Three-point functions in the SU(2) sector at strong coupling
UUT-Komaba 13-16December, 2013
Three-point functions in the SU(2) sectorat strong coupling
Yoichi Kazama † and Shota Komatsu ‡ Institute of Physics, University of Tokyo,Komaba, Meguro-ku, Tokyo 153-8902 Japan
Abstract
Extending the methods developed in our previous works ( , ),we compute the three-point functions at strong coupling of the non-BPS states withlarge quantum numbers corresponding to the composite operators belonging to the so-called SU(2) sector in the N = 4 super-Yang-Mills theory in four dimensions. This isachieved by the semi-classical evaluation of the three-point functions in the dual stringtheory in the AdS × S spacetime, using the general one-cut finite gap solutions as theexternal states. In spite of the complexity of the contributions from various parts inthe intermediate stages, the final answer for the three-point function takes a remarkablysimple form, exhibiting the structure reminiscent of the one obtained at weak coupling.In particular, in the Frolov-Tseytlin limit the result is expressed in terms of markedlysimilar integrals, however with different contours of integration. We discuss a naturalmechanism for introducing additional singularities on the worldsheet without affectingthe infinite number of conserved charges, which can modify the contours of integration. † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] J a n ontents EAdS × S and classical solutions 14 EAdS × S spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 Sigma model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Pohlmeyer reduction for a string in S . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 One-cut finite gap solutions in S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Basic one-cut solution and “reconstruction” formula . . . . . . . . . . . . . . . . . 212.2.2 One-cut solutions from multi-cut solutions . . . . . . . . . . . . . . . . . . . . . . . 252.3 Some properties of the eigenvectors of the monodromy matrix . . . . . . . . . . . . . . . . 28 S part 30 S part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Symmetry structure of the vertex operators and the classical solutions . . . . . . . 474.3.2 Construction of the wave function for the right sector . . . . . . . . . . . . . . . . 504.3.3 Contribution of the left sector and complete wave function for the S part . . . . . 544.3.4 Correspondence with the gauge theory side . . . . . . . . . . . . . . . . . . . . . . 56 S part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.1.1 Contributions from the convolution integrals . . . . . . . . . . . . . . . . . . . . . 836.1.2 Contributions from the singular part of the Wronskians . . . . . . . . . . . . . . . 856.1.3 Result for the S part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2 The EAdS part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2.1 Contribution from the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2.2 Contribution from the wave function . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.3 Total contribution from the EAdS part . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Complete expression for the three-point function . . . . . . . . . . . . . . . . . . . . . . . 92 A.1 Parameters of one-cut solutions in terms of the position of the cut . . . . . . . . . . . . . 110A.2 Pohlmeyer reduction for one-cut solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.3 Computation of various integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
B Pohlmeyer reduction 114C Relation between the Pohlmeyer reduction and the sigma model formulation 116
C.1 Reconstruction formula for the Pohlmeyer reduction . . . . . . . . . . . . . . . . . . . . . 116C.2 Relation between the connections and the eigenvectors . . . . . . . . . . . . . . . . . . . . 117
D Details of the WKB expansion 119
D.1 Direct expansion of the solutions to the ALP . . . . . . . . . . . . . . . . . . . . . . . . . 119D.2 Born series expansion of the Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Introduction
After more than fifteen years since the advent of the AdS/CFT correspondence [1–3],we now have a large number of examples of this type of duality in various dimensions.In the majority of these examples, the bulk and the boundary theories share the same(super)conformal symmetry, showing the obvious importance of such a symmetry. Onthe other hand, the behaviors of the theories in each specific correspondence are actuallyrather different, especially in different dimensions. This evidently is due to differentdynamics and it in turn urges us to understand the common dynamical structure atthe root of the duality which is represented in different fashions in the bulk and on theboundary in various examples.With such a purpose in mind, in this article we shall study the three point functionsof certain semiclassical non-BPS states in the strong coupling regime in the context of theduality between the string theory in
AdS × S and the N = 4 super Yang-Mills theory infour dimensions. More specifically, we will deal with the string theory in the EAdS × S subspace, where EAdS stands for the Euclidean AdS . It should be dual to the so-calledSU(2) sector of the super Yang-Mills theory. This should certainly be of great interestin view of the fact that the results for the corresponding quantities at weak coupling haverecently become available [4–7]. Detailed comparison of the results in two regimes mayallow us to identify the common non-trivial structure beyond kinematics.As it will be evident, the computation of the three-point functions of non-BPS statesin string theory in a curved spacetime is quite non-trivial even at the leading semi-classicallevel. In the first of such attempts [8], the contribution from the AdS part was evaluatedfor the string in AdS × S k , where the string is assumed to be rotating only in S k . Sincethe contribution from the sphere part was not computed in [8], the complete answer forthe three-point function was not given. In this context, our present work can be regardedas (the extended version of) the completion of the work initiated by [8].At about the same time, computation of the three-point functions for different type ofheavy external states was attempted by the present authors [9]. We took as the externalstates the so-called Gubser-Klebanov-Polyakov (GKP) strings [10] spinning within AdS with large spins. In this work, the contribution to the three-point function from theaction evaluated on the saddle point configuration was computed by a method similarto the one in [8]. However, unlike the case of [8], the GKP string is not point-like onthe boundary, and hence the contributions from the non-trivial vertex operators were Actually the global symmetry of this sector is SO(4) = SU(2) × SU(2), as we will emphasize later.
EAdS × S . However, the applications of our general meth-ods developed in [9, 11] to the present case are not quite straightforward. One difficulty isthat the external states in the S sector, which are taken to be general one-cut finite gapsolutions [12–15] for the purpose of making comparison with the weak coupling result,are much more structured than the large spin limit of the GKP solutions. In particular,this makes the analysis of the analyticity property of the basic quantities on the spectralcurve considerably more complex. Another new ingredient concerns the logic of the de-termination of the internal wave functions for the S sector, which look rather differentfrom those for the AdS part. For the AdS part, the wave functions explicitly depend onthe positions on the boundary at which the external string states land and the form of thedependence is well-known from the conformal symmetry, namely the appropriate powerof the difference of these positions. For the S part, such landing positions do not existand one must reconsider how to determine the proper wave functions. We shall develop aunified method with which one can construct the wave function for a string in a generalspacetime. Moreover, our method cleanly factorizes the kinematical and the dynamicalcontributions to the wave functions. This feature is important both conceptually andpractically. Also, it should be mentioned that this is the first time where we have tocombine the contributions from the two different sectors, S and EAdS , which neverthe-less are interconnected through the Virasoro constraints. We shall see that when thesecontributions are put together, considerable simplifications occur, showing the intimateinterrelation between them, as expected.The end result of our rather involved computation is a remarkably simple formulafor the three-point function, which exhibits intriguing features. First, one recognizes theexpressions to be quite analogous to those that appear in the weak coupling result, evenbefore taking any special limits. A priori it is not obvious why the result in the strongcoupling limit should resemble the weak coupling answer so closely. This resemblancebecomes more conspicuous upon taking the so-called Frolov-Tseytlin limit, where theangular momentum J for the S rotation is quite large so that the ratio √ λ/J , where λ is the ’t Hooft coupling, is small. In this limit the integrands of the integrals expressing5he answer become almost identical. However, the integration contours do not quitematch. This is not immediately a contradiction since there is no rigorous argument whythree-point functions should agree exactly in that limit. Nevertheless, it is of interestto look for a possible mechanism to modify the contours. One important fact to benoted in this regard is that, in addition to the ordinary one-cut solutions we used for ourexternal states, there exist different types of one-cut solutions which can be obtained bytaking certain degeneration limits of multi-cut solutions. Since the values of the infinitenumber of conserved charges do not change in this limiting procedure, these solutionsshould be considered on equal footing with the corresponding ordinary one-cut solutions.The important difference, however, is that such a “degenerate solution” has one or moreadditional singularities on the worldsheet. Since the determination of the contours ofintegration depends crucially on the analytic structure of the saddle point configuration,this phenomenon provides an example of a natural mechanism by which the contourof integration in the formula for the three-point function can be modified. This issue,however, should be studied further in future investigations.Now as this article has become rather lengthy due to various steps of somewhat in-volved analyses, it should be helpful to give a brief preview of the basic procedures andexhibit the main result. The next subsection will be devoted to this purpose. The three-point function we wish to compute in the semi-classical approximation has thefollowing structure: G ( x , x , x ) = e − S [ X ∗ ] (cid:15) (cid:89) i =1 V i [ X ∗ ; x i , Q i ] (cid:15) . (1.1)It consists of the contribution of the action and that of the vertex operators, evaluatedon the saddle point configuration denoted by X ∗ . The subscript (cid:15) signifies a small cut-offwhich regulates the divergences contained in S and V i . As we shall show, these divergencescancel against each other and the total three-point function is completely finite. Thevertex operator V i [ X ∗ ; x i , Q i ] (cid:15) is assumed to carry a large charge Q i of order O ( √ λ ) andis located at x i on the boundary of the AdS space.In the case of a string in EAdS × S , the action and the vertex operators are splitinto the EAdS part and the S part. Their contributions are connected solely throughthe Virasoro constraint T ( z ) EAdS + T ( z ) S = 0 (and its anti-holomorphic counterpart).In the semi-classical approximation, an external state is characterized by the asymptotic6ehavior of a classical solution, which should be the saddle point configuration for itstwo-point function. However, conformally invariant vertex operator which creates sucha state is practically impossible to construct at present. Moreover, even if one had thevertex operator, it is of no use since the explicit saddle point solution X ∗ on which toevaluate the vertex operator (and the action) cannot be obtained by existing technology.Such difficulties, although seemingly insurmountable, can be overcome with the aid ofthe integrable and analytic structure of the system. For this purpose, it is convenient toformulate the string theory in question as a non-linear sigma model. Since the treatmentof the S part and the EAdS part are essentially the same in this regard, we shall focusprimarily on the S part in this summary. The basic information is then contained in theright-current j ≡ Y − d Y and the left-current l ≡ d YY − , where Y is the 2 × Y I ( I = 1 , , ,
4) of S inthe manner Y = (cid:18) Z Z − ¯ Z ¯ Z (cid:19) , Z = Y + iY , Z = Y + iY . (1.2) Y transforms under the global symmetry group SO(4)= SU(2) L × SU(2) R as Y → V L Y V R ,with V L ∈ SU(2) L , V R ∈ SU(2) R . The equation of motion for Y I can then be expressed inthe Lax form, with the complex spectral parameter x , as (cid:2) ∂ + J rz , ¯ ∂ + J r ¯ z (cid:3) = 0 , J = J rz dz + J r ¯ z d ¯ z , J rz = j z − x , J r ¯ z = j ¯ z x , (1.3)which makes the classical integrability of the system manifest. The information of the infi-nite number of conserved charges is encoded in the monodromy matrix Ω( x ) = P exp( − (cid:72) J ( x )),the eigenvalues of which are given by Ω( x ) ∼ diag( e ip ( x ) , e − ip ( x ) ), where p ( x ) is the quasi-momentum. One can then define the spectral curve Γ by det( y − Ω( x )) = 0, whichdescribes a two-sheeted Riemann surface in the variable x with a number of cuts withadditional singularities. To each such curve corresponds a classical “finite gap solu-tion” [12–15], which can be constructed in terms of the solutions of the so-called (rightand left) auxiliary linear problems, to be abbreviated as ALP throughout, given by( i ) (cid:18) ∂ + j z − x (cid:19) ψ = 0 , (cid:18) ¯ ∂ + j ¯ z x (cid:19) ψ = 0 , (1.4)( ii ) (cid:18) ∂ + xl z − x (cid:19) ˜ ψ = 0 , (cid:18) ¯ ∂ − xl ¯ z x (cid:19) ˜ ψ = 0 . (1.5)The solutions ψ and ˜ ψ are expressed in terms of the Riemann theta functions and theexponential functions, which depend on the data of the curve such as the location of thebranch points and other singularities. We will be interested in the “one-cut solution”,7he curve for which has a single square root branch cut of finite size , since the vertexoperators producing such solutions should correspond to the composite operators in theSU(2) sector in N = 4 super Yang-Mills theory.Now with this setup, let us sketch how one can compute the three point functions withthe above one-cut solutions as external legs. First consider the evaluation of the action part. As described in [9] for the GKP stringand will be detailed for the case of our interest, the action integral can be written in theform S ∼ (cid:82) ˜Σ (cid:36) ∧ η , where (cid:36) and η are, respectively a holomorphic 1-form and a closed1-form defined on the double cover ˜Σ of the worldsheet. By using the Stokes theorem, thiscan be rewritten as a contour integral S ∼ (cid:82) ∂ ˜Σ Π η , where ∂ ˜Σ is the boundary of ˜Σ andthe function Π = (cid:82) z (cid:36) is single-valued on ˜Σ. This expression for the action can be furtherrewritten, using a generalization of the Riemann bilinear identity developed in [9], into asum of products of certain contour integrals. The important point is that the contours ofthese integrals interconnect the vertex insertion points z , z , z , thereby correlating thebehaviors around these points. Therefore, to compute the integral it is natural to studythe behavior of the eigenfunctions of the ALP around z i and more importantly along thepaths connecting z i and z j .Although we do not know the exact saddle point solution for the three-point function,we do know the behavior in the vicinity of each z i since it should be the same as the one-cut solution discussed above. This provides the form of the currents needed to analyzethe ALP around z i . Clearly there are two independent solutions around each z i and onecan compute the local monodromy matrix Ω i belonging to SL(2,C), which mixes thesesolutions upon going around z i . Then one can take the basis of the solutions of ALPat z i to be the eigenvectors of Ω i , denoted by i ± , belonging to the eigenvalues e ± ip i ( x ) of Ω i . These eigenvectors are normalized with respect to the SL(2,C) invariant product (cid:104) ψ, χ (cid:105) ≡ det( ψ, χ ), to be refereed to as Wronskian throughout this article, as (cid:104) i + , i − (cid:105) = 1.To gain information about the solution of ALP valid in the entire worldsheet, one canmake the “WKB expansion” with ζ = (1 − x ) / (1+ x ) as the small parameter correspondingto (cid:126) . One then finds that the same contour integrals with which the action is expressedappear in the WKB expansion of the Wronskians (cid:104) i ± , j ± (cid:105) . Therefore our task is reduced As already mentioned in the introduction, this class can contain solutions which are obtained from m -cut solutions by shrinking m − When it is not confusing, we use one-cut solution to refer either to the one-cut solution of ALP orthe solution of the original equation of motion reconstructed in terms of such solutions. For the normalization of each eigenvector, see section 2.3.
8o their computation.The crucial information about such Wronskians is contained in the global consistencycondition of the monodromy matrices given byΩ Ω Ω = 1 . (1.6)Since Ω i ’s cannot in general be diagonalized simultaneously, this serves as a highly non-trivial constraint. In fact this condition allows one to express certain products of twoWronskians in terms of the local quasi-momenta p i ( x )’s, an example of which is given by (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) = sin p ( x )+ p ( x )+ p ( x )2 sin − p ( x ) − p ( x )+ p ( x )2 sin p ( x ) sin p ( x ) . (1.7)It turns out that the knowledge of the Wronskians, such as (cid:104) + , + (cid:105) , between the eigen-functions at different insertion points is of utmost importance. All the basic quantities,namely the contour integrals giving the contribution of the action and the wave functions,to be discussed shortly, can be expressed in terms of the Wronskians.Therefore the crucial task is to separate out, from the relations such as (1.7), theindividual Wronskian (cid:104) i ± , j ± (cid:105) . This can be achieved if we know which of the two factorsis responsible for each zero and the pole on the spectral curve, produced by the expressionon the right hand side. This information dictates the analyticity property of the individualWronskian in x and by solving the appropriate Riemann-Hilbert problem we can obtainthe Wronskians.As an example, consider the poles produced by the zeros of sin p ( x ) on the right handside of (1.7), namely at p ( x pole ) = nπ . These are the singular points of the spectral curvewhere the monodromy matrix Ω ( x pole ) takes the form of a Jordan block and the biggerof the two eigenvectors 1 ± diverges. This means that the Wronskian on the left handside of (1.7) involving such a “big solution” must be responsible for these poles. Nowwhich eigenvector is big and which is small near z i depends on the value of x . In the caseof the ordinary one-cut solution its explicit form tells us that it is dictated by the signof Re q i ( x ). This means that across the line Re q i ( x ) = 0, the analytic property of theeigenfunctions i ± changes. We can then extract the regular part of the Wronskian between“small solutions” by using the well-known technique of Wiener-Hopf decomposition, whichtakes the form of a convolution integral with the contour along the line Re q i ( x ) = 0. Dueto the two-sheeted nature of the spectral curve, the kernel of the decomposition formulamust be appropriately generalized.Now the remaining analysis, namely that of the zeros of the right hand side of (1.7), issimilar in spirit but is much more complicated because it involves the interplay betweenthe three local quasi-momenta p ( x ) , p ( x ) , p ( x ) and requires a certain knowledge of9he global properties of the solutions of the ALP on the spectral parameter plane. Toproperly deal with this problem, we will introduce a notion of the “exact WKB curve”.Also, since each p i ( x ) is double valued, the convolution kernel will be defined on an eight-sheeted Riemann surface. Moreover it turns out that the contour of integration must bedetermined not just by Re q i ( x ) = 0 for each i but also by certain global “connectivityconditions” expressed in terms of the quantity N i ≡ | Re p i ( x ) | . Despite such technicalcomplexities, we will be able to compute the desired Wronskians in terms of the quasi-momenta p i ( x ).With the procedures described above, one obtains the contribution from the actionfor the S part. Further, in an analogous manner, the corresponding contribution fromthe EAdS part can be computed. Let us now turn to the computation of the contribution of the vertex operators. To thisend, we extend the powerful method developed in our previous work [11] for the GKPstring to more general string. It is based on the state-operator correspondence and theconstruction of the corresponding wave function in terms of the action-angle variables. Ifone can construct the action-angle variables ( S i , φ i ), the wave function can be constructedsimply as Ψ[ φ ] = exp (cid:32) i (cid:88) i S i φ i − E ( { S i } ) τ (cid:33) , (1.8)where E ( { S i } ) is the worldsheet energy . Although the construction of such variablesfor a non-linear system is prohibitively hard in general, for integrable systems of thepresent type there exists a beautiful method [13–15], based on the Sklyanin’s separationof variables [16], which allows us to construct them from the Baker-Akhiezer eigenvec-tor ψ , which is the solution of ALP satisfying the monodromy equation of the formΩ( x ; τ, σ ) ψ ( x ; τ, σ ) = e ip ( x ) ψ ( x ; τ, σ ). More precisely, the dynamical information is en-coded in the function n · ψ ( x, τ ), where n = ( n , n ), to be specified later, is referred toas the “normalization vector”. It is known that for an m -cut solution n · ψ ( x, τ ) as afunction of x has m zeros at certain positions x = { γ , γ , . . . , γ m } and the dynamicalvariables z ( γ i ) and p ( γ i ), where z = √ λ ( x + x − ) / (4 π ), can be shown to form canonicalconjugate pairs. Then by making a suitable canonical transformation, one can constructthe action-angle variables ( S i , φ i ), where, in particular, the angle variables are given by The sum of such energies of course vanishes for the total system due to the Virasoro constraint. φ i = 2 π m (cid:88) j =1 (cid:90) γ j x ω i , i = 1 , , . . . , m . (1.9)Here, ω i are suitably normalized holomorphic differentials (with certain singularities de-pending on the specific problem) and x is an arbitrary base point. In the case of theone-cut solution of our interest, we have one angle variable φ R associated with the rightALP shown in (1.4) and one left angle variable φ L associated with the left ALP describedin (1.5). Hereafter we will only refer to the “right sector” for brevity of explanation.Now as we shall describe in section 2.2, we can write down a simple formula whichreconstructs the classical string solution from the Baker-Akhiezer vector ψ . Therefore,with a choice of the normalization vector n , one can associate the angle variable φ R ( n ) toa classical solution, through the (zeros of the) quantity n · ψ .Let Y denote the form of the three-point saddle solution near the vertex insertionpoint z i . We will call this part of the solution the i th leg. As we have to normalize thethree-point function by the two-point function for each leg, what we wish to computeis the angle variable φ R ( n ) associated to Y relative to the one φ ref R ( n ) associated to the“reference two-point solution” Y ref which is created by the same vertex operator at z i .Now the vertex operators of our interest are those which correspond to the gauge-invariant composite operators in the SU(2) sector of the super Yang-Mills theory. Aswe discuss in detail in section 4, the basic operators of that category are the charge-diagonal operators which are “highest weight” with respect to the global symmetry groupSU(2) R × SU(2) L (cid:39) SO(4). Focusing just on the SU(2) R property, one can characterizesuch an operator by what we call a “polarization spinor”, in this case n diag = (1 , t , whichis annihilated by the raising operator of SU(2) R . More general operator of our interest canthen be obtained from such a diagonal operator by an SU(2) R ( × SU(2) L ) rotation andis characterized by the polarization spinor n , obtained from n diag by the correspondingSU(2) R rotation. In this way, each vertex operator is associated with such a spinor n .What is important is that this “polarization spinor” n can be shown to be identicalto the “normalization vector” n which determines the angle variable φ R ( n ) through thequantity n · ψ . As was elaborated in our previous work [11], once the normalization vector n is specified, the relative shift ∆ φ R = φ R − φ ref R of the angle variable for the three-pointsolution Y around z i from that for the two-point reference solution Y ref can be computedfrom the knowledge of the transformation matrix V ∈ SL(2,C) which connects Y and Y ref Precisely speaking, we need the Baker-Akhiezer vectors for both the left and the right ALP, but weignore such a detail here. The same vertex operator can produce slightly different semi-classical behavior around it, dependingon whether it resides in the two-point function or in the three point function.
11n the manner Y = Y ref V in the vicinity of z i . As it will be shown in section 4, the allowedform of V can be deduced from the property that both Y and Y ref are produced from thesame vertex operator characterized by the polarization spinor n .Then, by using the master formula developed in [11], we can express ∆ φ R in terms of n ,the solutions of the Baker-Akhiezer functions corresponding to Y ref , and the parametersdescribing V . Applying this procedure to each leg of the three-point function and bymaking use of a relation between the normalization vector n and the value of ψ at x = ∞ ,we can express the wave function (for the right sector) in terms of the Wronskians asΨ S R = (cid:89) { i,j,k } (cid:18) (cid:104) j − , k − (cid:105)(cid:104) i − , j − (cid:105)(cid:104) k − , i − (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:104) n i , n j (cid:105)(cid:104) n k , n i (cid:105)(cid:104) n j , n k (cid:105) (cid:19) R i + R j − R k . (1.10)Here n i is the polarization spinor associated with the vertex operator V i at z i and R i isthe absolute value of the SU(2) R charge carried by V i . Note that the kinematical partexpressed in terms of (cid:104) n i , n j (cid:105) is clearly separated from the dynamical part, which againis composed of the Wronskians of the solutions of the ALP. The wave function for the EAdS part can be obtained in a similar fashion. In that case, the Wronskians (cid:104) n i , n j (cid:105) can be expressed in terms of the difference of the landing positions of the three legs onthe boundary of EAdS and yield the familiar coordinate dependence of the three-pointfunctions. We now have all the ingredients for the evaluation of three-point functions. Substitutingthe explicit expressions of the Wronskians (cid:104) i ± , j ± (cid:105) into the action and the wave functionand assembling the contributions from the S part and the EAdS part together, we findthat remarkable simplifications take place in the sum. The final result for the generalone-cut external states is thus found to be (cid:104)V V V (cid:105) = 1 N C | x − x | ∆ +∆ − ∆ | x − x | ∆ +∆ − ∆ | x − x | ∆ +∆ − ∆ × (cid:104) n , n (cid:105) R + R − R (cid:104) n , n (cid:105) R + R − R (cid:104) n , n (cid:105) R + R − R × (cid:104) ˜ n , ˜ n (cid:105) L + L − L (cid:104) ˜ n , ˜ n (cid:105) L + L − L (cid:104) ˜ n , ˜ n (cid:105) L + L − L , (1.11)where the prefactor 1 /N comes from the string coupling constant g s and the logarithm ofthe structure constant C is given by 12 n C = (cid:90) M uuu −−− z ( x ) ( dp + dp + dp )2 πi ln sin (cid:18) p + p + p (cid:19) + (cid:90) M uuu −− + z ( x ) ( dp + dp − dp )2 πi ln sin (cid:18) p + p − p (cid:19) + (cid:90) M uuu − + − z ( x ) ( dp − dp + dp )2 πi ln sin (cid:18) p − p + p (cid:19) + (cid:90) M uuu + −− z ( x ) ( − dp + dp + dp )2 πi ln sin (cid:18) − p + p + p (cid:19) − (cid:90) ˆ M uuu −−− z ( x ) ( d ˆ p + d ˆ p + d ˆ p )2 πi ln sin (cid:18) ˆ p + ˆ p + ˆ p (cid:19) − (cid:90) ˆ M uuu −− + z ( x ) ( d ˆ p + d ˆ p − d ˆ p )2 πi ln sin (cid:18) ˆ p + ˆ p − ˆ p (cid:19) − (cid:90) ˆ M uuu − + − z ( x ) ( d ˆ p − d ˆ p + d ˆ p )2 πi ln sin (cid:18) ˆ p − ˆ p + ˆ p (cid:19) − (cid:90) ˆ M uuu + −− z ( x ) ( − d ˆ p + d ˆ p + d ˆ p )2 πi ln sin (cid:18) − ˆ p + ˆ p + ˆ p (cid:19) − (cid:88) j =1 (cid:90) Γ uj − z ( x ) dp j πi ln sin p j + 2 (cid:88) j =1 (cid:90) ˆΓ uj − z ( x ) d ˆ p j πi ln sin ˆ p j + Contact . (1.12) The notations used in the above expressions are as follows. In the equation (1.11), ∆ i isthe conformal dimension of the i -th vertex operator V i and n i and ˜ n i are the polarizationspinors for V i with respect to SU(2) R and SU(2) L . In the expression for ln C , p i and ˆ p i are the quasi-momenta for the i -th leg for the S part and the EAdS part respectively. z ( x ) is the Zhukovsky variable given in (2.36). The symbols M uuu ±±± and Γ uj − denote thecontours of integration for the S part and ˆ M uuu ±±± and ˆΓ uj − are the contours for the EAdS contribution. The last term Contact stands for some special terms which depend on thedetail of the external states. It should be noted that the result above for the three-point function for the operators corresponding to general one-cut solutions is alreadyreminiscent of the expression in the weak coupling regime. In section 7, we demonstratethat our formula gives the correct result for the case of three BPS operators and that itreduces to the two-point function in the limit when the charge of one of the operatorsbecomes negligibly small. Further, we analyze the Frolov-Tseytlin limit for the case ofone non-BPS and two BPS operators and find that the integrals giving the three-pointcoupling take extremely similar forms, except for different contours of integration. Forthis issue, we point out the existence of a natural mechanism by which the contours canbe modified.Now we briefly indicate the organization of the rest of this article: In section 2, webegin with the description of the string in
EAdS × S spacetime and discuss the one-cutsolutions we will consider in this work. In section 3, we will study the contribution ofthe action for the S part to the three-point function and show that the action can bere-expressed in terms of certain contour integrals. In section 4, we describe the evaluationof the wave functions for the S part. Characterizing the vertex operator by a polarizationspinor and identifying it with the normalization vector determining the angle variable, weapply the master formula for the shift of the angle variables developed in our previouswork to construct the wave functions. Section 5 will be devoted to the explicit evaluationof the Wronskians. The main task is to find the analyticity property of the Wronskian13rom the improved WKB analysis of the ALP. Using this information, we can projectout the individual Wronskian from the expression of the product of Wronskians in termsof the quasi-momenta p i ( x ) by the use of the Wiener-Hopf decomposition. In section 6,all the results obtained up to this point are put together to produce the final result forthe three-point functions of the general one-cut external states. In section 7, in additionto some basic checks of our result, we present the analysis of the Frolov-Tseytlin limitand discuss its outcome. Finally, in section 8 we make some important comments on ourpresent work and indicate possible future directions. Several appendices are provided tosupply some additional details. EAdS × S and classical solutions We begin by setting up the formalism to deal with the strings in
EAdS × S in subsection2.1 and describe the classical solutions we will use as the external states of the three-pointfunctions in subsection 2.2. We then give a brief account on the basic set-up of the three-point function in subsection 2.3. EAdS × S spacetime In this article, we will exclusively deal with the string propagating in the product spaceof the Euclidean
AdS subspace of AdS (to be denoted by EAdS ) and the sphere S .If we describe the AdS in terms of the embedding coordinates by X M η MN X N = − M is taken to run as M = − , , , , , η MN = diag ( − , − , , , , EAdS subspace is defined by setting X = X = 0. Therefore we will parametrize the EAdS × S space in the following way : EAdS : X µ X µ ≡ X µ η µν X ν = − , µ, ν = − , , , , (2.1) η µν = diag ( − , , , ,S : Y I Y I ≡ Y I δ IJ Y J = 1 , I, J = 1 , , , . (2.2)The Poincar´e coordinates ( x r , z ) = ( x , x , x , x , z ) of AdS are defined in the usual wayas X − + X = 1 z , X − − X = z + x r x r z , X r = x r z , (2.3) In order to conform to the standard convention, we have chosen the range of the S embeddingcoordinates Y I to be I = 1 , , ,
4, some of which coincide in name to the range of the
EAdS coordinates.We believe this will not cause confusion. z = 0 corresponds to the boundary of AdS . When restricted to the EAdS subspace , its boundary is the Euclidean plane parametrized by ( x , x ).The action of a string in this space is given by S = √ λπ (cid:90) d z (cid:16) ∂X µ ¯ ∂X µ + Λ( X µ X µ + 1) + ∂Y I ¯ ∂Y I + ˜Λ( Y I Y I − (cid:17) , (2.4)where Λ and ˜Λ are Lagrange multiplier fields. Upon eliminating them the equations ofmotion become ∂ ¯ ∂X µ − ( ∂X ν ¯ ∂X ν ) X µ = 0 , (2.5) ∂ ¯ ∂Y I + ( ∂Y J ¯ ∂Y J ) Y I = 0 . (2.6)For physical configurations, we must in addition impose the Virasoro constraints, whichrequire that the sum of the stress-energy tensors for the AdS part and the sphere partmust vanish. Namely, T AdS ( z ) + T S ( z ) = 0 , ¯ T AdS (¯ z ) + ¯ T S (¯ z ) = 0 , (2.7) T AdS ( z ) = ∂X µ ∂X µ , T S ( z ) = ∂Y I ∂Y I , (2.8)¯ T AdS (¯ z ) = ¯ ∂X µ ¯ ∂X µ , ¯ T S (¯ z ) = ¯ ∂Y I ¯ ∂Y I . (2.9)For the AdS part, we shall take the external states to be those without the two-dimensionalspins. Then near the vertex insertion point the saddle point solution should approach thetwo-point solution, which is known to be point-like. The forms of T AdS ( z ) and ¯ T AdS (¯ z ) forsuch a two-point solution are uniquely determined by their transformation properties asa (2 ,
0) and a (0 ,
2) tensor respectively and are given in terms of the conformal dimension∆ of the vertex operator as T AdS, ( z ) = κ z , ¯ T AdS, = κ ¯ z , κ = ∆2 √ λ . (2.10)Therefore, taking into account the Virasoro condition, near each vertex insertion point z i we must have T AdS ( z ) ∼ κ i ( z − z i ) , T S ( z ) ∼ − κ i ( z − z i ) as z → z i , (2.11)and similarly for the anti-holomorphic parts. In the case of three-point functions, theinformation of such asymptotic behaviors suffices to determine the form of the energy-momentum tensor exactly everywhere. For the EAdS , the holomorphic part takes theform T ( z ) = (cid:18) κ z z z − z + κ z z z − z + κ z z z − z (cid:19) z − z )( z − z )( z − z ) , (2.12) z ij ≡ z i − z j . (2.13) If desired, one can also deal with the
AdS subspace given by X = X = 0, with the Minkowskiboundary plane ( x , x ). AdS for the stress tensor for the AdS partand simply write T ( z ) and ¯ T (¯ z ) for T AdS ( z ) and ¯ T AdS (¯ z ).We now discuss the methods for constructing the solutions of the equations of motionwith the use of the classical integrability of the system. There exist two apparently differ-ent formalisms. One is the sigma model formulation [12, 17] and the other is the so-calledPohlmeyer reduction [18,19]. The former deals with variables which transform covariantlyunder the global symmetry transformations, whereas the latter employs invariant vari-ables. Because of this feature they have advantages and disadvantages depending on theproblem one would like to solve. We shall employ both. It should be remarked howeverthat they are actually connected by a “gauge transformation”, as shown in Appendix C.2. Consider first the sigma model formulation. We will focus on the S part, as the EAdS part can be treated similarly. The embedding coordinates { Y I } are conveniently assembledinto a 2 × Y = (cid:18) Z Z − ¯ Z ¯ Z (cid:19) , (2.14) Z = Y + iY , Z = Y + iY , (2.15)which transforms under the global symmetry group SO (4) = SU(2) L × SU(2) R as Y (cid:48) = U L Y U R , U R ∈ SU(2) R , U L ∈ SU(2) L . (2.16)The quantities of central importance are the “right” and the “left” currents (or connec-tions) j and l respectively, defined by j ≡ Y − d Y , l ≡ d YY − . (2.17)Evidently, j and l are related by l = Y j Y − . Under the transformation (2.16) theytransform covariantly as j → U − R jU R and l → U L lU − L . Now reflecting the classicalintegrability of the system these equations can be extended to one parameter family ofequations called Lax equations given by[ ∂ + J rz ( x ) , ¯ ∂ + J r ¯ z ( x )] = 0 , J rz ( x ) ≡ j z − x , J r ¯ z ( x ) ≡ j ¯ z x , (2.18)[ ∂ + J ll ( x ) , ¯ ∂ + J l ¯ z ( x )] = 0 , J lz ( x ) ≡ xl z − x , J l ¯ z ( x ) ≡ − xl ¯ z x , (2.19)16here x is the complex spectral parameter. The two connections J r = J rz dz + J r ¯ z d ¯ z and J l = J lz dz + J l ¯ z d ¯ z are related by the gauge transformation of the form Y ( d + J r ) Y − = d + J l . It is useful to note that the energy-momentum tensors and hence the Virasoroconditions can be expressed in terms of the currents in a concise way. We have, in thecylinder coordinate, T S ( z ) = −
12 Tr ( j z j z ) = − κ , ¯ T S (¯ z ) = −
12 Tr ( j ¯ z j ¯ z ) = − κ . (2.20)Central to the construction and the analysis of the solutions of the equations of motionare the right and the left auxiliary linear problems, to be abbreviated as ALP, which arecoupled linear differential equations for vector functions:right ALP : ( ∂ + J rz ( x )) ψ = 0 , ( ¯ ∂ + J r ¯ z ( x )) ψ = 0 , (2.21)left ALP : ( ∂ + J lz ( x )) ˜ ψ = 0 , ( ¯ ∂ + J l ¯ z ( x )) ˜ ψ = 0 . (2.22)Compatibility of the system of ALP implies the original equations of motion. Upondeveloping ψ and ˜ ψ from a point z along a closed spacelike curve, we obtain the rightand the left monodromy matrices Ω( x ) and (cid:101) Ω( x ) respectively asΩ( x ; z ) = P exp (cid:18) − (cid:73) J r (cid:19) = P exp (cid:18) − (cid:73) (cid:18) j z dz − x + j ¯ z d ¯ z x (cid:19)(cid:19) , (2.23) (cid:101) Ω( x ; z ) = P exp (cid:18) − (cid:73) J l (cid:19) = P exp (cid:18) − (cid:73) (cid:18) xl z dz − x − xl ¯ z d ¯ z x (cid:19)(cid:19) = Y Ω( x ; z ) Y − . (2.24)By virtue of the flatness of the connection, expansion of Ω( x ) as a function of x aroundany point yields an infinite number of conserved charges as coefficients. In particular,expansions around x = ∞ and x = 0 yield, in the leading behavior, the Noether chargesfor the global SU(2) R and SU(2) L , respectively, defined by Q R ≡ √ λ π (cid:73) ∗ j = √ λ π (cid:90) π dσj τ , (2.25) Q L ≡ √ λ π (cid:73) ∗ l = √ λ π (cid:90) π dσl τ . (2.26)Indeed, expanding Ω( x ; z ) around x = ∞ and x = 0 and using the definitions above, weget Ω( x ; z ) = 1 − x π √ λ Q R + O ( x − ) , ( x → ∞ ) , (2.27) Y Ω( x ; z ) Y − = 1 + x π √ λ Q L + O ( x ) , ( x → . (2.28)By diagonalizing Ω( x ; z ), we can obtain a quantity independent of z . Since det Ω( x ; z ) =1, its eigenvalues must be of the structure u ( x ; z )Ω( x ; z ) u ( x ; z ) − = (cid:18) e ip ( x ) e − ip ( x ) (cid:19) , (2.29)17here p ( x ) is called the quasi-momentum. Comparing with the diagonalized form of theexpressions (2.27) and (2.28), the behaviors of p ( x ) around x = ∞ and x = 0 are of theform p ( x ) − p ( ∞ ) = − x π √ λ R + O ( x − ) , ( x → ∞ ) , (2.30) p ( x ) − p ( ∞ ) = 2 πm + x π √ λ L + O ( x ) , ( x → , (2.31)where m is an integer and the right and the left charges R and L are the (positive)eigenvalues of Q R and Q L respectively.For the study of the ALP and construction of the finite gap solutions of our interest,the analytic property of the quasi-momentum is of critical importance. Such a structureis encoded in the spectral curve defined byΓ : Γ( x, y ) = det( y − Ω( x ; z )) = 0 , (2.32)which is equivalent to ( y − e ip ( x ) )( y − e − ip ( x ) ) = 0. In the present case, it can be regardedas a two-sheeted Riemann surface with various singularities. From the definition of themonodromy matrix (2.23) and (2.24) and the constraints (2.20), it is clear that p ( x ) haspoles at x = ± − πκ . Since p ( x ) lives on atwo-sheeted surface, we specify its branch by defining the signs at these singularities. Weshall employ the definition p ( x ) ∼ − πκx − O (( x − ) , ( x → + ) , (2.33) p ( x ) ∼ − πκx + 1 + O (( x + 1) ) , ( x → − + ) . (2.34)where the + superscript on 1 + signifies that the point is on the first sheet. Similarly,we shall use − superscript for points on the second sheet. We will give a more detaileddiscussion of the structure of p ( x ) for the one-cut solutions of our interest in subsection2.2.From the structure of the spectral curve and the quasi-momentum p ( x ) defined uponit, one can extract important information. For this purpose, we first define the a - and b -cycles in the usual way. For the hyperelliptic curve of our interest, an a -cycle is definedas a cycle which goes around the cut on the same sheet. On the other hand, a b -cycle isdefined as the one which starts from a point on the first sheet, goes into the second sheetthrough the cut and eventually comes back to the same point on th first sheet. Clearly,around an a -cycle, we have (cid:72) a i dp = 0. In contrast, the integral along the b -cycle does notvanish in general and gives (cid:72) b i dp = 2 πn i , where n i is an integer called the mode number.Now using the a -type cycles, one can define a set of conserved charges called the filling ractions as S i ≡ i π (cid:73) a i pdz (cid:18) = (cid:73) a i zdp πi (cid:19) , (2.35)where z ≡ √ λ π (cid:18) x + 1 x (cid:19) (2.36)is the Zhukovsky variable. In particular, the filling fractions S ∞ and S defined withthe contours a ∞ and a , which encircle the point at ∞ and 0 respectively, are of specialimportance since they are related to the global SU(2) R and SU(2) L charges in the followingway, as can be checked using (2.30) and (2.31): S ∞ = − R , S = L . (2.37) S The sigma model formulation we have sketched above is convenient for analyzing theproperty of the system under the global symmetry transformations. Hence it will be usedas the basis of the construction of the wave function corresponding to the vertex operatorsin section 4. On the other hand, for the analysis of the contribution of the action, whichis invariant under the global transformation, the formalism of the
Pohlmeyer reduction will be more convenient.The essential idea of the Pohlmeyer reduction is to describe the motion of the stringin a suitably defined moving frame. This then leads to the Lax equations in terms of theconnections which are invariant under the global symmetry transformations. Below weshall only sketch the procedures and then summarize the basic equations we will needlater. Further details will be given in Appendix B.In what follows we shall denote a 4-component field A I simply as A and use thenotations A · B = A I B I , A = A I A I . The basic moving frame of 4-component fields, tobe called q i , ( i = 1 , , , q ≡ Y, q ≡ a∂Y + b ¯ ∂Y, q ≡ c∂Y + d ¯ ∂Y and q ≡ N , where N is the unit vector orthogonal to Y, ∂Y and ¯ ∂Y , and the (field-dependent)coefficients a, b, c, d are chosen so that the simple conditions q · q = − , q = q = 0 aresatisfied. (Note that since Y = 1, we automatically have q = 1 , q · q = q · q = 0.) Letus define an SO(4)-invariant field γ by the relation ∂Y · ¯ ∂Y = (cid:112) T ¯ T cos 2 γ . (2.38)Then, the coefficients a, b, c, d can be expressed in terms of T, ¯ T and γ , giving q and q
19f the form q = − i sin 2 γ (cid:20) e iγ √ T ∂Y + e − iγ √ ¯ T ¯ ∂Y (cid:21) , (2.39) q = i sin 2 γ (cid:20) e iγ √ ¯ T ¯ ∂Y + e − iγ √ T ∂Y (cid:21) . (2.40)Once the moving frame is prepared, one can compute the derivatives of q i and expressthem in terms of q i again. The result can be assembled into the following equations ∂W + B Lz W + W B Rz = 0 , ¯ ∂W + B L ¯ z W + W B R ¯ z = 0 , (2.41)where W is given by W = 12 (cid:18) q + iq q q q − iq (cid:19) , (2.42)and B L,Rz, ¯ z are matrices whose components are expressed in terms of T, ¯ T and γ . (Explicitforms are given in Appendix B.) From the equations (2.41) one deduces that the left andthe right connections B L and B R , given in (B.21)–(B.24), are flat, namely[ ∂ + B Lz , ¯ ∂ + B L ¯ z ] = 0 , [ ∂ + B Rz , ¯ ∂ + B R ¯ z ] = 0 . (2.43)These relations give the equations of motion for the invariant fields in the form ∂ ¯ ∂γ + √ T ¯ T γ + 2 ρ ˜ ρ √ T ¯ T γ = 0 ,∂ ˜ ρ + 2 ¯ ∂γ sin 2 γ ρ = 0 , ¯ ∂ρ + 2 ∂γ sin 2 γ ˜ ρ = 0 , (2.44)where ρ and ˜ ρ are defined by ρ ≡ N · ∂ Y , ˜ ρ ≡ N · ¯ ∂ Y . (2.45)Just as in the case of the sigma model formulation, the integrability of the systemallows one to introduce a spectral parameter ζ , related to x by ζ = 1 − x x , (2.46)without spoiling the flatness conditions. The Lax equation so obtained is given by[ ∂ + B z ( ζ ) , ¯ ∂ + B ¯ z ( ζ )] = 0 , (2.47)where B z ( ζ ) ≡ Φ z ζ + A z , B ¯ z ( ζ ) ≡ ζ Φ ¯ z + A ¯ z , Φ z ≡ (cid:32) − √ T e − iγ − √ T e iγ (cid:33) , Φ ¯ z ≡ (cid:32) √ ¯ T e iγ √ ¯ T e − iγ (cid:33) ,A z ≡ (cid:32) − i∂γ ρe iγ √ T sin 2 γρe − iγ √ T sin 2 γ i∂γ (cid:33) , A ¯ z ≡ (cid:32) i ¯ ∂γ ρe − iγ √ ¯ T sin 2 γ ˜ ρe iγ √ ¯ T sin 2 γ − i ¯ ∂γ (cid:33) . (2.48)20ne can consider the auxiliary linear problem also for the Pohlmeyer connections(2.47), ( ∂ + B z ( ζ )) ˆ ψ = 0 , (cid:0) ¯ ∂ + B ¯ z ( ζ ) (cid:1) ˆ ψ = 0 , (2.49)where ˆ ψ denotes the solution in this formulation. As shown in Appendix C.2, thePohlmeyer connections (2.47) are actually related to the connections in the sigma modelformulation, (2.21) and (2.22), by gauge transformations. Correspondingly, the solutionsto the ALP are also related by gauge transformations as ψ = G − ˆ ψ , ˜ ψ = ˜ G − ˆ ψ , (2.50)where ψ and ˜ ψ are the solutions to the right and the left ALP respectively and G and ˜ G are the gauge transformations, the explicit form of which are given in Appendix C.2. Hereand hereafter, we shall often refer to the use of the Pohlmeyer formulation as choosingthe Pohlmeyer gauge . S We now describe a particular class of solutions to the equations of motion and theVirasoro constraints, which can be constructed by the so-called finite gap integrationmethod [13–15]. These solutions describe the local behaviors of the saddle point solutionfor the three-point function in the vicinity of the vertex insertion point. The class of ourinterest is characterized by the associated spectral curve having one square-root branchcut of finte size and will be referred to as a one-cut solution . We will first consider the“basic” one-cut solutions, which are customarily referred to as genus 0 solutions, andstudy their properties in detail. Then, we describe another class of one-cut solutionswhich are obtained from multi-cut solutions by certain degeneration procedure. We showthat they contain additional singularities on the worldsheet, which may play an importantrole when we compare the three point functions at strong and weak couplings in section7.
A powerful method for constructing a large class of classical solutions in the sigma modelformulation is the so-called finite gap integration method. (For a comprehensive review,see [15].) The method consists of two steps. As the first step, the solutions to the leftand the right ALP, called the Baker-Akhiezer functions, are constructed by treating theproblems as Riemann-Hilbert problems on a finite genus Riemann surface. Namely, byproving that the function satisfying all the required analytic properties is unique, one21onstructs such a function in terms of the Riemann theta functions and the exponentialfunctions. Then, as the second step, one develops the “reconstruction” formula , whichconstructs the solutions to the original equations of motion from the knowledge of theBaker-Akhiezer functions. In this subsection, we will describe the simplest class of solu-tions corresponding to the case of genus zero Riemann surface, or a two-sheeted surfacewith one square-root branch cut. Such solutions will be referred to as the basic one-cutsolutions.Consider first the right ALP given in (1.4) and let ψ ± ( x, z, ¯ z ) be the Baker-Akhiezervector which are at the same time the eigenvectors of the monodromy matrix Ω( x ) corre-sponding to the eigenvalues e ± ip ( x ) respectively. According to the general theory of finitegap integration, ψ ± corresponding to the one-cut solution are given by simple exponentialfunctions as ψ + ( x ; τ, σ ) = (cid:18) c +1 exp (cid:0) iσ π (cid:82) x ∞ + dp + τ π (cid:82) x ∞ + dq (cid:1) c +2 exp (cid:0) iσ π (cid:82) x ∞ − dp + τ π (cid:82) x ∞ − dq (cid:1) (cid:19) , (2.51) ψ − ( x ; τ, σ ) = ψ + (ˆ σx ; τ, σ ) . (2.52)where c + i are constants, ˆ σx denotes the point x on the opposite sheet, and ∞ + ( ∞ − ) isthe point at infinity on the first (resp. second) sheet. The quantity dp is the differentialof the quasi-momentum p ( x ), while dq is the differential of the quasi-energy q ( x ). Justlike p ( x ), the quasi-energy q ( x ) is defined by the pole behavior at x = ± + of the form q ( x ) ∼ − πκx − O (( x − ) , ( x → + ) , (2.53) q ( x ) ∼ +2 πκx + 1 + O (( x + 1) ) , ( x → − + ) . (2.54)The structure and the signs of the residue at x = ± q ( x ) are determined so that theholomorphicity of the solution (2.51) at x (cid:39) ± x = 1 the holomorphic part of the ALP is dominating and hence the Baker-Akhiezervector should be holomorphic. This is in fact realized since p ( x ) = q ( x ) near x = 1 andhence the exponent of ψ ± is a function of the combination z = τ + iσ . In the same way,at x = − ψ ± becomes anti-holomorphic as desired.Now for the left ALP, the Baker-Akhiezer eigenvectors, denoted by ˜ ψ ± ( x, z, ¯ z ), aregiven by ˜ ψ + ( x ; τ, σ ) = (cid:18) c − exp (cid:0) iσ π (cid:82) x + dp + τ π (cid:82) x + dq (cid:1) c − exp (cid:0) iσ π (cid:82) x − dp + τ π (cid:82) x − dq (cid:1) (cid:19) , (2.55)˜ ψ − ( x ; τ, σ ) = ˜ ψ + (ˆ σx ; τ, σ ) , (2.56) Although it is usually referred to as the “reconstruction” formula, in practice it is used as a solution-generating formula. This is because the Baker-Akhiezer functions are constructed not by solving ALPwith specific known connections but by more generic methods. u and its complexconjugate ¯ u on the spectral curve. Such a cut is described by a factor of the form y ( x ) ≡ (cid:112) ( x − u )( x − ¯ u ) . (2.57)We define the branch of y ( x ) to be such that the sign of y ( x ) is +1 at x = 1 + . Then p ( x )and q ( x ) satisfying the prescribed analyticity properties are fixed to be p ( x ) = − πκy ( x ) (cid:18) | − u | x − (cid:15) | u | x + 1 (cid:19) , (2.58) q ( x ) = − πκy ( x ) (cid:18) | − u | x − − (cid:15) | u | x + 1 (cid:19) , (2.59) (cid:15) = (cid:26) +1 for | Re u | > − | Re u | < . (2.60)Here we fixed p ( x ) and q ( x ) such that they vanish at the branch points although theanalyticity properties only determine the differential dp and dq . This choice is suitablefor the purpose of this paper since the solutions to the ALP in the Pohlmeyer gauge. Theforms of p ( x ) and q ( x ) depend on whether the cut is placed to the right or to the leftof x = 1. Substituting these forms into the formulas for ψ ± and ˜ ψ ± we get the one-cutsolutions for the ALP.Let us now describe the second step, the (re)construction of the solutions of the equa-tions of motion from the Baker-Akhiezer vectors. Although this has been discussed inthe literature [13–15], we present below a more transparent formula. Let us form a 2 × ψ ± satisfyingthe right ALP as Ψ = ( ψ + ψ − ) and consider the quantity˜Ψ ≡ Y Ψ . (2.61)Then, by using the definitions l z = ∂ YY − and j z = Y − ∂ Y , we can easily show that (cid:18) ∂ + xl z − x (cid:19) ˜Ψ = Y (cid:18) ∂ + j z − x (cid:19) Ψ = 0 , (2.62) (cid:18) ¯ ∂ − xl ¯ z x (cid:19) ˜Ψ = Y (cid:18) ∂ + j ¯ z x (cid:19) Ψ = 0 . (2.63)If we express ˜Ψ in terms of two column vectors ˜ ψ ± as ˜Ψ = ( ˜ ψ + ˜ ψ − ), the above equationsshow that ˜ ψ ± are actually two independent solutions to the left ALP. This means thatthere exist solutions ψ ± and ˜ ψ ± to the right and the left ALP respectively so that Y canbe expressed as Y = ˜ΨΨ − . (2.64)23his general relation by itself, however, is not useful since even if we provide a solutionΨ explicitly, finding ˜Ψ which satisfies (2.64) tantamounts to finding Y itself. Now theformula (2.64) turns into a genuine reconstruction formula when we consider the specialvalues of the spectral parameter x . If we set x = 0, it is evident from the form ofALP redisplayed above in (2.62) and (2.63) that the left ALP equations for ˜Ψ reduce to ∂ ˜Ψ = ¯ ∂ ˜Ψ = 0, and hence ˜Ψ( x = 0) becomes a constant matrix. Therefore the solution Y isreconstructed from the right ALP solution Ψ as Y ( z, ¯ z ) = ˜Ψ( x = 0)Ψ − ( z, ¯ z ; x = 0), wherethe constant matrix ˜Ψ( x = 0) represents the freedom of making a global transformationfrom left. Similarly, by setting x = ∞ , we can make the right ALP equations trivial,namely ∂ Ψ = ¯ ∂ Ψ = 0. Then Ψ( x = ∞ ) becomes a constant matrix and Y can bereconstructed from the left ALP solution ˜Ψ as Y ( z, ¯ z ) = ˜Ψ( z, ¯ z ; x = ∞ )Ψ − ( x = ∞ ).Summarizing, we have two types of simple reconstruction formulas Y ( z, ¯ z ) = ˜Ψ(0)Ψ − ( z, ¯ z ; 0) , (2.65) Y ( z, ¯ z ) = ˜Ψ( z, ¯ z ; ∞ )Ψ − ( ∞ ) . (2.66)By using the reconstruction formula given above, one can write down the general basicone-cut solution explicitly. It can be written in the form [12, 15] Y = (cid:18) cos θ e ν τ + im σ sin θ e ν τ + im σ − sin θ e − ν τ − im σ cos θ e − ν τ − im σ (cid:19) , (2.67)where the parameters ν i , m i and θ must satisfy the following conditions expressing theequations of motion and the Virasoro conditions: ν − m = ν − m , (2.68)4 κ = ( ν + m ) cos θ ν + m ) sin θ , (2.69) ν m cos θ ν m sin θ . (2.70)Applying the reconstruction formula (2.65) with the constant matrix ˜Ψ(0) taken to bethe identity matrix and using the form of ψ + given in (2.51), we easily find that theparameters m i and ν i can be expressed in terms of p ( x ) and q ( x ) as m = 12 π (cid:90) ∞ + + dp , ν = 12 π (cid:90) ∞ + + dq , (2.71) m = 12 π (cid:90) ∞ − + dp , ν = 12 π (cid:90) ∞ − + dq . (2.72)The right and the left Noether charges R and L can be computed directly from the solution242.67) and are given in terms of the parameters ν i , m i and θ in a universal manner as R √ λ = 12 (cid:18) − ν cos θ ν sin θ (cid:19) , (2.73) L √ λ = 12 (cid:18) − ν cos θ − ν sin θ (cid:19) . (2.74)Explicit expressions of R and L in terms of the position of the cut are given in AppendixA.1. As a result, we find that the charges R and L are positive irrespective of the positionof the cut. This means that they should be regarded not as the charges themselves but astheir absolute magnitudes . On the other hand, the relative magnitude of R and L dependson the position of the cut as R < L for | Re u | > , (2.75) R > L for | Re u | < . (2.76)In section 4.3.4, we will see that the difference in the relative magnitude corresponds tothe difference of the class of vertex operators for which the solution is the saddle point ofthe two-point function. We now discuss a more general type of “one-cut” solutions, namely the ones with addi-tional cuts of infinitesimal size besides a cut of finite size. As we shall discuss in section7.5.2, this type of solutions may play an important role in the comparison of the three-point functions at strong and weak couplings. Besides such specific reason, as theseinfinitesimal cuts do not contribute to any of the (infinite number of) conserved charges,they should, on general grounds, be considered on an equal footing with the correspond-ing solutions carrying the same charges. As a matter of fact, it is much more natural toconsider solutions with infinite number of infinitesimal cuts, as they correspond to theinfinite number of angle variables which must exist for a string theory even when theirconjugate action variables have vanishing values . Now adding an infinitesimal cut tothe genus g Riemann surface is equivalent to shrinking a cut in the genus g + 1 surface .As we shall see, depending on the choice of the parameters we either get back an ordinarygenus g finite gap solution or we obtain a new solution with additional singularities.In contrast to the one-cut solution corresponding to genus zero we have been consid-ering, for a genus g finite gap solution with g ≥ As already emphasized in [11], in order to construct a three-point solution in the framework of thefinite gap method, which is tailored for construction of two-point solutions, inclusion of infinite numberof small infinitesimal cuts is necessary as one has to produce an additional singularity corresponding tothe third vertex operator. A similar discussion of this process can be found in [20]. z ) in addition to the exponential part: ψ = h + ( x ) Θ( A ( x ) + kσ − iωτ − ζ γ − (0) )Θ( A ( ∞ + ) − ζ γ − (0) )Θ( A ( x ) − ζ γ − (0) )Θ( A ( ∞ + ) + kσ − iωτ − ζ γ − (0) ) exp (cid:18) iσ π (cid:90) x ∞ + dp + τ π (cid:90) x ∞ + dq (cid:19) , (2.77) ψ = h − ( x ) Θ( A ( x ) + kσ − iωτ − ζ γ + (0) )Θ( A ( ∞ − ) − ζ γ + (0) )Θ( A ( x ) − ζ γ + (0) )Θ( A ( ∞ − ) + kσ − iωτ − ζ γ + (0) ) exp (cid:18) iσ π (cid:90) x ∞ − dp + τ π (cid:90) x ∞ − dq (cid:19) . (2.78)As it is not our purpose here to review the details of the finite gap construction, below wewill only explain the minimum of the ingredients and refer the reader to a review articlesuch as [15]. Also, for simplicity and clarity, we will focus on the case of the degenerationfrom g = 1 to g = 0. This suffices to explain the essence of the construction and thegeneralization to the case of higher genus is straightforward.For a g = 1 two-cut solution, the Riemann theta function Θ( z ) reduces to the elliptictheta function θ ( z ) defined by θ ( z ) ≡ (cid:88) m ∈ Z exp (cid:0) imz + πi Π m (cid:1) , (2.79)where Π is the period given by the integral of the holomorphic differential w over the b -cycle of the torus Π = (cid:73) b w . (2.80)As usual, w is normalized by the integral over the a -cycle as (cid:72) a w = 1. A ( x ) appearing inthe argument of the Θ-functions is the Abel map defined by A ( x ) = 2 π (cid:90) x ∞ + w . (2.81) h ± ( x ) are normalization constants and k and ω are the “momentum” and the “energy”defined by the integrals k ≡ π (cid:73) b dp , ω ≡ π (cid:73) b dq . (2.82)A quantity of importance is the constant ζ γ ± (0) defined by ζ γ ± (0) ≡ A ( γ ± (0)) + K , (2.83)In this formula, K is the “vector of Riemann constants”, which for a torus is simply anumber proportional to the period Π as K = π Π . (2.84) For its definition for a general genus g surface, see for example [21]. γ ± (0) are certain points on the Riemann surface, which determine the initialconditions for the solution.Let us now study what happens when we pinch the a -cycle. In order to keep the nor-malization condition (cid:72) a w = 1 intact, w must behave near the position of the infinitesimalcut x c as w ∼ (cid:26) πi x − x c for x on the first sheet − πi x − x c for x on the second sheet . (2.85)This means that the imaginary part of the period Π defined by the integral over the b -cycle approaches positive infinity in the mannerΠ = (cid:73) b w ∼ πi (cid:90) x c + (cid:15)x c − (cid:15) dxx − x c ∼ − iπ ln (cid:15) → + i ∞ . (2.86)Now writing the θ -function as θ ( z ) = (cid:88) m ∈ Z exp (cid:0) imz + πi (Re Π) m (cid:1) · exp (cid:0) − π Im Π m (cid:1) , (2.87)we see that the last factor vanishes as Im Π → ∞ , except for m = 0. Therefore in thislimit we get θ ( z ) → z = kσ − iωτ in the formulas for ψ i given in (2.77) and (2.78), thearguments of the θ -functions containing z are actually of the form z − a , with a constantshift a given by a = ζ γ ± (0) + · · · . What is important is that ζ γ ± (0) diverges as we pinchthe a -cycle. First, obviously Im K diverges as π Im Π. Second, if γ ± (0) is at the positionof the shrunk cut x c , Im A ( γ ± (0)) diverges just like π Im Π: A ( γ ± (0)) = 2 π (cid:90) x c + (cid:15) ∞ + dw ∼ π πi ln (cid:15) ∼ iπ Im Π → i ∞ . (2.88)Since A ( γ ± (0)) is finite otherwise, we must distinguish two cases: case (a) Im ζ γ ± (0) ∼ π Im Π for γ ± (0) = x c and case (b) Im ζ γ ± (0) ∼ π Im Π for γ ± (0) (cid:54) = x c . Therefore let uswrite a = l Im π Π + c , where l = 2 or l = 1 and c is a finite constant. Then the θ -functionwith this shift can be written as θ ( z − a ) = (cid:88) m ∈ Z exp (cid:0) im ( z − π Re Π − c ) + πi (Re Π) m (cid:1) · exp (cid:0) − π Im Π( m − lm ) (cid:1) . (2.89)First consider the case (a). It is easy to see that terms with negative m all vanish inthe limit Im Π → ∞ . On the other hand, the terms with m = 0 and m = 2 are finite Precisely speaking, γ ± (0) are certain divisors γ ± ( t ) depending on the infinite set of higher times t = ( t , t , t , . . . ) evaluated at t = 0. For a detailed definition, see [15]. m ≥ m = 1diverges. In other words, θ ( z − a ) → ˆ Ce iz , ˆ C → ∞ . (2.90)As the θ -functions occur in pairs in the numerator and the denominator in ψ i , their ratiogoes to a z -independent finite constant in the degeneration limit and we get back theusual g = 0 one-cut solution. In fact, by repeating this type of process, one can producea finite gap solution from an infinite gap solution, which must be the generic situation fortheories with infinite degrees of freedom, such as string theory.Next consider the case (b). For l = 1, two terms in the series survive in the limitIm Π → ∞ , namely m = 0 and m = 1. Therefore we obtain a non-trivial function of theform θ ( z − a ) → Ce iz = 1 + Ce ikσ + ωτ , (2.91)where C is a constant. In particular, this function can vanish at certain points, thenumber of which depend on the magnitude of k . Such a θ -function in the denominatorof the expressions for ψ i gives rise to additional simple poles on the worldsheet. Indistinction to the singularity due to a vertex operator, these singularities do not carryany charges (including infinitely many higher charges) because the solution is obtainedwithout changing the form of p ( x ).Although we will not explicitly make use of the degenerate multi-cut solutions dis-cussed above in the bulk of our investigation, they will be recognized in section 7.5.2 asproviding an example of a concrete mechanism by which extra singularities can be nat-urally produced. Existence of such singularities can modify the contours of the integralsthat express the three-point coupling and may play an important role in the interpretationof our final result. As described in the preceding subsections, the solutions of the ALP play the central rolein the construction of the two-point solutions to the equations of motion. Now for theconstruction of the three-point functions, to be discussed starting from the next section,what will be of vital importance are the special linear combinations of the solutions ofALP, namely the eigenvectors of the local monodromy matrix Ω i , defined around eachvertex insertion point z i . We will denote such eigenvectors and eigenvalues as i ± and e ± ip i ( x ) , which satisfy the relations Ω i i ± = e ± p i ( x ) i ± . (2.92)28n what follows, we will describe some important properties of i ± and related states.Of crucial importance in the computation of three-point functions will be the SL(2,C)invariant product for i ± and j ± given by (cid:104) i ± , j ± (cid:105) ≡ det ( i ± , j ± ) . (2.93)In the rest of the paper, we shall refer to this skew-product as Wronskian . Since theWronskians are invariant under gauge transformations, we can use the results in variousgauges interchangeably. For example, from the relation (2.50) between the eigenvectorsin the sigma model formulation and the Pohlmeyer formulation, we have the equalities (cid:104) i ± , j ± (cid:105) = (cid:104) ˜ i ± , ˜ j ± (cid:105) = (cid:104) ˆ i ± , ˆ j ± (cid:105) . (2.94)For later convenience, let us fix the normalization of the eigenvectors i ± . We will firstimpose the usual condition (cid:104) i + , i − (cid:105) = 1 . (2.95)This, however, does not fully fix the normalization of the individual eigenfunctions, as wecan rescale i ± as i + → ai + and i − → a − i − , without violating the condition (2.95). Todetermine the normalization completely, we will make use of the asymptotic behavior of i ± around the puncture z i . For this purpose, it is convenient to employ the Pohlmeyer gauge,as it is invariant under the global symmetry transformation. Now although the explicitform of the solution for the three-point function is not known, it can be approximated bythe solution for the two-point function in the vicinity of the vertex operators. Therefore,we can determine the normalization of ˆ i ± by demanding that they coincide with thecorresponding two-point functions at the insertion point of the vertex operator:ˆ i ± ( x ; τ ( i ) , σ ( i ) ) −→ ˆ i ± ( x ; τ = τ ( i ) , σ = σ ( i ) ) . (2.96)In this formula, ( τ ( i ) , σ ( i ) ) are the local cylinder coordinates around z i , defined by τ ( i ) + iσ ( i ) = ln (cid:18) z − z i (cid:15) i (cid:19) . (2.97)Here we have chosen the origin of τ ( i ) to be such that τ ( i ) = 0 on the small circle | z − z i | = (cid:15) i ,which will serve to separate the contributions from the action and the wave function insubsequent sections. Using the results of Appendix A, the eigenvectors for the two-pointfunction ˆ i ± can be computed asˆ i ( x ; τ, σ ) = e πi/ √ (cid:16) x − ¯ u i x − u i (cid:17) / (cid:16) ¯ u i − u i − (cid:17) / e πi/ √ (cid:16) x − u i x − ¯ u i (cid:17) / (cid:16) u i − u i − (cid:17) / exp (cid:18) q i ( x ) τ + ip i ( x ) σ π (cid:19) , (2.98)ˆ i − ( x ; τ, σ ) = e − πi/ √ (cid:16) x − ¯ u i x − u i (cid:17) / (cid:16) ¯ u i − u i − (cid:17) / − e − πi/ √ (cid:16) x − u i x − ¯ u i (cid:17) / (cid:16) u i − u i − (cid:17) / exp (cid:18) − ( q i ( x ) τ + ip i ( x ) σ )2 π (cid:19) , (2.99)29here u i and ¯ u i are the positions of the branch points of the quasi-momentum p i ( x ) forthe i -th puncture. The conditions (2.96), (2.98) and (2.99) determine the normalizationof i ± completely. The important property of the eigenvectors so normalized is that theytransform in the following way when they cross the branch cut :ˆ i + ( x ) (cid:12)(cid:12)(cid:12) on 2nd sheet = ˆ i − ( x ) (cid:12)(cid:12)(cid:12) on 1st sheet , ˆ i − ( x ) (cid:12)(cid:12)(cid:12) on 2nd sheet = − ˆ i + ( x ) (cid:12)(cid:12)(cid:12) on 1st sheet . (2.100)This relation will be used in section 5.5 to determine the normalization of certain Wron-skians. S part Let us now start our study of the three-point functions. In what follows, we will denotetheir structure as (cid:104)V V V (cid:105) = exp ( F S + F EAdS ) , (3.1)where F S = F action + F vertex , (3.2) F EAdS = ˆ F action + ˆ F vertex . (3.3)In this section, we focus on the contribution of the action for the S part, namely F action .First, in subsection 3.1, we rewrite the action as a boundary contour integral using theStokes theorem and then apply the generalized Riemann bilinear identity derived in [9]to bring it to a more convenient form. Next we turn in subsection 3.2 to the analysisof the WKB expansion of the auxiliary linear problem. We then find that the samecontour integrals we used to rewrite the action appear also in the WKB expansion of theWronskians of the solutions to the ALP. Using this relation, we re-express the action interms of the Wronskians in subsection 3.3. The resultant expression will be used for theexplicit evaluation of the contribution of the action in section 6. For the three-point function of our interest, the (regularized) action for the S part of thestring is given by S S = √ λπ (cid:90) Σ \{ (cid:15) i } d z∂Y I ¯ ∂Y I , (3.4) Note that the extra minus sign is necessary in the second equation of (2.100) in order to retain thecondition (2.95). \{ (cid:15) i } denotes the worldsheet for the three-point function, which isa two-sphere with a small disk of radius (cid:15) i cut out at each vertex operator insertionpoint z i . Such a point will often be referred to as a puncture also. In [8] and [22],such worldsheet cut-offs are related to the spacetime cut-off in AdS in order to obtainthe spacetime dependence of the correlation functions without introducing the vertexoperators. In contrast, as we shall separately take into account the contribution of thevertex operators, (cid:15) i ’s can be taken to be arbitrary in our approach, as long as they aresufficiently small and the same for the S part and the EAdS part.As the action is invariant under the global symmetry transformations, it is natural toexpress (3.4) in terms of the quantities used in the Pohlmeyer reduction. From (2.38), wecan indeed write S S = √ λπ (cid:90) Σ \{ (cid:15) i } d z (cid:112) T ¯ T cos 2 γ . (3.5)We further rewrite (3.5) by introducing the following one-forms: (cid:36) ≡ √ T dz , (3.6) η ≡ − (cid:112) ¯ T cos 2 γd ¯ z + 2 √ T (cid:18) − ( ∂γ ) + ρ T (cid:19) dz . (3.7)The second term on the right hand side of (3.7) is added to make η closed, as one canverify using the relation (2.44). With these one-forms, we can re-express the action (3.5)as a wedge product of the form S S = i √ λ π (cid:90) Σ \{ (cid:15) i } (cid:36) ∧ η , (3.8)where an extra prefactor i/ dz ∧ d ¯ z = − i d z . Then denoting the integral of (cid:36) ( z ) asΠ( z ) = (cid:90) zz (cid:36) ( z (cid:48) ) dz (cid:48) , (3.9)the action can be rewritten, using the Stokes theorem, as a contour integral along aboundary ∂ ˜Σ of a certain region ˜Σ (see figure 3.1): S S = i √ λ π (cid:90) ˜Σ (cid:36) ∧ η = i √ λ π (cid:90) ˜Σ d (Π η ) = i √ λ π (cid:90) ∂ ˜Σ Π η . (3.10)To determine the proper region of integration ˜Σ, we need to know the analytic structureof Π( z ), which in turn is dictated by that of T ( z ). As already explained in section 2, in31igure 3.1: The coutour ∂ ˜Σ which forms the boundary of the double cover of the world-sheet ˜Σ, dipicted as the union of the first and the second sheet. There are three logarithmicbranch cuts attached to the puncture in the middle, in addition to one square-root branchcut, shown as a wavy segment, through which the two sheets are connected.the case of three-point functions the information of the asymptotic behavior of T ( z ) ateach puncture z i is sufficiently restrictive to determine T ( z ) exactly to be of the form T ( z ) = (cid:18) κ z z z − z + κ z z z − z + κ z z z − z (cid:19) z − z )( z − z )( z − z ) , (3.11) z ij ≡ z i − z j . From this, one can show that Π( z ) has three logarithmic branch cuts running from thepunctures z i , and one square-root branch cut connecting two zeros of T ( z ), to be denotedby t and t . Therefore, we should take ˜Σ to be the double cover ( y = T ( z )) of theworldsheet Σ with an appropriate boundary ∂ ˜Σ, so that Π( z ) is single-valued on thewhole integration region. In what follows, we will consider the case where the branch cutis located between z and z as depicted in figure 3.2. In such a case, the branch of thesquare-root of T ( z ) can be chosen so that it behaves near the punctures on the first sheetas (cid:112) T ( z ) ∼ κ i z − z i as z → z i ( i = 1 , , ∼ − κ z − z as z → z . (3.12)Although the discussion to follow is tailored for this particular case, the final result for thethree-point function, to be obtained in section 6, will turn out to be completely symmetricunder the permutation of the punctures.At this point, we shall apply the generalized Riemann bilinear identity, derived in [9],to the integral (3.10). As the derivation is lengthy, we refer the reader to [9] for details32nd just present the result . It can be written as (cid:90) ˜Σ (cid:36) ∧ η = Local + Double + Global + Extra , (3.13)where the definition of each term will be given successively below . The first term, Local ,denotes the contribution from the product of contour integrals, each of which is justaround the puncture and hence called “local”. It is of the form
Local = (cid:88) i (cid:73) C i (cid:36) (cid:73) C i η + (cid:88) i Double = − (cid:88) i (cid:73) C i η (cid:90) zz ∗ i (cid:36) . (3.15)The third term, Global , denotes the contribution from the product of contour integrals,one of which is along a contour connecting two different punctures. It is given by Global = (cid:18)(cid:73) C + C ¯2 −C (cid:36) (cid:90) (cid:96) η + (cid:73) C ¯2 + C −C (cid:36) (cid:90) (cid:96) η + (cid:73) C + C −C ¯2 (cid:36) (cid:90) (cid:96) ¯31 η (cid:19) − ( (cid:36) ↔ η ) . (3.16)More precisely, (cid:96) ij denotes the contour connecting z ∗ i and z ∗ j , where z ∗ i is the point nearthe puncture z i satisfying z ∗ i − z i = (cid:15) i . The barred indices indicate the points on thesecond sheet of the double cover y = T ( z ). For instance, C ¯ i is a contour encircling thepoint z ¯ i , which is on the second sheet right below z i . Finally, the term Extra denotesadditional terms which come from the integrals around the zeros of √ T , to be denotedby t k , at which η becomes singular, and is given by Extra = (cid:88) k (cid:73) D k Π η . (3.17)Here D k is the contour which encircles t k twice as depicted in figure 3.2.Among these four terms, Local and Double are expressed solely in terms of the inte-grals around the punctures and are easy to compute. The explicit results, computed in By decomposing the contours (cid:96) ij ’s in (3.13) into d and (cid:96) i ’s defined in [9], we arrive at the formuladerived in [9]. In [8] and [22], the ordinary Riemann bilinear identity was applied to derive an expression similar to(3.13) but without the terms Local and Double . In their cases, Local and Double vanish and the use ofthe ordinary Riemann bilinear identity is justified. On the other hand, these two terms do not vanish inour case and we must use the generalized Riemann bilinear identity. (cid:73) C i (cid:36) = 2 πiκ i , (cid:73) C i η = 2 πiκ i Λ i , (3.18) (cid:73) C i η (cid:90) zz ∗ i (cid:36) = − πκ i Λ i , for i = 1 , ¯2 , . (3.19)Here Λ i ’s are given in terms of γ i and ρ i , defined in (A.21) and (A.22) respectively, asΛ i = cos 2 γ i + 2 ρ i κ i . (3.20)It is important to note that Local and Double are real since κ i and g i are all real.Therefore they contribute exclusively to the imaginary part of the action (3.10) and henceonly yield an overall phase of the three-point functions. We shall neglect such quantitiesin this paper.Figure 3.2: Definitions of the contours used to rewrite the action: The contours whichenclose the punctures ( C i ) are shown in the left figure and the ones which connect twopunctures ( (cid:96) ij ) are shown in the right figure. In both figures, the portions of the contourson the second sheet are drawn as dashed lines. Also depicted in the right figure are thestarting points and the end points of the contours, z ∗ i ’s.Among the remaining two types of terms, Extra can be explicitly evaluated as follows.Since the worldsheet is assumed to be smooth except at the punctures, the quantity √ T ¯ T cos 2 γ , which is the integrand of the action integral given in (3.5), should not vanisheven at the zeros of T ( z ). This in turn implies that γ is logarithmically divergent at suchpoints in the manner γ ∼ ± i | z − t k | as z → t k . (3.21) The one-forms (cid:36) and η flip the sign under the exchange of two sheets. Therefore (3.18) is oddwhereas (3.19) is even under such sheet-exchange. In (3.19), κ i for i = ¯2 is set to be equal to κ . T ( z ) as T ( z ) ∼ c ( z − t k ) around t k , we can write down theleading singular behavior of η around t k as η ∼ − √ T ( ∂γ ) d ¯ z ∼ d ¯ z √ c ( z − t k ) / . (3.22)Thus the integral along D k can be computed as (cid:73) D k Π η = (cid:73) D k √ c ( z − t k ) / d ¯ z √ c ( z − t k ) / = − πi . (3.23)Since there exist two zeros, Extra is twice this integral and hence is given by Extra = − πi . (3.24)For later convenience, we shall derive another expression for the action using a differentset of one-forms given by¯ (cid:36) = (cid:112) ¯ T d ¯ z , (3.25)˜ η = −√ T cos 2 γdz + 2 √ T (cid:18) − ( ¯ ∂γ ) + ρ ¯ T (cid:19) d ¯ z , (3.26)and then consider the average of the two expressions. Using the forms above, the actioncan be written as S S = − i √ λ π (cid:90) ˜Σ ¯ (cid:36) ∧ ˜ η . (3.27)As compared to (3.10), the expression (3.27) has an extra minus sign, which is due tothe property dz ∧ d ¯ z = − d ¯ z ∧ dz . Applying the generalized Riemann bilinear identity to(3.27), we get − (cid:90) ˜Σ ¯ (cid:36) ∧ ˜ η = − (cid:0) Local + Double + Global + Extra (cid:1) , (3.28)where Local , Double and Global are given respectively by (3.14), (3.15) and (3.16) with (cid:36) and η replaced by ¯ (cid:36) and ˜ η . The integrals of ¯ (cid:36) and ˜ η around the punctures are given by (cid:73) C i ¯ (cid:36) = − πiκ i , (cid:73) C i ˜ η = − πiκ i ¯Λ i , (3.29) (cid:73) C i ˜ η (cid:90) zz ∗ i ¯ (cid:36) = − πκ i ¯Λ i , for i = 1 , (cid:98) , i ’s are given in terms of γ i and ˜ ρ i , defined in Appendix A.2, as¯Λ i = cos 2 γ i + 2 ˜ ρ i κ i . (3.31) (3.29) is odd and (3.30) is even under the exchange of the first and the second sheets, as in the caseof the integrals of (cid:36) and η given in (3.18) and (3.19). Local and Double are real and they contribute only to the overall phase. On theother hand, Extra can be evaluated just like Extra and yields + πi/ 3. Thus, by aver-aging over the two expressions (3.13) and (3.28) and neglecting terms which contributeexclusively to the overall phase, we arrive at the following more symmetric expression:12 (cid:18)(cid:90) ˜Σ (cid:36) ∧ η − (cid:90) ˜Σ ¯ (cid:36) ∧ ˜ η (cid:19) = − πi (cid:0) Global − Global (cid:1) . (3.32)The quantity (3.32) consists of various integrals along the contours C i and (cid:96) ij . Amongthem, the ones along C i can be easily computed using (3.18) and (3.29). The integral of (cid:36) along (cid:96) ij can also be computed in principle as we know the explicit form of (cid:36) . Thusthe major nontrivial task is the evaluation of (cid:82) (cid:96) ij η and (cid:82) (cid:96) ij ˜ η . In the rest of this section,we will see how these integrals are related to the Wronskians of the form (cid:104) i ± , j ± (cid:105) , where i ± are the Baker-Akhiezer eigenvectors at z i of the ALP, corresponding to the eigenvalues e ± ip i ( x ) . We now perform the WKB expansion of the auxiliary linear problem and observe thatthe contour integrals of our interest, (cid:82) (cid:96) ij η and (cid:82) (cid:96) ij ˜ η , appear in the expansion of theWronskians between the eigenvectors of the monodromy matrices.Let us first consider the WKB expansion of the solutions to the ALP. For this purpose,it is convenient to use the ALP of the Pohlmeyer reduction (2.49). The use of (2.49) hastwo main virtues. First, as Φ’s are given explicitly in terms of T ( z ) and ¯ T (¯ z ), it is easierto perform the expansion around ζ = 0 or around ζ = ∞ . Second, since the connection(2.47) is expressed solely in terms of the quantities invariant under the global symmetrytransformation, we can directly explore the dynamical aspect of the problem setting asideall the kinematical information.We shall first perform the expansion around ζ = 0. To facilitate this task, it isconvenient to perform a further gauge transformation and convert (2.49) to the “diagonalgauge”, where the ALP take the form (cid:18) ∂ + 1 ζ Φ dz + A dz (cid:19) ˆ ψ d = 0 , (cid:0) ¯ ∂ + ζ Φ d ¯ z + A d ¯ z (cid:1) ˆ ψ d = 0 . (3.33)In the above, ˆ ψ d in the diagonal gauge is defined byˆ ψ d ≡ √ (cid:18) e iγ/ − e − iγ/ e iγ/ e − iγ/ (cid:19) ˆ ψ , (3.34)36nd Φ d ’s and A d ’s are given byΦ dz = √ T (cid:18) − (cid:19) , Φ d ¯ z = √ ¯ T (cid:18) − cos 2 γ i sin 2 γ − i sin 2 γ cos 2 γ (cid:19) ,A dz = (cid:32) − ρ √ T cot 2 γ iρ √ T − i∂γ − iρ √ T − i∂γ ρ √ T cot 2 γ (cid:33) , A d ¯ z = − ˜ ρ √ ¯ T sin 2 γ (cid:18) − (cid:19) . (3.35)Note that the leading terms in the ALP equations as ζ → 0, namely Φ dz for the firstequation and A d ¯ z for the second, have been diagonalized. Because of this feature, theleading exponential behavior of the two linearly independent solutions around ζ ∼ ψ d ∼ (cid:18) (cid:19) exp (cid:20) ζ (cid:90) zz (cid:36) (cid:21) , ˆ ψ d ∼ (cid:18) (cid:19) exp (cid:20) − ζ (cid:90) zz (cid:36) (cid:21) , (3.36)By performing the WKB expansion around ζ ∼ ζ , as shown in Appendix D.1.The quantities of prime interest in the subsequent discussions are the Wronskians ofthe eigenvectors of the monodromy matrices. To perform the WKB expansion of suchWronskians, we need to have a good control over the asymptotics of the Wronskians (cid:104) i ± , j ± (cid:105) around ζ = 0. For this purpose, both of the eigenvectors in the Wronskian needto be small solutions since big solutions can contain a multiple of small solutions and henceare ambiguous [23–26]. When ζ is sufficiently close to zero, one can show that the plussolutions i + are the small solutions if Re ζ is positive whereas it is the minus solutions i − which are small if Re ζ is negative. Thus, the Wronskians that can be expandedconsistently around ζ = 0 are (cid:104) i + , j + (cid:105) ’s for Re ζ > (cid:104) i − , j − (cid:105) ’s for Re ζ < 0. Thedetailed form of the expansion can be determined by employing the Born series expansionexplained in Appendix D.2 and the results are given in the following simple form:For Re ζ > , (cid:104) + , + (cid:105) = exp ( − S → ) , (cid:104) + , + (cid:105) = exp ( − S → ) , (cid:104) + , + (cid:105) = exp ( − S ˆ3 → ) , (3.37)For Re ζ < , (cid:104) − , − (cid:105) = exp ( S → ) , (cid:104) − , − (cid:105) = exp ( S → ) , (cid:104) − , − (cid:105) = exp ( S ˆ3 → ) . (3.38)In these expressions, S i → j stands for the quantity S i → j = 12 ζ (cid:90) (cid:96) ij (cid:36) + (cid:90) (cid:96) ij α + ζ (cid:90) (cid:96) ij η + · · · , (3.39)where the one-form α is given in (D.41) in Appendix D.2. A remarkable feature of (3.39)is that the integral of our interest (cid:82) (cid:96) ij η makes its appearance in the exponent S i → j .37ow to make use of the averaging procedure described in the previous subsection, weneed the other type of integrals (cid:82) (cid:96) ij ˜ η which appear in Global . To obtain them, we needto expand the Wronskians this time around ζ = ∞ . Since the discussion is similar to theexpansion around ζ = 0, we will not elaborate on the details and simply give the results:For Re ζ > , (cid:104) + , + (cid:105) = exp (cid:16) − ˜ S → (cid:17) , (cid:104) + , + (cid:105) = exp (cid:16) − ˜ S → (cid:17) , (cid:104) + , + (cid:105) = exp (cid:16) − ˜ S ˆ3 → (cid:17) , (3.40)For Re ζ < , (cid:104) − , − (cid:105) = exp (cid:16) ˜ S → (cid:17) , (cid:104) − , − (cid:105) = exp (cid:16) ˜ S → (cid:17) , (cid:104) − , − (cid:105) = exp (cid:16) ˜ S ˆ3 → (cid:17) , (3.41)Here ˜ S i → j is defined by ˜ S i → j = ζ (cid:90) (cid:96) ij ¯ (cid:36) + (cid:90) (cid:96) ij ˜ α + 12 ζ (cid:90) (cid:96) ij ˜ η + · · · , (3.42)where ˜ α is a one-form given in (D.42) in Appendix D.2. Making use of these two types ofexpansions, we will be able to rewrite the action in terms of the Wronskians, as describedin the next subsection. We are now ready to derive an explicit expression of the action in terms of the Wronskians.As shown in the previous subsection, the integrals we used to rewrite the action, namely (cid:72) (cid:96) ij η and (cid:72) (cid:96) ij ˜ η , can be extracted from the Wronskians. For instance, consider the integral (cid:72) (cid:96) η , which appears in (cid:104) − , − (cid:105) . Differentiating ln (cid:104) − , − (cid:105) with respect to ζ using (3.38)and (3.39), we get ∂ ζ ln (cid:104) − , − (cid:105) = − ζ (cid:90) (cid:96) (cid:36) + 12 (cid:90) (cid:96) η + O ( ζ ) . (3.43)Therefore we can get the integral (cid:72) (cid:96) η by subtracting the first divergent term and thentaking the limit ζ → 0. Similarly (cid:72) (cid:96) ˜ η can be obtained from (cid:104) − , − (cid:105) in the ζ → ∞ limit. Such procedures can be compactly implemented if we use the variable x instead of ζ , which are related as in (2.46). Then, we can write (cid:73) (cid:96) η = − : ∂ x ln (cid:104) − , − (cid:105) : + , (cid:73) (cid:96) ˜ η = − : ∂ x ln (cid:104) − , − (cid:105) : − , (3.44)where the “normal ordering” symbol : A ( x ) : ± is defined by : A ( x ) : ± ≡ lim x →± [ A ( x ) − (double pole at x = ± . (3.45)38his precisely subtracts the divergent term mentioned above. Substituting such expres-sions to the definitions of Global and Global , we can express them in terms of the Wron-skians. Then, using (3.32), we arrive at the following expression for the contribution fromthe S part of the action F action : F action = − S S = √ λ A (cid:36) + A η . (3.46)The first term in (3.46) expresses the contributions of Extra and Extra . The second term A (cid:36) denotes the contribution of (cid:82) (cid:96) ij (cid:36) and (cid:82) (cid:96) ij ¯ (cid:36) in Global and Global and is given by A (cid:36) = √ λ (cid:18) ( κ Λ + κ Λ − κ Λ ) (cid:90) (cid:96) (cid:36) + ( κ Λ − κ Λ + κ Λ ) (cid:90) (cid:96) ˆ31 (cid:36) + ( − κ Λ + κ Λ + κ Λ ) (cid:90) (cid:96) (cid:36) (cid:19) + (cid:0) Λ i → ¯Λ i , (cid:36) → ¯ (cid:36) (cid:1) , (3.47)where Λ i and ¯Λ i are as given in (3.20) and (3.31) and (cid:0) Λ i → ¯Λ i , (cid:36) → ¯ (cid:36) (cid:1) in the lastline denotes the terms obtained by replacing Λ i and (cid:36) in the second line with ¯Λ i and ¯ (cid:36) respectively. The third term A η is the contribution of (cid:82) (cid:96) ij η and (cid:82) (cid:96) ij ˜ η , which is expressedin terms of the Wronskians in the following way: A η = √ λ (cid:2) ( κ + κ − κ ) ( : ∂ x ln (cid:104) − , − (cid:105) : + − : ∂ x ln (cid:104) + , + (cid:105) : − )+ ( κ − κ + κ ) ( : ∂ x ln (cid:104) − , − (cid:105) : + − : ∂ x ln (cid:104) + , + (cid:105) : − )+ ( − κ + κ + κ ) ( : ∂ x ln (cid:104) − , − (cid:105) : + − : ∂ x ln (cid:104) − , − (cid:105) : − ) (cid:3) . (3.48)The general formula (3.46) will later be used in section 6 to compute the three-pointfunctions. Having found the structure of the contribution of the action part, we shall now study thatof the vertex operators. Before plunging into the details of the analysis, let us describe in this subsection the basicidea and the framework, which includes a brief review of the methods developed in ourprevious work [11]. 39s explained in detail in [9], the precise form of the conformally invariant vertexoperator corresponding to a string solution in a curved spacetime, such as AdS discussedthere or EAdS × S of our interest in this paper, is in general not known. In particular,for a non-BPS solution with non-trivial σ dependence the corresponding vertex operatorwould contain infinite number of derivatives and is hard to construct. To overcome thisdifficulty, we have developed in [11] a powerful method of computing the contribution ofthe vertex operators by using the state-operator correspondence and the construction ofthe corresponding wave function in terms of the action-angle variables. Although it wasapplied in [11] to the case of the GKP string in AdS , the basic idea of the method isapplicable to more general situations, including the present one, albeit with appropriatemodifications and refinements.Let us briefly review the essential ingredients of the method. (For details, see section3 of [11].) The state-operator correspondence, in the semi-classical approximation, isexpressed by the following equation: V [ q ∗ ( z = 0)] e − S q ∗ ( τ< = Ψ[ q ∗ ] (cid:12)(cid:12) τ =0 . (4.1)Here q ∗ signifies the saddle point configuration, V [ q ∗ ( z = 0)] is the value of the vertexoperator inserted at the origin of the worldsheet z = e τ + iσ = 0, corresponding to thecylinder time τ = −∞ , the factor exp[ − S q ∗ ( τ < q ∗ ] (cid:12)(cid:12) τ =0 is the semi-classical wave function describing thestate on that circle. In particular, if we can construct the action-angle variables ( S i , φ i )of the system and use { φ i } as q , then the wave function evaluated at the cylinder time τ can be expressed simply asΨ[ φ ] = exp (cid:32) i (cid:88) i S i φ i − E ( { S i } ) τ (cid:33) , (4.2)where the action variables S i and the worldsheet energy E ( { S i } ) are constant.In the case of the classical string in R × S , the method for the construction ofthe action-angle variables was developed in [13–15], employing the so-called Sklyanin’sseparation of variables [16]. This method was adapted to the case of the GKP string in AdS in [11] and, as we shall see, can be applied to the present case of the string in EAdS × S with appropriate modifications. In this method, the essential dynamical informationis contained in the two-component Baker-Akhiezer vectors ψ ± , which satisfy the ALP forthe right sector and are the eigenvectors of the monodromy matrix ΩΩ( x ; τ, σ ) ψ ± ( x ; τ, σ ) = e ± ip ( x ) ψ ± ( x ; τ, σ ) . (4.3)More precisely, the dynamical information is encoded in the normalized Baker-Akhiezervector h ( x ; τ ), defined to be proportional to ψ ( x ; τ, σ = 0) (conventionally taken to be40 + ) and satisfying the normalization condition n · h = n h + n h = 1 , h = 1 n · ψ ψ , (4.4)where n is called the normalization vector . For a finite gap solution associated to agenus g algebraic curve, h ( x ; τ ) as a function of x is known to have g + 1 poles at thepositions x = { γ , γ , . . . , γ g +1 } and the dynamical variables z ( γ i ) and p ( γ i ), where z isthe Zhukovsky variable defined in (2.36), can be shown to form canonical conjugate pairs.Then by making a suitable canonical transformation, one can go to the action-angle pairs( S i , φ i ), where, in particular, the angle variable is given by the generalized Abel map φ i = 2 π (cid:88) j (cid:90) γ j x ω i . (4.5)Here, ω i are suitably normalized holomorphic differentials (with certain singularities de-pending on the specific problem) and x is an arbitrary base point. Now since one canreconstruct the classical string solution from the Baker-Akhiezer vector ψ (and ˜ ψ , whichis the solution of the left ALP) as shown in (2.65) and (2.66), with a choice of the nor-malization vector n one can associate a set of angle variables φ i to a classical solution.In fact, the angle variables can be thought to be determined by the quantity n · ψ , sincethe poles of the normalized vector h occur at the zeros of n · ψ , as is clear from (4.4).As we are actually dealing with a quantum system using semi-classical approximation, aclassical solution should be thought of as being produced by a quantum vertex operatorcarrying a large charge. Further, since in our framework the vertex operator is replaced bythe corresponding wave function, the angle variables defined through a classical solutionshould be used to describe the wave function of the corresponding semiclassical state.Now the serious problem is that we do not know the exact saddle point solution for thethree-point function. The only information we know is that in the vicinity of each vertexinsertion point z i , the exact three-point solution, to be represented by a 2 × Y given by Y = (cid:18) Z Z − ¯ Z ¯ Z (cid:19) , Z = Y + iY , Z = Y + iY , (4.6)which must be almost identical to the two-point solution produced by the same vertexoperator. Let us denote such a solution by Y ref and call it a reference solution. As wehave to normalize the three-point function precisely by such a two-point function for eachleg, what is important is the difference between Y and Y ref . Note that even if they areproduced by the same vertex operator, they are different because Y is influenced by thepresence of other vertex operators in the three-point function.Here and in what follows, the global isometry group G = SU(2) L × SU(2) R and itscomplexification G c = SL(2 , C) L × SL(2 , C) R play the central roles. Being the symmetry41roups of the equations of motion (and the Virasoro conditions), two solutions of theequations of motion are connected by the action of G and/or G c . The difference betweentheir actions are that (when expressed in terms of the Minkowski worldsheet variables)while G connects a real solution to a real solution, G c transforms a real solution to acomplex solution. Since the three-point interaction is inherently a tunneling process, thesaddle point solution for such a process must be complex. Therefore near z i the twosolutions Y and Y ref must be connected by an element of G c in the manner Y = (cid:101) V Y ref V , (cid:101) V ∈ SL(2 , C) L , V ∈ SL(2 , C) R (4.7)This means that the angle variables associated to Y , as defined relative to the ones asso-ciated to Y ref , should be computable from the knowledge of the transformation matrices (cid:101) V and V . This connection was made completely explicit in [11] and the master formulasgiving such shifts of the angle variables were obtained. Corresponding to the solutions ψ ( x ) and ˜ ψ ( x ) of the the right and the left ALP respectively, there are right angle variable φ R and the left angle variable φ L . Their shifts are given by ∆ φ R = − i ln (cid:18) ( n · ψ + ( ∞ ))( n · ψ ref − ( ∞ ))( n · ψ ref+ ( ∞ ))( n · ψ − ( ∞ )) (cid:19) , (4.8) ∆ φ L = − i ln (cid:32) (˜ n · ˜ ψ + (0))(˜ n · ˜ ψ ref − (0))(˜ n · ˜ ψ ref+ (0))(˜ n · ˜ ψ − (0)) (cid:33) (4.9)where n and ˜ n are the normalization vectors for the right and the left sector and ψ ± ( x )and ψ ref ± ( x ) are the Baker-Akhiezer eigenvectors corresponding to the solutions Y and Y ref respectively and are related by ψ ± = V − ψ ref ± , ˜ ψ ± = (cid:101) V ˜ ψ ref ± . (4.10)How V and (cid:101) V can be obtained will be described in detail in subsection 4.3.The remaining problem is to fix the normalization vectors n and ˜ n , relevant for theleft and the right sectors. In the case of the string which is entirely in AdS [11], we fixedthem by the following argument. Consider for simplicity the wave function correspondingto a conformal primary operator of the gauge theory sitting at the origin of the boundaryof AdS . Such an operator is characterized by the invariance under the special confor-mal transformation. Therefore the corresponding wave function and the angle variablescomprising it should also be invariant. Explicitly it requires that n · ψ and ˜ n · ˜ ψ must bepreserved under the special conformal transformation and this determined n and ˜ n . These equations are obtained from the fundamental formula (3.74) of [11] by substituting the defini-tion of the function f ( x ) given in (3.62) of the same reference and noting the expression of the function h ( x ) shown in (4.4) of this paper. EAdS × S studied in the present work is the same. However because the structures of the gaugetheory operators and the corresponding string solutions are more complicated, we needto generalize and refine the argument. As a result of this improvement, not only hasthe determination of the normalization vectors become more systematic but also theirphysical meaning has been identified more clearly. Moreover, the entire procedure ofthe constructions of the wave functions for the S part and the EAdS part has becomecompletely parallel and transparent. Below we shall begin the analysis first from thegauge theory side. As sketched above, in order to construct the wave functions expressing the effect of theinsertion of the vertex operators, we must study how to characterize the global symmetryproperties of the vertex operators and the classical configurations that they produce intheir vicinity.For this purpose, it is convenient to first look at the symmetry properties of the corre-sponding gauge theory operators. The three composite operators O ( x ) , O ( x ) , O ( x )making up the three-point functions in the so-called “SU(2) sector” are composed of thecomplex scalar fields Z ≡ Φ + i Φ , X ≡ Φ + i Φ and their complex conjugates ¯ Z and ¯ X ,where Φ I ( I = 1 , , , 4) are four of the six real hermitian fields in the adjoint represen-tation of the gauge group. Under the global symmetry group SO(4) = SU(2) R × SU(2) L ,these fields transform in the doublet representations of SU(2) R and SU(2) L with the rightand the left charges R and L given in Table 1:Table 1. The SU(2) R and SU(2) L chargesfor the basic scalar fields. R L Z +1 / / Z − / − / X − / / X +1 / − / O i .vacuum excitation O Z X O ¯ Z ¯ X O Z ¯ X These transformation properties are succinctly represented by the 2 × (cid:18) Z X − ¯ X ¯ Z (cid:19) , (4.11)which gets transformed as U L Φ U R , where U L ∈ SU(2) L , U R ∈ SU(2) R . In spite of thisSO(4) symmetry, in the existing literature [27] the operators O i are taken to be com-43osed of a special pair of fields indicated in Table 2. For example, O is of the formtr ( ZZ · · · XZZX · · · Z ). In the spin-chain interpretation, Z and X represent the up andthe down spin respectively so that O is a state built upon the all-spin-up vacuum statetr Z l on l sites by flipping some of the up-spins into the down-spins which represent ex-citations. Therefore at each site there is an SU(2) group acting on a spin, and accordingto Table 1 it is identified with SU(2) R for this case. For the entire operator O , what isrelevant is the total SU(2) R , the generator of which will be denoted by S iR .Let us now characterize the spin-chain states corresponding to the operators of thetype O from the point of view of this total SU(2) R . First, since the constituents Z and X carry definite spin quantum numbers, every state of type O carries a definiteright and left global charges. Second, every such state is actually a highest weight stateannihilated by the operator S + R = S R + iS R . For the vacuum state | Z l (cid:105) = | ↑ l (cid:105) it is obvious.As for the excited states, they can be written as the Bethe states (cid:81) i =1 B ( u i ) | ↑ l (cid:105) , where B ( u i ) is the familiar magnon creation operator carrying the spectral parameter u i . It iswell-known [28] that such a state is a highest weight state of the total SU(2) R and henceannihilated by the same S + R , provided that the Bethe state is “on-shell”, namely thatthe spectral parameters satisfy the Bethe ansatz equations. Therefore we have found thatkinematically all the operators of type O can be characterized as the highest weight stateof the total SU(2) R .Now in order to deal with other operators built upon a “vacuum state” different fromtr Z l , let us introduce the general linear combinations of Φ I as (cid:126)P · (cid:126) Φ = (cid:80) I =1 P I Φ I . Todiscuss the transformation property under SU(2) R × SU(2) L , it is more convenient to dealwith the matrix P ≡ (cid:18) P + iP P + iP − ( P − iP ) P − iP (cid:19) = P I Σ I , (4.12)Σ I ≡ (1 , iσ , iσ , iσ ) . (4.13)Then, we have the representation (cid:126)P · (cid:126) Φ = 12 tr (cid:0) σ P t σ Φ (cid:1) . (4.14)In this notation, P corresponding to Z, ¯ Z, X, ¯ X take the form P Z = 1 − σ , P ¯ Z = 1 + σ , P X = − ( σ − iσ ) , P ¯ X = σ + iσ .As we argued above, all the on-shell states built upon a common vacuum are annihi-lated by the same S + R . In other words as long as the global transformation property isconcerned, the vacuum state can be considered as the representative of all the states built The reason for this choice is that it is the simplest one that can produce non-extremal three-pointfunctions. s + R acting ona single spin state. As it will be slightly more convenient, instead of the annihilationoperator, we will use the “raising operator” K = exp( αs + R ), where α is any constant. Thevacuum is then characterized by the form of K that leaves its building block invariant .Let us explain this idea concretely for the operator Z , which is the building blockfor the simplest vacuum state tr Z l . In the general notation (4.14), we can express Z as Z = tr ( σ P tZ σ Φ) with P Z = 1 − σ . Now let us look for the raising operators K Z and˜ K Z for SU(2) R and SU(2) L respectively, which leave Z invariant. Since Φ transforms into˜ K Z Φ K Z , the invariance condition reads12 tr (cid:16) σ P tZ σ ˜ K Z Φ K Z (cid:17) = 12 tr (cid:0) σ P tZ σ Φ (cid:1) . (4.15)This is equivalent to the condition P Z = ˜ K − Z P Z K − Z . (4.16)It is easy to find the solutions , which read K Z = (cid:18) β (cid:19) = e βσ + , ˜ K Z = (cid:18) β (cid:19) = e ˜ βσ − , (4.17)where β and ˜ β are arbitrary constants.Next we consider a general case where the vacuum state is given by tr ( (cid:126)P · (cid:126) Φ) l , witharbitrary (cid:126)P . Since, in general, (cid:126)P · (cid:126) Φ does not carry a definite set of left and right chargesdefined as in Table 1, this state and the ones built upon it by some spin-chain typeexcitations are not charge eigenstates. Nevertheless, we can characterize this family ofstates again by the raising operators K and ˜ K which leave (cid:126)P · (cid:126) Φ invariant. Just as in(4.16), this condition is expressed as P = ˜ K − P K − . (4.18)where P corresponds to (cid:126)P . Since (cid:126)P · (cid:126) Φ can be obtained from Z by an SU(2) L × SU(2) R transformation, P can be obtained from P Z by a corresponding transformation of the form P = U L P Z U R . (4.19)Then combined with (4.18) we readily obtain the relation P Z = ( U − L ˜ K − U L ) P Z ( U R K − U − R ).Comparing this with (4.16) we can express the raising operators K and ˜ K in terms of theones for the operator Z given in (4.17) in the form K = U − R K Z U R , ˜ K = U L ˜ K Z U − L . (4.20) The most general solutions are of the form (cid:18) α β α − (cid:19) and (cid:18) α − β α (cid:19) . However since we areinterested in the raising type operators, it is sufficient to consider the operators of the form (4.17). n and ˜ n , which are left invariant under the following action of K and ˜ K respectively : K t n = n , ˜ K t ˜ n = ˜ n . (4.21)Since the overall factor for these vectors are inessential, we can normalize them to haveunit length as n † n = ˜ n † ˜ n = 1. We shall refer to them as polarization spinors , as theycharacterize, so to speak, the “direction of polarization” of the highest weight operator (cid:126)P · (cid:126) Φ. It should be noted that from the knowledge of n and ˜ n , one can reconstruct P which is invariant under the raising operators, as in (4.18). In fact, if we set P = − iσ ˜ nn t , (4.22)one can easily check that this P satisfies (4.18), with the use of the defining equations(4.21) and a simple formula σ U − σ = U t valid for any invertible 2 × U satisfyingdet U = 1.Let us illustrate these concepts by computing the polarization spinors for the operators Z and ¯ Z respectively. For the operator Z we already computed the right and the leftraising operators in (4.17). Then it is easy to see that the corresponding polarizationspinors n Z and ˜ n Z satisfying K tZ n Z = n Z and ˜ K tZ ˜ n Z = ˜ n Z are given by n Z = (cid:18) (cid:19) , ˜ n Z = (cid:18) (cid:19) . (4.23)As a check, from the formula (4.22), we immediately get P Z = (cid:18) (cid:19) , which is thedesired form. As for the operator ¯ Z , repeating the similar analysis, the raising operatorsleaving P ¯ Z = 1 + σ invariant can be readily obtained to be K ¯ Z = (cid:18) α (cid:19) , ˜ K ¯ Z = (cid:18) α (cid:19) , (4.24)with α and ˜ α being arbitrary constants. The corresponding polarization spinors can betaken to be n ¯ Z = (cid:18) (cid:19) , ˜ n ¯ Z = (cid:18) (cid:19) . (4.25)Finally consider the normalization spinors for a general operator (cid:126)P · (cid:126) Φ which is related to Z = (cid:126)P Z · (cid:126) Φ through the relation of the form (4.19). Since the raising operators for suchan operator are obtained from those for Z in the manner (4.20), the polarization vectors n and ˜ n are expressed in terms of n Z and ˜ n Z as n = U tR n Z , ˜ n = ( U tL ) − ˜ n Z . (4.26) Intentionally we are using the same letters n and ˜ n for the vectors introduced here as those usedpreviously for the normalization vectors. This is because they will be shown to be identical. 46s an application of this formula, let us re-derive n ¯ Z and ˜ n ¯ Z from this perspective. Since P ¯ Z = 1 + σ and P Z = 1 − σ , it is easy to see that they are related by an SU(2) L × SU(2) R transformation of the form P ¯ Z = U L P Z U R , U L = iσ , U R = − iσ . (4.27)In fact this transformation realizes the mapping ( Z, X ) → ( ¯ Z, − ¯ X ). Substituting theforms of U L and U R into the above formula (4.26), we obtain U tR n Z = (1 , t and( U tL ) − ˜ n Z = − (0 , t ∝ (0 , t , which agree with (4.25).Summarizing, we can say that, as far as the global symmetry properties are concerned,the operators of type O and O are characterized by the polarization spinors n Z and ˜ n Z ,while the operators of type O are associated with n ¯ Z and ˜ n ¯ Z . For more general operatorsbuilt upon the vacuum tr ( (cid:126)P · (cid:126) Φ) l , the corresponding polarization spinors are obtained from n Z and ˜ n Z by appropriate transformations which connect P with P Z as shown in (4.26).The importance of the above analysis is that, as we shall describe below, precisely thesame characterization scheme must be valid for the vertex operators in string theory whichcorrespond to the gauge theory composite operators like O i . Moreover, it will be shownthat the polarization spinors introduced purely from the group theoretic point of viewabove will be identified with the “normalization vectors” that appeared in (4.4), whichplay pivotal roles in the construction of the angle variables and hence the construction ofthe wave functions describing the contribution of the vertex operators. S part We now begin the explicit construction of the wave functions contributing to the three-point functions in string theory. As emphasized in the introduction, an essential ingredientfor the success of the computation of the three-point functions is the separation of thekinematical and the dynamical factors. Although the dynamics is quite different betweenthe gauge theory and the corresponding string theory, the kinematical symmetry proper-ties correspond quite directly between the gauge theory operators and the vertex operatorsof string theory. Therefore in this subsection we will describe how we can implement thescheme of the symmetry characterization of the operators developed in the preceding sub-section for the gauge theory operators to the vertex operators and the classical solutionsproduced by them. Since the analysis concerning the each factor of the symmetry groupSU(2) R × SU(2) L is completely similar and can be performed independently, after somegeneral discussions we will almost exclusively focus on the SU(2) R part of the symmetrytransformations and various corresponding quantities for clarity of presentations.47n the saddle point approximation scheme we are employing, we cannot directly dealwith the vertex operator: What we can deal with are the classical solutions produced bythe vertex operators carrying large charges. Therefore we need to extract the informationof the quantum vertex operators indirectly through such classical solutions.For definiteness, we first focus on a solution with diagonal SU(2) R × SU(2) L chargesdescribing a two-point function of an operator built on the tr ( Z l )-vacuum ( O and O in section 4.2) and its conjugate . In what follows, we shall denote such a solution by Y diag . Then we can associate a pair of polarization spinors n Z and ˜ n Z and the raisingoperators (4.17) to the vertex operator that produces the solution. For convenience, wedisplay them again with appropriate renaming: n diag = (cid:18) (cid:19) , ˜ n diag = (cid:18) (cid:19) , (4.28) K diag ( β ) = (cid:18) β (cid:19) , ˜ K diag ( ˜ β ) = (cid:18) β (cid:19) . (4.29)All the solutions describing a two-point function of mutually conjugate operators, (cid:104)OO(cid:105) ,can be obtained from this basic solution Y diag by an SU(2) R × SU(2) L transformation.Since a normalized three-point function in the gauge theory can be obtained by dividingan unnormalized one by (cid:104)OO(cid:105) -type two-point functions as (cid:104)O i O j O k (cid:105) (cid:113) (cid:104)O i O i (cid:105)(cid:104)O j O j (cid:105)(cid:104)O k O k (cid:105) , (4.30)the aforementioned solutions, to be denoted by Y ref , can be used as reference solutions todetermine the normalization of the wave function. An important feature of such solutionsis that they are real-valued when expressed in terms of the Minkowski worldsheet variables.This qualification will be extremely important since the equation of motion is actuallyinvariant under a larger group SL(2 , C) R × SL(2 , C) L and its action can produce “complex”solutions which signify tunneling. Such a tunneling process is necessary for the three-pointinteractions to take place, as we shall see.From now on till the end of this subsection, we shall suppress all the left transforma-tions and display only the right transformations. The results for the left transformationswill be summarized in subsection 4.3.3.Now consider a three-point function produced by vertex operators, corresponding tothe gauge theory operators, inserted at z i on the worldsheet. We will take the operatorsto be those obtained by SO(4) rotations of the operators built on the tr ( Z l )-vacuum.This suffices for the present purpose since such three-point functions include the ones What is meant by “conjugation” is the usual complex conjugation of the fields, Z → ¯ Z and X → ¯ X . Note that O and O in section 4.2 are built on the tr ( Z l )-vacuum while O can be obtained fromthe operator built on tr ( Z l ) by an SO(4) rotation (4.27), which effects ( Z, X ) → ( ¯ Z, − ¯ X ). z i by Y . Asymptotically as z → z i such a configuration must be well-approximated by a two-point reference solution Y ref ,which is produced by the same vertex operator. Even if they are produced by the samevertex operator, Y and Y ref are different since Y is influenced non-trivially and dynamicallyby the other two vertex operators present. We write the transformation between them at z (cid:39) z i as Y ( z (cid:39) z i ) = Y ref V ( z → z i ) , V ∈ SL(2 , C) R . (4.31)This relative difference is the quantity of interest since we need to normalize the three-point function by the two-point functions. In general V belongs to SL(2 , C) R ⊃ SU(2) R ,since the three-point interaction is necessarily a tunneling process. In contrast the refer-ence solution Y ref can be obtained from Y diag by a transformation belonging to SU(2) R inthe form Y ref = Y diag U ref , U ref ∈ SU(2) R . (4.32)The relation between Y , Y ref and Y diag is sketched in figure 4.1.Figure 4.1: A schematic picture which explains the relation between the local three-pointsolution Y and the two-point solutions Y diag and Y ref . Y ref is obtained from Y diag by a real global transformation U ref , while Y and Y ref , which are produced by the same vertexoperator, are related through a complexified global transformation V .Now just as we did already for the solution Y diag , we can associate to the solution Y ref the polarization spinor n ref and the raising transformation K ref which leaves it invariant.Then from the general formula (4.26) and (4.20) we can express them in terms of the Note that Y ref is the solution for the two-point function, expressed globally in terms of the cylindercoordinate. Thus we need to express Y in terms of the local coordinate (cid:0) τ ( i ) , σ ( i ) (cid:1) given in (2.97) tocompare two solutions. n ref = ( U ref ) t n diag , (4.33) K ref ( β ) = ( U ref ) − K diag ( β ) U ref . (4.34)By the same token we can associate the polarization spinor n and the raising transforma-tion K to the local solution Y . However since Y is produced by the same vertex operatoras Y ref , we must have n = n ref . As for K , just as in (4.34), under the transformation V which produces Y from Y ref , the raising operator K ref ( β ) transforms into K ( β ) in themanner K ( β ) = V − K ref ( β ) V . Since this operator must leave n and hence n ref invariant,we must have V − K ref ( β ) V = K ref ( β (cid:48) ) (4.35)for some β (cid:48) . Substituting the relation (4.34), we get( V (cid:48) ) − K diag ( β ) V (cid:48) = K diag ( β (cid:48) ) ,V (cid:48) ≡ U ref V ( U ref ) − . (4.36)This means that the operator V (cid:48) transforms a raising operator into a raising operator forthe diagonal solution. It is not difficult to show that the general form of such an operatoris (cid:18) a b a − (cid:19) . Note that this contains a scale transformation which is in SL(2 , C) R butnot in SU(2) R . From this result we can solve for V and its inverse and obtain the followinguseful representations V = ( U ref ) − (cid:18) a b a − (cid:19) U ref , (4.37) V − = ( U ref ) − (cid:18) a − − b a (cid:19) U ref . (4.38)At this stage we need not know the actual values of a and b in these formulas. b will turnout to be irrelevant and a will be expressed in terms of certain Wronskians. We are now ready for the construction of the wave function for the right sector using theformula for the shift of the angle variable φ R given in (4.8).First we need to fix the normalization vector n appearing in that formula. As we shallshow, the answer is that it coincides precisely with the polarization spinor n introducedfrom the group theoretical point of view in (4.21) in subsection 4.2. Recall that in theformalism developed in [13–15], the zeros of n · ψ ( x ), where ψ is the Baker-Akhiezervector and n is the normalization vector, determines the angle variables. When one50akes a global SL(2,C) R transformation V R on the string solution Y like Y → Y V R , theBaker-Akhiezer vector transforms like ψ → V − R ψ . In particular, take V R to be the raisingoperator K under which the vertex operator producing the solution Y is invariant. Thenthe wave function corresponding to the vertex operator and hence the angle variablescomprising it must also be invariant. This means that the zeros of n · ( K − ψ ) = ( K t n ) · ψ must coincide with the zeros of n · ψ and hence we must have K t n ∝ n . However since K is similar to K diag , it is clear that the constant of proportionality can only be unity and n must satisfy K t n = n . This, however, is nothing but the definition of the polarizationspinor given in (4.21). In other words, the proper choice of the normalization vector forconstructing the wave function is precisely the polarization spinor associated to the vertexoperator to which the wave function corresponds.Having found the proper choice of the normalization vector in the formula (4.8) forthe shift of the angle variable φ R , what remains to be understood is how to evaluate theinner products n · ψ ± ( ∞ ) and n · ψ ref ± ( ∞ ). Corresponding to the relation (4.31), in thevicinity of z i , ψ ± and ψ ref ± are related by the constant transformation V as ψ ± ( z (cid:39) z i ) = V − ψ ref ± ( z (cid:39) z i ). Now recall the form of the ALP for the right sector given in (1.4). Wesee that for x = ∞ the coefficients of the connections j z and j ¯ z vanish and hence thesolutions ψ ± ( x = ∞ ) and ψ ref ± ( x = ∞ ) themselves become constant. Combining thesepieces of information, we obtain the relation ψ ± ( ∞ ) = V − ψ ref ± ( ∞ ) . (4.39)The right hand side can be evaluated using the representation (4.38) as ψ ± ( ∞ ) = ( U ref ) − (cid:18) a − − b a (cid:19) U ref ψ ref ± ( ∞ ) = ( U ref ) − (cid:18) a − − b a (cid:19) ψ diag ± ( ∞ ) , (4.40)where ψ diag ± ( x ) is the Baker-Akhiezer vector for Y diag , which is related to ψ ref ± ( x ) by ψ ref ± ( x ) = (cid:0) U ref (cid:1) − ψ diag ± ( x ) . (4.41)We now need to know ψ diag ± ( ∞ ), which are the eigenstates of the monodromy matrixnear x = ∞ corresponding to the eigenvalues e ± ip ( x ) . For a charge-diagonal solution Y diag , the monodromy matrix near x = ∞ is diagonal and hence is either of the form(a) diag ( e ip ( x ) , e − ip ( x ) ) or (b) diag ( e − ip ( x ) , e ip ( x ) ), depending on the solution. For the case( a ) the eigenvectors are ψ diag+ ( ∞ ) = (1 , t , ψ diag − ( ∞ ) = (0 , t , while for the case (b) theirforms are swapped. Since Y diag is produced by the vertex operator with the definitepolarization spinor specified in (4.28), there should be a definite answer. To determinethe proper choice of (a) or (b), we need to construct the wave function for each choiceand see if it has the same transformation property as the corresponding operator in the51auge theory. As it will be checked later in this subsection, it turned out that the case(b) is the correct choice. Therefore we will take ψ diag+ ( ∞ ) = (cid:18) (cid:19) , ψ diag − ( ∞ ) = (cid:18) (cid:19) . (4.42)Substituting them into (4.40), we obtain the important relations ψ + ( ∞ ) = ( U ref ) − (cid:16) aψ diag+ ( ∞ ) − bψ diag − ( ∞ ) (cid:17) = aψ ref+ ( ∞ ) − bψ ref − ( ∞ ) , (4.43) ψ − ( ∞ ) = ( U ref ) − a − ψ diag − ( ∞ ) = a − ψ ref − ( ∞ ) . (4.44)As for the polarization spinor, observe that by inspection the following relation holds: n diag = ( − iσ ) ψ diag − ( ∞ ) . (4.45)This relation is actually universal in the following sense. Let us act ( U ref ) t from left. Thenthe relation becomes (cid:0) ( U ref ) t n diag = (cid:1) n ref = ( U ref ) t ( − iσ ) ψ diag − ( ∞ )= ( − iσ )( U ref ) − ψ diag − ( ∞ )= − iσ ψ ref − ( ∞ ) , (4.46)where we used the identity σ ( U ref ) t σ = ( U ref ) − . Thus, exactly the same form ofrelation holds for the reference solution and in fact for any solution related by an SU(2) R transformation. Together with the formula (4.44) we get the relation n = − iaσ ψ − ( ∞ ) , (4.47)which will be extremely important.Let us now recall the formula (4.8) for the shift of the angle variable φ R . Displayingit again for convenience, it is of the form ∆ φ R = − i ln (cid:18) ( n · ψ + ( ∞ ))( n · ψ ref − ( ∞ ))( n · ψ ref+ ( ∞ ))( n · ψ − ( ∞ )) (cid:19) . (4.48)From (4.43) and (4.46), we can write n · ψ + ( ∞ ) = an · ψ ref+ ( ∞ ). As for n · ψ − ( ∞ ), useof (4.44) gives n · ψ − ( ∞ ) = a − n · ψ ref − ( ∞ ). Now due to the relation (4.46), the quantity n · ψ − ( ∞ ) = n ref · ψ − ( ∞ ), which appears both in the numerator and the denominator ofthe formula (4.48), vanishes. Therefore we must first regularize n slightly to make thequantity n · ψ − ( ∞ ) finite, cancel them in the formula and then remove the regularization.As for the same quantity appearing in n · ψ + ( ∞ ), we can safely set it to zero from thebeginning since n · ψ ref+ ( ∞ ) is non-vanishing. In this way we find that n · ψ ref ± ( ∞ )’s allcancel out and we are left with an extremely simple formula for ∆ φ R given by ∆ φ R = − i ln a . (4.49)52ote that the shift depends only on the quantity a , which parametrizes the scale trans-formation not belonging to SU(2) R , showing the tunneling nature of the effect.Let us now write the formula (4.47) for the operator at z i with a subscript i as n i = − ia i σ i − ( ∞ ). Then, from the definition of the Wronskian we obtain (cid:104) n i , n j (cid:105) = a i a j (cid:104) i − , j − (cid:105) (cid:12)(cid:12) ∞ . Writing out all the relations of this form and forming appropriate ratios,we can easily extract out each a i . The result can be written in a universal form as a i = (cid:104) j − , k − (cid:105)(cid:104) i − , j − (cid:105)(cid:104) k − , i − (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:104) n i , n j (cid:105)(cid:104) n k , n i (cid:105)(cid:104) n j , n k (cid:105) . (4.50)Then substituting this expression into the formula (4.49) we obtain the shift of the anglevariable φ R at the position z i as e i ∆ φ R,i = (cid:104) j − , k − (cid:105)(cid:104) i − , j − (cid:105)(cid:104) k − , i − (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:104) n i , n j (cid:105)(cid:104) n k , n i (cid:105)(cid:104) n j , n k (cid:105) . (4.51)This formula is remarkable in that it cleanly separates the kinematical part composed of (cid:104) n i , n j (cid:105) and the dynamical part described by (cid:104) i − , j − (cid:105) (cid:12)(cid:12) ∞ .As the last step of the construction of the wave function, we need to pay attention tothe convention of [15] that we are adopting. In that work, the Poisson bracket is definedto be { p, q } = 1 for the usual momentum p and the coordinate q . In this convention thePoisson bracket of the action angle variables was worked out to be given by { φ, S } = 1. Inother words the action variable S corresponds to q and the angle variable φ correspondsto p . Therefore upon quantization in the angle variable representation, we must set S = i∂/∂φ . This means that the wave function that carries charge S is given by e − iSφ , not by e iSφ .Recalling the relation (2.37) between the action variable S ∞ and the right charge R , namely S ∞ = − R , and employing the formula (4.51), the contribution to the wavefunction from the right sector is obtained asΨ S R = exp (cid:32) − i (cid:88) i =1 ( − R i ) ∆ φ R,i (cid:33) = (cid:89) { i,j,k } (cid:32) (cid:104) n i , n j (cid:105)(cid:104) i − , j − (cid:105) (cid:12)(cid:12) ∞ (cid:33) R i + R j − R k , (4.52)where { i, j, k } denotes the cyclic permutations of { , , } .At this stage, let us confirm that the wave function so constructed indeed carriesthe correct charge. To see this, it suffices to consider the U(1) transformation whichcorresponds to the diagonal right-charge rotations. Let us examine the case of the charge-diagonal operator built upon the Z -type vacuum, such as O or O in section 4.2. In sucha case the reference state is the charge-diagonal state itself, hence U ref = 1. Then if weset a = e iθ/ , b = 0 in the formula (4.37), the SU(2) R transformation matrix V becomesdiag ( e iθ/ , e − iθ/ ), which is a U(1) transformation under which Z and ¯ Z , carrying the53ight charge 1 / − / Z → e iθ/ Z and ¯ Z → e − iθ/ . Nowaccording to (4.49), under such a transformation the wave function acquires the phase e − i ( − R ) ln a = e iRθ . This shows that the wave function has the same (positive) charge R as the operator of the form tr ( Z R ). This proves that the choice of ψ diag ± ( ∞ ) we madein (4.42) is the correct one. If we had made the other choice, the wave function wouldhave acquired the phase e − iRθ , which contradicts the fact that the corresponding operatorin the gauge theory is built on the tr ( Z l )-vacuum. Similar argument can be made forthe left sector and again one can check that the wave function (4.52) carries the correctcharges. S part We now briefly describe the analysis for the left sector, to complete the construction ofthe wave function for the S part.The procedure is exactly the same as for the right sector but there are a couple ofnotable differences. First, the transformation matrices act from the left and consequentlyin various formulas the matrices are replaced by their inverses. In particular, the formulascorresponding to (4.37) and (4.39) for the transformation (cid:101) V that connects three-pointsolution and the reference solution in the manner Y = (cid:101) V Y ref take the form (cid:101) V = ˜ U ref (cid:18) a b a − (cid:19) ( ˜ U ref ) − , (4.53)˜ ψ ± (0) = (cid:101) V ˜ ψ ref ± (0) , (4.54)where ˜ U ref ∈ SU(2) L is the matrix effecting the connection Y ref = ˜ U ref Y diag . Second, theraising matrix for the diagonal solution is now lower triangular, namely˜ K diag ( β ) = (cid:18) β (cid:19) . (4.55)Thirdly, the polarization spinor for Z is ˜ n diag = (1 , t , as discussed in (4.25). Lastly,because of the form of the ALP for the left sector, the Baker-Akhiezer vector becomescoordinate-independent at x = 0 instead of at x = ∞ .Let us now list the basic results for the left sector, omitting the intermediate details.Just as for the right sector, the formulas below are valid for any type of operator. ψ diag+ = (cid:18) (cid:19) , ψ diag − = (cid:18) (cid:19) , (4.56)˜ ψ + (0) = a − ˜ ψ ref+ (0) + b ˜ ψ ref − (0) , ˜ ψ − (0) = a ˜ ψ ref − (0) , (4.57)˜ n = aiσ ˜ ψ + (0) , ∆ φ L = − i ln a − . (4.58)54sing these formulas, we obtain the contribution to the wave function from the left sectoras Ψ S L = exp (cid:32) − i (cid:88) i =1 L i ∆ φ L,i (cid:33) = (cid:89) { i,j,k } (cid:32) (cid:104) ˜ n i , ˜ n j (cid:105)(cid:104) i + , j + (cid:105) (cid:12)(cid:12) (cid:33) L i + L j − L k , (4.59)where we used the gauge invariance of the Wronskians and replaced (cid:104) ˜ i + , ˜ j + (cid:105) with (cid:104) i + , j + (cid:105) .Together with Ψ S R obtained in (4.52) we now have the complete wave function for the S part. It is of the structure e F vertex = Ψ S L Ψ S R e V energy , F vertex = V kin + V dyn + V energy . (4.60)Let us explain each term (4.60) in order. The first term V kin stands for the kinematicalpart composed of the Wronskians (cid:104) n i , n j (cid:105) and (cid:104) ˜ n i , ˜ n j (cid:105) , V kin =( R + R − R ) ln (cid:104) n , n (cid:105) + ( R + R − R ) ln (cid:104) n , n (cid:105) + ( R + R − R ) ln (cid:104) n , n (cid:105) + ( L + L − L ) ln (cid:104) ˜ n , ˜ n (cid:105) + ( L + L − L ) ln (cid:104) ˜ n , ˜ n (cid:105) + ( L + L − L ) ln (cid:104) ˜ n , ˜ n (cid:105) . (4.61)The second term V dyn refers to the dynamical part consisting of the Wronskians (cid:104) i − , j − (cid:105) (cid:12)(cid:12) ∞ and (cid:104) ˜ i + , ˜ j + (cid:105) (cid:12)(cid:12) , V dyn = − ( R + R − R ) ln (cid:104) − , − (cid:105) (cid:12)(cid:12) ∞ − ( R + R − R ) ln (cid:104) − , − (cid:105) (cid:12)(cid:12) ∞ − ( R + R − R ) ln (cid:104) − , − (cid:105) (cid:12)(cid:12) ∞ − ( L + L − L ) ln (cid:104) + , + (cid:105) (cid:12)(cid:12) − ( L + L − L ) ln (cid:104) + , + (cid:105) (cid:12)(cid:12) − ( L + L − L ) ln (cid:104) + , + (cid:105) (cid:12)(cid:12) . (4.62)The last term V energy denotes the contribution involving the worldsheet energy shown inthe last term of (4.2). Such a term is necessary for the following reason. As explainedbelow (2.97) and at the beginning of section 3.1, we evaluate our wave function on thecircle defined by τ ( i ) = 0, corresponding to ln | z − z i | = ln (cid:15) i . On the other hand, thewave function introduced through the state operator mapping in (4.1) is defined on theunit circle described by ln | z − z i | = 0. The term V energy is needed to fill this gap. As theenergy of the each external state is given by 2 √ λκ i , V energy can be evaluated explicitlyas V energy = 2 √ λ (cid:88) i =1 κ i ln (cid:15) i . (4.63) The energy can be computed from the behavior of the stress-energy tensor around the puncture(2.11). { Y , ∂ τ Y } → { φ i , S i } . However, insection 7.3, it will be checked that our result for the three-point function reproduces thenormalized two-point function in an appropriate limit. Therefore we can a posterioriconfirm that the wave function is properly normalized and the additional contributionsare absent. Second, one recognizes that the power of (cid:104) n i , n j (cid:105) , namely R i + R j − R k , isthe familiar combination, made out of conformal weights and spins, for the coordinatedifferences in the three-point functions of a conformal field theory, except for the overallsign . In the next subsection, we will elaborate on this structure of the power from thepoint of view of the dual gauge theory. Also in section 6.2, where we construct the wavefunction for the EAdS part, the above difference in the overall sign will be explained.Summarizing, the product of (4.52) and (4.59) gives the general form of the wave func-tions for the three-point function. It is expressed in terms of the two types of Wronskians.One type is the Wronskians between the solutions of the ALP around vertex insertionpoints. They will be evaluated in section 5. The others are the Wronskians between thepolarization spinors associated with the vertex operators, which are of purely kinematicalnature and hence should be common to the string and the gauge theory sides. We shall now examine our formula for the wave function from the point of view of corre-spondence with the gauge theory side.First consider the question of how to distinguish the different types of gauge theoryoperators O i from their corresponding wave functions in string theory. The wave functionconstructed above is expressed in terms of the polarization spinors, which depend onlyon the type of the vacuum on which the corresponding gauge theory operator is built,the eigenvectors of the ALP in the vicinity of the insertion point z i , and the chargescarried by the vertex operators. A natural question is how we can distinguish the type ofvertex operators involved from these data. Operators of O and O in section 4.2 can bedistinguished by the structure of their polarization spinors because the vacuum on whichthey are built are different. On the other hand, operators of O and O , which are builton the same type of the vacuum, are characterized by the same polarization spinors andhence it appears that one cannot distinguish them from the formula for the wave function.Since these operators differ only in the types of excitations, X or ¯ X , the question is howthis is reflected. The answer is in the relation between the absolute magnitude of the56harges R and L , which are given by R and L respectively. Because the charges carriedby the operator X are ( R , L ) = ( − / , / O operator built upon Z -vacuum with X as excitations must satisfy the inequality R < L . Similarly, the magnitudes of the total charges for the operator of type O alsoobey R < L . On the other hand, for the operator of type O , we have R > L .Such distinction is reflected not only on the charges but also on the dynamical propertyof the eigenstates i ± appearing in the wave function formula. As discussed in (2.75) and(2.76), the relative magnitude of R and L for a one-cut solution is determined by theposition of the cut in the quasi-momentum p ( x ): When the real part of the position ofthe branch cut is in the interval [ − , 1] in the spectral parameter space such a solutionhas R > L and hence corresponds to the operator of type O . Contrarily the operatorof type O having R < L corresponds to a solution with the cut outside the aboveinterval. Conceptually this is quite intriguing. From the spin-chain perspective, O and O form distinct types of spin chains, which cannot be transformed into each other byan SU(2) R × SU(2) L transformation. On the other hand, in string theory the solutionscorresponding to these distinct spin chains are described in a more unified way. It wouldbe interesting to realize such a unified treatment on the gauge theory side as well.Let us next examine the role and the meaning of the kinematical factor V kin from thepoint of view of the dual gauge theory. In this regard, note that the quantity (cid:104) n i , n j (cid:105) ,being a skew product, vanishes when n i and n j coincide. This in fact happens for thecase of the operators O and O discussed in section 4.2, which are built upon the same Z -vacuum and hence carry the same polarization spinors. There are three possibilities.If the power R i + R j − R k is positive, then the wave function and hence the three-pointfunction vanishes. This would express a selection rule. On the other hand, if it is negative,the three-point function diverges. For an internal symmetry such as SU(2) R this shouldnot occur. The last possibility is that the power is exactly zero. In this case, we shouldregularize (cid:104) n i , n j (cid:105) slightly away from zero and then apply the vanishing power to get theresult, which is unity.Let us see which of these cases is actually realized for the set of operators O , O and O in section 4.2. Let l i be the total length of the operator O i and M i be the number ofexcitations. The number of “vacuum fields” is then given by l i − M i . There are two obviousconservation laws for these numbers if all the fields and anti-fields of the set {O i } are fullycontracted to form propagators. One is the conservation concerning excitations, i.e. , thetotal number of X ’s should equal the total number of ¯ X ’s. The other is the conservationconcerning the vacuum fields, i.e. the total number of Z ’s should equal the total numberof ¯ Z ’s. From the structure of O i ’s it is easy to find that these two conservation laws are57xpressed as ( i ) M = M + M , ( ii ) l + l − l = 2 M . (4.64)Now consider the right and the left charges carried by O i . From Table 1, and the com-positions of O i , we get, for example, R = l − M , L = l , etc.. Then, computing thepowers of interest we get R + R − R = M + M − M = 0 , (4.65) L + L − L = 12 ( l + l − l ) − M = 0 . (4.66)Therefore precisely due to the conservation laws, ( i ) and ( ii ) above, of the number ofcontracting fields, the powers that occur for the vanishing Wronskians (cid:104) n , n (cid:105) and (cid:104) ˜ n , ˜ n (cid:105) are zero. Hence, in the computation of the three-point function of O i ’s such factors simplyproduce unity.Up to this point we have obtained the general formulas for the contribution of theaction part and the wave function part, both of which are expressed in terms of theWronskians of the form (cid:104) i ± , j ± (cid:105) . In the next section we will evaluate these quantities tosubstantiate the general formulas. In the previous two sections, we have shown that both the contribution of the action andthat of the vertex operators are expressible in terms of the Wronskians (cid:104) i ± , j ± (cid:105) betweenthe eigenvectors of the monodromy matrices. The goal of this section is to evaluatethose Wronskians. First, in section 5.1, we show that certain products of Wronskians areexpressed in terms of the quasi-momenta. Next, in sections 5.2 and 5.3, we determine theanalytic properties ( i.e. poles and zeros) of each Wronskian as a function of the spectralparameter x . With such a knowledge, we apply, in section 5.4, a generalized version of theWiener-Hopf decomposition formula to the products of the Wronskians and determine theindividual factor. Finally, in section 5.5, we compute the singular part and the constantpart of the Wronskian, which cannot be determined by the Wiener-Hopf method. To obtain the information of the Wronskian (cid:104) i ± , j ± (cid:105) between the eigenvectors of the ALPat different points, we need some condition which governs the global property of suchWronskians. As we shall see, such a condition is provided by the global consistency58ondition for the product of the local monodromy matrices Ω i associated with the vertexinsertion points z i . Since the total monodromy must be trivial upon going around theentire worldsheet, we must have Ω Ω Ω = 1 . (5.1)Although this appears to be a rather weak condition, it is sufficiently powerful to determinethe forms of certain products of the Wronskians in terms of the quasi-momenta p i ( x ), asdiscussed in [8, 9]. Let us quickly reproduce those expressions. Take the basis in whichΩ is diagonal, namely Ω = (cid:18) e ip e − ip (cid:19) . (5.2)Since the set of eigenvectors j ± at z j form a complete basis, one can expand the eigenvec-tors i ± at z i in terms of them in the following way: i ± = (cid:104) i ± , j − (cid:105) j + − (cid:104) i ± , j + (cid:105) j − . (5.3)Making use of this formula, Ω can be expressed in the Ω -diagonal basis asΩ = M (cid:18) e ip e − ip (cid:19) M , (5.4)where the matrix M ij , effecting the change of basis, is given by M ij = (cid:18) −(cid:104) i − , j + (cid:105) −(cid:104) i − , j − (cid:105)(cid:104) i + , j + (cid:105) (cid:104) i + , j − (cid:105) (cid:19) . (5.5)Now owing to the constraint (5.1), Ω and Ω must satisfy the following relation:tr (Ω Ω ) = tr Ω − = 2 cos p . (5.6)Substituting the equations (5.2) and (5.4) into (5.6), we obtain an equation for (cid:104) ± , ± (cid:105) of the formcos ( p − p ) (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) − cos ( p + p ) (cid:104) + , − (cid:105)(cid:104) − , + (cid:105) = cos p . (5.7)This equation, together with the Schouten identity for 1 ± and 2 ± given by (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) − (cid:104) + , − (cid:105)(cid:104) − , + (cid:105) = (cid:104) + , − (cid:105)(cid:104) + , − (cid:105) = 1 , (5.8) The general form of the Schouten identity is given by (cid:104) i , j (cid:105)(cid:104) k , l (cid:105) + (cid:104) i , k (cid:105)(cid:104) j , l (cid:105) + (cid:104) i , l (cid:105)(cid:104) j , k (cid:105) = 0. Itcan be proven directly from the definition of the Wronskians. (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) and (cid:104) + , − (cid:105)(cid:104) − , + (cid:105) .In a similar manner, products of certain other Wronskians can also be obtained, whichare summarized as the following set of equations : (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) = sin p + p + p sin p + p − p sin p sin p , (5.9) (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) = sin p + p + p sin − p + p + p sin p sin p , (5.10) (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) = sin p + p + p sin p − p + p sin p sin p , (5.11) (cid:104) + , − (cid:105)(cid:104) − , + (cid:105) = sin p − p + p sin p − p − p sin p sin p , (5.12) (cid:104) + , − (cid:105)(cid:104) − , + (cid:105) = sin p + p − p sin − p + p − p sin p sin p , (5.13) (cid:104) + , − (cid:105)(cid:104) − , + (cid:105) = sin − p + p + p sin − p − p + p sin p sin p . (5.14)What we need for the computation of the three-point functions, however, are theindividual Wronskians and not just the products given in (5.9)–(5.14). Such a knowledgewill be extracted based on the analytic properties of the Wronskians regarded as functionsof the complex spectral parameter x . We will analyze such properties in the next twosubsections. An individual Wronskian, viewed as a function of x , is almost uniquely determined by itsanalytic properties, namely the positions of the poles and the zeros. From the expressionsexhibited in (5.9)–(5.14), we know that the products of Wronskians have poles at sin p i = 0and zeros at sin (( ± p ± p ± p ) / 2) = 0. Therefore the question is which factor of theproduct is responsible for such a pole and/or a zero. In this subsection, we will describehow to analyze the structure of the poles.To illustrate the basic idea, we will consider the Wronskians (cid:104) + , + (cid:105) and (cid:104) − , − (cid:105) asexamples, for which the product is given by (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) = sin p + p + p sin p + p − p sin p sin p . Note that the equations (5.9)–(5.14) appear slightly different in form from those derived in [9]. Thisis because 2 + in this paper corresponds to 2 − in [9] and 2 − in this paper corresponds to − + in [9]. As we will discuss later, the Wronskian also contains essential singularities at x = ± 1. In addition,an overall proportionality constant cannot be determined by the positions of zeros and poles. Theseambiguities will be fixed in section 5.5. p = 0 and denote the position of the poleby x pole . There are two types of points at which sin p vanishes, the branch points andthe “singular points”. First consider the case where x pole is a singular point, at whichthe two eigenvalues of the monodromy matrix Ω degenerate to either +1 or − 1. This,however, does not mean that Ω is proportional to the unit matrix for the following reason:If Ω ∝ 1, the monodromy condition Ω Ω Ω = 1 forces p to be equal to + p or − p modulo π . However, since p , p and p can be chosen completely independently, there isno reason for such special relation to hold. Thus, the only remaining possibility is thatthe monodromy matrix Ω takes the form of a Jordan-block at x = x pole , namely,Ω ( x pole ) ∼ ± (cid:18) c (cid:19) . (5.15)In this case, the eigenvectors 1 + and 1 − degenerate at x = x pole and we have one eigen-vector. To see what happens at x = x pole more explicitly, let us study the asymptoticbehavior of 1 ± near z . In the vicinity of each puncture, the saddle point solution forthe three-point function can be well-approximated by an appropriate solution for a two-point function. Consequently, the eigenvectors for the three-point function 1 ± can alsobe approximated near z by the eigenvectors for the two-point function 1 ± . As shown in(2.96), this structure can be seen most transparently in the Pohlmeyer gauge. Workingout the subleading corrections, we obtain the following expansion for the eigenfunctionsˆ1 ± : ˆ1 + = ˆ1 (cid:16) c ( σ (1) , x ) e a τ (1) + c ( σ (1) , x ) e a τ (1) + · · · (cid:17) , (5.16)ˆ1 − = ˆ1 − (cid:16) c ( σ (1) , x ) e ˜ a τ (1) + ˜ c ( σ (1) , x ) e ˜ a τ (1) + · · · (cid:17) . (5.17)Here τ (1) and σ (1) are the local coordinates near z given in (2.97) and c k and ˜ c k are 2 × σ (1) and x . The constants a k in the exponents are such thatsuccessive terms are becoming smaller by exponential factors as τ → −∞ . An importantobservation is that since ˆ1 ± are eigenfunctions corresponding to a two-point function,they are insensitive to the global monodromy constraint (5.1) on the three-point functionand hence non-degenerate at x = x pole . An apparent puzzle now is how exponentiallysmall corrections can produce the degeneracy of ˆ1 ± .The answer is the following. Since one of the solutions ˆ1 ± is exponentially increasing( i.e. big) and the other is decreasing ( i.e. small) as τ → −∞ , let us consider the casewhere ˆ1 is big and ˆ1 − is small. Now for ˆ1 ± to become degenerate at x = x pole ,logically there are three possibilities( a ) ˆ1 + = α ˆ1 − , α = finite , (5.18)( b ) ˆ1 + = β ˆ1 − , β → ∞ , (5.19)( c ) ˆ1 − = β ˆ1 + , β → ∞ . (5.20)61irst, since ˆ1 is much larger than ˆ1 − by assumption, the case ( a ) cannot occur. Nowconsider the case where x is slightly different from x pole . Then β is large but finite andthe relations ( b ) or ( c ) must be realized approximately. But it is obvious that ( b ) is theonly consistent relation since exponentially small solution can appear in the big solutionbut not the other way around. Therefore we must have the situationˆ1 + = ˆ1 + · · · + β ˆ1 − + · · · , (5.21)As x → x pole , β diverges and (5.21) goes over to the relation ( b ). The situation is thesame if ˆ1 − is the big solution: Always the big solution diverges at the degeneration point,while the small solution remains finite .Similar argument can be applied to the other Wronskians, making use of the generalasymptotic behavior of the eigenvectors in the Pohlmeyer gauge, which is of the formˆ i ± ∼ e ± q ( x ) τ ( i ) ( z ∼ z i ) . (5.22)It is clear from this expression that which one of the ˆ i ± diverges as z → z i is governed bythe sign of the real part of the quasi-energy q ( x ). Since the divergence of the eigenfunctionproduces a pole on the Wronskian containing it, we can determine which Wronskian ofthe product is responsible for the pole with the following general rule: At sin p i = 0, theWronskians behave asRe q ( x ) > ⇒ (cid:104) i + , ∗(cid:105) = finite , (cid:104) i − , ∗(cid:105) = ∞ , (5.23)Re q ( x ) < ⇒ (cid:104) i + , ∗(cid:105) = ∞ , (cid:104) i − , ∗(cid:105) = finite . (5.24)Hence, for Re q ( x ) > (cid:104) i − , ∗(cid:105) , while for Re q ( x ) < (cid:104) i + , ∗(cid:105) . Having determined the pole structure, let us next discuss the zeros of the Wronskians. Thedetermination of the zeros is substantially more difficult since, in contrast to the poleswhich are local phenomena, the zeros are determined by the global properties on theRiemann surface. As shown in previous works [24–26], the notion of the WKB curve [23]is one of the main tools to explore such global properties. However, as its name indicates,the WKB curve is useful only when the leading term in the WKB expansion is sufficientlyaccurate. For this reason, it is not powerful enough to fully determine the zeros of theWronskians in the whole region of the spectral parameter space. In this subsection weshall introduce an appropriate generalization of the WKB curve, to be called the exactWKB curve , to overcome this difficulty. Remark: This does not mean of course that there is only one solution at the degeneration point.There must exist another independent solution of new structure, namely the structure which is differentfrom ˆ1 ± . However, as long as we stick to this basis, what we see is that one of the solutions diverges anddisappears. .3.1 WKB approximation and WKB curves In order to motivate the generalized version, we shall first briefly review the ordinaryWKB curves defined in [23].When the expansion parameter ζ is sufficiently small, the leading term of the WKBexpansion for the solutions to ALP (3.33) around z i is given byˆ ψ ∼ exp (cid:32) ± ζ (cid:90) zz ∗ i √ T dz (cid:33) . (5.25)Of the two independent solutions given above, one is the small solution , which decreasesexponentially as it approaches z i and the other is the big solution , which increases ex-ponentially in the same limit. In order to make the variation of the magnitude of thesolution more precise, one defines the WKB curves as the curves along which the phaseof the leading term (5.25) in the WKB expansion is constant. More explicitly, they arecharacterized by the equation Im (cid:32) √ Tζ dz (cid:33) = 0 . (5.26)By analyzing the structure of (5.26), one finds the following three characteristic propertiesof the WKB curves. (i) At generic points on the worldsheet, the WKB curves are non-intersecting. (ii) At a puncture, the WKB curves radiate in all directions from thepuncture. (iii) At a zero of T ( z ), there are three special WKB curves which radiate fromthe zero and separate three different regions of the WKB curves. For details, see figure5.1.Along the WKB curve, the magnitude of the leading term in the WKB expansion (5.25)increases or decreases monotonically, until they reach a zero or a pole of T ( z ). Thus, iftwo punctures z i and z j are connected by a WKB curve and the spectral parameter ζ is sufficiently small, the small solution s i defined around z i will grow exponentially asit approaches the other puncture z j . In other words, the small solution s i behaves likethe big solution around z j . Therefore s i will be linearly independent of s j and hence theWronskian between these two small solutions (cid:104) s i , s j (cid:105) must be non-vanishing.With this logic, we conclude that the Wronskians (cid:104) i ± , j ± (cid:105) are non-vanishing if thefollowing three conditions are satisfied: (a) Two punctures z i and z j are connected by aWKB curve. (b) Two eigenvectors i ± and j ± are both small solutions. (c) The leadingWKB solutions (5.25) are sufficiently accurate.63ero PoleFigure 5.1: Sketch of WKB curves near a zero ( a red cross in the left figure) and a pole(a red circle in the right figure). There are exactly three WKB curves that emanate froma zero. In contrast, there are infinitely many WKB curves radiating from a pole in alldirections. Evidently, the analysis above is valid only in a restricted region of the spectral parameterplane where the approximation by the leading term of the WKB expansion is reliable.Actually, even if we improve the approximation by going to the next order approximation,we still cannot cover the entire spectral parameter plane because such an expansion isonly an asymptotic series. It is indeed possible that as we change x the small and thebig solutions interchange their roles. Such a phenomenon is clearly non-perturbative andcannot be captured by the usual expansion. So to understand the structure of the zeroson the whole spectral parameter plane, it is necessary to generalize the notion of WKBcurves in a non-perturbative fashion.In order to seek such an improvement, we need to look closely at the general structureof the conventional WKB expansion. Let us denote the components of the solution ˆ ψ d tothe ALP in the diagonal gauge (3.33) asˆ ψ d = (cid:18) ψ (1) ψ (2) (cid:19) . (5.27)By substituting (5.27) into the ALP (3.33), we obtain the equations for the components ψ (1) and ψ (2) . Then, upon eliminating ψ (2) in favor of ψ (1) , we get a second-order dif-ferential equation for ψ (1) . To solve this equation, we expand ψ (1) in powers of ζ in theform ψ (1) = (cid:115) ρT − ∂γ √ T exp (cid:20)(cid:90) zz (cid:18) W − ζ + W + ζW + · · · (cid:19)(cid:21) . (5.28)64ne can then determine the one-forms W n order by order recursively. This procedure isdescribed in Appendix D.1. As a result of such a computation, we find that the WKBexpansions for two linearly independent solutions to the ALP can be expressed in thefollowing form: (cid:32) f (1) ± f (2) ± (cid:33) exp (cid:18) ± (cid:90) zz W WKB ( z, ¯ z ; ζ ) (cid:19) . (5.29)Here W WKB ≡ W z WKB dz + W ¯ z WKB d ¯ z is the one-form defined as a power series in ζ , with theleading term given by √ T dz/ (2 ζ ). On the other hand, the functions f (1) ± and f (2) ± aredefined in terms of W z WKB by f (1) ± ≡ k WKB = (cid:115) ρ − √ T ∂γT W z WKB , (5.30) f (2) ± ≡ − i (cid:112) W z WKB (cid:34) ± W z WKB + (cid:32) √ T ζ − ρ cos 2 γ √ T + ∂ ln k WKB (cid:33)(cid:35) . (5.31)With this structure in mind, we now introduce an improved notion of the WKB curve,to be called the “exact WKB curve”, by writing the exact solutions to the ALP in theform ˆ ψ d = (cid:32) f (1)ex f (2)ex (cid:33) exp (cid:18)(cid:90) zz W ex ( z, ¯ z ; ζ ) (cid:19) , (5.32)where f (1)ex and f (2)ex are given by f (1)ex = (cid:115) ρ − √ T ∂γT W z ex , f (2)ex = − i (cid:112) W z ex (cid:34) W z ex + (cid:32) √ T ζ − ρ cos 2 γ √ T + ∂ ln f (1)ex (cid:33)(cid:35) . (5.33)Note that the expression (5.32) is identical in form to (5.29) with the plus sign chosen.However, there is an essential difference. While W WKB is given by the asymptotic seriesin powers of ζ and is hence ambiguous non-perturbatively, W ex on the other hand isunambiguous as it is defined directly by the exact solution ˆ ψ . Of course, if we expand W ex perturbatively in powers of ζ , the series will coincide with W WKB . In this sense, W ex can be regarded as the non-perturbative completion of W WKB . Now one of the virtues of theexpression (5.32) is that we can easily construct another solution satisfying (cid:104) ˆ ψ d , ˆ ψ (cid:48) d (cid:105) = 1by choosing the opposite the signs asˆ ψ (cid:48) d = (cid:18) f (cid:48) ex(1) f (cid:48) ex(2) (cid:19) exp (cid:18) − (cid:90) zz W ex ( z, ¯ z ; ζ ) (cid:19) , (5.34)where f (cid:48) ex(1) and f (cid:48) ex(2) are given by f (cid:48) ex(1) = (cid:115) ρ − √ T ∂γT W z ex , f (cid:48) ex(2) = − i (cid:112) W z ex (cid:34) − W z ex + (cid:32) √ T ζ − ρ cos 2 γ √ T + ∂ ln f (cid:48) ex(1) (cid:33)(cid:35) . (5.35)65sing the definition (5.32), let us now discuss the generalization of the WKB curves.The quantity √ T dz/ζ used to define the original WKB curves is proportional to theleading term in the expansion of W WKB . Therefore the most natural generalization of theWKB curves would be to use W ex , which is a non-perturbative completion of W WKB , todefine them as Im ( W ex ( z ; ζ )) = 0 . (5.36)Unfortunately, there is a problem with this definition. Since there are many exact solutionsto the ALP, a different choice of the solution ˆ ψ d leads to a different W ex and thus todifferent curves. We can avoid this problem by defining the curves in terms of the smallsolution s i (for a general value of ζ ) near each puncture z i . We shall call them the exactWKB curves and denote them by EWKB ( i ) .The precise definition is given as follows: The exact WKB curves associated to thepuncture z i are defined as the curves satisfying the equationIm (cid:0) W ( i )ex ( z ; ζ ) (cid:1) = 0 , (5.37)where W ( i )ex is the exponential factor for the solution s i , which is the smaller of the twoeigenvectors i + and i − . Explicitly, it is defined through the expression s i ∝ (cid:32) f (1)ex f (2)ex (cid:33) exp (cid:18)(cid:90) zz W ( i )ex ( z, ¯ z ; ζ ) (cid:19) . (5.38)Let us now make several comments. First, it is easy to see that this definition of theexact WKB curves reduces to that of the ordinary WKB curves when ζ is sufficientlysmall. Second, as in (5.34), with a flip of sign in the exponent, we can obtain anothersolution b i ≡ (cid:18) f (cid:48) ex(1) f (cid:48) ex(2) (cid:19) exp (cid:18) − (cid:90) zz W ( i )ex ( z, ¯ z ; ζ ) (cid:19) , (5.39)which is big near the puncture z i and satisfies (cid:104) s i , b i (cid:105) = 1. Such a solution b i , however,is not guaranteed to be an eigenvector since the eigenvector distinct from s i is in generalgiven by a linear combination of the form b i + cs i .Now the definition of EWKB ( i ) given above refers to a specific puncture from whichthe curves emanate. In order for the notion of the exact WKB curve to be valid for theentire worldsheet, we must guarantee that the definitions of EWKB ( i ) ’s for i = 1 , , s i as we follow an EWKB ( i ) . Along such a curve the phase of the66xponential factor of s i stays constant, while its magnitude increases monotonically ,until it reaches some endpoint. Consider the case in which this endpoint is the punctureat z j . In such a case, we know that s i grows exponentially as it approaches z j and in factbehaves like a big solution b j , up to an admixture of the exponentially small solution s j .Thus, with sufficient accuracy, s i can be expressed in the small neighborhood of z j as s i ∝ b j = (cid:18) f (cid:48) ex(1) f (cid:48) ex(2) (cid:19) exp (cid:18) − (cid:90) zz W ( j )ex ( z, ¯ z ; ζ ) (cid:19) . (5.40)But since the exponent of the small solution s j , which is used to define EWKB ( j ) , is thesame as that of b j except for the sign, we see that by definition the curve we have beenfollowing becomes an EWKB ( j ) curve in the vicinity of z j , when z i and z j are connectedby such a curve. Therefore the definitions of EWKB ( i ) and EWKB ( j ) are indeed globallyconsistent.Let us now make use of the exact WKB curves to determine the analytic properties ofthe Wronskians. First, by following exactly the same logic as in the case of the ordinaryWKB curves, we can immediately conclude that the Wronskian involving two small solu-tions s i and s j must be nonzero if two punctures z i and z j are connected by some exactWKB curves. Although this is an extremely useful information, the problem seems to bethat, unlike the ordinary WKB curves, we do not know the configurations of the EWKBcurves since the exact solutions to the ALP are not available.Nevertheless, we shall show below that by making use of a characteristic quantity de-fined locally around each puncture for the EWKB curves, it is possible to fully classify thetopology (connectivity) of the curves on the entire worldsheet. The quantity in question isthe “number density” of the EWKB curves emanating from a puncture at z i . To motivateits definition, consider two such curves which emanate from z i and end at z j and let theconstant phase of W ( i )ex along the two curves be φ and φ . Evidently the magnitude of thedifference | φ − φ | is the same around z i and around z j , that is, it is conserved. If thereis no singularity in the region between these lines, we can draw in more EWKB curvesconnecting z i and z j . Because of the property of the constancy of the phase differencenoted above, it is quite natural to draw the curves in such a way that the difference ofthe phases of the adjacent curves is some fixed unit angle. Going around z i and countingthe number of such lines, we can define the number (density) of the EWKB ( i ) curves as N i ≡ π (cid:73) C i | Im W ( i )ex | , (5.41) Strictly speaking the small eigenvector (5.38) also contains a prefactor in front of the exponential.This prefactor, however, does not play a significant role in our discussion since it drops out if we considerthe ratio of two solutions s i /b i . It is in fact sufficient to know the ratio in order to identify the smallsolution and the big solution. In (5.41), we have chosen a convenient normalization of N i . C i is an infinitesimal circle around z i . Although N i is not an integer in general, wewill call it “a number of lines”. Actually we can express N i in a more explicit way. Fromthe asymptotic behavior of i ± (2.96), we can obtain the form of W ( i )ex near z i as W ( i )ex ∼ ± (cid:0) q i ( x ) dτ ( i ) + ip i ( x ) dσ ( i ) (cid:1) as z → z i . (5.42)Here ( τ ( i ) , σ ( i ) ) is the local coordinate defined in (2.97), and + or − sign is chosen depend-ing on which of the solutions i ± is small. Substituting (5.42) into the definition (5.41),we obtain a simple expression N i ≡ | Re p i ( x ) | . (5.43)Since the phase around the puncture is governed by the local monodromy, it is naturalthat N i can be expressed in terms of p i ( x ).Before we make use of the concept of N i in a more global context, let us derive twoimportant properties of the EWKB ( i ) ’s which will be necessary for determining theirconfigurations.The first property will be termed the non-contractibility . It can be stated as follows:“All the exact WKB curves which start and end at the same puncture are non-contractible.”In other words, such curves must go around a different puncture at least once. The proofis simple. Recall that the Wronskians between small solutions should be nonzero if twopunctures are connected by an exact WKB curve. If we apply this statement to the samepuncture z i connected by an EWKB curve, we would conclude that (cid:104) s i , s i (cid:105) is non-zero,which is clearly false. The only way to be consistent with the general assertion above isthat the curve is non-contractible and the solution gets transformed by the non-trivialmonodromy Ω as it goes around other punctures. In this case the Wronskian is of theform (cid:104) s i , Ω s i (cid:105) , which need not vanish.The next property is concerned with the endpoints of the exact WKB curves. It canbe stated as follows:“ All but finitely many exact WKB curves terminate at punctures. ”The proof can be given as follows. As in the case of the ordinary WKB curves, the possibleendpoints are the zeros or the poles of W ( i )ex . Concerning the former, the number of exactWKB curves flowing into a zero is always finite, as shown in figure 5.1. On the other hand,a pole can be the endpoint of infinitely many curves and thus plays a crucial role in thestudy of the analyticity of the Wronskians. Now there are three different types of poles for68 ( i )ex . The first is a puncture, at which the vertex operator is inserted. The second type ofa pole corresponds to the situation where the small eigenvector s i develops a singularityat a position different from the puncture. Since we only consider the worldsheet withoutadditional singularities as mentioned in section 2.3, such a singularity in s i should notoccur. The last type of divergence for W ( i )ex occurs when s i develops a zero. Indeed, s i in general has several zeros on the Riemann surface. However, such points cannot be theendpoints of the exact WKB curves for the following reason: At the zeros of s i , the ratio s i /b i of the small and the big solutions must also vanish . But this contradicts the basicproperty of the exact WKB curve that such a ratio, determined by the exponential factorin (5.38), monotonically increases along the exact WKB curve as we move away from z i .From these considerations, we find that apart from a finite number of curves which canflow into zeros of W ( i )ex , the rest of the infinitely many exact WKB curves must end at thepunctures.The two properties we have proved above are extremely important for the followingreason. They provide certain global restrictions for the EWKB curves for all values ofthe spectral parameter, about which we only know the local behaviors explicitly in thevicinity of the punctures. Below, they allow us to show that there are essentially twodistinct classes of configurations for the exact WKB curves.These two classes are distinguished by whether the number of lines N i fully satisfy thetriangle inequalities or not . When N i ’s satisfy the relations N i + N j − N k > , (5.44)for all possible combinations of distinct i, j, k , we refer to such a configuration as symmet-ric . It is easy to show that if (5.44) is satisfied the number of lines connecting z i and z j cannot be zero. As this holds for all the interconnecting lines, the three punctures mustbe piece-wise connected to each other as in the left figure of figure 5.2.On the other hand, in the second case, which we shall call asymmetric , not all thetriangle inequalities are satisfied. For example, one is violated like N + N − N < . (5.45)In this case, one can readily convince oneself that, while all the curves emanating from z and z end at z , there must exist a non-contractible curve connecting z to itself. Thisis depicted in the right figure of figure 5.2. The big solution b i cannot vanish at such points so as to ensure the normalization condition (cid:104) s i , b i (cid:105) =1. In the case of the usual WKB curves, W ex ∼ (cid:112) T ( z ) dz and hence N i is proportional to κ i . Classifi-cation by the triangle inequalities for κ i already appeared in [9]. z z (a) Symmetric case ( N i + N j − N k > z z z (b) Asymmetric case ( N + N − N < N i = | Re p i ( x ) | . Note that N i depends on x . In fact it happensthat as x changes a symmetric configuration can turn into an asymmetric configurationand vice versa. In an application of the present idea to the classical three-point functionin Liouville theory [29], it was checked that such a transition must be taken into accountin order to obtain the correct result. Below, we will see explicitly how the patterns ofthe configurations of the exact WKB curves analyzed above can be used to determine thezeros of the Wronskians. As an example, let us focus on the factorsin (cid:18) p + p + p (cid:19) , (5.46)and determine which Wronskians develop a zero when this factor vanishes. (The logicbelow applies to all the other cases straightforwardly.) From the relations (5.9)–(5.14),we find that the products of Wronskians that become zero are (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) , (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) , (cid:104) + , + (cid:105)(cid:104) − , − (cid:105) . (5.47)For convenience, let us define the following two sets of eigenvectors, namely the set S + ≡{ + , + , + } and the set S − ≡ { − , − , − } . An important feature of the quantities shownin (5.47) is that only the Wronskians of the eigenstates in the same group, S + or S − ,appear. This is in fact a general feature and holds also for other situations.70ow, let us present two theorems, which will be useful in the determination of thezeros. The first theorem is the following assertion, which we have already proved: Theorem 1. When two punctures z i and z j are connected by an exact WKB curve,the Wronskian between the two small eigenvectors (cid:104) s i , s j (cid:105) is non-vanishing.The second theorem classifies the possibilities of the patterns of the zeros and is statedas follows: Theorem 2. There are only two distinct possibilities concerning the zeros of theWronskians in (5.47): Either (a) all the Wronskians among the members of S + arezero and those among S − are nonzero, or (b) all the Wronskians among S − are zeroand those among S + are nonzero.The proof is as follows. Let us first note that in each product of two Wronskians appearingin (5.47), only one of them vanishes. In fact if both factors become zero simultaneously,the product develops a double zero, which contradicts the fact that the zeros of (5.46) areall simple zeros. This property implies that in the list given in (5.47), at least two of theindividual Wronskians which actually vanish must be between the members belonging tothe same set, which can be S + or S − . Suppose they belong to S + . Since (cid:104) i + , j + (cid:105) = 0means that i + and j + are parallel to each other, vanishing of two such different Wronskiansbetween the states of S + implies that in fact all the three states in S + are proportional toeach other. Therefore the third Wronskian from the set S + must also vanish. Obviouslythe same logic applies to the S − case. This proves the theorem.We can now analyze the zeros of the Wronskians using these theorems. First considerthe symmetric case. Since one of the states i ± must be a small solution, either S + or S − must contain two small solutions. For a symmetric configuration, they must be connectedby an exact WKB curve. Then by theorem 1 the Wronskian between them must be non-vanishing. Theorem 2 further asserts that all the Wronskians for the members of that setare non-vanishing, while the ones for elements of the other set all vanish.Next, consider the asymmetric case. For simplicity, let us assume that N > N + N is satisfied . In such a case, there exist exact WKB curves which start from z , go around z (or z ), and return to z . To make use of the existence of such a curve, consider thefollowing Wronskians: (cid:104) + , Ω + (cid:105) , (cid:104) − , Ω − (cid:105) . (5.48) Generalization to other cases is straightforward. 71o compute them, we first note that 1 ± can be expressed in terms of 2 ± in the followingmanner 1 ± = (cid:104) ± , − (cid:105) + − (cid:104) ± , + (cid:105) − . (5.49)Then, applying Ω to (5.49) and substituting them to (5.48), we can express (5.48) interms of the ordinary Wronskians as (cid:104) + , Ω + (cid:105) = 2 i sin p (cid:104) + , − (cid:105)(cid:104) + , + (cid:105) , (5.50) (cid:104) − , Ω − (cid:105) = 2 i sin p (cid:104) − , − (cid:105)(cid:104) − , + (cid:105) . (5.51)Consider the case where 1 + is the small solution. Since Ω + can be obtained by parallel-transporting 1 + along the exact WKB curve which starts and ends at z , Ω + must behaveas the big solution around z . Therefore, the Wronskian (cid:104) + , Ω + (cid:105) is non-vanishing inthis case. Then from (5.50) it follows that (cid:104) + , + (cid:105) must also be non-vanishing. Applyingthe theorem 2, we conclude that the Wronskians between the members of S + are non-vanishing and those of S − all vanish. In an entirely similar manner, when 1 − is the smalleigenvector, we obtain the result where the roles of S + and S − are interchanged.Performing similar analyses for the other cases, we obtain the general rules summarizedbelow.Rule 1: Decomposition of the eigenvectors into two groups. When a factor of the form sin ( (cid:80) i (cid:15) i p i / 2) vanishes, the Wronskians which vanishare the ones among { (cid:15) , (cid:15) , (cid:15) } or the ones among { − (cid:15) , − (cid:15) , − (cid:15) } .Rule 2: Symmetric case. When the configuration of the exact WKB curves is symmetric, the Wronskiansfrom the group which contains two or more small solutions are nonzero whereas theWronskians from the other group are zero.Rule 3: Asymmetric case. When the configuration of the exact WKB curves is asymmetric and N i ’s satisfy N i > N j + N k , the Wronskians from the group which contains the smaller of the twosolutions i ± are nonzero whereas the Wronskians from the other group are zero.In the next subsection, we will utilize these rules to evaluate the individual Wronskians. Making use of the data for the analyticity of the Wronskians obtained in the previoussubsection, we now set up and solve a Riemann-Hilbert problem to decompose the product72f Wronskians and extract the individual Wronskians.The standard method for such aprocedure is known as the Wiener-Hopf decomposition, which extracts from a complicatedfunction a part regular on the upper half plane and the part regular on the lower half plane.The typical set up is as follows. Suppose F ( x ) is a function which decreases sufficientlyfast at infinity and can be written as a sum of two components F ( x ) = F ↑ ( x ) + F ↓ ( x ),where F ↑ ( x ) is regular on the upper half plane while F ↓ ( x ) is regular on the lower halfplane. Then, each component, in the region where it is regular, can be extracted from F ( x ) as F ↑ ( x ) = (cid:90) ∞−∞ dx (cid:48) πi x (cid:48) − x F ( x (cid:48) ) (Im x > , (5.52) F ↓ ( x ) = − (cid:90) ∞−∞ dx (cid:48) πi x (cid:48) − x F ( x (cid:48) ) (Im x < . (5.53)These equations can be easily proven by first substituting F ( x (cid:48) ) = F ↑ ( x (cid:48) ) + F ↓ ( x (cid:48) ) on theright hand side and then closing the integration contour for F ↑ ( x (cid:48) ) ( F ↓ ( x (cid:48) )) on the upper(lower) half plane. Now when the argument x is not in the region specified in (5.52) and(5.53), we need to analytically continue the above formulas. For instance, F ↑ ( x ) in theregion where Im x < F ↑ ( x ) = F ( x ) − F ↓ ( x ) = F ( x ) + (cid:90) ∞−∞ dx (cid:48) πi x (cid:48) − x F ( x (cid:48) ) . (5.54)Note that the first term F ( x ) on the right hand side can be thought of as due to theintegral along a small circle around x (cid:48) = x .To apply this method to the case of our interest, namely to the equations (5.9)–(5.14),we take the logarithm and represent them in a general form asln (cid:104) i (cid:15) i , j (cid:15) j (cid:105) + ln (cid:104) i − (cid:15) i , j − (cid:15) i (cid:105) = ln sin (cid:18) (cid:15) i p i + (cid:15) j p j + p k (cid:19) + ln sin (cid:18) (cid:15) i p i + (cid:15) j p j − p k (cid:19) − ln sin p i − ln sin p j . (5.55)Here (cid:15) i denotes a + or − sign. In this process, we have neglected the contributions of theform ln( − p i ( x ) is defined on aRiemann surface with branch cuts, we need to generalize the Wiener-Hopf decompositionformula in an appropriate way, as discussed below.73 .4.1 Separation of the poles Let us first decompose the terms of the form − ln sin p i , which give rise to poles of theWronskians. As shown in the previous section, which Wronskian develops a pole is de-termined purely by the sign of the real part of the quasi-momentum q i ( x ). Therefore,we should be able to decompose the quantity − ln sin p i by using a convolution integralalong the curve defined by Re q i = 0. For the ordinary Wiener-Hopf decomposition, theconvolution kernel is given simply by 1 / ( x − x (cid:48) ). In the present case, however, we havea two-sheeted Riemann surface and hence we must make sure that the kernel has thesimple pole only when x and x (cid:48) coincide on the same sheet. When they are on top of eachother on different sheets, no singularity should occur. The appropriate kernel with thisproperty is given by (cid:98) K i ( x (cid:48) ; x ) ≡ x (cid:48) − x ) (cid:32)(cid:115) ( x − u i )( x − ¯ u i )( x (cid:48) − u i )( x (cid:48) − ¯ u i ) + 1 (cid:33) . (5.56)When x and x (cid:48) get close to each other but on different sheets, the square root factor tendsto − x (cid:48) tends to ∞ , the kernel (cid:98) K i ( x (cid:48) ; x ) decreases like ( x (cid:48) ) − , which is sufficientlyfast for our purpose.With such a convolution kernel, we can carry out the Wiener-Hopf decomposition inthe usual way. Namely the term − ln sin p i ( x ) can be decomposed into the contributionsof (cid:104) i + , j (cid:15) j (cid:105) ( x ) and (cid:104) i − , j − (cid:15) j (cid:105) ( x ) as (cid:104) i + , j (cid:15) j (cid:105) ( x ) (cid:51) (cid:73) Γ i + (cid:98) K i ∗ ( − ln sin p i ) , (5.57) (cid:104) i − , j − (cid:15) j (cid:105) ( x ) (cid:51) (cid:73) Γ i − (cid:98) K i ∗ ( − ln sin p i ) , (5.58)where the convolution integral is defined as (cid:90) A ∗ B ≡ (cid:90) dx (cid:48) πi A ( x (cid:48) ; x ) B ( x (cid:48) ) . (5.59)As for the contours of integration, Γ i + is defined by Re q i = 0 and Γ i − stands for − Γ i + .The direction of the contour Γ i + is defined such that (cid:104) i + , j (cid:15) j (cid:105) ( x ) does not contain polesin the region to the left of the contour .Now note that under the holomorphic involution x → ˆ σx , the quasi-momentum p i ( x )and the square-root contained in (5.56) simply flip sign. Making use of this property, we A typical form of the contour is depicted in figure 7.4 in section 7, where we study explicit examples. (cid:104) i + , j (cid:15) j (cid:105) ( x ) (cid:51) − (cid:73) Γ ui + K i ∗ ln sin p i , (5.60) (cid:104) i − , j − (cid:15) j (cid:105) ( x ) (cid:51) − (cid:73) Γ ui − K i ∗ ln sin p i . (5.61)Here, Γ ui ± denotes the portion of Γ i ± on the upper-sheet of the spectral curve and thekernel K i ( x (cid:48) ; x ) (without a hat) is defined by K i ( x (cid:48) ; x ) ≡ x (cid:48) − x (cid:115) ( x − u i )( x − ¯ u i )( x (cid:48) − u i )( x (cid:48) − ¯ u i ) . (5.62)Again we have neglected the factors of the form ln( − 1) arising from the sign flip of p i ( x ),as they only modify the overall phase of the Wronskians and the three-point functions.It is important to note that (5.57) and (5.58) are valid only when x is on the lefthand side of the contours, just as in the case of the ordinary Wiener-Hopf decomposition.When the argument x is on the right hand side of the contour Γ i ± , we must add − ln sin p i to (5.57) and (5.58), as explained in (5.54). Such effects can be taken into account alsoin (5.60) and (5.61), if x is on the upper sheet, by adding a small circle encircling x (cid:48) = x counterclockwise to the integration contours. In what follows, such contributions will bereferred to as contact terms . Next we shall discuss the decomposition of the first two terms on the right hand side of(5.55), which are responsible for the zeros of the Wronskians. To perform the decomposi-tion, again we need to determine the appropriate convolution kernel and the integrationcontour.Let us first discuss the convolution kernel. As the terms of our focus depend on allthe quasi-momenta p i ( x )’s, the appropriate convolution kernel must be a function on theRiemann surface which contains all the branch cuts of the p i ( x )’s. Such a kernel can beeasily written down as a generalization of the expression (5.56) and is given by (cid:98) K all ≡ x (cid:48) − x ) (cid:89) i =1 (cid:32)(cid:115) ( x − u i )( x − ¯ u i )( x (cid:48) − u i )( x (cid:48) − ¯ u i ) + 1 (cid:33) . (5.63)Since there are two choices of sign for each square root factor on the right hand sideof (5.63), (cid:98) K all is properly defined on the eightfold cover of the complex plane. In whatfollows, we distinguish these eight sheets as {∗ , ∗ , ∗} -sheet, where the successive entry ∗ 75s either “ u ” denoting upper sheet or “ l ” denoting lower sheet, referring to the two sheetsfor p ( x ), p ( x ) and p ( x ) respectively. It is clear that the kernel (5.63) has a pole witha residue +1 at x (cid:48) = x only when two-points are on the same sheet. Therefore it has adesired property for the Wiener-Hopf decomposition.Let us next turn to the contour of integration. As discussed in the previous section,the zeros of the Wronskians are determined by the following two properties: (i) The con-nectivity of the exact WKB-curves and (ii) the relative magnitude of the eigenvectors i ± .Therefore, curves across which these two properties change can be the possible integra-tion contours. Corresponding to the properties (i) and (ii) above, there are two typesof integration contours; the curves defined by Re q i ( x ) = 0 and the curves defined by N i = N j + N k . An important point to bare in mind is that in general only some portionsof these curves will be the proper integration contours, since in some cases the analyticityof the Wronskians does not change even when we cross these curves. In order to determinethe correct integration contours explicitly, we need to apply the general rules derived inthe previous section. However, as the form of the contours determined through such aprocedure depends on the specific details of the choice of the external states, we will post-pone such an analysis until section 7, where we work out some specific examples. Thus,in what follows we will denote the integration contours without specifying their explicitforms as M ±±± , where M (cid:15) (cid:15) (cid:15) denotes the contour we use to determine the contributionof the factor sin ( (cid:80) i (cid:15) i p i ) to (cid:104) i (cid:15) i , j (cid:15) j (cid:105) . They are defined such that they flip the orientationif we flip the signs of three indices, for example M +++ = −M −−− Employing the kernel and the contours given above, let us perform the decompositionof the product of Wronskians, taking that of (cid:104) + , + (cid:105) and (cid:104) − , − (cid:105) as a representativeexample. Applying the Wiener-Hopf decomposition to the relation (5.55) with i = 1 , j = 2and (cid:15) = + , (cid:15) = +, we obtain (cid:104) + , + (cid:105) (cid:51) (cid:73) M +++ (cid:98) K all ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:73) M ++ − (cid:98) K all ∗ ln sin (cid:18) p + p − p (cid:19) , (5.64) (cid:104) − , − (cid:105) (cid:51) (cid:73) M −−− (cid:98) K all ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:73) M −− + (cid:98) K all ∗ ln sin (cid:18) p + p − p (cid:19) . (5.65)As in the case of the ordinary Wiener-Hopf decomposition, the expressions (5.64) and(5.65) are valid only when x is located to the left of the integration contour. Additionalterms, to be discussed shortly, are needed when x is on the other side of the contour.Let us now show that the kernel (cid:98) K all used in (5.64) and (5.65) can be effectivelyreplaced by simpler combinations of the form ( K i + K j ) / 8. To explain the idea, consider76he following integral as an example: (cid:73) M +++ dx (cid:48) πi (cid:98) K all ( x (cid:48) ; x ) ln sin (cid:18) p + p + p (cid:19) ( x (cid:48) ) . (5.66)As the first step, we make a change of integration variable from x (cid:48) to ˆ σ x (cid:48) , where ˆ σ i denotesthe holomorphic involution with respect to p i , namely the operation that exchanges thetwo sheets associated with p i . Although this clearly leaves the value of the integral intact,the form of the integral changes. One can easily verify that the following transformationformulas for the integrand and the contours hold:ln sin (cid:18) p + p + p (cid:19) (ˆ σ x (cid:48) ) = ln sin (cid:18) p + p − p (cid:19) ( x (cid:48) ) , (5.67) (cid:98) K all (ˆ σ x (cid:48) ; x ) = (cid:98) K (3)all ( x (cid:48) ; x ) , (5.68) (cid:73) M +++ d (ˆ σ x (cid:48) ) = (cid:73) M ++ − dx (cid:48) . (5.69)In the second line (5.68), the “sign-flipped kernel” (cid:98) K (3)all is defined by (cid:98) K (3)all ≡ x (cid:48) − x ) (cid:32) − (cid:115) ( x − u )( x − ¯ u )( x (cid:48) − u )( x (cid:48) − ¯ u ) + 1 (cid:33) (cid:89) (cid:96) =1 , (cid:32)(cid:115) ( x − u (cid:96) )( x − ¯ u (cid:96) )( x (cid:48) − u (cid:96) )( x (cid:48) − ¯ u (cid:96) ) + 1 (cid:33) . (5.70)Making such transformations, we can re-express the integral (5.66) as (cid:73) M ++ − dx (cid:48) πi (cid:98) K (3)all ( x (cid:48) ; x ) ln sin (cid:18) p + p − p (cid:19) ( x (cid:48) ) . (5.71)Performing similar analysis for all the possible sign-flips, we obtain 2 different expressionsfor (5.66). Then averaging over all the 2 expressions, we find that the final expressionsare given in terms of the kernels K i as follows: (cid:104) + , + (cid:105) (cid:51) (cid:18)(cid:73) M +++ ( K + K ) ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:73) M ++ − ( K + K ) ∗ ln sin (cid:18) p + p − p (cid:19) + (cid:73) M + − + ( K − K ) ∗ ln sin (cid:18) p − p + p (cid:19) + (cid:73) M − ++ ( −K + K ) ∗ ln sin (cid:18) − p + p + p (cid:19)(cid:19) , (5.72) (cid:104) − , − (cid:105) (cid:51) (cid:18)(cid:73) M −−− ( K + K ) ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:73) M −− + ( K + K ) ∗ ln sin (cid:18) p + p − p (cid:19) + (cid:73) M − + − ( K − K ) ∗ ln sin (cid:18) p − p + p (cid:19) + (cid:73) M + −− ( −K + K ) ∗ ln sin (cid:18) − p + p + p (cid:19)(cid:19) . (5.73)77ust as before, we neglected the contributions of the form ln( − 1) as leading to purephases. Also, the same remarks made below equations (5.64) and (5.65) on the positionof x relative to the contour lines apply to the expressions (5.72) and (5.73) above.Finally, for later convenience, let us further re-write the above expressions as integralsperformed purely on the { u, u, u } -sheet. Each contour M (cid:15) (cid:15) (cid:15) has parts on the eightdifferent sheets denoted by M u,u,u(cid:15) (cid:15) (cid:15) , M u,u,l(cid:15) (cid:15) (cid:15) , etc., where the superscripts indicate therelevant sheet in an obvious way. Consider for example the first integral in (5.72) alongthe contour M +++ . The form as given is for the portion M uuu +++ . For the portion denotedby M ulu +++ for example, if we wish to express its contribution in terms of an integral onthe { u, u, u } -sheet, we need to change the sign of K and p . Then the integral becomesidentical to that of the first term in the second line of (5.72), except along M uuu + − + . Insimilar fashions we can re-express the contributions from the eight parts of M +++ interms of the integrals on the { u, u, u } -sheet. After repeating the same procedure for therest of the three terms in (5.72), one finds that the net effect is that each term of (5.72)is multiplied by a factor of eight, with each contour restricted to the { u, u, u } -sheet. Inthis way we obtain the representations (cid:104) + , + (cid:105) (cid:51) (cid:32)(cid:73) M uuu +++ ( K + K ) ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:73) M uuu ++ − ( K + K ) ∗ ln sin (cid:18) p + p − p (cid:19) + (cid:73) M uuu + − + ( K − K ) ∗ ln sin (cid:18) p − p + p (cid:19) + (cid:73) M uuu − ++ ( −K + K ) ∗ ln sin (cid:18) − p + p + p (cid:19)(cid:33) , (5.74) (cid:104) − , − (cid:105) (cid:51) (cid:32)(cid:73) M uuu −−− ( K + K ) ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:73) M uuu −− + ( K + K ) ∗ ln sin (cid:18) p + p − p (cid:19) + (cid:73) M uuu − + − ( K − K ) ∗ ln sin (cid:18) p − p + p (cid:19) + (cid:73) M uuu + −− ( −K + K ) ∗ ln sin (cid:18) − p + p + p (cid:19)(cid:33) . (5.75)The results obtained in this subsection and the previous subsection are both expressedin terms of certain convolution integrals on the spectral curve. Thus, in what follows, wewill denote their sum by Conv (cid:104) i ± , j ± (cid:105) .Before ending this subsection, let us make one important remark. Although eachconvolution integral obtained so far is divergent at x = ± 1, the divergence cancels inthe sum Conv (cid:104) i ± , j ± (cid:105) . Thus the contribution singular at x = ± One can confirm this by expanding the convolution integrals around x = ± .5 Singular part and constant part of the Wronskians In addition to the main non-trivial parts determined by the Wiener-Hopf decompositiondescribed above, there are two further contributions to the Wronskians. One is the con-tribution singular at x = ± 1, coming from such structure in the connections used in ALP.The other is the possibility of adding a constant function on the spectral curve. In thissubsection, we will determine these two contributions.Let us first focus on terms singular at x = 1. To determine such terms, we will needthe WKB expansions around x = 1 for all the Wronskians, not just the ones that werediscussed in section 3.2, namely (cid:104) i + , j + (cid:105) and (cid:104) i − , j − (cid:105) . This is because of the followingreason: Although the formulas we obtained for the contribution of the action and thatof the wave function appear to contain Wronskians of the type (cid:104) i + , j + (cid:105) and (cid:104) i − , j − (cid:105) only, we must understand their behavior when they are followed into the second sheetas well in order to know the analyticity property on the entire Riemann surface. Asshown in (2.100), when we cross the branch cut associated with p i ( x ) into the lower sheet,the eigenfunctions i + and i − behave like i − and − i + on the upper sheet, respectively, .Therefore the behavior of (cid:104) i + , j + (cid:105) on the { u, l, ∗} -sheet can be obtained from the behaviorof (cid:104) i + , j − (cid:105) on the { u, u, ∗} -sheet, etc.Now the WKB expansions of the Wronskians of the type (cid:104) i + , j − (cid:105) can be obtainedfrom those of (cid:104) i + , j + (cid:105) by the use of the following Schouten identities: (cid:104) i + , j − (cid:105)(cid:104) j + , k + (cid:105) + (cid:104) i + , j + (cid:105)(cid:104) j − , k + (cid:105) + (cid:104) i + , k + (cid:105)(cid:104) j − , j + (cid:105) = 0 . (5.76)Indeed these identities can be regarded as the equations for the six unknown Wronskiansof the form (cid:104) i + , j − (cid:105) . If we consider all the combinations of i, j and k in (5.76), we obtainthree independent equations. Combining them with the equations (5.12)–(5.14) for theproducts of the Wronskians, we can completely determine (cid:104) i + , j − (cid:105) ’s in terms of (cid:104) i + , j + (cid:105) 79n the following form: (cid:104) + , − (cid:105) = e − i ( p + p − p ) / sin (cid:0) p − p − p (cid:1) sin p (cid:104) + , + (cid:105)(cid:104) + , + (cid:105) , (5.77) (cid:104) − , + (cid:105) = e i ( p + p − p ) / sin (cid:0) p − p − p (cid:1) sin p (cid:104) + , + (cid:105)(cid:104) + , + (cid:105) , (5.78) (cid:104) + , − (cid:105) = e − i ( − p + p + p ) / sin (cid:0) − p + p − p (cid:1) sin p (cid:104) + , + (cid:105)(cid:104) + , + (cid:105) , (5.79) (cid:104) − , + (cid:105) = e i ( − p + p + p ) / sin (cid:0) p + p − p (cid:1) sin p (cid:104) + , + (cid:105)(cid:104) + , + (cid:105) , (5.80) (cid:104) + , − (cid:105) = e − i ( p − p + p ) / sin (cid:0) − p − p + p (cid:1) sin p (cid:104) + , + (cid:105)(cid:104) + , + (cid:105) , (5.81) (cid:104) − , + (cid:105) = e i ( p − p + p ) / sin (cid:0) − p + p + p (cid:1) sin p (cid:104) + , + (cid:105)(cid:104) + , + (cid:105) . (5.82)From these expressions, we can obtain the WKB-expansion for every Wronskian using theresults for (cid:104) i + , j + (cid:105) .The singular term of the Wronskians is given simply by the leading term in the WKBexpansion. For instance, the singular terms for (cid:104) i + , j + (cid:105) and (cid:104) i − , j − (cid:105) at x = 1 on the { u, u, u } -sheet is determined from the expansion (3.37) and (3.38) asln (cid:104) + , + (cid:105) x ∼ ∼ − x (cid:90) (cid:96) √ T dz , ln (cid:104) − , − (cid:105) x ∼ ∼ − x (cid:90) (cid:96) √ T dz , (5.83)ln (cid:104) + , + (cid:105) x ∼ ∼ − x (cid:90) (cid:96) √ T dz , ln (cid:104) − , − (cid:105) x ∼ ∼ − x (cid:90) (cid:96) √ T dz , (5.84)ln (cid:104) + , + (cid:105) x ∼ ∼ − x (cid:90) (cid:96) ˆ31 √ T dz , ln (cid:104) − , − (cid:105) x ∼ ∼ − x (cid:90) (cid:96) √ T dz . (5.85)Then by using (5.77)–(5.82) we can determine the singular terms for (cid:104) i + , j − (cid:105) on the { u, u, u } -sheet asln (cid:104) + , − (cid:105) x ∼ ∼ πi ( κ + κ − κ )1 − x + 21 − x (cid:90) (cid:96) ˆ2ˆ3 + (cid:96) ˆ31 √ T dz , (5.86)ln (cid:104) − , + (cid:105) x ∼ ∼ πi ( − κ − κ + κ )1 − x + 21 − x (cid:90) (cid:96) + (cid:96) ˆ3ˆ2 √ T dz , (5.87)ln (cid:104) + , − (cid:105) x ∼ ∼ πi ( − κ + κ + κ )1 − x + 21 − x (cid:90) (cid:96) + (cid:96) √ T dz , (5.88)ln (cid:104) − , + (cid:105) x ∼ ∼ πi ( κ − κ − κ )1 − x + 21 − x (cid:90) (cid:96) ˆ31 + (cid:96) √ T dz , (5.89)ln (cid:104) + , − (cid:105) x ∼ ∼ πi ( κ − κ + κ )1 − x + 21 − x (cid:90) (cid:96) + (cid:96) √ T dz , (5.90)ln (cid:104) − , + (cid:105) x ∼ ∼ πi ( − κ + κ − κ )1 − x + 21 − x (cid:90) (cid:96) + (cid:96) √ T dz . (5.91)80n order to determine the singular terms completely, we also need to understand thesingular behavior on other sheets. As already described, this can be done by utilizing thefact that i + and i − transform into i − and − i + respectively as one crosses a branch cutassociated to p i ( x ). For instance, applying this rule we can easily find that the singularterm for (cid:104) + , + (cid:105) must behave in the following way on each sheet: (cid:104) + , + (cid:105) x ∼ ∼ − x (cid:90) (cid:96) √ T dz (on the { u , u , ∗} -sheet) , (5.92) (cid:104) + , + (cid:105) x ∼ ∼ πi ( κ + κ − κ )1 − x + 21 − x (cid:90) (cid:96) ˆ2ˆ3 + (cid:96) ˆ31 √ T dz (on the { u , l , ∗} -sheet) , (5.93) (cid:104) + , + (cid:105) x ∼ ∼ πi ( − κ − κ + κ )1 − x + 21 − x (cid:90) (cid:96) + (cid:96) ˆ3ˆ2 √ T dz (on the { l , u , ∗} -sheet) , (5.94) (cid:104) + , + (cid:105) x ∼ ∼ − x (cid:90) (cid:96) √ T dz (on the { l , l , ∗} -sheet) . (5.95)Combining all these results, it is possible to write down the expression on the entireRiemann surface which gives the correct singular behavior on the respective sheet. It isgiven by Sing + [ (cid:104) + , + (cid:105) ] = 11 − x (cid:115) ( x − u )( x − ¯ u )(1 − u )(1 − ¯ u ) (cid:32) πi ( κ + κ − κ ) + 2 (cid:90) (cid:96) ˆ1ˆ2 + (cid:96) ˆ2ˆ3 + (cid:96) ˆ31 √ T dz (cid:33) + 11 − x (cid:115) ( x − u )( x − ¯ u )(1 − u )(1 − ¯ u ) (cid:32) πi ( − κ − κ + κ ) + 2 (cid:90) (cid:96) + (cid:96) + (cid:96) ˆ1ˆ2 √ T dz (cid:33) . (5.96)Here and hereafter, we will use the notation Sing ± [ f ( x )] to denote the singular term of f ( x ) around x = ± 1. In an entirely similar manner, we can determine the terms singularat x = − Sing − [ (cid:104) + , + (cid:105) ] = 11 + x (cid:115) ( x − u )( x − ¯ u )(1 − u )(1 − ¯ u ) (cid:32) πi ( − κ − κ + κ ) + 2 (cid:90) (cid:96) ˆ1ˆ2 + (cid:96) ˆ2ˆ3 + (cid:96) ˆ31 (cid:112) ¯ T d ¯ z (cid:33) + 11 + x (cid:115) ( x − u )( x − ¯ u )(1 − u )(1 − ¯ u ) (cid:32) πi ( κ + κ − κ ) + 2 (cid:90) (cid:96) + (cid:96) + (cid:96) ˆ1ˆ2 (cid:112) ¯ T d ¯ z (cid:33) . (5.97)Singular terms for other Wronskians at x = ± i ± that i + ( i − ) transforms into i − ( − i + ) as it crosses the branch cut of p i . This leadsto the following constraint for the Wronskians (cid:104) i + , j + (cid:105) (ˆ σ i ˆ σ j x ) = (cid:104) i − , j − (cid:105) ( x ) . (5.98)It turns out that all the results obtained so far satisfy (5.98). Since this property getslost upon adding a constant to the logarithm of the Wronskian, it shows that our resultsare already complete and we should not add any constant functions. Up to the last section, we have developed necessary methods and acquired the knowledgeof the various parts that make up the three-point functions of our interest. Now we areready to put them together and see that they combine in a non-trivial fashion to producea rather remarkable answer.First in subsection 6.1, we obtain the complete result for the S part by puttingtogether the contribution of the action and that of the vertex operators. These twocontributions combine nicely to produce a simple expression in terms of integrals onthe spectral curve. Then, adapting the methods developed for the S part, we evaluatein subsection 6.2 the EAdS part of the three-point function. Our focus will be on thedifferences between the S and EAdS contributions. Finally in subsection 6.3, we presentthe full answer by combining the contributions of the S part and the EAdS part. Wewill see that the structure of the final answer closely resembles that of the weak couplingresult. Detailed comparison for certain specific cases will be performed in section 7. S part Before we begin the actual computations, let us summarize the structure of the contri-butions from the S part to the logarithm of the three-point function, which we denoteby F S . As was already indicated in section 2.3, F S consists of the contribution of theaction and that of the vertex operators, namely F S = F action + F vertex . (6.1)Each contribution can be further split into several different pieces as F action = √ λ A (cid:36) + A η , F vertex = V kin + V dyn + V energy . (6.2)Among these terms, A (cid:36) , V kin and V energy have already been evaluated respectively in(3.47), (4.61) and (4.63). Thus, our main task will be to compute A η and V dyn . As shown82n (3.48) and (4.62), A η is given by the normal ordered derivatives of the Wronskians, : ∂ x ln (cid:104) i + , j + (cid:105) : ± , whereas V dyn is given by the Wronskians evaluated at x = 0 and x = ∞ ,ln (cid:104) i + , j + (cid:105)| ∞ and ln (cid:104) i − , j − (cid:105)| . From the discussion in section 5, we know the Wronskiansare comprised of two different parts, the convolution-integral part Conv [ln (cid:104) i ∗ , j ∗ (cid:105) ] and thesingular part Sing ± [ln (cid:104) i ∗ , j ∗ (cid:105) ]. They both contribute to A η and V dyn . In what follows, weexamine these two parts separately and evaluate their contributions to A η and V dyn . We begin with the computation of the convolution integrals. To illustrate the basicidea, let us study Conv [ln (cid:104) + , + (cid:105) ] | ∞ , Conv [ln (cid:104) − , − (cid:105) ] | and : ∂ x Conv [ln (cid:104) + , + (cid:105) ] : ± asrepresentative examples.To compute the first two quantities, we need to know on which side of the integrationcontours the points x = 0 and x = ∞ are located. This is because the convolutionintegrals derived in subsection 5.4 are valid only when x is on the left hand side of thecontours. When x is on the right hand side of the contours, we must include the contactterms, which originate from the integration around x (cid:48) = x . Unfortunately, the form ofthe contours depend on the specific details of the solutions we use and hence we cannotgive a general discussion. We will therefore postpone the discussion of the contact termsuntil we study several explicit examples in the next section.Apart from such contact terms, Conv [ (cid:104) + , + (cid:105) ] | ∞ and Conv [ (cid:104) − , − (cid:105) ] | can be ob-tained directly from (5.60), (5.61), (5.74) and (5.75) by setting the value of x in theconvolution kernels K i ( x (cid:48) ; x ) to be 0 and ∞ respectively.Next, consider the evaluation of the normal-ordered derivative : ∂ x Conv [ln (cid:104) + , + (cid:105) ] : ± .This quantity does not receive contributions from the contact terms since the integrationcontours pass right through x = ± : ∂ x Conv [ln (cid:104) + , + (cid:105) ] : ± alwayson the left hand side of the contour. In addition, since the convolution integrals arenonsingular at x = ± 1, as discussed at the end of section 5.4, the normal ordering is infact unnecessary. Thus, : ∂ x Conv [ln (cid:104) + , + (cid:105) ] : ± can be obtained from (5.60) and (5.74) bysimply replacing K i ( x (cid:48) ; x ) with their derivatives ∂ x K i ( x (cid:48) ; x ) | x = ± .Applying similar analyses to other Wronskians and using the formulas (3.48) and(4.62), we can obtain the contributions of the convolution integrals to A η and V dyn , which83ill be denoted by Conv [ A η ] and Conv [ V dyn ]. They are given by Conv [ A η ] = √ λ (cid:34)(cid:90) M uuu −−− (cid:68)(cid:68) κ i ∂ x K i | + − κ i ∂ x K i | − (cid:69)(cid:69) ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:90) M uuu −− + (cid:68)(cid:68) κ i ∂ x K i | + − κ i ∂ x K i | − (cid:69)(cid:69) ∗ ln sin (cid:18) p + p − p (cid:19) + (cid:90) M uuu − + − (cid:68)(cid:68) κ i ∂ x K i | + − κ i ∂ x K i | − (cid:69)(cid:69) ∗ ln sin (cid:18) p − p + p (cid:19) + (cid:90) M uuu + −− (cid:68)(cid:68) κ i ∂ x K i | + − κ i ∂ x K i | − (cid:69)(cid:69) ∗ ln sin (cid:18) − p + p + p (cid:19) − (cid:88) j =1 (cid:90) Γ uj − (cid:0) κ j ∂ x K j | + − κ j ∂ x K j | − (cid:1) ∗ ln sin p j (cid:35) , (6.3) Conv [ V dyn ] = (cid:90) M uuu −−− (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) ∗ ln sin (cid:18) p + p + p (cid:19) + (cid:90) M uuu −− + (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) ∗ ln sin (cid:18) p + p − p (cid:19) + (cid:90) M uuu − + − (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) ∗ ln sin (cid:18) p − p + p (cid:19) + (cid:90) M uuu + −− (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) ∗ ln sin (cid:18) − p + p + p (cid:19) − (cid:88) j =1 (cid:90) Γ uj − (cid:0) S j ∞ K j | ∞ + S j K j | (cid:1) ∗ ln sin p j . (6.4)To simplify the expressions, we have introduced the double bracket notation (cid:68)(cid:68) (cid:63) (cid:69)(cid:69) , todenote sum of three terms with designated combinations of signs, defined as (cid:68)(cid:68) a i (cid:69)(cid:69) = a + a + a , (cid:68)(cid:68) a i (cid:69)(cid:69) = a + a − a , etc. , (6.5)Also, we have employed the abbreviated symbols ∂ x K i | ± , K i | ∞ and K i | , which are definedby ∂ x K i | ± ≡ ∂ x K i ( x (cid:48) ; x ) | x = ± , K i | ∞ ≡ K i ( x (cid:48) ; ∞ ) , K i | ≡ K i ( x (cid:48) ; 0) . (6.6)It turns out that the two contributions (6.3) and (6.4) combine to give a remarkablysimple expression displayed below. This is due to the crucial relation of the form √ λκ i ∂ x K i | + − √ λκ i ∂ x K i | − + S i ∞ K i | ∞ + S i K i | = z ( x (cid:48) ) dp i ( x (cid:48) ) dx (cid:48) , (6.7)where z ( x ) on the right hand side is the Zhukovsky variable, defined in (2.36). Althoughthis equality can be verified by a direct computation using the explicit form of p i ( x ) for84he one-cut solutions given in (2.58), it is important to give a more intuitive and essentialunderstanding. Note that the right hand side of (6.7) is proportional to the integrandof the filling fraction given in (2.35). Therefore when integrated over appropriate a -typecycles, it produces the corresponding conserved charges. In other words, it is characterizedby the singularities associated with such charges. Now observe that the left hand sideprecisely consists of terms which provide such singularities. The first two terms areresponsible for the singularities at x = ± 1, while the last two terms contain the polesat x = ∞ and x = 0 associated with the charges S i ∞ and S i respectively. Furthermore,it should be emphasized that the formula above unifies the contributions in two senseof the word. First, it unites the contributions from the action, represented by the firsttwo terms, and those from the vertex operators, represented by the last two terms. Onlywhen they are put together one can reproduce all the singularities of the right hand side.Second, the expression obtained on the right hand side is universal in that all the specificdata shown on the left hand side, namely κ i , S i ∞ and S i , are contained in one quantity p i ( x ). As we shall discuss in section 6.2, this feature allows us to write down the sameform of the result (except for an overall sign) given by the right hand side of (6.7) forthe contributions from the EAdS part, using the quasi-momentum for that part of thestring.Now, applying (6.7) we can rewrite the sum T conv ≡ Conv [ A η ] + Conv [ V dyn ] into thefollowing compact expression: T conv = (cid:90) M uuu −−− z ( x ) ( dp + dp + dp )2 πi ln sin (cid:18) p + p + p (cid:19) + (cid:90) M uuu −− + z ( x ) ( dp + dp − dp )2 πi ln sin (cid:18) p + p − p (cid:19) + (cid:90) M uuu − + − z ( x ) ( dp − dp + dp )2 πi ln sin (cid:18) p − p + p (cid:19) + (cid:90) M uuu + −− z ( x ) ( − dp + dp + dp )2 πi ln sin (cid:18) − p + p + p (cid:19) − (cid:88) j =1 (cid:90) Γ uj − z ( x ) dp j πi ln sin p j + Contact . (6.8)In the last line, we included the possible contributions from the contact terms, denotedby Contact . We now turn to the computation of the singular part Sing ± [ln (cid:104) i ∗ , j ∗ (cid:105) ]. By substitutingthe expressions for the singular part of the Wronskians, such as (5.96) and (5.97), into85he formulas (3.48) and (4.62), we can evaluate the contributions of the singular part in astraightforward manner. From this calculation, we find that a part of the terms contributeonly to the overall phase of the three-point functions. For instance, the first and the thirdterm in (5.96), which are proportional to ± πi ( κ + κ − κ ), will only yield an overallphase owing to the factor of πi . Just as before, we will ignore such contributions in thiswork. Then the contributions of Sing + [ln (cid:104) i ∗ , j ∗ (cid:105) ] to A η and V dyn , denoted by Sing + [ A η ]and Sing + [ V dyn ], are obtained as Sing + [ A η ] = √ λ (cid:20)(cid:68)(cid:68) κ i : ∂ x K i (1; x ) : + − κ i ∂ x K i (1; x ) | − (cid:69)(cid:69) (cid:90) (cid:96) (cid:36) + (cid:68)(cid:68) κ i : ∂ x K i (1; x ) : + − κ i ∂ x K i (1; x ) | − (cid:69)(cid:69) (cid:90) (cid:96) (cid:36) + (cid:68)(cid:68) κ i : ∂ x K i (1; x ) : + − κ i ∂ x K i (1; x ) | − (cid:69)(cid:69) (cid:90) (cid:96) ˆ31 (cid:36) (cid:35) , (6.9)and Sing + [ V dyn ] = (cid:20)(cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) (cid:90) (cid:96) (cid:36) + (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) (cid:90) (cid:96) (cid:36) + (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) (cid:90) (cid:96) ˆ31 (cid:36) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (cid:48) =+1 . (6.10)Note that in the present case, in contrast to the case of : ∂ x Conv [ln (cid:104) i ∗ , j ∗ (cid:105) ] : ± discussedpreviously, the normal ordering in : ∂ x K i (1; x ) : + is necessary since ∂ x K i (1; x ) is singularat x = 1. In an entirely similar manner, the contributions of Sing − [ln (cid:104) i ∗ , j ∗ (cid:105) ] to A η and V dyn , denoted by Sing − [ A η ] and Sing − [ V dyn ], are computed as Sing − [ A η ] = −√ λ (cid:20)(cid:68)(cid:68) κ i ∂ x K i ( − x ) | + − κ i : ∂ x K i ( − x ) : − (cid:69)(cid:69) (cid:90) (cid:96) ¯ (cid:36) + (cid:68)(cid:68) κ i ∂ x K i ( − x ) | + − κ i : ∂ x K i ( − x ) : − (cid:69)(cid:69) (cid:90) (cid:96) ¯ (cid:36) + (cid:68)(cid:68) κ i ∂ x K i ( − x ) | + − κ i : ∂ x K i ( − x ) : − (cid:69)(cid:69) (cid:90) (cid:96) ˆ31 ¯ (cid:36) (cid:35) , (6.11)and Sing − [ V dyn ] = − (cid:20)(cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) (cid:90) (cid:96) ¯ (cid:36) + (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) (cid:90) (cid:96) ¯ (cid:36) (cid:68)(cid:68) S i ∞ K i | ∞ + S i K i | (cid:69)(cid:69) (cid:90) (cid:96) ˆ31 ¯ (cid:36) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (cid:48) = − . (6.12)Now just as we did for Conv [ A η ] + Conv [ V dyn ], we can make use of the relation (6.7)86o rewrite the sum Sing ± [ A η ] + Sing ± [ V dyn ] into much simpler forms. The results are Sing + [ A η ] + Sing + [ V dyn ] = : z ( x ) (cid:18) dp dx + dp dx − dp dx (cid:19) : + (cid:90) (cid:96) (cid:36) + : z ( x ) (cid:18) dp dx − dp dx + dp dx (cid:19) : + (cid:90) (cid:96) ˆ31 (cid:36) + : z ( x ) (cid:18) − dp dx + dp dx + dp dx (cid:19) : + (cid:90) (cid:96) (cid:36) , (6.13)and Sing − [ A η ] + Sing − [ V dyn ] = − : z ( x ) (cid:18) dp dx + dp dx − dp dx (cid:19) : − (cid:90) (cid:96) ¯ (cid:36) − : z ( x ) (cid:18) dp dx − dp dx + dp dx (cid:19) : − (cid:90) (cid:96) ˆ31 ¯ (cid:36) − : z ( x ) (cid:18) − dp dx + dp dx + dp dx (cid:19) : − (cid:90) (cid:96) ¯ (cid:36) . (6.14)The expressions : z ( x ) dp i /dx : ± in (6.13) and (6.14) above can be evaluated using theexplicit form of the quasi-momentum, given in (2.58), as : z ( x ) dp i dx : + = − πκ i − πκ i Λ i , : z ( x ) dp i dx : − = 2 πκ i + πκ i ¯Λ i . (6.15)This provides fairly explicit forms for the expressions Sing ± [ A η ] + Sing ± [ V dyn ]. S part We can now combine the results obtained so far and obtain the net contribution of the S part. Recall that the general structure of the S part of the three-point functions wehave computed is of the form F S = √ λ √ λ (cid:88) i =1 κ i ln (cid:15) i + A (cid:36) + V kin + Conv [ A η ] + Conv [ V dyn ]+ Sing + [ A η ] + Sing + [ V dyn ] + Sing − [ A η ] + Sing − [ V dyn ] . (6.16)Among the various terms shown above, those which can be expressed in terms of thecontour integrals of (cid:36) or ¯ (cid:36) can be combined and evaluated using the explicit form of : z dp i /dx : ± given in (6.15). The result is T sing ≡A (cid:36) + Sing + [ A η ] + Sing + [ V dyn ] + Sing − [ A η ] + Sing − [ V dyn ]= − √ λ (cid:34) ( κ + κ − κ ) (cid:90) (cid:96) ( (cid:36) + ¯ (cid:36) ) + ( κ − κ + κ ) (cid:90) (cid:96) ˆ31 ( (cid:36) + ¯ (cid:36) )+( − κ + κ + κ ) (cid:90) (cid:96) ( (cid:36) + ¯ (cid:36) ) (cid:21) . (6.17) Definitions of Λ i and ¯Λ i are given in (3.20) and (3.31). (cid:36) and ¯ (cid:36) behave near the punctures as (cid:36) → κ i z − z i , ¯ (cid:36) → κ i ¯ z − ¯ z i , ( z → z i ) for i = 1 , ¯2 , , (6.18)the expression (6.17) diverges in the following fashion as the regularization parameters (cid:15) i ’s tend to zero: T sing → − √ λ (cid:88) i =1 κ i ln (cid:15) i = −V energy . (6.19)Notice, however, that this divergence is precisely canceled by the second term of (6.16).Therefore, the quantity (6.16) as a whole is finite in the limit (cid:15) i → 0. This is as expectedfor correctly normalized three-point functions.Let us summarize the final result for the logarithm of the three-point functions comingfrom the S part. It can be written in the form F S = √ λ V energy + T sing + V kin + T conv , (6.20)where V kin is the kinematical factor depending only on the normalization vectors given in(4.61), T conv is the sum of the contributions from the convolution integrals (6.8), and T sing ,which is given in (6.17), represents the sum of A (cid:36) defined in (3.47) and the contributionsfrom the singular parts of the Wronskians. EAdS part We now discuss the contributions from the EAdS part. Since the logic of the evaluation isalmost entirely similar, we will not repeat the long analysis we performed for the S part.In fact it suffices to explain which part of the analysis for the S part can be “copied”and which part has to be modified. Let us begin with the contribution from the action integral. Since EAdS and S areformally quite similar, the computation of the action integral can be performed in exactlythe same manner. There is, however, a simple but crucial difference. It is the overall signof the integral. For EAdS , the counterpart of the matrix Y shown in (2.14) is given by X ≡ (cid:18) X + XX − ¯ X (cid:19) , (6.21)where X ± ≡ X − ± X , , X ≡ X + iX , ¯ X ≡ X − iX . (6.22)88he right current is then defined asˆ j ≡ X − d X = ˆ j z dz + ˆ j ¯ z d ¯ z . (6.23)Now compare the expressions of the stress tensors and the action integrals for S and EAdS , expressed in terms of the respective right current. They are given by T ( z ) ≡ T AdS ( z ) = 12 tr (ˆ j z ˆ j z ) = κ , T S ( z ) = − 12 tr ( j z j z ) = − κ , (6.24) S EAdS = √ λ π (cid:90) d z tr (ˆ j z ˆ j ¯ z ) , S S = − √ λ π (cid:90) d z tr ( j z j ¯ z ) . (6.25)This shows that while we have the equality tr (ˆ j z ˆ j z ) = tr ( j z j ¯ z ) = κ , the signs in frontof the action integrals are opposite. Therefore all the results for the action integral areformally the same as those for the S case, but with opposite signs. This will lead tovarious cancellations with the contributions from the S part, as we shall see shortly. As for the evaluation of the contribution from the wave function, the basic logic of the for-malism developed in section 4 for the S still applies. However, there are a few importantmodifications, as we shall explain below.As discussed in our previous work [11], in the case of a string in EAdS the globalsymmetry group is SL(2 , C) R × SL(2 , C) L and hence the the raising operators with respectto which we define the highest weight state are the left and the right special conformaltransformations given by V sc R = (cid:18) β R (cid:19) , V sc L = (cid:18) β L (cid:19) , (6.26)where β R and β L are constants. Applying our general argument for the determination ofthe polarization spinors, we readily find( V sc R ) t n diag = n diag , n diag = (cid:18) (cid:19) , (6.27)( V sc L ) t ˜ n diag = ˜ n diag , ˜ n diag = (cid:18) (cid:19) . (6.28)It should be noted that, compared to the S case given in (4.23), n diag here for the rightsector is the same as ˜ n Z for the left sector there and similarly ˜ n for the left sector inthe present case is identical to n Z for the right sector for the S case. Now the algebraicmanipulations for the construction of the wave functions are the same as for the S caseup to the computation of the factor e i ∆ φ . Therefore, for the right sector, we get the same89esult for the Z -type operator in the left sector, given in (4.58). For example at z wehave e i ∆ φ R, = a − = (cid:104) + , + (cid:105)(cid:104) + , + (cid:105)(cid:104) + , + (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ∞ (cid:104) n , n (cid:105)(cid:104) n , n (cid:105)(cid:104) n , n (cid:105) (6.29)This is the inverse of the result for S obtained in (4.51) with i − replaced by i + . Theresult for the left sector is similar. What this means is that the wave function for the EAdS is obtained from the one for the S case by (i) reversing the sign of the powersand (ii) exchanging i + and i − . Abusing the same notations for the polarization spinorsand the eigenvectors as in the S case, we getΨ EAdS R = (cid:89) i (cid:54) = j (cid:54) = k (cid:32) (cid:104) n i , n j (cid:105)(cid:104) i + , j + (cid:105) (cid:12)(cid:12) ∞ (cid:33) − ( R i + R j − R k ) , (6.30)Ψ EAdS L = (cid:89) i (cid:54) = j (cid:54) = k (cid:32) (cid:104) ˜ n i , ˜ n j (cid:105)(cid:104) ˜ i − , ˜ j − (cid:105) (cid:12)(cid:12) (cid:33) − ( L i + L j − L k ) , (6.31)where R i and L i here are the combinations of the conformal dimension ∆ i and the spin S i given by R i = 12 (∆ i − S i ) , L i = 12 (∆ i + S i ) . (6.32)This reversal of power relative to the S case is what is desired. Effectively it is equivalentto employing e + iSφ as the form of the wave function, which is what we adopted in theprevious work [11] for the three-point function of the GKP string in EAdS and lead tothe power structure given in (6.30) and (6.31). As we shall show below, correctness of thispower structure becomes obvious when we relate the Wronskian (cid:104) n i , n j (cid:105) to the differenceof the coordinates x i and x j , where x i is the position of the i -th vertex operator on theboundary of EAdS .Recall that the embedding coordinates of EAdS are taken to be X µ ( µ = − , , , SO (1 , 3) with signature ( − , + , + , +), while the Poincar´e coordinatesare given by z = 1 / ( X − + X ), x r = zX r , ( r = 1 , X − − X is expressed as z + ( (cid:126)x /z ). Consider approaching a point on the boundary z = 0 with finite values of x r .Then the term z in X − − X becomes negligible compared to (cid:126)x /z and X µ approaches anull vector, with large components. Such a vector can be parametrized, up to an overallscale, by the boundary coordinates (cid:126)x = ( x , x ) as X − = 12 (1 + (cid:126)x ) , X = 12 (1 − (cid:126)x ) , ( X , X ) = (cid:126)x , (6.33) (cid:126)x = x r η rs x s = x r x r , η rs = (+ , +) , r, s = 1 , . (6.34)As usual, one can map X µ to the matrix X µ ˆΣ µ , with ˆΣ µ = (1 , σ , σ , σ ), which transformsfrom left under SL (2 , C ) and from right under SL (2 , C ) ∗ . Then, it is well-known that for90 null vector X µ the matrix elements of X µ ˆΣ µ can be written as a product of spinors (ortwistors) as ( X µ ˆΣ µ ) α ˙ α = (cid:18) x ¯ x (cid:126)x (cid:19) = ( σ ˜ n ) α n ˙ α , (6.35)where x ≡ x + ix , ¯ x ≡ x − ix , (6.36) n = (cid:18) x (cid:19) , ˜ n = (cid:18) ¯ x (cid:19) . (6.37)These spinors can be identified precisely as the polarization spinors characterizing a vertexoperator which is placed at (cid:126)x on the boundary for the following reasons. First theytransform in the correct way: Under the global transformation X µ ˆΣ µ → V L ( X µ ˆΣ µ ) V R ,we have ( σ ˜ n ) α → ( V L σ ˜ n ) α and n ˙ α → ( nV R ) ˙ α . This is equivalent to ˜ n → V tL ˜ n and n → V tR n , which are the right transformation laws. Second, these spinors coincide withthe polarization spinors given in (6.27) and (6.28) when we bring the point (cid:126)x to the originof the boundary by the translation by the vector − (cid:126)x . This is effected by the right andthe left translation matrices given by V tr R ( − x ) = (cid:18) − x (cid:19) , V tr L ( − ¯ x ) = (cid:18) − ¯ x (cid:19) . (6.38)Then we get ( V tr R ) t ( − x ) n = (cid:18) (cid:19) , ( V tr L ) t ( − ¯ x )˜ n = (cid:18) (cid:19) . (6.39)Therefore n and ˜ n can be identified with the polarization spinors for the vertex operatorat (cid:126)x on the boundary. Now let n (cid:48) and ˜ n (cid:48) be similar polarization spinors corresponding toa vertex operator at (cid:126)x (cid:48) on the boundary. Then we immediately get (cid:104) n, n (cid:48) (cid:105) = x (cid:48) − x , (cid:104) ˜ n, ˜ n (cid:48) (cid:105) = ¯ x (cid:48) − ¯ x , (6.40) (cid:104) n, n (cid:48) (cid:105)(cid:104) ˜ n, ˜ n (cid:48) (cid:105) = ( x (cid:48) − x )(¯ x (cid:48) − ¯ x ) = ( x (cid:48) − x ) . (6.41)In this way, for the EAdS the Wronskians formed by the polarization spinors producethe difference of the boundary position vectors. Therefore the relevant part of the wavefunction becomes (cid:89) { i,j,k } (cid:104) n i , n j (cid:105) − ( R i + R j − R k ) (cid:104) ˜ n i , ˜ n j (cid:105) − ( L i + L j − L k ) = (cid:89) { i,j,k } ( x i − x j ) − ( R i + R j − R k ) (¯ x i − ¯ x j ) − ( L i + L j − L k ) . (6.42)91n particular, for the case of spinless configurations that we are considering, this becomes (cid:89) { i,j,k } | x i − x j | ∆ i +∆ j − ∆ k , (6.43)which exhibits the familiar coordinate dependence for the three-point function in such acase. EAdS part As we have seen, the structure of the contribution from the EAdS part is essentially thesame as that from the S case, except for the important reversal of signs in the powers inthe contributing factor (or the terms contributing to the the logarithm of the three-pointcoupling.) This change of sign occurred both for the action and for the wave function.As we compute the basic Wronskians in exactly the same way as before and use them tocompute the contributions to the logarithm of the three-point function from the actionpart and the wave function part, we again obtain the expression of the form of the lefthand side of (6.7), with the overall sign reversed. Therefore, we can use the identity (6.7)again to obtain the result − z ( x (cid:48) ) d ˆ p i ( x (cid:48) ) /dx (cid:48) , where ˆ p i denotes the quasi-momentum forthe EAdS part of the string. One can check that in fact this rule of correspondence,namely p i ( x ) → ˆ p i ( x ) and the reversal of sign for the convolution integrals, applies to allthe contributions. Thus, combining all the results for the AdS part, the contribution tothe logarithm of the three-point function is given by the following expression: F EAdS = − √ λ V energy + ˆ T sing + ˆ V kin + ˆ T conv . (6.44)Here, ˆ V energy and ˆ T sing are equal to −V energy and −T sing respectively, ˆ V kin is the kinematicalfactor given in (6.43), and ˆ T conv is the convolution integrals obtained from the unhattedcounterpart for the S case with the substitution rule described above. We are finally ready to put together the contributions from the S part summarized in(6.20) and those from the EAdS part given in (6.44) and present the full answer for thethree-point function. As we have already discussed, the divergent terms cancel with eachother for the S part and the EAdS part separately. On the other hand, the constantterms proportional to √ λ/ S and EAdS contributions. Thus we are leftwith the kinematical factors and the contributions from the convolution integrals whichare of the same structure except for the overall sign. Therefore, factoring the kinematical92tructure as (cid:104)V V V (cid:105) = 1 N C | x − x | ∆ +∆ − ∆ | x − x | ∆ +∆ − ∆ | x − x | ∆ +∆ − ∆ × (cid:104) n , n (cid:105) R + R − R (cid:104) n , n (cid:105) R + R − R (cid:104) n , n (cid:105) R + R − R × (cid:104) ˜ n , ˜ n (cid:105) L + L − L (cid:104) ˜ n , ˜ n (cid:105) L + L − L (cid:104) ˜ n , ˜ n (cid:105) L + L − L , (6.45)the logarithm of the structure constant C is finally given by ln C = (cid:90) M uuu −−− z ( x ) ( dp + dp + dp )2 πi ln sin (cid:18) p + p + p (cid:19) + (cid:90) M uuu −− + z ( x ) ( dp + dp − dp )2 πi ln sin (cid:18) p + p − p (cid:19) + (cid:90) M uuu − + − z ( x ) ( dp − dp + dp )2 πi ln sin (cid:18) p − p + p (cid:19) + (cid:90) M uuu + −− z ( x ) ( − dp + dp + dp )2 πi ln sin (cid:18) − p + p + p (cid:19) − (cid:90) ˆ M uuu −−− z ( x ) ( d ˆ p + d ˆ p + d ˆ p )2 πi ln sin (cid:18) ˆ p + ˆ p + ˆ p (cid:19) − (cid:90) ˆ M uuu −− + z ( x ) ( d ˆ p + d ˆ p − d ˆ p )2 πi ln sin (cid:18) ˆ p + ˆ p − ˆ p (cid:19) − (cid:90) ˆ M uuu − + − z ( x ) ( d ˆ p − d ˆ p + d ˆ p )2 πi ln sin (cid:18) ˆ p − ˆ p + ˆ p (cid:19) − (cid:90) ˆ M uuu + −− z ( x ) ( − d ˆ p + d ˆ p + d ˆ p )2 πi ln sin (cid:18) − ˆ p + ˆ p + ˆ p (cid:19) − (cid:88) j =1 (cid:90) Γ uj − z ( x ) dp j πi ln sin p j + 2 (cid:88) j =1 (cid:90) ˆΓ uj − z ( x ) d ˆ p j πi ln sin ˆ p j + Contact , (6.46) where Contact stands for the contribution from the contact terms. We find it trulyremarkable that, in spite of the complexity of both the analysis and the intermediateexpressions, the final answer takes such a simple form. Moreover, it exhibits essentialsimilarity to the form of the weak coupling result [4–7] even before taking any furtherlimits. In the next section, we shall evaluate the structure constant (6.46) more explicitly,including the quantity Contact , for several important examples and compare with theweak coupling results more closely. The results obtained in the previous section are quite general and applicable to three-point functions of arbitrary one-cut solutions on EAdS × S . In this section we focuson several explicit examples, make some basic checks and discuss the relation with theresults at weak coupling.In subsection 7.1, we first explain the basic set-up, which will be used throughoutthis section. Then, in subsection 7.2, we study the correlation functions of three BPSoperators and see that the contributions from the S part and the EAdS part completelycancel out in this case. The results thus obtained fully agree with the results obtainedin the gauge theory. In subsection 7.3, we study the behavior of the three-point function93nder the limit where the charge of one of the operators becomes negligibly small while theother two operators become identical. We confirm that the result reduces to that of thetwo-point function, as expected. Next, in subsection 7.4, we study three-point functions ofone non-BPS and two BPS operators, which were studied on the gauge theory side in [4].We will focus on certain explicit examples and show that the full three-point functionscan be expressed in terms of simple integrals which resemble the semi-classical limit ofthe results at weak coupling [4–7]. Then, in subsection 7.5, we discuss the Frolov-Tseytlinlimit of such three-point functions. In this limit, the integrands in the final expressionapproximately agree with the ones in the weak coupling, whereas the integration contoursare rather different. Lastly, we discuss the possible origin and the implication of thismismatch. Before starting the detailed analysis, let us clarify the basic set-ups to be used in thissection.The three-point functions studied extensively on the gauge theory side are those ofthe following three types of operators (see also Table 2.): O = tr (cid:0) Z l − M X M (cid:1) + · · · , O = tr (cid:0) ¯ Z l − M ¯ X M (cid:1) + · · · , O = tr (cid:0) Z l − M ¯ X M (cid:1) + · · · . As explained in section 4.3.4, such three-point functions vanish unless the conservationlaws for the charges, (4.64), are satisfied. Due to these conservation laws, one cannot ingeneral take the operators to be simple BPS states, such as tr ( Z l ) or tr ( ¯ Z l ), which are thehighest-weight vectors of the global SU(2) R × SU(2) L symmetry. Instead, we need to usedescendants of the global symmetry to satisfy the conservation laws when we study three-point functions involving BPS operators [4, 27]. While this can be done without problemson the gauge theory side, it leads to certain difficulty on the string theory side. Thisis because all the classical solutions of string are known to (or believed to) correspondto some highest-weight states. To circumvent this difficulty, below we will utilize theglobal transformations to make all three operators to be built on different “vacua”. Onthe string theory side, this corresponds to taking the polarization vectors of the threeoperators, n i ’s and ˜ n i ’s, to be all distinct. Then no conservation laws will be imposedand we can safely take the limit where some of the operators become BPS while keepingthem to be of highest-weight. Since the correlation functions involving descendants canbe obtained from the correlation functions involving the highest-weight states by simplegroup theoretical manipulations, knowledge of the three-point functions for the highest As we have shown in section 4, such conservation laws can be derived also on the string theory side. V kin , of our result and the dynamicalparts of three-point functions, which are main subjects of study in this section, will notbe affected.After making the global transformations, the operators O , O and O can be treatedalmost on the same footing. However, there is an important difference between O andthe other two in string theory: As explained in section 4.3.4, the quasi-momenta for theoperators O and O contain branch cuts in the | Re x | > O contains a branch cut in the | Re x | < Let us first study the correlation functions of three BPS operators. In order to applythe general formula for the three-point functions of one-cut solutions obtained in theprevious section, we need the explicit forms of p ( x ) and q ( x ) for the BPS operators,which in particular determine the integration contours. Within the bosonic sector, thecharacteristic feature of a BPS state is that, as it should correspond to a supergravitymode, it is “point-like”, meaning that its two-point function is σ -independent. In thelanguage of the spectral curve, it means the absence of a branch cut, since a branch cutcorresponds to a non-trivial string mode with σ -dependence.Now in fixing the forms of p ( x ) and q ( x ), there is a subtle problem with the configu-ration without a branch cut. In the case of one-cut solutions corresponding to non-BPSoperators, the constant parts of p ( x ) and q ( x ) are fixed in such a way that they vanish atthe branch points. Obviously, for configurations without a branch cut, this prescriptioncannot be applied. One natural remedy would be to start with a non-BPS solution, applythe usual method above to fix the constants and then shrink the cut to obtain a BPSsolution. This idea, however, still does not cure the problem since the resultant p ( x ) and q ( x ) depend on the points on the spectral curve at which we shrink the branch cut. Theexistence of such an ambiguity possibly implies that the semi-classical three-point func-tions are affected by the presence of infinitesimal branch cuts. Although such an assertionsounds counter-intuitive, it is not totally inconceivable since similar effects were alreadyobserved in the study of “heavy-heavy-light” three-point functions in [30].Below we shall fix the ambiguity by employing a prescription which is quite naturalfrom the viewpoint of the correspondence with the spin chain on the gauge theory side. In [30], such effects were called back reactions . x = 0 or at x = ∞ in producingBPS operators. This choice is based on the following fact: In gauge theory, adding asmall number of Bethe roots at x = 0 or x = ∞ correspond to performing a small globaltransformation and keeps the operator to be BPS, whereas adding a small number ofBethe roots at generic points on the spectral curve creates nontrivial magnon excitationsand makes the operator non-BPS.Having identified the classical solutions corresponding to BPS operators, let us nowdetermine the integration contours. First we focus on the S -part of three-point functions.As discussed in section 4.3.4, for O and O , p i ( x ) and q i ( x ) can have branch cuts onlyin the the | Re x | > x = ∞ . Then from the general form of the one-cut solution given in (2.58) and (2.59),we get p i ( x ) = − πκ i (cid:18) x − x + 1 (cid:19) , q i ( x ) = − πκ i (cid:18) x − − x + 1 (cid:19) , (7.1)which vanish at x = ∞ , as desired. On the other hand, for O , since the branch cuts canonly be in the | Re x | < x = 0. Thenfrom (2.58) and (2.59) we get p ( x ) = − πκ (cid:18) xx − − xx + 1 (cid:19) = − πκ (cid:18) x − x + 1 (cid:19) (7.2) q ( x ) = − πκ (cid:18) xx − xx + 1 (cid:19) . (7.3)These expressions vanish at x = 0.As discussed in detail in section 5, the contours for the convolution integrals consistof two types of curves. The first type are those defined by Re q i ( x ) = 0, across which therelative magnitude of i + and i − changes. They determine the integration contours Γ ui − defined in section 5.4 and are depicted in figure 7.1. Note that in the present case, thecontours Γ u − and Γ u − coincide since q ( x ) = q ( x ). The second type are the curves definedby N i = N j + N k , across which the connectivity of the exact WKB curves changes. Now fora BPS operator, N i = | Re p i ( x ) | is given by a common function | Re (( x + 1) − + ( x − − ) | times the factor − πκ i , as shown above. Since κ i ’s satisfy the triangular inequalities, thismeans that N i = N j + N k cannot be satisfied. Hence the second type of curves are absentand the integration contours are determined solely by the first type of curves.With this knowledge, we can now apply the general rules given at the end of section5.3 to determine the integration contours M uuu ±±± . As an example, consider the contour M uuu −−− , which is used for the convolution integral involving sin ( − p ( x ) − p ( x ) − p ( x )).From the Rule 1, either Wronskians among S − = { − , − , − } vanish or those among96 + = { + , + , + } vanish. Then we must apply Rule 2, since the triangle inequalities aresatisfied in the present case. It states that if two of the members of S − (resp. S + ) are smallsolutions, then the Wronskians for the members of S + (resp. S − ) vanish. Now considerthe curve Γ u − . From its definition, it is along Re q ( x ) = 0 with the direction such thatto the left of this curve 1 − is the small solution. The curve Γ u − is identical, as we alreadyremarked. These curves are depicted in the left figure of figure 7.1, together with thestates which are small in the three regions separated by these curves. Together with therules mentioned above, we see explicitly that the analyticity of Wronskians change acrosssuch curves and hence we can identify Γ u − (= Γ u − ) as the contour M uuu −−− . Similarly, thecurve Γ u − , identified as M uuu + −− , is shown in the right figure of figure 7.1. In this way, wefind the contours M uuu ±±± to be given by M uuu −−− = Γ u − (= Γ u − ) , M uuu + −− = Γ u − , M uuu − + − = Γ u − , M uuu −− + = Γ u − (= Γ u − ) . (7.4)Figure 7.1: The contours Γ ui − , defined by Re q i = 0. In each region, we showed which ofthe eigenvectors is the small solution.Let us next consider the effects of the contact terms. As argued in section 6, suchcontribution must be taken into account when x = 0 ( x = ∞ ) is on the left (right) handside of the integration contours. The effect is most conveniently done by adding a smallcircle around x = 0 ( x = ∞ ) to the contour for each integration in (6.8). However, inthe case of BPS operators, the integration contours terminate right at x = 0 or x = ∞ .Therefore we need to first regularize them by putting a small branch cut slightly awayfrom x = 0 or x = ∞ and then take the limit where the branch cut shrinks to x = 0 or x = ∞ . An example of such a procedure is depicted in figure 7.2. Since the sine-functionsin the convolution integrals (6.8) turn out to vanish only on the real axis in the case ofBPS operators, we can further deform the contours into those on the unit circle. As a97esult, we find that the S -part of the three-point function is given by (cid:73) U z ( dp + dp + dp )2 πi ln sin (cid:18) p + p + p (cid:19) + (cid:73) U z ( dp + dp − dp )2 πi ln sin (cid:18) p + p − p (cid:19) + (cid:73) U z ( dp − dp + dp )2 πi ln sin (cid:18) p − p + p (cid:19) + (cid:73) U z ( − dp + dp + dp )2 πi ln sin (cid:18) − p + p + p (cid:19) − (cid:88) j =1 (cid:90) U z dp j πi ln sin p j , where U denotes the contour which goes around the unit circle clockwise.(a) Putting a small branch cutaway from x = 0. (b) Shrinking the cut anddeforming the contour.Figure 7.2: An example of the contour deformation. The contour depicted in (b) can befurther deformed into the contour on the unit circle.Next consider the EAdS -part of the three-point function. The quasi-momenta andthe quasi-energies for the operators without spin in AdS are given in [8] by ˆ p i ( x ) = − πκ i (cid:18) x − x + 1 (cid:19) , ˆ q i ( x ) = − πκ i (cid:18) x − − x + 1 − (cid:19) . (7.5)Then, performing a similar analysis as in the case of S -part, we find that the result isagain given by the integrals along the unit circle. As the quasi-momenta p i ( x ) for the S -part and the ones ˆ p i ( x ) for the EAdS -part coincide in the case of BPS operators, wesee from the general formula (6.46) that the contributions form these two parts canceleach other completely. Therefore, the three-point function for three BPS operators isgiven purely by the kinematical factors as (cid:104)V V V (cid:105) = 1 | x − x | ∆ +∆ − ∆ | x − x | ∆ +∆ − ∆ | x − x | ∆ +∆ − ∆ × (cid:104) n , n (cid:105) R + R − R (cid:104) n , n (cid:105) R + R − R (cid:104) n , n (cid:105) R + R − R × (cid:104) ˜ n , ˜ n (cid:105) L + L − L (cid:104) ˜ n , ˜ n (cid:105) L + L − L (cid:104) ˜ n , ˜ n (cid:105) L + L − L , (7.6)This is consistent with the result in the gauge theory that the three-point functions of BPSoperators are tree-level exact and have no dependence on the ’t Hooft coupling constant λ . The spectral parameter x used in (7.5) is related to the spectral parameter ξ used in [8] by ξ =( x − / ( x + 1). .3 Limit producing two-point function Having seen that the BPS three-point functions are correctly reproduced from our generalformula, let us next discuss the limit where the three-point functions are expected toreduce to two-point functions. As an example, we take two of the operators O and O to have identical quasi-momenta and quasi-energy, while O is a BPS operator withvanishingly small charge .( a ) ( b ) ( c )Figure 7.3: The curves which determine the integration contours in the limit where three-point functions reduce to two-point functions. In the left and the middle figures, thecontours Γ ui − , determined by Re q i ( x ) = 0 are depicted. The segment represented by awavy line is the branch cut. In the rightmost figure, the curve defined by N = N + N isdrawn in blue. For convenience, we redisplayed the curves in figures ( a ) and ( b ) as dottedlines.To understand what happens in such a limit, let us draw the two types of curves,namely Re q i = 0 and N i = N j + N k . The first type of curves are depicted in the first andthe second figures of figure 7.3. As for the second type, the only curve we need to consideris the curve given by N = N + N . This is because the inequalities N + N ≥ N and N + N ≥ N are always satisfied since N = N in the present case. When the operator O is sufficiently small, the curve defined by N = N + N almost vanishes and we canpractically ignore the effects of such a curve. Thus the integration contours are givenpurely by Re q = Re q = 0. Applying the rules given in the previous section and takinginto account the contact terms, we find that the convolution integrals for the S -part are Although the case considered here appears similar to the one studied in the gauge theory [30] with O taken to be small but nonvanishing, there is a difference: In [30], O and O must have slightly differentquasi-momenta in the presence of O , due to the conservation law for the magnons. In the present case,however, as we performed the global transformation, no conservation law is imposed and we can take O and O to have identical quasi-momenta. (cid:90) Γ u − + C ∞ z ( dp + dp + dp )2 πi ln sin (cid:18) p + p + p (cid:19) + (cid:90) Γ u − + C ∞ z ( dp + dp − dp )2 πi ln sin (cid:18) p + p − p (cid:19) + (cid:90) Γ u − + C z ( dp − dp + dp )2 πi ln sin (cid:18) p − p + p (cid:19) + (cid:90) Γ u − z ( − dp + dp + dp )2 πi ln sin (cid:18) − p + p + p (cid:19) − (cid:88) j =1 (cid:90) Γ uj − + C ∞ z dp j πi ln sin p j − (cid:90) Γ u − + C z dp πi ln sin p , (7.7)where C ∞ is the contour encircling x = ∞ counterclockwise and C is the contour encircling x = 0 clockwise. Setting p = p and p = 0 in this formula, we see that in this limitall the terms in (7.7) completely cancel out with each other. Similar cancellation occursalso for the EAdS -part. Therefore the structure constant C of the three-point functionin this limit becomes unity and the result correctly reproduces the correctly normalizedtwo-point function given by (cid:104) n , n (cid:105) R (cid:104) ˜ n , ˜ n (cid:105) L | x − x | . (7.8)Here, ∆, R and L are, respectively, the conformal dimension, the (absolute values of the)right and the left global charges, which are common to O and O . Having checked that our formula correctly reproduces the known results in simple limits,let us now study more nontrivial examples. In this subsection, we take up the three-pointfunctions of one non-BPS and two BPS operators, which were studied on the gauge-theoryside in [4]. As in [4], we take O to be non-BPS and O and O to be BPS. In this case,the typical forms of the curves corresponding to Re q i = 0 and N i = N j + N k , are givenin figure 7.4.To perform a more detailed analysis, we need to specify the properties of the operatorsmore explicitly, since the precise form of the integration contours depend on such details.As we wish to analyze the so-called Frolov-Tseytlin limit and make a comparison withthe results in the gauge theory in the next subsection, we will take as a representativeexample the following set of operators carrying large conformal dimensions: O : BPS , πκ = 2500 , O : non-BPS , πκ = 3250 ,p ( u ) − p ( ∞ + ) = − π , p (0 + ) − p ( ∞ + ) = − π , O : BPS , πκ = 3000 . (7.9)100igure 7.4: Typical configuration of the curves produced by the conditions Re q i = 0and N i = N j + N k , for the three-point functions of one non-BPS operator and two BPSoperators. In the left figure, Re q = 0, Re q = 0 and Re q = 0 are drawn respectivelyin black, orange and brown. In the right figure, N = N + N is drawn in blue and N = N + N is drawn in green.Here u denotes the position of an end of the branch cut for the non-BPS operator O .For these operators, the curves defined by Re q i = 0 and those defined by N i = N j + N k are depicted respectively in figure 7.5 and figure 7.6.As in the case of the three BPS operators, we must now apply the general rules of sec-tion 5 to determine the integration contours. As an example, consider the contour M uuu −−− in the region where | Re x | (cid:29) 1. Focus first on the left figure of figure 7.5. Compared tothe typical configuration shown in the left figure of figure 7.4, the curve determined byRe q = 0 (shown in brown in figure 7.4) is depicted here as a point in the middle sincewe are considering the region where | Re x | (cid:29) 1. Since the inside of the shrunken regionis where 3 + is small, we have 3 − as the small solution everywhere in this figure. From thedirection of the curves Γ u − and Γ u − , we can easily tell which of the states 1 ± and 2 ± arethe small solutions in each of the region separated by these curves.Figure 7.5: The contours Γ ui − , defined by Re q i = 0. The left figure shows the configurationof contours in the | x | (cid:29) | x | < a ) ( b ) ( c )Figure 7.6: The curves defined by N i = N j + N k . The figure ( a ) shows the configuration ofcurves in | x | (cid:29) b ) shows the configuration of N = N + N in | x | < c ) shows the configuration of N = N + N in | x | < N = N + N does not exist.Figure 7.7: Magnified view of a part of the figure ( a ) of figure 7.6, with data necessaryfor determining the contour of integration. In each region separated by lines and/or thecut (wavy line), the set of “small” eigenvectors are indicated. The green circle separatesthe symmetric (S) and the assymetric (A) regions, to which different rules of analysisapply. The result is that across the boundary of the shaded area, the analyticity of theWronskian changes. For details of the analysis using this figure, see the explanation inthe main text. 102ow, in distinction to the case of three BPS operators, we must also take into accountthe possible change of the analyticity of the Wronskians as we cross the lines defined by N i = N j + N k . Thus, we must analyze relevant curves drawn in figure 7.6 ( a ), where theone in green corresponds to N = N + N and the one in blue represents N = N + N .Across these lines the configuration changes from symmetric to asymmetric. Accordingly,the rule to find the non-vanishing set of Wronskians changes from Rule 2 to Rule 3. Letus focus on the green curve, which is re-drawn in figure 7.7, with additional information.It turns out that the configuration is symmetric inside the green circles and asymmetricoutside, indicated by the letters S and A respectively. Now in the region outside of thearc of the large green circle bordered by the lines representing Γ u − , shown in figure 7.7 bythe red straight lines, 1 − , + , − are the small solutions, as indicated in the figure. As thisis the asymmetric region we apply the Rules 1 and 3 and conclude that the Wronskiansamong the states { + , + , + } are non-vanishing. As we cross the arc into the shadedregion inside of the green circle where the configuration is symmetric, still 1 − , + , − arethe small solutions but now we must apply the Rules 1 and 2. Then we learn that theWronskians among the states { − , − , − } are non-vanishing instead. In other words, theanalyticity property of the Wronskians change across this portion of the green curve andhence it serves as a part of the contour for the convolution integral. This explains theportion of the contour along the arc of the large circle shown in the left-most figure infigure 7.8. Now consider what happens when this contour meets the Γ u − line. Acrossthis line, the small solution changes from 1 − to 1 + . Thus when we cross this line frominside the large circle, the set of small solutions change from 1 − , 2 + and 3 − to 1 + , 2 + and 3 − as shown in figure 7.7. As we are still in the symmetric region, the Rules 1 and2 apply and hence we learn that set of non-vanishing Wronskians change across this line.Therefore this portion must constitute a part of the contour. This explains the straightred line starting from the the point of intersection with the large circle. In this fashion,we can uniquely obtain the integration contour M uuu −−− , shown in the leftmost figure offigure 7.8, across which the analyticity property of the Wronskians change. All the othercontours M uuu ±±± can also be determined in an entirely similar manner, the result of whichare depicted in figure 7.8 and figure 7.9.The contours shown in figure 7.8 and figure 7.9 can be simplified by continuous defor-mation as long as we do not make them pass through the singularities of the integrands.We can determine the positions of the singularities numerically and find that most of thesingularities lie on the real axis. Avoiding them, we can deform each contour into a sumof the contour along the unit circle and the one which is far from the unit circle. The103igure 7.8: The integration contours M uuu ±±± in the region | x | (cid:29) 1. From left to right, M uuu −−− , M uuu −− + , M uuu − + − and M uuu + −− .Figure 7.9: The integration contours M uuu ±±± in the region | x | < 1. From left to right, M uuu −−− , M uuu −− + , M uuu − + − and M uuu + −− .results of this deformation are summarized as M uuu −−− (cid:55)−→ (cid:0) M uuu −−− (cid:1) (cid:48) + U , M uuu −− + (cid:55)−→ (cid:0) Γ u − (cid:1) (cid:48) + U , M uuu − + − (cid:55)−→ (cid:0) M uuu − + − (cid:1) (cid:48) + U , M uuu + −− (cid:55)−→ (cid:0) Γ u − (cid:1) (cid:48) + U , Γ u − (cid:55)−→ U , Γ u − (cid:55)−→ (cid:0) Γ u − (cid:1) (cid:48) + U , Γ u − (cid:55)−→ U , (7.10)where, as before, U denotes the unit circle and the primed contours are as depicted infigure 7.10.Figure 7.10: The contours obtained after the deformation. On the left figure, we depicted (cid:0) Γ u − (cid:1) (cid:48) . On the right figure, we depicted (cid:0) M uuu −−− (cid:1) (cid:48) in black and (cid:0) M uuu − + − (cid:1) (cid:48) in blue.Let us make a remark on the separation of the integration contours into the unit circle104nd the large contours. It is intriguingly reminiscent of the expressions for the one-loopcorrection to the spectrum of a classical string [31]. In that context, the integration alongthe unit circle is interpreted as giving the dressing phase and the finite size corrections.Since our results do not include one-loop corrections, it is not at all clear whether ourresults can be interpreted in a similar way. However, the apparent structural similaritycalls for further study. We are now ready to discuss the Frolov-Tseytlin limit of the three-point function andcompare it with the weak coupling result. Let us briefly recall how such a limit arises.As shown in [32], the dynamics of the fluctuations around a fast-rotating string on S can be mapped to the dynamics of the Landau-Lifshitz model, which arises as a coherentstate description of the XXX spin chain. In such a situation, the angular momentum J of the S rotation can be taken to be so large that the ratio √ λ/J becomes vanishinglysmall, even when λ is large. For the spectral problem, it has been demonstrated that sucha limit is quite useful in comparing the strong coupling result with the weak couplingcounterpart. We would like to see if it applies also to the three point functions. For thispurpose, we need to know how such a limit is taken at the level of the quasi-momenta.Since the SO(4) charges of the external states are proportional to κ i , the appropriate limitis to scale all the κ i to infinity while keeping the mode numbers (cid:72) b i dp finite. As alreadyindicated, we have chosen the example in the previous subsection to be such that we canreadily take such a limit.Upon taking the Frolov-Tseytlin limit, two simplifications occur in our formula. First,since the branch points are far away from the unit circle, we can approximate p ( x ) onthe unit circle by a quasi-momentum for a BPS operator, namely p ( x ) (cid:39) p BPS2 ( x ) = − πκ (cid:18) x − x + 1 (cid:19) . (7.11)Now recall that the contribution from the EAdS part is such that it precisely canceledthe S part in the case of the three BPS operators. Since the EAdS part is unchangedfor the present case, again the same exact cancellation takes place as far as the integralsover the unit circles are concerned. Therefore we can drop such integrals and obtain105 ( M uuu −−− ) (cid:48) + C ∞ z ( dp + dp + dp )2 πi ln sin (cid:18) p + p + p (cid:19) + (cid:90) (cid:16) Γ u − (cid:17) (cid:48) + C ∞ z ( dp + dp − dp )2 πi ln sin (cid:18) p + p − p (cid:19) + (cid:90) ( M uuu − + − ) (cid:48) + C ∞ z ( dp − dp + dp )2 πi ln sin (cid:18) p − p + p (cid:19) + (cid:90) (cid:16) Γ u − (cid:17) (cid:48) + C ∞ z ( − dp + dp + dp )2 πi ln sin (cid:18) − p + p + p (cid:19) − (cid:90) (cid:16) Γ u − (cid:17) (cid:48) + C ∞ z dp πi ln sin p , (7.12) Second simplification occurs because on the large contours the integration variable x isof order κ i . This is precisely the situation where we can approximate the quasi-momentaof the classical strings by the corresponding quantities for the spin-chains. Indeed, asexplained in [12], the quasi-momentum for the string can be identified with that of theLandau-Lifshitz model, which describes the spin-chain on the gauge theory side in theabove limit. More precisely, we can use the following identification of the quasi-momentaon the large contour: p string ( x ) (cid:39) p spin ( z ( x )) . (7.13)The use of the Zhukovsky variable z ( x ) on the right hand side is motivated by the fact thatin the all-loop asymptotic Bethe ansatz equation [33, 34], the rapidity of the spin-chainon the gauge theory side is identified with the Zhukovsky variable on the string theoryside. In the present situation, however, since z ( x ) (cid:39) x for large x , the quasi-momenta in(7.12) can be replaced simply with the quasi-momenta for the corresponding spin-chainstates at the same value of x .With such a replacement, the expression (7.12) already appears rather similar tothe weak-coupling result. To make the resemblance more conspicuous, we can regardthe integral of sin (( − p + p + p ) / 2) along (Γ u − ) (cid:48) on the upper sheet as the integral ofsin (( p + p − p ) / 2) along the reversed contour on the lower sheet for p , which we de-note by (Γ l − ) (cid:48) . Combining this with the integral of sin (( p + p − p ) / 2) along (Γ u − ) (cid:48) already present and defining (cid:0) Γ − (cid:1) (cid:48) to be the sum of (Γ u − ) (cid:48) and (Γ l − ) (cid:48) , we can write(7.12) as (cid:90) ( Γ − ) (cid:48) + C ∞ z ( dp + dp − dp )2 πi ln sin (cid:18) p + p − p (cid:19) − (cid:90) ( Γ − ) (cid:48) + C ∞ z dp πi ln sin p + Mismatch , (7.14)where Mismatch is given by Mismatch = (cid:90) ( M uuu −−− ) (cid:48) + C ∞ z ( dp + dp + dp )2 πi ln sin (cid:18) p + p + p (cid:19) + (cid:90) ( M uuu − + − ) (cid:48) + C ∞ z ( dp − dp + dp )2 πi ln sin (cid:18) p − p + p (cid:19) . (7.15)106ow the corresponding weak-coupling result obtained in [4] can be re-cast into the fol-lowing form by the use of integration by parts, (cid:90) −A z (cid:16) dp spin1 + dp spin2 − dp spin3 (cid:17) πi ln sin (cid:32) p spin1 + p spin2 − p spin3 (cid:33) − (cid:90) −A z dp spin2 πi ln sin p spin2 , (7.16)where A is the contour which encircles the branch cut of p counterclockwise. Comparing(7.14) and (7.16), one notes the following: (i) The terms denoted by Mismatch in the strongcoupling result are not present in the weak coupling expression. (ii) The integrands ofthe rest of the terms are precisely of the same form as for the weak coupling result, butthe contours of integrations are different. This makes a difference in the answer since indeforming the contours from those for the strong coupling to those for the weak couplingpicks up non-vanishing contributions from the singularities of the integrands. Concerningthe three-point functions, there is no firm argument that the Frolov-Tseytlin limit must beuniversal for all the observables. Therefore the discrepancies that we found above do notimmediately imply the breakdown of the duality. However, it is certainly of importanceto clarify the origin of these differences. As a part of the possible understanding, belowwe shall offer a natural mechanism which can change the contours of integration. The mechanism that we wish to point out is based on the possibility of having extrasingularities on the worldsheet. To see this, let us first recall that in the derivation ofthe important rules which determine the analyticity of the Wronskians, we have made animportant assumption that the only singularities on the worldsheet of the solutions of theALP occur at the positions of the vertex insertion points. This in turn means that if thereexist extra singularities this assumption breaks down and affects the rules for determiningthe contours of the convolution integrals . Depending on the number and the positionsof the extra singularities, the contours can be modified in various ways and it might bepossible to obtain the contour which appear in the weak coupling result.Now we can provide some arguments which indicate that indeed the existence of ad-ditional singularities is not uncommon. First, recall that the usual finite gap methodis capable of constructing solutions which correspond to the saddle point configurationsfor two-point functions. As such they contain only two singularities, normally placed at τ = ±∞ in the cylinder coordinates. In such a formalism designed to deal with two-pointfunctions, description of three-point solutions would require additional singularities. In A similar mechanism of changing the integration contour by the extra singularities is discussed inthe context of the so-called ODE/IM correspondence [35]. In this paper, we have succeeded in computing the three-point functions at strong couplingof certain non-BPS states with large charges corresponding to the composite operatorsin the SU(2) sector of the N = 4 super Yang-Mills theory. As we have already given asummary of the main result in section 1.1, we shall not repeat it here. Instead, below wewould like to give some comments and indicate some important issues to be clarified inthe future.One conspicuous feature of our result is that even for rather general external statesthe integrands of the integrals expressing the structure constant exhibit structures quitesimilar to the corresponding result at weak coupling. This is quite non-trivial since theweak coupling result in the relevant semi-classical regime is obtained from the determinantformula for the inner product of the Bethe states, which is so different from the methodemployed for strong coupling. This suggests that we should seek better understandingby reformulating the weak coupling computation in a more “physical” way. As a steptoward such a goal, an attempt was made in [36], where the inner product of the Bethestates is re-expressed in terms of an integral over the separated dynamical variables. Asthe notion of the wave function is clearly visible in this formulation, it may give a hint for108he common feature of the strong and the weak coupling regimes, if an efficient methodto identify the semi-classical saddle point can be developed.In contrast to the similarity of the integrands, there is a rather clear difference inthe contours of the integrals expressing the three-point coupling in the weak and thestrong coupling computations. This is not just a quantitative difference but rather aqualitative one. Reflecting the fact that the determinant formula deals with the Betheroots, the contour of integration in the weak coupling case is around a cut formed bythe condensation of such Bethe roots. Information of such a cut is contained in thequasi-momentum p ( x ). On the other hand, the principal quantity which determines theintegration contour is the real part of the quasi-energy q ( x ), which is conjugate to theworldsheet time τ . Apparently, this notion is not present in the weak coupling formulation.Together with the possible extra singularities on the worldsheet discussed in section 7.5.2,the question of the contour requires better understanding.There are a couple of further interesting questions that one should study concerningour result. One is about the limit of our formula where one of the operators is muchsmaller than the other two. Such three-point functions were first studied on the string-theory side in [37–39] assuming that the light operator does not change the saddle-pointconfiguration of the other two operators. However, a systematic study on the gauge-theory side [30] reveals that the light operator in some cases modifies the saddle-pointsubstantially. By examining the limit of our formula, it would be possible to understandin detail when and how such a “back-reaction” occurs. Another important problem is tounderstand the physical meaning of the integration along the unit circle in our formulaand clarify if it can be interpreted as the contribution from the dressing phase and thefinite size correction as in the case of the one-loop spectrum of a classical string [31].Finally, let us go back once again to the rather simple structure of the integrand wefound, similar to the weak coupling result. The simplicity of such a result suggests thatthere should perhaps be a better more intrinsic formulation for computing the three-pointfunctions. In the existing literature, including this work, the calculation of the three-pointfunction in the strong coupling regime is divided into the computation of the contributionof the action part and that of the vertex operator (wave function) part. As we have seenin section 6, in the process of putting these separate contributions together there occursa substantial simplification, besides the usual cancellation of divergences. This stronglyindicates that such a separation is not essential and one should rather seek relationswhich reflect the structure of the entire three-point function based on some dynamicalsymmetry of the theory including the integrable structure. This is of utmost importancesince the true understanding of the AdS/CFT duality lies not just in the comparisonof the calculations of various physical quantities in the strong and the weak coupling109egimes itself but rather in identifying the common principle behind such computationsand agreements.To make the above remark somewhat more concrete, let us recall that the most im-portant ingredient in the computation performed in this paper at strong coupling is theglobal consistency relations for the monodromy of the solutions of the auxiliary linearproblem around three vertex insertion points. Together with the analyticity property inthe spectral parameter, the important quantity (cid:104) i ± , j ± (cid:105) , which relates the behavior of thesolutions around different insertion points, is extracted and serves as the building blockfor the three-point coupling. On the other hand, in the weak coupling computations sofar performed, the computation of the three-point coupling is reduced to those of theinner products of the Bethe states and their combinations. Although this is an efficientmethod, it is based basically on the picture of the two-point function and not on someprinciple which governs the entire three-point function. Therefore we believe that an ex-tremely important problem is to find some functional equations (or differential equations)satisfied by the three-point function, from which one can determine the coupling constantmore or less directly. We hope to discuss this type of formulation elsewhere. Acknowledgment We would like to thank Y. Jiang, I. Kostov, D. Serban and P. Vieira for discussions.S.K. would like to acknowledge the hospitality of the Perimeter Institute for TheoreticalPhysics, where part of this work was done. The research of Y.K. is supported in part bythe Grant-in-Aid for Scientific Research (B) No. 25287049, while that of S.K. is supportedin part by JSPS Research Fellowship for Young Scientists, from the Japan Ministry ofEducation, Culture, Sports, Science and Technology. A Details on the one-cut solutions In this appendix, we will provide some further details on the one-cut solutions. A.1 Parameters of one-cut solutions in terms of the position ofthe cut In section 2.2 we have given generic expressions for the parameters which characterizethe one-cut solutions in terms of the integrals involving p ( x ) and q ( x ). If we now use theexplicit forms of p ( x ) and q ( x ) given in (2.58) and (2.59), one can evaluate the parameters ν i , m i and θ in terms of the position of the cut specified by u . The results take severaldifferent forms depending on the region where the cut is located. It is convenient to110xpress them in universal forms by introducing two additional sign factors η and η , .Together with the factor (cid:15) already introduced in (2.60), we give their definitions in thefollowing table:Table 3. Sign factors to distinguish between the positions of the cut.Re u < − − < Re u < < Re u < < Re u(cid:15) + − − + η + + + − η , + + − +Then, ν and ν are obtained as ν = κ (cid:20) − η + η , | u || u − | + (cid:15) η − η , | u || u + 1 | (cid:21) , (A.1) ν = κ (cid:20) η − η , | u || u − | − (cid:15) η + η , | u || u + 1 | (cid:21) = (cid:15)ν ( u → − u ) . (A.2)As for m i , we can immediately obtain them form ν i by the substitution (cid:15) → − (cid:15) , because,as seen in (2.58) and (2.59), this interchanges q ( x ) and p ( x ): m = ν ( (cid:15) → − (cid:15) ) , (A.3) m = ν ( (cid:15) → − (cid:15) ) . (A.4)Now cos ( θ / 2) and sin ( θ / 2) can be deduced from the Virasoro condition (2.70) ascos θ | u | − η η , Re u | u | , sin θ | u | + η η , Re u | u | . (A.5)The right and the left charges are obtained from (2.30) and (2.31) to be R = − κ √ λη (cid:18) Re u − | u − | + (cid:15) Re u + 1 | u + 1 | (cid:19) , (A.6) L = κ √ λη , | u | (cid:18) | u | − Re u | u − | + (cid:15) | u | + Re u | u + 1 | (cid:19) . (A.7)From the definition of R and L as the Noether charges, they must be expressed in termsof the parameters ν i and θ in a universal manner independent of the position of thecut. Indeed by using the formulas already obtained for the parameters and the chargesin terms of u , we can check the universal expressions R √ λ = 12 (cid:18) − ν cos θ ν sin θ (cid:19) , (A.8) L √ λ = 12 (cid:18) − ν cos θ − ν sin θ (cid:19) . (A.9)111inally, let us discuss the signs and the relative magnitudes of the parameters and thecharges. The signs and the relative magnitude of ν i depend on u . From the formulas for ν i we can check that | Re u | > ν < ν < , (A.10) | Re u | < ν < < ν , ( | ν | < ν ) . (A.11)As for the angles, we always have cos θ > sin θ . (A.12)The signs of R and L can be checked to be always positive. ( R for the case | Re u | > L for the case | Re u | < R and L can be deduced easily from the difference1 √ λ ( R − L ) = 2 ν sin θ . (A.13)As the sign of ν has already been obtained in (A.10) and (A.11), we immediately get R < L for | Re u | > , (A.14) R > L for | Re u | < . (A.15) A.2 Pohlmeyer reduction for one-cut solutions Let us next consider the variables appearing in the Pohlmeyer reduction, ρ , ˜ ρ and γ forone-cut solutions. From their definitions, we can express them in terms of the parametersof the one-cut solution ascos 2 γ = ν − m κ = ν − m κ , (A.16) ρ = 18 cos θ θ (cid:0) ( ν + m ) − ( ν + m ) (cid:1) , (A.17)˜ ρ = 18 cos θ θ (cid:0) ( ν − m ) − ( ν − m ) (cid:1) , (A.18)where we used z = τ + iσ coordinate when we compute these quantities .Using the results in the previous subsection, we can re-express (A.16), (A.17) and(A.18) in terms of the branch points u and ¯ u . They are given bycos 2 γ = (cid:15) | u | − | u − | , sin 2 γ = 2Im u | u − | , (A.19) ρ = − κ Im u | u − | , ˜ ρ = κ Im u | u + 1 | . (A.20) Note that γ is invariant under the coordinate change z → z (cid:48) = f ( z ), whereas ρ and ˜ ρ transformrespectively as ρ → ρ (cid:48) = ρ/ ( ∂f ) and ˜ ρ → ˜ ρ (cid:48) = ˜ ρ/ ( ¯ ∂f ) . γ i = 12 arcsin (cid:18) u i | u i − | (cid:19) , (A.21) ρ i = − κ Im u i | u i − | , (A.22)˜ ρ i = κ Im u i | u i + 1 | . (A.23)They will be used in the computation of three-point functions. A.3 Computation of various integrals Using the above results, let us compute various integrals which appear in Local and Double in section 3. Around a puncture, one can approximate the behavior of the world-sheetby that of the two-point functions. Thus, when three string states are semi-classicallydescribed 1-cut solutions, we expect the following asymptotic behavior of the one-forms: λ z → z i ∼ κ i dw i , ω z → z i ∼ − κ i cos 2 γ i d ¯ w i + 2 ρ i κ i dw i , (A.24)where w i is the local coordinate w i ≡ τ ( i ) + iσ ( i ) around the puncture z i .Using (A.24), one can evaluate various integrals. First, the contour integrals of λ and ω along C i ’s are given by (cid:73) C i λ = 2 πiκ i , (cid:73) C i ω = 2 πi (cid:18) κ i cos 2 γ i ρ i κ i (cid:19) i = 1 , ¯2 , . (A.25)On the other hand, the double contour integral, which appears in Double can be computedas follows: (cid:73) C i ω (cid:90) zz ∗ i λ = (cid:90) σ =2 πσ =0 (cid:18) − κ i cos 2 γ i d ¯ w i + 2 ρ i κ i dw i (cid:19) (cid:90) σ (cid:48) = σσ (cid:48) =0 κ i dw (cid:48) i = − (cid:90) π dσ (cid:18) κ i cos 2 γ i ρ i κ i (cid:19) κ i σ = − π (cid:18) κ i cos 2 γ i ρ i κ i (cid:19) κ i . (A.26)These results are used in section 3.1 to explicitly evaluate Local and Double .113 Pohlmeyer reduction In this appendix, we will give some details of the Pohlmeyer reduction for the string on S . In terms of the embedding coordinate Y I ( I = 1 , . . . , S is realized as a hypersurfacein R satisfying (cid:80) I Y I = 1. The basic idea of the Pohlmeyer reduction is to describe thedynamics of the string in terms of a moving frame in R consisting of four basis vectors { Y I , ∂Y I , ¯ ∂Y I , N I } , which satisfy the following properties: N I N I = 1 , N I Y I = N I ∂Y I = N I ¯ ∂Y I = 0 . (B.1)Then, using the equation of motion, ∂ ¯ ∂Y I + (cid:0) ∂Y J ¯ ∂Y J (cid:1) Y I = 0 and the Virasoro con-straints, ∂Y I ∂Y I = − T ( z ) and ¯ ∂Y I ¯ ∂Y I = − ¯ T (¯ z ), we can express the derivatives of thesebasis vectors, ∂N I , ∂ Y I , etc . again in terms of the basis vectors: ∂N I = 2 ρT sin γ ∂Y I + 2 cos 2 γρ √ T ¯ T sin γ ¯ ∂Y I , (B.2)¯ ∂N I = 2 ρ ¯ T sin γ ¯ ∂Y I + 2 cos 2 γ ˜ ρ √ T ¯ T sin γ ∂Y I , (B.3) ∂ Y = T Y I + ∂ ln (cid:0) T ¯ T sin γ (cid:1) ∂Y I + (cid:114) ¯ TT ∂γ sin 2 γ ¯ ∂Y I + 2 ρN I , (B.4)¯ ∂ Y = ¯ T Y + ¯ ∂ ln (cid:0) T ¯ T sin γ (cid:1) ∂Y I + (cid:114) T ¯ T ∂γ sin 2 γ ∂Y I + 2 ˜ ρN I , (B.5) ∂ ¯ ∂Y = − (cid:112) T ¯ T cos 2 γY , (B.6)where ρ , ˜ ρ and γ are defined by ∂Y I ¯ ∂Y I = (cid:112) T ¯ T cos 2 γ , ρ ≡ N I ∂ Y I , ˜ ρ ≡ N I ¯ ∂ Y I . (B.7)Using the equation of motion, one can also show that γ , ρ and ˜ ρ satisfy the generalizedsin-Gordon equation, which is given in (2.44).Let us next derive a flat connection associated with the system of equations (B.2)–(B.6). For this purpose, it is convenient to introduce the following orthonormal basis: q ≡ Y , q ≡ − i sin 2 γ (cid:20) e iγ √ T ∂Y + e − iγ √ ¯ T ¯ ∂Y (cid:21) , (B.8) q ≡ i sin 2 γ (cid:20) e iγ √ ¯ T ¯ ∂Y + e − iγ √ T ∂Y (cid:21) , q ≡ N , (B.9)which satisfy the following normalization conditions: q = q = 1 , q q = − . (B.10)114ith these orthonormal vectors, (B.2)–(B.6) can be re-expressed as the following set ofequations, ∂q = √ T (cid:2) e iγ q + e − iγ q (cid:3) , (B.11) ∂q = e − iγ √ T q + i∂γq − iρ √ T sin 2 γ e iγ q , (B.12) ∂q = e iγ √ T q − i∂γq + 2 iρ √ T sin 2 γ e − iγ q , (B.13) ∂q = iρe − iγ √ T sin 2 γ q − iρe iγ √ T sin 2 γ q , (B.14)¯ ∂q = − √ ¯ T (cid:2) e − iγ q + e iγ q (cid:3) , (B.15)¯ ∂q = − e iγ (cid:112) ¯ T q − i ¯ ∂γq − i ˜ ρ √ ¯ T sin 2 γ e − iγ q , (B.16)¯ ∂q = − e − iγ (cid:112) ¯ T q + i ¯ ∂γq + 2 i ˜ ρ √ ¯ T sin 2 γ e iγ q , (B.17)¯ ∂q = i ˜ ρe iγ √ ¯ T sin 2 γ q + i ˜ ρe − iγ √ ¯ T sin 2 γ q . (B.18)By expressing the basis in a matrix form, W ≡ (cid:18) q + iq q q q − iq (cid:19) , (B.19)we can convert the above equations into the following form: ∂W + B Lz W + W B Rz = 0 , ¯ ∂W + B L ¯ z W + W B R ¯ z = 0 , (B.20)where B L,Rz, ¯ z are matrices defined by B Lz ≡ (cid:32) − i∂γ ρe iγ √ T sin 2 γ − √ T e − iγρe − iγ √ T sin 2 γ − √ T e iγ i∂γ (cid:33) , (B.21) B Rz ≡ (cid:32) i∂γ − ρe iγ √ T sin 2 γ − √ T e − iγ − ρe − iγ √ T sin 2 γ − √ T e iγ − i∂γ (cid:33) , (B.22) B L ¯ z ≡ i ¯ ∂γ ρe − iγ √ ¯ T sin 2 γ + √ ¯ T e iγ ˜ ρe iγ √ ¯ T sin 2 γ + √ ¯ T e − iγ − i ¯ ∂γ , (B.23) B R ¯ z ≡ − i ¯ ∂γ − ˜ ρe − iγ √ ¯ T sin 2 γ + √ ¯ T e iγ − ˜ ρe iγ √ ¯ T sin 2 γ + √ ¯ T e − iγ i ¯ ∂γ . (B.24)(B.20) is equivalent to the flatness conditions of the connections B L and B R , ∂B L ¯ z − ¯ ∂B Lz + [ B Lz , B L ¯ z ] = 0 , ∂B R ¯ z − ¯ ∂B Rz + [ B Rz , B R ¯ z ] = 0 . (B.25)115wing to the classical integrability of the string sigma model, we can “deform” the aboveconnection without spoiling the flatness by introducing a spectral parameter ζ = (1 − x ) / (1 + x ) as B z ( ζ ) ≡ Φ z ζ + A z , B ¯ z ( ζ ) ≡ ζ Φ ¯ z + A ¯ z . (B.26)Φ’s and A ’s are defined by Φ z ≡ (cid:32) − √ T e − iγ − √ T e iγ (cid:33) , Φ ¯ z ≡ (cid:32) √ ¯ T e iγ √ ¯ T e − iγ (cid:33) , (B.27) A z ≡ (cid:32) − i∂γ ρe iγ √ T sin 2 γρe − iγ √ T sin 2 γ i∂γ (cid:33) , A ¯ z ≡ (cid:32) i ¯ ∂γ ρe − iγ √ ¯ T sin 2 γ ˜ ρe iγ √ ¯ T sin 2 γ − i ¯ ∂γ (cid:33) . (B.28)The deformed connection (B.26) evaluated at ζ = 1 or ζ = − B L,R in the following way: B L = B ( ζ = 1) , (cid:0) B R (cid:1) t = σ B ( ζ = − σ . (B.29)Furthermore (B.26) is related to the usual left/right connection by an appropriate gaugetransformation as will be shown in Appendix C. C Relation between the Pohlmeyer reduction andthe sigma model formulation In this appendix, we explain how the Pohlmeyer reduction and the sigma model formula-tion are related. C.1 Reconstruction formula for the Pohlmeyer reduction In section 2.2 we presented the simple formulas (2.65) and (2.66) which reconstruct thesolution Y of the equations of motion from the eigenfunctions of the ALP in the sigmamodel formulation. We now describe a similar formula for the Pohlmeyer reduction andby comparing such reconstruction formulas we can relate the two formulations. Considerthe left and the right ALP associated with the Pohlmeyer reduction, (cid:0) d + B L (cid:1) ψ L = 0 , (cid:0) d + B R (cid:1) ψ R = 0 , (C.1) (B.26) is equivalent in form to the SL(2)-Hitchin system. However, the boundary conditions weimpose around the punctures are different from the ones used in the usual analysis of the Hitchin system. ψ L,R and ψ L,R be two linearly independent solutions satisfying the normalizationconditions det (cid:0) ψ L , ψ L (cid:1) = 1 , det (cid:0) ψ R , ψ R (cid:1) = 1 . (C.2)Then, similarly to the sigma model case, the embedding coordinates Y can be recon-structed by the formula Y = q = (cid:18) Z Z − ¯ Z ¯ Z (cid:19) = (cid:0) Ψ L (cid:1) t Ψ R , (C.3)where Ψ L,R are 2 × L ≡ (cid:0) ψ L , ψ L (cid:1) , Ψ R ≡ (cid:0) ψ R , ψ R (cid:1) . (C.4)Concerning the property under the global symmetry transformations, we should notethe following. Since the Pohlmeyer connections B L and B R in the equation (C.1) areinvariant, Ψ L and Ψ R must also be invariant under such transformations acting from left.However, as for transformations from right, they may transform non-trivially. In fact, aswe shall see shortly, they must transform covariantly from right so that the solutions ofthe ALP for the Pohlmeyer and the sigma model formulations are connected consistentlyby a gauge transformation.Furthermore, one can check that the quantities q and q , which consist of the deriva-tives of Y , can be reconstructed as q = (cid:0) Ψ L (cid:1) t (cid:18) (cid:19) Ψ R , q = (cid:0) Ψ L (cid:1) t (cid:18) (cid:19) Ψ R . (C.5)From these formulas the derivatives of Y can be obtained as ∂ Y = √ T (cid:2) e iγ q + e − iγ q (cid:3) , ¯ ∂ Y = − √ ¯ T (cid:2) e − iγ q + e iγ q (cid:3) . (C.6)Note that, in distinction to the case of the sigma model, the reconstruction formulasfor the Pohlmeyer reduction does not use the eigenvectors of the monodromy matrices,namely ˆ ψ ± . The solutions ψ L,Ri used are simply two linearly independent solutions to theALP, which are not necessarily the eigenvectors of Ω. C.2 Relation between the connections and the eigenvectors We now discuss the relation between the connections and the eigenvectors of the thePohlmeyer reduction and those of the sigma model.117irst consider the relation to the right connection of the sigma model. From theformulas for ∂ Y and ¯ ∂ Y given in (C.6), we can form the right connection j as j z = √ T (cid:0) Ψ R (cid:1) − (cid:18) e iγ e − iγ (cid:19) Ψ R , j ¯ z = − (cid:112) ¯ T (cid:0) Ψ R (cid:1) − (cid:18) e − iγ e iγ (cid:19) Ψ R . (C.7)Then, comparing (C.7) with (B.26)–(B.28), we find that the following gauge transforma-tion connects the flat connections of the two formulations:11 − x j z = G − B z ( ζ ) G + G − ∂ G , (C.8)11 + x j ¯ z = G − B ¯ z ( ζ ) G + G − ¯ ∂ G , (C.9)where G = iσ Ψ R . (C.10)The eigevectors ψ ± of the sigma model formulation and those of the Pohlmeyer reduction,denoted by ˆ ψ ± , are related as ψ ± = G − ˆ ψ ± . (C.11)Note that the factor of i in (C.10) is needed to reproduce the correct normalizationcondition (cid:104) ψ + , ψ − (cid:105) = 1. Under the global SU(2) R transformation U R , ψ ± transform as ψ ± → U − R ψ ± . From the above formulas (C.10) and (C.11) we see that this correspondsto the transformation Ψ R → Ψ R U R , as remarked previously.In an exactly similar manner, we can construct the left current l ’s by l z = √ T (cid:0) Ψ L (cid:1) t (cid:18) e iγ e − iγ (cid:19) (cid:104)(cid:0) Ψ L (cid:1) t (cid:105) − , l ¯ z = − (cid:112) ¯ T (cid:0) Ψ L (cid:1) t (cid:18) e − iγ e iγ (cid:19) (cid:104)(cid:0) Ψ L (cid:1) t (cid:105) − , (C.12)Comparing (C.12) with (B.26)–(B.28), we find that the following gauge transformationconnects the two connections: x − x l z = ˜ G − B z ( ζ ) ˜ G + ˜ G − ∂ ˜ G , (C.13) − x x l ¯ z = ˜ G − B z ( ζ ) ˜ G + ˜ G − ¯ ∂ ˜ G , (C.14)where ˜ G = (cid:2) (Ψ L ) t ( − iσ ) (cid:3) − = i Ψ L σ . (C.15)The eigenvectors are related as ˜ ψ ± = ˜ G − ˆ ψ ± . (C.16)Using (C.11) and (C.16), one can show the equivalence between the reconstruction for-mulas (2.65), (2.66) and (C.3). 118 Details of the WKB expansion In this appendix, we explain the details of the WKB expansion for the solutions to theALP. We will describe two approaches, each of which has its own merit. First in subsectionD.1, we will perform a direct expansion in the small parameter ζ , which is useful forclarifying the general structure of the expansion. This method, however, turned out to benot quite suitable for deriving the explicit formulas for the expansion of the Wronskians.Therefore, in subsection D.2, we take a slightly different approach based on the Bornseries expansion. This allows us to derive the expressions for the Wronskians up to the O ( ζ ) terms with relative ease, with the results given in (3.37), (3.38), (3.40) and (3.41). D.1 Direct expansion of the solutions to the ALP In this subsection, we will perform a direct expansion of the ALP in the “diagonal gauge”introduced in section 3.2. In this gauge the ALP equations become (cid:18) ∂ + 1 ζ Φ dz + A dz (cid:19) ˆ ψ d = 0 , (cid:0) ¯ ∂ + ζ Φ d ¯ z + A d ¯ z (cid:1) ˆ ψ d = 0 . (D.1)Denoting the components of ˆ ψ d as ˆ ψ d ≡ (cid:18) ψ (1) ψ (2) (cid:19) , (D.2)and substituting the expressions for Φ dz , A dz , etc. given in (3.35), the ALP equations abovetake the form ∂ψ (1) + √ T ζ ψ (1) − ρ √ T cot 2 γψ (1) + i (cid:18) ρ √ T − ∂γ (cid:19) ψ (2) = 0 , (D.3) ∂ψ (2) − √ T ζ ψ (2) + ρ √ T cot 2 γψ (2) − i (cid:18) ρ √ T + ∂γ (cid:19) ψ (1) = 0 , (D.4)and ¯ ∂ψ (1) − ζ √ ¯ T cos 2 γ ψ (1) − ˜ ρ √ ¯ T sin 2 γ ψ (1) + i √ ¯ T sin 2 γ ψ (2) = 0 , (D.5)¯ ∂ψ (2) + ζ √ ¯ T cos 2 γ ψ (2) + ˜ ρ √ ¯ T sin 2 γ ψ (2) − i √ ¯ T sin 2 γ ψ (1) = 0 . (D.6)Let us examine the first two equations (D.3) and (D.4). To perform the WKB expan-sion, it is useful to introduce a coordinate w defined by dw = √ T dz . (D.7)119y this coordinate transformation we can absorb the factor √ T and bring the equationsto the simplified form ∂ w ψ (1) + 12 ζ ψ (1) − ρT cot 2 γψ (1) + i (cid:16) ρT − ∂ w γ (cid:17) ψ (2) = 0 , (D.8) ∂ w ψ (2) − ζ ψ (2) + ρT cot 2 γψ (2) − i (cid:16) ρT + ∂ w γ (cid:17) ψ (1) = 0 . (D.9)Let us express ψ (2) in terms of ψ (1) using (D.8). We get ψ (2) = − i (cid:16) ρT − ∂ w γ (cid:17) − (cid:20) ∂ w ψ (1) + (cid:18) ζ − ρT cos 2 γ (cid:19) ψ (1) (cid:21) . (D.10)Substituting (D.10) into (D.9), we obtain a second order differential equation for ψ (1) ofthe form ∂ w ψ (1) − ∂ w ln (cid:16) ρT − ∂ w γ (cid:17) ∂ w ψ (1) − Aψ (1) = 0 , (D.11)where A is given by A = (cid:18) ζ − ρT cot 2 γ (cid:19) + ∂ w (cid:16) ρT cot 2 γ (cid:17) + ∂ w ln (cid:16) ρT − ∂ w γ (cid:17) (cid:18) ζ − ρT cot 2 γ (cid:19) + ( ∂ w γ ) − (cid:16) ρT (cid:17) . (D.12)We now make the WKB expansion of ψ (1) in powers of ζ in the form, ψ (1) = (cid:114) ρT − ∂ w γ exp (cid:20) W − ζ + W + ζW + · · · (cid:21) , (D.13)and substitute it into (D.11). Then, at order ζ − , we get the equation( ∂ w W − ) = 14 , (D.14)with the solutions given by ∂ w W − = ± / 2. At the next order, we get the equation ∂ w W − + 2 ∂ w W − ∂ w W = 12 ∂ w ln (cid:16) ρT − ∂ w γ (cid:17) − ρT cot 2 γ . (D.15)From this ∂ w W is determined as ∂ w W = ± (cid:20) ∂ w ln (cid:16) ρT − ∂ w γ (cid:17) − ρT cot 2 γ (cid:21) , (D.16)where the plus sign is for ∂ w W − = +1 / ∂ w W − = − / ∂ w W as ∂ w W = ± (cid:20) ( ∂ w γ ) − (cid:16) ρT (cid:17) + ∂ w (cid:16) ρT cot 2 γ (cid:17) − ∂ w ln (cid:16) ρT − ∂ w γ (cid:17)(cid:21) − ∂ w ln (cid:16) ρT − ∂ w γ (cid:17) , (D.17)120here the choice of the sign should be the same as in (D.16). Continuing in this fashionusing (D.5) and (D.6), we can determine ¯ ∂W − , ¯ ∂W and ¯ ∂W to be¯ ∂W − = 0 , ¯ ∂W = ± (cid:20) 12 ¯ ∂ ln (cid:16) ρT − ∂ w γ (cid:17) − ˜ ρ √ ¯ T sin 2 γ (cid:21) , ¯ ∂W = ± (cid:20) η − 12 ¯ ∂∂ w ln (cid:16) ρT − ∂ w γ (cid:17)(cid:21) − 12 ¯ ∂∂ w ln (cid:16) ρT − ∂ w γ (cid:17) . (D.18)The results obtained above can be reorganized into a compact form. In fact we canwrite the expansion (D.13) as ψ (1) = exp [ W odd + W even ] , (D.19)where W odd (resp. W even ) denotes terms which (do not) change sign under the sign-flip of ∂ w W − . Then, by substituting (D.19) into (D.11) and extracting the terms odd under theabove flip of sign, we can obtain the following simple equation expressing W even in termsof W odd : W even = − 12 ln ∂ w W odd . (D.20)As is clear from the analysis above, the WKB expansion of W odd is given in terms of theintegrals of certain functions of the worldsheet variables, such as γ , ρ and ˜ ρ . On the otherhand, the even part W even , which depends only on the derivatives of W odd , is expressedpurely in terms of the local values of the worldsheet variables. With such classifications,we can recast the WKB expansion of the two linearly independent solutions of the ALPinto the following form:ˆ ψ d = (cid:32) f (1) ± f (2) ± (cid:33) exp (cid:18) ± (cid:90) zz W WKB ( z, ¯ z ; ζ ) (cid:19) . (D.21)Here we renamed W odd to W WKB and the functions f (1) ± and f (2) ± are defined in terms of W z WKB by f (1) ± ≡ k WKB = (cid:115) ρ − √ T ∂γT W z WKB , (D.22) f (2) ± ≡ − i (cid:112) W z WKB (cid:34) ± W z WKB + (cid:32) √ T ζ − ρ cos 2 γ √ T + ∂ ln k WKB (cid:33)(cid:35) . (D.23) D.2 Born series expansion of the Wronskians In this subsection, we will derive the explicit form of the expansion for the Wronskians upto O ( ζ ) using the Born series method, which turned out to be more convenient compared121o the direct expansion described above. In particular, with this method it is much easierto take into account the normalization conditions of the eigenvectors i ± given in (2.96).Although the method has been described in Appendix B of [22], we will spell out thedetails of the derivation since several additional considerations are necessary in our case.To illustrate the basic idea, let us take the Wronskian (cid:104) + , + (cid:105) as an example anddiscuss its expansion. To compute (cid:104) + , + (cid:105) , we need to parallel-transport the eigenvector1 + , which is defined originally in the neighborhood of z , to the neighborhood of z usingthe flat connection and compute the Wronskian with 2 + . In the diagonal gauge, thisprocedure can be implemented in the following way: (cid:104) ˆ2 d + , ˆ1 d + (cid:105) = (cid:104) ˆ2 d + ( z ∗ ) , P exp (cid:20) − (cid:90) dt (cid:18) ζ H ( t ) + V ( t ) (cid:19)(cid:21) ˆ1 d + ( z ∗ ) (cid:105) . (D.24)In this expression t parametrizes the curve joining z ∗ (at t = 0) and z ∗ (at t = 1) and H and V are defined in terms of the connection in the diagonal gauge, given in (3.35), as H ( t ) ≡ ˜Φ z ˙ z , V ( t ) ≡ ˜ A z ˙ z + ˜ A ¯ z ˙¯ z + ζ ˜Φ ¯ z ˙¯ z , (D.25)with ˙ z and ˙¯ z standing for dz/dt and d ¯ z/dt respectively. The equation (D.24) is similar inform to the transition amplitude in quantum mechanics, where H ( t ) /ζ is the unperturbedHamiltonian and V ( t ) is the time-dependent perturbation. Therefore we can derive theexpansion of (D.24) by applying the familiar Born series expansion.As the first step toward this goal, let us determine the expansion of the “initial states”,ˆ1 d + ( z ∗ ) and ˆ2 d + ( z ∗ ). As explained in section 2.3, the eigenvectors can be well-approximatednear the puncture by those of the corresponding two-point functions. Thus, the expansionof the initial states can be obtained from the explicit form of ˆ i ± given in (2.98) and (2.99)as ˆ1 d + ( z ∗ ) ∼ ˆ1 ,d + = (cid:18) O ( ζ )1 + O ( ζ ) (cid:19) , ˆ2 d + ( z ∗ ) ∼ ˆ2 ,d + = (cid:18) O ( ζ ) O ( ζ ) (cid:19) . (D.26)Let us now study the leading terms ( i.e. the O ( V ) terms) in the Born series expansionof (D.24). They can be expressed as1 (2)+ ( z ∗ )2 (1)+ ( z ∗ ) (cid:104) e | e − (cid:82) H dt/ζ | e (cid:105) − (1)+ ( z ∗ )2 (2)+ ( z ∗ ) (cid:104) e | e − (cid:82) H dt/ζ | e (cid:105) , (D.27)where | e (cid:105) and | e (cid:105) stand for the unit vectors | e (cid:105) = (cid:18) (cid:19) , | e (cid:105) = (cid:18) (cid:19) , (D.28)and i (1) ± and i (2) ± are the upper and the lower component of ˆ i d ± respectively, which can beexpressed as ˆ i d ± = i (1) ± | e (cid:105) + i (2) ± | e (cid:105) . (D.29)122sing (D.26), we can evaluate the expression (D.27) explicitly as (cid:0) O ( ζ ) (cid:1) exp (cid:18)(cid:90) (cid:96) ζ (cid:36) (cid:19) − O ( ζ ) exp (cid:18) − (cid:90) (cid:96) ζ (cid:36) (cid:19) , (D.30)where (cid:96) is the contour that connects z ∗ and z ∗ , defined in section 3.1. Note that thesecond term in (D.30), which has an overall O ( ζ ) factor can be safely neglected only whenRe (cid:16)(cid:82) (cid:96) (cid:36)/ζ (cid:17) is positive so that the exponential exp (cid:16) − (cid:82) (cid:96) (cid:36)/ζ (cid:17) becomes vanishinglysmall. The positivity of Re (cid:16)(cid:82) (cid:96) (cid:36)/ζ (cid:17) is guaranteed when the following two conditionsare satisfied:1. The eigenvectors, 1 + and 2 + , are small solutions.2. z and z are connected by a WKB curve z ( s ) defined to be satisfying the conditionIm (cid:18) √ T dzds (cid:19) = 0 , (D.31)where s parameterizes the curves.This can be deduced in the following way: First, from the definition (D.31), one can showthat the real part of the integral (cid:82) (cid:36)/ζ monotonically increases or decreases along theWKB curve. Second, when 1 + and 2 + are both small solutions, Re (cid:0)(cid:82) (cid:36)/ζ (cid:1) increasesas we move away from z in the vicinity of z while it increases as we approach z inthe vicinity of z . From these two observations, one can conclude that Re (cid:16)(cid:82) (cid:96) (cid:36)/ζ (cid:17) is positive when both of the eigenvectors are small and the punctures are connectedby a WKB curve. Actually, in practice the second condition above is inessential. This isbecause all the punctures are always connected with each other by WKB curves, except atdiscrete values of Arg ( ζ ), due to the triangular inequalities, ∆ i < ∆ j +∆ k (or equivalently κ i < κ j + κ k ), which hold in all the cases we study in this paper.Let us now move on to the study of the O ( V ) contributions. When 1 + and 2 + aresmall solutions, the O ( V ) terms in the Born series expansion are given by − (2)+ ( z ∗ )2 (1)+ ( z ∗ ) (cid:90) dt (cid:104) e | e − (cid:82) t H dt/ζ V ( t ) e − (cid:82) t H dt/ζ | e (cid:105)− (1)+ ( z ∗ )2 (1)+ ( z ∗ ) (cid:90) dt (cid:104) e | e − (cid:82) t H dt/ζ V ( t ) e − (cid:82) t H dt/ζ | e (cid:105) + 1 (2)+ ( z ∗ )2 (2)+ ( z ∗ ) (cid:90) dt (cid:104) e | e − (cid:82) t H dt/ζ V ( t ) e − (cid:82) t H dt/ζ | e (cid:105) . (D.32)Note that we have omitted the terms of the form, (cid:104) e | ∗ | e (cid:105) , since they are proportionalto the factor exp (cid:16)(cid:82) (cid:96) (cid:36)/ζ (cid:17) , which, as discussed above, is exponentially small when 1 + + are small solutions. Since | e (cid:105) and | e (cid:105) are the eigenvectors of H , we can evaluate(D.32) as − (2)+ ( z ∗ )2 (1)+ ( z ∗ ) e (cid:82) (cid:96) (cid:36)/ (2 ζ ) (cid:90) dt (cid:104) e | V ( t ) | e (cid:105)− (1)+ ( z ∗ )2 (1)+ ( z ∗ ) e (cid:82) (cid:96) (cid:36)/ (2 ζ ) (cid:90) dt (cid:104) e | V ( t ) | e (cid:105) e − (cid:82) t (cid:36)/ζ + 1 (2)+ ( z ∗ )2 (2)+ ( z ∗ ) e (cid:82) (cid:96) (cid:36)/ (2 ζ ) (cid:90) dt (cid:104) e | V ( t ) | e (cid:105) e − (cid:82) t (cid:36)/ζ . (D.33)In the limit ζ → 0, the integral over t in the second term will be exponentially suppressedby the factor exp (cid:16) − (cid:82) t (cid:36)/ζ (cid:17) , except when the interval is short, i.e. < t < O ( ζ ).Thus, to O ( ζ ), one can take (cid:36) in (cid:82) t (cid:36)/ζ to be constant and replace V ( t ) with V (0).We can thus approximate the second term in (D.33) as − ζ (1)+ ( z ∗ )2 (1)+ ( z ∗ ) e (cid:82) (cid:96) (cid:36)/ (2 ζ ) (cid:104) e | V (0) | e (cid:105) (cid:16)(cid:112) T ( z ∗ ) ˙ z ( t = 0) (cid:17) − . (D.34)Since the factor 1 (1)+ ( z ∗ ) is of O ( ζ ), (D.34) as a whole is of O ( ζ ) and thus can be neglectedto the order of our approximation. Similarly, one can also show that the third term of(D.33) is of O ( ζ ). Thus, up to O ( ζ ), the contribution comes only from the first termproportional to − e (cid:82) (cid:96) (cid:36)/ (2 ζ ) (cid:90) dt (cid:104) e | V ( t ) | e (cid:105) . (D.35)Lastly let us examine the O ( V ) terms. The only term which contributes at O ( ζ ) is1 (2)+ ( z ∗ )2 (1)+ ( z ∗ ) (cid:90) dt (cid:90) t dt (cid:104) e | e − (cid:82) t H dt/ζ V ( t ) e − (cid:82) t t H dt/ζ V ( t ) e − (cid:82) t H dt/ζ | e (cid:105) . (D.36)Inserting the identity 1 = | e (cid:105)(cid:104) e | + | e (cid:105)(cid:104) e | , this quantity can be computed as1 (2)+ ( z ∗ )2 (1)+ ( z ∗ ) e (cid:82) (cid:96) (cid:36)/ (2 ζ ) (cid:32) (cid:20)(cid:90) dt (cid:104) e | V ( t ) | e (cid:105) (cid:21) + (cid:90) dt (cid:90) t dt e − (cid:82) t t (cid:36)/ζ (cid:104) e | V ( t ) | e (cid:105)(cid:104) e | V ( t ) | e (cid:105) (cid:19) . (D.37)As in the discussion of the O ( V ) terms, we can take (cid:36) in (cid:82) t t (cid:36)/ζ to be constant andreplace V ( t ) with V ( t ) in the second term of (D.37), thanks to the suppression factorexp (cid:16) − (cid:82) t t (cid:36)/ζ (cid:17) . Then (D.37) can be evaluated as e (cid:82) (cid:96) (cid:36)/ (2 ζ ) (cid:32) (cid:20)(cid:90) dt (cid:104) e | V ( t ) | e (cid:105) (cid:21) + ζ (cid:90) dt (cid:104) e | V ( t ) | e (cid:105)(cid:104) e | V ( t ) | e (cid:105) ˙ z √ T (cid:33) . (D.38)124utting together the expressions (D.30), (D.35) and (D.38), we find that the result canbe grouped into an exponential in the following way: (cid:104) + , + (cid:105) ∼ exp (cid:18) ζ (cid:90) (cid:96) (cid:36) − (cid:90) dt (cid:104) e | V ( t ) | e (cid:105) + ζ (cid:90) dt (cid:104) e | V ( t ) | e (cid:105)(cid:104) e | V ( t ) | e (cid:105) ˙ z √ T (cid:19) . (D.39)Thus we have obtained the expansion of (cid:104) + , + (cid:105) to be given by (cid:104) + , + (cid:105) = exp (cid:18) − ζ (cid:90) (cid:96) (cid:36) − (cid:90) (cid:96) α − ζ (cid:90) (cid:96) η + O ( ζ ) (cid:19) , (D.40)where the one-form α is given by α = − ρ √ T cot 2 γdz − ˜ ρ √ ¯ T sin 2 γ d ¯ z . 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