sin p i = 0 and so are the Wronskians. This means that in this limit the zeros of the Wronskians must cancel the pole singularities. In order for such cancellations to occur, all the zeros on the [u, u, u]-sheet must belong to i + , j + when p k is very small. Now, since all the operators are BPS and there are no branch cuts connecting different sheets, those zeros cannot leave the [u, u, u]-sheet even if we increase the value of p k (see figure 4). We therefore conclude that all the zeros on the [u, u, u]-sheet must always belong to i + , j + , not to i − , j − , when the three operators are BPS. Next we consider the Wronskians with opposite signs i + , j − and i − , j + . Also in this case, the determination of the poles are straightforward since they are associated with the eigenvectors themselves. By the same reasoning as above, we conclude that the poles at sin p i = 0 belong to i + , j − whereas the poles at sin p j belong to i − , j + . On the other hand, the determination of zeros is more complicated and we need to resort to the argument given in [14]. As reviewed in appendix G, it leads to the following general rules: 1. When a factor sin ( i i p i /2) vanishes, the Wronskians which vanish are the ones among 2. On the [u, u, u]-sheet, the Wronskians from the group which contains more than one + eigenvectors all vanish whereas the Wronskians from the other group do not.  Applying these rules, we can determine the analyticity on the [u, u, u]-sheet as given in tables 1 and 2. As already mentioned, the analyticity on other sheets can be deduced from the relations (4.3).

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Extension to non-BPS. Now we turn to non-BPS correlators. Their analytic properties can be inferred from those of the BPS correlators if we make the following physically reasonable assumption: Continuity assumption. When all the branch cuts of the quasi-momenta p i (x) shrink to zero size, the classical solution for the non-BPS correlator smoothly goes over to those for the BPS correlators.
This assumption implies in particular that the Wronskians for the BPS and the non-BPS cases are also smoothly connected. Now, let us consider the three BPS correlator discussed above and insert a very small cut to make it non-BPS. Because of the continuity assumption, the zeros and the poles of the Wronskians cannot move to a different sheet as long as the cut is sufficiently small and therefore the analyticity of Wronskians for such non-BPS correlators must be the same as the one for the BPS correlators. If we gradually increase the sizes of the cuts, at some point the zeros and the poles start crossing the branch cuts and move over to a different sheet, leading to a change in the analytic property of the Wronskians. A simple way to take such effects into account is to first compute the correlators with small cuts and then analytically continue the final results with respect to the sizes of the cuts. See figure 5. This analytic continuation to larger cuts will be discussed in section 6.2, and we will comment on how it affects the integration contours. 20 Thus, until then, we will restrict ourselves to the spectral curves with small cuts. 20 A similar phenomenon was observed in the context of one-loop corrections to the classical energy both at weak [35] and strong coupling [32,33]. In both cases, as the sizes of the cuts become bigger, some of the physical excitations cross the cuts. At weak coupling, this leads to the change in the distribution of the Bethe roots. Nevertheless, the final result turns out to be a smooth function of the sizes of the cuts, and we expect that this is also the case for three-point functions. Figure 5. The analyticity of the Wronskians for the non-BPS correlators. Under the continuity assumption, the analyticity remains the same as the one for the BPS correlators as long as the cuts are sufficiently small (left figure). As we increase the size of the cut, the zeros and the poles start to move and, at some point, they cross the cut and cause the change in the analyticity (right figure). Such effects affect the integration contours of the final result as we shall see in section 6.2.

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Spurious zeros and poles. In addition to the structures we discussed so far, there are extra spurious zeros and poles of the eigenvectors (η n and δ n in (4.19)). Owing to the normalization condition i + , i − = 1, whenever i + has such a zero i − has a pole, and vice versa. Thus, these extra zeros and poles cancel out if we consider the products of the Wronskians appearing on the left hand side of (5.7). Nevertheless we should keep in mind that such poles and zeros are present as we solve for the individual Wronskians below. In the next section, we first subtract such zeros and poles from the Wronskians and study the Riemann-Hilbert problem for these subtracted quantities.

Solving the Riemann-Hilbert problem
Let us now use the analytic properties to set up and solve the Riemann-Hilbert problem to determine the Wronskians. Here we will only discuss i + , j + and i − , j − since these are the Wronskians relevant for the computation of the structure constant. (The argument below can be straightforwardly generalized to other Wronskians but we will not elaborate on it here.) In what follows, we analyze the logarithmic derivative of the relation (5.7), namely Since the Wronskians contain extra zeros and poles which are absent on the right hand side of (5.8) as mentioned above, it is convenient to consider the following quantities from which extra zeros or poles are subtracted: Here e (i) k i is a one-form given by (4.36) in section 4.3. As explained there, it depends on the choice of the cut C (i) k i , which we are perturbing. In terms of W ij ±± , (5.8) can be written as To uniquely characterize the solution to the Riemann-Hilbert problem, we have to specify its period integrals in addition to its analyticity. In the case at hand, the period integrals of W ij ±± are given by These properties can be shown as follows: since i ± , j ± is a single-valued function on the Riemann surface, the integral of ∂ u ln i ± , j ± along any closed curve gives nπ where n is an integer. As we are considering the spectral curves with small cuts, which are continuously connected to the curves without cuts, n must be zero in such a case. Together with the property of e (i) k i given in (4.36), this leads to (5.11). This property will be later utilized in checking the correctness of the expressions of W ij ±± we shall construct. (5.12) Using (5.12), it is easy to verify that f + is regular on the upper-half plane and f − is regular on the lower-half plane. When x is not in the region specified in (5.12), we need to analytically-continue these formulas. This leads, for instance, to the following expression for f + (x) on the lower-half plane (Im Here the first term f (x) is produced by the integral along a small circle around x = x as shown in figure 6. An important feature of this method is that the contour of integration separates domains with different analyticity structures. This is true also in a more complicated situation where functions are defined on a multi-sheeted Riemann surface as in (5.10).

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For later convenience, let us also present the version of (5.14) obtained by integration by parts: (Im x < 0) . (5.14) Here . It is this form of the Wiener-Hopf decomposition that will be generalized below in order to deal with the multi-sheeted Riemann surface on which p i (x)'s are defined.
Decomposition of the poles. We first study the factors ∂ u ln sin p i and ∂ u ln sin p j , which give rise to the poles of the Wronskians. Below we focus on ∂ u ln sin p i since the case for ∂ u ln sin p j is completely similar. As in the standard Wiener-Hopf decomposition, we should be able to decompose it by considering a convolution integral whose contour separates the domains with different analyticity. As summarized in table 1, the poles of 1/ sin p i belong to i + , j + when the rapidity is on the first sheet of p i while they belong to i − , j − when it is on the second sheet of p i . Obviously, these two regions are separated by the branch cuts of p i and so the contour should be taken to go around the cuts. Now what we must properly deal with is the choice of the convolution kernel, as we have a two-sheeted Riemann surface instead of a simple complex plane. The natural generalization of the kernel (5.14) in the present case is given by the bidifferential characterized by the properties listed below, which is often called the Bergman kernel in physics literature. 21 To define the Bergman kernel, we must first pick a basis of cycles. Then, the Bergman kernel B(p, q) is a differential in both p and q and is uniquely specified by the following properties; where {a j } is the basis of a-cycles.
3. Analyticity: B(p, q) is meromorphic in p with only a double pole at p = q with the following structure: Here ζ is a local coordinate around p q. 21 This quantity is introduced by J. Fay [36] as the bidifferential made from the prime form and is called "the normalized bidifferential of the second kind" (see also [37]). Although in mathematics the Bergman kernel, strictly speaking, refers to slightly broader notion, we shall follow the physics nomenclature. We thank M. Jimbo and A. Nakayashiki for useful information on these matters.

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In addition to these properties, when the curve is hyperelliptic, it satisfies

Involution property:
B(σp,σq) = B(p, q) . (5.18) This is because the kernel B(σp,σq) satisfies all three properties listed above, which specify the Bergman kernel uniquely. In the present case, we can define the Bergman kernel for each of the three spectral curves and we denote them by Here the subscript k i designates our choice of the basis of the cycles; namely we choose the a-cycles as the cycles that surround each cut except C (i) k i . 22 Now, using these kernels, one can decompose ∂ u ln sin p i as follows: (5.20) Here rest represents the terms coming from decomposition of the rest of terms on the right hand side of (5.10). The integration contour Γ i is on the 1st sheet of p i and goes along the cuts C (i) s as depicted in figure 7-(a). Let us now make a remark on the contour: unlike other poles, the poles at the branch points are shared equally by ∂ u ln i + , j + and ∂ u ln i − , j − , since each eigenvector has a square-root singularity as shown in (4.4). To realize this structure, one must average over different ways of avoiding the branch points as shown in figure 7-(b). Apart from this small subtlety, these formulas are natural generalization of (5.14) and more importantly they are consistent with the property of W ij ±± (5.11), thanks to the normalization of the Bergman kernel (5.16). As in the standard Wiener-Hopf method, the expressions in the other regions can be obtained by analytic continuation.
Before proceeding, let us rewrite (5.20) in a form where the action of the holomorphic involution is more clearly seen. For this purpose, we first make a change of the integration variable from u toσ i u , withσ i being the holomorphic involution for the spectral curve of p i . This, of course, leaves the value of the integral intact, but its form gets slightly modified. For instance, the integrand is transformed as Here, the new kernelB (i) k i (p, q) is defined by (5.22) and has a pole when p =σ i q,  where ζ is a local coordinate around p andσ i q. Similarly, the integration contour is modified as follows (see figure 7-(c) for more explanation): From these transformation rules, we can rewrite the integral appearing in (5.20) as Here and below we neglect the term coming from ln(−1) since it changes the structure constant only by an overall phase. Now, by averaging two sides of (5.25), we arrive at the following expressions for W ij ±± : k i * ln sin p i + rest (u ∈ 1st sheet of p i ) .

(5.26)
Here A (i) k i is the "anti-symmetrized" kernel defined by 27) and the notation C F * f denotes the convolution integral The kernel A (i) k i is odd under the holomorphic involution of each of the arguments: The first equality follows immediately from the definition whereas the second equality can be derived using the property of the Bergman kernel (5.18). This property is used in section 6 when we write down the expression for the semi-classical structure constant.

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Decomposition of the zeros. We now decompose ∂ u ln sin(p i + p j + p k )/2 and ∂ u ln sin(p i + p j − p k )/2, which are responsible for the zeros of the Wronskians. Since these quantities are defined on the 2 3 -sheeted Riemann surface, both the integration contour and the convolution kernel must also be defined on the same eight-sheeted surface. Let us first specify the integration contour. As in the previous case, the contour should be taken such that it separates the domains with different analyticity. As stated in the rules in section 5.2, when sin ( i i p i ) vanishes, the Wronskians that vanish are the ones among {1 1 , 2 2 , 3 3 } or the ones among {1 − 1 , 2 − 2 , 3 − 3 }. Depending on which of the two groups contain vanishing Wronskians, the eight-sheeted Riemann surface is divided into two regions. Then the integration contour, denoted by Γ 1 2 3 , will be placed at the boundary of the two regions. For instance, for the case of sin(p 1 + p 2 + p 3 )/2, the two regions and the contour are depicted in figure 8. To find the analyticity structure and the contour of other factors, one just needs to exchange the sheets appropriately, thanks to the property (4.3). For example, the analyticity structure and the contour of sin(p 1 + p 2 − p 3 )/2 are given by those in figure 8 with [ * , * , u]-sheets and [ * , * , l]-sheets swapped.
We next determine the convolution kernel. To carry out the desired decomposition, we use the kernel B all (p, q), which satisfies the first and the third properties, (5.15) and (5.17) respectively, of the Bergman kernel and the following slightly different normalization condition: Using this kernel, we can, for instance, decompose ∂ u ln(p i + p j + p k )/2 in the following way: figure 8) .

(5.31)
Again, in virtue of the normalization condition (5.30), this is consistent with the property of W ij ±± (5.11). The decomposition of ∂ u ln sin(p i + p j − p k ) can be performed in a similar manner.
Let us make a clarifying remark. Although the existence of the kernel B all with the properties listed above has not been explicitly proven, the convolution integral (5.31) can be rewritten entirely in terms of the standard Bergman kernel, the existence of which is firmly established. To show this, we again make use of the holomorphic involution. To illustrate the idea, let us consider the following terms that appear in the expression for W 12 ++ and W 12 −− : Figure 8. The analyticity structure and the integration contour for the decomposition of sin(p i + p j + p k )/2. In the region depicted in white, the Wronskians among {1 + , 2 + , 3 + } have zeros, whereas in the region depicted in gray, the Wronskians among {1 − , 2 − , 3 − } have zeros. The integration contour Γ +++ , denoted in red, separates the white region from the gray region.
If we change the integration variables from u toσ 3 u, the integrand and the contour of (5.32) transform as ln sin where the new kernel, B all (p, q), has a pole at p =σ 3 q. Using this transformation rule, we can re-express the integral (5.32) as all (u, u ) ≡ B all (u,σ 1σ2σ3 u ) .

(5.36)
Now the kernel K(p, q) has four double poles as shown in table 3. As is clear from table 3, the analytic properties of K(p, q) are identical to those of A (1) k 2 , which are expressed in terms of the usual Bergman kernels. Thus we can replace K(p, q) in (5.35) with A (1) Table 3. The structure of the poles of the kernels K, A

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k1 and A k2 . The first row designates the position of the double pole as a function of p and the numbers within the table are the coefficient of each pole. One can easily see that K and A Performing similar analysis to other 4 terms, we obtain the following expression for W 12 ++ : k 2 ) * ln sin k 2 ) * ln sin As in the standard Wiener-Hopf decomposition, this integral expression is valid in the region where W 12 ++ does not have any poles, which in this case correspond to the [l, l, * ]-sheets. 23 Finally, let us discuss the simplification of the integration contours. The contours of (5.37) are defined on the eight-sheeted Riemann surface. However, for comparison with the results in the literature, it is more convenient to convert them into integrals defined purely on the [u, u, u]-sheet. This can be achieved again by making use of the holomorphic involution: for instance, take the integral along Γ +++ in (5.37) and consider the portion of the integral on the [u, l, u]-sheet. If we perform the holomorphic involutionσ 2 to the integrated variable u , this contribution becomes identical to the third term in (5.37) evaluated on the [u, u, u]-sheet. Repeating the same analysis for the other relevant integrals, we arrive at the expression k 2 ) * ln sin k 2 ) * ln sin k 2 ) * ln sin where γ 1 2 3 is a portion of Γ 1 2 3 on the [u, u, u]-sheet. It is clear from figure 8 that Γ +++ does not have any portion on the [u, u, u]-sheet, and thus γ +++ = ∅. The other contours are along the cuts of some of the quasi-momenta as shown below:

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Here Γ i 's are the contours given in figure 7-(a). Substituting (5.39) into (5.38) and restoring the terms coming from the decomposition of poles, we finally obtain k 2 ) * ln sin k 2 ) * ln sin Similarly, we can write down the expression for W 12 −− , which is valid when the spectral parameter is on the [u, u, * ]-sheets: k 2 ) * ln sin The expressions for the other W ij ++ 's and W ij −− 's can be obtained from (5.40) and (5.41) by the permutation of the indices.

Results at weak coupling
Now we combine the results of sections 2, 4 and 5 and write down the explicit integral expression for the structure constant.

Integral expression for the semi-classical structure constant
As shown in (4.35), the variation of the semi-classical structure constant is given in terms of the Wronskians. To compute those Wronskians, we integrate the results obtained in the previous section (5.40) and (5.41). The net effect of integration is to replace the kernels A (i) . This is, however, still ambiguous since the initial point of the u-integration v 0 is not fixed. To determine v 0 , we impose the following condition which comes from the normalization of the eigenvectors (4.3): In terms of the logarithm of the Wronskians, this reads As in the previous analyses, we have neglected the minus sign in (6.1), which only affects the overall phase of the final result. We shall now show that (6.2) is satisified if we choose v 0 to be the branch point of C (i) k i , which we denote by b k i (see figure 9). Under this choice,

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the Wronskians are given by Here, the precise integration contour is the one given in figure 9.
Having determined the Wronskians, we can now compute the angle variable by evaluating (6.4) at u = ∞ and substituting them into (4.35). It turns out that the terms E (i) k i (u = ∞) precisely cancel the last term in (4.35). Thus, as anticipated, the contribution from extra zeros and poles do not appear in the final expression, which takes the form (6.8) 24 For instance, the expression for E (i) k i can be derived as follows:  Here the one formᾱ From the properties of the Bergman kernel, one can show thatᾱ (i) k i has the following analytic properties: Now, it is easy to check that the one form 25 also satisfies the same analytic properties. Since (6.10) specifies the one form uniquely, this means thatᾱ (i) k i is identical to (6.11). Using this fact and the identity, we can integrate the relation ∂ ln C 123 /∂S (i) k i to get the following integral expression: ln Li 2 (e 2ip k ) . (6.13) 25 One can show (6.11) using the argument similar to the one given in section 3.2: to perturb S (i) k i , one needs to add to pidu a one form whose period integral does not vanish only along the cycle at infinity and the cycle around C (i) k i . By comparing the residues carefully, we arrive at (6.11). Figure 10. Two examples of type I-I-II three-point functions. In both figures, the fields denoted by black letters correspond to the vacuum and the fields denoted by red letters correspond to the magnons. (a) The configuration studied in most of the literature (see e. g. [11]). It amounts to choosing the polarization vectors as n 1 = n 3 =ñ 1 =ñ 3 = (1, 0) t and n 2 =ñ 2 = (0, 1) t (b) The configuration used in [16].Z andỸ are given byZ = Z +Z + Y −Ȳ andỸ =Ȳ −Z respectively. The polarization vectors in this case are given by n 1 =ñ 1 = (1, 0) t , n 2 =ñ 2 = (0, 1) t and n 3 =ñ 3 = (1, 1) t .

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Here the summation in the first line denotes the sum over the cyclic permutation (abbreviated as "cperm") of {1, 2, 3} and M i is the total number of magnons in p i . Note that (6.13) is the contribution from the SU(2) R sector only. For the complete result for the structure function for the distinct types of three-point functions, to be analyzed in the next section, it must be combined with the contribution from the SU(2) L sector as well.

Results and comparison with the literature
The operators forming the three-point functions we are studying transform under a single group SO(4) = SU(2) L × SU(2) R . For such correlators, there are two distinct classes, as discussed in [14].
Type I-I-II three-point function. The first class of such three-point functions is called type I-I-II. These are the ones for which two of the operators have magnon excitations in the same SU(2), whereas the magnons for the third operator are in the other SU (2). Examples of such configurations are depicted in figure 10. This class of three-point functions were studied extensively in the literature and it was shown in [18,38] that they can be expressed as a product of two Izergin-Korepin determinants [39,40]. From such exact expressions, the semi-classical limit was extracted in [19][20][21]. In what follows, we shall reproduce it from our result (6.13).
Let us, for simplicity, consider the case where O 1 and O 2 belong to SU(2) R and O 3 belongs to SU(2) L . The structure constant factorizes into the left and the right parts as explained in section 2 and each part can be expressed in terms of integrals of the type given in (6.13). To get an explicit expression for C 123 from (6.13), we also need to know the BPS L i + L j − L k 2 (ln n i , n j + ln ñ i ,ñ j ) . (6.14) Using this expression, we can write down the result for the type I-I-II three-point function as where each part is given by Here K denotes the contribution determined purely by kinematics and the SU(2) L,R global charges l i and r i are given bỹ (6.20) The second and the third terms L and R contain the dynamical information of the threepoint functions and come from SU(2) L and SU(2) R respectively. The last term N is the part corresponding to the norms of the Bethe states in the exact quantum expression (see for instance [11]). To make a direct connection with the results in [21], we now rewrite the second and the third terms in R by pushing the contour onto the second sheet as we did in figure 7-(c). Then the two terms read Li 2 e −(−ip 1 +ip 2 +iL 3 /2u) . (6.21) Now using the dilogarithm identity, we can show 26 that (6.21) is identical to the first term in (6.18). By performing similar manipulation, we can also show that the first and the second terms in (6.17) are equivalent.
In this way, we can obtain the following alternative expression for L + R: Together with the norm part N , this perfectly agrees with the result in [21]. 26 The terms coming from the second and the third terms in the identity (6.22) vanish. Figure 11. An example of the type I-I-I three-point functions studied in the [16].Ỹ in the figure represents Y +Z. The polarization vectors are given by n 1 =ñ 1 = (1, 0) t , n 2 =ñ 2 = (0, 1) t and n 3 =ñ 3 = (1, 1) t .

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Type I-I-I three-point function. Let us now turn to the case where all the three operators have magnons in the same SU(2)-sector. They are called type I-I-I in [24]. An example of this class of correlators is given in figure 11. As compared to the type I-I-II correlators, they have much more complicated structure and the exact results known at weak coupling are given either in terms of the sum of the triple product of determinants [24] or in terms of the multiple-integral expression based on the separation of variables [41]. Both of these expressions are hard to deal with and their semi-classical limit has not been computed. Despite such complications for the exact result, the semiclassical result we derive below turned out to take a remarkably simple form. It would certainly be a challenging future problem to reproduce it from the expressions given in [24] and [41].
For definiteness, let us consider the case where all the operators belong to SU(2) R . In this case, there is no dynamical contribution from SU(2) L and we can write down the full expression using (6.13) as ln C 123 = K + R + N , (6.24) with each part given by Here the definitions of l i and r i are modified from (6.20) in the following manner: (6.28) As advertized, the expression above for the structure constant is as simple as the one for the I-I-II type.
Remark on the integration contour. So far, we have been assuming that the cuts in p i are sufficiently small. In particular, we used this assumption when we derive the analyticity of the Wronskians. Let us briefly explain what we expect when we gradually increase the sizes of the cuts in the integral expression (6.13).
Since the dilogarithm Li 2 (x) has a branch cut emanating from x = 1, the integrands of (6.13) contain infinitely many branch-point sigularities at e i(p i +p j −p k ) = 1 and e 2ip i = 1. These correspond to the zeros and the poles of the Wronskians respectively. As we increase the size of the cut, at some point, they start crossing the cut. When this happens, we need to deform the contour as depicted in figure 12 in order to keep the final result continuous with respect to the size of the cut. Thus, if we consider the operators with large cuts, the integration contours will be substantially deformed and will no longer be given by the ones around the cuts. This would explain the observation made in [21] that one must deform the contours appropriately in order to reproduce the value obtained by numerics. It would be important to perform detailed numerical computation and confirm the claim we made here.

Application to the strong coupling
One of the important findings of the present work is that, as far as the semi-classical behaviors are concerned, the same structure and the logic underlie the three point functions both at weak and strong couplings. In this section we shall apply the machineries developed so far to the computation at strong coupling.

Classical integrability of string sigma model on S 3
Let us first give a brief review 27 of the classical integrability of the string sigma model on S 3 emphasizing the similarity to and the difference from the Landau-Lifshitz model discussed in section 3.
For the string sigma model on S 3 , we can define two sets of Lax pairs as Here x is the spectral parameter and the currents j andj are defined using the embedding coordinate Y i (i = 1, . . . 4) as For each Lax pair, we have an auxiliary linear problem and a monodromy matrix: Note that the two sets of quantities defined above are related with each other byj = GjG −1 , ψ = Gψ andΩ = GΩG −1 . As with the Landau-Lifshitz model, the quasi-momentum p(x) is given by the logarithm of the eigenvalue of the monodromy matrix Ω ∼Ω ∼ diag(e ip , e −ip ). The spectral curve is defined also in a similar way as The asymptotic behavior of the quansi-momentum around 0 and ∞ encodes the information of the global charges 28 as where Q andQ are the charges of the SU(2) R and SU(2) L respectively. We should note that, unlike the Landau-Lifshitz model, the quasi-momentum does not have a pole at x = 0. Instead, it has poles at x = ±1 with residues given by the worldsheet 29 energy E and momentum P: Owing to this pole structure, the singular points of the spectral curve accumulate to x = ±1 as shown in figure 13. 27 For a more detailed account, see [14,23,29,30,34]. 28 In the most general situation, the quasi-momentum around x = 0 behaves as p(x) ∼ 2πm + xQ/g + · · · , where m is an integer called the winding number. Here we are considering the m = 0 case for simplicity. 29 E and P defined here are the energy and the momentum of the S 3 sigma model in the conformal gauge. They therefore do not have definite physical meaning. In particular E is in general different from the lightcone energy of the string sigma model in AdS5 × S 5 . Figure 13. The structure of the spectral curve at strong coupling. In addition to the branch cuts, it has infinitely many singular points, denoted by black dots, which accumulate to x = ±1. Those singular points should be regarded as degenerate branch points.

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As in the Landau-Lifshitz sigma model, the filling fractions are given by contour integrals on the spectral curve. However, their explicit forms are slightly modified: Here u(x) is the rapidity variable, given by u(x) = g x + 1 x , (7.9) and the integration contour goes around the k-th branch cut 30 C k counterclockwise on the first sheet.

SU(2) L and SU(2) R excitations at strong coupling
One of the conspicuous differences from the Landau-Lifshitz model is that the filling fractions given by (7.8) can be negative at strong coupling, and it turns out that the signs of the filling fractions are tied to whether the state has excitations in the SU(2) L sector or in the SU(2) R sector.
To understand this point, let us consider the perturbation around the BMN vacuum. It was shown in [32,33] that the quasi-momentum receives the following correction when an infinitesimal cut is inserted at x = x * : (7.10) 30 As in the Landau Lifshitz sigma model, we should consider the (infinitely many) singular points satisfying e 2ip(x) = 1 also as (degenerate) branch cuts.

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Here n is the filling fraction inserted at x = x * and the factor dx/du in (7.10) is necessary due to the definition of S k in (7.8). We can also compute the energy shift using the results 31 in [32,33] as In (7.11), the quantity 1/(x 2 * −1) is positive when |x * | > 1, while it is negative when |x * | < 1. Since all the physical excitations around the BMN vacuum must have the positive energy shift, 32 this means that n must be positive if |x * | > 1 whereas it must be negative when |x * | < 1. This is in marked contrast with the situation in the Landau-Lifshitz model, where we always needed to take n to be positive to describe the physical states. Physically, this is because the Bethe roots in the region |x * | < 1 correspond to anti-particles: in order to construct a physical state from the anti-particles, we need to insert "holes" just as in the Dirac's fermi sea.
We can show more generally that the filling fraction defined by (7.8) must be positive whenever the cut is outside the unit circle whereas they must be negative whenever the cut is inside the unit circle. Now, to understand the physical meaning of these two types of cuts, let us consider the relation 33 between the global charges and the filling fractions: For the BMN vacuum, all the filling fractions are zero and Q andQ are equal. Now, if we insert cuts outside the unit circle, which have positive filling fractions, we must either decrease Q or increaseQ in order to satisfy (7.13). However, since the BMN vacuum has the maximal 34 Q andQ, the only way to achieve this is to decrease Q. This clearly tells us that those states correspond to the ones with excitations in SU(2) R . By a similar argument, we can show that the states with cuts inside the unit circle correspond to the states with SU(2) L excitations. For a summary, see figure 14. In appendix F, we provide an interpretation of the SU(2) L -and SU(2) R -sectors from the point of view of the full spectral curve of the AdS 5 × S 5 sigma model.

Angle variables, ln C 123 and Wronskians at strong coupling
With this knowledge, we now construct the angle variables which compute the derivative of ln C 123 , and express them in terms of the Wronskians. Below we shall treat the SU(2) Rsector and the SU(2) L -sector separately. 31 The argument roughly goes as follows: as is clear from (7.10), the perturbation modifies the behavior around x = ±1. Owing to the Virasoro constraint, the AdS quasi-momentump around x = ±1 must also be deformed in the same way. Once we understand howp is modified, we can then read off the energy shift from its asymptotic behavior at x = ∞. 32 In other words, one cannot lower the energy starting from the BMN vacuum. 33 (7.13) follows from the fact that r and l can be expressed as SU(2) R -sector. Let us first discuss the states with SU(2) R excitations. To construct the angle variables, we should study the normalized eigenvectors of the monodromy matrix as in the Landau-Lifshitz model. One important difference in the present situation is that we now have two sets of linear problems and monodromy matrices. For the SU(2) R , the appropriate one to use is (7.3). This is because (7.4) is invariant under the SU(2) R transformations and is therefore insensible to the SU(2) R excitations. As in section 3.2, the separated variables can be constructed from the poles γ i of the normalized eigenvector h(x), h(x) ≡ 1 n , ψ + ψ + . (7.14) Here ψ ± are the solutions to the auxiliary linear problem (7.3) satisfying As shown in [30], a pair of canonically conjugate variables at strong coupling is given not by (γ i , −ip(γ i )) but by (u(γ i ), −ip(γ i )), where u(x) is the rapidity defined by (7.9). This explains the form of the filling fraction given in (7.8). Now, to construct the angle variables, we need to consider the generating function of the canonical transformation (3.19) and then differentiate it with respect to S k . As explained in the previous subsection, we should simultaneously decrease the global charge Q when we vary S k . This amounts to adding to p(x)du(x) a one form whose integral does not vanish only for the cycle around C k and the cycle at infinity. As a result, we get where ω k is the one form satisfying

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Let us next express the derivative of ln C 123 in terms of the angle variables. The arguments leading to (2.38) are by and large applicable also to the present case, except for one important point. At strong coupling, in addition to the contribution from the S 3 part of the sigma model, we should also include the contribution from the AdS part. In particular, whenever we perturb the filling fraction in the S 3 part, we inevitably change the conformal dimension ∆ i , which is a global charge in AdS. This leads to the following modification of (2.38): ∂ ln C 123 ∆ is the angle variable conjugate to ∆ i , whose definition is given in appendix H. Now, following the argument in section 4.3, we can express the angle variable φ (i) k i in terms of the Wronskians and the result takes the same form as (4.35). We can perform similar analysis also to the AdS part (see appendix H for details) to get the following expression of the angle variable φ Here x i denotes the position of the operator O i and the eigenvectors i + 's andĩ + 's are the solutions to the auxiliary linear problems of the AdS 3 sigma model.

SU
(2) L -sector. For the SU(2) L -sector, the linear problem we should consider is (7.4), as it is the one that transforms nontrivially under the SU(2) L transformation. In this case, the separated variables in the SU(2) L sector can be constructed from the polesγ i of the normalized eigenvectorh(x), Hereψ + is the solution to the auxiliary linear problem (7.4) satisfying Then the separated variables can be constructed from the poles atγ i as (u(γ i ), −ip(γ i )). From the separated variables, we can construct the angle variables in the same manner as for the SU(2) L -sector. The only modification in the present case is that, when we change the filling fraction S k , we need to change l but not r as discussed in section 7.2. This can be achieved by adding to p(x)du(x) a one form whose integral does not vanish only for the cycle around C k and the cycle around x = 0. Then we get the expression, where ω k is the one form satisfying 35 35 Here the contour for the second integral goes around x = 0 on the first sheet counterclockwise.

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Using these angle variables, we can express the derivative of ln C 123 as 36 Here, as in the previous relation (7.18), φ (i) ∆ is the AdS angle variable (7.19). Let us next express the angle variables in terms of the Wronskians. Although the basic logic in section 4.3 applies also to the present case, we have to modify (4.29) and (4.35) appropriately as follows: (7.25) Here the one formsω P Q;k and e k are defined by (4.26) and (4.28) respectively.
To express (7.25) in terms of Wronskians, we use the highest weight condition again. In this case, we should study the behavior of the monodromy matrixΩ(x) around x = 0 on the first sheet,Ω Applying the argument similar to the one in section 4.3, we arrive at the following form of the eigenvectors at x = 0 (on the first sheet): ψ + (0) = añ ,ψ − (0) = a −1 iσ 2ñ + bñ . (7.27) Using (7.25) and (7.27), we finally get the expression for the angle variables in terms of the Wronskians: Here the Wronskians are evaluated on the first sheet andφ (i) k i denotes the angle variable of the operator O i associated with the k i -th cut whereasñ i is the SU(2) L polarization vector for O i . To derive (7.28), we used the fact that the Wronskians among i + 's are equivalent to the Wronskians amongĩ + 's, i + , j + = ĩ + ,j + . This is because two sets of eigenvectors are related with each other by the similarity transformation,ĩ + = G i + , and the Wronskians are invariant under such transformation. Figure 15. The positions of the separated variables for two-point functions at strong coupling. From the orthogonality of on-shell states (7.29), we conclude that the separated variables are either at the singular points outside the unit circle on the first sheet, or at the singular points inside the unit circle on the second sheet.

Semi-classical orthogonality of on-shell states at strong coupling
The key ideas for determining the analyticity of the Wronskians in the Landau-Lifshitz model were the requirement of the semi-classical orthogonality between two different onshell states and the assumption of the continuity between the BPS correlators and the non-BPS correlators. Here we apply these two ideas to the analysis at strong coupling.
Just as for weak coupling, one can construct, at strong coupling, a different on-shell state by introducing an infinitesimal cut at the position of the singular point. We should, however, be careful about whether the perturbation is physical or not: as explained in section 7.2, in order to obtain a physical state, we should insert a positive filling fraction when the singular point is outside the unit circle, whereas we should insert a negative filling fraction when the singular point is inside the unit circle. With this in mind, we now impose the orthogonality condition ψ|ψ + δψ = 0 . (7.29) Here δψ must correspond to a physical perturbation in the sense explained above. Now it is not so hard to verify that the argument in section 4.2 applied to the present case leads to the conclusion that the separated variables are at the singular points outside the unit circle on the first sheet, or at the singular points inside the unit circle on the second sheet (see figure 15). Then, repeating the argument 37 given in section 5.2, with the above modification taken into account, we can determine the poles and the zeros of the Wronskians. The results are summarized in table 4. Now, using these analyticity properties, we can solve the Riemann-Hilbert problem and determine the individual Wronskians as described in section 5.3. The main difference in the present case is that the Wronskians change the analyticity when they cross |x| = 1. 37 Since the monodromy relation at strong coupling takes exactly the same form as (2.32), the equation (5.7) holds without modification.

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1/ sin p i 1/ sin p j sin This leads to extra integration contours around the unit circle. Once the Wronskians are determined, we can compute the angle variable and then determine the structure constants using (7.18) and (7.24). The results will be given explicitly in the next subsection.

Results and discussions
We now write down the results for the three-point functions at strong coupling explicitly and compare them with the results in [14].
Type I-I-II three-point functions. Let us first consider the type I-I-II three-point functions. Below we assume that O 1 and O 2 belong to SU(2) R while O 3 belongs to SU(2) L . For such a three-point function, the result has the following structure: Here K is the kinematical part given by  Figure 16. Integration contours for the type I-I-II three-point functions. Γ 1 and Γ 2 encircle counterclockwise the branch cuts of p 1 and p 2 respectively, whereas Γ 3 goes around the branch cuts of p 3 clockwise. U is the contour which goes counterclockwise around the unit circle.
and each factor is given as follows: Li 2 e −ip 1 +ip 2 +ip 3 , Li 2 e 2ip k , Li 2 e 2ip k . (7.34) The contours of integration are depicted in figure 16 andp i is the AdS quasi-momentum given byp A few remarks are in order. Firstly, as shown in (7.33), the integrals along the unit circle are multiplied by the extra factor of 2 (denoted by 2U ) as compared to the integrals along the cuts. 38 This factor can be deduced by carefully applying the argument given in section 6 to the strong coupling analysis. Roughly speaking, this is because the integrals along the unit circle exist on every sheet of the eight-sheeted Riemann surface whereas the integrals along the cuts exist only on some (roughly the half) of the sheets (see figure 8). Secondly, each integral along U is actually divergent owing to the poles in p i at x = ±1. However, such divergences cancel out when we combine all the terms in (7.30). To illustrate this point, let us consider the integral U du 2π Li 2 e i(p 1 +p 2 +p 3 ) .

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Since we are interested in the behavior around x = ±1, where the integrand develops singularities, we can approximate the quasi-momenta by their asymptotic form just as in (7.35), namely p i (x) ∼ ∆ i x/(2g(x 2 − 1)). To see the behavior in the vicinity of x = ±1 on the unit circle, we parametrize the Zhukowsky variable as x = e iθ near x = +1 and as x = −e −iθ near x = −1 and expand the expression for p i (x) above with respect to θ. In both cases, the result reads Plugging this expression into the dilogarithm, we obtain Since Li 2 (0) is finite, there will be no singularity when θ approaches zero from below (i.e. when the integration variable is on the lower semi-circle). On the other hand, when θ approaches zero from above, the argument of the dilogarithm diverges and we need to use its asymptotic expression to obtain Here the sign in front of πi depends on the choice of the branch of the logarithm. However, as we see below, the final result does not depend on the choice of this sign. As can be seen clearly, this expression contains a double pole and a single pole with respect to θ. Now if we combine all the terms contained in (7.30), we get which add up to the finite result −(π 2 )/2. This confirms the absence of the singularities in the full expression (7.30). Thirdly, as in the weak coupling, the integrals along the cuts can be re-expressed by pushing some of the contours onto the second sheet: Here the first term can be interpreted as the contribution from the SU(2) R whereas the second term can be regarded as coming from the SU(2) L . However, such factorization is not complete at strong coupling since the integrals along the unit circles cannot be rewritten in a similar manner.

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Relation with the hexagon form factor. Let us make a comment on the relation with the hexagon form factor approach. As given in [16], the result from the hexagon form factor consists of two parts: one is the asymptotic part, which is given by the sum over partitions of the physical rapidities, and the other is the wrapping correction, which is the contribution from the mirror particles. In [16], they showed in simple cases that the integration along the branch cuts arises from the asymptotic part whereas the integration along the unit circle contains the first leading wrapping correction. More recently, it was demonstrated in [42] that, by partially resumming the mirror particle contributions, one could get an integral of the dilogarithm along the unit circle and correctly reproduce a part of our results (7.30). It would be an very interesting future problem to try to resum all the hexagon form factors at strong coupling and reproduce our full result, which would account for various more complicated processes involving the mirror particles.
BPS limit and Frolov-Tseytlin limit. We now study several limits of the result (7.30) and perform the consistency checks. Let us first consider the three-point functions of the BMN vacuum. As the quasi-momentum for the BMN vacuum does not have any branch cuts, we only have integrals around the unit circle in that case. Furthermore, since the quasi-momenta in S 3 and AdS 3 coincide for the BMN vacuum, the two dynamical factors become identical, i.e. D S = D AdS , and cancel out in (7.30). Therefore we only have a contribution from the kinematical part in the final answer. This is consistent with the fact that the BPS three-point function does not receive quantum corrections.
Let us next study the Frolov-Tseytlin limit [26] by taking the charges r and l to be much larger than the coupling constant g while keeping the mode numbers of the cuts to be finite. In terms of the spectral curve, this amounts to pushing the branch cuts far away from the unit circle. More precisely, the branch cuts for p 1 and p 2 are pushed out into the region |x| 1 whereas the branch cuts of p 3 are confined to the region |x| 1. In such a limit, we can approximate the quasi-momenta on the unit circle by the quasi-momenta of the BMN vacuum. As explained above, for the BMN vacuum, integrals along the unit circle cancel out between S 3 and AdS 3 . Thus, in the Frolov-Tsyetlin limit, the integrals along the unit circle become negligible.
To study the remaining contributions, it is convenient to express the result (7.30) in terms ofp 3 defined byp As explained in appendix F,p can be interpreted as the quasi-momentum defined on a different sheet in the full eight-sheeted spectral curve and the relation (7.43) is nothing but the Z 4 automorphism of the string sigma model in AdS 5 × S 5 . It isp 3 that is connected to the quasi-momentum for the SU(2) L -sector at weak coupling. Now, to write down the expression in the Frolov-Tseytlin limit, we need to know the limiting forms of the quasi-momenta and the rapidity variable. In the region |x| x , (7.45) in the region |x| 1. As for the rapidity variable, it takes the following asymptotic form: Using these asymptotic forms and replacing the global charges Q i andQ i with the spinchain variables as given in (6.20), we can verify that (7.30) in the Frolov-Tseytlin limit coincides with the result at weak coupling (6.15). One can also study the next-leading order correction to the Frolov-Tseytlin limit and compare it with the results in [43]. In [43], based on the previous results at strong coupling [14], they concluded that the next-leading order correction to the Frolov-Tseytlin limit agrees with the one-loop structure constant at weak coupling except for integration contours. Since we now have contours 39 which coincide with the weak coupling ones in the Frolov-Tseytlin limit, the results match also at this order. For details, see section 6 of [43].
Type I-I-I three-point functions. Next we consider the Type I-I-I three-point functions. As in section 6.2, we consider the case where all the operators belong to SU(2) R . Also in this case, the result can be expressed as ln C 123 = K + D S − D AdS . (7.47) Here K and D AdS are given by the same expressions as before, namely (7.31), (7.32) and (7.34). On the other hand, D S for the Type I-I-I three-point function is given by with Li 2 e 2ip k . (7.49) The integration contours in (7.49) are depicted in figure 17. We can study the Frolov-Tseytlin limit also in this case and the result again matches with the result at weak coupling (6.24). Comparison with the result in [14]. Before ending this section, let us comment on the relation with the previous results for the three-point functions at strong coupling [14].

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In [14], we determined the analyticity of the Wronskians assuming that the saddle-point configuration of the worldsheet is smooth except at the positions of the vertex operators. The integration contours obtained under this assumption are more complicated than what we have found in this paper and the result in the Frolov-Tseytlin limit did not quite agree with the result at weak coupling. This implies that the assumption of smoothness is not quite correct and the saddle-point configuration has extra singularities. Although counterintuitive it may seem at first thought, such extra singularities are not so uncommon as already pointed out in [14]. For instance, consider the finite gap solution for the twopoint function whose spectral curve contains more than one cuts. Such a solution is given in terms of the ratio of the theta functions defined on the higher-genus Riemann surface. Although those ratios are free of singularities on the Lorentzian worldsheet, they have infinitely many poles 40 on the Euclidean worldsheet, which is more appropriate for studying the correlation functions. Such extra poles, if present, can affect the argument in [14] and change the integration contours. By contrast, the logic presented in this paper is based on the orthogonality of the on-shell states, which is the exact quantum property of the system, and therefore would be more universal and reliable.

Conclusion and prospects
In this paper, we studied the semi-classical three-point function in the SU(2)-sector of N = 4 super Yang-Mills theory in four dimensions at weak coupling. The key idea was to express it as a saddle-point value of the coherent-state path integral and utilize the classical integrability of the Landau-Lifshitz model. This revealed the nature of the semi-classical structure constant as a generating function of the angle variables. For the computation of the angle variables, many of the machineries developed for the strong coupling analysis could be transplanted, the most important among which are the expression of the angle variables in terms of the Wronskians and the functional equation for the Wronskians. To solve the functional equation, we developed a new logic to determine the analyticity, which 40 Such poles do not correspond to the insertion of vertex operators and do not affect the monodromy relation.

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is based on the orthogonality of two different on-shell states. The final results agree with the results in the literature and also make predictions for as-yet-unknown semi-classical structure constants for certain types of three-point functions.
We then re-examined the strong coupling analysis based on our new logic. It led to a modification of the integration contours of the result obtained in [14] and rendered the result in the Frolov-Tseytlin limit to be in agreement with the weak coupling one. In addition, the new result is consistent with the recent hexagon form factor approach [16].
As for the prospects, of paramount importance is to further explore the implication of the cognate integrability structure at weak and strong coupling, which we discussed in this paper. Given the importance of the monodromy relation at the tree level and at strong coupling, a natural next step is to study it at higher loops. This may lead to a first-principle derivation of the integrable structure for three-point functions at finite coupling. Another important structure worth mentioning in this regard is the striking similarity between our functional equations (5.7), which are the direct consequence of the monodromy relation, and the relations 41 constraining the lightcone string vertex in the pp-wave background [17]. It would be interesting to figure out the reason for this similarity. More generally, clarifying the integrable structure threading gauge and string theories would be a cornerstone for deeper understanding of the AdS/CFT correspondence. It may also yield practical merit if it leads to a new formulation of integrability for structure constants, which is more powerful than the existing approaches.
Apart from such challenging and far-reaching questions, there are numerous future directions that could be explored with the results and the techniques developed in this paper. Below we briefly address some of them.
Semi-classical limit of type I-I-I three-point functions. In this paper, we made predictions for the semi-classical limit of type I-I-I three-point functions at weak coupling (6.24). It would be an interesting problem to reproduce it from the exact quantum expression given in [24,41]. Since the result in [24,41] has a more complicated structure than the type I-I-II three-point function, we probably need to develop new tools for studying it.
Resummation of the hexagon form factor at strong coupling. Another interesting direction of research is to analyze the strong-coupling semiclassical limit using the hexagon form factor formalism. It was shown in [16,42] that a part of our result can be reproduced from the resummation of the hexagon form factor at strong coupling. It is important to further push this line of research and try to obtain the full strong coupling result from the hexagon form factor. This would be a litmus test for the hexagon form factor approach.
One-loop corrections at strong coupling. In the spectral problem, the power of the classical integrability and the associated spectral curve was not limited to the leading strong coupling limit. It also provided an efficient framework to study one-loop corrections around the classical solution [32,33]. The main idea there was to describe fluctuations as infinitesimal cuts inserted in the classical spectral curve. In this paper, we employed JHEP10(2016)042 a very similar idea to determine the analytic properties of the Wronskians. It would then be extremely interesting, by extending our argument, to try to include the one-loop corrections. As a first step, it may be simpler to first analyze the weak coupling limit since the next-leading correction in the semi-classical limit was already computed by other means [44].
Application to other quantities. It would also be interesting to apply the method discussed here to other quantities in N = 4 SYM. Of particular interest among them is the four-point function. The four-point function at the tree-level was studied in the paper [45] using integrability. However, even at that level, the resultant expression is rather involved owing to the complicated combinatorics of Wick contractions. In order to uncover a hidden structure, it might be helpful to study their semi-classical limit using our formalism. Such a structure is already known at strong coupling where it was shown that the four-point function of semi-classical operators can be described by the functional equation called χsystem [46]. It would be interesting to try to construct the weak-coupling counterpart of the χ-system. In addition, it might also be possible to use our framework to study non-planar observables such as the non-planar dilatation operator.
Entanglement entropy in integrable spin chains and field theories. Another interesting possibility is to apply the ideas and the techniques of this paper to the computation of entanglement entropy in general integrable spin chains and field theories. To compute the entanglement entropy, one must first construct a reduced density matrix. In the case of spin chains, this can be achieved by preparing two identical states, cutting them into two halves and gluing the left (or the right) halves. This procedure is similar to the tailoring method for the three-point function [11]. Thus, it may be possible to study the entanglement entropy of the semi-classical state, which contains a large number of long wave-length excitations, using the formalism developed in this paper. This would be of particular interest since the entanglement entropy for such a highly excited state is difficult to compute by other methods.
We hope to revisit some of these questions in the future.

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the monopole harmonics does not simply rotate into a linear combination of monopole harmonics. This is because, under the rotation, while the open sets with respect to which the monopole harmonics is defined get rotated into different regions, the gauge connection A µ (x) is not changed. Therefore in order to recover the same relative configuration of the open sets and the form of the connection one must make a suitable gauge transformation of A µ (x). This produces an extra U(1) phase factor of the form exp(iΦ(n, n )q), where Φ(n, n ) is the area of the triangle on the unit sphere the vertices of which are defined by n, n and the vector n 0 . It is clear from the preceding discussions that this phase, to be called the Wess-Zumino phase, is an essential ingredient in realizing the spin 1 2 representation in terms of the coherent states |n .
An important quantity in which this phase appears is the inner product of the coherent states. One can show by direct calculation that where tan Φ(n , n) 2 = (n × n) · n 0 1 + n 0 · n + n 0 · n + n · n . (A.5) More intuitive expression of the Wess-Zumino phase will also be given shortly. Before leaving this subsection, let us record two basic relations we will use. One is the (over)completeness relation which reads This can be readily verified by substituting the explicit form of |n given in (A.3) and performing the integration. One then obtains that the r.h.s. is indeed equal to | ↑ ↑ | + | ↓ ↓ | = 1. Another basic relation of use is n| S|n = 1 2 n, which can also be checked with ease.
Brief derivation of the Landau-Lifshitz model. Making use of the coherent state representation of the SU(2) spin 1/2 state explained above, we now briefly describe how the Landau-Lifshitz model arises from the Heisenberg spin chain in the semiclassical limit.
Let us denote by | n = |n 1 ⊗ · · · ⊗ |n L a coherent state of the spin chain and consider the transition amplitude n final |e −iHt | n initial from the initial state to the final state, through the Hamiltonian of the Heisenberg spin chain given (up to a convenient constant) by By the standard procedure, namely by performing the time evolution in infinitesimal steps with the insertions of the completeness relation (A.6) at each step, one obtains the coherent JHEP10(2016)042 state path-integral representation n final |e −iHt | n initial = D n(t)e iS , (A. 8) with the action S given by By taking the continuum limit of this expression, we obtain the well-known action of the Landau-Lifshitz model.
The first term on the r.h.s. represents the Wess-Zumino phase produced through the inner product as given in (A.4) and (A.5).
Just as in the Wess-Zumino-Novikov-Witten model, for example, such a Wess-Zumino term has a representation in terms of an integral one dimension higher (in this case as a three dimensional integral) of the form where s-dependence of n is defined such that n(s = 1) = (0, 0, 1) and n(s = 0) = n. The expression (A.11) has a rather intuitive meaning. Since n is a unit vector, ∂ t n and ∂ s n are perpendicular to n. Therefore the exterior product ∂ t n × ∂ s n is in the direction of n and n · (∂ t n × ∂ s n) dtds is nothing but the infinitesimal area element. Hence the integration gives the area and together with the factor of 1/2, which is the value of q = eg discussed in the previous subsection, we get the exponent of the Wess-Zumino phase factor.
B Poisson brackets and the r-matrix for the Landau-Lifshitz model As described in section 3.2, the classical r-matrix for the Landau-Lifshitz model can be obtained quickly as the classical limit of the well-known form of the quantum R-matrix of the Heisenberg spin chain. However, it would be of interest to supply the first principle derivation of the r-matrix from the computation of the Poisson brackets among the coherent state variables n i (σ, τ ). Below we give a sketch of such a derivation.
Poisson brackets. First we derive the Poisson (Dirac) bracket structure of the Landau-Lifshitz model. The most straightforward way is to start from the action (3.1), regard n as the fundamental variable and derive the Dirac brackets, taking into account the constraints n 2 = 1. However, in practice it turned out to be much easier to first parametrize the 2sphere by θ and φ and then compute the Dirac brackets. In terms of these angle variables, the action of the Landau-Lifshitz sigma model takes the form S = − dτ dσ 1 4 (cos θ∂ τ φ + φ sin θ∂ τ θ) + g 2 2 ∂ σ θ∂ σ θ + sin 2 θ∂ σ φ∂ σ φ . (B.1)

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In this expression, r(u) is the so-called classical r-matrix, which in this case is given by r(u) = P u . (B.10) The symbol P denotes the operator which permutes the two spaces in the tensor product: It is well-known 44 that when the Poisson bracket between the Lax matrices can be expressed in terms of the classical r-matrix as in (B.9), the Poisson bracket between the monodromy matrices can also be expressed by the classical r-matrix as C Highest weight condition on the semi-classical wave function Here we study constraints from the highest weight condition on the semi-classical wave function and show that the constant vector n appearing in the normalization condition (3.15) must be equal to the polarization vector. As explained in section 2.2, the states constructed on the rotated vacuum characterized by the polarization vector n = (n 1 , n 2 ) t satisfy the highest weight condition S + |Ψ = 0 , (C.1) with S + given in (2.22). To understand the consequence of this condition in the semiclassical limit, let us recall the form of the semi-classical wave function (in the action-angle basis), As explained in section 3.2, the angle variables φ k can be constructed from the poles of the factor n , ψ + , where n is a constant vector which defines the normalization condition. 45 Thus, in order to gurantee the highest weight property of the semi-classical wave function, we need to choose n such that n , ψ + is invariant under the transformation generated by S + . For this purpose, let us first go back to the Heisenberg spin chain. In the Heisenberg spin chain, the Lax operator is given by By the straightforward computation, one can show that it transforms under e aS + as e aS + L(u)e −aS + =AL(u)A −1 , (C.4) 44 A proof of (B.11) below can be found in page 106-107 of [50]. 45 Thus in literature this vector is usually referred to as the normalization vector.

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where the matrix A is given by Now, since the Landau-Lifshitz sigma model is obtained by taking the continuum limit of the Heisenberg spin chain, (C.4) implies the following transformation rule of the Lax matrix of the Landau-Lifshitz sigma model: This means that a solution ψ + to the auxiliary linear problem transforms as ψ + → Aψ + in order to compensate for the transformation (C.6). Thus the Wronskian n , ψ + gets transformed as n , ψ + → n , Aψ + = A −1 n , ψ + , (C.7) where the equality follows from the SL(2) invariance of the skew-symmetric product. It is then easy to see that the invariance under the transformation requires n = A −1 n and this leads to the identification n = n.

D Construction of the separated variables
In this appendix, we will describe how the separated variables are obtained for the Landau-Lifshitz model. These basic relaitions will be utilized in what follows.

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Separated variablesà la Sklyanin. Having displayed the explicit expression for the Poisson brackets, we now construct the separated canonically conjugate variables by the so-called Sklyanin's magic recipe [31]. In this method, such variables are obtained as associated to the poles of the normalized eigenvector h of the monodromy matrix, defined in the following way: 46 Ω(u)h(u) = e ip(u) h(u) , n , h = 1 .

(D.3)
Here n = (n 1 , n 2 ) t is the polarization vector. To simplify the construction it turns out to be convenient to first transform the monodromy matrixΩ(x) by a similarity transformation into the formΩ (x) ≡ n 2 n 1 −n 1 n 2 Ω(x) In what follows, we make use of these relations to derive the commutation relations between γ i 's and p(γ i )'s. We start from the analysis of {B(u) ,B(v)} = 0. SinceB has zeros at γ i and γ j (i = j), it can be expressed in the formB(u) = (u − γ i )B (u) orB(u) = (u − γ j )B (u). The functions B (u) and B (u) are not known but what is important is that they have the properties B (γ i ) = 0 and B (γ j ) = 0. Then the commutation relation betweenB(u) and B(v) can be rewritten as (D. 6) Now at this stage, we can safely take the limit u → γ i and v → γ j . Then the first three terms vanish the last term gives the relation 46 In Sklyanin's original formulation, the normalization condition is expressed in terms of the ordinary inner product as n · h = 1. Here we are instead using the skew-symmetric inner product in order to make connection with the Wronskian. It is equivalent to the original formulation under the identification of n with iσ2n. 47 To see this, it is helpful to consider the relation between the normalized eigenvector h and the unnormalized eigenvector ψ+. The normalized eigenvector can be constructed from the unnormalized eigenvector by h = ψ+/ n , ψ+ . Therefore the poles of the normalized eigenvector arise when the unnormalized eigenvector satisfy n , ψ+ = 0. Thus, at the poles of the normalized eigenvector, the vector parallel to n becomes the eigenvector of the monodromy matrix. Then, it is easy to see that (D.5) follows.

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Next consider the commutation relation betweenÃ(u) andB(v). Here again, we should substitute the expansionÃ(u) =Ã(γ i ) + (u − γ i )A (u) as well as the ones for B and B . Then similarly to the previous case, the limit u → γ i and v → γ j can be easily taken and, making use of the relation (D.7), we can deduce the important relation {Ã(γ i ) , γ j } =Ã(γ i )δ ij .

E Baker-Akhiezer vectors for the two-point functions
In the case of two-point functions, the explicit solutions can be constructed by the finite gap method [29]. For the general spectral curve with genus g, the solutions to the auxiliary linear problem evaluated at (τ, σ) = (0, 0) reads 48 whereσ is the holomorphic involution and the functions k − (u) and k + (u) are characterized uniquely by their divisors and the normalization at infinity: j , k − (∞ + ) = 1 .

(E.2)
Here γ i are the initial values of the separated variables parametrizing the moduli for twopoint functions, and γ i andγ i are the divisors satisfying 49 As noted in [29], the solutions (E.1) describe the highest weight eigenstate of S 3 . This means that the corresponding polarization vector is n = (1, 0) t . The solutions for more general rotated vacua can be obtained by the global rotation.

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The solutions (E.1) do not satisfy the normalization conditions ψ 0 + , ψ 0 − = 1. To normalize the solutions, we need to divide them by ψ 0 + , ψ 0 − as in (4.6). After the division, we obtain with C(u) given by Now, owing to (E.2), C(u) contains 2(g + 1) square-root zeros atγ i andσγ i . In addition, as argued in section 4.1, it must contain the square-root singularity at the positions of the branch points b k . Thus the divisor of C(u) is given by Combined with (E.2), this determines the divisor of the factor n , ψ + to be This shows that n , ψ + has spurious zeros and poles atγ j andσγ j unless we chooseγ j to be invariant under the holomorphic involution. For the genus 0 solutions including the ones corresponding to the BPS operators, we can confirm that it is always possible to chooseγ j to be invariant underσ by analyzing the explicit form of the solution. On the other hand, the situation for the higher genus solutions is less obvious since it is in general not clear if we can chooseγ j to be invarint underσ without violating the relation (E.3). However, when the cuts are sufficiently small, the solution would be very close to the BPS one, and, therefore from the continuity argument similar to the one used in section 5, we expect that it is possible to chooseγ j to be invariant under the involution (at least for some appropriate choices 50 of γ j .)

F Quasi-momentum in the full spectral curve
In this appendix, we shall clarify the relation (7.43).
For this purpose, consider the monodromy matrix for the full AdS 5 ×S 5 is a (4|4)×(4|4) matrix given by Ω AdS 5 ×S 5 (x) ∼ diag e ip 1 , e ip 2 , e ip 3 , e ip 4 |e ip 1 , e ip 2 , e ip 3 , e ip 4 . (F.1) Herep i andp i denote the quasi-momenta for the S 5 part and for the AdS 5 part respectively. 50 Different choices of γ j in the moduli of two-point functions only change the overall phase of the structure constant. Thus, for the computation of the three-point functions, we can choose a convenient one.